Let \(G\) be the set of all \(n \times n\) matrices such that each row and each column has one 1 and \((n-1)\) zeros. Assuming that matrix multiplication is associative, prove that \(G\) forms a group under multiplication and that it has \(n!\) elements.
Let \(N\) be the set of positive integers and \(f\) a function from \(N\) to \(N\). Define, for \(k \in N\) and \(n \geq 0\), \(f^n(k) = k\), \(f^{n+1}(k) = f(f^n(k))\). If \(k\) and \(l\) are such that there are \(n \geq 0\) and \(m \geq 0\) with \(f^n(k) = f^m(l)\), let \(d(k, l)\) be the least possible value of \(n + m\) for such a pair; otherwise set \(d(k, l) = +\infty\).
Distinct points \(A\), \(B\) are on the same side of a plane \(\pi\). Find a point \(P\) in \(\pi\) such that the sum of the distances \(PA\), \(PB\) is a minimum, and prove that \(P\) has this property.
Find the coordinates of the mirror image of the point \((h, k)\) in the line \[lx + my + n = 0.\] Show that the rectangular hyperbola \[xy = c^2\] touches the rectangular hyperbola \[xy - 2c(x + y) + 3c^2 = 0,\] and that each is the mirror image of the other in the common tangent.
Show that the curve defined by \[x = (t-1)e^{-t}, \quad y = tx, \quad -\infty < t < \infty,\] has a loop and find the area it encloses.
The function \(f\) is defined by \[f(x) = \frac{1-\cos x}{x^2} \quad (x \neq 0),\] \[= \frac{1}{2} \quad (x = 0).\] Determine the maxima and minima of \(f\) in the range \(-2\pi < x < 2\pi\).
The square wave function \(f_0(x)\) is defined by \[f_0(x) = 1 \quad \text{if} \quad 2n < x < 2n + 1\] \[= -1 \quad \text{if} \quad 2n + 1 < x < 2n + 2 \quad \text{(for } n = 0, 1, 2, \ldots\text{),}\] and functions \(f_j(x)\) are defined by \(f_j(x) = f_{j-1}(2x)\), for \(j = 1, 2, \ldots\) Evaluate \[\int_0^1 f_{j_1}(x)\ldots f_{j_r}(x)\,dx,\] where \(1 \leq j_1 < \ldots < j_r\), and \(k_1, \ldots, k_r\) are non-negative integers. Show that \[\int_0^1 \left(\sum_{i=1}^{n} a_i f_i(x)\right)^4 \,dx \leq 3\left(\int_0^1 \left(\sum_{i=1}^{n} a_i f_i(x)\right)^2 \,dx\right)^2,\] where \(a_1, \ldots, a_n\) are any real numbers.
On the basis of an interview, the \(N\) candidates for admission to a college may be ranked in order of excellence. The candidates are interviewed in random order; that is, each possible ordering is equally likely.
A basket contains \(N\) eggs, a proportion \(P\) of which are rotten. It is decided to estimate \(P\) by \(R/n\), where \(R\) is the number of rotten eggs in a sample of \(n\) eggs chosen randomly from the basket. Prove that the mean of this estimate is \(P\) and its variance is \((N-n)P(1-P)/n(N-1)\).
A lamina of mass \(m\) with centre of mass \(G\) moves in its own plane. The velocity of \(G\) has components \(u\) and \(v\) relative to some fixed external rectangular axes, and the lamina has angular velocity \(\Omega\). Find the position relative to \(G\) of the instantaneous centre of rest, \(I\). If \(L\) is the sum of the moments about \(I\) of the external forces, show that \[L = \frac{1}{2\Omega}\frac{d}{dt}[m(r^2 + k^2)\Omega^2],\] where \(r = GI\) and \(k\) is the radius of gyration about \(G\). Hence, or otherwise, show that, if the body starts from rest, the initial value of the angular acceleration is \[\dot{\Omega} = \frac{L}{mK^2},\] where \(K\) is the radius of gyration about \(I\).