Problems

Obtain the conditions for the general equation of the second degree \[ ax^2+2hxy+by^2+2gx+2fy+c=0 \] to be an ellipse, a parabola, or a hyperbola. Show that if the equation represents a rectangular hyperbola then referred to the asymptotes as co-ordinate axes \(O'\xi, O'\eta\), its equation will be \[ 2(h^2-ab)\xi\eta = \Delta, \] where \(\Delta = \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix}\).

(i) Eliminate \(\theta\) from the equations \[ a\tan\theta+\sec\theta=h, \quad a\cot\theta+\csc\theta=k. \] (ii) Sum the infinite series \[ \frac{1}{2!} + \frac{4}{3!} + \frac{9}{4!} + \dots + \frac{n^2}{(n+1)!} + \dots. \]

If \(\theta=2\pi/7\), prove that \begin{align*} \sin\theta+\sin2\theta+\sin4\theta &= \sqrt{7}/2, \\ \tan^2\theta+\tan^2 2\theta+\tan^2 4\theta &= 21. \end{align*}

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.)