Problems

Determine \(\lambda\) so that the equation in \(x\) \[ \frac{2A}{x+a} + \frac{\lambda}{x} - \frac{2B}{x-a} = 0 \] may have equal roots; and if \(\lambda_1, \lambda_2, x_1, x_2\) be the two values of \(\lambda\) and the two corresponding values of \(x\), prove that \[ x_1 x_2 = a^2, \quad \lambda_1 \lambda_2 = (A-B)^2. \]

If \(n\) is a positive integer, prove that \(3 \cdot 5^{2n+1} + 2^{3n+1}\) is divisible by 17 and \(3^{2n+2}-8n-9\) is divisible by 64.

Evaluate the integrals:

1. \(\int \frac{x-1}{x^2}e^x dx\); \quad (ii) \(\int \frac{(2x^3+1)dx}{x(x^3+1)\sqrt{x^6(1+x^3)^2+1}}\).

Draw the graph of the curve \[ (x^2-1)(x^2-4)y^2 - x^2 = 0 \] and find the area bounded by the line \(x=1\) and the two arcs of the curve passing through the origin.

(i) Prove that if \[ H_n(x) = e^{x^2} \frac{d^n e^{-x^2}}{dx^n}, \] then \[ \frac{dH_n(x)}{dx} + 2nH_{n-1}(x) = 0. \] (ii) Find the \(n\)th derivative of the function \(y = \frac{x}{x^2-1}\).

Two planes are inclined at an angle \(\theta\). A straight line makes angles \(\alpha\) and \(\beta\) with the normals to the two planes. Prove that if its projections on the two planes are perpendicular, then \[ 1 - \cos^2\alpha - \cos^2\beta \pm \cos\alpha\cos\beta\cos\theta = 0. \]

Find the limits of

1. \(n\{e-(1+\frac{1}{n})^n\}\); \quad (ii) \(n\left( \frac{\gamma^n-1}{\log n} \right)\)

Find the two nearest points on the curves \(y^2-4x=0\), \(x^2+y^2-6y+8=0\), and evaluate their distance.

A plane curve is referred to polar coordinates \(r, \theta\). The perpendicular from the origin upon the tangent to the curve is \(p\). If \(u=1/r\), prove that \[ \frac{1}{p^2} = u^2 + \left( \frac{du}{d\theta} \right)^2, \] and that \(\rho\), the radius of curvature, is given by \[ \frac{1}{\rho} = u^3 \frac{u + \frac{d^2u}{d\theta^2}}{\left\{ u^2 + \left( \frac{du}{d\theta} \right)^2 \right\}^{\frac{3}{2}}}. \]

(i) Find \(\sum_{n=1}^N (n+1)\sin n\alpha\). (ii) Find \(x\cos\theta + \frac{1}{2}x^2 \cos 2\theta + \frac{1}{3}x^3 \cos 3\theta + \dots\), when \(x=\cos\theta\). (iii) If \(y=(\sin^{-1}x)^2\), show that \[ (1-x^2)y'' - xy' - 2 = 0, \] and hence \[ (\sin^{-1}x)^2 = \frac{2x^2}{2!} + \frac{2^2}{4!} 2x^4 + \frac{2^2 \cdot 4^2}{6!} 2x^6 + \frac{2^2 \cdot 4^2 \cdot 6^2}{8!} 2x^8 + \dots. \]