\(x^3+ax+b = 0\) has real roots \(\alpha_1, \alpha_2, \alpha_3\) where \(\alpha_1 \leq \alpha_2 \leq \alpha_3\). Similarly \(x^3 + cx + d = 0\) has real roots \(\gamma_1, \gamma_2, \gamma_3\) where \(\gamma_1 \leq \gamma_2 \leq \gamma_3\). Show that if \[\frac{\alpha_1}{\gamma_1} \leq \frac{\alpha_2}{\gamma_2} \leq \frac{\alpha_3}{\gamma_3} \quad (d \neq 0), \text{ then } \left(\frac{a}{c}\right)^3 = \left(\frac{b}{d}\right)^2.\]
Let \(S\) be the set of all real numbers of the form \(\pm (a^2 + b^2)^{\frac{1}{2}}\), where \(a\) and \(b\) are rational. (i) Show that the non-zero elements of \(S\) form a group under multiplication. (ii) Show that there are elements of \(S\) which are not rational, and that \(S\) is not closed under addition.
Evaluate the \(n \times n\) determinant \[\begin{vmatrix} -2 & 1 & 0 & 0 & \ldots & 0 \\ 1 & -2 & 1 & 0 & \ldots & 0 \\ 0 & 1 & -2 & 1 & \ldots & 0 \\ 0 & 0 & 1 & -2 & \ldots & 0 \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ 0 & 0 & 0 & 0 & \ldots & -2 \end{vmatrix}\]
A point moves in the plane so that its distances from a fixed point \(P\) and a fixed line \(l\) (not through \(P\)) are in the ratio \(\lambda\) to 1. Describe the locus of the point and draw a sketch of the loci obtained for varying \(\lambda\), indicating the effect as \(\lambda\) increases and the locus for \(\lambda = 1\). Do the same where \(P\) and \(l\) are replaced by (a) two fixed points, and (b) two fixed (intersecting) lines. [Detailed arguments are not required for (a) or (b).]
Two circles \(\Gamma\) and \(\gamma\) (lying inside \(\Gamma\)) of radii \(R\) and \(r\), respectively, whose centres are a distance \(d\) apart, have the property that if \(A\) is any point of \(\Gamma\) and the tangents from \(A\) to \(\gamma\) meet \(\Gamma\) again in \(B\) and \(C\), then \(BC\) touches \(\gamma\). Show that \(d^2 = R^2 - 2Rr\).
Find the local maxima of \(e^{ax}\sin x\) in \([0, 4\pi]\). Let \(m(a)\) be the maximum value of \(e^{ax}\sin x\) in \([0, 4\pi]\). Show that for \(a > 0\) there is a unique point \(g(a)\) in \([0, 4\pi]\) such that \[m(a) = e^{ag(a)}\sin g(a),\] and show that \(2\pi < g(a) < 3\pi\). Establish a similiar result for \(a < 0\). Deduce that there is no continuous function \(g(a)\) defined for all \(a\), satisfying \((*)\). Determine \(m(a)\) and show that it is continuous.
Find the general solution of the differential equation \[y\frac{d^2y}{dx^2} = y\frac{dy}{dx} + \left(\frac{dy}{dx}\right)^2.\]
By writing \(\lambda^2+b\lambda+c = (\lambda+A)^2+B\), or otherwise, show that \(\lambda^2+b\lambda+c \geq 0\) for all real \(\lambda\) if and only if \(b^2 \leq 4c\). By considering \(\int_0^1 (f(x) + \lambda g(x))^2dx\), or otherwise, show that if \(f\) and \(g\) are real functions then \[\left(\int_0^1 f(x)g(x)dx\right)^2 \leq \int_0^1 (f(x))^2dx \int_0^1 (g(x))^2dx.\] Deduce that \[\sqrt{\int_0^1 (f(x)+g(x))^2dx} \leq \sqrt{\int_0^1 (f(x))^2dx} + \sqrt{\int_0^1 (g(x))^2dx}.\] Show that if \(a_1, a_2, \ldots, a_n\) and \(b_1, b_2, \ldots, b_n\) are real then \[\left(\sum_{r=1}^{n} a_rb_r\right)^2 \leq \left(\sum_{r=1}^{n} a_r^2\right)\left(\sum_{r=1}^{n} b_r^2\right).\]
A table tennis championship is arranged for \(2^n\) players. It is organised as a 'knockout' tournament with a draw for opponents before each round except the final. (Only the winners of a round proceed to the next.) Two players are chosen at random before the draw for the first round. What are the probabilities that they meet (i) in the first round? (ii) in the final? (iii) in any round of the tournament?
Sixty cars are chosen at random from all those of makes \(A\), \(B\) and \(C\) with a mileage of approximately 10,000. Each is evaluated for tyre wear and classified as good, average or poor.