UFM Pure

Sum the series \[ \sum_{n=1}^N \frac{3n-1}{n(n+1)(n+3)}. \]

Discuss the behaviour of the function \[ \frac{\log(1+x) - \frac{1}{x}(10-3x-4\cos x)}{x \sin x - x^2} \] as \(x \to 0\) and as \(x \to \infty\).

Discuss the convergence of the series \[ 1+z+z^2+\dots+z^n+\dots, \] where \(z\) may be real or complex.

(a) Find the limit, as \(x\) tends to zero, of (i) \((b^x - a^x)/x\) where \(a\) and \(b\) are positive; (ii) \(x \sin x / \log \cos x\). (b) If \(n\) is a positive integer, show that \[ \left(1+\frac{1}{n}\right)^n \le 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \dots + \frac{1}{n!} < 3. \]

Discuss the convergence of the series \[ 1+z+z^2+\dots+z^n+\dots, \] where \(z\) may be real or complex.

The function \(f(x)\) is ``bounded as \(x\to 0\) through positive values'' if and only if there exist positive constants \(K, \delta\) such that \(|f(x)| < K\) for \(0< x< \delta\). Show that if \(f(x), g(x)\) are bounded as \(x\to 0\) through positive values then so are \(f(x)+g(x)\), \(f(x)g(x)\), \(\int_x^\delta f(t)\,dt\). Show that \[ \int_x^1 \frac{e^{-t}}{t}dt + \log x \] is bounded as \(x\to 0\) through positive values.

Show that \[ \sum_{r=0}^n r(r+1)\dots(r+k-1) = \frac{1}{k+1}n(n+1)\dots(n+k). \] Deduce that, if \(a_{r+1} = a_r/(1+ra_r)\), then \[ \sum_{r=0}^n \frac{1}{a_ra_{r+2}} = \frac{n+1}{a_0} + \frac{n+1}{3a_0^2}(n^2+2n+3) + \frac{1}{20}(n-1)n(n+1)(n+2)(n+3). \]

What do you mean by (a) a finite limit and (b) an infinite limit? Evaluate the following limits:

1. \(\left( \frac{1}{\sqrt{(n^2+1)}} + \frac{1}{\sqrt{(n^2+2)}} + \dots + \frac{1}{\sqrt{(n^2+n)}} \right)\) as \(n\to\infty\) (\(n\) an integer),
2. \(\frac{\sin x + x}{\sin x - x}\) as \(x \to 0\),
3. \((x^{1/x}-1)\log x\) as \(x \to \infty\).

Sum the series \[ 1^3 - 2^3 + 3^3 - 4^3 + \dots - (2n)^3 + (2n+1)^3 \] and \[ \sum_{n=1}^\infty \frac{2.5.8. \dots (3n-1)}{3.6.9. \dots 3n} \left(\frac{1}{3}\right)^n. \]

Find the least value of the expression \[ y = \frac{1}{n} \sum_{r=0}^{n-1} \left( x - \sin \frac{r\pi}{n} \right)^2 \] for real values of \(x\). \newline If the minimum is \(y_n\), attained for \(x=x_n\), prove that, when \(n \to \infty\), \[ x_n \to \frac{2}{\pi}, \quad y_n \to \frac{1}{2} - \frac{4}{\pi^2}. \]

Evaluate the following limits: \[ \frac{\sqrt[3]{x} - \sqrt[3]{a}}{\sqrt[4]{x} - \sqrt[4]{a}} \quad \text{as } x \to a \quad (a>0), \] \[ (\pi - 2x)\tan x \quad \text{as } x \to \tfrac{1}{2}\pi \quad (x < \tfrac{1}{2}\pi), \] \[ \frac{n}{n^2} + \frac{n+1}{n^2} + \frac{n+2}{n^2} + \dots + \frac{2n}{n^2} \quad \text{as } n \to \infty. \]

Find the limit of

1. [(i)] \(\dfrac{\sqrt{1+x}-1}{1-\sqrt{1-x}}\) as \(x \to 0\),
2. [(ii)] \(\dfrac{3^n + (-2)^n}{3^n - 2^n}\) as \(n \to \infty\),
3. [(iii)] \(\dfrac{n^2}{2^n}\) as \(n \to \infty\),
4. [(iv)] \(\sqrt[n]{n}\) as \(n \to \infty\),

Evaluate the limits as \(x\) tends to 1 of the expressions:

1. [(i)] \(\dfrac{1}{x-1} \left\{ \operatorname{cosec} \pi x + \dfrac{1}{\pi(x-1)} \right\}\);
2. [(ii)] \(\dfrac{\cos^2 \frac{1}{2}\pi x}{e^x-ex}\);
3. [(iii)] \(\dfrac{x \log x}{(2x-1)\log(2x-1)}\).

Explain what is meant by the statement that ``\(f(n)\) tends to the limit \(l\) as \(n\) tends to infinity'' where \(n\) is a positive integer. \newline A positive quantity \(a_n\) satisfies the relationship \[ a_n = \frac{1}{2}\left(a_{n-1} + \frac{a^2}{a_{n-1}}\right), \] where \(n\) is a positive integer greater than unity and \(a\) is positive. Prove that, provided \(a_1>0\),

1. [(i)] \(a_n \ge a\) for \(n \ge 2\).
2. [(ii)] \(a_{n-1} \ge a_n\) for \(n \ge 3\).
3. [(iii)] \(a_n - a \le \frac{1}{2}(a_{n-1}-a)\) for \(n \ge 3\).

Evaluate the limits as \(x\) tends to infinity of the following expressions: \[ \sqrt{x^2+1}-x, \quad x^{-a}\log x \quad (a>0), \quad \text{and} \quad (x/\sqrt{x^2+1})^x. \]

Explain what you understand by a convergent series. Investigate for what ranges of values of \(x\) the following series are convergent:

1. \(e^x + e^{2x} + e^{3x} + \dots\),
2. \(\dfrac{1}{2} + \dfrac{1.3}{2.5}x + \dfrac{1.3.5}{2.5.8}x^2 + \dfrac{1.3.5.7}{2.5.8.11}x^3 + \dots\).

When \(x=a\), the functions \(f(x)\), \(g(x)\), \(f'(x)\) and \(g'(x)\) have the values \(0, 0, b\) and \(c\) respectively. Prove that, if \(c \ne 0\), \(f(x)/g(x)\) tends to \(b/c\) as \(x\) tends to \(a\). Evaluate the limits as \(x\) tends to \(0\) of \[ \text{(i) } \frac{3^x - 3^{-x}}{2^x - 2^{-x}}, \quad \text{(ii) } \frac{1}{x} \int_0^x \sqrt{(3t^2+4)} \, dt. \]

Prove that \[ \cos \theta \cos \theta + \cos^2 \theta \cos 2\theta + \dots + \cos^n \theta \cos n\theta = \cos^{n+1} \theta \cot \theta \sin n\theta. \]

Find the limits of \(\frac{x^3+y^3}{x-y}\) as \(x\) and \(y\) tend to zero

1. [(a)] along the curve \(y = x - x^2\);
2. [(b)] along the curve \(y = x - x^3\);
3. [(c)] along the curve \(y = x - x^4\).

Prove the identities

1. [(i)] \(\displaystyle\sum_{v=1}^n (2v-1)\sin(2v-1)\theta = \frac{\cos\theta\sin 2n\theta - 2n\sin\theta\cos 2n\theta}{2\sin^2\theta}\).
2. [(ii)] \(\displaystyle\sum_{\alpha,\beta,\gamma} \sin^2\alpha\sin(\beta-\gamma)\cos(\theta-\beta)\cos(\theta-\gamma) = -\sin(\beta-\gamma)\sin(\gamma-\alpha)\sin(\alpha-\beta)\cos^2\theta\).

Give an account of various methods of finding the sum of \(n\) terms of series of the form \(\sum a_n, \sum a_n x^n\), (1) when \(a_n\) is a polynomial in \(n\), (2) when the coefficients are connected by a relation of the form \[ a_{n+k} + p_1 a_{n+k-1} \dots + p_k a_n = 0, \] \(p_1, p_2 \dots\) being constants, i.e. independent of \(n\). To what extent do the two classes of series (1) and (2) overlap? Find \(1^4+2^4 \dots + n^4\); and shew without carrying out the work in detail how the sum can be found by each of your other methods as far as applicable.

Find the scale of relation, of the form \(u_{n+2}+pu_{n+1}+qu_n=0\), and the sum of the first \(n\) terms of the recurring series \[ 1+5+19+65+\dots. \]

If \[ S_n(\theta) = \sum_{r=1}^n \cos^r\theta \sin r\theta \] prove (by induction or otherwise) that, if \(0<\theta<\pi\), \[ S_n(\theta) = \cot\theta(1-\cos^n\theta \cos n\theta). \] Prove also that \(S_n(0) = 0\). If \(n\) is fixed, what is the limit of \(S_n(\theta)\) as \(\theta\to 0\)? Shew that the series converges as \(n\to\infty\) for all values of \(\theta\), but that its sum is not continuous.

The numbers \(u_1, u_2, u_3, \dots\) are connected by the relation \(u_n - 2u_{n+1}\cos\theta + u_{n+2}=0\). \(n=1, 2, \dots\), and \(\theta\) is not an integral multiple of \(\pi\). Shew that \(u_n = A \cos n\theta + B \sin n\theta\), and express \(A, B\) in terms of \(u_1, u_2\), and \(\theta\).

Prove that, if \[ -1 < x < 1, \] then \(x^n n^s\) tends to zero as the positive integer \(n\) tends to infinity. Sum to \(n\) terms the series whose \(r\)th term is \(rx^r\), and hence find for what values of \(x\) the infinite series is convergent, and find its sum. Sum to infinity, the series whose \(r\)th term is \(r x^r \cosh r\theta\), stating for what values of \(x, \theta\) the series is convergent.

A sequence of numbers \(u_1, u_2, u_3 \ldots\) is defined by the relations \begin{align*} u_1 &= a+b\\ u_n &= a+b-\frac{ab}{u_{n-1}}, \end{align*} where \(a+b \neq 0\). Show that if \(a \neq b\) then \[u_n = \frac{a^{n+1}-b^{n+1}}{a^n-b^n},\] and when \(a > b > 0\) determine the limit to which \(u_n\) tends as \(n\) tends to infinity. Find a formula for \(u_n\) when \(a = b\), and determine the limit to which \(u_n\) tends as \(n\) tends to infinity.

Let the sequence \((x_n)\) of positive numbers be defined by $$(1) \quad x_1 = 6, \quad \text{and} \quad (2) \quad x_{n+1} = \sqrt{8x_n - 15}.$$ Show that \(5 < x_{n+1} < x_n\) for all \(n\), and that \(x_n \to 5\) as \(n \to \infty\). Discuss what happens when (1) is replaced by \(x_1 = 4\).

A sequence of functions \(P_n(x)\), \(n = 0, 1, 2, \ldots\), is defined by setting \begin{align*} P_0(x) &= 1, \quad P_1(x) = x,\\ nP_n(x) &= (2n-1)xP_{n-1}(x) - (n-1)P_{n-2}(x) \end{align*} and requiring \begin{equation*} P_n(x) = \sum_{r=0}^{n} A(n, r)x^r. \end{equation*} if \(n \geq 2\). Show that \(P_n(x)\) is a polynomial of degree \(n\), say Construct a flow diagram for the evaluation of the coefficients in \(P_N(x)\) for a given value of \(N \geq 2\).

Let \(a\) and \(b\) be real numbers with \(a > 0\). Successive terms in the sequence \(\{f_n\}\) of real numbers are related by \[f_{n+1} = af_n + b\]

1. [(i)] If \(r\) is any real root of the polynomial \(x^3 - ax - b\), prove that \(f_n - r\) has the same sign for all values of \(n\).
2. [(ii)] Now suppose that \(x^3 - ax - b\) has three real roots \(r_1, r_2, r_3\) with \(r_1 < r_2 < r_3\). Prove that \(\{f_n\}\) is an increasing sequence if \(f_1 < r_1\) or \(r_2 < f_1 < r_3\) but is decreasing or constant otherwise.

The ``logistic'' difference equation is \begin{equation*} x_{n+1} = ax_n(1 - x_n), \end{equation*} where \(1 < a < 4\). Show that if either \(x_1 < 0\) or \(x_1 > 1\), then \(x_n \to -\infty\) as \(n \to \infty\), but if \(0 < x_1 < 1\), then \(0 < x_n < 1\) for all \(n\). Show further that if \(x_n\) tends to a finite limit \(x\) as \(n \to \infty\), then \(x = 0\) or \(x = 1 - 1/a\). By writing \(x_n = x + \epsilon_n\), and considering \(\epsilon_{n+1}/\epsilon_n\), or otherwise, show that sequences \(x_n\) with \(x_1\) sufficiently close to \(1 - 1/a\) get steadily closer to \(1 - 1/a\) provided \(a < 3\).

Let \(u_1\) be an odd positive integer greater than 1. For \(n > 1\), \(u_n\) is defined by the relation \begin{equation*} u_n = u_{n-1}^2 - 2. \end{equation*} Show that, for \(n > 1\), 1 is the highest common factor of \(u_n\) and \(u_m\) for \(1 \leq m \leq n-1\). Show further that 1 is the highest common factor of \(u_n\) and \(u_m - 1\) for \(1 \leq m \leq n\).

Two numbers \(a\) and \(b\) are given such that \(a > b > 0\). Two sequences \(a_n\) and \(b_n\) (\(n = 0, 1, 2, \ldots\)) are defined by the rules:

1. [(i)] \(a_0 = a\), \(b_0 = b\);
2. [(ii)] \(a_{n+1}\) is the arithmetic mean, and \(b_{n+1}\) is the harmonic mean, between \(a_n\) and \(b_n\), i.e. $$2a_{n+1} = a_n + b_n, \quad \frac{2}{b_{n+1}} = \frac{1}{a_n} + \frac{1}{b_n} \quad (n \geq 0);$$

(i) Find $$\lim_{n\to\infty} \{\sqrt{n^2+n+1}-n\}.$$ (ii) Positive numbers \(x_0\) and \(y_0\) are given. \(x_1\) and \(y_1\) are the arithmetic and geometric means of \(x_0\) and \(y_0\); \(x_2\) and \(y_2\) are the arithmetic and geometric means of \(x_1\) and \(y_1\); and so on. Show that \(x_n\) and \(y_n\) tend to finite limits as \(n\) tends to infinity, and that these limits are equal.

A set of functions \(y_n(x)\), \((n = 0, 1, 2, \ldots)\) is defined by $$y_n(x) = \cos(n \cos^{-1} x).$$ Show that

1. [(a)] \(y_{n+1} - 2xy_n + y_{n-1} = 0\);
2. [(b)] \(y_n\) is a polynomial in \(x\) of degree \(n\);
3. [(c)] \(y_n\) satisfies the differential equation $$(1-x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} + n^2y = 0;$$
4. [(d)] the roots of the equation \(y_n(x) = 0\) all lie in the range \(-1 < x < 1\).

The series of polynomials \(f_n(x)\) for \(n=0, 1, 2, \dots\) are defined by \[ f_n(x) = x^{2n+2}e^{1/x}(d/dx)^{n+1}e^{-1/x}. \] Prove that for \(n \ge 1\) \[ f_n(x) = -(2nx-1)f_{n-1}(x)+x^2f_{n-1}'(x). \] Hence show by induction that \(f_n(x)\) is a polynomial in \(x\) of degree \(n\), and find the coefficient of the highest term. Prove further that the equation \(f_n(x)=0\) has \(n\) distinct real roots.

A sequence of non-negative numbers \(u_0, u_1, u_2, \dots\) is defined by the recurrence relations \[ u_n^2 = 3u_{n-1}-2 \quad (n \ge 1) \] in terms of the first member of the sequence \(u_0\). It is given that \(1 < u_0 \le 2\). Show that \(u_n \ge u_0\) and that \[ 0 \le 4 - u_n^2 \le \left(\frac{3}{2+u_0}\right)^n (4-u_0^2) \] for all \(n \ge 0\). Hence, or otherwise, prove that \(u_n\) tends to a definite limit as \(n\) tends to infinity and evaluate this limit for each \(u_0\).

Explain briefly the theory of recurring series, shewing that if \(2r\) terms of the series are given it can in general be continued as a recurring series of the \(r\)th order in one way only. Find the \((n+1)\)th term of the recurring series \[ -2+2x+14x^2+50x^3+\dots. \]

The solid angle subtended at a point \(O\) by a plane area may be defined as the area cut off on a sphere of unit radius whose centre is \(O\) by the straight lines joining \(O\) to the perimeter of the plane area. Find the solid angle subtended by a circle at a point on the line through its centre and perpendicular to its plane in terms of \(\alpha\), the angle subtended at the point by a radius of the circle. \par Shew also that a rectangle of sides \(2a, 2b\) subtends a solid angle \[ 4\sin^{-1}\frac{ab}{\sqrt{(a^2+h^2)(b^2+h^2)}} \] at a point on the line through its centre and perpendicular to its plane, where \(h\) is the perpendicular distance of the point from the plane.

Show that, if \(f(x)\) is an increasing positive function for \(0 \leq x \leq 1\), then \[\frac{1}{n} \sum_{r=0}^{n-1} f\left(\frac{r}{n}\right) \leq \int_0^1 f(x) dx \leq \frac{1}{n} \sum_{r=1}^n f\left(\frac{r}{n}\right).\] Deduce that, for \(k \geq 0\), \[\left|n^{-k-1} \sum_{r=0}^n r^k - \frac{1}{k+1}\right| \leq \frac{1}{n}.\] Use similar arguments to show that \[\left|\sum_{r=n}^{2n} \frac{1}{r} - \log 2\right|\] can be made as small as we like by taking \(n\) large enough.

By using diagrams or otherwise, explain why \[\sum_{r=n}^{\infty} r^{-2} > \int_{n}^{\infty} x^{-2}dx > \sum_{r=n+1}^{\infty} r^{-2}.\] If we write \(A = \sum_{r=1}^{\infty} r^{-2}\), show that \[n^{-1} > A - \sum_{r=1}^{n} r^{-2} > (n+1)^{-1}.\] How large must we take \(n\) to ensure that \(\sum_{r=1}^{n} r^{-2}\) approximates \(A\) with an error of less than \(10^{-4}\)? Show that, for the same \(n\), \[(n+1)^{-1} + \sum_{r=1}^{n} r^{-2}\] approximates \(A\) with an error of less than about \(10^{-8}\).

Let \(f\) be a positive function of \(x\) with a negative first derivative for \(x \geq 1\). Show that \[\sum_{m=2}^{n} f(m) \leq \int_{1}^{n} f(x)dx \leq \sum_{m=1}^{n-1} f(m)\] where \(n > 2\) is an integer. Hence, or otherwise, show that (i) \(\sum_{m=1}^{n} \frac{1}{m^2} \leq 2\), for all \(n \geq 1\), (ii) \(\sum_{m=1}^{n} \frac{1}{m}\) is unbounded as \(n \to \infty\), (iii) \(0 \leq \left(\sum_{m=1}^{n} \frac{1}{m}\right) - \ln n \leq 1\), for all \(n \geq 1\). Show also that \[\sum_{m=4}^{n} \frac{1}{m\ln^2 m}\] is bounded as \(n \to \infty\).

Prove that, if \(x > 0\) and \(N\) is a positive integer, then \[\frac{1}{2^x} + \frac{1}{3^x} + \cdots + \frac{1}{(N+1)^x} < \int_1^{N+1} \frac{dx}{x^x} < 1 + \frac{1}{2^x} + \cdots + \frac{1}{N^x}.\] Deduce, or prove otherwise, that \(\sum_{n=1}^{\infty} n^{-x}\) is convergent when \(x > 1\) and divergent when \(x < 1\). Find the set of values of the real number \(\beta\) for which the infinite series \[\sum_{n=1}^{\infty} \frac{n^{\beta}}{n^{2\beta} - n^{\beta} + 1}\] is convergent, and the set of values of \(\beta\) for which it is divergent.

Let \(f(x)\) be a continuous decreasing function of \(x\) for \(x > 0\), and \(m\) and \(n\) be positive integers with \(m < n\). Show that $$\int_m^{n+1} f(x) \, dx < \sum_{r=m}^n f(r) < \int_{m-1}^n f(x) \, dx,$$ and hence that $$1.19 < \sum_{r=1}^{\infty} \frac{1}{r^3} < 1.22.$$

Let \(f(1) = 0\) and \[f(n) = 1 + \frac{1}{2} + \ldots + \frac{1}{n-1} - \log_e(n), \quad (n = 2, 3, \ldots).\] By interpreting \(f(n)\) as an area, or otherwise, show that \(f(n)\) is an increasing sequence, and that \(f(n) < 1\) for all \(n\). Show also that \[\frac{1}{2n(n+1)} < f(n+1) - f(n) < \frac{1}{n(n+1)}.\]

Show that, for \(r \geq 10\), \[(r-\frac{1}{2})(r+\frac{1}{2}) < r^2 < (r-\frac{39}{80})(r+\frac{41}{80}).\] Deduce that \[\frac{80}{761} \leq \sum_{r=10}^{\infty} \frac{1}{r^2} \leq \frac{2}{19}.\]

For \(a \leq x \leq b\) the function \(f(x)\) is positive and decreasing, and the graph of \(y = f(x)\) is concave upwards. Prove that \((b-a)f(b) < \int_a^b f(x) \, dx < \frac{1}{2}(b-a)\{f(a) + f(b)\}.\) If \(S(n,k) = \frac{1}{k+1} + \frac{1}{k+2} + \cdots + \frac{1}{nk},\) where \(k\) and \(n\) are positive integers, show, by splitting the range of integration of \(\int_1^x \frac{dx}{x}\) into suitable parts, that \(0 < \log n - S(n,k) < \frac{n-1}{2nk}.\) Deduce that \(\log n - \left( \frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{n^2} \right) \to 0\) as \(n \to \infty\).

Prove that, if \(y > x > 0\) and \(k > 0\), then \(x^k (y-x) < \int_x^y t^k dt < y^k (y-x).\) Hence show that \(\lim_{n \to \infty} \{n^{k-1}(1^k + 2^k + \ldots + n^k)\} = \frac{1}{k+1}.\)

Prove that $$\log \frac{n}{n-1} - \frac{1}{n} = \int_0^1 \frac{t}{(n-t)^n} dt \quad (n = 2, 3, \ldots).$$ Denoting the right-hand side by \(u_n\), prove that $$0 < u_n < \frac{1}{2(n-1)^n},$$ and that the series \(\sum_{n=2}^\infty u_n\) is convergent, with a sum \(U\) satisfying \(0 < U < \frac{1}{2}\). Deduce (or prove otherwise) that $$\sum_{n=1}^N \frac{1}{n} - \log N$$ tends to a limit \(\gamma\) as \(N \to \infty\), and that \(\frac{1}{2} < \gamma < 1\).

Prove that $$\int_1^n \log x \, dx < \sum_{r=2}^n \log r < \int_1^n \log x \, dx + \log n.$$ Hence, or otherwise, evaluate $$\lim_{n \to \infty} \frac{(n!)^{1/n}}{n}.$$

Show that the series \[ 1 + \frac{1}{2^k} + \frac{1}{3^k} + \dots \] is convergent if \(k>1\) but divergent if \(k=1\). Discuss the convergence of the series \[ 1 - \frac{1}{2^k} + \frac{1}{3^k} - \dots \] for real values of \(k\).

If \(m>1\), prove that \[ \int_m^{m+1} \frac{dt}{t} < \frac{1}{m} < \int_{m-1}^m \frac{dt}{t}. \] Hence, or otherwise, prove that, if \(n\) is a positive integer, \[ \log 2 < \frac{1}{n} + \frac{1}{n+1} + \dots + \frac{1}{2n-1} < \frac{1}{2n} + \log 2. \]

Using the fact that \begin{align} \lim_{n\to\infty}\left(\frac{b-a}{n}\sum_{m=1}^{n}f(a+m[b-a]/n)\right) = \int_{a}^{b}f(x)dx \end{align} show that \begin{align} \lim_{n\to\infty}\left(\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{n+n}\right) = \ln 2 \end{align} Evaluate \begin{align} \lim_{n\to\infty}\left(\frac{n}{n^2+1}+\frac{n}{n^2+4}+...+\frac{n}{n^2+n^2}\right) \end{align}

Find the limit of \[\left(\frac{\beta x^{\beta-1}}{x^\beta - a^\beta} - \frac{1}{x-a}\right)\] as \(x \to a\).

1. [(i)] Evaluate the limits
   1. [(ii)] By using the fact that \(u \ln u \to 0\) as \(u \to 0\), or otherwise, evaluate \begin{equation*} \lim_{x \to \pi/2}(\sin x)^{\tan x} \end{equation*}

If $$f(x) = \frac{(1+x)^{\frac{1}{2}} - 1}{1-(1-x)^{\frac{1}{2}}},$$ find (i) \(\lim_{x \to 0} f(x)\), and (ii) \(\lim_{x \to 0} \frac{df}{dx}\).

Define \(\int_a^b f(x)dx\) as the limit of a sum; using the integral expression for \(\log x\) or otherwise prove that \begin{align} \sum_{r=1}^n \frac{1}{8n + r} \to 2\log 3 - 3\log 2 \end{align} as \(n \to \infty\).

By considering \(\int_1^2 \log x dx\) evaluate the limit, as \(n\) tends to infinity, of $$\left[\left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right)\left(1+\frac{3}{n}\right)...\left(1+\frac{n-1}{n}\right)\right]^{\frac{1}{n}}.$$

(i) Explain how definite integrals may be obtained as the limit of suitable sums. Illustrate by obtaining \(\int_0^3 x^2 dx\) without making use of the relation \(dx^3/dx = 3x^2\). [You may assume that \(\displaystyle \sum_{n=1}^{N} n^2 = \frac{1}{6}N(N+1)(2N+1)\)] (ii) Assuming that \(\log n = \int_1^n x^{-1} dx\), find $$\lim_{n \to \infty} \left(\frac{1}{2n} + \frac{1}{2n+1} + \ldots + \frac{1}{3n}\right).$$

Find the limits of the following expressions \[\frac{x - \sin x}{x^3} \quad \text{and} \quad \frac{1 - \frac{1}{2}x^2 - \cos x}{x^4}\] as \(x \to 0\). Find also the limits of \[\frac{\cos \frac{1}{2}x}{x^2 - \pi^2}\] as \(x \to \pi\), and as \(x \to \infty\).

Find the following limits: $$\lim_{x \to 0} \frac{2\sin x - \sin 2x}{x^3}, \quad \lim_{x \to 0} x \sin\left(\frac{1}{x}\right), \quad \lim_{x \to 0} \frac{\cos ec x - \cot x}{x}, \quad \lim_{x \to 1} \frac{\cos \frac{1}{2}\pi x}{\log x}.$$

If \[f(x) = (\sin x - \sin a)^{-1} - (x - a)^{-1}\sec a\] evaluate \[\frac{d}{da}\left[\text{Lt}_{x \to a} f(x)\right] - \text{Lt}_{x \to a} f'(x).\]

(i) \(a\), \(b\), \(c\), \(d\) are positive numbers, \(c\) and \(d\) not being equal. Find the limit of $$\frac{(a^x - b^x)}{(c^x - d^x)}$$ as \(x\) tends to \(0\). (ii) If \(a\), \(b\), \(c\) are positive numbers, show that \(a^{\log_b c} = c^{\log_b a}\).

Find the limit, as \(x\) tends to zero, of \[ \frac{x\cos x - \sin x}{x^3}. \] Sketch the curve \[ y = \frac{\sin x}{x} \] and discuss briefly its form for small values of \(x\). Show that, for large values of \(x\), there are maxima and minima near the points \(x=(n+\frac{1}{2})\pi\), where \(n\) is an integer. Obtain a closer approximation to the values of \(x\) at the maxima and minima, with an error of order \(\dfrac{1}{n^3}\).

Find the limits as \(n\) tends to infinity of

1. [(i)] \(\displaystyle \frac{(n+1)^n}{n^{n+1}}\),
2. [(ii)] \(\displaystyle \int_0^1 x^n \cos x \,dx\),
3. [(iii)] the smallest positive solution of the equation \(x \tan nx = 1\).

Starting from some (stated) definition of \(\log x\), prove from first principles that \((\log x)/x \to 0\) as \(x \to \infty\). Investigate the limit of \(x^{1/x}\) when (i) \(x \to \infty\), (ii) \(x \to 0\) by positive values. Sketch the graph of \(y=x^{1/x}\) for \(x>0\).

Prove that \[ \int_1^x \frac{dt}{t+\alpha} \le \log x \le \int_1^x \frac{dt}{t-\alpha}, \] where \(x>0\) and \(\alpha>0\). Hence show that \[ \log x = \lim_{n\to\infty} n(\sqrt[n]{x}-1). \] Use the above expression for \(\log x\) to prove that \[ \log(x^m) = m \log x \] for positive integral values of \(m\).

If \[ f(x) = \int_0^\infty \frac{e^{-x^2t}}{1+t} dt \quad (x\neq 0), \] establish the inequalities \[ f(x) < \frac{1}{x^2}, \quad \text{and} \quad f(x) < \frac{1}{1-x^2}\log\frac{1}{x^2} \quad (x \neq 1). \] [Hint. For the second inequality use \(e^{-\alpha} \le 1/(1+\alpha)\) for \(\alpha \ge 0\).] Which inequality is the stronger (i) when \(|x|\) is very large, (ii) when \(|x|\) is very small? Prove that \[ x^2f(x) \to 1 \quad \text{as} \quad |x| \to \infty. \] Show that \[ e^{-x^2}f(x) = \int_{x^2}^1 \frac{du}{u} - \int_{x^2}^1 \frac{1-e^{-u}}{u} du + \int_1^\infty \frac{e^{-u}}{u} du; \] and deduce that \[ \frac{f(x)}{\log(1/x^2)} \to 1 \quad \text{as} \quad x \to 0. \]

Starting from any (stated) definition of the natural logarithm of a positive number \(x\), prove that

1. [(i)] \(\log xy = \log x + \log y\);
2. [(ii)] \(\frac{\log x}{x^{1/k}} \to 0\) as \(x \to \infty\), where \(k\) is any fixed positive integer.

Find the limits, as \(n \to \infty\), of

1. [(i)] \(\frac{\log a_1 + \log a_2 + \dots + \log a_n}{n \log n}\) where \(a_1, a_2, \dots, a_n\) are any \(n\) numbers included between \(n\) and \(2n\);
2. [(ii)] \(\frac{\sqrt{n^2+1} + \sqrt{n^2+2} + \dots + \sqrt{n^2+n}}{n^2}\).

Prove that, if \(f(x)\) is continuous for \(a \leq x \leq b\), then \[ \frac{1}{n} \sum_{\nu=0}^{n-1} f\left\{a+\frac{\nu}{n}(b-a)\right\} \] tends to a limit as \(n\to\infty\). Taking \(a=0, b=1\), and \(f(x)=x\log x\), deduce that \[ 1^1 2^2 \dots (n-1)^{n-1} = n^{\frac{1}{2}n^2} e^{-\frac{1}{4}n^2(1+\epsilon_n)}, \] where \(\epsilon_n \to 0\) as \(n \to \infty\).

Let \(p_r, q_r\) (\(r = 1, 2, \ldots\)) be two sequences such that \(p_r = q_{r+1} - q_r\) for all \(r \geq 1\). Evaluate \(\sum_{r=1}^N p_r\). Hence or otherwise evaluate

1. \(\sum_{r=1}^N \frac{1}{4r^2 - 1}\),
2. \(\sum_{r=1}^N \frac{1}{r(r+1)(r+2)}\),
3. \(\sum_{r=1}^N \sin r\theta\) [hint: first multiply by \(\sin \frac{1}{2}\theta\)].

For any fixed angle \(\theta\) with \(\sin \frac{1}{2}\theta \neq 0\), write \(S_N = \sum_{n=1}^{N} \sin n\theta.\) Prove that there exist numbers \(\varepsilon_M\) (which may depend on \(\theta\)) such that \(\varepsilon_M \to 0\) as \(M \to \infty\) and \(\left| \frac{1}{M+1} \sum_{n=N}^{M+N} S_n - \frac{1}{2}\cot \frac{1}{2}\theta \right| < \varepsilon_M\) for all positive \(M\), \(N\). For what values of \(\theta\) does the series \(\sum_{n=1}^{\infty} \sin n\theta\) converge?

Find \(\displaystyle \sum_{n=0}^N n\cos n\theta\). Prove that this series does not converge as \(N\) tends to infinity, for any given real value of \(\theta\).

By using the identity $$\frac{1}{y+1} = \frac{1}{y-1} - \frac{2}{y^2-1},$$ or otherwise, determine for what real values of \(x\) the series $$\sum_{n=1}^{\infty} \frac{2n}{x^{2n} + 1}$$ is convergent and show that when it is convergent the series has sum \(\frac{2}{x^2-1}\).

(i) Given that \(\sum_{n=1}^{\infty} n^{-2} = S_{\infty}\), find \(\sum_{n=1}^{\infty} n^{-2}(n+1)^{-2}\). (ii) Find (a) \(\sum_{n=1}^{\infty} nr^n\) and (b) \(\sum_{n=1}^{\infty} n^2 r^n\). What can be concluded about the 'sum to infinity' \(\sum_{n=1}^{\infty} n^2 r^n\)?

Show that $$1^2 - 2^2 + 3^2 - \ldots + (-)^{n-1}n^2 = (-)^{n-1}(n^2 + n)/2.$$ Find also the sum of $$1^3 - 2^3 + 3^3 - \ldots + (-)^{n-1}n^3,$$ distinguishing between the cases when \(n\) is odd and even.

\(a_1, a_2, \ldots, a_n\) are distinct numbers, and \(b_1 > b_2 > \cdots > b_n\). If \(\rho\) is a permutation of \((1, 2, \ldots, n)\), so that \(i\) becomes \(\rho(i)\), the number \(F(\rho)\) is defined by $$F(\rho) = a_{\rho(1)}b_1 + a_{\rho(2)}b_2 + \cdots + a_{\rho(n)}b_n.$$ Show that \(F(\rho)\) attains a maximum value (as \(\rho\) varies) when \(\rho\) is chosen so that $$a_{\rho(1)} > a_{\rho(2)} > \cdots > a_{\rho(n)}.$$ \((z_1, z_2, \ldots, z_{n1}, \ldots)\) is a sequence of positive integers, and \(x_i = x_j\) if and only if \(i = j\). Show that $$\sum_{i=1}^{\infty} \frac{1}{x_i(i + 1)}$$ is convergent. What is the largest possible value that this sum can take?

Prove that the infinite series \(\sum \frac{z^n}{n!}\) is convergent for all values of \(z\), real or complex. State any general theorems on series used in the proof. Sum the series \[ \sum_{n=1}^{\infty} \frac{n^3 z^{n-1}}{(n-1)!} \] Hence or otherwise sum the infinite series \[ 1 - \frac{3^3x^2}{2!} + \frac{5^3x^4}{4!} - \frac{7^3x^6}{6!} + \dots. \]

Sum the infinite series

1. \(\sum_{n=1}^\infty \frac{1}{n(n+2)n!}\);
2. \(\sum_{n=1}^\infty \frac{(-1)^{n+1}}{(3n-2)(3n-1)3n(3n+1)(3n+2)}\).

By considering the inequalities \[ \frac{1}{r(r+1)} < \frac{1}{r^2} < \frac{1}{r^2-1}, \] prove that \[ \frac{m}{ (m+1)(2m+1)} < \sum_{r=m+1}^{2m} \frac{1}{r^2} < \frac{m}{(m+1)(2m+1)} + \frac{3m+1}{4m(m+1)(2m+1)}. \] Hence find the value of \(\sum_{r=101}^{200} \frac{1}{r^2}\) with an error of less than \(2.10^{-5}\).

If \(f(n) = \sum_{r=1}^n \csc^2\frac{(2r-1)\pi}{4n}\), prove (by using the identity \(\csc^2\theta + \sec^2\theta = 4\csc^2 2\theta\) or otherwise) that \(f(2n)=4f(n)\), and hence evaluate \(f(2^k)\). Assuming that \(\sin\theta < \theta < \tan\theta\) when \(0 < \theta < \frac{1}{2}\pi\), prove that \[ f(n) > \frac{16n^2}{\pi^2}\sum_{r=1}^n \frac{1}{(2r-1)^2} > f(n)-n. \] Deduce that \[ \lim_{n\to\infty} \sum_{r=1}^{2^n} \frac{1}{(2r-1)^2} = \frac{\pi^2}{8}. \]

Using the notation \begin{align*} f(x) \ll g(x) \quad &\text{if } f(x)/g(x) \to 0 \text{ as } x \to \infty, \\ f(x) \prec g(x) \quad &\text{if } f(x)/g(x) \to \text{a limit between 0 and 1}, \\ f(x) \sim g(x) \quad &\text{if } f(x)/g(x) \to 1, \end{align*} arrange the following functions in order by means of the symbols \(\ll, \prec, \sim\). \[ 3^x, \quad x e^x, \quad \int_1^x t e^t dt, \quad \int_1^x 3^t dt. \]

Sum the series \[ 1^3 + 3^3 + 5^3 + \dots + (2n-1)^3. \]

Sum, for any positive integer \(n\),

1. [(i)] \(\sin\theta + \sin 2\theta + \sin 3\theta + \dots + \sin n\theta + \frac{1}{2}\sin(n+1)\theta\),
2. [(ii)] \(\sin\frac{\theta}{2} + \sin\frac{3\theta}{2} + \sin\frac{5\theta}{2} + \dots + \sin\frac{2n+1}{2}\theta\);

Sum the series \[ \sum_{r=1}^n \frac{1}{r(r+1)(r+2)}, \quad \sum_{r=1}^\infty \frac{r}{2^r}, \quad \sum_{r=1}^\infty \frac{1.3.5 \dots (2r-1)}{3.6.9 \dots 3r}. \]

Find the sums of the infinite series

1. \(\sin\theta + r\sin 2\theta + \dots + r^{n-1}\sin n\theta + \dots\)
2. \(\sin\theta + r\sin 3\theta + \dots + r^{n-1}\sin(2n-1)\theta + \dots\),

Sum the series

1. [(i)] \(\frac{2^3}{1!} + \frac{3^3}{2!} + \frac{4^3}{3!} + \dots\) to infinity,
2. [(ii)] \(\frac{1}{1.2.3} + \frac{1}{2.3.4} + \frac{1}{3.4.5} + \dots\) to \(n\) terms.

Find the sum of \(n\) terms of the series \(1^3+2^3+3^3+\dots\). Find also the sum to \(n\) terms of the series \[ \frac{x}{1+x} + \frac{x^2}{1+x+x^2+x^3} + \dots + \frac{x^{2^{n-1}}}{1+x+\dots+x^{2^n-1}}; \] and shew that the sum to infinity is \(x\) or 1 according as \(x\) is numerically less than or greater than 1.

Prove that the infinite series whose \(n\)th terms are (i) \(\frac{n^2}{2^n}\), (ii) \(\frac{n+2}{n(n+1)(n+3)}\), are convergent, and find their sums.

Shew that

1. [(i)] \(1+2(\cos\alpha+\cos 2\alpha+\dots+\cos n\alpha) = \sin(n+\frac{1}{2})\alpha \operatorname{cosec} \frac{\alpha}{2}\).
2. [(ii)] \(1+2(\cos\alpha\cos\theta+\cos 2\alpha\cos 2\theta+\dots+\cos n\alpha\cos n\theta) = \frac{\cos n\alpha \cos(n+1)\theta - \cos(n+1)\alpha\cos n\theta}{\cos\theta-\cos\alpha}\).

Sum to \(n\) terms the series

1. [(i)] \(\cos\alpha.\cos\alpha + \cos^2\alpha.\cos 2\alpha + \cos^3\alpha.\cos 3\alpha + \dots\),
2. [(ii)] \(\sec^2\frac{\pi}{4n} + \sec^2\frac{3\pi}{4n} + \sec^2\frac{5\pi}{4n} + \dots\).

Prove that the series \[ 1^2 + 2^2x + 3^2x^2 + 4^2x^3 + \dots \] is convergent when \(x\) lies between \(-1\) and \(+1\). \par Shew that if \(x=0.9\) the sum to infinity is 54,100.

Prove that if \(p_n/q_n\) is the \(n\)th convergent of \(\displaystyle\frac{a_1}{b_1+}\frac{a_2}{b_2+}\frac{a_3}{b_3+}\dots\), then \[ p_n = b_np_{n-1}+a_np_{n-2}. \] Find the value of \[ \frac{1}{1+}\frac{x}{1-x+}\frac{x}{2-x+}\dots\frac{x}{n+1-x}. \]

Sum the series:

1. \(\dfrac{1}{1.2} + \dfrac{1}{2.3} + \dfrac{1}{3.4} + \dots\) to \(n\) terms;
2. \(1^2+2^2x+4^2x^2+5^2x^3+7^2x^4+8^2x^5+\dots\) to infinity.

If \(|x|<1\), sum to infinity the series whose \(n\)th terms are

1. [(i)] \(n^2 x^n\),
2. [(ii)] \(\log(1+x^{2^n})\).

By use of the identity \[(1+y)(1-y+y^2-\ldots+(-y)^n) \equiv 1-(-y)^{n+1},\] or otherwise, prove that, for any \(n > 0\), the value of \(\tan^{-1} x\) in the range \((-\frac{1}{2}\pi, \frac{1}{2}\pi)\) lies between \(s_n\) and \(s_{n+1}\), where \[s_n = \sum_{r=1}^n \frac{(-1)^{r+1} x^{2r-1}}{2r-1}.\] Show that \[\sum_{n=1}^\infty \frac{(-1)^{n+1}}{(2n-1) \cdot 3^n}\] converges to \(\pi/(6\sqrt{3})\).

For \(n > 2\), prove by induction that $$(1-a_1)(1-a_2)\ldots(1-a_n) > 1-(a_1+a_2+\ldots+a_n),$$ where \(a_1, a_2, \ldots, a_n\) are positive numbers less than unity. Expanding \((1+x/n)^n\) by the binomial theorem, show that, for \(n\) a positive integer greater than 2, and \(x\) positive, $$S_n - \frac{x^2}{2n}S_{n-2} < \left(1+\frac{x}{n}\right)^n < S_n,$$ where $$S_n = \sum_{r=0}^n x^r/r!.$$ Given that, as \(n \to \infty\), \(S_n\) approaches a finite limit (dependent on \(x\)), show that \((1+x/n)^n\) approaches the same limit.

If \(f(x) = \sin(a\sin^{-1}x)\), \(-1 \leq x \leq 1\), show that \begin{equation*} (1-x^2)f''(x) - xf'(x) + a^2f(x) = 0. \end{equation*} Use Leibnitz' theorem to show that \begin{equation*} f^{(n+2)}(0) = (n^2-a^2)f^{(n)}(0), \end{equation*} and hence find \(\sin(5\sin^{-1}x)\) as a polynomial in \(x\). (Assume that there is such a polynomial.)

