Problems

Prove that necessary and sufficient conditions that the points representing in the Argand diagram the roots of the equation \[ z^4+4az^3+6bz^2+4cz+d=0 \] shall form a square are \(b=a^2, c=a^3\).

Find any maxima, minima and points of inflexion on the curve \[ y = \frac{|x-1|}{x^2+1}. \]

Obtain the equation of the circle of curvature of the curve \(y=1-\cos x\) at the origin. If \((x, y_1)\) and \((x, y_2)\) are respectively points of the curve and the circle of curvature near the origin, prove that, as \(x\) tends to 0, \[ \frac{y_2-y_1}{x^4} \to \frac{1}{6}. \]

Discuss the convergence of the series \[ 1+z+z^2+\dots+z^n+\dots, \] where \(z\) may be real or complex.

It is given that \(u_{n+1}=\frac{1}{2}(u_n + A^2/u_n)\), where \(n=1, 2, 3,\dots\), and \(0 < A \le u_1\). Prove that

1. \(u_{n+1} \ge A\) and \(u_{n+1} \le u_n\);
2. \(d_{n+1}=d_n^2\), where \(d_n=(u_n - A)/(u_n+A)\);
3. as \(n\) tends to infinity, \(u_n\) tends to \(A\).

Taking \(A^2=99, u_1=10\), calculate \(\sqrt{11}\) correct to four places of decimals.

Prove that the increment in the angle \(A\) of a triangle due to small increments in the sides is given by the equation \[ bc \sin A\, \delta A = -a(\cos C\,\delta b + \cos B\,\delta c - \delta a). \] The measurement of any side is liable to a small error of \(\pm\mu\) per cent. Prove that, if \(B\) and \(C\) are acute, the calculated value of \(A\) is liable to an error of about \[ \pm 1.15 \frac{\mu a^2}{bc \sin A} \text{ degrees}. \] Find an expression for the possible error in \(A\) if \(B\) is obtuse.

Find an integral value of \(x\) such that \[ \frac{e^x}{x^{12}} > 10^{20}. \] (Your answer need not be the smallest possible value of \(x\), but it must not exceed that value by more than ten per cent. Tables may be used.) Enunciate and prove a general statement of which the existence of an \(x\) satisfying the above inequality is a particular consequence. (You may start from any definition of \(e^x\), to be specified.)

If any of the following expressions are meaningless, explain why. Evaluate each of the integrals which has a meaning: \[ \int_0^1 \frac{dx}{e\sqrt[3]{x(1-x)^2}}; \quad \int_{-4}^{-2} \frac{dx}{2x+1}; \quad \int_0^{3\pi/4} \tan\theta\,d\theta. \]

Sketch the curve whose equation in polar coordinates is \[ r = \sin 3\theta - 2\sin\theta. \] Find any maximum or minimum values of \(r\). Prove that each of the smaller loops of the curve has area less than 0.005.

It is given that, for all \(x>0, y>0\), \[ \int_1^{xy} f(t)\,dt = \phi(y), \] where \(\phi(y)\) is independent of \(x\). Write down the results of differentiating this equation partially with respect to \(x\) and with respect to \(y\). Find the most general form of the function \(f(t)\).