Prove that, for any positive integer \(n\), \[ (1+x)^n = 1+nx+\binom{n}{2}x^2+\dots+\binom{n}{r}x^r+\dots+x^n, \] where \[ \binom{n}{r} = \frac{n!}{(n-r)!r!}. \] Deduce the identity \[ 1+\sum_{r=1}^{m-1} (-1)^r \binom{2m-1}{r} = \frac{(-1)^{m+1}}{2} \left\{ \binom{2m}{m} - \binom{2m}{m-1} \right\}, \] where \(m\) is any positive integer.
\(D_n\) is the \((n \times n)\) determinant \[ \begin{vmatrix} \operatorname{cosec} 2\alpha & \tan\alpha & 0 & \dots & 0 & 0 \\ \cot\alpha & \operatorname{cosec} 2\alpha & \tan\alpha & \dots & 0 & 0 \\ 0 & \cot\alpha & \operatorname{cosec} 2\alpha & \dots & 0 & 0 \\ \dots & \dots & \dots & \dots & \dots & \dots \\ 0 & 0 & 0 & \dots & \operatorname{cosec} 2\alpha & \tan\alpha \\ 0 & 0 & 0 & \dots & \cot\alpha & \operatorname{cosec} 2\alpha \end{vmatrix} \] where \(0 < \alpha < \pi/2\). Find a relation connecting \(D_n, D_{n-1}\) and \(D_{n-2}\), and hence evaluate \(D_n\).
State and prove De Moivre's theorem about \((\cos\theta+i\sin\theta)^r\), where \(r\) is a rational number (i.e. a number of the form \(p/q\), where \(p\) and \(q\) are integers and \(q\neq 0\)). Using this theorem, or otherwise, find the sum to \(n\) terms of the series \[ \cos\theta \sin\theta + \cos^2\theta \sin 2\theta + \dots + \cos^m\theta \sin m\theta + \dots + \cos^n\theta \sin n\theta. \]
A pyramid consists of a square base and four equal triangular faces meeting at its vertex. If the total surface area is kept fixed, show that the volume of the pyramid is greatest when each of the angles at its vertex is \(36^\circ 52'\).
Evaluate the integrals: \[ \int \frac{dx}{x^4+4}, \quad \int e^{ax}\cos bxdx \quad (a\neq 0, b\neq 0), \quad \int \frac{dx}{x+(x^2-1)^{\frac{1}{2}}}. \]
Find a reduction formula for \[ I_n = \int \frac{dx}{(5+4\cos x)^n} \] in terms of \(I_{n-1}\) and \(I_{n-2}\) (\(n \ge 2\)), and use it to show that \[ \int_0^{2\pi/3} \frac{dx}{(5+4\cos x)^2} = \frac{1}{81}(5\pi-6\sqrt{3}). \]
A curve is defined by the parametric equations \[ x=\frac{1}{t(t+1)}, \quad y=\frac{1}{t(t+3)}. \] Find its asymptotes and trace the curve. What is its form near the origin? Obtain the algebraic relation connecting \(x\) and \(y\) which is satisfied at each point of the curve.
(i) The variables \(x\) and \(y\) satisfy the equation \(f(x,y)=0\) which may be regarded as defining \(y\) as a function of \(x\). Show that \[ \frac{d^2y}{dx^2} = -\frac{f_{xx}f_y^2 - 2f_{xy}f_xf_y+f_{yy}f_x^2}{f_y^3}. \] (ii) The variables \(x, y\) and \(z\) satisfy the two equations \(f(x,y,z)=0, g(x,y,z)=0\). By eliminating \(z\) between these equations it is possible to obtain a relation connecting \(x\) and \(y\) which defines \(y\) as a function of \(x\). Show that \[ \frac{dy}{dx} = -\frac{f_xg_z-f_zg_x}{f_yg_z-f_zg_y}. \]
Find all the solutions of the equations \begin{align*} x+2y+4z &= 12, \\ xy+2xz+4yz &= 22, \\ xyz &= 6. \end{align*}
If \(f(x)\) is a polynomial and \(f'(x)\) its derivative, state, without proof, what you can deduce about the roots of the equation \(f(x)=0\) from a knowledge of the roots of the equation \(f'(x)=0\). \newline Prove that the equation \[ 1-x+\frac{x^2}{2} - \frac{x^3}{3} + \dots + (-1)^n \frac{x^n}{n} = 0 \] has one real root if \(n\) is odd, and no real root if \(n\) is even. \newline Hence or otherwise find the number of real roots of \[ 1 - \frac{x}{1 \cdot 2} + \frac{x^2}{2 \cdot 3} - \dots + (-1)^n \frac{x^n}{n(n+1)} = 0. \]