Problems

A smooth cylinder of radius \(a\) is rigidly fixed along one of its generators to a horizontal plane. A particle which is initially at rest on the top of the cylinder is slightly disturbed and allowed to slide down. Shew that the particle strikes the plane at a distance \[ \frac{5}{27}(\sqrt{5}+4\sqrt{2})a \] from the line of contact of the cylinder and the plane.

The bob of a simple pendulum is executing small oscillations, and when it is 1 cm from its equilibrium position its velocity is 1 cm. per second. If the length of the pendulum is 1 metre and the mass of the bob is 10 grams, shew that the maximum rate at which work is done during the motion is approximately 170 ergs per second.

A uniform rod of length \(2a\) is held at an angle of \(\frac{1}{3}\pi\) to the vertical and dropped from rest without rotation. After it has fallen a distance \(\frac{1}{2}a\) the upper end of the rod is suddenly fixed. Shew that, when the rod becomes vertical, its angular velocity is \[ \frac{5}{8}\sqrt{\frac{3g}{a}}. \]

If the equation \[ x^5 + 5qx^3+5rx^2+t=0 \] has two equal roots, prove that either of them is a root of the quadratic \[ 3rx^2-6q^2x-4qr+t=0. \]

If \(a, b, c\) and \(d\) are all real, and if \((a^2+b^2+c^2)(b^2+c^2+d^2) = (ab+bc+cd)^2\), prove that \(a, b, c\) and \(d\) are in geometrical progression.

Prove that

1. [(i)] \(\dfrac{1}{2^3 \cdot 3!} - \dfrac{1 \cdot 3}{2^4 \cdot 4!} + \dfrac{1 \cdot 3 \cdot 5}{2^5 \cdot 5!} - \dots \text{ to infinity} = \dfrac{23}{24} - \dfrac{2}{3}\sqrt{2}\);
2. [(ii)] \(1+\dfrac{n}{m}+\dfrac{n(n-1)}{m(m-1)}+\dfrac{n(n-1)(n-2)}{m(m-1)(m-2)}+\dots \text{ to } n+1 \text{ terms} = \dfrac{m+1}{m-n+1}\), provided that \(m\) is not less than \(n\).

Prove that the integral part of \((\sqrt{3}+1)^{2n+1}\) is \((\sqrt{3}+1)^{2n+1} - (\sqrt{3}-1)^{2n+1}\). \item[(i)] Solve the equation \[ \sin 2x + \cos 2x + \sin x = \cos x. \] \item[(ii)] If the equation \[ p\cos 4\theta + q\sin 4\theta = r \] has solutions \(\alpha, \beta, \gamma, \delta\) none of which differ by a multiple of \(\pi\), prove that \[ \operatorname{cosec} 2\alpha + \operatorname{cosec} 2\beta + \operatorname{cosec} 2\gamma + \operatorname{cosec} 2\delta = 0. \]

The diagonals \(2a, 2b\) of a rhombus subtend angles \(\theta, \phi\) at a point whose distance from the centre of the rhombus is \(x\); prove that \[ b^2(x^2-a^2)^2 \tan^2\theta + a^2(x^2-b^2)^2 \tan^2\phi = 4a^2b^2x^2. \]

An infinite right circular cone of semi-vertical angle \(\alpha\) cuts a sphere in two circles; the diameter of the larger circle subtends an angle \(2\theta\) at the centre of the sphere. If \(c\) is the radius of the larger circle, prove that the area of the portion of the surface of the sphere lying outside the cone is \[ 4\pi c^2 \cos\alpha \cos(\theta-\alpha)\operatorname{cosec}^2\theta. \] \item[(i)] Sum to \(n\) terms \[ \sin^3\theta + \frac{1}{3}\sin^3 3\theta + \frac{1}{9}\sin^3 9\theta + \frac{1}{27}\sin^3 27\theta + \dots. \] \item[(ii)] Sum to infinity \[ \cos\theta - \frac{1}{2}\cos 3\theta + \frac{1 \cdot 3}{2 \cdot 4}\cos 5\theta - \frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6}\cos 7\theta + \dots. \]

If \(\alpha=\pi/2n\), prove that \[ \frac{\sin 2\alpha \sin 4\alpha \sin 6\alpha \dots \sin(2n-2)\alpha}{\sin\alpha \sin 3\alpha \sin 5\alpha \dots \sin(2n-1)\alpha} = n. \]