Solve the recurrence relation \[u_{n+2}-2\alpha u_{n+1}+(\alpha^2+\lambda^2)u_n = 0, \quad u_0 = 0, \quad u_1 = 1,\] in the following two cases.
Let \(q\) be an integer. If \(q > 1\) show that every positive real number \(x\) has an expansion to the base \(q\), that is \[x = \sum_{r=0}^{N} a_r q^r + \sum_{s=1}^{\infty} b_s q^{-s}\] where \(N\) is finite, and for each \(r\) and \(s\), \(a_r\) and \(b_s\) are integers satisfying \(0 \leq a_r < |q|\) and \(0 \leq b_s < |q|\). Is this result still true if \(q = -2\)?
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.