Problems

Prove that, if \(x > 0, 0 < p < 1\), then \[ (1+x)^p < 1+px. \] Hence show that, if \(a>0, b>0\) and if \(n\) is a positive integer, then \[ (a^n+b)^{1/n} < a + \frac{b}{na^{n-1}}. \] Show further by using the identity \[ (a^n+b) - a^n = b \] \[ (a^n+b)^{\frac{n-1}{n}} + a(a^n+b)^{\frac{n-2}{n}} + \dots + a^{n-1} \] or otherwise, that \[ a + \frac{b}{na^{n-1}+\frac{1}{2}(n-1)\frac{b}{a}} < (a^n+b)^{1/n}. \]

Prove that

1. [(i)] \(\dfrac{1}{2.3.4.5} + \dfrac{4}{3.4.5.6} + \dfrac{9}{4.5.6.7} + \dots = \dfrac{5}{36}\),
2. [(ii)] \(\cos\theta - \sin 2\theta \frac{\cos^2\theta}{2} - \cos 3\theta \frac{\cos^3\theta}{3} + \sin 4\theta \frac{\cos^4\theta}{4} + \cos 5\theta \frac{\cos^5\theta}{5} - \dots = \cot^{-1}(1+\tan\theta+\tan^2\theta)\).

Prove that if \(\alpha, \beta, \gamma\) are the roots of the equation \[ x^3 - 3px^2 - 3(1-p)x + 1 = 0, \] then \[ \beta(1-\gamma) = \gamma(1-\alpha) = \alpha(1-\beta) = 1 \] or \[ \beta(1-\alpha) = \gamma(1-\beta) = \alpha(1-\gamma) = 1. \] Show that \(\alpha, \beta, \gamma\) are real if \(p\) is real.

Prove that the radius of the nine-points circle of a triangle is half the radius of the circumcircle. \par The distances between the centres of the escribed circles of a triangle being \(\alpha, \beta, \gamma\), prove that \[ 4R = \frac{\alpha^2}{r_2+r_3} = \frac{\beta^2}{r_3+r_1} = \frac{\gamma^2}{r_1+r_2} = \frac{(r_2+r_3)(r_3+r_1)(r_1+r_2)}{r_2r_3+r_3r_1+r_1r_2} = \frac{\alpha\beta\gamma}{2\sqrt{\{\sigma(\sigma-\alpha)(\sigma-\beta)(\sigma-\gamma)\}}}, \] where \(2\sigma = \alpha+\beta+\gamma\).

\(y\) is the implicit function of two variables \(x, \alpha\) defined by the equation \[ y = x + x\phi(y). \] If \(u\) is a function of \(y\) and \(F(u)\) a function of \(u\), prove that \[ \frac{\partial}{\partial x}\left\{F'(u)\frac{\partial u}{\partial \alpha}\right\} = \frac{\partial}{\partial \alpha}\left\{F'(u)\frac{\partial u}{\partial x}\right\}, \] and that \[ \frac{\partial u}{\partial x} - \phi(y)\frac{\partial u}{\partial \alpha} = \frac{\partial}{\partial x}\left[\{\phi(y)\}^2 \frac{\partial u}{\partial \alpha}\right], \] and, generally, \[ \frac{\partial^n u}{\partial x^n} = \frac{\partial^{n-1}}{\partial x^{n-1}}\left[\{\phi(y)\}^n \frac{\partial u}{\partial \alpha}\right]. \]

If \(p(x)\) is a polynomial of the \(k\)th degree and if \[ H_n(x) = e^{p(x)}\frac{d^n e^{-p(x)}}{dx^n}, \] prove that \[ H_n(x) = q_{k-1}(x)H_{n-1}(x) + q_{k-2}(x)H_{n-2}(x) + \dots + q_0 H_{n-k}(x), \] where \(q_i(x)\) is a polynomial of the \(i\)th degree in \(x\). \par Find the actual equation for the case in which \(p(x)=x^2\).

Evaluate

1. [(i)] \(\displaystyle\int \sin^{n-1}x \cos\{(n+1)x+\alpha\}dx\),
2. [(ii)] \(\displaystyle\int x^2\sqrt{x^2+x^4}dx\)

Prove that of all the quadrilaterals with sides of given lengths the one which can be inscribed in a circle has the largest area.

Prove that all the curves represented by the equation \[ \frac{x^{n+1}}{a} + \frac{y^{n+1}}{b} = \left(\frac{ab}{a+b}\right)^n \] for different positive values of \(n\), touch each other at the point \[ \left(\frac{ab}{a+b}, \frac{ab}{a+b}\right). \] Prove that the radius of curvature at the point of contact is equal to \[ \frac{(a^2+b^2)^{3/2}}{n(a+b)^2}. \]

Trace the curve \[ b^3y^2(2-by)-x^2=0, \] and show that its area is \(\dfrac{5\pi}{4b^2}\).