Prove that if \(u\) and \(v\) are functions of \(x\) and if \(n\) is a positive integer then \begin{equation*} \frac{d^n}{dx^n}(uv) = \sum_{r=0}^{n} \binom{n}{r} \frac{d^r u}{dx^r} \frac{d^{n-r} v}{dx^{n-r}} = \frac{d^n u}{dx^n}v + \binom{n}{1} \frac{d^{n-1} u}{dx^{n-1}} \frac{dv}{dx} + \cdots + \binom{n}{r} \frac{d^{n-r} u}{dx^{n-r}} \frac{d^r v}{dx^r} + \cdots + u\frac{d^n v}{dx^n}, \end{equation*} where \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\). The functions \(L_n\) are defined by \begin{equation*} L_n(x) = e^x \frac{d^n}{dx^n}(e^{-x}x^n). \end{equation*} Show that \(L_n\) is a polynomial of degree \(n\). By considering \begin{equation*} \frac{d^{n+1}}{dx^{n+1}}(e^{-x}x^n), \end{equation*} prove that \begin{equation*} L_{n+1}(x) = (n+1-x)L_n(x) + x\frac{d}{dx}L_n(x). \end{equation*}

Let \(y(x) = \sin^{-1}x\), and write \(y^{(r)}(x)\) for the value of the \(r\)th derivative \(\frac{d^r y}{dx^r}\) at the point \(x\). Prove that \((1-x^2)y^{(2)}(x) - xy^{(1)}(x) = 0\), and deduce that for all \(n \geq 0\) \((1-x^2)y^{(n+2)}(x) - (2n+1)xy^{(n+1)}(x) - n^2y^{(n)}(x) = 0\). Hence show that for all \(r \geq 0\) \(y^{(2r)}(0) = 0\), \(y^{(2r+1)}(0) = \left[\frac{(2r)!}{2^r r!}\right]^2\).

(i) Show that if \(|x| < 1\) then \[(1+x)(1+x^2)(1+x^4)(1+x^8)\ldots(1+x^{2^n}) \to \frac{1}{1-x}\] as \(n \to \infty\). (ii) Show that \(\sum_{r=1}^{\infty} \frac{x^{2^r-1}}{1-x^{2^r}}\) converges to \(\frac{x}{1-x}\) if \(|x| < 1\) and to \(\frac{1}{1-x^{-1}}\) if \(|x| > 1\).

Let \(\displaystyle L(x) = \int_1^x \frac{ds}{s}\) for \(x > 0\).

1. [(i)] Prove that \(L(xy) = L(x) + L(y)\).
2. [(ii)]Show that, for \(t \neq 1\), \[\frac{1}{1-t} = 1 + t + t^2 + \ldots + t^{n-1} + \frac{t^n}{1-t},\] and deduce that \[L(1-x) = -x - \frac{x^2}{2} - \ldots - \frac{x^n}{n} - \int_0^x \frac{t^n}{1-t}dt.\]
3. [(iii)] For \(|x| < 1\), show that \[\left|\int_0^x \frac{t^n}{1-t}dt\right| \leq \frac{1}{n+1} \frac{|x|^{n+1}}{1-|x|},\] and deduce that \[L(1-x) = -x - \frac{x^2}{2} - \ldots - \frac{x^n}{n} - \ldots\]

A sequence \(a_0, a_1, a_2, \ldots\) is defined by the following recurrence relation: \begin{equation*} a_n = a_0a_{n-1} + a_1a_{n-2} + \ldots + a_{n-1}a_0, \quad a_0 = 1 \end{equation*} Setting \(f(x) = \sum_{n=0}^{\infty}a_nx^n\), show that \(f(x)\) satisfies the equation \begin{equation*} xf(x)^2 - f(x) + 1 = 0. \end{equation*} Deduce that \(a_n = \frac{1}{n+1}\binom{2n}{n}\). [The convergence of series may be assumed.]

If two variables \(x\) and \(z\) are related by \[z = x + \lambda g(z)\] where \(\lambda\) is a constant, then any smooth function \(F(z)\) satisfies Lagrange's Identity \[F(z) \equiv F(x) + \lambda g(x)\frac{dF(x)}{dx} + \sum_{n=2}^{\infty} \frac{\lambda^n}{n!}\frac{d^{n-1}}{dx^{n-1}}\left(\{g(x)\}^n\frac{dF(x)}{dx}\right),\] which you may use without proof. (i) By using Lagrange's identity, or otherwise, show that one root of the equation \[4z = 2 + z^3\] is given by \[z = \sum_{n=0}^{\infty} \frac{(3n)!}{(2n+1)!n!}\frac{1}{2^{4n+1}}\] (ii) By considering \(z = x + \lambda(z^2 - 1)\), or otherwise, prove the identity \[0 \equiv x^3 + \sum_{n=2}^{\infty} \frac{x^n}{n!}\frac{d^{n-1}}{dx^{n-1}}(x^2-1)^n\] [You may assume all series converge.]

Prove that if \(|x| < 1\) then \(\sum_{n=1}^{\infty} x^n\) is convergent. Prove that, if \(0 < \theta < 1\), \(\sum_{n=1}^{\infty} \sin(\theta^n)\) is convergent to sum \(S\), where $$\frac{\sin\theta}{1-\theta} < S < \frac{\theta}{1-\theta}.$$ Is \(\sum \cos(\theta^n)\) also convergent?

1. [(i)] Evaluate $$\int_1^{\infty} \frac{dx}{x\sqrt{1 + x^2}}.$$
2. [(ii)] If \(x\) is small, find (to the first order in \(x\)) the error in the approximation $$(1 + x)^{1/x} \simeq e.$$

A sequence of functions \(f_n(x)\), \(n=0, 1, 2, \dots\), is defined by \[ \begin{cases} f_0(x) = 1 \\ f_{n+1}(x) = (1+x)^{f_n(x)} \end{cases} \] Assuming that \(f_n(x)\) can be expanded in powers of \(x\), show that \[ f_n(x) = 1+x+x^2+\frac{3}{2}x^3+\dots \text{ for } n\ge3. \] Show that \(f_{n+1}(x)=f_n(x)+x^{n+1}+\text{higher powers of } x\). Deduce that the coefficient of \(x^m\) in the expansion of \(f_n(x)\) in powers of \(x\) is independent of \(n\) for \(n \ge m\).

Obtain in its simplest form the derivative of \[ f(x) = \tfrac{1}{2}x + \sin x + \tfrac{1}{2}\sin 2x + \dots + \tfrac{1}{n}\sin nx + \frac{\cos(n+\tfrac{1}{2})x-k}{(2n+1)\sin\tfrac{1}{2}x}. \] Prove that, if \(k > 1\), the function \(f(x)\) attains its greatest value in the interval \(0 < x < 2\pi\) for the value \(x=\pi\). Prove that, if \(k < -1\), \(f(x)\) takes its least value in \(0 < x < 2\pi\) for \(x=\pi\). Deduce that the infinite series \[ \sin x + \tfrac{1}{2}\sin 2x + \tfrac{1}{3}\sin 3x + \dots \] is convergent for \(0 < x < 2\pi\), and find its sum.

(i) Find \(\lim_{x \to 1} \frac{x^K-1}{x-1}\), when \(K\) is a positive integer; deduce the result for \(K\) a positive rational number. (ii) Find \[ \lim_{n \to \infty} \left(\frac{1^p+2^p+\dots+n^p}{n^{p+1}}\right). \]

Differentiate \[ \tan^{-1} \frac{1+x}{1-x}, \quad \log (\tan x + \sec x). \] Find the \(n\)th differential coefficients of \(\sin x\) and \(\sin^3 x\).

Prove that \[ \frac{1}{3\left(1 - \frac{1}{2^2}\right)} - \frac{1}{4\left(1 + \frac{1}{3^2}\right)} + \frac{1}{5\left(1 - \frac{1}{4^2}\right)} - \dots = \frac{1}{2}. \]

By taking logarithms, or otherwise, find the limits of the positive value of \(\left(1+\frac{1}{x}\right)^x\) as \(x\) tends (a) to zero, (b) to infinity, positive values only of \(x\) being considered throughout. Draw roughly the graph of the function.

Prove by differentiation (or otherwise) that if \(x>0\), \(\log_e(1+x)\) lies between the sums to \(n\) and \(n+1\) terms of the series \[ x - \frac{x^2}{2} + \frac{x^3}{3} - \dots. \] Deduce that if \(0 < x \le 1\), \(\log_e(1+x)\) is equal to the sum of the series to infinity.

By repeated integration by parts, or otherwise, shew that \[ f(x) = f(0) + \frac{x}{1!}f'(0) + \dots + \frac{x^n}{n!}f^{(n)}(0) + \int_0^x \frac{(x-t)^n}{n!} f^{(n+1)}(t)dt. \] Hence prove that, for \(-1 < x < 1\), \[ \log_e(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots \text{ ad inf.} \]

Prove that a function which vanishes with \(x\), is continuous, and has a differential coefficient positive for all positive values of \(x\), is itself positive for all positive values of \(x\). \par Prove that each of the functions \[ 1-\cos x, \quad x-\sin x, \quad \tfrac{1}{2}x^2-1+\cos x, \quad \tfrac{1}{6}x^3-x+\sin x, \quad \tfrac{1}{24}x^4-\tfrac{1}{2}x^2+1-\cos x, \] is positive for all positive values of \(x\). Hence obtain expansions of \(\cos x\) and \(\sin x\) in powers of \(x\) valid for all real values of \(x\).

Prove that

1. [(i)] \(x - \frac{1}{3}x^3 + \frac{1}{5}x^5 - \frac{1}{7}x^7 + \dots = \tan^{-1}x\);
2. [(ii)] \(\sin 2x - \frac{1}{2}\sin 4x + \frac{1}{3}\sin 6x - \dots = x\),

Obtain an expression for \(\sin x\) as a power series in \(x\), and give an expression for the remainder after \(n\) terms.

Shew that \[ \sum_{m=0}^{N} \frac{\cos m\phi}{\cos^m \theta} = \frac{\cos^2 \theta - \cos\theta\cos\phi}{\cos^2\theta-2\cos\theta\cos\phi+1} - \frac{\cos(N+1)\phi}{\cos^{N-1}\theta} + \frac{\cos N\phi}{\cos^N\theta}. \] Indicate how the value of \(\sum_{m=0}^{N} \frac{m\sin(m-1)\phi}{\cos^{m+1}\phi}\) could be found from the above equation.

If \(0 < x < 1\), shew that \(n^2x^n \to 0\), as \(n \to \infty\). Find the limit as \(n \to \infty\) of \[ \frac{x^{2n}}{n+x^{2n}}; \] distinguish between the two cases.

Prove that the series \(\sum_0^\infty x^n \sinh(n+1)\alpha\) is convergent if \(x\) is numerically less than \(e^{-\alpha}\), \(\alpha\) being assumed positive and the sum is \(\sinh\alpha / (1 - 2x \cosh\alpha + x^2)\); but that the series \(\sum_0^\infty x^n \sin(n+1)\alpha\) is convergent provided \(x\) is numerically less than unity, the sum being \(\sin\alpha / (1-2x\cos\alpha+x^2)\).

Let \(f(x) = 2\cos x^2 - \frac{1}{x^2}\sin x^2\) for \(x \geq 1\). Find \(\int_1^t f(x)dx\) for \(t \geq 1\), and show that it tends to a limit as \(t \to \infty\) but that \(f(x)\) does not tend to zero as \(x \to \infty\). Give an example of a function \(h(x) \geq 0\), defined for \(x \geq 1\) and such that \(\int_1^t h(x)dx \to \infty\) as \(t \to \infty\) but \(h(x) \to 0\) as \(x \to \infty\). Give an example of a function \(l(x) \geq 0\), defined for \(x \geq 1\), and such that \(dl/dx > 0\) for all \(x \geq 1\) but \(l(x) \leq 1\) for all \(x \geq 1\). [It may be helpful to sketch such an \(l(x)\); but an explicit formula should be given.]

The following are three properties that may or may not belong to a sequence \((a_n)\) of strictly positive real numbers: \begin{align} (P_1) \quad & a_n \to 0 \text{ as } n \to \infty.\\ (P_2) \quad & \sum_{n=1}^\infty a_n \text{ converges}.\\ (P_3) \quad & \text{There is a constant } C \text{ such that } na_n < C \text{ for all values of } n. \end{align} For each pair of integers \((i,j)\) with \(1 \leq i \leq 3\), \(1 \leq j \leq 3\), \(i \neq j\), establish whether the statement `\(P_i\) implies \(P_j\)' is true or false. [You may quote without proof the behaviour of standard series.]

Explain carefully what is meant by the statement that a function of a real variable \(x\) is continuous at a particular value \(x_0\) of \(x\). The function \(f(x)\) of the real variable \(x\) takes the value 0 whenever \(x\) is irrational, and the value \(x^2(1-x^2)\) whenever \(x\) is rational. Find all values of \(x\) at which \(f(x)\) is continuous. Determine which (if either) of the following statements defines a real number, and find each number so defined:

1. [(i)] `\(k\) is the largest real number such that \(f(x) = k\) for some \(x\).'
2. [(ii)] `\(m\) is the smallest real number such that \(f(x) \leq m\) for all \(x\).'

\(a_0, a_1, a_2, \ldots\) is a sequence of real numbers. Explain carefully what the following statements mean: \begin{align} (i) \quad & a_n \to a \text{ as } n \to \infty;\\ (ii) \quad & \sum_{n=0}^{\infty} a_n = b. \end{align} Show from first principles that if \(|x| < 1\) then \(x^n \to 0\) as \(n \to \infty\), and $$\sum_{n=0}^{\infty} x^n = (1-x)^{-1}.$$ [You may find the substitution \(|x| = 1/(1+y)\) helpful.]

The function \(f(x)\) is defined on the interval \(0 < x < 1\) as follows: (a) if \(x\) is rational, and \(x = p/q\) in lowest terms, then \(f(x) = 1/q\); (b) if \(x\) is irrational, \(f(x) = 0\). Show that if \(\epsilon\) is greater than \(0\), there are only finitely many points of the interval for which \(f(x) \geq \epsilon\). Deduce that \(f\) is continuous at the irrational points of the interval, and is discontinuous at the rational points.

Sketch the curves described by the following equations:

1. [(i)] \(y^2 = x(x-2)^3\);
2. [(ii)] \(y = \lim_{n \to \infty} \frac{x^{2n} \sin \frac{1}{2}\pi x + x^2}{x^{2n} + 1}\).

Define \(f_n(x) = n^2 x (1-x)e^{-nx}\) for \(0 \leq x \leq 1\), \(n = 0, 1, 2 \ldots\). Show that, for each \(x\) such that \(0 < x \leq 1\), \[\lim_{n \to \infty} f_n(x) = 0.\] Show also that \[\lim_{n \to \infty} \int_0^1 f_n(x) dx \neq \int_0^1 \lim_{n \to \infty} f_n(x) dx.\] [You may assume that \(\lim_{n \to \infty} nb^{-n} = 0\) for \(b > 1\).]

Let \(a_1\), \(a_2\), ... be an infinite sequence of real numbers. For each positive integer \(n\) let \(k(n)\) be the largest integer \(\leq n\) satisfying \(a_{k(n)} \geq a_j\) for \(j = 1, ..., n\); thus \(a_{k(n)}\) is the largest of the numbers \(a_1, ..., a_n\) and \(k(n)\) is the last place in which it occurs. Prove that either (i) \(k(n)\) tends to \(+\infty\) as \(n \to \infty\) or (ii) \(k(m) = k(m+1) = k(m+2) = ...\) for some integer \(m\). Deduce that there are integers \(m_1\), \(m_2\), ... with \(m_1 < m_2 < ...\) and either \[a_{m_1} \leq a_{m_2} \leq ... \quad \text{or} \quad a_{m_1} \geq a_{m_2} \geq ... \]

The function \(f(x)\) is defined, for \(x > 0\), by the formula $$f(x) = \int_0^{\pi/2} \frac{d\theta}{x + \cos\theta}.$$ Evaluate \(f(x)\), distinguishing the cases (i) \(0 < x < 1\), (ii) \(x = 1\), (iii) \(x > 1\), and expressing the results in cases (i) and (iii) in terms of the variable \(u\) defined by $$u^2 = \frac{|x-1|}{x+1}, \quad u > 0.$$ Prove from your results, or from the original definition, that \(f(x) \to f(1)\) when \(x \to 1\) from below and from above.

Starting with any definition you please, establish the principal properties of the function \(\log x\), including a proof that \((\log x)/x^k\) tends to zero as \(x\) tends to infinity, for any positive number \(k\).

Under what circumstances is a function \(f(x)\) said to be continuous at \(x=k\)? The constants \(a\) and \(b\) are both greater than 1; \(f(x)\) is any function such that \(f(ax) = bf(x)\) for all values of \(x\); \(f(0)=0\). If for \(-1 \le x \le 1\), \(|f(x)|

A plane area is formed of the circle \(r=a\) and the portions of the four loops of the curve \(r=2a\sin 2\theta\) exterior to the circle. Shew that the whole area of the figure is \((\frac{2}{3}\pi + \sqrt{3})a^2\).

Prove that the series \[ 1 + \frac{1}{2^a} + \frac{1}{3^a} + \frac{1}{4^a} + \dots + \frac{1}{n^a} + \dots \] is divergent if \(a = 1\) or \(a < 1\), and convergent if \(a > 1\). Prove in another way that the series \[ S=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\dots+\frac{1}{n}+\dots \] is divergent by shewing that the sum of each group of fractions whose denominators contain the same number of digits exceeds \(9/10\). Deduce that, if all fractions in which the figure 0 occurs are omitted from the series \(S\), the sum of those fractions that remain whose denominators contain \(r\) digits is less than \(9/10^{r-1}\), and that the fractions that remain form a convergent series. Shew also that, if all fractions are omitted from \(S\) in which any other digit than 0, e.g. 7, occurs, the fractions that remain form a convergent series.

Define limit and convergent series. Taking \(a\) to be positive, discuss the limits of \[ \text{(i) } a^n, \quad \text{(ii) } \frac{a^n-1}{a^n+1} \] as the positive integer \(n\) tends to infinity. Discuss as completely as you can the convergence of the series \[ 1 + z + z^2 + z^3 + \dots \quad (\text{\(z\) real or complex}). \]

(i) Shew that, if \(x > 0\), then \(x^{1/n} \to 1\) as \(n \to \infty\). \par (ii) Shew that, if \(a>0\) and \(a_n \to a\), then \(\sqrt{a_n} \to \sqrt{a}\).

What is meant by the statements (i) that a sequence \(s_n\) tends to a limit as \(n \to \infty\), (ii) that an infinite series is convergent? Prove

1. [(i)] \(x^n \to 0\) as \(n \to \infty\), if \(-1 < x < 1\),
2. [(ii)] \(\sum_0^\infty x^n\) is convergent, if \(-1 < x < 1\).

Define "convergent sequence of real numbers." Prove that, if \(a_n \to a\) and \(b_n \to b\) as \(n\to\infty\), then \(a_n b_n \to ab\). If \(c_n\) is a sequence of positive numbers and if \[\frac{c_{n+1}^2}{c_n^2} \to c^2,\] shew that \[c_n^{1/n} \to |c|.\] If \(a_n < b_n\) and if \(a_n \to a\), \(b_n \to b\), shew that \[a \le b.\]

Explain in precise language what you mean by the statement that \(u_n\) tends to a limit \(l\) as \(n\) tends to infinity, and evaluate \(\lim_{n\to\infty} x^n\) when it exists, considering the special cases which may arise for various values of \(x\). \par Prove that \[ x\sin\alpha+x^2\sin 2\alpha+\dots+x^n\sin n\alpha = \frac{x\sin\alpha}{1-2x\cos\alpha+x^2} - \frac{\sin(n+1)\alpha - x\sin n\alpha}{1-2x\cos\alpha+x^2}x^{n+1}, \] and examine fully the convergence of the series as \(n\) tends to infinity.

Define a convergent series. State and prove the theorem used in discussing the convergency of such series as \[ \frac{1}{1} - \frac{1}{4} + \frac{1}{7} - \frac{1}{10} + \frac{1}{13} - \dots \] and \[ 2 - \frac{5}{4} + \frac{8}{7} - \frac{11}{10} + \frac{14}{13} - \dots. \] Prove that the second series can be made convergent by bracketing the terms in pairs.

Explain and illustrate the concept of convergence in connexion with infinite series. Discuss the convergence or otherwise of the series

1. [(i)] \(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\dots\),
2. [(ii)] \(\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\dots\),
3. [(iii)] \(\frac{1}{1} + \frac{1}{1+2} + \frac{1}{1+2+3} + \frac{1}{1+2+3+4} + \dots\).

What is meant by the statement that the series \(u_1+u_2+u_3+\dots\) is convergent? Discuss the convergence of the series

1. [(i)] \(\sum_1^\infty \frac{1}{n^{1+a}}\),
2. [(ii)] \(\sum_1^\infty \frac{(-1)^n}{1+n^a}\).

Prove that an infinite series \(u_1+u_2+u_3+\dots\) is convergent or divergent according as when \(n\) tends to infinity \(u_{n+1}/u_n\) tends to a limit less than or greater than unity. State and prove a test for the case in which the limit of \(u_{n+1}/u_n\) is unity. Examine the convergency or divergency of the series whose \(n\)th terms are \(n^4/n!, (n!)^2 x^n/3n!\).

Show that the series \[ 1 - \frac{1}{1+x} + \frac{1}{2} - \frac{1}{2+x} + \frac{1}{3} - \frac{1}{3+x} + \dots \] is convergent, provided that \(x\) is not a negative integer.

Shew that the series \[ \frac{1}{1^p}+\frac{1}{2^p}+\frac{1}{3^p}+\dots \] is convergent only when \(p>1\). Discuss the convergency of the series \[ \frac{1^p}{2^q}+\frac{2^p}{3^q}+\frac{3^p}{4^q}+\dots \] where \(p\) and \(q\) are positive numbers.

Shew that the series \(\frac{1}{1^{1+\kappa}}+\frac{1}{2^{1+\kappa}}+\frac{1}{3^{1+\kappa}}+\dots\) converges only if \(\kappa > 0\). Discuss the convergence of the series whose \(n\)th term is \(\frac{n^a}{(n+1)^b}\), where \(a, b\) are given positive numbers.

Show that, if $$f(x) = \sum_{n=1}^{\infty} a_n \sin nx, \qquad (1)$$ then $$a_n = \frac{2}{\pi} \int_0^{\pi} f(x) \sin nx \, dx. \qquad (2)$$ Assuming that the function $$f(x) = x(\pi - x) \qquad (0 \leq x \leq \pi)$$ can be expressed as an infinite series $$\sum_{n=1}^{\infty} a_n \sin nx,$$ and that the coefficients are still given by the formula (2), show that in this case $$a_{2m} = 0, \quad a_{2m+1} = \frac{8}{\pi(2m+1)^3},$$ and hence sum the series $$1 - \frac{1}{3^3} + \frac{1}{5^3} - \frac{1}{7^3} + \cdots$$

Prove that \[\int_0^{2\pi} \sin nx \sin mx\, dx = 0\] when the positive integers \(n\) and \(m\) are not equal, and evaluate the integral for the case when \(n = m\). Let \(f(x)\) be a periodic function with period \(2\pi\), which may be expressed in the form \[f(x) = \sum_{n=1}^{\infty} a_n \sin nx\] for some constants \(a_n\). Use the results of the first part to obtain expressions for the \(a_n\) in terms of \(f(x)\) by multiplying by \(\sin mx\) and integrating term by term. [You may assume that this procedure is justified.] We now seek, for fixed \(N\), to choose \(a_n\) so that the sum \[\sum_{n=1}^{N} a_n \sin nx\] approximates \(f(x)\) as closely as possible, in the sense that \[\int_0^{2\pi} \left\{f(x) - \sum_{n=1}^{N} a_n \sin nx\right\}^2\, dx\] is minimal. By differentiating with respect to each \(a_n\) separately, show that the solution is given by \(a_n = \hat{a}_n\).

Suppose, if possible, that \(\pi^2 = a/b\), where \(a\) and \(b\) are positive integers. Let \[f(x) = \frac{x^n(1-x)^n}{n!},\] \[G(x) = b\pi \sum_{r=0}^{n} (-1)^r\pi^{2n-2r}f^{(2r)}(x),\] where \(f^{(2r)}(x)\) denotes the \((2r)\)th derivative of \(f(x)\), and \(n\) is a positive integer. Prove that \[\frac{d}{dx}\{G'(x)\sin\pi x - \pi G(x)\cos\pi x\} = \pi^2 a^n \sin\pi x \cdot f(x),\] and deduce that \[\pi\int_0^1 a^n\sin\pi x \cdot f(x)dx = G(0) + G(1).\] Prove that the right-hand side of this equation is an integer. Show also that, by choice of \(n\) sufficiently large, the left-hand side can be made to lie strictly between 0 and 1. Establish a contradiction to the original supposition that \(\pi^2\) is rational.

The function \(f\) satisfies the equation \[f(x) = \frac{1}{4}\left(f\left(\frac{x}{2}\right)+f\left(\frac{x+\pi}{2}\right)\right)\] for \(0 < x < \pi\). Show that if there is a constant \(M\) such that \(|f(x)| < M\) for \(0 < x < \pi\), then \(f(x) = 0\) whenever \(0 < x < \pi\). Given that \[\sum_{n=1}^{\infty} \frac{1}{(x-n\pi)^2} < 1\] for \(|x| \leq \frac{\pi}{2}\) and \(\textrm{cosec}^2 x - \frac{1}{x^2} < 1\) for \(0 < x < \frac{\pi}{2}\), prove that \[\textrm{cosec}^2 x = \sum_{n=-\infty}^{\infty} \frac{1}{(x-n\pi)^2}\] whenever \(x\) is not a multiple of \(\pi\).

Show that if \(m\) and \(n\) are integers with \(m \geq n \geq 1\), then \(1/m! \leq n^{n-m}/n!\). Deduce that, if \(n \geq 3\), then \[0 < n! \sum_{r=n+1}^{\infty} \frac{1}{r!} < 1.\] Shoe that if \[e = \sum_{r=0}^{\infty} \frac{1}{r!}\] then \(n!e\) is not an integer, and so \(e\) is irrational (ie \(e\) is not of the form \(p/q\) for any integers \(p\) and \(q\)). Show that if \[\cosh x = \sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!}\] then (i) \(\cosh 1\) is irrational, (ii) \(\cosh \sqrt{2}\) is irrational.

Prove that, if \(f(x)\) is a polynomial with integral coefficients, then the sum of the infinite series \[f(0) + f(1)/1! + f(2)/2! + \ldots\] is an integral multiple of \(e\).

Let \(f(x)\) be a real differentiable function defined for \(a < x < b\) and suppose that $$f(a) = f(b) = 0.$$ Show that there is at least one number \(\xi\) in \(a < \xi < b\) for which $$|f'(\xi)| \geq \frac{4}{(b-a)^2} \left| \int_a^b |f(x)| dx \right|.$$

Either by showing that \(n!e\) is never an integer (for \(n = 1, 2, \ldots\)), or in any way, prove that $$e = 1 + \frac{1}{1!} + \frac{1}{2!} + \cdots$$ is irrational (that is, it cannot be expressed in the form \(p/q\) with \(p\) and \(q\) integral).

The function equal to \(e^{-x}\) when \(|x| \leq 1\), and equal to 0 when \(|x| > 1\), is denoted by \(f(x)\). \(x_1\), \(\ldots\), \(x_n\) and \(a_1\), \(\ldots\), \(a_n\) are real numbers such that $$\sum_{r=1}^n a_r f(x-x_r) > 0$$ for all real numbers \(x\). Prove that $$\sum_{r=1}^n a_r > 0.$$

Prove that, when \(x > -1\), $$\log(1 + x) = \frac{2x}{2 + x} + \frac{2x^3}{3(2 + x)^3} + \frac{2x^5}{5(2 + x)^5} + \cdots$$ Hence, or otherwise, show (i) that \(a_n = [1 + (1/n)]^{n+p}\) decreases as \(n\) increases when \(p > 1\); (ii) that, provided \(n\) is sufficiently large, \(a_n\) increases as \(n\) increases when \(p < 1\); (iii) that \(a_n\) increases throughout as \(n\) increases when \(p \leq 0\).

Explain what is meant by ``\(a_n \to a\) as \(n \to \infty\).'' Prove that, if \(a_n \to a\), \(b_n \to b\) as \(n \to \infty\), then

1. [(i)] \(a_n b_n \to ab\),
2. [(ii)] \(\frac{a_n b_{2n} + a_{n+1} b_{2n-1} + \dots + a_{2n} b_n}{n} \to ab\).

Prove that, if \(s_n = a_1+a_2+\dots+a_n\), where \(a_1, a_2, \dots\) are positive, and \[ t_n = a_1 + \frac{a_2}{s_1} + \frac{a_3}{s_2} + \dots + \frac{a_n}{s_{n-1}}, \] then \[ t_n > \log s_n. \] Deduce that the infinite series \(\sum \frac{1}{n}\) is divergent.

Write a short essay on the theory of the convergence of series of positive terms, starting from the beginning and proceeding far enough for the proof of the following propositions:

1. If \(\Sigma a_n^2\) and \(\Sigma b_n^2\) are convergent, then \(\Sigma a_n b_n\) is convergent;
2. If \(\Sigma a_n^2, \Sigma b_n^2,\) and \(\Sigma c_n^2\) are convergent, then \(\Sigma a_n b_n c_n\) is convergent.

Examine the convergence of the series whose \(n\)th term is \(\frac{x^n}{x^{2n}+x^n+1}\) for any value of \(x\). Find the sum of the series \(\sum_{n=2}^\infty \frac{n^3 x^n}{n^4-1}\), when convergent.

Prove that, if two infinite series of positive terms \(\sum u_n, \sum v_n\) are such that \(u_n/v_n\) tends to a finite limiting value, not zero, when \(n\) tends to infinity, the series are both convergent or both divergent. Deduce a rule for the convergence or divergence of \(\sum u_n\), when \(u_{n+1}/u_n\) tends to a limit \(k\). Examine for different positive or negative values of \(z\) and \(p\) the convergence or divergence of the series whose \(n\)th terms are \[ \text{(i) } z^n, \quad \text{(ii) } \frac{\sqrt{n+1}-\sqrt{n}}{n^p}. \]

Show that if \(u_n>0\) and \(\frac{u_{n+1}}{u_n} < \rho < 1\), then \(\sum_{n=1}^\infty u_n\) is convergent. Show that \(\sum n^p r^n\) and \(\sum n! r^{n^2}\) are convergent if \(0

Discuss completely the convergence of the logarithmic series for different real values of the variable, and prove that, if \(n>1\), \[ \frac{1}{n} + \frac{1}{2n(n-1)} > \log\left(\frac{n}{n-1}\right) > \frac{2}{2n-1}. \]

Shew that, if \(a_n \to a\) and \(b_n \to b\) as \(n \to \infty\), then

1. [(i)] \(\text{Max}\,(a_n, b_n) \to \text{Max}\,(a,b)\), where \(\text{Max}\,(a,b)\) is equal to \(a\) if \(a \geq b\), and is equal to \(b\) if \(b > a\),
2. [(ii)] if \(b \neq 0\), then \(\frac{a_n}{b_n} \to \frac{a}{b}\),
3. [(iii)] if \(P(x)\) and \(Q(x)\) are polynomials and if \(Q(a) \neq 0\), then \(\frac{P(a_n)}{Q(a_n)} \to \frac{P(a)}{Q(a)}\).

If \(b_1, b_2, b_3, \dots, b_n\) are numbers such that \[ b_1 \ge b_2 \ge b_3 \ge \dots \ge b_n, \] and \(a_1, a_2, a_3, \dots, a_n\) are such that \[ a_1+a_2+\dots+a_r \le S \quad (r=1,\dots,n) \] shew that \[ a_1b_1+a_2b_2+\dots+a_nb_n \le Sb_1. \] If \(b_1, b_2, b_3, \dots\) is a decreasing sequence such that \[ \lim_{n\to\infty} b_n = 0, \] shew that \[ \sum_1^\infty b_n\cos nx \] converges if \(x \neq 2m\pi\), where \(m\) is an integer.

Prove that \[ 1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n-1}-\log n \] tends to a finite limit, as \(n\to\infty\). If \(p=2^q\), where \(q\) is an integer, prove that \[ \sum_{r=2}^p \frac{1}{r \log r} > \frac{\log(q+1)}{2 \log 2}. \]

Shew that if \(n\) can be found so that \(\frac{v_m}{u_m}\) is finite whenever \(m>n\), and the series \(u_1+u_2+\dots\) converges, then the series \(v_1+v_2+\dots\) converges. \par Examine the convergence of the series whose \(n\)th term is \(\frac{x^n}{n^2-x^{2n}}\) for all values of \(x\).

State any tests that you know for the convergence of series that are not absolutely convergent. Discuss completely the convergence of \[ \Sigma \frac{x^n}{n^\alpha \log n}, \] where \(\alpha\) is real, and \(x\) real or complex, distinguishing between absolute and conditional convergence.

Explain what is meant by the uniform convergence of a series and give an example of a series which converges in \(0 < x\le 1\) and yet is not uniformly convergent in any interval \(0\le x \le \delta\), however small the positive number \(\delta\) may be. It is given that \[ f(x)=f_1(x)+f_2(x)+\dots+f_n(x)+\dots \] for all values of \(x\) in some neighbourhood of \(a\), and that \(f_1(x),f_2(x),\dots\) are differentiable in this neighbourhood. Prove that, if the series \[ f'_1(x)+f'_2(x)+\dots+f'_n(x)+\dots \] converges uniformly in some neighbourhood of \(a\), then \(f(x)\) is differentiable at \(a\) and \[ f'(a)=f'_1(a)+f'_2(a)+\dots+f'_n(a)+\dots. \]

Define the upper and lower limits of a function of an integral variable. If \(f(n)n_0\), and \(f(n)>A\) for an infinite number of values of \(n\), prove that \(\varlimsup f(n)\) exists, and that \(A \le \varlimsup f(n) \le B\). If \(G(y)\) is the upper bound of \(\phi(x)\) in \(a

State and prove the Heine-Borel theorem for one variable. Deduce that if \(f(x)\) is continuous in \(a\le x\le b\), then, corresponding to any positive \(\epsilon\), there is an \(\eta\) such that in any interval contained in \(ab\) and of length less than \(\eta\) the oscillation of \(f(x)\) is less than \(\epsilon\).

(i) Given that \(\alpha\) and \(\beta\) are the roots of \[ x^2 - px + q = 0, \] form the equation whose roots are \(\alpha^3 - \frac{1}{\beta^3}\), \(\beta^3 - \frac{1}{\alpha^3}\). (ii) Given that the equation \[ x^n - ax^2 + bx - c = 0, \quad (c \neq 0, n > 2), \] has a thrice repeated root \(\xi\), establish the relations \[ \xi = \frac{(n-1)b}{2(n-2)a} = \frac{2nc}{(n-1)b}, \] and \[ \xi^n = \frac{2c}{(n-1)(n-2)}. \]

Show by comparison with the identity \(4\cos^3\alpha - 3\cos\alpha - \cos 3\alpha = 0\) that the cubic equation \(x^3-3qx-r=0\) can be solved in terms of cosines provided that \(4q^3 > r^2\). If \(\alpha\) is defined by the equation \(\cos 3\alpha = r/2q^{3/2}\), show that \(2q^{1/2}\cos\alpha\) is a root, and find the other two roots. Use the method to solve the equation \[ x^3 - 6x^2 + 6x + 8 = 0. \]

Prove that the sum of the roots of the equation \[ \begin{vmatrix} x & h & g \\ h & x & f \\ g & f & x \end{vmatrix} = 0 \] is zero, and that the sum of the squares of the roots is \[ 2 (f^2+g^2+h^2). \] Taking \(f, g, h\) to be real, and assuming that the roots are then all real, prove that no root exceeds \[ 2\sqrt{\tfrac{1}{3}(f^2+g^2+h^2)} \] in absolute value. In what circumstances (if any) can a root be equal to this in absolute value?

Find the condition on the coefficients \(p, q, r, s\) of the equation \[ x^4+px^3+qx^2+rx+s=0 \] for two of its roots \(\alpha, \beta\) to satisfy the equation \(\alpha+\beta=0\). Show that the equation \[ x^4 - 2x^3 + 4x^2 + 6x - 21 = 0 \] satisfies this condition, and solve it completely.

If the polynomial \[ ax^3+x^2-3bx+3b^2 \] has two coincident zeros show that, in general, it is a perfect cube. Hence, or otherwise, show that if \[ x^4+4ax^3+2x^2-4bx+3b^2 \] has three equal zeros then the fourth is identical with them, and find for what values of \(a\) and \(b\) this is the case.

The roots of the cubic equation \[ ax^3+bx^2+cx+d=0 \quad (a\neq 0, d\neq 0) \] are \(\alpha, \beta, \gamma\). Find the equations whose roots are (i) \(\displaystyle\frac{1}{\alpha}, \frac{1}{\beta}, \frac{1}{\gamma}\); (ii) \(\alpha^2, \beta^2, \gamma^2\); (iii) \(\beta\gamma/\alpha, \gamma\alpha/\beta, \alpha\beta/\gamma\). Deduce a necessary and sufficient condition that the product of two of the roots of the original equation should be equal to the third root.

(i) Show that, if \begin{align*} x^3+px+q &= 0, \\ x^3+rx+s &= 0 \end{align*} have a common root, then \[ (q-s)^3 = (ps-qr)(p-r)^2. \] (ii) If \(\alpha, \beta, \gamma\) are the roots of \(x^3+px+q=0\), find the equation with roots \(\alpha^3, \beta^3, \gamma^3\).

Prove that if the two equations \begin{align*} ax^2+2bx+c &= 0 \\ a'x^2+2b'x+c' &= 0 \end{align*} have a single common root, then \[ 4(b^2-ac)(b'^2-a'c') - (ac'+ca'-2bb')^2=0. \] Show that the condition \[ 4(b^2-ac)(b'^2-a'c') - (ac'+ca'-2bb')^2 \ge 0, \] is necessary for the fraction \[ \frac{ax^2+2bx+c}{a'x^2+2b'x+c'} \] to assume all real values for real values of \(x\), and that if this condition is not fulfilled, the range of inadmissible values of the fraction will either be entirely between or entirely outside the roots of the equation \[ x^2(b'^2-a'c') + x(ac'+ca'-2bb') + b^2-ac = 0. \]

Show that in any algebraic equation \[ x^n - p_1x^{n-1} + p_2x^{n-2} - \dots + (-1)^n p_n = 0 \] the coefficient \(p_r\) is the sum of all the products formed by taking the roots together \(r\) at a time. If the roots are \(x_1, x_2, \dots, x_n\), prove that \[ (1-p_2+p_4-\dots)^2 + (p_1-p_3+p_5-\dots)^2 = (1+x_1^2)(1+x_2^2)\dots(1+x_n^2). \]

Three roots of the quartic equation \[ (x^2+1)^2 = ax(1-x^2)+b(1-x^4) \] satisfy the equation \[ x^3+px^2+qx+r=0. \] Prove that \[ p^2-q^2-r^2+1=0. \]

Prove that the sum of the roots of the equation \[ \begin{vmatrix} a_1 - x & b_1 & c_1 \\ a_2 & b_2 - x & c_2 \\ a_3 & b_3 & c_3 - x \end{vmatrix} = 0 \] is \(a_1 + b_2 + c_3\). Express the sum of the squares of the roots in terms of \(a_1, b_1, \dots, c_3\).

Prove that, if \(h(x)\) is the H.C.F. of two polynomials \(f(x), g(x)\), then polynomials \(A(x), B(x)\) exist such that \[ A(x)f(x) + B(x)g(x) = h(x). \] Obtain an identity of this form when \[ f(x) = x^{10} + 1, \quad g(x) = x^6 + 1. \]

For what values of \(r\) does the equation \[ x^3 - 3x + r = 0 \] have three distinct real roots? Solve completely the equation \[ 4x^3 - 27a^2(x-a) = 0. \]

Find all the solutions of the equations \begin{align*} x+2y+4z &= 12, \\ xy+2xz+4yz &= 22, \\ xyz &= 6. \end{align*}

If \(a, b\) and \(c\) are the roots of the equation \(x^3=px+q\), express \(a^2+b^2+c^2\), \(a^3+b^3+c^3\) and \(a^4+b^4+c^4\) in terms of \(p\) and \(q\). If \[ \Delta = \begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2 \end{vmatrix}, \] show that \(\Delta = (a-b)(b-c)(c-a)\) and that \(\Delta^2 = 4p^3-27q^2\). If \(x, y\) and \(z\) are real numbers, prove that \[ 2(x-y)^2(y-z)^2(z-x)^2 \le [x^2+y^2+z^2-\frac{1}{3}(x+y+z)^2]^3 \] and that the equality sign holds if and only if the three numbers are in arithmetic progression.

Show that, if \(x\) is a root of the equation \(x^4-6x^2+1=p(x^3-x)\), then \(\frac{1+x}{1-x}\) is also a root, and find the other two roots in terms of \(x\). Hence show that if \(p\) is real all the four roots of the equation are real.

Solve: \begin{align*} x+y+z &= 1, \\ x^2+y^2+z^2 &= 21, \\ x^3+y^3+z^3 &= 55. \end{align*}

The equation \(x^4+ax^3+bx^2+cx+d=0\) is such that the sum of two of its roots is equal to the sum of the remaining two. Shew that \(a^3-4ab+8c=0\). \newline If, in particular, \(a=2, b=-1, c=-2, d=-3\), find all the roots.

Find the condition that the two equations \begin{align*} x^2+2ax+b^2 &= 0, \\ x^3+3p^2x+q^3 &= 0 \end{align*} should have a common root. Verify your condition from first principles

1. [(i)] when \(a=b\);
2. [(ii)] when \(q=0, b \ne 0\).

(i) Given that the product of two of the roots is 2, solve the equation \[ x^4+2x^3-14x^2-11x-2=0. \] (ii) Show that if \(a\) is a root of the equation \[ x^4+3x^3-6x^2-3x+1=0 \] then so also is \(\frac{a-1}{a+1}\). Express the remaining roots in terms of \(a\), and hence, or otherwise, solve the equation completely.

Prove that if \[ 1+c_1x+c_2x^2+c_3x^3+\dots = (ax^2+2bx+1)^{-1}, \] then \[ 1+c_1^2x+c_2^2x^2+c_3^2x^3+\dots = \frac{1+ax}{1-ax}\{a^2x^2+2(a-2b^2)x+1\}^{-1}. \]

Prove that, if \(\alpha, \beta, \gamma\) are the roots of \[ x^3 + qx + r = 0, \] then \[ \alpha^2 (\beta + \gamma) + \beta^2 (\gamma + \alpha) + \gamma^2 (\alpha + \beta) = 3r \] and \[ \alpha^3 (\beta + \gamma) + \beta^3 (\gamma + \alpha) + \gamma^3 (\alpha + \beta) = -2q^2. \]

A family of parabolas have a given point as vertex, and all pass through another given point. Prove that the locus of their foci is a cubic curve.

