The cubic polynomial \(f(x) = x^3+bx+1\) has the roots \(\alpha, \beta, \gamma\). Find, in terms of \(b\), the coefficients of the polynomial \[ g(x) = \{(\beta+\gamma)x - \beta\gamma\}\{(\gamma+\alpha)x - \gamma\alpha\}\{(\alpha+\beta)x - \alpha\beta\}. \]
The quartic equation \[ x^4 + ax^3 + bx^2 + cx + d = 0 \] has four real roots. Prove that
(i) If \(\alpha+\beta+\gamma = \frac{1}{2}\pi\), prove that \[ (\sin\alpha+\cos\alpha)(\sin\beta+\cos\beta)(\sin\gamma+\cos\gamma) = 2(\sin\alpha\sin\beta\sin\gamma + \cos\alpha\cos\beta\cos\gamma). \] (ii) If \(\alpha+\beta+\gamma=\pi\), \(\lambda+\mu+\nu=0\), prove that \[ \frac{\lambda}{\cos^2\alpha} + \frac{\mu}{\cos^2\beta} + \frac{\nu}{\cos^2\gamma} = 2 \tan\alpha\tan\beta\tan\gamma \left( \frac{\lambda}{\sin 2\alpha} + \frac{\mu}{\sin 2\beta} + \frac{\nu}{\sin 2\gamma} \right). \]
Evaluate \[ S_N = \sum_{\nu=1}^N e^{\frac{\nu x}{N}} \cos \frac{\nu y}{N}. \] Find the limit, as \(N\) tends to infinity, of \(\frac{1}{N}S_N\), and compare its value with \(\int_0^1 e^{tx} \cos(ty) dt\).
Find all maxima and all minima of the two functions \[ y = e^{-\sqrt{3}x} \sin^3 x \] and \[ y = \int_x^\infty \frac{\sin\xi}{\xi(1+\xi^2)} d\xi. \]
(i) Determine \[ \lim_{x \to 0} \frac{\log(e^x+e^{-x}-1)}{\log \cos x}. \] (ii) Determine \[ \lim_{x \to \infty} \frac{e^{ax}-x}{e^{ax}+x} \] for all real values of \(a\).
Determine constants \(A, B, C, D\) such that \[ \frac{x^4+1}{(x^2+1)^4} = \frac{d}{dx} \frac{Ax^5+Bx^3+Cx}{(x^2+1)^3} + \frac{D}{x^2+1}. \] Hence or otherwise, prove that \[ 0.502 < \int_0^1 \frac{x^4+1}{(x^2+1)^4} dx < 0.503. \]
Find the volume of the body
\[ (\sqrt{x^2+y^2}-a)^2 < b^2 - z^2 \]
for \(0
If \(0
The equation \(z = F(x,y)\) is obtained by eliminating \(u\) between the equations \(y=f(u,x)\) and \(z=g(u,x)\). Prove that \[ \frac{\partial f}{\partial u} \frac{\partial F}{\partial x} = \frac{\partial f}{\partial u} \frac{\partial g}{\partial x} - \frac{\partial f}{\partial x} \frac{\partial g}{\partial u}, \] \[ \frac{\partial f}{\partial u} \frac{\partial F}{\partial y} = \frac{\partial g}{\partial u}. \]