Problems

State and prove a theorem on the effect on the value of a determinant of interchanging two rows or two columns. If \(a_1, a_2, \dots, a_n\), are in arithmetical progression, shew that the ratio of \[ \begin{vmatrix} a_1, & a_2, & a_3, & \dots & a_n \\ a_n, & a_1, & a_2, & \dots & a_{n-1} \\ a_{n-1}, & a_n, & a_1, & \dots & a_{n-2} \\ \vdots & & & & \vdots \\ a_2, & a_3, & a_4, & \dots & a_1 \end{vmatrix} \] to the sum of these \(n\) quantities is expressible solely in terms of \(n\) and the common difference.

(i) Prove the formula \(\frac{1}{r}\frac{dp}{dr}\) for the curvature at a point of a plane curve. (ii) Investigate the curvature at the point \((1,2)\) of the curve \[ (y-2)^2 = x(x-1)^2. \]

Shew that the number of real roots of the algebraic equation \(f(x)=0\) cannot exceed by more than unity that of its derived equation \(f'(x)=0\). Find the necessary and sufficient condition for the cubic equation \(x^3+3ax+b=0\) to have three real roots. \(a\) and \(b\) are real.

Determine the surface area and volume of the solid figure obtained by revolving the curve \(r=a(1+2\cos\theta)\) about its axis of symmetry.

State and prove Leibnitz' Theorem on the \(n\)th differential coefficient of the product of two functions. If \(y_n = \frac{d^n}{dx^n}(x^n f(x))\), where \(f(x)\) is a differentiable function of \(x\), shew that

1. [(i)] \(y_n = x\frac{d}{dx}y_{n-1} + n y_{n-1}\),
2. [(ii)] \(y_n = x^{n-r}\left(\frac{d}{dx}\right)^{n-r} y_r + n(n-r)x^{n-r-1}\left(\frac{d}{dx}\right)^{n-r-1}y_r + \dots\) \[ + \frac{n!(n-r)!}{s!(n-s)!(n-r-s)!} x^{n-r-s} \left(\frac{d}{dx}\right)^{n-r-s} y_r + \dots + \frac{n!}{r!}y_r. \]

Find the integral \[ \int (1-x^2)^{\frac{3}{2}} dx, \] and evaluate \[ \int_2^3 \frac{dx}{[(x-1)(3-x)]^{\frac{3}{2}}} \quad \text{and} \quad \int_{a+b}^{a+2b} \frac{dx}{2a+b - \sqrt{(x-a)(x-b)}}. \]

Describe some method of investigating the behaviour of the function \(\frac{f(x)}{\phi(x)}\) as \(x\) tends to \(a\), where \(f(a)=\phi(a)=0\). Determine the limiting values as \(x \to 0\) of:

1. [(i)] \(\frac{1}{x} - \frac{1}{x^2}\log(1+x)\),
2. [(ii)] \(\left(\frac{1}{x}\right)^{\sin x}\).

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 \(-1 < x < 1\).

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 that the locus of a point \(P\), such that the tangents from \(P\) to the two conics \[ S \equiv x^2+y^2+z^2=0, \quad S' \equiv ax^2+by^2+cz^2=0 \] form a harmonic pencil, is the conic \[ F \equiv a(b+c)x^2 + b(c+a)y^2 + c(a+b)z^2=0. \] Shew that \(\lambda^2 S + \lambda F + abc S' = 0\), where \(\lambda\) is a parameter, is the equation of a conic touching the four common tangents of \(S\) and \(S'\). Hence shew that the equation of the four common tangents is \(F^2 - 4abcSS' = 0\).