If \(\alpha, \beta, \gamma\) are the roots of \[ x^3 - 6x^2 + 18x - 36 = 0, \] prove that \begin{align*} \alpha^2 + \beta^2 + \gamma^2 &= 0, \\ \alpha^3 + \beta^3 + \gamma^3 &= 0. \end{align*}

Shew that, if \[ (b-c)^2(x-a)^2 + (c-a)^2(x-b)^2 + (a-b)^2(x-c)^2 = 0, \] and no two of \(a, b, c\) are equal, then \[ x = \tfrac{1}{3} \{a+b+c \pm (a^2+b^2+c^2-bc-ca-ab)^{\frac{1}{2}}\}. \] Shew that one root of the equation \(x^3 = 100(x-1)\) is approximately 1.0103, and determine the other roots, correct to two places of decimals.

Show that, if the cubic equation \(x^3 - a_1 x^2 + a_2 x - a_3 = 0\) has roots \(\alpha\), \(\beta\), \(\gamma\) and if \(a_3 \neq 0\), then \(\frac{a_2}{a_3} = \frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma}\). Deduce that the equation \(x^3 - ax^2 + 11bx - 4b = 0\) cannot have three strictly positive integer roots. Find a value of \(a\) such that \(x^3 - ax^2 + 96x - 108 = 0\) does have three positive, integer roots.

The roots of the equation \(x^3 + ax^2 +bx+ c = 0\) are distinct and form a geometric progression. Taken in another order, they form an arithmetic progression. Find \(b\) and \(c\) in terms of \(a\).

Suppose that \(a\), \(b\) and \(c\) are real numbers such that the equation \[x^3-ax^2+bx-c=0\] has three distinct real roots, which are in geometric progression. Prove that \(abc > 0\) and that \[\left|\frac{a}{c}-1\right| > 2.\]

The equation \(x^3 + ax^2 + bx + c\) (\(c \neq 0\)) has three distinct roots which are in geometric progression and whose reciprocals may be rearranged to form an arithmetic progression. Find \(b\) and \(c\) in terms of \(a\).

Let \(b\) and \(c\) be real numbers. The cubic equation \(x^3 + 3x^2 + bx + c = 0\) has three distinct real roots which are in geometric progression. Show that there are unique values of \(b\) and \(c\) such that the roots of this equation are integers, and find this equation and its roots.

1. [(i)] If all the roots of the equation \(x^3 + px^2 + qx + r^3 = 0\) are positive, show that \(p \leq 3r\) and \(q \geq 3r^2\).
2. [(ii)] The numbers \(a\), \(b\), \(c\) are positive, and $$d = (b + c - a)(c + a - b)(a + b - c).$$ By considering \(d^2\), or otherwise, show that \(d \leq abc\).

Find the conditions that the roots of the equation \[ x^3+3px^2+3qx+r=0 \] should be (i) in arithmetic progression, (ii) in geometric progression, (iii) in harmonic progression.

If \(a\) and \(b\) are real numbers, show that the equation \[ x^4 + ax^3 + (b-2)x^2 + ax + 1 = 0 \] has four real roots if and only if one of the following two sets of conditions is satisfied:

1. \(b \le 0, a^2 < 16b^2\);
2. \(b \ge 8, 4(b-4) < a^2 < \frac{1}{4}b^2\).

Find for what values of \(a\) and \(b\) the roots of the equation \[ x^4 - 4x^3 + ax^2 + bx - 1 = 0 \] are in arithmetical progression.

Prove that if \(\tan \alpha, \tan \beta, \tan \gamma\) are in arithmetic progression, then so are \(\cot (\alpha - \beta)\), \(\tan \beta\), \(\cot (\gamma - \beta)\).

\(PSP'\), \(QSQ'\) are any two focal chords of a parabola. Shew that the common chord of the circles described on \(PP'\), \(QQ'\) as diameters passes through the vertex of the parabola.

Form the equation whose roots are the sum and product of the reciprocals of the roots of the equation \[ x^2 + \lambda x + \mu = 0. \] If the equation thus formed is identical with the original quadratic equation, prove that \[ \lambda^2 = (1 - \lambda)^2 \] and \[ \mu^3 = \mu^2 + 1. \]

Find the condition that the equations \[ ax^2+2bx+c=0, \quad a'x^2+2b'x+c'=0 \] may have a common root. Prove that, if \(a, c, a', b', c'\) are given so that \(b'^2 > a'c'\), two real values \(b_1, b_2\) of \(b\) can be found to ensure a common root; and form the equation whose roots are the other roots of the equations \[ ax^2+2b_1x+c=0, \quad a'x^2+2b_2x+c'=0. \]

The quartic equation \[ 4x^4 + \lambda x^3 + 35x^2 + \mu x + 4 = 0 \] has its roots in geometric progression. What real values might be taken for the ratio of the progression?

Find the conditions that the roots of \[ x^3-ax^2+bx-c=0 \] shall be (i) in G.P., (ii) in A.P., (iii) in H.P. Show that if the roots are not in A.P. then there are in general three transformations of the form \(x=y+\lambda\) such that the transformed cubic in \(y\) has its roots in G.P.

State and prove the harmonic properties of a quadrilateral. \(P\) is a variable point upon a conic which circumscribes the triangle \(ABC\). \(AP, BC\) meet in \(Q\); \(AB, PC\) in \(R\). Shew that \(QR\) always passes through a fixed point.

Explain how \(\sqrt{13}\) can be expanded as a simple continued fraction. Shew that, if \(p_n/q_n\) is the \(n\)th convergent, \(p_4/q_4=48/13\); and prove the relations \[ p_{2n+2}=10p_{2n}-p_{2n-2}, \quad q_{2n+2}=10q_{2n}-q_{2n-2}. \]

(i) Prove that \(x=2\sin 10^\circ\) is a root of the equation \(x^3-3x+1=0\), and find the other two roots. (ii) If \(c = \cos^2\theta - \frac{1}{3}\cos^3\theta\cos 3\theta + \frac{1}{5}\cos^5\theta\cos 5\theta - \dots\) to infinity, prove that \[ \tan 2c = 2\cot^2\theta. \]

Solve the equation \[ 81x^4 + 54x^3 - 189x^2 - 66x + 40 = 0, \] given that the roots are in arithmetic progression.

Prove that if \(n\) is a positive integer, \[ \cos nx - \cos n\theta = 2^{n-1}\prod_{r=0}^{n-1}\left\{\cos x - \cos\left(\theta+\frac{2r\pi}{n}\right)\right\}. \] Deduce a product for \(\sin n\theta\). Also show that \[ \cos\frac{\pi}{n}\cos\frac{2\pi}{n}\dots\cos\frac{(2n-1)\pi}{n} = \frac{(-1)^n-1}{2^{2n-1}}. \]

(i) Solve \[ \frac{x^2-a^2}{(x-a)^3} - \frac{x^2-b^2}{(x-b)^3} + \frac{x^2-c^2}{(x-c)^3} = 0 \] \[ \frac{(x+a)^3}{(x+a)^3} - \frac{(x+b)^3}{(x+b)^3} + \frac{(x+c)^3}{(x+c)^3} = 0 \] for \(x\), where \(a,b,c\) are unequal. [Note: The second equation appears to have typos in the source; it's transcribed as written, but likely intended to be different.] (ii) Shew that if the roots of the equation \(ax^4+bx^3+cx^2+dx+e=0\) are in harmonic progression, then \(d^3=4cde-8be^2\), and \(25ad^2e = (cd-eb)(11eb-cd)\). Verify these conditions in the case of \(40x^4-22x^3-21x^2+2x+1=0\) and solve for \(x\).

Prove that the roots of the equation \[ x^3+3px^2+3qx+r=0 \] will be in geometrical progression if \[ p^3r=q^3; \] but will be in harmonical progression if \[ 2q^3 = r(3pq-r). \]

Prove that in general three normals can be drawn to a parabola through a given point. ABC is an equilateral triangle inscribed in a parabola of which A is the vertex. The normals at two points P and Q on the parabola meet in B. Prove that the length of PQ is twice the latus-rectum of the parabola and that the orthocentre of the triangle BPQ lies in AC.

Show that, if the polynomial \[f(x) = x^3+3ax+b \quad (a \neq 0)\] can be expressed in the form \[A(x-p)^3+B(x-q)^3,\] where \(A\) and \(B\) are constants, then \(p \neq q\), and \(p, q\) are the roots of the equation \[at^2+bt-a^2 = 0.\] Prove, conversely, that if this equation has unequal roots then \(f(x)\) can be written in the form (1). Hence or otherwise find the real root of the equation \[x^3+54x+54 = 0.\]

The quartic equation \(x^4 - s_1 x^3+s_2x^2-s_3x+s_4 = 0\) has roots \(\alpha\), \(\beta\), \(\gamma\), \(\delta\). Find the cubic equation with roots \(\alpha\beta + \gamma\delta\), \(\alpha\gamma + \beta\delta\), \(\alpha\delta + \beta\gamma\). Supposing that methods of solving quadratic and cubic equations are known, describe a procedure for solving a quartic equation.

The cubic equation \[x^3 + ax^2 + bx + c = 0\] has roots \(\alpha, \beta, \gamma\). Find a cubic with roots \(\alpha^3, \beta^3, \gamma^3\), its coefficients being expressed in terms of \(a, b\) and \(c\).

The equation $$ax^4 + bx^3 + cx^2 + dx + e = 0,$$ where \(a\) and \(e\) are not zero, has roots \(\alpha, \beta, \gamma, \delta\). Show how it is possible to obtain \(\alpha^n + \beta^n + \gamma^n + \delta^n\) in terms of the coefficients \(a, b, c, d, e\) for all values of \(n\), where \(n\) is a positive or negative integer. Obtain the equations whose roots are

1. [(i)] \(\alpha^2, \beta^2, \gamma^2, \delta^2\);
2. [(ii)] \(\alpha - 3, \beta - 3, \gamma - 3, \delta - 3\).

If \(a\), \(b\) and \(c\) are the roots of the equation $$x^3 + px^2 + qx + r = 0,$$ form the equation with the roots \(a^3 - bc\), \(b^3 - ca\), and \(c^3 - ab\). Prove that one of the roots is the geometric mean of the other two if, and only if, \(rp^3 = q^3\). Find in a similar way a condition for one root to be the arithmetic mean of the other two. What can be said about \(a\), \(b\) and \(c\) if both these conditions hold?

Given that the roots of the equation $$y^8 + 3y^2 + 2y - 1 = 0$$ are the fourth powers of the roots of an equation $$x^8 + ax^2 + bx + c = 0$$ with rational coefficients \(a, b, c\), find suitable values for \(a, b, c\).

(i) Find the equation whose roots are the cubes of the roots of the equation \[x^3 + ax^2 + bx + c = 0.\] (ii) Show how to obtain the equation whose roots are the roots of a given algebraic equation multiplied by the same constant quantity. Hence, or otherwise, prove that an algebraic equation with integer coefficients and unit coefficient for the highest power cannot have a real rational root which is not integral.

Find the equation whose roots are the squares of the roots of the cubic equation \(a_0x^3+a_1x^2+a_2x+a_3=0\). If the values of the coefficients \(a_0, a_1, a_2, a_3\) are given numerically, and the roots \(\alpha, \beta, \gamma\) of the equation are real, and such that \(|\alpha| > |\beta| > |\gamma|\), show that the continued repetition of this process will yield an approximate value of \(|\alpha|\). Suggest a method of finding the other roots.

Show that the result of eliminating \(y\) and \(z\) between the three equations \begin{align} y^2+2ay+b=0, \tag{1} \\ z^2+2cz+d=0, \tag{2} \\ x=yz, \nonumber \end{align} is the quartic equation whose roots are \(\alpha\beta, \alpha\beta', \alpha'\beta, \alpha'\beta'\), where \(\alpha, \alpha'\) are the roots of the quadratic equation \((1)\) and \(\beta, \beta'\) are the roots of the quadratic equation \((2)\). Find the quartic equation whose roots are \[ (\alpha-\beta), (\alpha'-\beta), (\alpha-\beta'), (\alpha'-\beta'), \] and hence write down the equation whose roots are \[ (\alpha-\beta)^2, (\alpha'-\beta)^2, (\alpha-\beta')^2, (\alpha'-\beta')^2. \]

Find the equation whose roots are less by 2 than the squares of the roots of \[ x^3+qx+r=0. \] Examine the particular case \[ x^3-3x+1=0, \] and interpret the result.

The equation \[ x^3+px^2+qx+r=0 \] has roots \(\alpha, \beta, \gamma\). Find the equations with roots (i) \(\beta+\gamma, \gamma+\alpha, \alpha+\beta\), (ii) \(\beta\gamma, \gamma\alpha, \alpha\beta\). Hence or otherwise determine necessary and sufficient conditions for the equation \[ x^3+px^2+qx+r=0 \] to have two roots (i) whose sum is \(a\), (ii) whose product is \(b\).

1. [(i)] If \(a_1, a_2, a_3\) are the roots of \[ x^3+px+q=0, \] where \(p+q+1 \ne 0\), find the equation with roots \[ 1/(1-a_i) \quad (i=1, 2, 3). \]
2. [(ii)] Solve the equations \begin{align*} x+y+z &= 7, \\ x^2+y^2+z^2-4z &= 5, \\ xyz-2xy &= 4. \end{align*}

Let \(z_1, \dots, z_n\) be the zeros of \[ f(z) = z^n+c_1z^{n-1}+\dots+c_{n-1}z+c_n \] and let \(f'(z)\) be the derivative. Show that \[ (-1)^{\frac{1}{2}n(n-1)} \prod_{1\le i < j\le n} (z_i-z_j)^2 = \prod_{i=1}^n f'(z_i). \] Hence, or otherwise, show that if \(y_1, \dots, y_n\) are the zeros of \[ 1+\frac{x}{1!} + \frac{x^2}{2!} + \dots + \frac{x^n}{n!} \] then \[ \prod_{1\le i < j\le n} (y_i-y_j)^2 = (-1)^{\frac{1}{2}n(n-1)}(n!)^n. \]

The cubic equation \(x^3+px+q=0\) has roots \(\alpha, \beta, \gamma\). Find the cubic equation whose roots are \((\beta-\gamma)^2, (\gamma-\alpha)^2, (\alpha-\beta)^2\). Hence or otherwise deduce the condition for the cubic equation \[ a_0x^3+3a_1x^2+3a_2x+a_3=0 \] to have a pair of equal roots in the form \[ (a_0^2a_3+2a_1^3-3a_0a_1a_2)^2+4(a_0a_2-a_1^2)^3=0. \]

Find the equation whose roots are the squares of the roots of the cubic equation \[ x^3 - ax^2 + bx - 1 = 0. \] Find all pairs of values of \(a\) and \(b\) for which the two equations are the same.

If \(a_r = x+(r-1)y\), show that \[ \begin{vmatrix} a_1 & a_2 & a_3 & \dots & a_n \\ a_n & a_1 & a_2 & \dots & a_{n-1} \\ a_{n-1} & a_n & a_1 & \dots & a_{n-2} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_2 & a_3 & a_4 & \dots & a_1 \end{vmatrix} = (-ny)^{n-1}\{x+\frac{1}{2}(n-1)y\}. \]

Form the equation whose roots are the reciprocals of the roots of the equation \[ x^3+ax^2+bx-c=0. \] Hence solve the equation \[ 35x^3 - 18x^2 + 1 = 0. \]

Prove that the geometric mean of \(n\) positive numbers is less than or equal to their arithmetic mean. Shew that, if the equation \[ x^n - nb_1 x^{n-1} + \dots + \frac{n(n-1)\dots(n-r+1)}{r!}(-b_r)^r x^{n-r} + \dots + (-b_n)^n = 0, \] where \(b_1, b_2, \dots, b_n\) are all real and greater than zero, has \(n\) real roots, then \(b_r \le b_{r-1}\) for \(r=1,2,\dots,n-1\).

Shew that \[ \frac{d^n}{dx^n} (\tan^{-1} x) = (-1)^{n-1} (n-1)! r^{-n} \sin n\phi, \] where \[ r \cos \phi = x, \quad r \sin \phi = 1. \]

Reduce the equation \(x^3+3px^2+3qx+r=0\) to the form \(y^3+3y+m=0\) by assuming \(x=\lambda y + \mu\); and solve this equation by assuming \(y=z-\frac{1}{z}\). Hence prove the condition for equal roots to be \[ 4(p^2-q)^3 = (2p^3-3pq+r)^2. \]

If \begin{align*} \frac{x}{a+\lambda} + \frac{y}{b+\lambda} + \frac{z}{c+\lambda} &= 1, \\ \frac{x}{a+\mu} + \frac{y}{b+\mu} + \frac{z}{c+\mu} &= 1, \\ \frac{x}{a+\nu} + \frac{y}{b+\nu} + \frac{z}{c+\nu} &= 1, \end{align*} prove that, for all values of \(\xi\) (except \(-a, -b\) and \(-c\)), \[ \frac{x}{a+\xi} + \frac{y}{b+\xi} + \frac{z}{c+\xi} = 1 + \frac{(\lambda-\xi)(\mu-\xi)(\nu-\xi)}{(a+\xi)(b+\xi)(c+\xi)}, \] and that \[ x = \frac{(a+\lambda)(a+\mu)(a+\nu)}{(a-b)(a-c)}. \]

If \(u_0=2\), \(u_1=2\cos\theta\), and \[ u_n = u_1 u_{n-1} - u_{n-2}, \quad (n>1) \] prove that \(u_n=2\cos n\theta\). By successive applications of the relation express \(u_5\) as a polynomial in \(u_1\), say \(f(u_1)\). Then by considering the equation \(f(u_1)=1\) shew that \(2\cos\frac{\pi}{15}\) is a root of the quartic equation \[ x^4 + x^3 - 4x^2 - 4x + 1 = 0. \] What are the other three roots?

(i) If \(u=xyz\), where \(x, y, z\) are connected by the relations \[ yz+zx+xy=a, \quad x+y+z=b \quad (a, b \text{ being constants}), \] prove that \[ \frac{du}{dx} = (x-y)(x-z). \] (ii) If \(\xi, \eta\) are functions of \(x, y\) such that \(\xi=e^x \cos y, \eta=e^x \sin y\), and \(x,y\) are functions of \(r, \theta\) such that \(x=e^r \cos\theta, y=e^r \sin\theta\), where \(r\) is a function of \(\theta\), prove that \[ \frac{d\xi}{d\eta} = \frac{\frac{dr}{d\theta}-\tan(y+\theta)}{1+\frac{dr}{d\theta}\tan(y+\theta)}. \]

Two pairs of points \(A, B\) and \(A', B'\) lie on an axis \(Ox\), and their abscissae are given by the equations \(ax^2+2bx+c=0\) and \(a'x^2+2b'x+c'=0\) respectively. Find an equation with rational coefficients which has \(AA' \cdot BB'\) for one of its roots. Give the geometrical interpretation of the relations obtained by equating the various coefficients in the equation to zero.

Prove that \[ \frac{1}{0!2n!} - \frac{1}{1!3!2n-1!} + \frac{1}{2!4!2n-2!} - \dots + (-)^{n+1} \frac{1}{n-1!n+1!} = \frac{1}{n-1!n+1!2n!}. \]

The complex numbers \(\alpha, \beta, \gamma, \delta\) are all non-zero and are also such that $$s_1 = s_5 = s_8 = 0,$$ where \(s_n\) (\(n = 1, 2, 3, \ldots\)) is defined by the equation $$s_n = \alpha^n + \beta^n + \gamma^n + \delta^n.$$ Prove that \(s_3 = 0\).

A triangle inscribed in the parabola \(y^2 = x\) has fixed centroid \((\xi, \eta)\) (where \(\eta^2 < \xi\)). Show that the area of the triangle is greatest when one vertex is at the point \((\eta^2, \eta)\). [The area of the triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), \((x_3, y_3)\) is given by the absolute value of \[\frac{1}{2}\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}.\] The product of the differences of the roots of the cubic equation \(t^3 + \alpha t^2 + \beta t + \gamma = 0\) is the square root of \(\alpha^2\beta^2 - 4\beta^3 + (18\alpha\beta - 4\alpha^3)\gamma - 27\gamma^2\).]

If \(\alpha, \beta, \gamma\) are the roots of the equation \begin{equation*} x^3 - s_1x^2 + s_2x - s_3 = 0, \end{equation*} show that either \(\alpha\beta\gamma = 0\), or \(\frac{s_2}{s_3} = \frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma}\). Deduce that the equation \(x^3 - ax^2 + 7bx - 2b = 0\) cannot have three strictly positive integer roots. Find three pairs of numbers \(a, b\) such that for each pair, the equation \(x^3 - ax^2 + bx - b = 0\) has three strictly positive integer roots.

Let \(z_1\), \(z_2\), \(z_3\) be complex numbers, and suppose that \(z_1^k+z_2^k+z_3^k\) is real for \(k = 1, 2, 3\). Show that at least one of the numbers \(z_1\), \(z_2\), \(z_3\) is also real.

Let \(S_n(a, b)\) be the sum of the \(n\)th powers of the roots of the cubic equation \begin{align*} x^3 + ax^2 + bx + 1 = 0. \end{align*} Evaluate \(S_0(a, b)\), \(S_1(a, b)\), \(S_2(a, b)\). Prove (i) \(S_n(a, b) = S_{-n}(b, a)\) (ii) \(S_n(a, b) = -aS_{n-1}(a, b) - bS_{n-2}(a, b) - S_{n-3}(a, b)\). Find by direct calculation the least \(m > 1\) such that \(S_m(0, 1) = 0\), and deduce that the \(m\)-th powers of the roots of \(x^3 + x + 1 = 0\) satisfy an equation of the form \(y^3 + ky + 1 = 0\). Deduce that \(\theta^{22} + \theta^{-11}\) is an integer, where \(\theta\) is a root of \(x^3 + x + 1 = 0\), and calculate its value.

Find all the solutions of the equations \begin{align} x + y + z + w &= 2,\\ x^2 + y^2 + z^2 + w^2 &= 4,\\ xyzw &= -1,\\ \frac{1}{x} + \frac{1}{y} + \frac{1}{z} + \frac{1}{w} &= 2. \end{align}

Show that the simultaneous equations \begin{align} x + y + z &= 3, \\ x^2 + y^2 + z^2 &= 3, \\ x^3 + y^3 + z^3 &= c, \end{align} have no solution in real numbers \(x\), \(y\), \(z\) unless \(c = 3\).

If \(x_i\) (\(i=1, 2, 3, \dots n\)) are the \(n\) roots of the equation \(f(x)=0\), when \(f(x)\) is a polynomial of degree \(n\), show that \[ \frac{f'(x)}{f(x)} = \sum_{i=1}^n \frac{1}{x-x_i}. \] If \(S_k\) is the sum of the \(k\)th powers of the roots of the equation \(x^4-4x^3-2x^2+1=0\), prove that, for any integer \(k\) (positive, zero, or negative), \(S_k\) is an integer. Find \(S_3, S_4, S_{-4}\).

The roots of the equation \[ x^3+3qx+r=0 \] are \(\alpha, \beta, \gamma\). Express \(P^2\) as a polynomial in \(q\) and \(r\), where \(P=(\beta-\gamma)(\gamma-\alpha)(\alpha-\beta)\). Explain why \(P\) cannot be expressed in this form. From your expression for \(P^2\), or otherwise, obtain necessary and sufficient conditions for the given equation to have

1. a repeated root,
2. three distinct real roots,
3. one real and two non-real roots,

Let \[ f(x) = (x-\alpha_1)\dots(x-\alpha_n) = x^n+a_1x^{n-1}+\dots+a_n \quad \text{and} \quad S_r = \alpha_1^r+\alpha_2^r+\dots+\alpha_n^r, \] where \(r \ge 1\). Prove the identity \[ f'(x) = \frac{f(x)-f(\alpha_1)}{x-\alpha_1} + \frac{f(x)-f(\alpha_2)}{x-\alpha_2} + \dots + \frac{f(x)-f(\alpha_n)}{x-\alpha_n}, \] where \(f'(x)\) is the derivative of \(f(x)\). Hence, or otherwise, show that \[ S_r + a_1S_{r-1} + \dots + a_{r-1}S_1 + ra_r = 0 \] if \(r \le n\). Show that if \(f(x)=x^n+ax+b\) then \(S_r=0\) for \(1 < r \le n-2\). Find \(S_{n-1}, S_n\) and \(S_{n+1}\).

If \(f(x)=0\) is an algebraic equation of integral degree, show that the sum of the \(m\)th powers of its roots is the coefficient of \(x^{-m}\) in the expansion of \(xf'(x)/f(x)\). Find the sum of the cubes of the roots of the quartic equation \[ x^4+x^3-2x^2-7=0. \]

The roots of the cubic equation \(x^3-3qx-pq=0\) are \(\alpha, \beta, \gamma\). Express \(\alpha^{-3}+\beta^{-3}+\gamma^{-3}\) in terms of \(p\) and \(q\). Show that the semi-symmetric functions of the roots \[ \alpha^2\beta+\beta^2\gamma+\gamma^2\alpha \quad \text{and} \quad \alpha^2\gamma+\beta^2\alpha+\gamma^2\beta \] are the roots of the quadratic equation \[ x^2+3pqx+9p^2q^2-27q^3=0. \]

If \(a, b, c\) are the roots of the equation \[ x^3 + px^2 + qx + r = 0, \] prove that \[ (a^2-bc)(b^2-ca)(c^2-ab) = rp^3 - q^3. \] If \(p, q, r\) are real, \(a\) is real and \(b, c\) are complex, prove that \(a\) is numerically greater or less than \(|b|\) according as \(rp^3-q^3\) is positive or negative.

The cubic equation \[ x^3 + px^2 + qx + r = 0, \quad \text{(I)} \] has roots \(\alpha\), \(\beta\) and \(\gamma\). Form the cubic equation with roots \(2\alpha - \beta - \gamma\), \(-\alpha + 2\beta - \gamma\), \(-\alpha - \beta + 2\gamma\), expressing the coefficients in terms of \(p\), \(q\) and \(r\). Find the condition that the roots of (I) should be in arithmetical progression. Find also the condition that the roots of (I) should be in geometrical progression.

If \(\alpha\) and \(\beta\) are the roots of \(y^2-qy+p^3=0\), where \(p\) and \(q\) are real, show how to determine the roots of \(x^3 - 3px + q = 0\) in terms of \(\alpha\) and \(\beta\). Prove that, if \(\alpha\) and \(\beta\) are real and distinct, the cubic has only one real root, and that, if \(\alpha\) and \(\beta\) are imaginary, it has three real roots. What happens if \(\alpha\) and \(\beta\) are coincident?

Prove that if \(a+b+c+d=0\):

1. [(i)] \(\frac{a^5+b^5+c^5+d^5}{5} = \frac{a^3+b^3+c^3+d^3}{3} \times \frac{a^2+b^2+c^2+d^2}{2}\);
2. [(ii)] \(\frac{a^7+b^7+c^7+d^7}{7} = \frac{a^3+b^3+c^3+d^3}{3} \times \left\{ \left(\frac{a^2+b^2+c^2+d^2}{2}\right)^2 - abcd \right\}\).

The roots of the equation \(x^3 + px - q = 0\) are \(\alpha, \beta, \gamma\), and \(s_n = (\alpha^n + \beta^n + \gamma^n)/n\). Express \(s_5, s_7, s_9, s_{11}\) in terms of \(s_2\) and \(s_3\).

Having given that \begin{align*} x + y + z &= 1, \\ x^2 + y^2 + z^2 &= 2, \\ x^3 + y^3 + z^3 &= 3, \end{align*} prove that \[ x^4 + y^4 + z^4 = 4\frac{1}{6}. \]

Having given that \(\alpha, \beta, \gamma\) are the roots of the equation \[x^3 + ax^2 + bx + c = 0,\] find \(\alpha^4+\beta^4+\gamma^4\) in terms of \(a, b, c\).

Obtain a cubic equation whose roots are the values of \(x, y, z\) given by \begin{align*} x+y+z &= 3, \\ x^2 + y^2 + z^2 &= 5, \\ x^3 + y^3 + z^3 &= 7. \end{align*} Prove that \[ x^4 + y^4 + z^4 = 9. \]

If \(\alpha, \beta, \gamma\) are the roots of \(x^3 + px + q = 0\), prove that \[ \frac{\alpha^5 + \beta^5 + \gamma^5}{5} = \frac{\alpha^3 + \beta^3 + \gamma^3}{3} \frac{\alpha^2 + \beta^2 + \gamma^2}{2}. \]

Denoting by \(x_1, x_2, x_3\) the roots of the equation \(x^3 + px + q = 0\), find the value of the sum \[ x_1 (x_2^3 + x_3^3) + x_2 (x_3^3 + x_1^3) + x_3 (x_1^3 + x_2^3). \]

If \begin{align*} \alpha + \beta + \gamma &= a, \\ \alpha^2 + \beta^2 + \gamma^2 &= b, \\ \alpha^3 + \beta^3 + \gamma^3 &= c, \end{align*} find \(\alpha\beta\gamma\) and \(\alpha^4+\beta^4+\gamma^4\) in terms of \(a\), \(b\) and \(c\). Verify that when \(a=0\), they are respectively \(\frac{1}{6}c\) and \(\frac{1}{2}b^2\).

1. Find the sum of the fourth powers of the roots of the equation \[x^3 + x - 1 = 0.\]
2. Find necessary and sufficient conditions to be satisfied by the coefficients \(p\) and \(q\) in order that the equation \[x^3 + px + q = 0\] may have two positive roots.

\(n\) quantities are given. \(s_r\) denotes the sum of the products of all combinations of the quantities \(r\) at a time. Prove that the sum of the products of all combinations of the squares of the quantities \(m\) at a time is given by \[ s_m^2 - 2s_{m-1}s_{m+1} + 2s_{m-2}s_{m+2} + \dots + (-)^{m-1}2s_1s_{2m-1} + (-)^m 2s_{2m} \] where we suppose \(2m \le n\).

The four roots \(\alpha\), \(\beta\), \(\gamma\), \(\delta\) of \(x^4 -px^2+qx-r = 0\) satisfy \(\alpha\beta+\gamma\delta = 0\); by considering the two quadratic equations satisfied by \(\alpha\beta\), \(\gamma\delta\) and by \(\alpha+\beta\), \(\gamma+\delta\), or otherwise, prove that \(q^2 = 4pr\). Solve \(x^4-12x^2 + 12x- 3 = 0\).

An equation has the property that if \(x\) is any (real or complex) root then \(1/x\) and \(1-x\) are also roots. What other expressions in \(x\) are also roots? A quintic equation without repeated roots has the above property. Determine the roots.

The equation \(x^4-8x^3+ax^2-28x+12\) has the property that the sum of a certain pair of roots is equal to the sum of the remaining two roots. Determine \(a\) and find all the roots.

\(x^3+ax+b = 0\) has real roots \(\alpha_1, \alpha_2, \alpha_3\) where \(\alpha_1 \leq \alpha_2 \leq \alpha_3\). Similarly \(x^3 + cx + d = 0\) has real roots \(\gamma_1, \gamma_2, \gamma_3\) where \(\gamma_1 \leq \gamma_2 \leq \gamma_3\). Show that if \[\frac{\alpha_1}{\gamma_1} \leq \frac{\alpha_2}{\gamma_2} \leq \frac{\alpha_3}{\gamma_3} \quad (d \neq 0), \text{ then } \left(\frac{a}{c}\right)^3 = \left(\frac{b}{d}\right)^2.\]

The cubic equation \[x^3 + 3qx + r = 0 \quad (r \neq 0)\] has roots \(\alpha\), \(\beta\) and \(\gamma\). Verify that the sextic equation \[r^2(x^2 + x + 1)^3 + 27q^3x^2(x + 1)^2 = 0\] is satisfied by \(\alpha/\beta\). Comment on this result in relation to the roots of the cubic in the cases (i) \(q = 0\) and (ii) \(4q^3 + r^2 = 0\).

State the relations between the roots \(\alpha\), \(\beta\), \(\gamma\) of the equation \(ax^3 + bx^2 + cx + d = 0\) and the coefficients \(a\), \(b\), \(c\), \(d\). Prove that \(\{0, i, -i\}\) (where \(i^2 = -1\)) is the only set of three distinct numbers (real or complex) such that each is equal to the sum of the cubes of the other two.

If \(\alpha\) is a complex fifth root of unity, prove that \(\alpha - \alpha^4\) is a root of the equation $$\alpha^4 + 5\alpha^2 + 5 = 0.$$ Express the other roots of this equation in terms of \(\alpha\).

Prove that there cannot exist four (real or complex) numbers, all different, such that the square of each of them is equal to the sum of the cubes of the other three. Prove also that there is one (and only one) set of four distinct numbers such that the square of each is equal to the sum of the fourth powers of the other three. Find the equation whose roots are these numbers.

Two numbers \(p\), \(q\) are given. It is required to form a cubic equation such that, if the roots are \(\alpha\), \(\beta\), \(\gamma\) (not necessarily distinct) then \(p\alpha + q\), \(p\beta + q\), \(p\gamma + q\) are also the roots. Find the cubic equation (i) for general values of \(p\), \(q\), (ii) when \(p = +1\), (iii) when \(p = -1\).

The complex numbers \(a, b, c\) satisfy the equations \[ a+b+c=3, \quad abc=2, \quad \begin{vmatrix} a & b & c \\ c & a & b \\ b & c & a \end{vmatrix} = 0. \] Find the cubic equation of which they are the roots, and hence or otherwise determine their values.

If \(\alpha, \beta, \gamma, \delta\) are roots of the equation \[x^4+qx^2+rx+s=0,\] prove that \[ \Sigma \alpha^2 = -2q, \quad \Sigma \alpha^2\beta^2 = q^2+2s, \quad \Sigma \alpha^2\beta^2\gamma^2 = r^2-2qs. \] Show that the equation with roots \(\alpha\beta+\gamma\delta, \alpha\gamma+\beta\delta, \alpha\delta+\beta\gamma\) is \[ x^3 - qx^2 - 4sx - r^2 + 4qs = 0, \] and find the equation with roots \((\alpha\beta-\gamma\delta)^2, (\alpha\gamma-\beta\delta)^2, (\alpha\delta-\beta\gamma)^2\). Hence or otherwise show that, if \(\alpha, \beta, \gamma, \delta\) are four numbers whose sum is zero, a necessary and sufficient condition that the product of one pair is equal to the product of the other pair is \(\Sigma \alpha\beta\gamma=0\). Mark in an Argand diagram the points representing four such numbers.

Find a condition in terms of \(a_0, a_1, a_2, a_3\) that the cubic equation \[ a_0x^3+3a_1x^2+3a_2x+a_3=0 \] should have two repeated roots. Hence or otherwise show that the turning values of the polynomial \[ a_0x^3+3a_1x^2+3a_2x+a_3 \] are the roots of the equation \[ a_0^2 y^2 - 2y(a_0^2a_3 - 3a_0a_1a_2 + 2a_1^3) + (a_0a_2-a_1^2)^2 - 4(a_1a_3-a_2^2)(a_0a_2-a_1^2)=0. \]

Show that an algebraical equation \(f(x)=0\) can at most have only one more real root than the derived equation \(f'(x)=0\). In the case when \(f(x)=x^4+(a-b)x^3+(a+b)x-1\), where \(a\) and \(b\) are both real and non-negative, prove that the equation \(f(x)=0\) has at least two real roots, and that \[ 4(a+b) > (b-a)^3 \] is a sufficient condition for it to have only two real roots.

Three complex numbers whose product is unity, are such that their sum \(p\) is equal to the sum of their reciprocals. Prove that one of the numbers must be unity, and calculate the values of the other two in terms of \(p\). If \(p\) is real, determine the restrictions on its value to ensure that all three original numbers are real. In the case of four complex numbers having the same general property, show that they must consist of two reciprocal pairs.

Prove that, if \(x, y, z\) are positive numbers such that \begin{align*} x+y+z &= 6, \\ x^2+y^2+z^2 &= 18, \end{align*} then none of \(x, y, z\) can exceed 4.

If \(y = (kx+d)/(x+k)\), evaluate \(y-x\) and \(y^2-d\) in terms of \(d, x, k\). Suppose now that \(d\) is a positive integer which is not a perfect square, and that \(l, m\) are positive integers such that \(l/m\) is an approximation to \(\sqrt{d}\). Prove that, if \(k\) is the integer next greater than \(\sqrt{d}\), a better approximation to \(\sqrt{d}\) is given by \((kl+dm)/(l+km)\). Prove also that the two approximations are both greater or both less than \(\sqrt{d}\).

Prove that any two positive numbers \(a\) and \(b\), of which \(a\) is the greater, can be expressed in the form \[ a = m (x + 1) \div (x - 1), \quad b = m (x - 1) \div (x + 1), \] where \(x\) is a number greater than unity. If \(a_1\) and \(b_1\) are the arithmetic and harmonic means of \(a\) and \(b\), \(a_2\) and \(b_2\) the arithmetic and harmonic means of \(a_1\) and \(b_1\), \(a_3\) and \(b_3\) the corresponding means of \(a_2\) and \(b_2\), and so on, then \[ a_n = \sqrt{(ab)} \frac{(\sqrt{a}+\sqrt{b})^{2^n} + (\sqrt{a} - \sqrt{b})^{2^n}}{(\sqrt{a}+\sqrt{b})^{2^n} - (\sqrt{a} - \sqrt{b})^{2^n}}. \]

The equation \(x^2 + ax + b = 0\) has real roots \(\alpha, \beta\). Form the quadratic equation with roots \(\alpha^2 - k^2, \beta^2 - k^2\), and show that \(\alpha, \beta\) are outside the interval \((-k, k)\) if \[ (b + k^2)^2 > k^2a^2 > 2k^2 (b + k^2). \] Find the conditions that the roots of the equation \[ x^4 + px^3 + qx^2 + px + 1 = 0 \] may be all real and unequal.

Shew that in any triangle \[ 4Rr(a\cos B + b\cos C + c\cos A) = abc - (a-b)(b-c)(c-a). \]

Prove that, if \[ u_2 = u_1^2 - 1, \quad u_1u_3 = u_2^2 - 1, \quad u_2u_4 = u_3^2 - 1, \quad u_3u_5 = u_4^2 - 1, \dots \] then \[ u_1+u_3 = u_1u_2, \quad u_2+u_4 = u_1u_3, \quad u_3+u_5 = u_1u_4, \quad u_4+u_6 = u_1u_5, \dots. \]

The coefficients \(a, b, c, a', b', c'\) are real in the quadratic expressions \[ f(x) = ax^2+bx+c, \quad \phi(x) = a'x^2+b'x+c' \] and a value of \(\lambda\) is taken so that the roots \(x_1, x_2\) of \(f(x)-\lambda\phi(x)=0\) are real. Prove (1) that, if \(f(x_1), f(x_2)\) have opposite signs, the like result holds whatever other such value \(\lambda\) has and moreover the roots are real for all real values of \(\lambda\): (2) that, if \(f(x_1), f(x_2)\) have the same sign, the like result holds for all such pairs of roots but the values of \(\lambda\) giving real roots are restricted.

If \(x > 1\) and \(m\) is a positive integer greater than 1, prove that \[ \frac{x^m-1}{m} - \frac{x^{m-1}-1}{m-1} > 0, \] and hence that \(x^m - 1 > m(x-1)\). Generalise this result to the case in which \(m\) is a rational number greater than 1. Prove that, if \(\alpha, \beta\) are positive rational numbers whose sum is 1, and \(a,b\) are positive, then \[ a^\alpha b^\beta < \alpha a + \beta b, \] unless \(a=b\).

The roots of the equation \[ x^3 - ax^2 + bx - c = 0, \] are the lengths of the sides of a triangle. \par Shew that the area of the triangle is \(\frac{1}{4}\{a(4ab-a^3-8c)\}^{\frac{1}{2}}\), and that if the triangle is right-angled then \[ a(4ab-a^3-8c)(a^2-2b)-8c^2=0. \]

If one root of the equation \(x^3+ax+b=0\) is twice the difference of the other two, prove that one root is \(\frac{13b}{3a}\).

Prove that, if the equation \(\sqrt{(ax+b)} + \sqrt{(cx+d)}=e\) has equal roots, they are given by \((ax+b)/(cx+d)=1\) or \(a^2/c^2\).

Let \(C_1\) be the plane curve whose polar equation is \(r\theta = 1\), \(\theta \geq \pi\) and let \(C_2\) be the curve whose equation is \(r(\theta + \pi) = 1\), \(\theta \geq \pi\). Show that these curves do not cross. Find the spiral-shaped area enclosed by \(C_1\), \(C_2\) and the line segment \(\theta = \pi\), \(1/2\pi \leq r \leq 1/\pi\). Find the area of the snail-shaped region bounded by the arc \(\pi \leq \theta \leq 3\pi\) of \(C_1\) and the line segment \(\theta = \pi\), \(1/3\pi \leq r \leq 1/\pi\).

(i) Explain why the transformation from Cartesian coordinates \((x, y)\) to polar coordinates \((r, \theta)\) given by \(r = (x^2 + y^2)^{\frac{1}{2}}\), \(\theta = \tan^{-1}(y/x)\) needs further comment. (ii) Let \(a\), \(b\), \(c\) and \(d\) be real numbers with \(ad - bc \neq 0\), and let \(T(x) = \frac{ax + b}{cx + d}\). Assuming that \(T(x) \neq x\), find a necessary and sufficient condition in terms of \(a\), \(b\), \(c\) and \(d\) for the identity \(T(T(x)) \equiv x\) to hold. Show that if \(T\) satisfies this condition and \(c \neq 0\) then there are two distinct solutions (possibly complex) of \(T(x) = x\).

Sketch the plane curve \(C\) whose polar equation is \(r = a\textrm{cosec}^2\frac{1}{2}\theta\), where \(0 < \theta < 2\pi\). Calculate: (i) the length of the arc \(C_1\) consisting of those points of \(C\) such that \(\frac{1}{2}\pi \leq \theta \leq \pi\); (ii) the area enclosed by the arc \(C_1\) and the radii \(\theta = \frac{1}{2}\pi\) and \(\theta = \pi\).

Sketch on the same diagram the curves given in polar co-ordinates \((r, \theta)\) by the equations \(r = \frac{1}{2}a(1 + \cos \theta)\) and \(r = a\theta\) (\(a > 0\), \(0 \leq \theta \leq 2\pi\)). Find the area of the region consisting of all those points \((r,\theta)\) such that \(\frac{1}{3}\pi \leq \theta \leq 2\pi\) and \(\frac{1}{2}a(1 + \cos \theta) \leq r \leq a\theta\).

Sketch and describe the three curves given in polar coordinates by \begin{align*} (i)&~ r = \sin\theta \quad (0 < \theta < \pi);\\ (ii)&~ r^{-1} = \sin\theta \quad (0 < \theta < \pi);\\ (iii)&~ r^{-2} = \sin 2\theta \quad (0 < \theta < \pi/2). \end{align*} [No credit will be given for solutions obtained by numerical methods alone.]

