A uniform sphere, of radius \(a\), is projected with velocity \(V\) down a rough plane of inclination \(\alpha\). The sphere also has an angular velocity \(\Omega\) about a horizontal diameter, in such a sense as to tend to cause rolling up the plane. Prove that it will never stop slipping unless the coefficient of friction \(\mu\) is greater than \(\frac{2}{5} \tan \alpha\), and that, if \(bV < 2a\Omega\), it will move back up the plane if $$\mu > \frac{\tan \alpha}{1 - \frac{5V}{2a\Omega}}.$$
Solve the simultaneous equations \begin{align} x + y + z &= 6, \\ (y + z)(z + x)(x + y) &= 60, \\ \begin{vmatrix} x & y & z \\ y & z & x \\ z & x & y \end{vmatrix} &= -18. \end{align}
Show that if an integer of the form \(4n + 3\) is expressed as a product of integers, then one at least of these integers is also of the form \(4n + 3\). Show that each pair of integers \(x_i\), \(x_j\) \((i \neq j)\) chosen from the sequence \(x_1\), \(x_2\), \(\ldots\), defined by \[x_1 = 1, \quad x_{n+1} = 4x_1x_2\ldots x_n + 3 \quad (n \geq 1)\] are coprime (that is, have highest common factor 1). Deduce that there are an infinity of prime numbers of the form \(4n + 3\).
The numbers \(c_0\), \(c_1\), \(\ldots\), \(c_n\) are defined by the identity \[(1 + x)^n = c_0 + c_1x + \ldots + c_nx^n;\] prove that \(\sum c_ic_j\) summed over all integer pairs \(i\), \(j\) such that \(0 \leq i < j \leq n\) is equal to \[2^{2n-1} - \frac{(2n-1)!}{n!(n-1)!}.\]
\(n\) different names are placed in a hat. One name is drawn at random, read out, and replaced in the hat. This is repeated until \(m\) names in all have been read out. What is the probability that no name has been read out twice? If \(r\) is a given integer, how large must \(m\) be in order to ensure that one name at least is read out \(r\) times?
Obtain a reduction formula for \[I_n = \int x^n \cos rx\,dx \quad (r \neq 0).\] If \[u_n = \int_0^{1\pi} x^n \sin^2 x\,dx,\] prove that, for \(n \geq 2\), \[u_n = \frac{(\tfrac{1}{2}\pi)^{n+1}}{2n+2} + \frac{n(\tfrac{1}{2}\pi)^{n-1}}{4} - \tfrac{1}{4}n(n-1)u_{n-2}.\]
Obtain a series expansion of \(\log_e\{1 + (1/x)\}\) in ascending powers of \(1/(2x+1)\). For what ranges of values of \(x\) is this expansion valid? Prove that if \(x\) is strictly positive, for what \[\frac{2x+1}{2x(x+1)} > \log_e\left(1 + \frac{1}{x}\right) > \frac{2}{2x+1}.\]
If \[f(x) = (\sin x - \sin a)^{-1} - (x - a)^{-1}\sec a\] evaluate \[\frac{d}{da}\left[\text{Lt}_{x \to a} f(x)\right] - \text{Lt}_{x \to a} f'(x).\]
The coordinates of a general point of a plane curve are given in parametric form as \(x(t)\), \(y(t)\). Prove that the coordinates \((\xi, \eta)\) of the centre of curvature at \((x, y)\) are given by \[x - \xi : y - \eta : x'^2 + y'^2 = y' : -x' : x'y'' - x''y',\] where the accents denote differentiation with respect to \(t\). For the cycloid \(x = t - \sin t\), \(y = 1 - \cos t\), where \(t\) ranges from \(0\) to \(2\pi\), show that the locus of the centre of curvature is coincident with the original curve displaced without rotation through a certain vector distance, to be found.
In a sphere of radius \(a\) is inscribed a right circular cylinder. Show that if its maximum height is \(2a/\sqrt{3}\). Find the height of the cylinder if its whole surface area, including the end faces, is a maximum.