Problems

Solve the simultaneous equations: \begin{align*} 4x + 2y - z &= 0, \\ 5x + y - 2z &= 0, \\ 4x^2 - 4(1-a)y^2+a^2z^2+3x+3y-4az+4a(1-a) &= 0. \end{align*} Determine whether there are values of \(a\) for which the equations have (i) only one set of (finite) solutions; (ii) an infinite number of sets of solutions.

The quartic equation \[ 4x^4 + \lambda x^3 + 35x^2 + \mu x + 4 = 0 \] has its roots in geometric progression. What real values might be taken for the ratio of the progression?

Two determinants \(|a_{rs}|, |b_{rs}|\), each of the fourth order, are given by the relations \begin{alignat*}{2} a_{1s} &= x_s^2+y_s^2; \quad & a_{2s} = -2x_s; \quad a_{3s} = -2y_s; \quad a_{4s} = 1; \\ b_{1s} &= 1; \quad & b_{2s} = x_s; \quad b_{3s} = y_s; \quad b_{4s} = x_s^2+y_s^2, \end{alignat*} for \(s=1,2,3,4\). By evaluating the product \(|a_{rs}||b_{rs}|\), prove that the distances \(l_{pq}\) (\(p,q=1,2,3,4\)) connecting four concyclic points satisfy the relation \(|l_{pq}^2|=0\).

Express \(\sin 7\theta\) in terms of \(\sin\theta\), and determine the values of \(\theta\) for which \(7\sin\theta > \sin 7\theta\). Illustrate your answer graphically for values of \(\theta\) between \(0\) and \(2\pi\).

Express \(\tan 5\theta\) in terms of \(\tan\theta\). By considering the values of \(\theta\) for which \(\tan 5\theta = \sqrt{3}\), solve the equation \[ x^4 - 6\sqrt{3}x^3 + 8x^2 + 2\sqrt{3}x - 1 = 0, \] giving your answers in trigonometrical form. Prove that \[ \tan\frac{\pi}{15} \tan\frac{2\pi}{15} \tan\frac{4\pi}{15} \tan\frac{8\pi}{15} + 1 = 0. \]

1. If \(f(u)\) is a function of \(u=ax^2+2hxy+by^2\), and \(f(u)\), when expressed in terms of \(x\) and \(y\), takes the form \(g(x,y)\), prove that \[ x\frac{\partial g}{\partial x} + y\frac{\partial g}{\partial y} = 2u \frac{df}{du}. \]
2. If \(f(u,v)\) is a function of \(u=(x\sin\alpha - y\cos\alpha)^2\) and \(v=(x\cos\alpha+y\sin\alpha)^3\), and \(f(u,v)\), when expressed in terms of \(x\) and \(y\), takes the form \(g(x,y)\), prove that \[ x\frac{\partial g}{\partial y} - y\frac{\partial g}{\partial x} = 2u\frac{\partial f}{\partial u} + 2v\frac{\partial f}{\partial v}, \] and that \[ \frac{\partial^2 g}{\partial x^2} + \frac{\partial^2 g}{\partial y^2} = 2\left(\frac{\partial f}{\partial u} + \frac{\partial f}{\partial v}\right) + 4\left(u \frac{\partial^2 f}{\partial u^2} + v^2 \frac{\partial^2 f}{\partial v^2}\right). \]

Find the third differential coefficient of \(\sin x/x\), and deduce, or find otherwise, the limit as \(x \to 0\) of \[ \frac{(6-3x^2)\sin x - x(6-x^2)\cos x}{x^4}. \] Prove also that the function \(\sin x/x\) has a minimum between \(x=\pi\) and \(x=\frac{3}{2}\pi\), and that a closer approximation to the position of the minimum is \(x = \frac{3\pi}{2} - \frac{2}{3\pi}\).

A particle moves in a plane so that its position at time \(t\), referred to fixed rectangular cartesian axes, is given by \(x = a \sin 2pt\), \(y = a \sin pt\). Sketch the path traced out by the particle, and find the radii of curvature at the points where the particle is moving in a direction parallel to one or other of the axes.

1. Find the indefinite integrals \[ \int \frac{(1+x^2) \, dx}{x^2(1-x)}, \quad \int \cos^4 x \, dx. \]
2. Evaluate the integral \[ \int_a^b x^2\sqrt{\{(x-a)(b-x)\}} \, dx, \] where \(0

\(P\) is a variable point \((at^2, 2at)\) and \(K\) is the fixed point \((ak^2, 2ak)\) of the parabola \(y^2=4ax\). The foot of the perpendicular from \(P\) to \(OK\) is \(M\), where \(O\) is the origin. Prove that, if \(OM=r\), \[ \frac{dr}{dt} = \frac{2(kt+2)a}{\sqrt{(4+k^2)}}. \] Prove also that, if the arc \(OK\) is rotated about \(OK\), the volume of the solid generated is \(2\pi k^5 a^3/15\sqrt{(4+k^2)}\).