Problems

State and prove the rule for the multiplication of two determinants. \par Hence shew that the product \[ (x^3+y^3+z^3-3xyz)(a^3+b^3+c^3-3abc) \] may be expressed in the form \[ A^3+B^3+C^3-3ABC. \]

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.