A closed curve is given in polar coordinates by the equation $$r = a(1 - \cos \theta).$$ Show that the tangent at the point \(\theta\) is inclined at an angle \(\psi = \frac{1}{2}\theta\) to the axis \(\theta = 0\). Find the radius of curvature at the point \(\theta\).

Sketch the curve whose equation, in polar coordinates, is \begin{equation*} \frac{l}{r} = 1+e\cos\theta, \end{equation*} \(e\) being a positive constant; distinguish between the cases \(e < 1\), \(e = 1\), \(e > 1\). By using the substitution \begin{equation*} \cos\phi = \frac{\cos\theta+e}{1+e\cos\theta} \quad (0 \leq \theta \leq \pi), \end{equation*} or otherwise, find the area enclosed by the curve when it is closed.

Sketch the `\(2m\)-rose' defined in polar coordinates by \(r = |\sin m\theta|\), for \(m = 1, 2, 3\). Show that for all integers \(m > 0\) the total area of the petals is independent of \(m\), and evaluate this area.

A curve is given parametrically in plane polar coordinates by \((r, \theta) = (e^t, 2\pi t)\) \((0 \leq t < \infty)\). Sketch the section of the curve for \(n \leq t \leq n + 1\), where \(n\) is an integer. Calculate the length of this section, and the area enclosed by it and the line \(\theta = 0\), \(r^n \leq r \leq r^{n+1}\).

Sketch the curve whose equation in polar coordinates is \[r = 1 - \frac{5}{6} \sin \theta.\] Find the range of real values of \(b\) for which the simultaneous equations \[(x^2 + y^2 + \frac{5}{6}y)^2 = x^2 + y^2\] \[y = b\] have a real solution.

Sketch the curve given by the equations \begin{align*} x &= a(\theta + \sin\theta)\\ y &= a(1 - \sin\theta), \quad a > 0. \end{align*} Find the area under the curve between two successive points where \(y = 0\).

A curve is given in polar coordinates by \(r(\theta)\) for \(0 \leq \theta \leq \pi\), and it is rotated about the axis \(\theta = 0\) to form a solid of revolution. Derive a formula for the surface area of the solid. Calculate the area of the surface so formed in the case \(r(\theta) = ae^{k\theta}\).

At time \(t = 0\), 4 insects \(A\), \(B\), \(C\) and \(D\) stand at the corners of a square of side \(a\). For time \(t > 0\) each insect crawls with constant speed \(v\) in the direction of the next insect in cyclic order (that is, \(A\) crawls towards \(B\), \(B\) towards \(C\), and so on). Show that the insects meet after a time \(a/v\) and that they encircle the centre of the square an infinite number of times. [Hint: Use polar coordinates.]

A circle of radius \(a\) rolls without slipping around the outside of a circle of radius \(2a\). Show that the arc length of a curve traced out by a point on its circumference is \(24a\).

A mouse runs along a straight line \(y = 0\) with uniform speed \(V_1\). When the mouse is at the point \(x = 0\), \(y = 0\) it is spotted by a cat at the point \(x = 0\), \(y = b\) which immediately gives pursuit. The cat runs with constant speed \(V_2\) (\(> V_1\)) and is always directed at the fleeing mouse. Make a qualitative sketch of the path of the cat (i) relative to a fixed frame of reference, and (ii) relative to a frame of reference moving with the mouse. Let \((r, \theta)\) be polar coordinates in this latter frame of reference, the mouse being at \(r = 0\) and \(\theta\) being measured from its direction of motion. Show that the differential equation of the path of the cat is \[\frac{1}{r}\frac{dr}{d\theta} = -\frac{V_2}{V_1}\csc\theta-\cot\theta.\] Integrate this, and show that if \(V_2 = 2V_1\) the cat catches the mouse after a time of pursuit \(2b/3V_1\).

A solid cone is described by the following equations (in cylindrical polar coordinates \((r, \phi, z)\)): \begin{align} r &\leq -z\tan\alpha \quad (\alpha < \pi/2)\\ z &\leq 0\\ z &\geq mr\sin\phi-a \quad (m < \cot\alpha) \end{align} \(\alpha\), \(m\) and \(a\) are constants with \(a > 0\). Sketch the cone. The cone is placed on its side on a plane, with its vertex at a point \(O\). The cone is in contact with the plane along the line segment \(OP\), which is initially of length \(a\sec\alpha\). The cone now rolls on the plane, the vertex remaining at \(O\). Obtain the polar equation of the locus of the point \(P\) in terms of the distance \(R\) from \(O\) to \(P\), and the angle between the line \(OP\) and its initial direction. What conditions are required on \(\alpha\) to ensure that the curve is closed?

A string of length \(\pi\) is attached to the point \((-1, 0)\) of the circle \(x^2 + y^2 = 1\), and is wrapped round the circle so that its other end is at the point \((1, 0)\). The string is unwound, being kept taut, and is wound up again the other way; if \(S\) is the path of the end of the string, show that \(S\) has length \(2\pi^2\), and find the area enclosed by \(S\).

A man is unwinding a string wrapped round a smooth closed convex curve \(ABCD\) on a piece of paper. When \(AB\) (of length \(a\)) is unwound, that part of the string becomes impregnated with ink. Prove that when \(ABC\) has been unwound, the area of the ink blot is \(\pi \int_B^C s \, d\psi\), where \(s\) is the arc-length measured from the mid-point of \(AB\), and \(\psi\) is the angle between the tangent and some fixed direction.

Sketch the curve \(C\) whose equation in polar coordinates is $$r^2 = a^2\cos 2\theta,$$ where \(a > 0\) and it is understood that \(r\) may take negative as well as positive values. Show that the perimeter \(s\) of \(C\) is given by $$s = 4a \int_0^{\pi/4} \frac{d\theta}{\sqrt{\cos 2\theta}}.$$ By means of the substitution \(t = \tan^4\theta\), or otherwise, express \(s\) in terms of the function \(B(p, q)\) defined by $$B(p, q) = \int_0^1 t^{p-1}(1-t)^{q-1} dt \quad (p > 0, q > 0).$$

Sketch the curve \(r = a(1 + \cos\theta)\) and find its total length. Find also the perpendicular distance between the origin and the tangent to the curve at the point \((r, \theta)\).

Calculate the volume of the solid of revolution formed by rotating the cardioid \(r = a(1-\cos\theta)\) about the line \(\theta = 0\).

\(P_1, P_2, \dots, P_N\) are \(N\) points lying on a straight line \(l\). For \(n=1, 2, \dots, N\), the polar coordinates measured from \(P_n\) as vertex and \(l\) as axis are \(r_n, \theta_n\), and \(a_1, a_2, \dots, a_N\) are constants. Show that every curve of the family \[ \sum_{n=1}^N \frac{a_n}{r_n} = \text{constant} \] cuts every curve of the family \[ \sum_{n=1}^N a_n \cos\theta_n = \text{constant} \] orthogonally.

Let \((r,\theta)\) denote polar coordinates in the plane. (i) Find the area lying within both the circle \(r=1\) and the cardioid \(r=1+\sin\theta\). (ii) Find the area lying within both the circle \(r=\sqrt{2}\sin\theta\) and the lemniscate \(r^2=\cos 2\theta\).

Give a rough sketch of the curve \[ y^2 - 2x^3 y + x^7 = 0, \] and find the area of the loop.

Sketch the curve \[ y^2(1+x^2) = (1-x^2)^2, \] and find the area of its loop.

The rectangular hyperbola \(xy = k^2\) is met by a circle passing through its centre \(O\) in four points \(A_1\), \(A_2\), \(B_1\), \(B_2\). The lengths of the perpendiculars from \(O\) to \(A_1A_2\) and \(B_1B_2\) are \(a\) and \(b\). Prove that $$ab = k^2.$$

Two circles \(C_1, C_2\) of radii \(r_1\) and \(r_2\), each touch the parabola \(y^2 = 4ax\) in two points. Show that the centres of the circles lie on the axis of the parabola, and that, if \(C_1\) and \(C_2\) touch each other, then the difference between their radii is \(4a\).

\(Q, R\) are two points on a rectangular hyperbola subtending a right angle at a point \(P\) of the curve. Prove that \(QR\) is parallel to the normal at \(P\).

If \(l_1 = 0, l_2 = 0\) are the equations of two lines, and if \(S = 0\) is the equation of a conic, interpret the equation \(S + \lambda l_1 l_2 = 0\), where \(\lambda\) is a parameter. Hence, or otherwise, show that if a circle meets an ellipse in four points, the joins of these points in pairs are equally inclined to the axis of the ellipse. Circles are drawn to meet the ellipse \(x^2/a^2 + y^2/b^2 = 1\) in four points lying in pairs on two lines through the fixed point \((x_1, y_1)\). Show that the circles form a co-axial system, and find the equation of the radical axis.

\(C\) is a circle whose centre is a point \(P\) on a rectangular hyperbola \(R\), and which passes through the centre \(O\) of \(R\). \(T\) is the diameter of \(C\) which is tangent to \(R\) at \(P\), and \(N\) is the diameter which is normal to \(R\) at \(P\). Show that the endpoints of \(T\) lie on the asymptotes of \(R\), while those of \(N\) lie on the axes of symmetry.

If \(a\) and \(b\) are real positive constants, show that the equation $$\pm\sqrt{\left(\frac{x}{a}\right)} \pm \sqrt{\left(\frac{y}{b}\right)} = 1$$ represents a conic section, and by considering the behaviour of the curve for large values of \(x\) and \(y\), that this conic section is a parabola. Find the direction of the axis of this parabola, and sketch the curve, indicating the sections of it which correspond to the various possible combinations of the \(\pm\) signs.

The normals at the points \(A\), \(B\), \(C\) of a parabola meet in a point \(P\), and \(H\) is the orthocentre of the triangle \(ABC\). Prove that

1. the circumcircle of \(ABC\) passes through the vertex of the parabola;
2. the projection of \(PH\) on the axis of the parabola is of constant length.

Prove that the normals to a parabola at the points \(Q\), \(R\) intersect on the curve if and only if \(QR\) passes through a certain fixed point. Suppose that this condition is satisfied, and let the normals at \(Q\), \(R\) meet at \(P\). If \(P'\) is the intersection of the parabola with the line parallel to the axis passing through the common point of \(QR\) and the directrix, show that \(PP'\) passes through the focus.

Interpret the equation \(S + \lambda T^2 = 0\), where \(S = 0\) and \(T = 0\) are the equations of a conic and one of its tangents, and \(\lambda\) is a constant. Hence or otherwise find the equations of the circles of curvature of the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) at the ends of the major axis. Show that these circles touch each other if \(a^2 = 2b^2\), and find the condition that each should be touched by the circle with the minor axis as diameter.

The point \((x', y')\) is exterior to the ellipse \[\frac{x^2}{a^2}+\frac{y^2}{b^2}-1=0.\] Establish a basic geometric property of the line \[\frac{xx'}{a^2}+\frac{yy'}{b^2}-1 = 0.\] Show that \[\lambda \left(\frac{x^2}{a^2}+\frac{y^2}{b^2}-1 \right) - \left(\frac{xx'}{a^2}+\frac{yy'}{b^2}-1 \right)^2 = 0\] when \[\lambda = \frac{x'^2}{a^2}+\frac{y'^2}{b^2}-1\] is the equation of a conic which touches the ellipse at two points. Identify this conic.

A parabola rolls symmetrically on an equal fixed parabola. Find the locus of its focus.

A, B, C, D are four points on a parabola. The lines through B and D parallel to the axis of the parabola meet CD and AB, respectively, in E and F. Prove that AC is parallel to EF.

\(A_1\), \(A_2\), \(A_3\), \(A_4\) are four points of the rectangular hyperbola whose general point is \((d, c/t)\). If the normals at these points are concurrent, prove that each of the points is the orthocentre of the triangle formed by the other three. Show also that the conic through \(A_1\), \(A_2\), \(A_3\), \(A_4\) and the centre of the hyperbola is a second rectangular hyperbola whose axes are parallel to the asymptotes of the first.

Show that the four points \((at_i^2, 2at_i)\), for \(i = 1,2,3,4\), of the parabola \(y^2 = 4ax\) are concyclic if, and only if, \(t_1 + t_2 + t_3 + t_4 = 0\). \(PP'\) is a chord of a parabola perpendicular to the axis. A circle touches the parabola in \(P\) and meets it again in \(Q\) and \(R\). Show that \(QR\) is parallel to the tangent at \(P'\).

\(P\) is a variable point that moves so that the sum of its distances from fixed points \(S, S'\) is constant. By finding the equation of the locus of \(P\), or otherwise, show that the tangent to this locus at \(P\) bisects the angle \(SPS'\) externally.

Two adjacent corners \(A\), \(B\) of a rigid rectangular lamina \(ABCD\) slide on the \(x\)-axis and the \(y\)-axis respectively, and all the motion is in one plane. Prove that the locus of \(C\) is an ellipse, and find the area of the ellipse in terms of \(a = AD\) and \(b = AB\). [The area of an ellipse is \(\pi \times\) the product of the lengths of the semi-axes.]

What is the equation of the chord of the parabola \(y^2 = 4a(x - k)\) joining the points \((at^2+k, 2at)\) and \((as^2+k, 2as)\)? What is the equation of the tangent at $(at^2+k, 2at)$? Show that the chords joining \((at^2 + k, 2at)\) to \((as^2 + k, 2as)\), where \(s = t + \lambda\) (\(\lambda\) fixed, \(t\) varying), all touch another parabola, and find its equation.

A mirror has the form of the ellipse \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.\] Light rays are emitted, in the plane of the ellipse, from the focus \((ae, 0)\) and are reflected at the mirror. Show that they all pass through the other focus, \((-ae, 0)\).

The surface of a lawn is a plane inclined to the horizontal at an angle \(\alpha\). A sprinkler is embedded in the surface, and emits droplets of water in all directions, the speed of projection being \(v\). Show that the region watered is an ellipse with area \(\pi v^4/g^2\cos^2\alpha\).

A triangle \(ABC\) is said to be self-conjugate with respect to a circle if \(A\) is the pole of \(BC\), \(B\) is the pole of \(CA\), and \(C\) is the pole of \(AB\). Show that if the triangle \(ABC\) has an obtuse angle there is just one circle with respect to which it is self-conjugate, but that otherwise there is no such circle.

A circle touches the ellipse \(x^2/a^2 + y^2/b^2 = 1\) at its intersections with the line \(x = c\). Find its centre and radius. Interpret your results when \(c\) is formally put equal to (i) \(a\), (ii) a value strictly between \(a\) and \(a/e\), (iii) \(a/e\), where \(e\) is the eccentricity of the ellipse.

Two lines in the plane are perpendicular. An ellipse in the plane moves so that it always touches both lines. Describe the locus of the centre of the ellipse.

A point moves in the plane so that its distances from a fixed point \(P\) and a fixed line \(l\) (not through \(P\)) are in the ratio \(\lambda\) to 1. Describe the locus of the point and draw a sketch of the loci obtained for varying \(\lambda\), indicating the effect as \(\lambda\) increases and the locus for \(\lambda = 1\). Do the same where \(P\) and \(l\) are replaced by (a) two fixed points, and (b) two fixed (intersecting) lines. [Detailed arguments are not required for (a) or (b).]

Prove that the straight line \[ty = x+at^2\] touches the parabola \(y^2 = 4ax\) (\(a \neq 0\)), and find the coordinates of the point of contact. The tangents from a point to the parabola meet the directrix in points \(L\) and \(M\). Show that, if \(LM\) is of a fixed length \(l\), the point must lie on \[(x+a)^2(y^2-4ax) = l^2x^2.\]

Suppose \(a > b > 0\). Show that the circle of curvature of the ellipse \begin{align*} x^2/a^2 + y^2/b^2 = 1 \end{align*} at the point \((0, -b)\) is \begin{align*} b^2x^2 + (by + b^2 - a^2)^2 = a^4. \end{align*} The tangent to the ellipse at \((a \cos \theta, b \sin \theta)\) meets the circle at \(P\) and \(Q\), and the lines from \((0, -b)\) to \(P\) and \(Q\) meet the \(x\)-axis at \(X_1\) and \(X_2\). Show that the distance from \(X_1\) to \(X_2\) is equal to the distance between the foci of the ellipse.

(i) Eliminate \(\theta\) from the equations \[ a\tan\theta+\sec\theta=h, \quad a\cot\theta+\csc\theta=k. \] (ii) Sum the infinite series \[ \frac{1}{2!} + \frac{4}{3!} + \frac{9}{4!} + \dots + \frac{n^2}{(n+1)!} + \dots. \]

Assume that, if a function of \(x\) vanishes for two values of \(x\), its derivative vanishes for an intermediate value of \(x\). If \[ \phi(x) = \int_x^b f(t) dt - \tfrac{1}{2}(b-x)\{f(x)+f(b)\} + \tfrac{1}{12}(b-x)^3 R, \] where the constant \(R\) is so chosen that \(\phi(a)\) vanishes, show that \(\phi'(x)\) vanishes for \(x=\alpha\) and \(x=b\), where \(a< \alpha< b\). Deduce that \[ \int_a^b f(t) dt = \tfrac{1}{2}(b-a)\{f(a)+f(b)\} - \tfrac{1}{12}(b-a)^3 f''(\beta), \] where \(a< \beta < b\). Hence show that the difference between \(\int_a^{a+nh} f(t) dt\) and \[ \tfrac{1}{2}h\{f(a)+f(a+nh)\} + h\{f(a+h)+f(a+2h)+ \dots + f(a+\overline{n-1}h)\} \] is less than \(\frac{1}{12}nh^3M\), where \(M\) is the greatest value of \(|f''(t)|\) in \(a< t< a+nh\).

By the use of Maclaurin's theorem, or otherwise, prove that \[ \sin x \sinh x = \frac{2x^2}{2!} - \frac{2^3x^6}{6!} + \frac{2^5x^{10}}{10!} - \dots. \]

If \[ f(x) = \frac{d^n}{dx^n}(x^2-1)^n \] and \(p(x)\) is any polynomial of degree less than \(n\), prove that \[ \int_{-1}^1 f(x)p(x)dx=0, \] and hence or otherwise show that \(f(x)\) vanishes for exactly \(n\) values of \(x\) between \(+1\) and \(-1\). If \(F(x)\) is polynomial of degree \(n\) such that \[ \int_{-1}^1 F(x)p(x)dx=0 \] for every polynomial \(p(x)\) of degree less than \(n\), show that \(F(x)\) is a constant multiple of \(f(x)\).

Show that \[ (1+x)^\lambda = 1 + \lambda x + \frac{\lambda(\lambda-1)}{2!}x^2 + \dots + \frac{\lambda(\lambda-1)\dots(\lambda-n+1)}{n!}x^n \] \[ + \frac{\lambda(\lambda-1)\dots(\lambda-n)}{n!}(1+x)^\lambda \int_0^x t^n (1+t)^{-\lambda-1} dt \] for \(x>-1\), and \(\lambda\) rational. Find the first four terms in the expansion of \(\left(\dfrac{1}{1+x}\right)^\lambda\) in powers of \(x\).

If \(y\) is defined as a function of \(x\) by the equation \(y\sqrt{1+x^2}=\log[x+\sqrt{1+x^2}]\), prove that \[ (1+x^2)y' + xy = 1 \] and express \(y\) as a series in ascending powers of \(x\). Hence show that the sum of the series \[ 1 - \frac{1}{3!} + \frac{(2!)^2}{5!} - \frac{(3!)^2}{7!} + \dots \] is \(\dfrac{4}{\sqrt{5}}\log\tfrac{1}{2}(1+\sqrt{5})\).

Find an integral value of \(x\) such that \[ \frac{e^x}{x^{12}} > 10^{20}. \] (Your answer need not be the smallest possible value of \(x\), but it must not exceed that value by more than ten per cent. Tables may be used.) Enunciate and prove a general statement of which the existence of an \(x\) satisfying the above inequality is a particular consequence. (You may start from any definition of \(e^x\), to be specified.)

Obtain an explicit formula for \((\frac{d}{dx})^n \tan^{-1}x\). Show that for \(x=0\) its value is zero for \(n\) even, and \(\pm (n-1)!\) if \(n\) is odd and of the form \(4p\pm 1\) where \(p\) is an integer. Hence write down the power series for \(\tan^{-1}x\).

State exactly what the statement "\(y^n e^{-y}\) tends to the limit 0 as \(y\) tends to \(+\infty\)" means. (It may be assumed true without proof.) A function \(f(x)\) of the real variable \(x\) is defined as follows: \begin{align*} f(x) &= e^{-1/x^2} \quad \text{if } x \neq 0, \\ f(x) &= 0 \quad \text{if } x = 0. \end{align*} When \(x \neq 0\), show that its \(n\)th derivative can be written \[ f^{(n)}(x) = G_n(1/x) \cdot e^{-1/x^2}, \] where \(G_n\) is a polynomial of degree \(3n\), and that the coefficients in \(G_n\) are all smaller than \(n!3^n\) in absolute magnitude. (Use mathematical induction.) Hence show that \[ |f^{(n)}(x)| < \frac{(n+1)!3^{n+1}}{|x|^{3n}} e^{-1/x^2} \] if \(0<|x|<1\). Prove that \(f^{(n)}(0)=0\) for all values of \(n\). Is there anything remarkable about this conclusion?

By use of the series for \(\log(1+z)\), or otherwise, prove for a range of values of \(r\) to be specified that \[ r \sin \theta + \frac{1}{2}r^2 \sin 2\theta + \frac{1}{3}r^3 \sin 3\theta \dots = \arctan\left(\frac{r \sin \theta}{1-r \cos \theta}\right), \] making clear how the many-valued function \(\arctan\) is to be interpreted.

Find an expression for \(\frac{d^n}{dx^n} \tan^{-1}x\). \newline Prove that when \(x=0\) its value is zero if \(n\) is even and \(\pm(n-1)!\) if \(n\) is of the form \(4p\pm 1\), where \(p\) is an integer. \newline Hence obtain the series expansion of \(\tan^{-1}x\).

(i) Find the limit of \((\cos x)^{\cot^2 x}\) as \(x \to 0\). \newline (ii) Determine constants \(a\) and \(b\) in order that \((1+a \cos 2x + b \cos 4x)/x^4\) may have a finite limit as \(x \to 0\), and find the value of the limit.

The quantity \(x\) (\(0 < x < 1\)) is determined by the equation \[ \cot(\lambda\sqrt{1-x}) = -\sqrt{\frac{x}{1-x}}, \] \(\lambda\) being positive. Discuss how the number of possible values of \(x\) depends on \(\lambda\) and show that if \(\lambda\) is such that only one value of \(x\) satisfies the equation, and if that value is small compared to unity, then approximately \[ \lambda^2 = \frac{\pi^2}{4} + \pi\sqrt{x} + \left(1+\frac{\pi^2}{4}\right)x. \]

Shew that an approximate solution of \(x \log x + x - 1 = \epsilon\), where \(\epsilon\) is small, is \[ x = 1 + \frac{\epsilon}{2} - \frac{\epsilon^2}{16}. \]

Prove that \(\log_e \{\log_e (1+x)^{1/x}\} = -\frac{1}{2}x + \frac{5}{24}x^2 - \frac{1}{8}x^3 - \dots\). Find, without using tables, the value of \(\log_e (\log_e 1.01)\) correct to five places of decimals, having given \(\log_e 10 = 2 \cdot 302585\).

Prove that \[ \log_e \frac{p}{q} = 2 \left\{ \frac{p-q}{p+q} + \frac{1}{3} \left( \frac{p-q}{p+q} \right)^3 + \frac{1}{5} \left( \frac{p-q}{p+q} \right)^5 + \dots \right\}, \] where \(p\) and \(q\) are both positive. Show that the error involved in stopping the series at the third term is certainly less than \[ \frac{(p-q)^7}{14pq(p+q)^6}. \]

An approximate value for the angle \(\phi\), measured in radians, is \(\displaystyle\frac{3 \sin\phi}{2 + \cos\phi}\), provided \(\phi\) is less than \(\frac{1}{2}\pi\). Establish this result when \(\phi\) is small, and shew that the error is approximately \(\displaystyle\frac{\phi^5}{180}\). Hence, express approximately the acute angles of a right-angled triangle in terms of the sides, and deduce that \[ \frac{6(a+b)c + 3c^2}{ab + 2(a+b)c + 4c^2} \] is nearly equal to \(\frac{1}{2}\pi\) for all positive values of \(a\) and \(b\), provided \(c^2 = a^2+b^2\).

If \[ y = (x+1)^\alpha (x-1)^\beta, \] prove that \[ \frac{d^n y}{dx^n} = (x+1)^{\alpha-n} (x-1)^{\beta-n} Q_n(x), \] where \(Q_n(x)\) is a polynomial of degree \(n\) (or lower) in \(x\), and shew that \[ Q_{n+1}(x) = \{(\alpha+\beta-2n)x - (\alpha-\beta)\}Q_n(x) + (x^2-1)Q_n'(x). \] Prove also that \[ (1-t)^n Q_n\left(\frac{1+t}{1-t}\right) = 2^n n! \sum_{v=0}^n \binom{\alpha}{v} \binom{\beta}{n-v} t^v, \] where \(\dbinom{\alpha}{0}=1\), \(\dbinom{\alpha}{v} = \dfrac{\alpha(\alpha-1)\dots(\alpha-v+1)}{v!}\) (\(v=1,2,\dots\)).

Shew that, if \[ e^x \sin x = a_0 + \frac{a_1}{1!}x + \frac{a_2}{2!}x^2 + \dots + \frac{a_n}{n!}x^n + \dots, \] then \(a_{4n}=0\), \(a_{4n+1} = (-1)^n 4^n\); and determine \(a_{4n+2}\) and \(a_{4n+3}\).

(i) Prove that an approximate solution of the equation \[ xe^{x-1} + x - 2 = \epsilon, \] where \(\epsilon\) is small, is \[ x = 1 + \tfrac{1}{3}\epsilon - \tfrac{1}{18}\epsilon^2. \] (ii) Assuming that \(\sin^{-1} x\) may be expanded in ascending powers of \(x\), find the first three non-zero terms in the expansion. The expansion of \(\sin x\) in powers of \(x\) may be assumed, if required.

Show that the equation \[ \sin x = \tanh x \] has infinitely many real roots and that, if \(n\) is a large positive integer, approximate values of a pair of roots are \[ x = (2n + \tfrac{1}{2}) \pi \pm 2e^{-(2n+\frac{1}{2})\pi}. \]

Prove that, if \(\alpha\) is small, one root of the equation \[ \alpha x^3 = x^2 - 1 \] is approximately \[ 1 + \tfrac{1}{2}\alpha + \tfrac{5}{8}\alpha^2, \] and find approximations to the other two roots.

Prove that \[ \left(\frac{d}{dx}\right)^n \tan^{-1}x = P_{n-1}(x)/(x^2+1)^n, \] where \(P_{n-1}\) is a polynomial in \(x\) of degree \(n-1\), and \[ P_{n+1} + 2(n+1)xP_n + n(n+1)(x^2+1)P_{n-1} = 0. \]

Show that \[ \phi(x) = \frac{3 \int_0^x (1+\sec y)\log\sec y\,dy}{\{x+\log(\sec x+\tan x)\}\log\sec x}\] is an even function of \(x\), and that for small \(x\) we have approximately \[ \phi(x) = 1+\frac{1}{420}x^4.\]

Discuss the nature of the contact of two given curves at a common point. Apply your results to shew that if the coordinates of a point on a curve are given functions of a parameter \(t\), (i) the coordinates \((\xi, \eta)\) of the centre of the osculating circle at the point \((x,y)\) are given by the equations \[ \frac{\xi-x}{-\dot{y}} = \frac{\eta-y}{\dot{x}} = \frac{\dot{x}^2+\dot{y}^2}{\dot{x}\ddot{y}-\ddot{x}\dot{y}}, \] (ii) the equation of the diameter of the parabola of closest contact which passes through the point is % The notation here is very condensed. Representing it with determinants. \[ \left| \begin{matrix} \ddot{x} & \dddot{y} \\ \dot{x} & \dot{y} \end{matrix} \right| - \left| \begin{matrix} \dot{x} & \dddot{y} \\ \dot{x} & \dot{y} \end{matrix} \right| (\xi-x, \eta-y) = 3 \left| \begin{matrix} \dot{x} & \ddot{y} \\ \dot{x} & \dot{y} \end{matrix} \right| \left| \begin{matrix} \dot{x} & \xi-x \\ \dot{y} & \eta-y \end{matrix} \right|, \] where dots indicate differentiation with regard to the parameter.

If \(y = \cos(m \sin^{-1} x)\), show that \begin{equation*} (1 - x^2)\left(\frac{dy}{dx}\right)^2 - m^2(1 - y^2) = 0 \end{equation*} \begin{equation*} (1 - x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} + m^2y = 0. \end{equation*} Using Leibniz' theorem, or otherwise, show that, for integer \(n \geq 0\), \begin{equation*} (1 - x^2)\frac{d^{n+2}y}{dx^{n+2}} - (2n + 1)x\frac{d^{n+1}y}{dx^{n+1}} + (m^2 - n^2)\frac{d^ny}{dx^n} = 0. \end{equation*} By considering the Taylor series of \(\cos(m \sin^{-1} x)\) about \(x = 0\), show that \begin{equation*} \cos mx - \cos(m \sin^{-1} x) = \frac{m^2x^4}{3!} + \text{higher order terms.} \end{equation*}

By applying the Taylor expansion to the function \(f(x) \equiv (x^2-1)^n\), or otherwise, prove that for all \(x\), and \(h \neq 0\), \[\left[\frac{(x^2-1) + 2hx + h^2}{h}\right]^n = \sum_{r=0}^{2n} \frac{h^{r-n}}{r!}\left(\frac{d}{dx}\right)^r[(x^2-1)^n].\] Write \((x^2-1)/h\) for \(h\) on each side of the above equation, and show that for \(1 \leq m \leq n\), \[\frac{1}{(n-m)!}\left(\frac{d}{dx}\right)^{n-m}[(x^2-1)^n] = \frac{1}{(n+m)!}(x^2-1)^m\left(\frac{d}{dx}\right)^{n+m}[(x^2-1)^n].\] Deduce that \[y(x) = \left(\frac{d}{dx}\right)^n[(x^2-1)^n]\] satisfies the differential equation \[\frac{d}{dx}\left[(x^2-1)\frac{dy}{dx}\right] - n(n+1)y = 0.\]

Expand in a power series in \(x\), as far as the term in \(x^3\), $$e \log \log(e + x) - x e^{-x/e},$$ where \(e\) is the base of natural logarithms.

Let $$f(x) = 1 + \frac{x}{a} + \frac{x^2}{a(a+1)} + \ldots + \frac{x^n}{a(a+1)\ldots(a+n-1)} + \ldots,$$ where \(|x| < 1\) and \(a\) is a constant satisfying \(0 < a < 1\). Show that \(f(x)\) can be expressed in a form using only the elementary functions and a finite number of operations of addition, subtraction, multiplication, division, integration and differentiation.

Let \[ y=f(x) = \frac{\sinh^{-1} x}{\sqrt{1+x^2}}. \] Prove that \[ (1+x^2)\frac{dy}{dx} + xy = 1. \] Show that the Maclaurin series for \(f(x)\) is \[ x - \frac{2^2}{3!}x^3 + \frac{2^4(2!)^2}{5!}x^5 - \frac{2^6(3!)^2}{7!}x^7 + \dots. \]

Obtain power series in increasing integral powers of \(x\) for \(\tan^{-1}x\), and \(\tanh^{-1}x\), where the principal value of the former function is to be taken. Verify from the series obtained that \(\tanh^{-1}ix=i\tan^{-1}x\).

Obtain the coordinates of the centre of curvature at any point of the curve \(x=f(t), y=g(t)\). Sketch the curve \(x=at\cos t, y=at\sin t\), and prove that the centre of curvature at any point lies inside the circle \(x^2+y^2=a^2\). Mark on your sketch the approximate position of the centre of curvature of a point given by a large value of \(t\).

Shew that if \[ e^{\tan^{-1} x} = a_0 + \frac{a_1}{1!} x + \frac{a_2}{2!} x^2 + \dots\dots + \frac{a_n}{n!} x^n + \dots\dots, \] then \[ a_{n+2} = a_{n+1} - n (n + 1) a_n. \]

Shew that \[ \frac{d^n e^{-x^2}}{dx^n} = (-1)^n e^{-x^2} \phi_n(x), \] where \(\phi_n(x)\) is a polynomial of degree \(n\) in which the coefficient of \(x^n\) is \(2^n\). Establish the relations \begin{align*} \phi_n'(x) &= 2n \phi_{n-1}(x), \\ \phi_n''(x) - 2x\phi_n'(x) + 2n\phi_n(x) &= 0. \end{align*}

Prove that, if \(c_n\) is the coefficient of \((x+1)^n\) in the expansion of \[ \frac{e^{x^2+2x}}{(x^2+2x+2)^2} \] in a series of positive powers of \((x+1)\), then \(c_n=0\) if \(n\) is odd, while \[ c_{2k} = \frac{1}{2e} \sum_{m=0}^k \frac{(-1)^m(m+1)(m+2)}{(k-m)!} \quad (k=0, 1, 2, \dots). \]

If \[y = \frac{\log \{x + \sqrt{(1+x^2)}\}}{\sqrt{(1+x^2)}},\] verify that \[(1+x^2)\frac{dy}{dx} + xy = 1.\] Assuming that \(y\) can be expanded in a series of ascending powers of \(x\), prove that the series is \[x - \frac{2}{3}x^3 + \frac{2.4}{3.5}x^5 - \dots + (-1)^n \frac{2.4\dots2n}{3.5\dots(2n+1)}x^{2n+1} + \dots.\]

Prove Leibnitz' formula for the \(n\)th differential coefficient of a product of two functions. \(y\) is a solution of the equation \[ 4x(1-x)\frac{d^2y}{dx^2} + 2(1-3x)\frac{dy}{dx}-y = 2+\log(1-x) \] for which \(\frac{dy}{dx}\) is finite when \(x=0\). Prove, by induction or otherwise, that \[ \left(\frac{d^n y}{dx^n}\right)_{x=0} = n!\left(\frac{1}{n+1} + \frac{1}{n+2} + \dots + \frac{1}{2n} + \frac{1}{2n+1}\right). \]

Find the \(n\)th derivative of the function \[ y = \frac{1}{x^2+c}, \] where \(c\) is a real constant, distinguishing the various cases that can arise. If \(c > 0\), prove that \[ \left| \frac{1}{n!} \frac{d^n y}{dx^n} \right| \le \frac{1}{c^{\frac{1}{2}n+1}} \] for all real values of \(x\).

Find the \(n\)th differential coefficient of (i) \(e^{ax}\cos bx\), (ii) \(\frac{\log x}{x}\). Prove that \[ \cosh ax \cos bx = 1 + \frac{x^2c^2\cos 2\alpha}{2!} + \dots + \frac{x^{2n}c^{2n}\cos 2n\alpha}{(2n)!} + \dots, \] where \(\tan\alpha = \frac{b}{a}\) and \(c^2=a^2+b^2\).

Prove that \(f(x+h) = f(x)+hf'(x+\theta h)\), for some value of \(\theta\) between 0 and 1, provided \(f(x)\) and its differential coefficient \(f'(x)\) satisfy certain conditions to be stated: and give a geometrical illustration of the theorem. If \(f(x)=e^x\), shew that \(\theta = \frac{1}{2} - \frac{h}{24} + \frac{h^2}{48}\) approximately, when \(h\) is small.

If \[ \left(\frac{d}{dx}\right)^n e^{-x^2} = \phi_n(x)e^{-x^2}, \] shew that \[ \phi_n + 2x\phi_{n-1} + 2(n-1)\phi_{n-2} = 0, \] and that \[ \phi_n'' - 2x\phi_n' + 2n\phi_n = 0. \] Shew also that \[ \phi_{2n}(0) = 1.3.5 \dots (2n-1).(-2)^n, \] and \[ \phi_{2n}'(0)=0. \]

A function \(\psi_n(x)\) is defined by the equation \[ \psi_n(x) = \frac{d^n}{dx^n}\frac{\sqrt{x}}{1+x} = \frac{\sqrt{x}}{(x+1)^{n+1}}\Psi_n(x). \] Shew

1. [(i)] that \(\Psi_n(x)\) is a polynomial of degree \(n\) in \(x\);
2. [(ii)] that \begin{align*} \Psi_{n+1}(x) &= x(1+x)\Psi_n'(x) - [(2n+\tfrac{1}{2})x+n-\tfrac{1}{2}]\Psi_n(x) \\ &= \frac{(-1)^n}{2^{n+1}} 1 \cdot 3 \dots (2n-1)(1+x)^{n+1} - (n+1)x\Psi_n(x); \end{align*}
3. [(iii)] that if \(n\) is even \(\Psi_n(x)\) has at least one root between \(-1\) and \(0\).

If \(Q_n(x) = (1+x^2)^{\frac{n}{2}+1} \frac{d^n y}{dx^n}\), where \(y=\frac{1}{\sqrt{(1+x^2)}}\), prove that \(Q_n(x)\) is a polynomial of degree \(n\) satisfying the following relations:

1. \(Q_{n+1} = (1+x^2)Q_n' - (2n+1)xQ_n\);
2. \(Q_{n+1} + (2n+1)xQ_n + n^2(1+x^2)Q_{n-1} = 0\);
3. \(Q_n' + n^2 Q_{n-1} = 0\);
4. \((1+x^2)Q_n'' - (2n-1)xQ_n' + n^2 Q_n = 0\).

State how to find the differential coefficient with respect to \(x\) of \[ \int_u^v f(x,t)dt, \] where \(u, v\) are functions of \(x\) and \(f(x,t)\) is a function of \(x,t\). If \[ \psi(x) = x^n \int_0^x \psi(t) dt, \] prove that \[ x \frac{d\psi(x)}{dx} = (n+x^{n+1})\psi(x), \] and hence find the form of \(\psi(x)\).

1. If \[ y = \tan^{-1} \frac{x\sin\alpha}{1+x\cos\alpha}, \] prove \[ \frac{d^n y}{dx^n} = \frac{(n-1)!}{\sin^n\alpha} \sin n(\alpha-y)\sin^n(\alpha-y). \] (The OCR is slightly different here, but this form seems more likely based on context. Original OCR: `sin n(y-a) sin^n(y-a)` and `sin^n a`. The image is clearer showing \(\alpha - y\)).
2. In the equation \[ f(x+h) = f(x) + hf'(x) + \frac{h^2}{2}f''(x+\theta h), \] find the first three terms of the expansion of \(\theta\) in ascending powers of \(h\), and calculate them for \(f(x) = \sin x\).

Find the \(n\)th differential coefficients of

1. [(i)] \((x+2)/(x^2-2x-3)\),
2. [(ii)] \(x^2\sin^3x\).

State McLaurin's theorem on the expansion of a function of \(x\) in ascending powers of \(x\). Prove that, if \(a_0+a_1x+a_2x^2+\dots\) is the expansion in ascending powers of \(x\) of \(\{\cosh^{-1}(1+x)\}^2\), \((n+1)(2n+1)a_{n+1} + n^2 a_n=0\).

Find the coefficients in the polynomial \(f_n(x)\) defined by \(f_n(x) = e^{-x} \frac{d^n}{dx^n} (x^n e^x)\). \par Prove the following identities, where \(n\) is a positive integer: \begin{align*} f_{n+1}(x) - (2n+1+x)f_n(x) + n^2 f_{n-1}(x) &= 0; \\ x f_n''(x) + (1+x)f_n'(x) - n f_n(x) &= 0. \end{align*}

If \[ f_n(x) = e^x \frac{d^n}{dx^n}(x^n e^{-x}), \] prove that \[ x\frac{d^2f_n(x)}{dx^2} + (1-x)\frac{df_n(x)}{dx} + n f_n(x) = 0. \]

Find the limit as \(x \to a\) of \((x^n-a^n)/(x-a)\) for commensurable values of \(n\), whether positive or negative, and apply the result to the differentiation of \(x^n\). Prove that \[ \frac{d^4}{dx^4}(x^a e^x) = a^x e^x. \]

Sketch the graph of the function \[\phi_n(x) = e^{-x} \left(1+x+\frac{x^2}{2!}+ \ldots +\frac{x^n}{n!}\right)-k,\] where \(k\) is a constant, \(0 < k < 1\); distinguish as you think fit between different values of \(n\). Show that there is just one positive value of \(x\) for which \[1+x+\frac{x^2}{2!}+ \ldots +\frac{x^n}{n!} = ke^x.\] Denoting this by \(x_n\), show that \(x_n < x_{n+1}\). [It may be assumed that, for any \(m\), \(x^me^{-x} \to 0\) as \(x \to \infty\).]

1. Show that \((1 + t)(1 - t + t^2 + \ldots + (-1)^n t^n) = 1 + (-1)^n t^{n+1}\).
2. Using this result for \(n = 1\) show that $$t - \frac{1}{2}t^2 < \ln(1 + t) < t \quad \text{for } t > 0.$$
3. Prove that $$\left|\ln(1 + t) - \sum_{r=1}^n (-1)^{r+1} \frac{t^r}{r}\right| < \frac{t^{n+1}}{n+1} \quad \text{for } t > 0.$$

Write down the expansions of \(e^x\) and \((1-x)^{-1}\) as power series in \(x\). Show that, for \(0 < a < \frac{1}{2}\), $$\int_0^a \frac{e^x-1}{x}dx < a + \frac{1}{4}a^2(1-\frac{1}{8}a)^{-1}.$$ Show also that $$1.80 < \int_0^1 \frac{e^x-1}{x}dx < 1.83.$$

Prove that if \(|x| \leq \frac{1}{2}\) then \(x \geq \log (1+x) \geq x-x^2\). By taking logarithms, or otherwise, show that for any positive integer \(k\) \[\left(1-\frac{1}{n^2}\right)\left(1-\frac{2}{n^2}\right)\ldots\left(1-\frac{kn}{n^2}\right) \to e^{-k^2/2}\] as \(n \to \infty\).

By considering the derivative of \(x - \sin x\) show that \(x \geq \sin x\) for all \(x \geq 0\). By considering the repeated derivatives of \(\sin x - x + x^3/3!\) show that \(\sin x \geq x - x^3/3!\) for all \(x \geq 0\). More generally, show that \[\sum_{r=0}^{2m} (-1)^r \frac{x^{2r+1}}{(2r+1)!} \geq \sin x \geq \sum_{r=0}^{2m-1} (-1)^r \frac{x^{2r+1}}{(2r+1)!}\] for all \(x \geq 0\) and \(m \geq 1\). Deduce that \[\left|\sum_{r=0}^{n-1} (-1)^r \frac{x^{2r+1}}{(2r+1)!} - \sin x\right| \leq \frac{x^{2n+1}}{(2n+1)!}\] for all \(x \geq 0\). [The power series expansion of \(\sin x\) must not be used.]

