Problems

If \(f(a)=0\) and \(\phi(a)=0\), shew how to find \(\lim_{x\to a}\frac{f(x)}{\phi(x)}\), where the functions \(f(x)\) and \(\phi(x)\) have continuous derivatives, and \(\phi'(a) \ne 0\). \par Evaluate

1. [(i)] \(\lim_{x\to 0}\frac{1-\sqrt{1-x^2}}{1-\cos 2x}\);
2. [(ii)] \(\lim_{x\to 0}\frac{(1-e^x)\tan x}{x\log(1+x)}\).

If \(z\) is a function of the independent variables \(x\) and \(y\), prove that \[ dz = \frac{\partial z}{\partial x}dx + \frac{\partial z}{\partial y}dy. \] The variables \(x\) and \(y\) are changed to \(r\) and \(\theta\), where \[ x=r\sec\theta; \quad y=r\tan\theta. \] Shew that \[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = \frac{\partial^2 u}{\partial r^2} + \frac{1}{r}\frac{\partial u}{\partial r} + \frac{\cos^2\theta}{r^2}\frac{\partial^2 u}{\partial\theta^2} - \frac{\sin\theta\cos\theta}{r^2}\frac{\partial u}{\partial\theta}. \]

Evaluate

1. [(i)] \(\int\sqrt[3]{\frac{a^3-x^3}{1-x^3}}x\,dx, \quad a>1\);
2. [(ii)] \(\int \log\frac{x(1+x^2)}{(1+x)(1+x^3)}\,dx\).

State Leibniz's Theorem. \par If \(y=x^n\log x\), shew that \[ x^2\frac{d^2y}{dx^2}-(2n-1)x\frac{dy}{dx}+n^2y=0, \] and that \[ x^2\frac{d^{p+2}y}{dx^{p+2}}+2(p-n)x\frac{d^{p+1}y}{dx^{p+1}}+(p-n)^2\frac{d^py}{dx^p}+x\frac{d^{p+1}y}{dx^{p+1}}=0. \]

Solve the differential equations \[ \sin x \cos x \frac{dy}{dx} + y = \cot x, \] \[ \frac{d^3y}{dx^3}+6\frac{d^2y}{dx^2}+11\frac{dy}{dx}+6y=e^{-x}. \]

Shew that if three of the four perpendiculars from the vertices of a tetrahedron on to the opposite faces pass through a point \(P\), the fourth also must pass through \(P\). \par In this case prove that the join of any pair of the feet of the perpendiculars intersects the join of the corresponding vertices, and that the three points of intersection lying in any face of the tetrahedron are collinear.

Two conics intersect in four points \(A, B, C, D\). Shew that if the tangents at \(A, B\) to the first conic meet on the second conic, so do the tangents to the first conic at \(C, D\).

Shew that if points in a straight line \(OX\) are connected in pairs \((P,Q)\) by the one-one relation \(axy+b(x+y)+c=0\), where \(x=OP, y=OQ\), then the cross ratio of four points \((P_1P_2P_3P_4)\) is equal to the cross ratio of the four corresponding points \((Q_1Q_2Q_3Q_4)\). \par Derive the existence of double points \((L, M)\) which are harmonic conjugates with respect to any pair \((P,Q)\).

Shew that for the conic given by the equation \(ax^2+by^2+2hxy+2gx+2fy+c=0\):

1. [(i)] The principal axes are given by the equation \[ \frac{\left(x-\frac{G}{C}\right)^2-\left(y-\frac{F}{C}\right)^2}{a-b} = \frac{\left(x-\frac{G}{C}\right)\left(y-\frac{F}{C}\right)}{h}, \] where \(C=ab-h^2, G=hf-bg, F=hg-af\).
2. [(ii)] If \(C\) is positive, the area enclosed by the curve is \(\frac{\pi D}{C^{3/2}}\), where \[ D = \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix}. \]

State the condition that the equation \(ax^2+by^2+2hxy+2gx+2fy+c=0\) shall represent two straight lines, and prove that it is both necessary and sufficient. \par The equation of the straight lines \(AB, AD\) is \(bx^2+ay^2=0\), and that of the straight lines \(CB, CD\) is \(ax^2+by^2+2hxy+2gx+2fy+c=0\). Find the equation of the two diagonals of the quadrilateral \(ABCD\) which do not pass through \(A\), and deduce that they will be perpendicular if \((a+b)(f^2+g^2)=2fgh+c(a^2+b^2-h^2)\).