(i) Find the simplest equation with integral coefficients which has \[ -\frac{1}{\sqrt{2}} + \sqrt{\frac{3}{2}} \quad \text{and} \quad -\sqrt{\frac{1}{2}} - \sqrt{-\frac{3}{2}} \] among its roots. What are the other roots of the equation? (ii) Prove that \[ 4(x^2+x+1)^3 - 27x^2(x+1)^2 = (x-1)^2(2x+1)^2(x+2)^2. \]
Prove that if \[ \frac{a}{l^2} + \frac{b}{m^2} + \frac{c}{n^2} = 0, \quad \frac{a}{x^2} + \frac{b}{y^2} + \frac{c}{z^2} = 0, \quad \frac{al}{x} + \frac{bm}{y} + \frac{cn}{z} = 0, \] where \(a, b, c\) are not zero and no two of \(\frac{x}{l}, \frac{y}{m}, \frac{z}{n}\) are equal, then \[ \frac{x}{l} + \frac{y}{m} + \frac{z}{n} = 0. \]
Prove that if \[ ax^2+by^2+cz^2+2fyz+2gzx+2hxy \] is the product of two factors linear in \(x, y\) and \(z\), then \[ \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix} = 0. \] Prove that in a triangle \(ABC\) \[ \begin{vmatrix} \sin 2A & \sin C & \sin B \\ \sin C & \sin 2B & \sin A \\ \sin B & \sin A & \sin 2C \end{vmatrix} = 0. \]
(i) Find all the values of \(\theta\) which satisfy the equation \[ \cos\theta + \cos 2\theta = \sin 3\theta. \] (ii) Find the sum of the series \[ \sin\theta\sin 2\theta + 2\sin 2\theta\sin 3\theta + \dots + n\sin n\theta\sin(n+1)\theta. \]
In a triangle \(ABC\), it is given that the line joining the orthocentre \(H\) and the circumcentre \(O\) is parallel to \(BC\). Prove that (i) the angle \(A\) cannot be less than \(60^\circ\), (ii) if the angle \(A\) is \(61^\circ\), the difference between the sides \(b\) and \(c\) is greater than the distance \(HO\) by less than \(2\frac{1}{2}\) per cent.
Examine the function \[ \frac{(x+1)^5}{x^5+1} \] for maxima and minima and sketch the general shape of its graph. Prove that if \(m\) does not lie between 0 and 16, the equation \[ (x+1)^5 = m(x^5+1) \] has no real root other than \(-1\).
Prove Leibniz's formula for the \(n\)th differential coefficient of a product. Prove that, if \(x = \sin\sqrt{y}\), then \[ (1-x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} = 2. \] Deduce that, if \(n\) is a positive integer, \[ (1-x^2)\frac{d^{n+2}y}{dx^{n+2}} - (2n+1)x\frac{d^{n+1}y}{dx^{n+1}} - n^2\frac{d^ny}{dx^n} = 0, \] and find the value of \(\frac{d^ny}{dx^n}\) when \(x=y=0\).
A circular disc of radius \(a\) is made to roll, without slipping, in contact with a fixed disc of the same size in the same plane. Prove that, with a suitable choice of axes, the equation of the tangent to the curve \(S\) traced out by a given point on the rim of the moving disc is \[ x\sin 3\theta - y\cos 3\theta = 3a\sin\theta, \] where \(\theta\) is half the angle through which the line of centres has turned. Prove that the radius of curvature of \(S\) is \(\frac{3}{4}a\sin\theta\).
Determine \(P, Q\) and \(R\) as functions of \(x\) such that the equation \[ \frac{d^2y}{dx^2} + P\frac{dy}{dx} + Qy = R \] may be identically satisfied by \(y=x\), \(y=x^2\) and \(y=x^3\). With these values of \(P, Q\) and \(R\), state what condition must be satisfied by the numerical coefficients \(a,b\) and \(c\) if the equation is also identically satisfied by \[ y = ax+bx^2+cx^3. \]
Obtain an equation connecting the integrals \[ \int \frac{x^m dx}{(1+x^2)^n} \quad \text{and} \quad \int \frac{x^{m-2}dx}{(1+x^2)^{n-1}}. \] Prove that the value of the integral \[ \int_0^\infty \frac{x^{2k}dx}{(1+x^2)^{k+1}} \] decreases as the positive integer \(k\) increases, and find the smallest value of \(k\) which makes the value of the integral less than \(0 \cdot 4\).