If \[y = \sin^{-1}x\] show that \[(1-x^2)y'' = xy',\] and hence using Leibniz' Theorem evaluate \(y^{(n)}(0)\). Write down the MacLaurin series for \(\sin^{-1}x\). By considering the series expansions of the two functions term by term, show that \[\sin^{-1}x < \frac{x}{1-x^2} \quad \text{for } 0 < x < 1.\]

Obtain a series expansion of \(\log_e\{1 + (1/x)\}\) in ascending powers of \(1/(2x+1)\). For what ranges of values of \(x\) is this expansion valid? Prove that if \(x\) is strictly positive, for what \[\frac{2x+1}{2x(x+1)} > \log_e\left(1 + \frac{1}{x}\right) > \frac{2}{2x+1}.\]

Using the equation \[ \tan^{-1}x = \int_0^x \frac{dt}{1+t^2} \] show that, if \(x>0\), \(\tan^{-1}x\) lies between \(x-\displaystyle\frac{x^3}{3}\) and \(x-\displaystyle\frac{x^3}{3}+\frac{x^5}{5}\). Use this result to evaluate \(\tan^{-1}\frac{1}{11}\) correct to five places of decimals.

Show that \[ \frac{\pi}{4} = 4 \arctan \frac{1}{5} - \arctan \frac{1}{239}. \] By using the series expansion of \(\arctan x\), or otherwise, evaluate \(\pi\) with an error of less than \(10^{-3}\).

Shew that \[ \frac{2n+(n+1)x}{2n+(n-1)x} < \sqrt[n]{(1+x)}, \] if \(x > 0\) and \(n > 1\). Shew also that the difference between the two functions, when \(x\) is small, is approximately \[ \frac{n^2-1}{12n^3}x^3. \]

Prove that if \(f(x)\) and its first two derivatives are continuous in \(0 \le x \le a\) (\(a>0\)), and \(x, x+h\) are any two points of this interval, then \[ f(x+h) = f(x) + hf'(x) + \frac{1}{2}h^2f''(x+\theta h), \] where \(\theta\) is some number satisfying \(0 < \theta < 1\). By taking \(h=-x\), or otherwise, prove that, if \(f(0)=0\), and \(f''(x)>0\) in \(0 < x < a\), then \(f(x)/x\) is an increasing function of \(x\) in this interval. Explain the geometrical significance of this result. Deduce that \((\sin x)/x\) decreases as \(x\) increases from \(0\) to \(\pi\).

A function \(f(x)\) and as many of its derivatives as are required are single valued and continuous for values of \(x\) in the neighbourhood of a given value \(a\). If \quad \(\psi(x) = f(a+h) - f(a+h-x) - xf'(a+h-x) - \dots - \frac{x^n}{n!}f^{(n)}(a+h-x)\), shew that \quad \(f(a+h) = f(a) + hf'(a) + \dots + \frac{h^n}{n!}f^{(n)}(a) + \psi(h)\), so that \(\psi(h)\) is the remainder in Taylor's theorem. Prove that \quad \(\psi'(x) = \frac{x^n}{n!} f^{(n+1)}(a+h-x)\), and deduce \quad \(\psi(h) = \frac{1}{n!} \int_0^h x^n f^{(n+1)}(a+h-x)\,dx\). Obtain this in the form \[ \frac{h^{n+1}}{n!} \int_0^1 f^{(n+1)}(a+th)(1-t)^n\,dt, \] and discuss its behaviour as \(n \to \infty\), when

1. [(i)] \(a=1, f(x)=x^m (-1 < x < 1)\);
2. [(ii)] \(a=0, f(x)=e^x\).

Assuming the logarithmic series, obtain superior and inferior limits for the remainder after \(n\) terms in the expansions in ascending powers of \(x\) of (i) \(\log_e (1+x)\), (ii) \(\log_e \{1/(1-x)\}\), (iii) \(\log_e \{(1+x)/(1-x)\}\). Prove that if these series are used to calculate \(\log_e(128/125)\) correct to ten places of decimals, six terms must be taken in each of the first two series, while two are sufficient in the third case; and, using the tables provided, obtain in each case the remainders correct to two significant figures.

Prove that, if \(f(x)\) is a function whose differential coefficient \(f'(x)\) is positive throughout a given interval, then \(f(x_2)>f(x_1)\), if \(x_2>x_1\), where \(x_1, x_2\) are any two values of \(x\) in the interval. \par Prove that \[ x - \frac{x^3}{6} + \frac{x^5}{120} > \sin x \] for all positive values of \(x\), and that \[ \left(1-\frac{x^2}{2}+\frac{x^4}{24}\right)\sin x > \left(x-\frac{x^3}{6}+\frac{x^5}{120}\right)\cos x \] when \(0

Prove that, if \(\cos\beta = \cos\theta\cos\phi+\sin\theta\sin\phi\cos\alpha\), and \(\sin\alpha = e\sin\beta\) \[ d\theta\{1-e^2\sin^2\phi\}^{\frac{1}{2}} + d\phi\{1-e^2\sin^2\theta\}^{\frac{1}{2}} = 0. \]

The function \(f(x)\) has a continuous second derivative \(f''(x)\) in the interval \([a,b]\); prove that, if \(a

Prove that, under certain conditions \[ f(x+h) = f(x) + hf'(x) + \frac{1}{2}h^2f''(x+\theta h), \quad 0 < \theta < 1. \] Give examples of cases in which the theorem does not hold. Expand \(y\) in terms of \(x\) by Maclaurin's Theorem, knowing that \((1-x^2)y'' - xy' - y = 0\) and that, when \(x=0\), \(y=1\) and \(y'=1\).

State Maclaurin's Theorem for the expansion of \(f(x)\). Apply this method to the expansion of \(\sin\left(x+\dfrac{\pi}{4}\right)\) in ascending powers of \(x\).

1. [(i)] Given that \(e^x = \sum_{r=0}^{\infty} \frac{x^r}{r!}\) \([0! = 1]\) prove that \(|e^{1.9} - 1\cdot 405| < 10^{-3}/2\).
2. [(ii)] The following is a table of the kinematic viscosity \(\nu\) of dry air at a pressure of one atmosphere at a given temperature \(T\), correct to three decimal places:
   \begin{tabular}{c|ccccc} \(T\) & 0 & 10 & 20 & 30 & 40 \\ \hline \(\nu\) & 0.132 & 0.141 & 0.150 & 0.160 & 0.159 \end{tabular}
   Would you be prepared to guess the value of \(\nu\) when \(T = 18\), \(T = 25\), \(T = 42\), \(T = 50\), \(T = 100\) or \(T = 500\)? What guess would you make if you did guess? How good would you expect your guesses to be? Give your reasons briefly but clearly.

Assume that for all \(x\) such that \(|x| < 1\), \(\sin^{-1}x = \sum_{r=0}^{\infty} \frac{(2r)!}{2^{2r}(r!)^2}\frac{x^{2r+1}}{2r+1}\). Writing \(u_r\) for the coefficient of \(x^{2r+1}\) in the above expansion, show that \(\frac{u_r}{u_{r-1}} = \frac{(2r-1)^2}{2r(2r+1)} < 1\), for all \(r \geq 1\). By quoting this series with \(x = \frac{1}{2}\), express \(\pi\) as the sum of a series of positive terms; hence construct a flow diagram to calculate \(\pi\), accumulating terms up to and including the first whose value is less than \(10^{-10}\). Prove that the value of \(\pi\) computed is correct to within \(\frac{1}{3} \cdot 10^{-10}\).

By repeated integration by parts, or otherwise, show that \[\frac{1}{n!} \int_0^1 (1-t)^n e^t dt = e - \sum_{r=0}^n \frac{1}{r!}.\] Prove that \(0 \leq \int_0^1 (1-t)^n e^t dt \leq 1\) (\(n \geq 1\)). Deduce that \[\left|e - \sum_{r=0}^n \frac{1}{r!}\right| \leq \frac{1}{n!}.\] [By convention, \(0! = 1\).]

Polynomials \(H_n(x)\) are defined by \begin{equation*} H_n(x) = (-1)^n e^{\frac{1}{2}x^2}\frac{d^n}{dx^n}(e^{-\frac{1}{2}x^2}). \end{equation*} Show that \(\sum_{n=0}^{\infty}\frac{1}{n!}H_n(x)y^n = \exp(xy - \frac{1}{2}y^2)\). Hence or otherwise show that \(\frac{dH_n}{dx}(x) = nH_{n-1}(x)\). [Taylor's theorem may be assumed. Questions of convergence need not be considered.]

(i) Let \(f(x) = e^{-1/x^2}\) for \(x \neq 0\), and \(f(0) = 0\). Prove that \(f^{(n)}(x)\) exists for all \(x\) and for all \(n\); calculate \(f^{(n)}(0)\). Comment upon Maclaurin's theorem applied to \(f(x)\). (ii) Let \(g(x) = x^2\sin \frac{1}{x}\) for \(x \neq 0\), \(g(0) = 0\). Determine, for every \(x\), the value of \(g'(x)\). What can you say about \(g''(x)\)?

1. [(i)] By considering the series expansion of \(e^{-x}(e^x - 1)^{n+1}\), or otherwise, show that $$n^n - (n+1)(n-1)^n + \frac{(n+1)n}{2!}(n-2)^n - \ldots(\text{to } n \text{ terms}) = 1.$$
2. [(ii)] Show that $$1^n - n2^n + \frac{n(n-1)}{2!}3^n - \ldots(\text{to } n+1 \text{ terms}) = (-1)^n n!.$$

Prove the expansion \[f(x+h) = f(x) + hf'(x) + \frac{h^2}{2}f''(x) + \ldots + \frac{h^{n-1}}{(n-1)!}f^{(n-1)}(x) + \frac{1}{(n-1)!}\int_0^h f^{(n)}(x+y)(h-y)^{n-1}\,dy\] by integrating the remainder term by parts. Assuming that the remainder term can be written \(h^n f^{(n)}(x + \theta h)/n!\), show that if \(f^{(n+1)}(x+y)\) is continuous in \(y\) at \(y = 0\) then \(\theta \to 1/(n+1)\) as \(h \to 0\).

If \(g(x)\) has a continuous \(n\)th derivative, and satisfies $$g(0) = g'(0) = g''(0) = \ldots = g^{(n-1)}(0) = 0,$$ prove that $$g(x) = \frac{1}{(n-1)!} \int_0^x g^{(n)}(t)(x-t)^{n-1} dt.$$ Deduce that, if \(f(x)\) has a continuous \(n\)th derivative, $$f(x) = f(0) + xf'(0) + \frac{x^2}{2!}f''(0) + \ldots + \frac{x^{n-1}}{(n-1)!}f^{(n-1)}(0) + R_n(x),$$ where $$R_n(x) = \frac{1}{(n-1)!} \int_0^x f^{(n)}(t)(x-t)^{n-1} dt.$$ If now \(f(x) = (1+x)^{1/2}\), show that \(|R_n(x)| < 1/(n-1)\) for all \(n > 2\) and all \(x\) such that \(-1 < x \leq 1\). What conclusion do you draw from this result?

If, for all \(x\) such that \(0 \leq x \leq h\) (\(h > 0\)), $$|c_0 + c_1x + c_2x^2 + \ldots + c_nx^n| \leq Ax^{n+1}$$ where \(A\) is a constant and \(n\) a given positive integer), show that the constants \(c_0, c_1, \ldots, c_n\) are all zero. A function \(f(x)\) is differentiable as many times as we wish, and in the interval \(0 \leq x \leq h\) its \((n-1)\)th derivative \(f^{(n-1)}(x)\) lies between \(\pm K_n\), for each \(n\). Prove by successive integration that, for \(0 \leq x \leq h\), $$f(x) - f(0) - xf'(0) - \ldots - \frac{x^n}{n!}f^{(n)}(0)$$ is between \(\pm K_n x^{n+1}/(n+1)!\). Deduce that if \(g(x) \equiv f(x^2)\) then \(g^{(2p+1)}(0) = 0\), \(g^{(2p)}(0) = (2p)!f^{(p)}(0)/p!\) for every positive integer \(p\). [It must not be assumed that the infinite Taylor series of \(f(x)\) converges to sum \(f(x)\).] Without assuming the binomial theorem for fractional indices, find the sixth derivative at \(x = 0\) of \((1-x^2)^{1/2}\).

Define the function \(f(x)\) for positive values of \(x\) by the equation \[f(x) = \int_x^{\infty} \frac{e^{x-t}}{t}dt.\] Prove that, for each positive integer \(n\), \[f(x) = \frac{1}{x} - \frac{1}{x^2} + \frac{2!}{x^3} - \frac{3!}{x^4} + \ldots + \frac{(-1)^{n-1}(n-1)!}{x^n} + (-1)^n n! \int_x^{\infty} \frac{e^{x-t}}{t^{n+1}} dt.\] Show that there is no value of \(x\) for which the infinite series \[\frac{1}{x} - \frac{1}{x^2} + \frac{2!}{x^3} - \ldots + \frac{(-1)^{n-1}(n-1)!}{x^n} + \ldots\] is convergent. Writing \[S_n(x) = \frac{1}{x} - \frac{1}{x^2} + \frac{2!}{x^3} - \frac{3!}{x^4} + \ldots + \frac{(-1)^{n-1}(n-1)!}{x^n},\] prove that \(f(x)\) always lies between \(S_n(x)\) and \(S_{n+1}(x)\). Use the series to obtain a value for \(f(10)\) with an error smaller than 1 part in 4000.

\(f(x)\) is a continuous function with continuous first, second and third derivatives, and \[ R(x) = \frac{1}{2} \int_0^x (x-t)^2 f'''(t) \,dt. \] Prove by integration by parts that \[ f(x) = f(0) + xf'(0) + \frac{x^2}{2}f''(0) + R(x). \] Show also that \[ R(x) = \frac{x^3}{3!}f'''(\theta x), \] where \(0 < \theta < 1\). State and prove a more general result applicable to a function with continuous derivatives up to and including the \(n\)th.

Show that \[ e^{a^2}\int_a^\infty e^{-x^2}\,dx = \frac{1}{2a}\left\{ 1 + \sum_{r=1}^n (-)^r \frac{1 \cdot 3 \cdot 5 \dots (2r-1)}{(2a^2)^r} \right\} + (-)^{n+1}R_n, \] where \[ R_n = \frac{1 \cdot 3 \cdot 5 \dots (2n+1)}{2^{n+1}} e^{a^2} \int_a^\infty \frac{e^{-x^2}}{x^{2n+2}}\,dx. \] Establish that, if \(a\) is positive, \(R_n\) is less than \[ \frac{1 \cdot 3 \cdot 5 \dots (2n-1)}{2^{n+1}} \frac{1}{a^{2n+1}}. \]

Prove, by integrating the inequality \(\cos\theta \le 1\), that \(\cos\theta\) lies between \[ \left(1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \dots + \frac{(-1)^n\theta^{2n}}{(2n)!}\right) \quad \text{and} \quad \left(1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \dots + \frac{(-1)^{n+1}\theta^{2n+2}}{(2n+2)!}\right). \] Deduce the infinite series for \(\cos\theta\). If an infinite series \(u_1+u_2+\dots+u_n+\dots\) of positive terms is convergent, shew that so also is \[ u_1^2+u_2^2+\dots+u_n^2+\dots. \]

Prove Taylor's Theorem, obtaining a form for the remainder after \(n\) terms. Apply the theorem to obtain an infinite series for \(\log(1+x)\) valid when \(-1

State and prove Taylor's Theorem with Lagrange's form of remainder. Shew that, if \(s\) is any positive number and \(n\) any positive integer, then \[ \int_0^\infty \frac{e^{-sx}dx}{\sqrt{1+x^2}} = \sum_{v=0}^{n-1} \frac{(-1)^v c_v}{s^{2v+1}} + \theta_n(s) \frac{(-1)^n c_n}{s^{2n+1}}, \] where \[ c_0=1, \quad c_v = 1^2 . 3^2 \dots (2v-1)^2 \quad (v \ge 1), \] and \(\theta_n(s)\) satisfies the inequality \(0 < \theta_n(s) < 1\). Give reasons for the existence of any integrals which you use.

For each integer \(n \geq 1\), write \(t_n\) for the number of ways of placing \(n\) people into groups (so that \(t_1 = 1\), \(t_2 = 2\), \(t_3 = 5\), etc.). Defining \(t_0 = 1\), show that \[t_{n+1} = \sum_{r=0}^{n} \binom{n}{r}t_{n-r},\] for \(n \geq 0\), and hence show that \(t_n/n!\) is the coefficient of \(x^n\) in the Maclaurin expansion of \(\exp(\exp x - 1)\), for each \(n \geq 1\).

A sequence of polynomials \(P_j(x)\) satisfies the relations \[P_j(x) = \frac{d}{dx}P_{j+1}(x)\] and \(P_j(x)\) is of degree \(j(j = 0, 1, 2, \ldots)\). Show that \(\sum_{j=0}^{m-1} \theta^j P_j(x)\) is identical with the first \(m\) terms of the expansion of \(A(\theta)e^{\theta x}\) in powers of \(\theta\), where \(A(\theta)\) is a certain polynomial in \(\theta\). Show further that \[\sum_{k=0}^j P_k(y)\frac{(x-y)^{j-k}}{(j-k)!}\] is independent of \(y\).

Show that if \(k\) is an integer greater than or equal to \(0\) then $$\sum_{n=0}^{\infty} \frac{n^k}{n!} = n_k e,$$ where \(n_k\) is an integer and \(e = \sum_{n=0}^{\infty} \frac{1}{n!}\). Show also that \(n_0 = 1\), and that $$n_k = \sum_{r=0}^{k-1} {}^k C_r n_r,$$ where \({}^k C_r\) is the coefficient of \(x^r\) in \((1+x)^k\) for \(k = 1, 2, \ldots\). (In this question both \(0^0\) and \(0!\) are to be taken to be \(1\).)

Let \(p_n\) be the number of ways in which a collection of \(n\) dissimilar objects may be divided into parts, each of which contains at least one object, no regard being paid to order. By convention, the 'division' of the collection into a single part, namely the whole collection, is to be included in this number; also \(p_0\) is to be taken as 1. Prove that \[p_{n+1} = \sum_{m=0}^n \binom{n}{m} p_{n-m},\] where \[\binom{n}{m} = \frac{n!}{m!(n-m)!}\] (0! being taken as 1), and hence, or otherwise, prove that \(p_n/n!\) is the coefficient of \(x^n\) in the expansion in ascending powers of \(x\) of \[e^{x-1}.\]

\begin{align} a(t) &= a_1 t + a_2 t^2/2! + \ldots + a_n t^n/n! + \ldots, \\ b(t) &= 1 + b_1 t + b_2 t^2/2! + \ldots + b_n t^n/n! + \ldots \end{align} are two power series such that \(b(t) = \exp\{a(t)\}\). Prove that the \(b_i\) are all integers if and only if the \(a_i\) all are.

By expanding the expression \((e^x-1)^n\) in two different ways, or otherwise, evaluate the sum \[ n^{n+2} - \binom{n}{1}(n-1)^{n+2} + \binom{n}{2}(n-2)^{n+2} - \dots + (-1)^{n-1} \binom{n}{n-1}, \] where \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\). Verify your result by direct computation for the case \(n=4\).

If \[ f_n(x, q) = \sum_{r=0}^{n-1} \frac{(1-q^{2n-2})(1-q^{2n-4})\dots(1-q^{2n-2r})}{(1-q^2)(1-q^4)\dots(1-q^{2r})} x^r, \] where the term \(r=0\) is to be interpreted as having the value 1, prove that \begin{align*} f_n(x, q) &= x^{n-1} f_n(\frac{1}{x}, q^2); \\ f_n(x, q) - f_{n-1}(x, q) &= x^{n-1} f_{n-1}(\frac{q^2}{x}, q). \end{align*} Deduce simple formulae for \(f_n(q, q)\) and \(f_n(-q, q)\).

Assuming the formula \[ \sin\theta = \theta \left(1-\frac{\theta^2}{\pi^2}\right)\left(1-\frac{\theta^2}{2^2\pi^2}\right)\left(1-\frac{\theta^2}{3^2\pi^2}\right)\dots, \] and the expansions of \(\sin\theta\) and \(\cos\theta\) in powers of \(\theta\), prove that \begin{align*} &\left(1+\frac{x}{2}\right)\left(1-\frac{x}{3}\right)\left(1+\frac{x}{5}\right)\left(1-\frac{x}{7}\right)\left(1+\frac{x}{9}\right)\left(1-\frac{x}{11}\right)\dots \\ &= 1 + \frac{\pi}{4}x - \frac{\pi^2}{4^2}\frac{x^2}{2!} - \frac{\pi^3}{4^3}\frac{x^3}{3!} + \frac{\pi^4}{4^4}\frac{x^4}{4!} + \frac{\pi^5}{4^5}\frac{x^5}{5!} - \dots, \end{align*} a change of sign occurring after each two terms.

By induction, or otherwise, prove the identity \[ \frac{(1-x^{n+1})(1-x^{n+2})(1-x^{n+3})\dots(1-x^{2n})}{(1-x)(1-x^3)(1-x^5)\dots(1-x^{2n-1})} = (1+x)(1+x^2)(1+x^3)\dots(1+x^n), \] where \(n\) is any positive integer. Prove that the expansions of \[ (1+x)(1+x^2)(1+x^3)\dots(1+x^n) \] and \[ (1-x)^{-1}(1-x^3)^{-1}(1-x^5)^{-1}\dots(1-x^{2n-1})^{-1} \] in ascending powers of \(x\) agree as far as the terms in \(x^n\). Hence, or otherwise, prove that the number of ways of expressing a positive integer \(n\) as a sum of one or more unequal positive integers is the same as the number of ways of expressing \(n\) as a sum of one or more odd (not necessarily unequal) positive integers.

Prove that, if \(u_1, u_2, \dots, u_n, \dots\) are connected by the relation \[ u_n = u_{n-1} + n^2 u_{n-2} \] for all positive integral values of \(n \ge 3\), and \(u_1=1, u_2=5\), then \[ u_n/(n+1)! = 1 - \frac{1}{2} + \frac{1}{3} - \dots + \frac{(-1)^n}{n+1}. \] (Note: The condition on \(n\) for the recurrence is inferred from mathematical consistency, as the original text is ambiguous.)

Show that, if \[ \frac{1}{1+u}e^{\frac{ux}{1+u}} = P_0(x) + P_1(x)\frac{u}{1!} + P_2(x)\frac{u^2}{2!} + \dots + P_n(x)\frac{u^n}{n!} \dots, \] then \[ P_n(x) = x^n - \frac{n^2}{1!}x^{n-1} + \frac{n^2(n-1)^2}{2!}x^{n-2} - \frac{n^2(n-1)^2(n-2)^2}{3!}x^{n-3} + \dots. \] By putting \(\frac{u}{1+u}=t\), deduce that \[ x^n = P_n(x) + \frac{n^2}{1!}P_{n-1}(x) + \frac{n^2(n-1)^2}{2!}P_{n-2}(x) + \dots. \]

If \(x\) and \(a\) are small and \(e^x \tan \frac{x}{2} = a\), prove by successive approximation that the first four terms in the expansion of \(x\) in powers of \(a\) give \(x = 2a - \frac{4}{3}a^2 + \frac{26}{9}a^3 - \frac{13}{3}a^4\).

Find \(a, b, c, d\) so that the coefficient of \(x^n\) in the expansion of \[ \frac{a+bx+cx^2+dx^3}{(1-x)^4} \] may be \(n^3\). Find the coefficient of \(x^n\) in the expansion of \((1+\lambda x+x^2)^n\), where \(n\) is a positive integer; and if \(\lambda=2\cos\theta\), deduce that \[ c_0^2 + c_1^2 \cos 2\theta + \dots + c_n^2 \cos 2n\theta = n! \cos n\theta \left\{\frac{(2\cos\theta)^n}{n!} + \frac{(2\cos\theta)^{n-2}}{1! (n-2)!} + \frac{(2\cos\theta)^{n-4}}{2! (n-4)!} + \dots \right\}, \] where the indices of \(\cos\theta\) are positive numbers or zero, and \((1+x)^n = c_0+c_1x+\dots+c_nx^n\).

Shew that the sum of the \(r\)th powers of the first \(n\) odd integers, when \(r\) is a positive integer, is the coefficient of \(x^r/r!\) in the expansion of \(e^{nx} \frac{\sinh nx}{\sinh x}\) and that, when \(r\) is odd, the sum can be expressed as a polynomial in \(n^2\). Obtain the sum in the case \(r=3\).

Find the number of homogeneous products of degree \(r\) in \(n\) letters, and show that if there are three letters \(a, b, c\), the sum of these products is \[ \Sigma a^{r+2}(b-c)/\Sigma a^2(b-c). \]

In the series \(u_0+u_1x+u_2x^2+\dots\) any three successive coefficients are connected by the relation \[ u_{r+1} + pu_r + qu_{r-1}=0; \] shew how to find the sum to \(n\) terms. Assuming that the series \[ 2+\frac{7}{5}x + \frac{91}{125}x^2 + \dots \] is of this type, find the \(n\)th term and the sum to infinity.

Shew how to sum the series \(a_0+a_1x+\dots+a_nx^n+\dots\), whose coefficients satisfy the relation \(a_n+pa_{n-1}+qa_{n-2}+ra_{n-3}=0\), \(p, q, r\) being given numbers. \par In the case where \(3a_n-7a_{n-1}+5a_{n-2}-a_{n-3}=0\), and \(a_0=1, a_1=8, a_2=17\), shew that \(2a_n = 20n-7+3^{2-n}\).

Prove by means of the expansions or otherwise that, when \(n\) is a positive integer and \(x\) is positive and less than \(n\), \[ \left(1+\frac{x}{n}\right)^n < e^x < \left(1-\frac{x}{n}\right)^{-n}. \] Deduce that the inequalities are also true when \(x\) is negative and numerically less than \(n\). Prove also that, when \(x\) is positive, \(\left(1+\frac{x}{n}\right)^n\) is nearer to \(e^x\) than \(\left(1-\frac{x}{n}\right)^{-n}\) is.

If \(p\) is small, so that \(p^3\) is negligible, prove that an approximation to a solution of the equation \(x^{2+p}=a^2\) is \[ x = a - \frac{1}{2}ap\log_e a + \frac{1}{8}ap^2(2+\log_e a)\log_e a. \]

If \(p(x)\) is a polynomial of the \(k\)th degree and if \[ H_n(x) = e^{p(x)}\frac{d^n e^{-p(x)}}{dx^n}, \] prove that \[ H_n(x) = q_{k-1}(x)H_{n-1}(x) + q_{k-2}(x)H_{n-2}(x) + \dots + q_0 H_{n-k}(x), \] where \(q_i(x)\) is a polynomial of the \(i\)th degree in \(x\). \par Find the actual equation for the case in which \(p(x)=x^2\).

Prove that the coefficient of \(x^n\) in the expansion of \[ \frac{1}{(1-x)(1-x^3)(1-x^6)} \] in powers of \(x\) is \[ \frac{1}{4} \left\{ (n+2)^2 - \frac{1 - (-1)^n}{2} \right\}. \]

Define a recurring series, its scale of relation, and generating function. Shew that the series whose \(n\)th term is \(2^n n^2 x^n\) is a recurring series and find its generating function. For what range of values of \(x\) does the latter represent the sum to infinity of the series?

Find the number of homogeneous products of \(n\) dimensions formed from \(r\) letters \(a,b,c,\dots,k\); and shew that the sum of such products is equal to \(\sum \frac{a^{n+r-1}}{(a-b)(a-c)\dots(a-k)}\).

Sum the series \[ 1+\frac{m}{1!}\frac{1}{2^2}+\frac{m(m-2)}{2!}\frac{1}{2^4}+\frac{m(m-2)(m-4)}{3!}\frac{1}{2^6}+\dots \] to infinity when \(m\) is an odd integer. Find the coefficient of \(x^{2n}\) in the expansion of \(\frac{1+x}{(1+x+x^2)^2}\) in ascending powers of \(x\).

If \(a_r\) is the coefficient of \(x^r\) in the expansion of \((1+x+x^2)^n\) in a series of ascending powers of \(x\), prove that

1. [(i)] \(a_0+a_1+a_2+\dots+a_{2n}=3^n\),
2. [(ii)] \(a_1+2a_2+3a_3+\dots+2na_{2n}=n\cdot3^{n-1}\),
3. [(iii)] \(a_0^2-a_1^2+a_2^2-\dots+a_{2n}^2=a_n\).

Let \(y_0(x) = x\), \(y_n(x) = 1 - \cos y_{n-1}(x)\) (\(n \geq 1\)). For fixed \(n\), find the limit of \(x^{-2^n}y_n(x)\) as \(x\) tends to zero.

A function \(f(x)\) has all its derivatives non-zero in some interval. It can be calculated with a maximum error \(\epsilon\), \(\epsilon\) being independent of \(x\). It is desired to evaluate its derivative \(f'(x)\) at some point \(x = x_0\) in that interval, using the result \(f'(x_0) \approx f_1(x_0)\), where \begin{equation*} f_1(x_0) \equiv \frac{f(x_0+h)-f(x_0-h)}{2h}. \end{equation*} By expanding \(f(x_0 \pm h)\) in Taylor series, and assuming that \(h\) is sufficiently small for each term of the Taylor series to be considerably smaller than the one before, show that the value of \(h\) which minimises the possible error in \(f'(x_0)\) is proportional to \(\epsilon^{\frac{1}{3}}\). Obtain the constant of proportionality in terms of an appropriate derivative of \(f\).

Let \[f(x) = \sum_{n=1}^{\infty} \frac{x}{n(n+x)}\] for real positive \(x\). Prove that \[2f(2x) - f(x) - f(x+\frac{1}{2}) = 2\log 2 - \frac{1}{(x+\frac{1}{2})}.\]

The bank of a river whose surface lies in the \((x, y)\)-plane is given by \(y = 0\). The surface current is in the \(x\)-direction and is given by \(ky\). A man who swims steadily at speed \(V\) starts from the point \((0, y_0)\) wishing to reach the point \((0, 0)\). Assuming that \(V > ky_0\), calculate the time it takes him to reach his destination

1. if he arranges to swim so that his path is a straight line;
2. if he swims towards the bank until he reaches it and then swims along the bank;
3. if he always points himself towards his destination.

A function \(y\) of \(x\) and \(\lambda\) is defined by the equation $$y = x^2 + \lambda x^2 y^{-\frac{1}{2}}$$ where \(\lambda\) is small. Assuming that \(y\) may be expressed in the form $$p(x) + \lambda q(x) + \lambda^2 r(x) + \lambda^3 s(x) + \ldots$$ find the functions \(p(x), q(x), r(x)\) and \(s(x)\). Convergence need not be discussed.

State Maclaurin's theorem for the expansion of a function \(y = f(x)\) in powers of \(x\). Use the theorem to obtain expansions in powers of \(x\) (up to terms in \(x^3\)) for

1. [(i)] \(\sin x\),
2. [(ii)] \(\cos x\),
3. [(iii)] \(\log_e(1 + x)\).

1. [(i)] Find the limit of \[ \frac{\sin(\theta \cos\theta)}{\cos(\theta \sin\theta)} \] as \(\theta \to \pi/2\).
2. [(ii)] Show that \(\frac{3\sin 2\phi}{2(2+\cos 2\phi)} - \phi\) is approximately zero if \(\phi\) is small, and estimate the order of magnitude of the error for small values of \(\phi\).

Prove, by taking logarithms or otherwise, that if \(k, l, m, n, p, q, r\) are positive numbers of the form \(n-3, n-2, n-1, n, n+1, n+2, n+3\), the ratio of \(l^6 n^9 q^6\) to \(k m^7 p^{15} r\) is \(1 + 120n^{-6} + 1260n^{-8} + \dots\).

Find the limiting values as \(x\) tends to \(0\) of

1. [(i)] \(\displaystyle \frac{\log(\cos x)}{x \sin x}\),
2. [(ii)] \(\displaystyle \frac{1}{e^{x^2}-1} \int_0^x \sin t^2 dt\),
3. [(iii)] \(\displaystyle \frac{1 - \sec(1-\cos x)}{1 - \cos(1-\sec x)}\).

Prove that, if \(m\) and \(n\) are fixed positive integers, then \[ \frac{m}{x^m-1} - \frac{n}{x^n-1} \] tends to a limit when \(x\) tends to 1, and find the limit. By putting \(y=x^\lambda\), or otherwise, prove that, if \(\lambda\) is a fixed positive rational number, then \[ \frac{y^\lambda-1}{y-1} \] tends to the limit \(\frac{1}{2}(1-\lambda)\) when \(y\) tends to 1. [The positive value of \(y^\lambda\) for \(y>0\) is to be taken.]

Obtain the expansion of \(\sin x\) in ascending powers of \(x\). For what values of \(x\) is this series convergent? \par A small arc \(PQ\) of a circle of radius 1 is of length \(x\). The arc \(PQ\) is bisected at \(Q_1\) and the arc \(PQ_1\) is bisected at \(Q_2\). The chords \(PQ\), \(PQ_1\) and \(PQ_2\) are of lengths \(c\), \(c_1\) and \(c_2\), respectively. Prove that \(\frac{1}{45}(c - 20c_1 + 64c_2)\) differs from \(x\) by a quantity of order \(x^7\).

If \(f(a), \phi(a)\) each equal to zero, explain how to find the limit of \(\frac{f(x)}{\phi(x)}\) when \(x \to a\), \(f(x), \phi(x)\) being continuous functions of \(x\). Shew that the limit when \(x \to a\) of \[ \frac{(2a^3x-x^4)^{\frac{1}{2}} - a(a^2x)^{\frac{1}{3}}}{a - (ax^2)^{\frac{1}{3}}} \text{ is } \frac{16}{9}a. \]

Evaluate the limit as \(x\) tends to infinity of \[ x\{\sqrt{(a^2+x^2)}-x\}. \]

Find \[ \lim_{x\to 0} \frac{(1+x)^{1/x}-e}{x}. \]

As \(x\) tends to \(a\), the functions \(f(x), g(x), f'(x)\) and \(g'(x)\) tend to the limits \(0, 0, b\) and \(c\) respectively. Prove that, if \(c \neq 0\), \(f(x)/g(x)\) tends to \(b/c\). \par Evaluate the following limits:

1. [(i)] \(\cot^2 x - \frac{1}{x^2}\), as \(x\) tends to 0;
2. [(ii)] \((1-x)^{-3} \int_1^x \log(3t-3t^2+t^3)dt\), as \(x\) tends to 1;
3. [(iii)] \(\frac{5^x-4^x}{3^x-2^x}\), as \(x\) tends to 0.

Arrange the following numbers in order so that as \(x\) increases without limit the ratio of each number to the preceding may tend to infinity: \[ x^2, 2^x, x^x, e^x, x^{\log x}, (\log x)^x, 2^{\log x}. \] Find the limiting values of \[ (\cos x)^{1/x}, \quad (\cos x)^{1/x^2}, \quad (\cos x)^{1/x^3} \] as \(x\) tends to zero through positive or negative values.

Find the limiting values of the expression \[ \frac{\sin x \sin y}{\cos x - \cos y} \] as the point \((x,y)\) approaches the origin along curves of the form (i) \(y=xk\), where \(k\) is positive, (ii) \(y=ax+bx^2\), where \(a\) and \(b\) have various constant values. Point out any cases in which the limits are infinite.

Find the limits, as \(x\) tends to \(\frac{1}{2}\), of the following expressions:

1. [(i)] \(\displaystyle \frac{(1-x)^m - x^m}{(1-x)^n - x^n}\), where \(m, n\) are positive integers;
2. [(ii)] \(\displaystyle \frac{(x-4x^4)^{\frac{1}{3}} - (\frac{1}{2})^\frac{1}{3}}{1-(8x^3)^{\frac{1}{2}}}\).

Assume the theorem: "If \(f'(x)\) exists for \(a \le x \le b\), then \[ f(b)-f(a)=(b-a)f'(\xi), \] where \(a<\xi

Give a definition of \(e^x\), and from your definition deduce (i) that \(\frac{e^x}{x^n} \to \infty\) as \(x \to \infty\), where \(n\) is a fixed positive integer, (ii) that \(1+x > xe^{1/x}\) for sufficiently large values of \(x\). Prove that, if \(a>1\) and \(b>0\) and \(y=x^b\), then \[ \frac{a^y}{x^n} \to \infty \text{ as } x \to \infty, \] and \[ \left(1+\frac{1}{x}\right)^{xy} \to \infty \text{ as } x \to \infty. \]

Shew how to find \(\lim_{x\to 0} \frac{f(x)}{g(x)}\), when \(f(0)=0\) and \(g(0)=0\). Find the limit as \(x \to 0\) of

1. \(\dfrac{9\sin x - 11x\cos x + 2x \cos 2x}{\sin x \{x \sin x - \log(1+x^3)\}}\),
2. \(\dfrac{x^2\sin\frac{1}{x}}{\sin x}\).

Expand in ascending powers of \(x\) the fraction \[ \frac{2x + (9+3x^2)^{1/2}}{3-x} \] as far as the fifth power of \(x\), and shew that, for small values of \(x\) it leads to a good approximation for \(e^x\). Deduce that \(e^{1/4} = 1.2840\dots\).

Prove that, if \(N\) and \(n\) are nearly equal, then \[ \left(\frac{N}{n}\right)^{1/3} = \frac{N+n}{n + \frac{1}{3}\frac{N+n}{4n-N}} \text{ approximately,} \] the error being approximately \(\frac{7}{81}\left(\frac{N-n}{n}\right)^3\).

By finding the fourth differential coefficient of \((\sin^2 x)/x^2\), or otherwise, shew that as \(x\) tends to zero the limit of \[ \frac{15}{x^5} - \frac{2x^4-18x^2+15}{x^6}\cos 2x + \frac{8x^2-24}{x^5}\sin 2x \] is \(\frac{4}{15}\).

Prove that when \(x\) is increased without limit the expression \((1+1/x)^x\) has a finite limit. Prove that \[ \frac{1}{3} + \frac{x}{4.1} + \frac{x^2}{5.1.2} + \frac{x^3}{6.1.2.3} + \dots = \frac{1}{x^2}\{e^x(x^2-2x+2)-2\}. \]

The section of the curve \(y = \cosh x\) between \(x = 0\) and \(x = a\) is rotated about the \(x\)-axis. Prove that the numerical value of the curved surface area thus obtained is twice that of the volume enclosed. The curve is now rotated about the \(y\)-axis. Calculate the ratio of the numerical values of volume to curved surface area, and show that in this case it depends on \(a\).

Show that \begin{equation*} \cosh x - \cosh y = 2\sinh\left(\frac{x+y}{2}\right)\sinh\left(\frac{x-y}{2}\right) \end{equation*} Show that the inverse hyperbolic function \begin{equation*} y = \sinh^{-1} x \end{equation*} satisfies the differential equation \begin{equation*} (x^2 + 1)\frac{d^2y}{dx^2} + x\frac{dy}{dx} = 0. \end{equation*}

A mapping of the \((X, Y)\) plane onto the \((x, y)\) plane is given by $$x = \sin X \cosh Y,$$ $$y = \cos X \sinh Y.$$ Find and sketch the curves in the \((x, y)\) plane which correspond under this mapping to the lines \(X = \text{const.}\) and \(Y = \text{const.}\) To which curves in the \((X, Y)\) plane do the lines \(x = 0, y = 0\) and \(x = y\) correspond?

Define the function \(e^y\), and deduce from your definition that, for all values of \(n\), \(y^n e^{-y} \to 0\) as \(y\to\infty\). Examine the behaviour of the following functions as \(x\) varies through real values, and in particular discuss their gradients for small positive and negative values of \(x\). Illustrate your results by sketch-graphs. \[ \text{(i) } \tanh\frac{1}{x}, \quad \text{(ii) } x\tanh\frac{1}{x}. \]

If \(u_0 = \sinh\alpha\), \(u_1=\sinh(\alpha+\beta)\) and \(u_{n+2}-2u_{n+1}\cosh\beta+u_n=0\) for all \(n \ge 0\), prove that \(u_n = \sinh(\alpha+n\beta)\). Sum the series \[ \sum_{r=0}^n \sinh(\alpha+r\beta) \] for all values of \(\beta\).

Give definitions of, and proofs of the simplest properties of, the hyperbolic functions \(\cosh x, \sinh x, \tanh x\). Draw the graphs of the functions and of the inverse functions; and express the inverse functions in terms of logarithms. Explain the parallelism between formulae involving the hyperbolic functions and the corresponding formulae involving the trigonometrical functions \(\cos x, \sin x, \tan x\).

If \(x\) is an acute angle and if \(y=\log\tan\left(\frac{\pi}{4}+\frac{x}{2}\right)\), prove that \(\cos x \cosh y=1\), and that \[ y = \sin x - \frac{1}{3}\sin 3x + \frac{1}{5}\sin 5x - \dots. \]

Define the hyperbolic functions and establish their most important properties, including the expressions for \(\sinh(u+v)\) and \(\cosh(u+v)\) in terms of \(\sinh u, \sinh v, \cosh u, \cosh v\). Prove that the sum of the series \[ \sinh\theta+\tan\theta\sinh 2\theta+\tan^2\theta\sinh 3\theta+\dots+\tan^{n-1}\theta\sinh n\theta \] is \[ \frac{\sinh\theta-\tan^n\theta\sinh(n+1)\theta+\tan^{n+1}\theta\sinh n\theta}{\sec^2\theta-2\cosh\theta\tan\theta}. \]

Prove that, if the circle of curvature at any point \(P\) on the cardioide \(r=a(1+\cos\theta)\), which has its cusp at \(O\) and \(OA\) for its axis, cuts the curve again in \(Q\), then \[ \frac{1}{OQ} - \frac{9}{OP} = -\frac{8}{OA}. \]

(i) Evaluate $$\int_0^1 \frac{dx}{1+x^3}.$$ (ii) If \(x\) is a function of \(t\) such that $$\frac{dx}{dt} = \sqrt{\frac{x}{1-x}}$$ and \(x = 0\) when \(t = 0\) find the value of \(t\) for which \(x = 1\).

Evaluate \begin{align*} (i)&\int\frac{dx}{x+\sqrt{(2x-1)}} \quad (x > \frac{1}{2});\\ (ii)&\int\frac{dx}{(a^2-x^2)^{3/2}} \quad (|x| < |a|). \end{align*}

Evaluate \[\int_0^1 \frac{u^{\frac{1}{2}}}{(1+u)^{\frac{1}{2}}}\,du.\]

(i) Integrate the function \[ \frac{1}{1+\sqrt{(1+e^x)}}. \] (ii) Show that the definite integrals \[ \int_0^1 \frac{\sin^{\frac{1}{2}}\pi x}{\sqrt{(x-x^2)}} dx, \quad \int_0^1 \frac{\cos^{\frac{1}{2}}\pi x}{\sqrt{(x-x^2)}} dx \] are equal. Hence, or otherwise, evaluate these integrals.

