Problems

If \(n\) is a positive integer, prove that the coefficient of \(x^n\) in the expansion of \(\dfrac{1+x}{(1+x+x^2)^3}\) in a series of ascending powers of \(x\) is \(\frac{1}{8}(n+1)(3n+2)\). If \(n\) is a positive integer, and if \((3\sqrt{3}+5)^{2n+1} = I+F\), where \(I\) is an integer and \(F\) a positive proper fraction, prove that \(F(I+F) = 2^{2n+1}\).

If \(n\) is a positive integer, prove that

1. [(i)] \(n^5 - 4n^3 + 5n^2 - 2n\) is divisible by 12,
2. [(ii)] \(n(n+1)(n+2)(n+3)\) is not a perfect square.

1. [(i)] If \(y\sqrt{1-x^2} = \cos^{-1}x\), prove that \[ (1-x^2)\frac{dy}{dx} - xy + 1 = 0. \]
2. [(ii)] If \(y=(x-1)^{n-1} \log x\), where \(n\) is a positive integer, prove that \[ \frac{d^n y}{dx^n} = \frac{(n-1)!}{x-1}\left(1 - \frac{1}{x^n}\right). \]

Find in polar coordinates an expression for the angle between the radius vector to a point on a curve and the tangent at that point. Prove that the locus of the centre of a circle passing through the pole and touching the curve \(r^m = a^m \cos m\theta\) is the curve \((2r)^n=a^n \cos n\theta\), where \(n(1-m)=m\).

\(AB\) is a diameter of a given circle, whose centre is \(O\), and \(CD\) is a chord parallel to \(AB\). Prove that the maximum value of the perimeter of the quadrilateral \(ABCD\) is five times the radius of the circle. If the figure is rotated about \(AB\) so that the circle generates a sphere, prove that the volume of the solid generated by the quadrilateral \(ABCD\) is a maximum when \[ 6 \cos AOC = \sqrt{13}-1. \]

If \(z\) is a function of two independent variables \(x\) and \(y\), prove that \(z\) has a stationary value when \(\dfrac{\partial z}{\partial x}=0\) and \(\dfrac{\partial z}{\partial y}=0\). If \(z=\left(\dfrac{p}{x}\right)^k + \left(\dfrac{x}{y}\right)^k + \left(\dfrac{y}{q}\right)^k\), where \(p, q\) and \(k\) are constants, prove that \(z\) has a stationary value when \(p, x, y, q\) are in geometrical progression.

Integrate

1. [(i)] \(\int x \tan^{-1} x dx\),
2. [(ii)] \(\int \frac{dx}{x\sqrt{x^2+2x-3}}\).

If \[ y = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4 + a_5 x^5, \] prove that \[ \int_{-1}^1 ydx = \frac{1}{9}(8y_0+5y_1+5y_2), \] where \(y_0, y_1, y_2\) are the values of \(y\) when \(x=0, \sqrt{\frac{3}{5}}, -\sqrt{\frac{3}{5}}\) respectively.

Find the equations of the tangents at the double point of the curve \[ x^2(a^2-x^2) = 8a^2y^2, \] and prove that the whole length of the curve is \(\pi a \sqrt{2}\).

Prove that the locus of the centre of a circle that bisects the circumferences of two given circles is a straight line.

A chord \(PQ\) of a parabola passes through the focus \(S\), and circles are described on \(SP\) and \(SQ\) as diameters. Prove that the tangent at the vertex \(A\) touches both circles, and that the length of a common tangent to the circles is a mean proportional between \(SA\) and \(PQ\).