Problems

Prove that, if \(n\) is a positive integer, \[ (\cos\theta+i\sin\theta)^n = \cos n\theta + i\sin n\theta. \] Deduce that \(\sin(2n-1)\theta\) can be expressed as a polynomial \(P(\sin\theta)\) of degree \(2n-1\) in \(\sin\theta\). Prove that, if \(\cos(2n-1)\alpha \ne 0\), the roots \(\beta_1, \dots, \beta_{2n-1}\) of \[ P(x) - \sin(2n-1)\alpha = 0 \] are \[ \beta_r = \sin\left(\alpha + \frac{2r\pi}{2n-1}\right), \quad \text{where } r=1, \dots, 2n-1. \] Deduce that, if \(n>1\), both \[ \sum_{r=1}^{2n-1} \sin\left(\alpha + \frac{2r\pi}{2n-1}\right) \quad \text{and} \quad \sum_{r=1}^{2n-1} \sin^2\left(\alpha + \frac{2r\pi}{2n-1}\right) \] are independent of \(\alpha\), and find the value of the first of them.

State, without proof, the binomial theorem for arbitrary real index. Express \(f(x) = \frac{(4-x)^2}{4(2+x)^2(1-x)}\) in partial fractions and show that, for \(0 < x < 1\), \(f(x) > \frac{1}{x} \log_e(1+x)\). (The expansion \(\log_e(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\dots\), for \(-1

Evaluate:

1. [(i)] \(\displaystyle\int_0^1 \frac{x(1-x^2)}{(1+x^2)^2} \, dx\);
2. [(ii)] \(\displaystyle\int_0^\infty e^{-x}\sin^2 x \, dx\);
3. [(iii)] \(\displaystyle\int_0^{\pi/2} \frac{dx}{4\cos x+3\sin x}\).

Show that the radius of curvature of a plane curve \(C\) at the point \(P\) is \(r \frac{dr}{dp}\), where \(r\) is the distance from \(P\) to a fixed point \(O\), and \(p\) is the perpendicular distance from \(O\) to the tangent to \(C\) at \(P\). Find the polar equation of the evolute (locus of centres of curvature) of the curve \(r=ae^{k\theta}\).

A cylindrical hole of radius \(r\) is bored through a solid sphere of radius \(a\), the axis of the hole being along a diameter of the sphere. Find the volume and total surface area of the remaining portion of the sphere, and show that, for fixed \(a\), its surface area is maximum when \(r=a/2\).

The function \(u \equiv f(x_1, x_2, \dots, x_n)\) satisfies the identity \[ f(kx_1, k^2 x_2, \dots, k^n x_n) = k^a f(x_1, x_2, \dots, x_n) \] for fixed \(a\), all \(x_1, x_2, \dots, x_n\), and all positive values of \(k\). Show that \[ x_1 \frac{\partial u}{\partial x_1} + 2x_2 \frac{\partial u}{\partial x_2} + \dots + nx_n \frac{\partial u}{\partial x_n} = au. \] If, further, \(u\) is defined as a function of \(\xi\) and \(\eta\) by the substitutions \[ x_r = \xi^r + \eta^r \quad (r=1, 2, \dots, n), \] show that \[ \xi\frac{\partial u}{\partial \xi} + 2\eta\frac{\partial u}{\partial \eta} = -au. \]