Find the indefinite integrals

1. [(i)] \(\int x^2 \tan^{-1} x \, dx\),
2. [(ii)] \(\int \frac{dx}{\sqrt{x} + \sqrt{1-x}}\).

Evaluate the following integrals (in which \(\sqrt{\phantom{x}}\) means the positive square root):

1. \(\displaystyle\int_{1/2}^X \frac{dx}{x\sqrt{(5x^2-4x+1)}}\);
2. \(\displaystyle\int_{-\pi/6}^{\pi/4} \sqrt{\sqrt{\frac{1}{2}-\frac{1}{2}\cos 4\theta} - \frac{1}{2}\sin 4\theta}\,\sec^4\theta\,d\theta\).

Integrate \[ \int_0^1 \frac{x^2 dx}{(x^2+1)^2}, \quad \int \frac{dx}{(x-a)\sqrt{x^2+1}}, \quad \int_0^{\frac{\pi}{4}} \tan^3 x\, dx. \]

Prove that \[ \int_0^a f(x) \,dx = \frac{1}{2} \int_0^a \{f(x)+f(a-x)\} \,dx \] and give a geometrical interpretation of the result. Evaluate \[ \int_0^1 \frac{dx}{(x^2-x+1)(e^{2x-1}+1)}. \]

Evaluate:

1. \(\displaystyle\int \frac{dx}{x(x-2)^3}\);
2. \(\displaystyle\int \frac{dx}{x+\sqrt{1-x^2}}\);
3. \(\displaystyle\int \frac{\log x\,dx}{x^n}\).

Find the indefinite integrals

1. \(\int \cosh^4 x\,dx\);
2. \(\int (1+x)^{-2} \sin^{-1} x\,dx\).

Evaluate

1. [(i)] \(\displaystyle\int_0^{\pi/2} \frac{dx}{2+\cos x}\),
2. [(ii)] \(\displaystyle\int_{1/3}^{1/2} \frac{dy}{y\sqrt{5y-1-6y^2}}\),
3. [(iii)] \(\displaystyle\int_0^\infty \frac{x \cdot dx}{x^6+1}\).

Evaluate:

1. [(i)] \(\int \sec^3 x dx\);
2. [(ii)] \(\int_0^\infty \frac{dx}{1+2\cosh x}\);
3. [(iii)] \(\int \frac{dx}{(x^2-1)\sqrt{x^2+1}}\).

The centre of a circular disc of radius \(r\) is \(O\), and \(P\) is a point on the line through \(O\) perpendicular to the plane of the disc such that \(OP=p\). Prove that the mean distance with respect to area of points of the disc from \(P\) is \[ \frac{2}{3r^2}\{(p^2+r^2)^{3/2}-p^3\}. \] Find the mean distance with respect to volume of the interior points of a sphere of radius \(a\) from a fixed external point at distance \(c\) from its centre.

Prove that \begin{align*} \int_0^\infty \frac{dx}{\cosh x + \cos \theta} &= \frac{\theta}{\sin \theta} \quad (0 < \theta < \pi), \\ \int_0^\pi \frac{d\theta}{\cosh x + \cos \theta} &= \frac{2}{\sinh x} \tan^{-1}(\tanh \frac{1}{2}x) \quad (x > 0), \end{align*} where the inverse tangent is to be taken between \(0\) and \(\frac{1}{2}\pi\). Discuss carefully the determinations of any inverse trigonometric functions you use. \newline Denoting the integrals by \(I(\theta)\) and \(J(x)\), respectively, verify that \[ \int_0^\pi I(\theta) d\theta = \int_0^\infty J(x) dx. \]

Prove that, if \[ \theta = \cot^{-1} x \quad (0 < \theta < \pi), \] then \[ \frac{d^n\theta}{dx^n} = (-1)^n (n-1)! \sin^n \theta \sin n\theta, \] where \(n\) is any positive integer. Show that the absolute value of \(d^n\theta/dx^n\) never exceeds \((n-1)!\) if \(n\) is odd, or \[ (n-1)! \cos^{n+1} \frac{\pi}{2(n+1)} \] if \(n\) is even.

Find a formula of reduction for the integral \[ \int_0^{\pi/2} \sin^m x \cos^n x \,dx \] reducing one of the indices by two; and evaluate \(\int_0^{\pi/2} \sin^2 x \cos^6 x \,dx\). Find the integrals \[ \int \frac{dx}{\sqrt{1-x^2}}, \quad \int \frac{\sqrt{1-x^2}}{x} \,dx. \]

Find the differential coefficient of \[ \tanh^{-1} \left\{ \frac{axp + b(x+p)+c}{qy} \right\}, \] where \(y^2 = ax^2 + 2bx + c\), \(q^2 = ap^2 + 2bp + c\). Hence, or otherwise, evaluate \[ \int \frac{dx}{(x-p)y}. \]

If \(u = \int_0^\theta \frac{d\theta}{\cos\theta}\), show that \(\theta = \int_0^u \frac{du}{\cosh u}\), and if \[ \int_0^\theta \frac{d\theta}{\cos\theta} + \int_0^\phi \frac{d\phi}{\cos\phi} = \int_0^\psi \frac{d\psi}{\cos\psi}, \] show that \[ \cos\psi = \frac{\cos\theta\cos\phi}{1+\sin\theta\sin\phi}. \]

Find \(\int \frac{dx}{x(1+x+x^2)}\), \(\int \frac{\sqrt{a^2-x^2}}{x^2}dx\), \(\int \frac{dx}{\sin x}\). The thickness of a circular disc of radius \(a\) at a distance \(r\) from the centre is \(2ap/(4a^2-r^2)^{\frac{1}{2}}\); find the average thickness of the disc.

Shew that in the range \(a < x < b\), \[ \frac{d}{dx}\left( -2\tan^{-1}\sqrt{\frac{b-x}{x-a}} \right) = \frac{1}{\sqrt{(b-x)(x-a)}}, \] and integrate with respect to \(t\) \[ \frac{1}{(1-kt)\sqrt{1-t^2}} \quad (0

In the theory of ``meridional parts,'' the function \(y\) corresponding to a given latitude \(\theta\) is defined by the equation \[ \frac{dy}{dx} = \frac{1}{\cos\theta} \frac{d\theta}{d\phi}, \] where \(x\) is the number of minutes of arc in the longitude (whose circular measure is \(\phi\)). Shew that, if \(y\) is zero when \(\theta=0\), \[ y = k \log_e \left( \frac{1+\sin\theta}{\cos\theta} \right), \] where \(k\) is a constant to be found. From the four-figure tables prove that in latitude 30\(^\circ\), \(y=1888\). [Take \(\log_e 10=2.303, \log_{10}\pi = 0.4971\).]

Perform the integrations \[ \int \frac{dx}{(x+1)^3(x-1)}; \quad \int \frac{dx}{\{(x+1)^3(x-1)\}^{1/2}}; \quad \int \sec x dx. \]

Prove the formulae for the radius of curvature of a plane curve \[ \frac{1}{\rho} = \frac{\frac{d^2y}{dx^2}}{\left(1+\left(\frac{dy}{dx}\right)^2\right)^{\frac{3}{2}}}, \quad \rho = r\frac{dr}{dp} = \frac{r^3}{r^2-p\frac{d^2r}{d\theta^2}} \cdot \] % Note: The second formula for rho is unusual and likely contains OCR errors. Standard formula is rho = r dr/dp. The second part is non-standard. Prove that the radius of curvature at a point \((r,\theta)\) of the curve \(r^n=a^n\cos n\theta\) subtends an angle \(\tan^{-1}\left(\frac{1}{n}\tan n\theta\right)\) at the pole.

Integrate the following expressions with respect to \(x\) \[ \frac{1}{\sqrt{(x^2-a^2)}}, \quad \frac{1}{2\sqrt{\{(2-x)(x-1)\}}}, \quad \frac{20}{25+9\cos x+12\sin x}. \]

Differentiate \(\sin^{-1}(\csc\theta\sqrt{\cos 2\theta})\), \(\tan^{-1}\{x/(1+\sqrt{1+x^2})\}\). Find the \(n\)th differential coefficient of \(x\tan^{-1}x\).

P is any point on an ellipse of which the foci are S and H. The distance SP is denoted by \(r\) and the angle HSP by \(\theta\). Show that the mean value of \(r\) with respect to arc is the semi-major axis \(a\), and that the mean value of \(r\) with respect to \(\theta\) is the semi-minor axis \(b\). If Q is any point in the interior of the ellipse, show that the mean value of the distance SQ with respect to area is \(a - \frac{b^2}{3a}\).

Prove that the mean distance of points on the surface of a sphere of radius \(a\) from an external point distant \(c\) from the centre is \(c+\frac{1}{3}\frac{a^2}{c}\). What is the value for an internal point?

Prove that the mean value with respect to area over the surface of a sphere centre \(O\) and radius \(a\) of the reciprocal of the distance from a fixed point \(C\) is equal to the reciprocal of \(OC\) if \(C\) is outside the sphere, but equal to the reciprocal of the radius \(a\) if \(C\) is inside the sphere.

Prove that, if \(f(x)\) is a function of \(x\) which has a derivative \(f'(x)\) for all values of \(x\) between \(a\) and \(b\) inclusive, and if \(f(a)=f(b)\), there is at least one value \(\xi\) between \(a\) and \(b\) for which \(f'(\xi)=0\). \newline Deduce from this theorem that, for some \(\xi\) between \(a\) and \(b\), \[ \text{(i)} \quad \frac{\phi(b)-\phi(a)}{\psi(b)-\psi(a)} = \frac{\phi'(\xi)}{\psi'(\xi)}, \] and that, for another \(\xi\), \[ \text{(ii)} \quad \frac{\phi(\xi)-\phi(a)}{\psi(b)-\psi(\xi)} = \frac{\phi'(\xi)}{\psi'(\xi)}, \] where in each case it is assumed that \(\phi'(x), \psi'(x)\) exist for all values of \(x\) between \(a\) and \(b\) inclusive, and that \(\psi'(x)\) does not vanish for any \(x\) between \(a\) and \(b\).

The position of a point moving in two dimensions is given by polar coordinates \(r, \theta\); find the component velocities and accelerations along and perpendicular to the radius vector. The velocities of a particle along and perpendicular to a radius vector from a fixed origin are \(\lambda r^2\) and \(\mu \theta^2\); find the polar equation of the path and the component accelerations in terms of \(r\) and \(\theta\).

Find the mean value of the distance of a point on the circumference of a circle of radius \(a\) from \(2n\) points arranged at equal distances along the circumference. Shew that when \(n \to \infty\) the mean is \(\dfrac{4a}{\pi}\).

\(AB\) and \(CD\) are perpendicular diameters of a circle. Find the mean value of the distance of \(A\) from points on the semicircle \(CBD\) and also the mean value of the reciprocal of that distance. Shew that the product of these means is \[ \frac{8\sqrt{2}\log_e(1+\sqrt{2})}{\pi^2}. \]

State (without proof) Rolle's theorem, and deduce that there is a number \(\xi\) between \(a\) and \(b\) such that \[ f(b) - f(a) = (b-a)f'(\xi), \quad (1) \] explaining what conditions must be satisfied by the function \(f(x)\) in order that the theorem may be valid. If \(f(x) = \sin x\), find all the values of \(\xi\) between \(a\) and \(b\) which satisfy the equation (1) when \(a=0\) and \(b=3\pi/2\). Illustrate the result with reference to the graph of \(\sin x\).

Define the mean value of \(f(x)\) with respect to \(x\) for values of \(x\) lying in an interval \((a,b)\). A point moves along a straight line in such a way that \[ v_t = v_s+ks, \] where \(v_t, v_s\) are the mean values of the velocity with respect to the distance travelled \(s\) and the time taken \(t\) respectively, and \(k\) is a constant. Shew that \(s,t\) satisfy the equation \[ \frac{ds}{s} = \frac{dt}{t}\{1+kt \pm \sqrt{kt(2+kt)}\}. \] Interpret this solution in the case \(k=0\), and shew on general grounds that a negative value of \(k\) is inadmissible.

What is meant by the Mean Value of a function \(f(x)\) with respect to a variable \(x\)? A point moves from rest along a straight line in such a way that its average velocity with respect to distance travelled bears a constant ratio \(k\) to that with respect to time elapsed. Shew that \(k > 1\).

The functions \(\phi(x)\) and \(\psi(x)\) are differentiable in the interval \(a < x < b\); and \(\psi'(x ) > 0\) for \(a < x < b\). Prove that there is at least one number \(\xi\) between \(a\) and \(b\) such that \[ \frac{\phi(\xi)-\phi(a)}{\psi(b)-\psi(\xi)} = \frac{\phi'(\xi)}{\psi'(\xi)}. \] If \(\phi(x)=x^2\) and \(\psi(x)=x\), find a value of \(\xi\) in terms of \(a\) and \(b\).

If \(r\) denotes distance from a focus of an ellipse, find the mean value of \(r\) with respect to angular distance from the major axis for points on the perimeter of the ellipse. Determine also for the ellipse the mean value of \(r\) with respect to area, stating the result in terms of the eccentricity \(e\) and the semi-latus rectum \(\lambda\).

The binary star \(\ast\) \(b\) of two positive integers is defined as follows: Put \(a_0 = a\) and \(b_0 = b\). For \(i \geq 1\), let \[a_i = 2a_{i-1}, \quad b_i = [b_{i-1}/2],\] where \([x]\) is the integer part of \(x\). Let \(n\) be the (unique) index such that \(b_n = 0\). For \(0 \leq i \leq n\) define \[c_i = a_i \quad \text{if} \quad b_i \quad \text{is odd,}\] \[= 0 \quad \text{if} \quad b_i \quad \text{is even.}\] Define \(a\) \(\ast\) \(b\) to be \(c_0 + c_1 + ... + c_n\). Identify \(a\) \(\ast\) \(b\), and justify your answer.

1. Find the equation of the line through the point \(\mathbf{a}\), which is perpendicular to the plane containing the non-collinear points \(\mathbf{b}\), \(\mathbf{c}\), \(\mathbf{d}\).
2. Find the shortest distance between the lines \(\mathbf{r} - \mathbf{a} = s\mathbf{u}\), \(\mathbf{r} - \mathbf{b} = t\mathbf{v}\), given that \(\mathbf{u}\) and \(\mathbf{v}\) are unit vectors with \(\mathbf{u} \times \mathbf{v} \neq \mathbf{0}\).

1. [(i)] Find a vector \(\mathbf{r}\) for which \begin{equation*} \mathbf{r} \times \mathbf{a} + \mathbf{r} = \mathbf{b} \end{equation*} where \(\mathbf{a}\) and \(\mathbf{b}\) are given vectors.
2. [(ii)] If \(\mathbf{u}(t)\) is a vector function of the variable \(t\) and if \(\mathbf{u}\) and \(\dot{\mathbf{u}}\) are unit vectors, show that \begin{equation*} \mathbf{u} \times (\dot{\mathbf{u}} \times \ddot{\mathbf{u}}) = -\dot{\mathbf{u}}. \end{equation*}
3. [(iii)] Evaluate the following expressions, where \(\mathbf{r} = \mathbf{r}(t)\) is a vector function of \(t\), \(r = |\mathbf{r}|\) and dots denote differentiation with respect to \(t\): \begin{equation*} (a) \int \dot{\mathbf{r}} \cdot \ddot{\mathbf{r}} dt, \quad (b) \int \mathbf{r} \times \ddot{\mathbf{r}} dt, \quad (c) \frac{d}{dt}\frac{\mathbf{r}}{r^3}. \end{equation*}

Find in terms of three non-zero vectors \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\), (such that \(\mathbf{a}\) is not perpendicular to \(\mathbf{b}\)) the most general vector \(\mathbf{r}\) which satisfies $$\mathbf{a} \times (\mathbf{b} \times (\mathbf{c} \times \mathbf{r})) = \mathbf{0},$$ examining carefully any configurations which give rise to exceptional cases.

\(P\), \(Q\), \(O\) and \(R\) are four distinct points which are not coplanar. Let \(a\) be the angle between the planes \(QOP\) and \(POR\). Define \(b\), \(c\), \(A\), \(B\) and \(C\) similarly by cyclic permutation of \(P\), \(Q\) and \(R\). Let \(\mathbf{p}\), \(\mathbf{q}\) and \(\mathbf{r}\) be the position vectors of \(P\), \(Q\) and \(R\) respectively with respect to \(O\) as origin, and let \(p\), \(q\), and \(r\) be the corresponding magnitudes of these vectors. By geometrical considerations, evaluate $$|(\mathbf{p} \times \mathbf{q}) \times (\mathbf{p} \times \mathbf{r})|$$ in terms of \(p\), \(q\), \(r\) and the angles \(a\), \(b\), \(c\), \(A\), \(B\) and \(C\). By expanding this repeated vector product, show also that $$\frac{|(\mathbf{p} \times \mathbf{q}) \times (\mathbf{p} \times \mathbf{r})|}{|(\mathbf{q} \times \mathbf{r}) \times (\mathbf{q} \times \mathbf{p})|} = \frac{p}{q}.$$ Deduce that $$\frac{\sin a}{\sin A} = \frac{\sin b}{\sin B} = \frac{\sin c}{\sin C}.$$ [It may be assumed without proof that, for any three vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\), $$(\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{b} \cdot \mathbf{c})\mathbf{a}$$ and $$(\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} = (\mathbf{b} \times \mathbf{c}) \cdot \mathbf{a} = (\mathbf{c} \times \mathbf{a}) \cdot \mathbf{b}.]$$

Prove that the plane bisecting the (interior) angle between the faces \(OAB\) and \(OAC\) of a tetrahedron \(OABC\) divides \(BC\) in the ratio of the areas of those two faces. Prove that this plane and the two planes which bisect the angles between the pairs of faces through \(OB\) and \(OC\) have a common line.

Spheres are described to touch two given planes and to pass through a given point. Prove that, in general, they all pass through a second fixed point and that the locus of their points of contact with either plane is a circle.

Two points \(P, Q\) lie inside a sphere of radius \(a\) and centre \(O\), and \(OP=p, OQ=q, \angle POQ=\theta\). The planes through \(P, Q\), perpendicular to \(OP, OQ\) respectively, intersect in a line \(l\). Prove that \(l\) cuts the sphere if \[ p^2+q^2-2pq\cos\theta < a^2\sin^2\theta, \] and find what length of the line \(l\) then lies inside the sphere.

Points on the surface of a sphere are projected from a vertex \(O\) of the surface onto a plane through the centre of the sphere and perpendicular to the radius to \(O\). Prove that in general circles on the sphere project into circles on the plane, mentioning any particular exceptions: and that the angle of intersection of two curves on the sphere is equal or supplementary to the angle between the corresponding curves on the plane. State the nature of the curve on the plane into which will project a curve on the sphere cutting the great circles through \(O\) at a constant angle.

The area of a triangle is to be calculated from measurements of the side \(a\) and of the angles \(B\) and \(C\), but each of these angles is over-estimated by \(1'\). If the angle \(A\) is very nearly \(90^\circ\), shew that the resulting error in the area of the triangle is approximately \(0.00015 a^2\).

The generalisation of metrical theorems by projection. Illustrate your account by finding the projective form of the theorem that the angle at the centre of a circle is double that at the circumference.

A curve \(C\) on the earth's surface (assumed to be a sphere of radius \(a\)) cuts the meridians at a constant acute angle \(\alpha\). Prove that \[ \cos\lambda \cosh m\phi = 1, \] where \(m=\cot\alpha\), and \((\lambda, \phi)\) are the latitude and longitude, respectively, of a variable point \(P\) of \(C\), \(\phi\) being measured continuously along \(C\) from the point where \(C\) crosses the equator. Find (i) the length of the arc of \(C\) between the points \(P_1(\lambda_1, \phi_1)\) and \(P_2(\lambda_2, \phi_2)\), and (ii) the area swept out by the projection of \(OP\) on the equatorial plane, when \(P\) describes the arc \(P_1P_2\) of \(C\). Give each result in terms of the latitudes \(\lambda_1\) and \(\lambda_2\).

If \(x=r\cos\theta\), \(y=r\sin\theta\), find the values of \(A, B, C, D\) such that \begin{align*} \delta x &= A\delta r + B\delta\theta + \epsilon (|\delta x| + |\delta\theta|), \\ \delta y &= C\delta r + D\delta\theta + \epsilon' (|\delta x| + |\delta\theta|), \end{align*} where \(\epsilon\) and \(\epsilon'\) both tend to zero as \(|\delta x| + |\delta\theta| \to 0\). \par If \(\phi(x,y)\) is a function of \(x\) and \(y\), then it may be regarded as a function of any two of the variables \(x, y, r\) and \(\theta\); find the values of \(\frac{\partial\phi}{\partial x}\) and \(\frac{\partial^2\phi}{\partial x^2}\) (i) when the independent variables are \(x\) and \(r\), (ii) when they are \(x\) and \(\theta\). These values are to be expressed in terms of the partial derivatives of \(\phi\) when the independent variables are \(x\) and \(y\).

From a point \((x',y')\) perpendiculars are drawn on the lines given by \[ ax^2+2hxy+by^2=0, \] the axes being at right angles; prove that the length of the perpendicular from \((x',y')\) on the line joining the feet of these perpendiculars is \[ \frac{ax'^2+2hx'y'+by'^2}{\{(a-b)^2+4h^2\}^{\frac{1}{2}}(x'^2+y'^2)^{\frac{1}{2}}}. \]

\(ABD, CAE, CBF\) are three circles touching each other at \(A, B, C\). The common tangent at \(C\) passes through \(D\), and \(DAE, DBF\) are straight lines. Prove that \(EF\) touches the circles at \(E\) and \(F\).

Prove that if \(\phi\) is the angle the radius vector of a plane curve makes with the tangent \[ \frac{dr}{ds} = \cos\phi, \quad r\frac{d\theta}{ds} = \sin\phi, \quad \frac{d^2r}{ds^2} = \frac{\sin^2\phi}{r} - \frac{\sin\phi}{\rho} \] where \(\rho\) is the radius of curvature. If the tangent at \(P\) to this curve is produced to \(P'\) at a distance from \(P\) equal to \(OP\), where \(O\) is the origin, prove that the angle \(\phi'\) between \(OP'\) and the tangent to the locus of \(P'\) is \(\tan^{-1}\frac{\rho r^2}{2r^3-\rho r^2}\), where \(\rho\) is the radius of curvature of the given curve at \(P\) and \(r'=OP'\).

Prove the formula \(\rho=r\frac{dr}{dp}\) for the radius of curvature of a curve given in terms of \(p\) the perpendicular on to the tangent from the origin, and \(r\) the radius vector. Obtain the \((p,r)\) equation of a parabola referred to its focus as origin in the form \(a.r=p^2\), and deduce that if the parabola is made to roll without slipping on a fixed straight line, its focus describes a curve whose radius of curvature is equal to the focal radius of the parabola at the corresponding instantaneous point of contact.

Prove that the length of the arc of the curve whose pedal \((p,r)\) equation is \(p=r-d\) between the points \(r=a, r=2a\) is \(a(\sqrt{3}-\frac{\pi}{3})\). Shew that the polar equation of this curve may be written in the form \[ 2r = a\sec^2\left(\frac{\sqrt{2ar-a^2}+\frac{a\pi}{2}-a\theta-a}{2a}\right). \]

Trace the curve \(y^2(a+x)=a^2(a-x)\), and show that the volume obtained by rotating it round the line \(x+a=0\) is \(2\pi^2a^3\).

Prove that the distance from the origin of the centre of curvature at any point of a curve is \(\left[ \left(\frac{dp}{d\psi}\right)^2 + \left(\frac{d^2p}{d\psi^2}\right)^2 \right]^{\frac{1}{2}}\), where \(\psi\) is the inclination of the tangent at the point to the initial line and \(p\) the perpendicular from the origin on the tangent. Find the value of \(p\) in terms of \(\psi\) for the cardioid \(r=a(1-\cos\theta)\) and determine the distance from the origin of the centre of curvature at the point \(\theta=\dfrac{\pi}{2}\).

Prove that the curvature \(\kappa\) of a twisted curve is given by \[ \kappa^2ds^4 = A^2+B^2+C^2, \] where \[ A = d^2ydz-d^2zdy, \quad B=d^2zdx-d^2xdz, \quad C=d^2xdy-d^2ydx. \] In what sense is this equation invariant? Find the curvature of the curve whose equations are \[ x=a\cos\theta, \quad y=b\sin\theta, \quad z=c\theta. \]

The "centre of mass," \(O\), of the electricity on a conductor, charged and alone in the field, is called the electric centre of the conductor. Prove that the potential at a point \(P\) in the field must lie between \[ \frac{E}{OP}\left(1+\frac{\sigma^2}{OP^2}\right) \quad \text{and} \quad \frac{E}{OP}\left(1-\frac{\sigma^2}{2OP^2}\right), \] where \(E\) is the total charge on the conductor, and \(\sigma\) is the greatest radius of the conductor from the electric centre \(O\). Also prove that if there are two conductors \(C, C'\) in the field, with electric centres \(O, O'\), and maximum radii \(a, a'\) measured from \(O, O'\), their mutual coefficient of potential is \(1/c'\), where \(c'\) cannot differ from \(OO'\) by more than \((a^2+a'^2)/OO'\).

Give definitions of the tangent, principal normal, binormal, curvature (\(1/\rho\)), torsion (\(1/\sigma\)), and centre of curvature of a twisted curve, explaining carefully any conventions of sign involved in your definitions. Shew that, if from a fixed point \(O\) lines \(OT, ON, OB\) are drawn parallel to the positive directions of the tangent, principal normal, and binormal, respectively, at a point \(P\) of a curve \(C\), then if \(P\) moves along \(C\) with unit velocity the triad \(OTNB\) has at any instant an angular velocity whose components about \(OT, ON, OB\) are respectively \(1/\sigma, 0, 1/\rho\). A curve \(C\) drawn on the surface of a right circular cone of semi-vertical angle \(\alpha\) cuts the generators at a constant angle \(\beta\). Shew that the curvature and torsion of \(C\) at a point \(P\) at distance \(r\) from the axis of the cone are given by \[ \frac{r}{\rho} = |\sin\beta|\sqrt{1-\cos^2\alpha\cos^2\beta}, \quad \frac{r}{\sigma} = \pm \cos\alpha\sin\beta\cos\beta, \] explaining the ambiguity in the second formula. Shew that as \(P\) describes \(C\), the centre of curvature describes a curve lying on the surface of a right circular cone and cutting the generators at a constant angle.

The members of a family of curves in the \(x,y\) plane satisfy the differential equation \begin{equation*} y\frac{dy}{dx} - y^2 = x^2 - x. \end{equation*} By multiplying this equation by a suitable function of \(x\) and integrating, or otherwise, obtain the curve which passes through the point \((0, 1)\). Show that this curve also passes through the point \((-a, 0)\) where \(a > 0\) and \(a = -\ln a\).

A family of parabolas is given by the equation $$(x-at)^2 = 4a(y-at^2), \quad (1)$$ where \(a\) is a positive constant and \(t\) is a real-valued parameter. Show that the number of members of the family passing through a given point \((x_0, y_0)\) is 0, 1 or 2 according as \(x_0^2\) is greater than, equal to, or less than \(5ay_0\). Show that, for each fixed value of \(t\), the function \(y\) of the variable \(x\) defined by the equation (1) satisfies the differential equation $$5x^2\left(\frac{dy}{dx}\right)^2 - 4ax\left(\frac{dy}{dx}\right) + x^2 - ay = 0. \quad (2)$$ Deduce a solution of (2) which cannot be obtained by giving any fixed value to \(t\) in (1). How many solutions of (2) are there for which \(dy/dx\) is everywhere continuous and \(y = 0\) when \(x = 0\)?

Show that the function $$f(x) = e^{-x} \int_{-\infty}^{x} e^{s} F(s) ds$$ satisfies the differential equation $$f'(x) + f(x) = F(x).$$ The function \(\phi(x)\) is defined as follows: $$\phi(x) = 0 \quad \text{for } x < 0$$ $$x \quad \text{for } x > 0$$ Given that $$f'(x) + f(x) = \phi(x) \quad \text{and} \quad f(-\infty) = 0,$$ find \(f(x)\). Show graphically the forms of the functions \(\phi(x)\) and \(f(x)\). Given that $$g'(x) + g(x) = \phi(x) - \phi(x-1) \quad \text{and} \quad g(-\infty) = 0,$$ find the function \(g(x)\) and show graphically the forms of the functions \(\phi(x) - \phi(x-1)\) and \(g(x)\).

(i) Evaluate $$\int_0^{\frac12\pi} x\left(\tfrac12\pi - x\right)\sin^2 x \, dx.$$ (ii) Find the general solution of the differential equation $$(x^2 \log x)y' + xy = (x^2 \log x - 1)\cos x$$ in the range \(x > 1\).

Find the general solution of the following equations for \(y\) as a function of \(x\):

1. [(i)] \(\sec x \frac{dy}{dx} + y = \sin x\),
2. [(ii)] \((x^2 + 1)\left(x\frac{dy}{dx} - y\right) = xy^4\).

Solve the equation \(\frac{dy}{dx}-2y=x+\cos x\).

\(a_0\), \(a_1\), \(\ldots\), \(a_{n-1}\) are complex numbers, and \(A_0\), \(A_1\), \(\ldots\), \(A_{n-1}\) are defined by $$A_s = \frac{1}{\sqrt{n}} \sum_{r=0}^{n-1} a_r \omega^{rs},$$ where \(\omega = e^{2\pi i/n}\). Prove that $$a_r = \frac{1}{\sqrt{n}} \sum_{s=0}^{n-1} A_s \omega^{-rs}$$ and that $$\sum_{s=0}^{n-1} |A_s|^2 = \sum_{r=0}^{n-1} |a_r|^2.$$

Prove that \[ (1+x)^n - (1-x)^n = 2nx \prod_{k=1}^m \left(1+x^2\cot^2\frac{k\pi}{n}\right), \] where \(n\) is any positive integer and \(m\) is the greatest integer less than \(\frac{1}{2}n\).

Express each of the polynomials \(x^m-1, x^n-1, x^{mn}-1\) as a product of linear factors involving the complex number \[ z = \cos \frac{2\pi}{mn} + i \sin \frac{2\pi}{mn}. \] Hence, or otherwise, prove that, if \(m, n\) are prime to one another (i.e. have no common factor greater than 1), then \[ (x-1)(x^{mn}-1) = (x^m-1)(x^n-1)P(x), \] \[ \frac{x^{mn}-1}{(x^m-1)(x^n-1)} = \frac{1}{x-1} + p(x), \] where \(P(x)\) and \(p(x)\) are polynomials.

(i) Prove that if \(n\) is an odd integer, \(\sin n\theta + \cos n\theta\) regarded as a rational integral function of \(\sin\theta\) and \(\cos\theta\) is divisible either by \(\sin\theta+\cos\theta\), or by \(\sin\theta-\cos\theta\). (ii) Prove that if \(m\) and \(n\) are two different odd integers, or two different even integers, \(m\sin n\theta - n\sin m\theta\) is divisible by \(\sin^3\theta\).

In the Argand diagram, the points \(P_0\) and \(P_1\) represent the complex numbers \(4+6i\) and \(10+2i\) respectively. Find the complex numbers which correspond to the other five vertices of the regular hexagon with centre \(P_0\) and one vertex at \(P_1\).

Obtain an expression for \(\tan 7\theta\) in terms of \(\tan\theta\), and find the value of \[ \cot\frac{\pi}{7}\cot\frac{2\pi}{7}\cot\frac{3\pi}{7}. \] Prove that \[ \cot\frac{\pi}{7}+\cot\frac{2\pi}{7}-\cot\frac{3\pi}{7} = \sqrt{7}. \]

Find the sum of the first \(n\) terms of each of the following series

1. \(\displaystyle\frac{1}{2\cdot5\cdot8} + \frac{1}{5\cdot8\cdot11} + \frac{1}{8\cdot11\cdot14} + \dots\);
2. \(\cos^3\theta - \frac{1}{3}\cos^3 3\theta + \frac{1}{9}\cos^3 9\theta - \frac{1}{27}\cos^3 27\theta + \dots\).

Sketch the curves \(\cosh x = \dfrac{y\cosh\alpha}{\sin y}\) for different values of the parameter \(\alpha\) (\(\alpha \ge 0\)), and for values of \(y\) between \(-\pi\) and \(\pi\). Show that, on the curve of parameter \(\alpha\), the function \[ \sinh(x+iy) - (x+iy)\cosh\alpha \] is purely real, and indicate its direction of increase along the curve.

The circumference of a circle, centre \(O\) and radius \(a\), is divided into \(2n+1\) equal arcs by points \(A_1, A_2, \dots, A_{2n+1}\), where \(n \ge 1\). Starting from \(A_1\), with the pencil never leaving the paper and moving always in an anticlockwise direction round \(O\), a sequence of chords is drawn which subtend at \(O\) successively the angles \(\theta, 2\theta, \dots, n\theta, \theta, 2\theta, \dots, n\theta, \theta, 2\theta, \dots\), where \((2n+1)\theta=2\pi\). Show that, if this construction is continued for sufficiently long, a polygonal line is drawn which begins and ends at \(A_1\) and contains each chord \(A_iA_j\) (\(1\le i < j \le 2n+1\)) exactly once; and find the total length of this line.

Show that \[ (\cos\theta+i\sin\theta)(\cos\phi+i\sin\phi), \] where \(i^2=-1\), depends only on the sum \(\theta+\phi\). Hence, or otherwise, show that \[ (2\cos\theta)^n - \binom{n}{1}(2\cos\theta)^{n-1}\cos\theta + \dots + (-1)^r \binom{n}{r}(2\cos\theta)^{n-r}\cos r\theta + \dots + (-1)^n \cos n\theta = \cos n\theta, \] where \(\binom{n}{r}\) denotes the binomial coefficient \(\dfrac{n!}{r!(n-r)!}\).

Two complex variables \(z=x+iy\), \(Z=X+iY\), are connected by the relation \[ Z = \sin(\tfrac{1}{2}\pi z). \] Show that to every point in the complex \(Z\)-plane there corresponds a point in the strip \(|x| \le \frac{1}{2}\) of the complex \(z\)-plane. Show also that the lines \(x=\text{constant}\), \(y=\text{constant}\) map into certain mutually orthogonal systems of ellipses and hyperbolae in the \(Z\)-plane.

Sum to \(N\) terms, and where possible to infinity, the series whose \(n\)th terms are \[ \text{(i)} \ (n+2)n, \quad \text{(ii)} \ (n+2)x^n, \quad \text{(iii)} \ (n+2)\cos n\theta. \]

Prove that \[ \tan \frac{\pi}{5} = \sqrt{5} \tan \frac{\pi}{10}, \] and hence, or otherwise, show that \[ \tan \frac{\pi}{20} = \frac{1}{4}(\sqrt{5}+1)\{4-\sqrt{(10+2\sqrt{5})}\}. \]

(i) Prove that \[ \sum_{r=1}^{r=n} \cos^r\theta \sin r\theta = \cot\theta(1-\cos^n\theta \cos n\theta). \] (ii) Without using tables show that \[ \cos \frac{\pi}{10} = \frac{1}{4}\sqrt{(10+2\sqrt{5})}. \]

Prove that \((e^{i\alpha}+e^{2i\alpha}+e^{4i\alpha})\) is one root of \(x^2+x+2=0\), where \(\alpha=2\pi/7\). Hence show that \begin{align*} \cos\alpha+\cos2\alpha+\cos4\alpha &= -\frac{1}{2}, \\ \sin\alpha+\sin2\alpha+\sin4\alpha &= \frac{1}{2}\sqrt{7}. \end{align*}

Prove that \[ \sum_{r=1}^n \frac{2(x-\cos r\alpha)}{x^2-2x \cos r\alpha+1} = \frac{(2n+1)x^{2n}}{x^{2n+1}-1} - \frac{1}{x-1}, \] where \((2n+1)\alpha=2\pi\). Verify that the two expressions have the same limit as \(x\) tends to 1.

Obtain the quadratic equation whose roots \(\eta\) and \(\bar{\eta}\) are given by \[ \eta = \omega + \omega^3 + \omega^4 + \omega^5 + \omega^9 \quad \text{and} \quad \bar{\eta} = \omega^{-1} + \omega^{-3} + \omega^{-4} + \omega^{-5} + \omega^{-9}, \] where \(\omega = \cos \frac{2\pi}{11} + i \sin \frac{2\pi}{11}\). Deduce that the absolute value of \(\eta\) is \(\sqrt{3}\), and that \[ \sin \frac{2\pi}{11} + \sin \frac{6\pi}{11} + \sin \frac{8\pi}{11} + \sin \frac{10\pi}{11} + \sin \frac{18\pi}{11} = \frac{\sqrt{11}}{2}. \]

The pairs of points \((R, P'; P, P'; \dots)\) and the pairs of points \((P', P''; P_1, P_1''; \dots)\) form two involutions on the same straight line. Shew that the necessary and sufficient condition that the pairs of points \((P_1, P_1''; P_2, P_2''; \dots)\) should also form an involution is that the double points of the two first involutions should form a harmonic range.

Prove that \[ x^{2n} - 2x^n y^n \cos n\theta + y^{2n} = \prod_{r=0}^{n-1} \left\{x^2 - 2xy \cos\left(\theta + \frac{2r\pi}{n}\right) + y^2\right\}. \] A regular polygon of \(n\) sides is inscribed in a circle. The projection on the plane of the circle of the line joining the centre to any point \(P\) in space makes an angle \(\alpha\) with the radius to one of the vertices, and \(r_1\) and \(r_2\) are the greatest and least distances of \(P\) from the circumference of the circle. Prove that the product of the distances of \(P\) from the vertices of the polygon is \((s^{2n} - 2s^n d^n \cos n\alpha + d^{2n})\), where \(r_1+r_2=2s, r_1-r_2=2d\).

Prove that, if \begin{align*} \sin(\beta+\gamma) + \sin(\gamma+\alpha) + \sin(\alpha+\beta) &= 0 \\ \text{and} \quad \alpha+\beta+\gamma &= \theta, \\ \text{then} \quad \frac{\sin\alpha+\sin\beta+\sin\gamma}{\sin\theta} = \frac{\cos\alpha+\cos\beta+\cos\gamma}{\cos\theta} &= \cos(\beta+\gamma) + \cos(\gamma+\alpha) + \cos(\alpha+\beta). \end{align*}

Sum the infinite series:

1. [(1)] \(\sin x - \frac{\sin 2x}{2} + \frac{\sin 3x}{3} - \dots\),
2. [(2)] \(\frac{\sin x}{\pi} - \frac{\sin 2x}{\pi^2} + \frac{\sin 3x}{\pi^3} - \dots\),
3. [(3)] \(\frac{\sin x}{1} - \frac{\sin 2x}{2!} + \frac{\sin 3x}{3!} - \dots\).

Prove that, if the equation \[ (a + \cos\theta) \cos(\theta-\gamma) = b \] is satisfied by \(\theta_1, \theta_2, \theta_3, \theta_4\), four different values of \(\theta\) which lie between 0 and \(2\pi\), \begin{align*} \cos\theta_1 + \cos\theta_2 + \cos\theta_3 + \cos\theta_4 &= -2a, \\ \sin\theta_1 + \sin\theta_2 + \sin\theta_3 + \sin\theta_4 &= 0, \\ \theta_1 + \theta_2 + \theta_3 + \theta_4 - 2\gamma &= 0 \text{ (or } 2r\pi). \end{align*}

Prove that, if \(x\) and \(y\) are real, \[ |\cot(x+iy)| < |\coth y|, \quad |\tan(x+iy)| < |\coth y|, \] where \(|a+ib|\) denotes as usual \(+\sqrt{(a^2+b^2)}\).

Assuming that the series \[ c(t) = 1 - \frac{t^2}{2!} + \frac{t^4}{4!} - \dots, \quad s(t) = t - \frac{t^3}{3!} + \frac{t^5}{5!} - \dots \] may be differentiated term by term, prove that the point with rectangular coordinates \(c(t), s(t)\) describes a circle with constant speed as the ``time'' \(t\) varies, and hence deduce that \[ c(t) = \cos t; \quad s(t) = \sin t. \]

Prove that if \(u\) is a complex number, and \(m\) and \(n\) are positive integers prime to one another, \(u^{m/n}\) has \(n\) values. Illustrate by a diagram. Shew that, if \[ r^2 = a^2+b^2+c^2, \quad \text{and} \quad \frac{b+ic}{r+a}=z, \] then \[ \frac{c+ia}{r+b} = i\frac{1-z}{1+z}, \quad \text{and} \quad \frac{a+ib}{r+c} = \frac{1+iz}{1-iz}. \] \item[] \hspace{1cm} Or Shew that when \(n\) is an integer \(\cos n\theta\) and \(\sin n\theta/\sin\theta\) are polynomials in \(\cos\theta\). Deduce that \[ \cos n\theta = \prod_{r=1}^{n} \left\{1 - \frac{\sin^2 \theta/2}{\sin^2 (2r-1)\pi/4n}\right\}, \] and shew that \[ \frac{\cos n\theta - \cos n\alpha}{1 - \cos n\alpha} = \prod_{1}^{n} \left\{1 - \frac{\sin^2\theta/2}{\sin^2(r\pi/n + \alpha/2)}\right\}. \]

The vertices \(A_1, A_2, A_3, A_4, A_5\) of a regular pentagon lie on a circle of unit radius with centre at the point \(O\). \(A_1\) is the mid-point of \(OP\). Prove that

1. [(i)] \(PA_1 \cdot PA_2 \cdot PA_3 \cdot PA_4 \cdot PA_5 = 31\);
2. [(ii)] \(\sum_{r=1}^{5} (PA_r)^2 = 25\);
3. [(iii)] \(A_1A_2 \cdot A_1A_3 \cdot A_1A_4 \cdot A_1A_5 = 5\).

\(P\) and \(Q\) are points of the plane outside the circumcircle of the regular polygon \(A_0 A_1 A_2 \ldots A_{n-1}\) whose centre is the point \(O\). The line-segment \(OP\) contains a vertex of the polygon, while the segment \(OQ\) perpendicularly bisects an edge. By representing this situation in the complex plane, or otherwise, show that the geometric mean of all the lengths \(QA_r\) exceeds the length \(QO\), while the length \(PO\) exceeds the geometric mean of all the lengths \(PA_r\).

Points \(A_1\), \(A_2\), \(\ldots\), \(A_n\) (where \(n \geq 3\)) are equally spaced round the circumference of a circle. Their distances from a line drawn through the centre are \(d_1\), \(d_2\), \(\ldots\), \(d_n\). Prove that $$d_1^2 + d_2^2 + \ldots + d_n^2$$ is the same for every direction of the line.

A closed polygon of \(2n\) sides, \(n\) of which are of length \(a\) and \(n\) of length \(b\), is inscribed in a circle. Show that the radius of the circle is independent of the arrangement of the sides, and find its value.

