Let \(f(x) = ax^2 + bx + c\) (\(a\), \(b\), \(c\) real, \(a > 0\)). Explain why the following statements are equivalent. (i) \(f(x) \leq 0\) for some real number \(x\). (ii) \(b^2 - 4ac \geq 0\). The real numbers \(a_1\), \(a_2\), ..., \(a_n\), \(b_1\), \(b_2\), ..., \(b_n\) are such that \(b_1^2 - b_2^2 - ... - b_n^2 > 0\). By considering the expression \((b_1 x - a_1)^2 - (b_2 x - a_2)^2 - ... - (b_n x - a_n)^2\), or otherwise, prove that \((a_1^2 - a_2^2 - ... - a_n^2)(b_1^2 - b_2^2 - ... - b_n^2) \leq (a_1 b_1 - a_2 b_2 - ... - a_n b_n)^2\).
Show that, if the cubic equation \(x^3 - a_1 x^2 + a_2 x - a_3 = 0\) has roots \(\alpha\), \(\beta\), \(\gamma\) and if \(a_3 \neq 0\), then \(\frac{a_2}{a_3} = \frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma}\). Deduce that the equation \(x^3 - ax^2 + 11bx - 4b = 0\) cannot have three strictly positive integer roots. Find a value of \(a\) such that \(x^3 - ax^2 + 96x - 108 = 0\) does have three positive, integer roots.
Let \(n\) be a positive integer, and consider the sequence \(\binom{n}{1}\), \(\binom{n}{2}\), ..., \(\binom{n}{n-1}\), where \(\binom{n}{r}\) denotes the binomial coefficient \(\frac{n!}{r!(n-r)!}\). (i) Show that no three consecutive terms of the sequence can be in geometric progression. (ii) Show that if there are three consecutive terms \(\binom{n}{r-1}\), \(\binom{n}{r}\), \(\binom{n}{r+1}\) in arithmetic progression, then \((n - 2r)^2 = n + 2\), and find an \(n\) for which there are three such terms. (iii) Show that it is never possible to have four consecutive terms of the sequence in arithmetic progression.
Let \(G\) be the set of all rational numbers which have an even numerator and an odd denominator, together with 0. Let the binary operation \(\circ\) on \(G\) be defined by \(x \circ y = x + y + xy\) (\(x\), \(y\) in \(G\)). Show that \((G, \circ)\) is a commutative group. Which, if either, of the following are subgroups: (i) the set of all non-negative numbers in \(G\), (ii) the set of all those elements of \(G\) which, in lowest terms, have numerator divisible by 3?
For elements \(a\), \(b\) of a multiplicative group \(G\), the element \(a^{-1}b^{-1}ab\) is written \([a, b]\). Show that if \(a\), \(b\) and \(c\) are in \(G\), then \([a, bc] = [a, c]c^{-1}[a, b]c\). Hence, or otherwise, show that if \([a, b]b = b[a, b]\) then \([a, b^n] = [a, b]^n\) for \(n = 1, 2, 3, ...\). If also \([a, b]a = a[a, b]\), prove, by considering \([a, b]^{-1}\), or otherwise, that \([a^n, b] = [a, b]^n\) for \(n = 1, 2, 3, ...\).
Define the product of two real \(2 \times 2\) matrices. Show that this multiplication is associative. A matrix \(A\) is said to commute with a matrix \(B\) if \(AB = BA\). Show that if \(A\) is a \(2 \times 2\) real matrix which commutes with every real \(2 \times 2\) matrix, then \(A = \begin{pmatrix} \lambda & 0 \\ 0 & \lambda \end{pmatrix}\), for some real number \(\lambda\).
Prove that if \(u\) and \(v\) are functions of \(x\) and if \(n\) is a positive integer then \begin{equation*} \frac{d^n}{dx^n}(uv) = \sum_{r=0}^{n} \binom{n}{r} \frac{d^r u}{dx^r} \frac{d^{n-r} v}{dx^{n-r}} = \frac{d^n u}{dx^n}v + \binom{n}{1} \frac{d^{n-1} u}{dx^{n-1}} \frac{dv}{dx} + \cdots + \binom{n}{r} \frac{d^{n-r} u}{dx^{n-r}} \frac{d^r v}{dx^r} + \cdots + u\frac{d^n v}{dx^n}, \end{equation*} where \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\). The functions \(L_n\) are defined by \begin{equation*} L_n(x) = e^x \frac{d^n}{dx^n}(e^{-x}x^n). \end{equation*} Show that \(L_n\) is a polynomial of degree \(n\). By considering \begin{equation*} \frac{d^{n+1}}{dx^{n+1}}(e^{-x}x^n), \end{equation*} prove that \begin{equation*} L_{n+1}(x) = (n+1-x)L_n(x) + x\frac{d}{dx}L_n(x). \end{equation*}
A square \(ABCD\) is made of stiff cardboard, and has sides of length \(2a\). Points \(P\), \(Q\), \(R\), \(S\) are taken inside the square, each at a distance \(xa\) from the centre; they are so placed that when the triangles \(APB\), \(BQC\), \(CRD\), \(DSA\) are cut away a single piece of cardboard remains, which can be folded about \(PQ\), \(QR\), \(RS\), \(SP\) so as to form the surface of a pyramid with \(A\), \(B\), \(C\), \(D\) coinciding at its apex. Show that the volume of the pyramid cannot exceed \begin{equation*} \frac{32\sqrt{2}}{75\sqrt{3}}a^3. \end{equation*}
A curve is given parametrically by \begin{align*} x &= a(\cos\theta + \log\tan\tfrac{1}{2}\theta)\\ y &= a\sin\theta, \end{align*} where \(0 < \theta < \frac{1}{2}\pi\) and \(a\) is constant. The points with parameters \(\theta, \frac{1}{2}\pi\) are denoted by \(P, A\) respectively; the tangent at \(P\) meets the \(x\)-axis at \(Q\). Prove that \(PQ = a\). Let \(C\) be the centre of curvature at \(P\) and let \(s\) be the arc length from \(A\) to \(P\). By considering \(ds/d\theta\), or otherwise, show that \(CQ\) is parallel to the \(y\)-axis.
Suppose that \(f\) is defined for \(a < x < b\), that \(a < c < b\), and that \(f'(c) = 0\). Show how one may, in general, determine whether \(f\) has a maximum, a minimum or neither at \(c\) by considering the sign of \(f'(x)\) in the neighbourhood of \(c\). Let \begin{equation*} f(x) = x^p(1-x)^q, \end{equation*} where \(p > 1\), \(q > 1\). Sketch the graph of \(f(x)\) for \(0 \leq x \leq 1\). Show by means of sketches how \(f\) behaves in this interval for other positive values of \(p\) and \(q\), distinguishing between different ranges of values of \(p\) and \(q\) so as to indicate the different types of curve that may occur.