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 evaluating the integral, sketch \begin{equation*} f(x) = \int_0^{\pi} \frac{\sin\theta\, d\theta}{(1 - 2x\cos\theta + x^2)^{1/2}} \end{equation*}
The sequence \(u_0, u_1, u_2, \ldots\) is defined by \(u_0 = 1\), \(u_1 = 1\), and \(u_{n+1} = u_n + u_{n-1}\) for \(n \geq 1\). Prove that \begin{equation*} u_{n+2}^2 + u_{n-1}^2 = 2(u_{n+1}^2 + u_n^2). \end{equation*} Using this result and induction, or otherwise, show that \begin{equation*} u_{2n} = u_n^2 + u_{n-1}^2 \quad \text{and} \end{equation*} \begin{equation*} u_{2n+1} = u_{n+1}^2 - u_{n-1}^2 \end{equation*}
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 \(a\) be a non-zero real number and define a binary operation on the set of real numbers by \begin{equation*} x * y = x + y + axy. \end{equation*} Show that the operation \(*\) is associative. Prove that \(x * y = -1/a\) if and only if \(x = -1/a\) or \(y = -1/a\) Let \(G\) be the set of all real numbers except \(-1/a\). Show that \((G, *)\) is a group.
Find all the stationary values of the function \(y(x)\) defined by \begin{equation*} \frac{ay + b}{cy + d} = \sin^2x + 2\cos x + 1 \end{equation*} where \(ad \neq bc\), \(a \neq 3c\) and \(a \neq -c\). Assume that \(a/c > 3\) or \(a/c < -1\) and show that \(y(x)\) is then a bounded function for all \(x\).
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 \(y = 0\).
Juggins enjoys playing the following game: he throws a die repeatedly. The game stops when he throws a 1; alternatively he can stop it after any throw. His score is the value of his last throw. How should Juggins play to maximise his expected score?
A die is thrown until an even number appears. What is the expected value of the sum of all the scores?