Problems

A point \(Q\) is taken on the \(x\)-axis. Give a careful discussion of the maximum or minimum values of the distance of \(Q\) from a variable point on the hyperbola \[ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1. \]

If \(y\) is a function of \(x\), and \(x=\xi \cos \alpha - \eta \sin \alpha\), \(y = \xi \sin \alpha + \eta \cos \alpha\), where \(\alpha\) is constant, express \(\dfrac{dy}{dx}\) and \(\dfrac{d^2y}{dx^2}\) in terms of \(\dfrac{d\eta}{d\xi}\) and \(\dfrac{d^2\eta}{d\xi^2}\). Deduce that \[ \frac{\dfrac{d^2y}{dx^2}}{\left[1+\left(\dfrac{dy}{dx}\right)^2\right]^{\frac{3}{2}}} = \frac{\dfrac{d^2\eta}{d\xi^2}}{\left[1+\left(\dfrac{d\eta}{d\xi}\right)^2\right]^{\frac{3}{2}}}, \] and interpret this result.

If \(y=\sin^{-1} x\), prove that \[ (1-x^2)\frac{d^2y}{dx^2} - x \frac{dy}{dx} = 0. \] Determine the values of \(y\) and its successive derivatives when \(x=0\), and hence expand \(y\) in a series of ascending powers of \(x\).

Prove that \[ \sum_{r=1}^n \frac{2(x-\cos r\alpha)}{x^2-2x \cos r\alpha+1} = \frac{(2n+1)x^{2n}}{x^{2n+1}-1} - \frac{1}{x-1}, \] where \((2n+1)\alpha=2\pi\). Verify that the two expressions have the same limit as \(x\) tends to 1.

Evaluate the limits as \(x\) tends to 1 of the expressions:

1. [(i)] \(\dfrac{1}{x-1} \left\{ \operatorname{cosec} \pi x + \dfrac{1}{\pi(x-1)} \right\}\);
2. [(ii)] \(\dfrac{\cos^2 \frac{1}{2}\pi x}{e^x-ex}\);
3. [(iii)] \(\dfrac{x \log x}{(2x-1)\log(2x-1)}\).

Evaluate

1. [(i)] \(\displaystyle\int_0^{\pi/4} (\sec x + \tan x)^2 dx\);
2. [(ii)] \(\displaystyle\int_{-\infty}^\infty x^4 e^{-x^2} dx\), given that \(\displaystyle\int_{-\infty}^\infty e^{-x^2} dx = \sqrt{\pi}\);
3. [(iii)] \(\displaystyle\int_2^3 \frac{x^2 dx}{\sqrt{\{(3-x)(x-2)\}}}\).

1. [(i)] Solve the equation \[ \frac{dy}{dx} = \frac{2y}{y-x-y^3}. \]
2. [(ii)] It is observed that, if the temperature of a cooling body at time \(t\) is \(\theta(t)\), then \[ \frac{\theta(t)-\theta(t+\tau)}{\theta(t+\tau)-\theta(t+2\tau)} \] depends upon \(\tau\) only. Obtain a differential equation for the function \(\theta(t)\), and hence, or otherwise, show that \(\theta(t)\) is of the form \(A+Be^{-kt}\), where \(A, B\) and \(k\) are constants.

1. [(i)] Three variables \(x, y, z\) satisfy the relation \(f(x,y,z)=0\). Prove that \[ \left(\frac{\partial z}{\partial x}\right)_y \left(\frac{\partial x}{\partial y}\right)_z \left(\frac{\partial y}{\partial z}\right)_x = -1, \] where \(\left(\dfrac{\partial z}{\partial x}\right)_y\) denotes the partial differential coefficient of \(z\), regarded as a function of \(x\) and \(y\), with respect to \(x\), and \(\left(\dfrac{\partial x}{\partial y}\right)_z\), \(\left(\dfrac{\partial y}{\partial z}\right)_x\) have similar meanings.
2. [(ii)] If \(f(x,y)=\phi(\xi, \eta)\), where \(\xi=x-y\) and \(\eta=x+y\), express \(\dfrac{\partial f}{\partial x}\) and \(\dfrac{\partial f}{\partial y}\) in terms of \(\dfrac{\partial \phi}{\partial \xi}\) and \(\dfrac{\partial \phi}{\partial \eta}\). Hence, find the general solution of the equation \[ \frac{\partial f}{\partial x} - \frac{\partial f}{\partial y} = 0. \] Find the solution such that \(f(x,y)=F(x)\), where \(F(x)\) is a given function, when \(y=0\).

Let \[ \Delta = \begin{vmatrix} x^2-y^2-z^2-t^2 & -2xy & 2xz & -2xt \\ 2xy & x^2-y^2-z^2-t^2 & 2xt & 2xz \\ -2xz & 2xt & x^2-y^2-z^2-t^2 & -2xy \\ 2xt & -2xz & 2xy & x^2-y^2-z^2-t^2 \end{vmatrix}. \] By expressing \(\Delta\) as the square of another determinant \(D\), and forming the square of \(D\) in a different way, or otherwise, prove that \(\Delta = (x^2+y^2+z^2+t^2)^4\).

If \(p, q\) and \(x\) are integers, and \(4q-p^2\) is a perfect square, prove that \(p\) is even and that \(y=x^2+px+q\) can be expressed as the sum of two perfect squares. Prove also that if \(p^2-4q\) is a perfect square and \(p\) is even, then \(y\) can be expressed as the difference of two perfect squares. What happens in this case when \(p\) is odd?