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 \(\lambda\).