Problems

Solve the equation \[ 81x^4 + 54x^3 - 189x^2 - 66x + 40 = 0, \] given that the roots are in arithmetic progression.

Prove that, if \(m\) is a positive integer, \[ (\cos x+i\sin x)^m = \cos mx + i\sin mx. \] Sum the infinite series \[ 1+x^2\cos 2x + x^4\cos 4x + \dots \quad (x^2<1). \]

Eliminate \(\theta\) and \(\phi\) between the equations \begin{align*} a\sec\theta+b\cosec\theta &= c \\ a\sec\phi+b\cosec\phi &= c \\ \theta+\phi &= 2\alpha. \end{align*}

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 \(C, D\).