Problems

Two particles whose masses are in the ratio 4:3 are connected by a light string of length \(\pi a\) and rest in equilibrium over a smooth horizontal cylinder of radius \(a\). If equilibrium is disturbed so that the heavier particle begins to descend, find at what point it will leave the surface, and shew that at that instant the pressure on the other particle is slightly greater than two-thirds of its weight.

An elastic string is stretched between two fixed points \(A\) and \(B\) in the same vertical line, \(B\) being below \(A\). Prove that if a particle is fixed to a point \(P\) of the string and released from rest in that position it will oscillate with simple harmonic motion of period \(t\sqrt{\mu}\) and of amplitude \(\mu a\), where \(t\) is the period and \(a\) the amplitude when \(P\) coincides with the mid-point of \(AB\), and \(\mu = 4AP.PB/AB^2\). The string may be assumed taut throughout.

Solve the equation \((x+b+c)(x+c+a)(x+a+b)+abc=0\). Eliminate \(x, y\) from \(x+y=a, x^3+y^3=b^3, x^5+y^5=c^5\).

Find the rationalized form of \(x^{1/r}+y^{1/r}+z^{1/r}=0\) in the cases \(r=3\) and \(4\).

If \((1+x+x^2)^n = 1+c_1x+c_2x^2+\dots+c_{2n}x^{2n}\), where \(n\) is a positive integer, prove that

1. \(c_n = 1-c_1^2+c_2^2-c_3^2+\dots+c_{2n}^2\),
2. \(c_r = {}_nC_r + \frac{r-1}{1!} {}_nC_{r-1} + \frac{(r-2)(r-3)}{2!} {}_nC_{r-2} + \dots\),

and find the last term, where \({}_nC_s\) is the coefficient of \(x^s\) in \((1+x)^n\).

Prove Wilson's theorem that if \(n\) is a prime number \(1+(n-1)!\) is divisible by \(n\). If \(n\) and \(n+2\) are both prime numbers, prove that \((n-2)\{(n-1)!\}-2\) is divisible by \(n(n+2)\).

If \(f'(x)\) is positive shew that \(f(x)\) is increasing. Prove that \(2x+x\cos x-3\sin x > 0\) if \(0 < x < \frac{\pi}{2}\).

If \(\{\cosh^{-1}(1+x)\}^2 = a_0+a_1x+a_2x^2+\dots\), find the values of \(a_0, a_1, \dots\) and shew that \[ (n+1)(2n+1)a_{n+1} + n^2 a_n = 0. \]

If \(x=r\cos\theta, y=r\sin\theta\), find \(\frac{\partial x}{\partial r}, \frac{\partial r}{\partial x}\) and interpret the results geometrically. \(R\), the radius of the circumscribed circle of a triangle \(ABC\), is expressed in terms of \(a, b\) and \(C\); find \(\frac{\partial R}{\partial a}\) and prove that \(\frac{\partial R}{\partial a} = R\cot A \cos B \text{cosec } C\).

Find the equation of the normal at any point of the curve given by \[ x/a = 3\cos t - 2\cos^3 t, \quad y/a = 3\sin t - 2\sin^3 t, \] and also find the equation of its evolute.