Problems

Sum the infinite series:

1. [(1)] \(\sin x - \frac{\sin 2x}{2} + \frac{\sin 3x}{3} - \dots\),
2. [(2)] \(\frac{\sin x}{\pi} - \frac{\sin 2x}{\pi^2} + \frac{\sin 3x}{\pi^3} - \dots\),
3. [(3)] \(\frac{\sin x}{1} - \frac{\sin 2x}{2!} + \frac{\sin 3x}{3!} - \dots\).

The resistance to an airship is proportional to the square of the speed. It is required to cover a fixed distance in a fixed time. Shew that the work done is a minimum when the speed is constant.

Define Simple Harmonic Motion, and establish its chief properties. Discuss the result of compounding simple harmonic motions (1) in the same straight line, (2) in perpendicular straight lines, the periods being equal, (3) in perpendicular straight lines, one period being twice the other.

One corner of a long rectangular strip of paper of breadth \(b\) is folded over so that it falls on the opposite edge, and so that the portion folded over is triangular. Shew that the minimum area of this portion is \(2\sqrt{3} b^2/9\).

Find the limits of \(\frac{x^3+y^3}{x-y}\) as \(x\) and \(y\) tend to zero

1. [(a)] along the curve \(y = x - x^2\);
2. [(b)] along the curve \(y = x - x^3\);
3. [(c)] along the curve \(y = x - x^4\).

For what positive value of \(\alpha\) does \(\frac{x^\alpha+y^\alpha}{x^2-y^2}\) tend to 1 as \(x\) and \(y\) tend to zero along the curve \(y = x-x^\alpha\)?

Three particles \(A, B, C\) each of the same mass rest on a smooth table at the corners of an equilateral triangle; \(AB\) and \(BC\) being tight inextensible strings. \(A\) is given a velocity \(v\) in the direction \(CB\). Shew that when the string \(AB\) again tightens \(C\) starts off with velocity \(\frac{v}{15}\).

Evaluate the integrals \[ \int_0^1 \sqrt{\frac{1+x}{1-x}} \,dx, \quad \int \frac{2x^2-2x-5}{2x^2-5x-3} \,dx, \quad \int_0^\pi \sin^5 x \,dx, \quad \int x \sin x \,dx. \]

Evaluate \(\int_0^2 \frac{dx}{(3-x)\sqrt{2x^2+4x+9}}\), the positive value of the root being taken. Indicate how you would proceed to evaluate the integral if \(3-x\) were replaced by \((3-x)^2\).

The ends of a bar of length \(l\) are fastened to studs which slide each in one of two communicating slots passing through \(O\) and forming a cross at right angles to each other. The centre of the bar is constrained to describe a circle round \(O\) with uniform speed. Shew that each extremity of the bar describes a simple harmonic motion, and that the velocity of a point on the bar distant \(a\) from one extremity is perpendicular to the line joining \(O\) to the point of the bar distant \(a\) from the other extremity.

A plane cuts off from a sphere a volume equal to \(\frac{7}{27}\) of the whole. Find the ratio in which the diameter perpendicular to the plane is divided by it.