Let $$I_{m,n} = \int_0^{\infty} \frac{x^m dx}{(1 + x^2)^n},$$ where \(m\), \(n\) are non-negative integers and \(3n > m + 1\). By first evaluating \(I_{0,n+1}\), show that Find a formula for \(I_{2n,n+1}\). $$I_{0,n+1} = \frac{2\pi}{3^{n+1}\sqrt{3}} \prod_{k=1}^{n} \left( 3 - \frac{1}{k} \right).$$
By taking \(xy, x + y\) as new variables, or otherwise, find how many values of \(x\) and \(y\) are for which the equations \begin{align} x^2 + y^2 &= 1 + x^2, \quad x^2 + y^3 = 1 + x^3 \end{align} have less than six distinct solutions.
If \(A + B + C = \frac{\pi}{2}\), prove that \begin{align} (\sin A + \cos A)(\sin B + \cos B)(\sin C + \cos C) = 2(\sin A \sin B \sin C + \cos A \cos B \cos C) \end{align}
What conditions on the real numbers \(a\), \(b\), \(c\) are needed to ensure that \begin{align} \frac{ax^2 + bx + c}{cx^2 + bx + a} = \lambda \end{align} has a real root \(x\) for every real \(\lambda\)?
Solution: \begin{align*} && \lambda &= \frac{ax^2 + bx + c}{cx^2 + bx + a} \quad \text{ has a real root for all }\lambda \\ \Leftrightarrow && 0 &= (a-\lambda c)x^2 + (b-\lambda b) x + (c-\lambda a) \quad \text{ has a real root for all }\lambda \\ \Leftrightarrow && 0 &\leq (b-\lambda b)^2 -4 (a-\lambda c) (c-\lambda a) \\ &&&= b^2(1-\lambda)^2 - 4(ac -\lambda(a^2+c^2) + \lambda^2 ac) \\ &&&= \lambda^2 (b^2-4ac) + \lambda (4a^2+4c^2 - 2b^2) + b^2 - 4ac \\ \Leftrightarrow && 0 &>(4a^2+4c^2-2b^2)^2 - 4(b^2-4ac)^2 \\ &&&= (4a^2+4c^2-2b^2-2b^2+8ac)(4a^2+4c^2-2b^2+2b^2-8ac) \\ &&&= 16(a^2+c^2-b^2+2ac)(a^2+c^2-2ac) \\ &&&= 16 ((a+c)^2-b^2)(a-c)^2 \end{align*} Therefore \(b^2 > (a+c)^2\)
If \(a_i(x)\), \(b_i(x)\), \(c_i(x)\) \((i = 1, 2, 3)\), are differentiable functions of \(x\), prove that \begin{align} \frac{d}{dx} \begin{vmatrix} a_1(x) & a_2(x) & a_3(x) \\ b_1(x) & b_2(x) & b_3(x) \\ c_1(x) & c_2(x) & c_3(x) \end{vmatrix} = \begin{vmatrix} a_1'(x) & a_2'(x) & a_3'(x) \\ b_1(x) & b_2(x) & b_3(x) \\ c_1(x) & c_2(x) & c_3(x) \end{vmatrix} \\ + \begin{vmatrix} a_1(x) & a_2(x) & a_3(x) \\ b_1'(x) & b_2'(x) & b_3'(x) \\ c_1(x) & c_2(x) & c_3(x) \end{vmatrix} + \begin{vmatrix} a_1(x) & a_2(x) & a_3(x) \\ b_1(x) & b_2(x) & b_3(x) \\ c_1'(x) & c_2'(x) & c_3'(x) \end{vmatrix} \end{align} Each of the functions \(u_1(x)\), \(u_2(x)\), \(u_3(x)\) is a solution, valid for all values of \(x\), of the differential equation \(y'' - xy' - \beta y + \gamma y = 0\), where \(\alpha\), \(\beta\), \(\gamma\) are constants. Find a first-order differential equation satisfied by the function \begin{align} f(x) = \begin{vmatrix} u_1 & u_2 & u_3 \\ u_1' & u_2' & u_3' \\ u_1'' & u_2'' & u_3'' \end{vmatrix} \end{align} and deduce that \(f(x)\) either vanishes identically or is non-zero for all values of \(x\).
At tennis the player serving has a probability \(\frac{3}{4}\) of winning any particular point, and his opponent has a probability \(\frac{1}{4}\). What is the probability that the player serving will win the game? [A game is finished as soon as one player has won at least four points and is at least two points ahead of his opponent.]
A certain hill has the following property. If a man stands anywhere on it and looks directly uphill, the horizontal distance from where he is to the furthest point of the hill that he can see depends only on his height and not on where he is on the hill. What is the shape of the hill? [It may be assumed that the hill is a surface of revolution.]
If \(|c| < 1\) and \begin{align} f(c) = \int_0^{\pi} \log(1 + c\cos x) dx, \end{align} prove that \begin{align} 2f(c) - f\left(\frac{c^2}{2-c^2}\right) = \pi \log(1 - \frac{1}{4}c^2). \end{align}
(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 \(x = 0\).
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 \(n \to \infty\).