Let \(k\), \(n\) be integers, \(k \geq 1\), \(n \geq 1\). Show that if \(n^2\) divides \((n+1)^k - 1\) then \(n\) divides \(k\), and deduce that if \((n+1)^k - 1 = n!\) and \(n \geq 6\), then \(n\) divides \(k\). [Hint: is \(n\) odd or even?] Hence find all pairs \((n, k)\) of positive integers such that \((n+1)^k - 1 = n!\)
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.
Let \(x_1,\ldots,x_n\) be distinct real numbers. Write down an expression for a polynomial \(e_k\), of degree \(n-1\), such that \begin{align*} e_k(x_l) = \begin{cases} 1 & (l = k),\\ 0 & (l \neq k). \end{cases} \end{align*} Given real numbers \(\alpha_1,\ldots,\alpha_n\), find a polynomial \(p\), of degree at most \(n-1\), for which \(p(x_k) = \alpha_k\) \((k = 1,\ldots,n)\). Show further, given numbers \(\beta_1,\ldots,\beta_n\), that there is a polynomial \(q\), of degree at most \(2n-1\), such that both \(q(x_k) = \alpha_k\) and \(q'(x_k) = \beta_k\) \((k = 1,\ldots,n)\). [It is sufficient to prove the existence of \(q\); you are not expected to find its coefficients in an explicit form. In the last part of the question, you may find it helpful firstly to find a polynomial \(\eta_k\) such that \begin{align*} \eta_k(x_l) = 0 \text{ } (l = 1,\ldots,n), \text{ } \eta'_k(x_k) = 1, \text{ } \eta'_k(x_l) = 0 \text{ } (l \neq k, l = 1,\ldots,n).] \end{align*}
\(C\) is a circle with centre \(O\) and radius \(R\), \(C'\) a circle with centre \(O'\) and radius \(r\) (\(< \frac{1}{2}R\)), and \(C'\) passes through \(O\). From a point \(T\) of \(C'\) a tangent is drawn meeting \(C\) at \(E\), \(F\). If the angle \(TO'O\) equals \(\alpha\), show that \begin{align*} TE^2 + TF^2 = 2[R^2 + 2r^2\cos\alpha(1-\cos\alpha)], \end{align*} and deduce that \begin{align*} 2R^2 - 8r^2 \leq TE^2 + TF^2 \leq 2R^2 + r^2. \end{align*}
\(P\) is a fixed point of a parabola, and \(l_1\), \(l_2\) are lines at right angles to each other passing through \(P\); \(l_1\), \(l_2\) meet the parabola again at \(P_1\) and \(P_2\) respectively. Show that \(P_1P_2\) passes through a fixed point \(P'\). For another point \(Q\) of the parabola, \(Q'\) is similarly defined. If \(P'Q'\) cuts the parabola at \(R\), \(S\), show that \(P\), \(Q\), \(R\), \(S\) lie on the circle with \(RS\) as diameter.
(i) Sketch the graph of \([e^x]\) for \(x \geq 0\); here \([y]\) means the integer part of \(y\). Evaluate \begin{align*} I = \int_0^{\log_e (n+1)} [e^x]dx \end{align*} and show that \(e^I = (n+1)^n/n!\). (ii) If \(f(x) = xg(\sin x)\), show that \begin{align*} f(x) + f(\pi-x) = \pi g(\sin x), \end{align*} and hence (or otherwise) that \begin{align*} \int_0^{\pi} \frac{x\sin x}{2-\sin^2 x}dx = \frac{\pi}{2}\int_{-1}^{1}\frac{du}{u^2+1} = \frac{\pi^2}{4}. \end{align*}
Let \begin{align*} I_n = \int_0^{\pi/4} \tan^n x dx. \end{align*} (i) Show that for \(n \geq 2\) \begin{align*} I_n = \frac{1}{n-1}I_{n-2} \end{align*} (ii) Show that \(\tan x \leq \frac{4x}{\pi}\) for \(0 \leq x \leq \pi/4\), and hence show that \(I_n \to 0\) as \(n\to\infty\). (iii) Hence show that \begin{align*} \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{2n-1} = \frac{\pi}{4} \text{ (Gregory's series)} \end{align*} and \begin{align*} \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{2n} = \frac{1}{2}\log_e 2. \end{align*}
For a curve defined parametrically by functions \(x(t)\), \(y(t)\), the radius of curvature is given by \begin{align*} \rho = \frac{(\dot{x}^2+\dot{y}^2)^{\frac{3}{2}}}{(\dot{x}\ddot{y}-\ddot{x}\dot{y})}. \end{align*} An ellipse is given by \begin{align*} x = a\cos t, \text{ } y = b\sin t. \end{align*} Find the parametric equations of the centre of curvature of the ellipse, and sketch its locus. Describe the shape carefully near the points corresponding to \(t = 0\), \(\pi/2\), \(\pi\), \(3\pi/2\).
Three points \(A\), \(B\) and \(C\) are placed independently and at random on the circumference of a circle (so that the angles made by the radii through each of \(A\), \(B\) and \(C\) with any fixed reference direction are uniformly distributed on \([0, 2\pi)\)). Show that the probability that the centre of the circle lies within the triangle \(ABC\) is \(\frac{1}{4}\).
Let \(X_1, X_2, ..., X_n\) be a random sample of size \(n\) drawn from a normal distribution with variance 1 and with unknown mean \(\beta\). Show how to use the sample mean to construct an interval which contains \(\beta\) with probability approximately 0.95. Now suppose that \(X_1, X_2, ..., X_n\) are not necessarily normally distributed, but merely that their common unknown distribution is continuous (so that \(P[X_i = x] = 0\) for any real \(x\)). Show that, if \(q_{\alpha}\) is the \(\alpha\)-quantile of the unknown distribution (i.e. if \(q_{\alpha}\) is such that \(P[X_i \leq q_{\alpha}] = \alpha\)), and if \(X_{(1)}, X_{(2)}, ..., X_{(n)}\) denotes the sample \(X_1, X_2, ..., X_n\) arranged in ascending order, then \(P[X_{(r)} < q_{\alpha} < X_{(r+1)}] = \binom{n}{r}\alpha^r(1-\alpha)^{n-r}\). Use this fact to construct, in the case when \(n = 6\), an interval within which the median \(q_{1/2}\) of the distribution will lie with probability at least 0.95. Evaluate both intervals when \((X_{(1)}, X_{(2)}, ..., X_{(6)}) = (-0.92, -0.77, 0.41, 0.47, 0.48, 0.99)\).