A particle of mud is thrown off from the ascending part of the tyre of a wheel (radius \(a\)) of a car travelling at constant speed \(V\). Show that the particle will be thrown clear of the wheel if its height above the hub at the instant when it leaves the tyre is greater than \(ga^2/V^2\).
Solve: \begin{align*} x^2-yz &= a^2, \\ y^2-zx &= b^2, \\ z^2-xy &= c^2, \end{align*} where \(a, b, c\) are real and different from one another. Prove that the values of \(x, y\) and \(z\) in any solution are real and different from one another.
If \(\alpha\) and \(\beta\) are the roots of \(y^2-qy+p^3=0\), where \(p\) and \(q\) are real, show how to determine the roots of \(x^3 - 3px + q = 0\) in terms of \(\alpha\) and \(\beta\). Prove that, if \(\alpha\) and \(\beta\) are real and distinct, the cubic has only one real root, and that, if \(\alpha\) and \(\beta\) are imaginary, it has three real roots. What happens if \(\alpha\) and \(\beta\) are coincident?
Solve: \begin{align*} x\cos\alpha + y\cos\beta + z\cos\gamma &= 1, \\ x\sin\alpha + y\sin\beta + z\sin\gamma &= 0, \\ x+y+z &= 1. \end{align*} Say what exceptional cases arise and whether the equations are then soluble.
Prove that \((e^{i\alpha}+e^{2i\alpha}+e^{4i\alpha})\) is one root of \(x^2+x+2=0\), where \(\alpha=2\pi/7\). Hence show that \begin{align*} \cos\alpha+\cos2\alpha+\cos4\alpha &= -\frac{1}{2}, \\ \sin\alpha+\sin2\alpha+\sin4\alpha &= \frac{1}{2}\sqrt{7}. \end{align*}
(i) Find \(\lim_{x \to 1} \frac{x^K-1}{x-1}\), when \(K\) is a positive integer; deduce the result for \(K\) a positive rational number. (ii) Find \[ \lim_{n \to \infty} \left(\frac{1^p+2^p+\dots+n^p}{n^{p+1}}\right). \]
Under what circumstances is a function \(f(x)\) said to be continuous at \(x=k\)? The constants \(a\) and \(b\) are both greater than 1; \(f(x)\) is any function such that \(f(ax) = bf(x)\) for all values of \(x\); \(f(0)=0\). If for \(-1 \le x \le 1\), \(|f(x)|
\(f(x)\) is continuous and has a derivative for \(a \le x \le b\); give the conditions that the largest value of \(f(x)\) in this interval occurs at a point where \(\frac{df(x)}{dx}=0\). What modifications must be made if, at a finite number of places in the interval, \(\frac{df(x)}{dx}\) does not exist? Find the largest and smallest values for \(-1 \le x \le 1\) of
Find a reduction formula for \(\int_0^{\pi/4} \tan^n x \,dx\). Prove that \(\lim_{n \to \infty} \int_0^{\pi/4} \tan^n x \,dx\) exists and is equal to zero.
A torus is the figure formed by rotating a circle of radius \(a\) about a line in its own plane at a distance \(h\) from its centre, \(h > a\). Find the volume and surface area of the torus. If Pappus' theorems are used, they must be proved.