By considering the sum of the roots of the equation \(z^5 = 1\), find an equation with integer coefficients which is satisfied by \(\cos \frac{2\pi}{5}\), and hence obtain an expression for \(\cos \frac{2\pi}{5}\). Prove the theorem (known to Euclid) that if a pentagon, a hexagon, and a decagon, regular and with sides \(a_5\), \(a_6\), \(a_{10}\) are inscribed in the same circle, then $$a_5^2 = a_6^2 + a_{10}^2.$$

Two regular polygons of \(n_1\) and \(n_2\) sides are inscribed in two concentric circles of radii \(r_1\) and \(r_2\) respectively. Prove that the sum of the squares on all the lines joining the vertices of one to the vertices of the other is \[n_1n_2(r_1^2 + r_2^2).\]

Let \[ \rho = \cos\frac{2\pi}{m} + i\sin\frac{2\pi}{m}, \] where \(m\) is a positive integer. For any integer \(r\) put \[ p_m(r) = \frac{\rho^r}{1-\rho} + \frac{\rho^{2r}}{1-\rho^2} + \dots + \frac{\rho^{(m-1)r}}{1-\rho^{m-1}}. \] By considering the differences \[ p_m(r+1) - p_m(r) \] and the sum \[ p_m(0)+p_m(1)+\dots+p_m(m-1), \] or otherwise, evaluate the \(p_m(r)\) for all \(m\) and \(r\). Show in particular that \[ p_m(0) = \frac{1}{2}(m-1). \]

Prove that \[ \sin 3\theta = 4 \sin \theta \sin(\theta + \tfrac{1}{3}\pi) \sin(\theta + \tfrac{2}{3}\pi). \] The trisectors of the angles of a triangle ABC meet in \(X, Y, Z\) (\(X\) being the point of intersection of the trisectors of B and C lying nearest to BC, and similarly for \(Y\) and \(Z\)). Express the ratio \(AY/AZ\) as simply as you can in terms of the angles of the triangle ABC, and hence find the angles of the triangle AYZ. Hence, or otherwise, prove that XYZ is an equilateral triangle.

Prove that \[ \sum_{r=0}^{n-1} \frac{1}{1-\cos\left(\phi+\frac{2r\pi}{n}\right)} = \frac{n^2}{1-\cos n\phi}. \]

Prove that \[ 2^{n-1} \sin\frac{\pi}{n} \sin\frac{2\pi}{n} \dots \sin\frac{(n-1)\pi}{n} = n. \] Hence or otherwise prove that \(\displaystyle\int_0^\pi \log \sin x dx = -\pi \log 2\).

Four real or complex numbers (other than zero) are such that their squares are the same numbers in the same or a different order; prove that each number is a root of unity.

Prove that for the continued fraction \(a_1 + \frac{1}{a_2+} \frac{1}{a_3+} \dots\) where the \(a\)'s are all positive, any convergent is intermediate in magnitude between the next two preceding ones. For the fraction \(a+\frac{1}{b+}\frac{1}{a+}\frac{1}{b+}\frac{1}{a+}\frac{1}{b+}\dots\), prove that if \(p_n/q_n\) is the \(n\)th convergent, \(p_{2n}=q_{2n+1}\) and \(bp_{2n-1}=aq_{2n}\).

Shew that the problem of determining the \(n\)th roots of 1 is equivalent to that of inscribing a regular polygon of \(n\) sides in a circle. If \(n\) denote an even integer, shew that the product \[ \left(x^2+\cot^2\frac{\pi}{2n}\right) \left(x^2+\cot^2\frac{3\pi}{2n}\right) \left(x^2+\cot^2\frac{5\pi}{2n}\right) \dots \left(x^2+\cot^2\frac{(n-1)\pi}{2n}\right) \] is equal to \(\frac{1}{2}\{(1+x)^n+(1-x)^n\}\).

Find the real linear and quadratic factors of \(z^n-1\) when \(n\) is an odd positive integer. Deduce that \[ \sin\frac{\pi}{n}\sin\frac{2\pi}{n}\dots\sin\frac{(n-1)\pi}{n} = \frac{n}{2^{n-1}}. \]

Prove that, if \(r\) is prime to \(n\) and \(\alpha = \cos\frac{2r\pi}{n} + i \sin\frac{2r\pi}{n}\), the \(n\)th roots of unity are \(1, \alpha, \alpha^2, \dots, \alpha^{n-1}\). Shew that, if \(p\) is prime to \(n\), \[ 1 + \alpha^p + \alpha^{2p} + \dots + \alpha^{(n-1)p} = 0. \]

If \(P_0, P_1, \dots, P_{n-1}\) are \(n\) equidistant points round a circle of unit radius, and \(a_r\) is the distance \(P_0P_r\), prove that \(a_1a_2\dots a_{n-1} = n\). Find also \(a_1+a_2+\dots+a_{n-1}\) and deduce that when \(n\) is large the average distance of the points from \(P_0\) is approximately \(4/\pi\).

If \(1, \alpha, \alpha^2, \alpha^3, \alpha^4\) are the fifth roots of unity, prove that \[ \alpha\tan^{-1}\alpha + \alpha^2\tan^{-1}\alpha^2 + \alpha^3\tan^{-1}\alpha^3 + \alpha^4\tan^{-1}\alpha^4 \] \[ = \pi\cos\frac{3\pi}{5} + \sin\frac{3\pi}{5}\log\left(\tan\frac{\pi}{20}\right) + \sin\frac{\pi}{5}\log\left(\tan\frac{3\pi}{20}\right). \]

If \(x\) is any complex root of the equation \(x^{11}-1=0\), and if \[ a=x+x^3+x^4+x^5+x^9, \quad b=x^2+x^6+x^7+x^8+x^{10}, \] prove that \((a-b)^2 = -11\). Show further that \[ (x^3+1)[a-b-2(x-x^{10})] = x^3-1, \] and deduce that \[ \tan\frac{3\pi}{11} + 4\sin\frac{2\pi}{11} = \sqrt{11}. \]

If \(x\) and \(\theta\) are real, and \(n\) is a positive integer, express \(x^{2n}-2x^n\cos n\theta+1\) as the product of \(n\) real quadratic factors.

Prove that, in a triangle \(ABC\), \[ \Sigma \sin^2 A \tan A = \tan A \tan B \tan C - 2\sin A \sin B \sin C. \]

Express \[ \begin{vmatrix} \sin^3\theta & \sin\theta & \cos\theta \\ \sin^3\alpha & \sin\alpha & \cos\alpha \\ \sin^3\beta & \sin\beta & \cos\beta \end{vmatrix} \] as the product of four sines and hence find all values of \(\theta\), in terms of \(\alpha\) and \(\beta\), for which the value of this determinant is zero.

By considering the expression for \(\cos 7\theta\) in terms of \(\cos\theta\), find the roots expressed in trigonometric form of the equation \[ 64x^6 - 112x^4 + 56x^2 - 7 = 0. \]

Shew how to determine the four fourth roots of a complex expression of the form \(a+ib\).

If \(\omega\) is one of the imaginary \(n\)th roots of unity, shew that \[ \sum_{r=1}^{n-1}\frac{1-\omega^r}{y-\omega^r} = \frac{n(y^{n-1}-1)}{y^n-1}. \] By the use of the calculus, or otherwise, prove that if \(x>1\), then \[ (n+1)^2(x+3)(x-1) > 4n^2\{x^{n+1}+(n+1)x^n-n-2\} - 4n(n+1)\sum_{r=1}^{n-1}(1-\omega^r)\log\frac{1}{1-\omega^r x^{-1}} \] \[ > 4(n+1)^2(x-1), \] where \(n-1\) is a positive integer, and \(x^n\) is real.

(i) Use de Moivre's theorem to express \(\cos 6\theta\) and \(\sin 6\theta\) in terms of powers of \(\cos\theta\) and \(\sin\theta\). (ii) Let the roots of the equation \(z^4 - 1 = i\sqrt{3}\) be \(z_r\) (\(r = 1, 2, 3, 4\)), where \(z_r\) lies in the \(r\)th quadrant of the complex plane. Show that \[(z_1 + z_3) = -(z_3 + z_4) = 2^{-\frac{1}{2}}(1 + i\sqrt{3}).\]

Prove that \(\tan^2(\pi/11)\) is a root of the equation $$x^5 - 55x^4 + 330x^3 - 462x^2 + 165x - 11 = 0,$$ and state what are its other roots. By expressing the left-hand side in terms of \(\tan(\pi/11)\), or otherwise, prove that $$\tan(3\pi/11) + 4\sin(2\pi/11) = \sqrt{11}.$$

(i) Solve the equation \[2\cos 5\theta + 10\cos 3\theta + 20\cos \theta - 1 = 0.\] (ii) Prove that if \(\theta_1\), \(\theta_2\), and \(\theta_3\) are values of \(\theta\) which satisfy the equation \[\tan(\theta + \alpha) = \kappa \tan 2\theta\] and are such that no two of them differ by an integral multiple of \(\pi\), then \(\theta_1 + \theta_2 + \theta_3\) is an integral multiple of \(\pi\).

Prove that \(\sum_{r=1}^{4} \cos^4 r\pi/9 = 19/16\). Find also the numerical value of \(\sum_{1}^{4} \sec^4 r\pi/9\).

Express \(\tan n\theta\) in terms of \(\tan \theta\), where \(n\) is a positive integer. If \(n\) is odd, prove that $$n \tan n\theta = 1 + 2 \sum_{r=1}^{(n-1)/2} \frac{\sec^2(2r-1)\alpha}{\tan^3(2r-1)\alpha - \tan^3 \theta},$$ where \(2n\alpha = \pi\).

Show that \(\cos(2n + 1)\psi\) may be expressed as a sum of odd powers of \(\cos\psi\) and that the coefficients of \(\cos\psi\) and \(\cos^3\psi\) in this expression are \((-1)^n(2n + 1)\) and \((-1)^{n+1}\frac{2}{3}n(n + 1)(2n + 1)\). By considering the roots of a suitable equation, show that $$\sum_{k=0}^{n-1} \sec^2 \left( \frac{2k + 1}{2n + 1} \frac{\pi}{2} \right) = \frac{2}{3}n(n + 1).$$

Prove, by induction or by using de Moivre's theorem or in any other way, that if \(n\) is a positive integer then \[\tan nx = \frac{U_n(t)}{V_n(t)},\] where \(t = \tan x\) and \(U_n(t)\), \(V_n(t)\) are even polynomials of respective degrees \(n-2\), \(n\) if \(n\) is even and \(n-1\), \(n-1\) if \(n\) is odd. Hence, or otherwise, show that if \(n\) is even, then the product \[\prod_{r=0}^{n-1} \tan \frac{r\pi + c}{n}\] has the same value for all those real numbers \(c\) for which it is defined.

Prove that, if \(\cos\theta=c\), and the \(a\)'s are constants, \[ \cos n\theta = a_n c^n + a_{n-2}c^{n-2} + \dots, \] the last term being \(a_1 c\) or \(a_0\) according as \(n\) is odd or even. Prove that \(a_n = 2^{n-1}\). Prove that a polynomial of degree \(n-2\) can be found which differs from \(x^n\) for \(-1\le x \le 1\) by at most \(1/2^{n-1}\), and find such a polynomial when \(n=6\).

Sum the series: \(\sin\theta - 2\cos 2\theta + 3\sin 3\theta - \dots - 2n\cos 2n\theta\).

Prove the identity \[ \sum_{s=0}^{N-1} \frac{1}{z-e^{is\theta}} = \frac{N}{z^N-1} - \frac{1}{2}\frac{z^{N-1}+1}{z^N-1}\cot\frac{\theta}{2} + \frac{i}{2}\frac{z^{N-1}+1}{z^N-1}. \] Hence, or otherwise, evaluate the sums \[ \sum_{s=0}^{N-1} \tan(s\theta), \quad \sum_{s=0}^{N-1} \tan^2(s\theta). \]

If \(\theta=2\pi/7\), prove that \begin{align*} \sin\theta+\sin2\theta+\sin4\theta &= \sqrt{7}/2, \\ \tan^2\theta+\tan^2 2\theta+\tan^2 4\theta &= 21. \end{align*}

Determine numbers \(A,B,\) and \(C\) such that for all \(\theta\) \[ A\sin^5\theta + B\sin^3\theta + C\sin\theta = \sin 5\theta. \] Hence show that \(\sin \pi/30\) is a root of the equation \[ 16x^4+8x^3-16x^2-8x+1=0, \] and give the remaining roots as sines of angles.

1. [(i)] Eliminate \(\theta\) from the equations: \begin{align*} 4x &= 3a\cos\theta+a\cos3\theta, \\ 4y &= 3a\sin\theta-a\sin3\theta. \end{align*}
2. [(ii)] Find the sum to \(n\) terms of the series: \[ \tan\theta\tan3\theta + \tan2\theta\tan4\theta + \dots + \tan r\theta \tan(r+2)\theta \dots. \]

If \(\alpha\) is a complex root of the equation \(x^7-1=0\), express the other six roots in terms of \(\alpha\). Show that \(\alpha+\alpha^2+\alpha^4\) is a root of a quadratic equation whose coefficients do not involve \(\alpha\). Prove that \[ \cos\frac{\pi}{7} - \cos\frac{2\pi}{7} + \cos\frac{3\pi}{7} = \frac{1}{2}, \quad -\sin\frac{\pi}{7} + \sin\frac{2\pi}{7} + \sin\frac{3\pi}{7} = \frac{\sqrt{7}}{2}. \]

Discuss the convergence of the series \[ 1+z+z^2+...+z^n+..., \] where \(z\) may be real or complex. Prove that, if \(0 \le r < 1\) and \(\theta\) is real, \[ \sum_1^\infty r^n \sin n\theta = \frac{r \sin \theta}{1-2r\cos\theta+r^2}. \]

Express \[ \frac{2nx}{(1+x)^{2n}-(1-x)^{2n}} \] in real partial fractions, where \(n\) is an integer greater than 1. Deduce that \[ \sum_{r=1}^{n-1} (-1)^{r-1} \left(\cos \frac{r\pi}{2n}\right)^{2n-2} = \frac{1}{2}. \]

State and prove De Moivre's theorem for a (positive or negative) rational index. Evaluate \[ 32\int_0^\infty e^{-x}\cos^6 x \,dx, \] giving the answer to two decimal places.

Prove that \(\tan^2(\pi/11)\) is a root of the equation \[ t^5 - 55t^4 + 330t^3 - 462t^2 + 165t - 11 = 0. \] What are the other roots? Hence, by expressing \[ u = \tan(3\pi/11) + 4\sin(2\pi/11) \] as a rational function of \(\tan(\pi/11)\), or otherwise, prove that \(u\) is equal to \(\sqrt{11}\).

Prove that, if \(n\) is a positive integer, \[ (\cos\theta+i\sin\theta)^n = \cos n\theta + i\sin n\theta. \] Deduce that \(\sin(2n-1)\theta\) can be expressed as a polynomial \(P(\sin\theta)\) of degree \(2n-1\) in \(\sin\theta\). Prove that, if \(\cos(2n-1)\alpha \ne 0\), the roots \(\beta_1, \dots, \beta_{2n-1}\) of \[ P(x) - \sin(2n-1)\alpha = 0 \] are \[ \beta_r = \sin\left(\alpha + \frac{2r\pi}{2n-1}\right), \quad \text{where } r=1, \dots, 2n-1. \] Deduce that, if \(n>1\), both \[ \sum_{r=1}^{2n-1} \sin\left(\alpha + \frac{2r\pi}{2n-1}\right) \quad \text{and} \quad \sum_{r=1}^{2n-1} \sin^2\left(\alpha + \frac{2r\pi}{2n-1}\right) \] are independent of \(\alpha\), and find the value of the first of them.

Prove that \[ \sin\theta \sum_{r=1}^n \sin(2r-1)\theta = \sin^2 n\theta. \] Hence, or otherwise, prove that \[ \sin^3\theta \sum_{r=1}^n (2r-1)^2 \sin(2r-1)\theta \] \[ = (4n^2+1)\sin^2\theta \sin^2 n\theta - 4n \sin\theta \sin n\theta (1-\cos\theta \cos n\theta) - 2(\sin n\theta - n\sin\theta)^2. \] By considering the case when \(\theta\) is very small, deduce the sum to \(n\) terms of the series \(1^3+3^3+5^3+\dots\).

Prove that, if \(i^2=-1\) and \(n\) is a positive integer, \[ \left(\frac{1+i\tan\theta}{1-i\tan\theta}\right)^n = \frac{1+i\tan n\theta}{1-i\tan n\theta}. \] Hence, or otherwise, express \(\tan n\theta\) in the form \(\dfrac{f(\tan\theta)}{g(\tan\theta)}\), where \(f(x)\) and \(g(x)\) are polynomials in \(x\). Evaluate \[ \text{(i) } \sum_{k=0}^{k=n-1} \cot \frac{(4k+1)\pi}{4n}; \quad \text{(ii) } \sum_{k=0}^{k=n-1} \cot^2 \frac{(4k+1)\pi}{4n}. \]

(a) Without using tables, obtain the value of cosine \(18^\circ\). Show carefully that your result is not cosine \(54^\circ\), and verify that your result is equivalent to \(\sqrt{(5+\sqrt{5})}/2\sqrt{2}\). (b) Prove that, if \(n\) is a positive integer, \[ 2^{2n} \cos^{2n+1}\theta = \cos(2n+1)\theta + {}_{2n+1}C_1 \cos(2n-1)\theta + {}_{2n+1}C_2 \cos(2n-3)\theta + \dots + {}_{2n+1}C_n \cos\theta. \]

State and prove De Moivre's theorem about \((\cos\theta+i\sin\theta)^r\), where \(r\) is a rational number (i.e. a number of the form \(p/q\), where \(p\) and \(q\) are integers and \(q\neq 0\)). Using this theorem, or otherwise, find the sum to \(n\) terms of the series \[ \cos\theta \sin\theta + \cos^2\theta \sin 2\theta + \dots + \cos^m\theta \sin m\theta + \dots + \cos^n\theta \sin n\theta. \]

Justify the statement that, if \(n\) is a positive integer or positive fraction, \[ (\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta. \] Prove that, if \(y = 2\cos\theta\), \[ 2\cos 7\theta = y^7 - 7y^5 + 14y^3 - 7y. \] Hence find the cubic with the roots \(4\cos^2\pi/14\), \(4\cos^2 3\pi/14\) and \(4\cos^2 5\pi/14\).

Prove that, if \(n\) angles of which no two differ by a multiple of \(\pi\) satisfy the relation \[ p_0 + p_1 \cot\theta + p_2 \cot^2\theta + p_3 \cot^3\theta + \dots p_n \cot^n\theta = 0, \] the cotangent of the sum of these angles is \[ - (p_0 - p_2 + p_4 - p_6 + \dots) \div (p_1 - p_3 + p_5 - p_7 + \dots). \] Hence or otherwise prove that the relation \[ \cot\theta = \frac{a + a_1 \operatorname{cosec}^2\theta + a_2 \operatorname{cosec}^4\theta + \dots a_r \operatorname{cosec}^{2r}\theta}{b + b_1 \operatorname{cosec}^2\theta + b_2 \operatorname{cosec}^4\theta + \dots b_s \operatorname{cosec}^{2s}\theta} \] is generally satisfied by either \(2r\) or \(2s+1\) values of \(\cot\theta\), whichever of these numbers is the greater: and that, if all these values are real, the cotangent of the sum of the corresponding angles is \(a/b\).

Prove that \begin{align} (X + Y + Z)(X + \omega Y + \omega^2 Z)(X + \omega^2 Y + \omega Z) = X^3 + Y^3 + Z^3 - 3XYZ, \end{align} where \(\omega = e^{2\pi i/3}\). Hence, or otherwise, find the roots of the equation \begin{align} x^3 - 6x + 6 = 0. \end{align}

Suppose that \(x\) and \(y\) are real and satisfy the equations \begin{align*} 2x^3\cos 3y + 2x^2\cos 2y + x\cos y &= -\frac14\\ 2x^3\sin 3y + 2x^2\sin 2y + x\sin y &= 0 \end{align*} Show that \(x^2 = \frac{1}{4}\) and find the possible values of \(y\).

Show that the equations in \(x_1, x_2, ..., x_n\) (with \(u, v\) constants): \[ux_1 x_2 + x_2 = v,\] \[ux_2 x_3 + x_3 = v,\] \[\vdots\] \[ux_{n-1} x_n + x_n = v,\] \[ux_n x_1 + x_1 = v,\] possess either one or two solutions with \(x_1 = x_2 = ... = x_n\), or else possess infinitely many solutions. Show that there are infinitely many solutions if and only if \(\sqrt{(1 + 4uv)}\) is a nonzero root of the equation \[(1+t)^n-(1-t)^n = 0,\] and hence if and only if \(uv = -\frac{1}{4} \sec^2 (\pi k/n)\) with \(1 \leq k \leq n - 1\).

Write down the (complex) factors of \(x^2 + y^2 + z^2 - yz - zx - xy\). If \(x\), \(y\), \(z\), \(a\), \(b\), \(c\) are real and \(ax + by + cz = 0\), prove that the product of $$\frac{x^2 + y^2 + z^2 - yz - zx - xy}{(x + y + z)^3} \text{ and } \frac{a^3 + b^3 + c^3 - bc - ca - ab}{(a + b + c)^3}$$ cannot be less than \(\frac{1}{4}\). Find the ratios \(x:y:z\) in terms of \(a\), \(b\), \(c\) if the product is equal to \(\frac{1}{4}\).

If \(x_1, x_2, \ldots, x_n\) denote the complex \(n\)th roots of unity, evaluate $$\prod_{i< j} (x_i - x_j).$$

Find expressions for the roots of the equation \[ z^6+z^5+z^4+z^3+z^2+z+1=0, \] and mark these roots on an Argand diagram. If \(\alpha=2\pi/7\), prove that \(\cos 3\alpha + \cos 2\alpha + \cos \alpha = -\frac{1}{2}\). Find the roots of the cubic equation \(\xi^3+\xi^2-2\xi-1=0\).

The roots of the cubic equation \(x^3-px+q=0\) are \(\alpha, \beta, \gamma\). Evaluate \(\alpha^7+\beta^7+\gamma^7\) in terms of \(p\) and \(q\). Hence, or otherwise, solve the equation \[ 64\sin^7\theta + \sin 7\theta = 0. \]

Solve completely the equations \[ \sin x = \sin 2y, \quad \sin y = \sin 2z, \quad \sin z = \sin 2x, \] each angle being restricted to be positive and less than \(\pi\).

Starting from the definitions of sine and cosine by means of the infinite series, \[ \sin x = x - \frac{x^3}{3!} + \dots, \qquad \cos x = 1 - \frac{x^2}{2!} + \dots, \] give an outline, without detailed proof, of the steps by which the chief results of the trigonometry of a real angle can be established.

Express \(\tan 5\theta\) in terms of \(\tan\theta\). (If a general formula is quoted, it must be proved.) Prove that the roots of the equation \[ t^5 - 5pt^4 - 10t^3 + 10pt^2 + 5t - p = 0, \] where \(p\) is real, are all real and distinct. Evaluate \(\tan \frac{\pi}{20}\).

Prove that \[ \cos(\alpha+\beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta, \] where \(\alpha, \beta\) are angles of any magnitude. Express \[ \cos^2 2\alpha + \cos^2 2\beta + \cos^2 2\gamma + 2\cos 2\alpha \cos 2\beta \cos 2\gamma - 1 \] as a product of four cosines.

Eliminate \(\theta\) and \(\phi\) between the equations \begin{align*} a\sec\theta+b\cosec\theta &= c \\ a\sec\phi+b\cosec\phi &= c \\ \theta+\phi &= 2\alpha. \end{align*}

If \[ \sin^2\theta = \sin(A-\theta)\sin(B-\theta)\sin(C-\theta), \] and \[ A+B+C=\pi, \] prove that \[ \cot\theta = \cot A + \cot B + \cot C. \]

Prove that, if \(\omega\) is an imaginary cube root of unity, then \(1+\omega+\omega^2=0\). Shew how to use the cube roots of unity to find the sum of a series obtained by picking out every third term from a known series; and prove that \[ 1+\frac{x^3}{3!} + \frac{x^6}{6!} + \frac{x^9}{9!} + \dots = \frac{1}{3}\left\{e^x+2e^{-x/2}\cos\frac{\sqrt{3}}{2}x\right\}. \]

\(\theta, \phi\) are the two unequal values of \(x\) which satisfy the equation \[ \sin^3\alpha \text{ cosec } x + \cos^3\alpha \sec x = 1 \] and which do not differ by a multiple of \(\pi\). Prove that \(\theta+\phi+2\alpha = (2n+1)\pi\), and \(2\cos\frac{1}{2}(\theta-\phi) = \sin 2\alpha\).

Express \(1-\cos^2\theta-\cos^2\phi-\cos^2\psi+2\cos\theta\cos\phi\cos\psi\) as the product of four sines. \par Solve the equation \(\cot^{-1}\frac{x}{a}+\cot^{-1}\frac{x}{b}+\cot^{-1}\frac{x}{c}+\cot^{-1}\frac{x}{d} = \frac{\pi}{2}\).

Find an expression for \(\tan n\theta\) in terms of \(\tan\theta\), where \(n\) is a positive integer. Prove that \[ \tan^2 20^\circ + \tan^2 40^\circ + \tan^2 80^\circ = 33, \] and that \[ \tan 20^\circ . \tan 40^\circ . \tan 80^\circ = \sqrt{3}. \]

Show that \[ 1+\frac{\cos\theta}{\cos\theta}+\frac{\cos 2\theta}{\cos^2\theta}+\dots+\frac{\cos(n-1)\theta}{\cos^{n-1}\theta} = \frac{\sin n\theta}{\sin\theta\cos^{n-1}\theta}, \] and that \[ \cos\theta\cos\theta+\cos^2\theta\cos 2\theta+\dots+\cos^n\theta\cos n\theta = \frac{\sin n\theta \cos^{n+1}\theta}{\sin\theta}. \]

Show that if \[e^x\sin x = a_0 + \frac{a_1}{1!}x + \frac{a_2}{2!}x^2 + \ldots + \frac{a_n}{n!}x^n + \ldots\] then \(a_0 = 0\), \(a_{4n+1} = (-1)^n 4^n\); and determine \(a_{4n+2}\) and \(a_{4n+3}\).

If \(y = e^{-x}\sin(x\sqrt{3})\), prove that \begin{align} \frac{d^n y}{dx^n} = (-2)^n e^{-x} \sin(x\sqrt{3} - \frac{1}{3}n\pi) \end{align} Hence, or otherwise, show that \begin{align} y = \frac{\sqrt{3}}{2}\sum_{n=0}^{\infty}\left(\frac{(2x)^{3n+1}}{(3n+1)!} - \frac{(2x)^{3n+2}}{(3n+2)!}\right) \end{align}

Show that the complex mapping \(w = z+z^{-1}\), where \(z = x+iy\), \(w = u+iv\) are complex numbers, maps straight lines through \(z = 0\) into confocal hyperbolas in the \(w\)-plane with foci at \(w = \pm 2\); and maps each of the circles \(|z| = r\), \(|z| = r^{-1}\) into the same ellipse, also with foci \(\pm 2\). Prove that this ellipse cuts each such hyperbola orthogonally.

Let \(\omega = e^{\pi i/k}\), where \(k\) is an integer greater than 1. Let \(T_0 = 0\) and \[T_j = \omega + \omega^2 +...+ \omega^j.\] Show that \(T_{2k} = 0\), and sketch the polygon \(T_0 T_1...T_{2k}\) in the Argand diagram. Now let \(S_0 = 0\) and \(S_j = \omega + \omega^2/2 +...+ \omega^j/j\). Express \(S_j\) in terms of \(T_0, ..., T_j\) and show that each of the numbers \(S_0, ..., S_{2k}\) lies within or on the polygon \(T_0 T_1...T_{2k}\).

By the use of complex numbers or otherwise, evaluate the sums \(\sum_{n=0}^{\infty} r^n \cos n\theta\) where \(0 < r < 1\). Hence write \[ \frac{(1 - r^2 \cos 2\theta) r \cos \theta - r^3 \sin \theta \sin 2\theta}{1 - 2r^2 \cos 2\theta + r^4} \] in the form \(\sum a_n r^n \cos n\theta\), where the \(a_n\) are constants to be determined.

Let \(\Gamma\) be an ellipse in the \((x, y)\) plane, whose axes are not necessarily parallel to the coordinate axes and whose centre is not necessarily at the origin. Let \(V\) be the set of points inside or on \(\Gamma\). Show that as \(z = x+iy\) varies over \(V\), with \(z_0\) a fixed complex number, \(|z-z_0|\) reaches its maximum value when \(z\) is on \(\Gamma\). If \(\Gamma\) is a circle of radius \(r\) with centre at the origin, find the point \(z\) of \(\Gamma\) such that \(|z-z_0|\) has its maximum value.

Complex numbers \(z = re^{i\theta}\) (\(r > 0\), \(\theta\) real) and \(w = u + iv\) (\(u\), \(v\) real) are connected by the relation $$2w = z + \frac{1}{z},$$ and \(z\) and \(w\) are represented by points in complex planes. Find the loci described by \(w\) when \(z\) describes the following curves:

1. [(i)] the circles \(r = R\), \(r = 1\), \(r = R^{-1}\) (\(R > 1\));
2. [(ii)] the ray \(\theta = \alpha\), \(r > 0\) (\(0 < \alpha < \frac{1}{2}\pi\)).

Prove by the use of complex numbers, or otherwise, that, if \(n\) is a positive integer, \(\cos n\theta\) can be expressed as a polynomial in \(\cos\theta\) of the form \[ \cos n\theta = p_0 \cos^n\theta - p_1 \cos^{n-2}\theta + p_2\cos^{n-4}\theta - \dots, \] where \(p_0, p_1, p_2, \dots\) are positive integers. (It is to be understood that the summation continues so long as the indices remain non-negative.) Show that \[ p_0+p_1+p_2+\dots = \tfrac{1}{2} \{ (1+\sqrt{2})^n + (1-\sqrt{2})^n \}. \]

(i) Solve the equation \[ \tan\theta + \sec2\theta = 1. \] (ii) Sum the infinite series \[ 1 - \frac{1}{2!}\cos2\theta + \frac{1}{4!}\cos4\theta - \dots. \]

Prove that

1. [(i)] \(\dfrac{1}{2^3 \cdot 3!} - \dfrac{1 \cdot 3}{2^4 \cdot 4!} + \dfrac{1 \cdot 3 \cdot 5}{2^5 \cdot 5!} - \dots \text{ to infinity} = \dfrac{23}{24} - \dfrac{2}{3}\sqrt{2}\);
2. [(ii)] \(1+\dfrac{n}{m}+\dfrac{n(n-1)}{m(m-1)}+\dfrac{n(n-1)(n-2)}{m(m-1)(m-2)}+\dots \text{ to } n+1 \text{ terms} = \dfrac{m+1}{m-n+1}\), provided that \(m\) is not less than \(n\).

Find the sum to infinity of the series \(1+2x\cos\theta + 2x^2\cos 2\theta + 2x^3 \cos 3\theta + \dots\) and deduce the series for \(2\cos n\theta\) in descending powers of \(2\cos\theta\).

By considering \((1-x)f(x)\), where \[ f(x)=c_0+c_1x+\dots+c_nx^n, \] where \(x\) is a complex number and the \(c\)'s are real numbers such that \[ c_0 > c_1 > \dots > c_n > 0, \] show that \(f(x)\) is not zero for \(|x|\le 1\).

If \(\sin(\xi+i\eta) = x \sin\alpha\) where \(x > 1\), find how \(\xi\) and \(\eta\) vary as \(\alpha\) varies from \(0\) to \(\pi\). Find the sum to infinity of the series \(\cos\alpha\cos\beta+\frac{1}{2}\cos^2\alpha\cos 2\beta+\frac{1}{3}\cos^3\alpha\cos 3\beta+\dots\).

Obtain the \(n\)th roots of \(a+b\sqrt{-1}\), where \(a\) and \(b\) are real. If \(\omega\) is one of the imaginary \(2n\)th roots of unity, prove that \[ \sin\theta+\omega\sin 2\theta+\omega^2\sin 3\theta+\dots+\omega^{2n-1}\sin 2n\theta = \frac{\sin\theta+\omega\sin 2n\theta - \sin(2n+1)\theta}{1-2\omega\cos\theta+\omega^2}. \] Deduce the sums of the series \[ \sin\theta+\omega^2\sin 3\theta+\omega^4\sin 5\theta+\dots+\omega^{2n-2}\sin(2n-1)\theta \] and \[ \sin 2\theta+\omega^2\sin 4\theta+\omega^4\sin 6\theta+\dots+\omega^{2n-2}\sin 2n\theta, \] and find all the values of \(\theta\) which make these two sums equal.

Obtain the cube roots of unity and establish their principal properties. Express in terms of the exponential function the sums of the infinite series

1. [(i)] \(1 \cdot \dfrac{a^6}{5!} + 2 \cdot \dfrac{a^{12}}{11!} + 3 \cdot \dfrac{a^{18}}{17!} + \dots\).
2. [(ii)] \(\sum_{r=0}^\infty \dfrac{a_r x^r}{r!}\), where \(a_r=0\) when \(r\) is of the form \(3n-1\), \(a_r=-1\) when \(r\) is of the form \(6n+1\), \(a_r=+1\) for other values of \(r\).

Prove that, if \(p\) and \(q\) are positive integers, \(e^{p/q} = 1 + \dfrac{p}{q} + \dfrac{p^2}{2q^2} + \dfrac{p^3}{3!q^3} + \dots\). If \(u = 1 + \dfrac{1}{1.3} + \dfrac{1}{1.3.5} + \dfrac{1}{1.3.5.7} + \dots\), and \(v = 1 - \dfrac{1}{1}\cdot\dfrac{1}{3} + \dfrac{1}{4}\cdot\dfrac{1}{2\cdot 5} - \dfrac{1}{8}\cdot\dfrac{1}{3\cdot 7} + \dfrac{1}{16}\cdot\dfrac{1}{4\cdot 9} - \dots\), prove that, to four places of decimals, \(u^2/v^2=e\).

If \[ (1+x)^n = c_0+c_1 x + c_2 x^2 + \dots \] prove that \[ c_0-c_2+c_4-\dots = 2^{\frac{n}{2}}\cos\frac{1}{4}n\pi. \] \(A_1 A_2 A_3 \dots A_n\) is a regular polygon of \(n\) sides inscribed in a circle of radius \(a\). \(P\) is any point on the circumference, \(O\) is the centre, and \(\angle POA_1=\theta\). Prove that \[ PA_1 \cdot PA_2 \cdot PA_3 \dots PA_n = 2a^n \sin\frac{1}{2}n\theta. \]

Prove that for all values of \(x\), real or complex, \[ \sin x = x \prod_{n=1}^\infty \left(1-\frac{x^2}{n^2\pi^2}\right). \]

Prove that if (1) \(f_n(z)\) is, for every positive integral value of \(n\), analytic in a region \(T\), and (2) \(\Sigma f_n(z)\) is uniformly convergent throughout any closed domain \(D\) interior to \(T\), then the sum \(f(z)\) of the series is analytic in \(T\). Deduce that the sum of a power series represents an analytic function inside its circle of convergence. Prove that the function \(\Sigma n^{-s}\), where \(s=\sigma+it\), is analytic for \(\sigma>1\).

Assuming that the elliptic functions sn, cn, dn have the usual periods, zeros and poles, and behave in the usual way in the immediate neighbourhood of their zeros and poles, show that \[ \text{cn } u \text{ cn }(u+v) + \text{sn } u \text{ dn } v \text{ sn }(u+v) - \text{cn }v = 0, \] and deduce the addition theorem for the cn function.

Solve completely the following differential equations:

1. \(y' = y + e^{-x}\);
2. \(x^3y'' + xy = 1\);
3. \(y' = \frac{2x+3y}{3x+8y}\).

A function \(z=f_m(x)\) is defined as the solution of the differential equation \[ \frac{dz}{dx} = m \frac{z}{x} \] (where \(m\) is constant) such that \(z=1\) when \(x=1\). Without solving the differential equation explicitly prove that \begin{align*} f_m(x) f_n(x) &= f_{m+n}(x), \\ f_m(x) f_m(y) &= f_m(xy). \end{align*} Deduce the values of \(f_m(1)\) and \(f_0(x)\), and prove that \[ f_{-m}(x) = f_m\left(\frac{1}{x}\right) = [f_m(x)]^{-1}. \]

Determine \(P, Q, R\) as functions of \(x\) such that the equation \[ \frac{d^2y}{dx^2} + P\frac{dy}{dx} + Qy=R \] may be satisfied by \(y=x\), \(y=1\) and \(y=1/x\) for all values of \(x\) (except \(x=0\) for \(y=1/x\)). With these values of \(P, Q, R\), state what condition must be satisfied by the numerical coefficients \(a,b,c\), if the equation is also satisfied, for all \(x\) except 0, by \[ y=ax+b+\frac{c}{x}. \]

If \(y=\psi_n(x)\) is a solution of the equation \[ \frac{d^2y}{dx^2} + \frac{2(n+1)}{x} \frac{dy}{dx} + y = 0, \] show that \(Y=\frac{1}{x}\frac{d}{dx}\{\psi_n(x)\}\) satisfies the equation \[ \frac{d^2Y}{dx^2} + \frac{2(n+2)}{x} \frac{dY}{dx} + Y=0. \] Hence show that, if \(n\) is a positive integer, \(\psi_n(x) = \left(\frac{1}{x}\frac{d}{dx}\right)^n \frac{\sin x}{x}\) satisfies the former differential equation.

Show that the solution of the equation \[ y'' + n^2 y = a \sin pt \] (where \(n\neq 0\) and \(p^2 \neq n^2\)), such that \(y=0\) and \(y'=0\) when \(t=0\), is \[ y = \frac{a}{n^2-p^2}\left(\sin pt - \frac{p}{n}\sin nt\right). \] Show also that, as \(p\) tends to \(n\), \(y\) tends to \[ \frac{a}{2n}\left(\frac{1}{n}\sin nt - t\cos nt\right), \] and verify that this is the solution when \(p\) is equal to \(n\).

By considering the differential equation \[ \frac{d^3y}{dx^3}=y \] with appropriate initial conditions, or otherwise, show that the sum of the infinite series \[ 1 + \frac{x^3}{3!} + \frac{x^6}{6!} + \frac{x^9}{9!} + \dots \] is \[ \frac{1}{3} \left\{ e^x + 2e^{-x/2} \cos\left(\frac{x\sqrt{3}}{2}\right) \right\}. \]

State Leibnitz's theorem for the \(n\)th differential coefficient of the product of two functions. If \[ y = (1-x^2)^{\frac{1}{2}m} \frac{d^m u}{dx^m}, \] where \(m\) is a positive integer, and \(u\) satisfies the equation \[ (1-x^2) \frac{d^2u}{dx^2} - 2x \frac{du}{dx} + n(n+1)u=0, \] prove that \(y\) satisfies the equation \[ (1-x^2) \frac{d^2y}{dx^2} - 2x \frac{dy}{dx} + \left\{ (n-m)(n+m+1) - \frac{m^2}{1-x^2} \right\} y = 0. \]

If \(y = \frac{\sin x}{x}\), show that \[ \frac{d^n y}{dx^n} = u_n \sin x + v_n \cos x, \] where \(u_n\) and \(v_n\) are polynomials in \(1/x\) satisfying the relations \begin{align*} x u_{2n+1} + (2n+1)u_{2n} &= 0, \\ x v_{2n+1} + (2n+1)v_{2n} &= (-1)^n. \end{align*} Prove also that \begin{align*} u_{2n+1} &= u'_{2n} - v_{2n}, \\ v_{2n+1} &= u_{2n} + v'_{2n}, \end{align*} where the dash denotes differentiation with respect to \(x\). Deduce that \(u=u_{2n}\) satisfies the equation \[ x^2 \frac{d^2u}{dx^2} + 2(2n+1)x \frac{du}{dx} + [x^2+2n(2n+1)]u = (-1)^n x. \]

A set of functions \(J_n(x)\), \(n=0, \pm 1, \pm 2, \dots\), satisfy the following equations: \begin{align*} J_{n-1}(x)+J_{n+1}(x) &= \frac{2n}{x}J_n(x), \\ J_{n-1}(x)-J_{n+1}(x) &= 2\frac{d}{dx}J_n(x). \end{align*} Show that \begin{align*} \left(\frac{1}{x}\frac{d}{dx}\right)^m x^n J_n(x) &= x^{n-m}J_{n-m}(x), \\ \left(\frac{1}{x}\frac{d}{dx}\right)^m x^{-n} J_n(x) &= (-)^m x^{-n-m}J_{n+m}(x). \end{align*} Also prove that \(x^2\dfrac{d^2J_n(x)}{dx^2} + x\dfrac{dJ_n(x)}{dx} + (x^2-n^2)J_n(x)=0\).

Solve

1. [(i)] \(\frac{d^2y}{dx^2} + 2\frac{dy}{dx} = x+2\).
2. [(ii)] \(y \sin x - \cos^3 x + \cos x \frac{dy}{dx} = 0\).

(i) Solve the equation \[ \frac{dy}{dx} \cos^2 x + y = \tan x, \] with the condition that \(y=0\) when \(x=0\). (ii) Solve the equation \[ \frac{d^2 y}{dx^2} + 2 \frac{dy}{dx} + y = e^{-x} \sin^2 x. \]

If \(y=\sin^{-1} x\), prove that \[ (1-x^2)\frac{d^2y}{dx^2} - x \frac{dy}{dx} = 0. \] Determine the values of \(y\) and its successive derivatives when \(x=0\), and hence expand \(y\) in a series of ascending powers of \(x\).

1. [(i)] If \(n \ge 2\), prove that \(y=\sec^{-1}x\) satisfies the equation \[ (x^2-x^n) \frac{d^n y}{dx^n} + \{(3n-4)x^2-n+1\} \frac{d^{n-1}y}{dx^{n-1}} + (n-2)(3n-5)x \frac{d^{n-2}y}{dx^{n-2}} + (n-2)^2(n-3)\frac{d^{n-3}y}{dx^{n-3}} = 0. \]
2. [(ii)] Find the general expression for \(y\) in terms of \(x\) which is such that \[ \frac{d^2 x}{dy^2} = -\frac{d^2 y}{dx^2}. \]

(i) If \(ax^2+2hxy+by^2+2gx+2fy+c=0\), show that \[ \frac{d^2y}{dx^2} = \Delta/(hx+by+f)^3, \] where \(\Delta = \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix}\). (ii) Verify by direct differentiation that if \(a\) is a constant the function \[ y(x) = \int_a^x (x-t)e^{x-t}f(t)dt \] is a solution of the equation \[ \frac{d^2y}{dx^2} - 2\frac{dy}{dx} + y = f(x). \] (iii) Prove that \[ \frac{d^3x}{dy^3}\left(\frac{dy}{dx}\right)^5 = 3\left(\frac{d^2y}{dx^2}\right)^2 - \frac{d^3y}{dx^3}\frac{dy}{dx}. \]

State and prove Leibniz' theorem for the \(n\)th derivative of a product of two functions. If \[ y_n(x)=e^x \frac{d^n}{dx^n}(x^ne^{-x}), \] prove that \[ y_n = x \frac{dy_{n-1}}{dx} + (n-x)y_{n-1} \] and \[ \frac{1}{n} \frac{dy_n}{dx} = \frac{dy_{n-1}}{dx} - y_{n-1}. \] Hence show that the polynomial \(y_n\) satisfies a certain linear differential equation of the second order.

1. Prove that, if \[ F(x) = e^{2x} \int_0^x e^{-2t} f(t) \,dt - e^x \int_0^x e^{-t} f(t) \,dt, \] then
   1. [(i)] \(F(0)=0\),
   2. [(ii)] \(F'(0)=0\),
   3. [(iii)] \(F''(x) - 3F'(x) + 2F(x) = f(x)\).
2. Find the most general function \(\phi(x)\), which is such that \[ \int_0^x t\phi(t)\,dt = x^2\phi(x). \]

Prove that, if \(y=\tan^{-1}x\), then \[ u = \frac{d^n y}{dx^n} = (n-1)! \cos^n y \cos\left[ny+\frac{1}{2}(n-1)\pi\right] \] for every positive integer \(n\). Deduce, or prove otherwise, that \(u\) satisfies the differential equation \[ (1+x^2)\frac{d^2u}{dx^2} + 2(n+1)x\frac{du}{dx} + n(n+1)u=0. \]

Assuming that a function \(f(x)\) satisfies the relation \[ f''(x) = \frac{n(n-1)}{x^2}f - f', \] and taking \[ g(x) = x^n \frac{d}{dx} \left( \frac{f(x)}{x^n} \right), \] find an expression for \(g''(x)\) in terms of \(g\). Hence, show how to calculate the sequence of functions \(F_n(x)\) in which \(F_0(0)=0\), \(F_0'(0)=1\), and \[ F_n''(x) - \frac{n(n+1)}{x^2}F_n(x) + F_n(x) = 0, \] and evaluate \(F_2(x)\) explicitly.

State and prove Leibniz's formula for \(\dfrac{d^n(uv)}{dx^n}\), where \(u\) and \(v\) are functions of \(x\). If \(y_n(x) = \dfrac{d^n}{dx^n}\{(x^2-1)^n\}\), prove the relations

1. [(i)] \(x\dfrac{dy_{n+1}}{dx} = (x^2-1)\dfrac{d^2y_n}{dx^2} + 2(n+2)x\dfrac{dy_n}{dx} + (n+1)(n+2)y_n\);
2. [(ii)] \(x^2\dfrac{dy_{n+1}}{dx} = 2(n+1)x\dfrac{dy_n}{dx} + 2(n+1)^2y_n\).

Shew that \[ \frac{d^n}{dx^n} \left(\frac{1}{x^2+2x+2}\right) = (-1)^n n! \sin(n+1)\theta \sin^{n+1}\theta,\] where \[\cot\theta = x+1.\]

If \(y = e^{ax^2}\) and \(u = \frac{d^n y}{dx^n}\), prove that \[ \frac{d^2u}{dx^2} - 2ax \frac{du}{dx} - (2n+2)au = 0. \] If \(u = e^{ax^2}v\), find a differential equation satisfied by \(v\). Shew that \(v\) is a polynomial of degree \(n\) in \(x\), and find the coefficient of \(x^{n-2}\).

A point \(Q\) is taken on the tangent at \(P\) to a plane curve \(\Gamma\) so that \(PQ\) is of fixed length. Prove that the normal at \(Q\) to the locus of \(Q\) when \(P\) moves along \(\Gamma\) passes through the centre of curvature of \(\Gamma\) at \(P\).

(i) If \(A\) and \(B\) are constants, obtain a differential equation, not involving \(A\) and \(B\), which is satisfied by \[ y = (A \sin x + B \cos x)/x. \] (ii) Evaluate: \[ \int_a^b \frac{dx}{\sqrt{(b-x)(x-a)}}, \quad \text{where } b > a; \qquad \int_0^1 x \sin^{-1} x \, dx. \]

Prove that, if \(y\) is equal to \(e^x\), or if \(y\) is equal to the sum of the first \(n+1\) terms of the expansion of \(e^x\) in ascending powers of \(x\), \[ x \frac{d^2y}{dx^2} - (n+x)\frac{dy}{dx} + ny = 0. \]

Prove that, if \(\alpha\) is a constant, the function \[ y = A \cos\alpha x + B \sin\alpha x + \frac{1}{\alpha}\int_0^x f(\xi)\sin\alpha(x-\xi)d\xi \] satisfies the equation \[ \frac{d^2y}{dx^2} + \alpha^2y = f(x). \]

The expenditures \(x(t)\) and \(y(t)\) on armaments at time \(t\) of two countries are governed by the equations \[\frac{dx}{dt} = -ax+by+k_1,\] \[\frac{dy}{dt} = -ay+bx+k_2,\] where \(a > 0\), \(b > 0\), \(k_1 > 0\), \(k_2 > 0\); also \(x(0) = y(0) = 0\). Show that the total expenditure \(x(t) + y(t)\) on armaments will increase without bound as \(t \to \infty\) if \(b \geq a\), but will tend to a limit if \(b < a\). Find \(x(t)\) and \(y(t)\) when \(b \neq a\).

The variables \(x\) and \(y\) satisfy the differential equations \begin{align} \frac{dx}{dt} &= 2x + y + e^t,\\ \frac{dy}{dt} &= x + 2y. \end{align} Solve these equations subject to the initial conditions \(x(0) = 0, y(0) = 1\). [You may find it helpful to set \(z = x + \lambda y\) and find the two values of \(\lambda\) such that \(z\) satisfies a first order differential equation which does not explicitly involve \(x\) or \(y\).]

Zarg's Law of space combat says that the rate of destruction of each side's battle cruisers is a constant \(k\) times the number of cruisers on the other side. In a battle there are always so many cruisers that their numbers can be treated as continuous variables. The Args start a battle with \(A_0\) cruisers, and the Bryds with \(B_0\) cruisers, with \(A_0 > B_0\). Show that when all the Bryd cruisers are destroyed there are \((A_0^2 - B_0^2)^{1/2}\) Arg cruisers left. However, before the battle the Bryds discover that \(A_0 < B_0\sqrt{2}\), and that they can split the Arg force into two parts. They can fight one part first and the other part later. How should they split the Arg force in order to win with the smallest losses? What happens if \(A_0 > B_0\sqrt{2}\)?

Two variables \(x\) and \(y\) are to be determined as functions of time \(t\). It is found that the rate of change of \(x\) is equal to the sum of \(k_1\) times the instantaneous value of \(x\) and \((-k_2)\) times the instantaneous value of \(y\). The rate of change of \(y\) is similarly equal to \((-k_3)\) times the value of \(x\) plus \(k_4\) times the value of \(y\). Here \(k_1\), \(k_2\), \(k_3\) and \(k_4\) are positive constants. Obtain a second-order differential equation for \(x(t)\) and show that if \(k_1 k_4 > k_2 k_3\) the solution is of the form \[x = A e^{\alpha t} + B e^{\beta t},\] where \(A\) and \(B\) are arbitrary constants and \(\alpha\) and \(\beta\) are positive.

The real-valued functions \(x(t)\) and \(y(t)\) satisfy the pair of coupled differential equations \begin{align*} \ddot{x} + M\dot{y} - \omega^2 x &= 0\\ \ddot{y} - 2\omega\dot{x} - \omega^2 y &= 0 \end{align*} where \(\omega\) is a real constant, and dot denotes differentiation with respect to \(t\). Obtain a differential equation satisfied by \(x + \lambda y\) for a suitable choice of \(\lambda\), not necessarily real, and hence find the general solution of \((*)\). Describe briefly the shape of the path of the point \((x,y)\) when \(t\) is very large and positive.

A village contains two shops, \(X\) and \(Y\), which compete with one another to supply its needs. A local schoolmaster, wishing to amuse himself by trying to forecast their prospects, supposes that each shop has allotted to it a number, called its 'prosperity rating', which varies with time; he denotes the prosperity ratings of \(X\), \(Y\) at time \(t\) by \(x\), \(y\) respectively. He then invents a theory which implies that the rates of change of \(x\), \(y\) are \[\lambda(N - x) + \mu(x - y), \lambda(N - y) + \mu(y - x)\] respectively, where \(\lambda\), \(\mu\), \(N\) are positive constants. Initially \(x\) has the higher prosperity rating. Show that, according to his theory, the prosperity rating of \(Y\) can never overtake that of \(X\), but that it almost does so in due course if a certain condition on \(\lambda\) and \(\mu\) is satisfied. Suppose now that (1) is replaced by \[\lambda(N - x) + \mu(x - y)^{\frac{1}{2}}, \lambda(N - y) + \mu(y - x)^{\frac{1}{2}}.\] By considering the sign of \(dz/dt\), where \(z = x - y\), discuss whether you would expect \(z\) to become small when \(t\) is large.

On a tropical island, there are only two species of animal. Both species feed on the abundant supplies of vegetation and species \(A\) also feeds on species \(B\), which reproduce at a higher rate than species \(A\). The numbers of individuals, \(N_A\) and \(N_B\), can be regarded as continuous functions of time, satisfying the differential equations \[\frac{dN_A}{dt} = 2N_A + N_B,\] \[\frac{dN_B}{dt} = 4N_B - 2N_A.\] At \(t = 0\), there are \(N_0\) animals in species \(A\) and \(4N_0\) in species \(B\). By obtaining a second order equation, or otherwise, find the numbers in each species for \(0 \leq t < \alpha\), where \(\tan \alpha = -2\) and \(\frac{\pi}{2} < \alpha < \pi\). What do you expect to happen to the species for \(t > \alpha\) ?

A substance \(A\) changes into a substance \(B\) at a rate of \(\alpha\) times the amount of \(A\) instantaneously present; \(B\) changes back into \(A\) at a rate of \(\beta\) times the amount of \(B\) present, and into \(C\) at a rate of \(\gamma\) times the amount of \(B\) present. If initially there is an amount \(X\) of \(A\) and no \(B\) or \(C\), show that the amount of \(C\) after a time \(t\) is $$X\left[1 - \frac{me^{mt} - ne^{nt}}{m-n}\right],$$ where \(m\) and \(n\) are the roots of the equation $$(z + \alpha)(z + \beta + \gamma) - \alpha\beta = 0.$$

It is agreed, in private, by two union leaders that ultimately the pay, \(x\), of a xerographer should be three quarters of the pay, \(y\), of a yogi. Publicly the xerographers' leader claims that, on the grounds of comparability, \(x\) should be governed by the equation \[\dot{x}+2x = 3y+t.\] The chief yogi, being more pessimistic about inflation, makes a claim for \[\dot{y}+2y = 3x+e^{2t}.\] The claims of these powerful unions are met in full. Has the gentleman's agreement been breached?

\(f(x), g(x)\) and \(h(x)\) are functions of \(x\) satisfying the equations \begin{align*} \frac{df}{dx} &= f+g+2h, \\ \frac{dg}{dx} &= 2g+2h, \\ \frac{dh}{dx} &= 7f+8g+24h. \end{align*} Show that \(f-g\) is of the form \(Ae^{\lambda x}\), where \(A\) is a constant. Find all the solutions of the form \[ f=f_0e^{\lambda x}, \quad g=g_0e^{\lambda x}, \quad h=h_0e^{\lambda x}, \] where \(\lambda\) is independent of \(x\), giving the possible values of \(\lambda\) and the corresponding ratios of the constants \(f_0, g_0, h_0\).

The coefficients \(a_1\) and \(a_2\) of the differential equation $$\frac{d^2y}{dx^2} + a_1 \frac{dy}{dx} + a_2y = 0$$ are real numbers. Write down the general real solution of the equation. It is given that every real solution of this equation is bounded for \(x \geq 0\) — that is, if \(f(x)\) is a real solution, there exists a constant \(M\) such that \(|f(x)| \leq M\) for all \(x \geq 0\). Show that \(a_1\) and \(a_2\) must be non-negative.

Find a solution of \(d^2y/dx^2 = y\) for which \(y = 0\) when \(x = l\), and \(y = a\) when \(x = 0\). Assuming a particular integral of the form \(x(A\cosh x + B\sinh x)\), or otherwise, solve $$\frac{d^2y}{dx^2} = y + 2\cosh(l-x),$$ given that \(y = 0\) when \(x = l\), and \(y = a\) when \(x = 0\).

If \(f(x) = e^{-ax}\sin(bx+c)\), \(a > 0\), and \(b > 0\), show that the values of \(x\) for which \(f(x)\) has either a maximum or a minimum form an arithmetic progression with difference \(\pi/b\). Show further that the values of \(f(x)\) at successive maxima form a geometric progression with ratio \(e^{-\pi a/b}\). Find the points of inflexion of \(f(x)\). Describe a physical problem for which \(f(x)\) might be a solution.

A measuring device has an indicator whose position satisfies the equation \[\frac{d^2x}{dt^2} + x = -2k\frac{dx}{dt}.\] Initially, \[x(0) = 1, \left.\frac{dx}{dt}\right|_{t=0} = -k.\] Find the solution \(x(t)\) when \(k > 0\), \(k \neq 1\). Sketch the graph of \(x(t)\) in the two cases \(k = 1\), \(k = 2\). Show that, for those values of \(k\) between 0 and 1 for which \(|x(2)| < 10^{-3}\), we have \(|x(3)| \geq 10^{-3}\).

In the differential equations \begin{align*} \frac{d^2y}{dx^2}+p\frac{dy}{dx}+qy &= 0, \quad (A)\\ \frac{d^2y}{dx^2}+p\frac{dy}{dx}+qy &= f(x), \quad (B) \end{align*} \(p\) and \(q\) are constants. Prove that

1. [(i)] the sum of any two solutions of \((A)\) is a solution of \((A)\);
2. [(ii)] the sum of any solution of \((A)\) and any solution of \((B)\) is a solution of \((B)\).

Show that the second order differential equation \[\frac{d^2y}{dx^2} + b \frac{dy}{dx} + cy = f(x),\] with \(b\) and \(c\) constants, can be written as the pair of equations \[\frac{dp}{dx} - m_1 p = f(x),\] \[\frac{dy}{dx} - m_2 y = p,\] where \(m_1\), \(m_2\) are constants to be determined. Hence, or otherwise, find the general solution of \[\frac{d^2y}{dx^2} + 2 \frac{dy}{dx} + y = e^{-x}\]

In the electric circuit below, the charge \(Q\) on the capacitor \(C\) is related to the applied electromotive force \(E\) by the differential equation $$L \frac{d^2Q}{dt^2} + 2R \frac{dQ}{dt} + Q/C = E,$$ where \(L > R^2C\), and \(E(t) = E_0 \cos \omega t\), where \(E_0\) and \(\omega\) are constants. Show that the current \(I(t) (= dQ/dt)\) is ultimately in phase with \(E(t)\) if and only if \(\omega^2 LC = 1\).

A commercial process is governed by the equation \(\ddot{x} + 3\dot{x} - 4x = 0\). At the first time \(T\) that \(|x(T)| = 100\) the process must be shut down at once. The total profit made on that run is then \(T\) thousand pounds. Unfortunately the initial values \(x(0)\), \(\dot{x}(0)\) cannot be controlled exactly and all that can be done is to ensure that \(|x(0)|, |\dot{x}(0)| \leq 1\). Estimate the minimum value of \(T\). A machine is available which would improve the accuracy with which \(x(0)\), \(\dot{x}(0)\) are controlled in such a way that \(|x(0)|, |\dot{x}(0)| \leq 1/10\) but this would cost an extra \(S\) thousand pounds per run. Would you advise the use of such a machine if \(S = 1\), if \(S = 4\) or if \(S = 10\) and why? (Clearly you have not got as much information as you might want in ideal circumstances but you do have enough information to come to a sensible decision.) [log\(_e\)10 = 2.30256.]

Prove that the solution of the differential equation \(\frac{dy}{dx} + ay = f(x),\) where \(a\) is constant, is \(y = y_0 e^{-ax} + \int_0^x f(u) e^{a(u-x)} du,\) where \(y_0 = y(0)\). Hence, or otherwise, solve \(\frac{d^2 y}{dx^2} + (a+b) \frac{dy}{dx} + aby = \begin{cases} 1, & (0 < x < 1) \\ 0, & (x > 1), \end{cases}\) where \(a\) and \(b\) are constants \((a \neq b)\), given that \(y = 0\), \(dy/dx = 0\) for \(x = 0\), and that \(y\) and \(dy/dx\) are continuous.

Prove, by substitution or otherwise, that the solution of the differential equation \(y'' + n^2y = f(x)\) with the conditions \(y(0) = y'(0) = 0\) is $$y(x) = \int_0^x \frac{1}{n}\sin(x-t)f(t)dt.$$ Solve the problem in the particular case \(f(x) = \sin nx\).

Solve the differential equation \[ \frac{d^2y}{dx^2} + 6\frac{dy}{dx} + 9y = 0 \] with the conditions \(y=2\) and \(\dfrac{dy}{dx}=-5\) at \(x=0\). Hence, or otherwise, find \(u_n\), given that \[ u_{n+2}+6u_{n+1}+9u_n = 0 \] for \(n\ge 0\), and \(u_0=2, u_1=-5\).

The sequence \(u_0, u_1, u_2, \dots\) is defined by \[ u_0 = 1, \quad u_1 = 2, \quad u_n = 2u_{n-1} - 5u_{n-2} \quad (n=2, 3, \dots). \] Obtain the general expression for \(u_n\). \[ u_n = \frac{1}{2} 5^{\frac{1}{2}(n+1)} \sin(n+1)\theta, \] where \(\theta\) is the acute angle defined by \(\tan\theta=2\).

Defining \(\cos x\) and \(\sin x\) as solutions of the differential equation \(\dfrac{d^2y}{dx^2} + y = 0\) with suitable given values of \(y\) and \(\dfrac{dy}{dx}\) at \(x=0\), and defining \(\dfrac{\pi}{2}\) as the least root of the equation \(\cos x = 0\), obtain, without the use of infinite series, the principal properties of \(\cos x\) and \(\sin x\).

Show that the differential equation \[x^3y'' + (x - 2)(xy' - y) = 0\] has a solution proportional to \(x^\alpha\) for some \(\alpha\). By making the substitution \(y = x^\tau\), or otherwise, find the general solution of this equation.

A second order linear differential equation for \(y\) is given by \[\frac{d^2y}{dx^2} + P(x)\frac{dy}{dx} + Q(x)y = 0.\] Make the substitution \(y = uv\), where \(u\) and \(v\) are functions of \(x\), and obtain a differential equation for \(u\) in terms of \(P\), \(Q\) and \(v\). What first order differential equation must \(v\) satisfy in order to eliminate the term in \(du/dx\)? Hence, or otherwise, solve \[\frac{d^2y}{dx^2} - \frac{1}{x}\frac{dy}{dx} + \left(1+\frac{3}{4x^2}\right) y = 0\] for \(x > 0\) when \(y\) satisfies the conditions \[y = (\frac{1}{2}\pi)^{\frac{1}{2}} \text{ at } x = \frac{1}{2}\pi,\] \[y=0 \text{ at } x = \pi.\]

Using the substitution \(x = e^t\) or otherwise solve \begin{align} x^2\frac{d^2y}{dx^2} - 4x\frac{dy}{dx} + 6y = 6\ln x - 5 \text{ for } x > 0 \end{align} given \(y(1) = 0\) and \(y(e) = 2\).

Show that \[x^2 y'' + 2x(x+2)y' + 2(x+1)^2 y = e^{-x}\] can be transformed to a second order linear differential equation with constant coefficients by using the substitution \(y = zx^n\) for a suitable value of \(n\). Find the general solution of the original equation.

Find the general solution, for \(x > 0\), of the differential equation \[x^2y'' - 4xy' + 6y = 0\] by searching for solutions of the form \(y = x^{\lambda}\). Find, similarly, a particular solution of the equation \[x^2y'' - 4xy' + 6y = Cx^k\] provided \(k \neq 2\), \(k \neq 3\). Hence find the general solution of (1), and the solution that satisfies \begin{align} y(1) = 1, \quad y'(1) = 0. \end{align} Write \(k = 3+\varepsilon\), and obtain a tentative solution to (1) and (2) in the exceptional case \(k = 3\) by carefully taking the limit of your last result as \(\varepsilon \to 0\). Verify that it does indeed satisfy both the differential equation (1) and conditions (2).

(i) Solve the differential equation $$\frac{d^2y}{dx^2} - \frac{dy}{dx} = e^x$$ subject to the conditions that \(y = d^2y/dx^2 = 2\) when \(x = 0\). (ii) Find the general solution of the differential equation $$\frac{1-x^2}{y^2}\frac{dy}{dx} + \frac{x}{y} + \sin^{-1}x = 0 \quad (-1 \leq x \leq 1)$$ by taking as a new variable a suitably chosen power of \(y\), or otherwise.

(i) By the substitution \(y = e^x\) or otherwise, solve the differential equation \begin{align} yy'' = y'^2 + yy'. \end{align} (ii) Find all the solutions of \(y'^2 = x^2\) for which \(y = 0\) at \(x = 0\).

Verify that the differential equation $$y'' = (x^2 - 1)y,$$ where dashes denote differentiations with respect to \(x\), is satisfied by \(y = e^{-\frac{1}{2}x^2}\). By writing \(y = ue^{-\frac{1}{2}x^2}\) and forming the differential equation for \(u\), or otherwise, obtain an expression for the general solution of the equation in \(y\). Show that the solution for which \(y = a\) and \(y' = b\) when \(x = 0\) may be written in the form $$y = ae^{-\frac{1}{2}x^2} + b \int_0^x e^{-\frac{1}{2}t^2} dt.$$

(i) Find the solution of the differential equation \(x dy/dx = 3y\) that takes the value 2 when \(x = 1\). (ii) Find a differential equation satisfied by the function \(g(x) = e^{-xf(x)}\) whenever \(f(x)\) is a function that satisfies the differential equation \(d^2f/dx^2 + xdf/dx - f = 0\).

1. [(i)] Using the substitution \(x = e^t\), or otherwise, solve the differential equation $$x^2 \frac{d^2y}{dx^2} + 3x \frac{dy}{dx} + y = 0.$$
2. [(ii)] Solve $$(x^2 - 1) \frac{dy}{dx} + 2(x + 3)y = 8x(x + 1)^6.$$

Given that any solution of the differential equation \[u'' + u = 0\] (where a dash denotes differentiation with respect to \(x\)) must be of the form \(A \sin x + B \cos x\), where \(A\) and \(B\) are constants, and that the function \(f(x)\) satisfies the differential equation \[y'' + \frac{2}{x} y' + y = 0,\] calculate \(\int_a^b xf(x) dx\) in terms of \(f(a)\) and \(f(b)\), indicating any special cases that may arise. Calculate \(\int_a^b x^2f(x) dx\) in terms of \(f'(a)\) and \(f'(b)\).

Show that, if \(u=x^2\), \[ x\frac{d^2f(x)}{dx^2} - \frac{df(x)}{dx} = 4x^3 \frac{d^2f(x)}{du^2}. \] Find a function satisfying the equation \[ x\frac{d^2f(x)}{dx^2} - \frac{df(x)}{dx} - 4x^3 f(x) = 0 \] and containing two arbitrary constants.

(i) Using the substitution \(y=xz\), or otherwise, obtain the general solution of the differential equation \[ x\frac{dy}{dx} = y + 2x\sqrt{(x^2+y^2)}. \] (ii) Find the solution of \[ \frac{d^2y}{dx^2}+\frac{dy}{dx}-2y=2 \] such that \(y=0\) and \(\dfrac{dy}{dx}=0\) when \(x=0\).

If \(y=u\) is known to be a solution of the differential equation \[ py''+qy'+ry=0, \] where \(p, q\) and \(r\) are given functions of \(x\), show that the solution of \[ py''+qy'+ry=s, \] where \(s\) is a given function of \(x\), can be reduced by means of the substitution \(y=uv\) to the solution of a first order linear differential equation for \(v'\). Given that \(y=(1+x^2)^{-1/2}\) is a solution of \[ (1+x^2)y''+2xy'+y/(1+x^2)=0, \] obtain the solution of \[ (1+x^2)y''+2xy'+y/(1+x^2)=x/(1+x^2) \] which vanishes at \(x=0\) and \(x=1\).

Find the solution of the equation \[ (x+2)^2 y'' - 2(x+2) y' + 2y = 0 \] which represents a curve touching the parabola \(y = x^2 + 1\) at the point \((1, 2)\).

(i) If \[ \frac{d^2y}{dx^2} + \frac{1}{x}\frac{dy}{dx} + y = 0, \] shew that \[ x^2\frac{d^3y}{dy^3} - \left(\frac{dx}{dy}\right)^2 - xy\left(\frac{dx}{dy}\right)^3 = 0. \] (ii) If \[ \frac{d^2y}{dx^2} + \left(\frac{l}{x-a}+\frac{m}{x-b}+\frac{n}{x-c}\right)\frac{dy}{dx} = 0, \] where \(l+m+n=2\), shew that \[ \frac{d^2y}{dX^2} + \left(\frac{l}{X-A}+\frac{m}{X-B}+\frac{n}{X-C}\right)\frac{dY}{dX} = 0, \] where \[ x = \frac{\alpha X + \beta}{\gamma X + \delta}, \quad a = \frac{\alpha A + \beta}{\gamma A + \delta} \text{ etc., and } \alpha\delta - \beta\gamma = 1. \]

If \(y = \sin(x \sin^{-1} x)\), prove \[(1-x^2) y'' - xy' + x^2 y = 0,\] where \(y'\) and \(y''\) represent the first and second derivatives of \(y\). Prove that the Maclaurin series of \(y\) is \[x\left[x + \frac{(1^2-x^2)x^3}{3!} + \frac{(3^2-x^2)(1^2-x^2)x^5}{5!} + \ldots\right].\]

Verify that \(y = \cos x \cosh x\) satisfies the relation $$\frac{d^2y}{dx^2} = -4y.$$ Hence or otherwise show that $$y = 1 + \sum_{n=1}^{\infty} (-4)^n \frac{x^{4n}}{(4n)!}$$ \([\cosh x = \frac{1}{2}(e^x + e^{-x})]\)

Show that the function \(y = \sin^2(m\sin^{-1}x)\) satisfies the differential equation \[(1-x^2)y'' = xy' + 2m^2(1-2y).\] Show that, at \(x = 0\), \[y^{(n+2)} = (n^2 - 4m^2)y^{(n)} \quad (n \geq 1)\] and derive the MacLaurin series for \(y\).

Show that, if \(y = \tanh^{-1} x\), then $$(1-x^2) \frac{d^2 y}{dx^2} - 2x \frac{dy}{dx} = 0.$$ Hence determine the value of the \(n\)th derivative of \(y\) at \(x = 0\). [You may use the theorem of Leibnitz.]

The polynomial \(T_n(x)\), where \(n\) is a non-negative integer, satisfies $$(1-x^2) \frac{d^2 T_n}{dx^2} - x \frac{dT_n}{dx} + n^2 T_n = 0;$$ $$T_0(1) = 1; \quad T_n(x) = (-1)^n T_n(-x).$$ By substituting \(x = \cos \theta\) and solving the transformed equation, obtain \(T_n(x)\) in simple form as a function of \(\theta\) and hence show that $$T_0(x) = 1, \quad T_1(x) = x,$$ and that $$T_{n+1}(x) = 2x T_n(x) - T_{n-1}(x) \quad \text{for } n = 1, 2, 3, \ldots$$

(i) Show that the general solution of \[(1 + ax) w'(x) + \frac{1}{2}aw(x) = 0\] is \[w(x) = A(1 + ax)^{-\frac{1}{2}}, \quad A \text{ constant}.\] (ii) Find polynomials \(p(x)\), \(q(x)\), \(r(x)\), \(s(x)\) such that \[(1-x^2)y''(x) - xy'(x) + \frac{1}{4}y(x) = \left[p(x)\frac{d}{dx}+q(x)\right]\left[r(x)\frac{d}{dx}+s(x)\right]y(x),\] for all \(y(x)\). (There are two possible answers; only one is required.) Hence, or otherwise, find the general solution of \[(1-x^2)y''(x) - xy'(x) + \frac{1}{4}y(x) = 0.\] [You are reminded that \[\left[f(x)\frac{d}{dx}+g(x)\right]h(x)\] is defined to be \(f(x)h'(x)+g(x)h(x)\) for any functions \(f(x)\), \(g(x)\) and \(h(x)\) of \(x\). You may assume if necessary that \[\int(1-ax)^{-\frac{1}{2}}(1+ax)^{-\frac{1}{2}}dx = a^{-1}(1+ax)^{\frac{1}{2}}(1-ax)^{-\frac{1}{2}}.\]

For the equation \[2x \frac{d^2y}{dx^2} + \frac{dy}{dx} + \frac{1}{2}y = 0, \quad x > 0,\] look for a solution in the form \[y = \sum_{r=0}^{\infty} a_r x^{r+\frac{1}{2}},\] and find a relation between successive coefficients in the sequence \(\{a_r\}\). Hence express this solution in terms of simple functions.

Verify that the differential equation $$x^2 y'' + [(n + \frac{1}{2})x + \frac{1}{2}](1-x^2)]y = 0,$$ where \(n\) is a positive integer, has the solution $$y = x^{\frac{1}{2}} e^{-\frac{1}{2}x} L_n(x),$$ where \(L_n(x)\) is the polynomial of degree \(n\) given by $$L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n}(x^n e^{-x}).$$

If \(y = (x^2-1)^n\), where \(n\) is a positive integer, prove that $$(1-x^2)\frac{dy}{dx} + 2nxy = 0.$$ By differentiating this equation \((n+1)\) times and using Leibniz' theorem, or otherwise, show that the function \(p_n(x)\), defined by $$p_n(x) = \frac{d^n}{dx^n}(x^2-1)^n,$$ satisfies the equation $$(1-x^2)\frac{d^2p_n}{dx^2} - 2x\frac{dp_n}{dx} + n(n+1)p_n = 0.$$ Show also that $$p_n(1) = (-1)^n p_n(-1) = 2^n n!$$

If \(y = \sin(k \sin^{-1} x)\), show that $$(1 - x^2) \frac{d^2 y}{dx^2} - x \frac{dy}{dx} + k^2 y = 0.$$ Assuming that \(y\) can be expanded in the form \(\sum_{n=0}^{\infty} a_n x^n\), find the coefficients \(a_n\). When does the expansion reduce to a polynomial?

Show that \(y = (x + (x^2 + 1)^{1/2})^k\) satisfies the differential equation \((x^2 + 1)y'' + xy' - k^2y = 0,\) Derive an equation connecting \(y^{(n)}(x)\), \(y^{(n+1)}(x)\) and \(y^{(n+2)}(x)\). Hence show that, if \(k\) is an integer, then \(y^{(k+1)}(x) = A(x^2 + 1)^{-k-\frac{1}{2}},\) where \(A\) is a constant, and find \(A\).

Obtain Leibniz's formula for the \(n\)th derivative of the product \(u(x)v(x)\). If \(y = \frac{1}{2}(\sinh^{-1} x)^2\), prove that $$(1+x^2)y'' + xy' - 1 = 0,$$ and deduce the value of \(y^{(n)}\) (the \(n\)th derivative of \(y\)) for \(x = 0\). Obtain the Maclaurin expansion for the function.

Suppose that \(u(x)\) and \(v(x)\) are polynomials in \(x\) of degrees \(n\) and \(n-1\) respectively, and that they satisfy identically the relation $$\sqrt{1 - [u(x)]^2} = v(x) \sqrt{1 - x^2}.$$ Prove that \(du/dx = \pm nv(x)\), and deduce that \(u(x)\) satisfies the differential equation $$(1 - x^2) \frac{d^2u}{dx^2} - x \frac{du}{dx} + n^2u = 0.$$ By making the change of variable \(x = \cos t\), or otherwise, deduce that $$u(x) = \pm \cos(n \cos^{-1} x), \quad v(x) = \pm \frac{\sin(n \cos^{-1} x)}{\sqrt{1 - x^2}}.$$

Assuming that the equation \[ x\frac{d^2y}{dx^2} + \frac{dy}{dx} - m^2xy = 0 \] is satisfied by a solution of the form \[ y = \sum_{r=0}^{\infty} c_r x^{\alpha+r}, \] where \(c_0\) is not zero, find the value of \(\alpha\), prove that \(c_{2r+1}=0\) and that \[ 4r^2c_{2r} = m^2c_{2r-2}, \] and write down the first four terms of the series.

Show that the equation \[ r\frac{\partial}{\partial r}\left(r\frac{\partial u}{\partial r}\right) + \frac{\partial^2 u}{\partial \theta^2} = 0 \] can be satisfied (identically) by taking \(u = r^n P\), where \(n\) is any positive integer and P is a certain polynomial of degree \(n\) in \(\cos\theta\), and that P must be a constant multiple of \[ 1 - \frac{n^2}{2!}\cos^2\theta + \frac{n^2(n^2-2^2)}{4!}\cos^4\theta - \frac{n^2(n^2-2^2)(n^2-4^2)}{6!}\cos^6\theta + \dots \] or of \[ \cos\theta - \frac{n^2-1^2}{3!}\cos^3\theta + \frac{(n^2-1^2)(n^2-3^2)}{5!}\cos^5\theta - \dots, \] according as \(n\) is even or odd, the summation continuing as far as the term in \(\cos^n\theta\). \par Verify that \(u=r^n \cos n\theta\) satisfies the above equation, and hence, or otherwise, express \(\cos n\theta\) as a polynomial in \(\cos\theta\).

Shew that if \(u=(1-x^2)^n\), \[ u'(1-x^2) + 2nxu = 0; \] and by differentiating this equation \(n+1\) times, shew that \[ (1-x^2)\frac{d^2P}{dx^2} - 2x\frac{dP}{dx} + n(n+1)P=0, \] where \[ P(x) = \left(\frac{d}{dx}\right)^n \{(1-x^2)^n\}. \] By proceeding similarly with \[ v=e^{-x^2}, \] shew that if \[ H(x)e^{-x^2} = \left(\frac{d}{dx}\right)^n e^{-x^2}, \] then \[ \frac{d^2H}{dx^2} - 2x\frac{dH}{dx} + 2nH = 0. \]

The function \(y=\sin x\) satisfies the differential equation \(\frac{d^2y}{dx^2}+y=0\). Assuming that \(\sin x\) can be expanded as a series in ascending powers of \(x\), deduce the series. Prove that, if \(y=\sin(n\sin^{-1}x)\), then \[ (1-x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} + n^2y = 0. \] Deduce the expansion of \(y\) as a series in ascending powers of \(x\), when such an expansion is possible. Discuss what happens when \(n\) is an integer.

Having given that \[ x(1-x)\frac{d^2y}{dx^2} - (3-10x)\frac{dy}{dx} - 24y = 0, \] prove that \[ x(1-x)\frac{d^{n+2}y}{dx^{n+2}} + \{n-3-2(n-5)x\}\frac{d^{n+1}y}{dx^{n+1}} - (n-3)(n-8)\frac{d^ny}{dx^n} = 0. \] Hence find, by Maclaurin's Theorem, that value of \(y\) which is zero when \(x=0\), and is such that its fourth differential coefficient is unity when \(x=0\).

Obtain, by the method of solution in series (using series of ascending powers of \(x\)), the complete primitive of the differential equation \[ (1-x^2)\frac{d^2y}{dx^2} + 2x\frac{dy}{dx} - 6y = 0. \] Taking \(x\) as real, examine carefully the range of validity of your solution.

Solve in series the equation \[ x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}+k^2y=0, \] giving special consideration to odd integral values of \(h\). If \(h\) is an even integer \(2m\), prove that the solution is \[ y = \left(x\frac{d}{dx}\right)^m (A\cos kx + B\sin kx). \] % Note: The question seems to have a typo, mixing h and k. I will assume k in the equation, and h in the condition.

Obtain the complete solution of the equation \[ x\frac{d^2y}{dx^2} + (\gamma+1)\frac{dy}{dx}-xy=0, \] in the form of series of ascending powers of \(x\), \(\gamma\) being any real number.

Shew that the integral \[ \int_0^\pi \cos(x\sin\phi-n\phi)d\phi \] is a solution of the equation \[ x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}+(x^2-n^2)y=0, \] if and only if \(n\) is an integer.

Obtain the solution of the equation \[ \frac{d^2 y}{dx^2} + \frac{1}{x}\frac{dy}{dx} + \left(1-\frac{n^2}{x^2}\right)y=0, \] where \(n\) is not an integer, in the form \[ A J_n(x) + B J_{-n}(x), \] where \[ J_n(x) = (\frac{1}{2}x)^n \sum_{v=0}^\infty \frac{(-1)^v (\frac{1}{2}x)^{2v}}{v!\Gamma(n+1+v)}. \] Show that \[ \frac{i e^{\frac{1}{2}(n+1)\pi i}}{\sin n\pi}\{e^{-n\pi i}J_n(ix) - J_{-n}(ix)\} \] is real for all real values of \(x\).

A particle moves in the \(x\)-\(y\) plane with the following equations of motion: \begin{align} \ddot{x} = y\dot{d}, \quad \ddot{y} = c - x\dot{d} \end{align} where \(c\) and \(d\) are constant. Show that the quantity \(\frac{1}{2}(\dot{x}^2 + \dot{y}^2) - cy\) is constant. At \(t = 0\) the initial conditions are \(x = 0\), \(y = 0\), \(\dot{x} = Q\) and \(\dot{y} = 0\). Show that the motion is along the \(x\)-axis if and only if \(Q\) has a certain value, which is to be determined.

The displacement \(x\) of a simple harmonic oscillator satisfies the differential equation $$\frac{d^2x}{dt^2} + \omega^2 x = 0.$$ Denoting by \(\mathbf{M}\) and \(\mathbf{M}'\) the functions \(x = a \cos \omega(t - \alpha)\) and \(x = a' \cos \omega(t - \alpha')\), show that the functions \(\mathbf{M}\) and \(\mathbf{M} + \mathbf{M}'\), defined by $$x = \lambda a \cos \omega(t - \alpha) \text{ and } x = a \cos \omega(t - \alpha) + a' \cos \omega(t - \alpha')$$ respectively, satisfy the differential equation. Prove that the solutions of the equation form a vector space. What is its dimension? The scalar product \(\mathbf{M} \cdot \mathbf{M}'\) is defined as \(aa' \cos \omega(\alpha - \alpha')\). Prove that $$(\mathbf{M} + \mathbf{M}') \cdot \mathbf{M}'' = \mathbf{M} \cdot \mathbf{M}'' + \mathbf{M}' \cdot \mathbf{M}''.$$ Writing \(\mathbf{M} = m_1 \mathbf{e}_1 + m_2 \mathbf{e}_2\), where \(\mathbf{e}_1\) and \(\mathbf{e}_2\) denote the functions \(x = \cos \omega t\) and \(x = \sin \omega t\) respectively, express \(\mathbf{M} \cdot \mathbf{M}'\) in terms of the components \((m_1, m_2)\) and \((m_1', m_2')\). If the time \(t\) is measured from the instant \(t = 0\), so that \(t = t - 0\), and \(\mathbf{M}\) is written as \(\tilde{m}_1 \tilde{\mathbf{e}}_1 + \tilde{m}_2 \tilde{\mathbf{e}}_2\), where \(\tilde{\mathbf{e}}_1\) and \(\tilde{\mathbf{e}}_2\) are the functions \(x = \cos \omega t\) and \(x = \sin \omega t\) respectively, find the relation between the components \((\tilde{m}_1, \tilde{m}_2)\) and \((m_1, m_2)\). What is the nature of the matrix involved?

A rigid sphere of density \(\rho\) and radius \(a\) is released from rest when its centre is at a height \(h\) above a large horizontal rubber sheet. Assuming that the part of the sheet outside the circle of contact with the sphere remains at the same horizontal level throughout the ensuing motion and that the tension \(T\) in the sheet is constant, show that if the vertical penetration \(x\) into the sheet is not greater than \(a\), then \(x\) satisfies the equation \begin{equation*} \frac{d^2x}{dt^2} = \frac{3T}{2\rho a^4}(x^2-2ax) + g, \end{equation*} where \(g\) is the acceleration due to gravity. Hence show that the sphere will penetrate to a depth \(ka\) (\(k < 1\)) if the tension is given by \begin{equation*} T = \frac{2g\rho a(h+ka-a)}{3k^2-k^3}. \end{equation*} [It may be assumed that the force per unit area exerted by the sheet at any point of contact is normal to it and of magnitude \(2T/a\).]

Find the general solution of the differential equation \[y\frac{d^2y}{dx^2} = y\frac{dy}{dx} + \left(\frac{dy}{dx}\right)^2.\]

Let \([x]\) denote the integer part of \(x\). Sketch the graph of \(\left[1 + \frac{x}{\pi}\right]\) for \(x \geq 0\). Find the solution \(y(x)\) for \(x \geq 0\) of the differential equation \[\frac{d^2y}{dx^2} + \left[1 + \frac{x}{\pi}\right]^2 y = 0,\] subject to the conditions \(y = 0\) and \(dy/dx = 1\) at \(x = 0\). You may assume that \(y\) and \(dy/dx\) are continuous everywhere. Show that \[\int_0^{\infty} y^2dx = \frac{\pi}{2} \sum_{n=1}^{\infty} \frac{1}{n^2}.\]

Find the most general solution of the 'differential equation' \[f'(x) = \lambda f(1-x),\] where \(\lambda\) is a real constant.

The function \(f\) satisfies \(f(-y) = -f(y)\) and is defined as follows for \(y \geq 0\). \[f(y) = y \quad \text{if } 0 \leq y \leq 1,\] \[f(y) = 2-y \quad \text{if } 1 \leq y \leq 2,\] \[f(y) = 0 \quad \text{if } y \geq 2.\] Solve the differential equation \(y'' + f(y) = 0\) with initial conditions \(y(0) = 0\), \(y'(0) = c\). Sketch the solutions corresponding to initial conditions \(y(0) = 0\), \(y'(0) = c\) for \(c = 1\), \(c = \frac{4}{3}\) and \(c = \frac{8}{3}\).

Find the relation that exists between \(P(x)\) and \(Q(y)\) if the equation \[\frac{d^2y}{dx^2} + P\frac{dy}{dx} + Qy = 0\] has two non-zero solutions one of which is the square of the other. Show that the condition \[y'' - \left(3x + \frac{1}{x}\right)y' + 2x^2y = 0,\] and hence obtain the complete solution of this equation.

Prove that if \[ax^2+2hxy+by^2+2gx+2fy+c=0,\] then \[\frac{d^2y}{dx^2} = \frac{abc+2fgh-af^2-bg^2-ch^2}{(hx+by+f)^3}.\]

Prove Liouville's theorem, that a bounded function regular at every point is necessarily a constant. Prove that \[ \wp(u-a) - \wp(u+a) = \frac{\wp'(u)\wp'(a)}{(\wp(u)-\wp(a))^2}. \]