Show that the determinant \[ D(a,b,x) = \begin{vmatrix} r_1+x & a+x & a+x & \dots & a+x \\ b+x & r_2+x & a+x & \dots & a+x \\ b+x & b+x & r_3+x & \dots & a+x \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ b+x & b+x & b+x & \dots & r_n+x \end{vmatrix} \] is linear in \(x\). Deduce that \[ D(a,b,0) = \frac{bf(a)-af(b)}{b-a}(a+b), \] \[ D(a,a,0) = f(a) - af'(a), \] where \(f(x) = (r_1-x)(r_2-x)\dots(r_n-x)\).
If \[ L_n(x) = e^x \frac{d^n}{dx^n}(x^n e^{-x}) \] show that:
Defining an infinite integral by the equation \(\int_0^\infty f(x)dx = \lim_{X\to\infty} \int_0^X f(x)dx\), show how to integrate an infinite integral by parts. By integration by parts, show that \[ \frac{4}{3} \int_0^\infty \frac{\sin^3 x}{x^3} dx = \int_0^\infty \frac{\sin^2 x}{x^2} dx = \int_0^\infty \frac{\sin x}{x} dx. \] (It may be assumed that these integrals exist.)
\(\alpha=0, \beta=0, \gamma=0, \delta=0\) are the equations of four lines, no three of which meet in a point. Show that, for all values of the parameters \(\lambda, \mu\), the conic \(\lambda^2\alpha^2 - 2\lambda\mu\gamma\delta + \mu^2\beta^2 = 0\) has double contact with each of the conics. \[ S = \alpha\beta+\gamma\delta=0 \quad \text{and} \quad S'=\alpha\beta-\gamma\delta=0. \] If the identical relation connecting \(\alpha, \beta, \gamma, \delta\) is \[ \alpha+\beta+\gamma+\delta=0, \] interpret the equation \(\alpha^2 - 2\gamma\delta + \beta^2 = 0\).
Show that the curve whose parametric equations referred to rectangular Cartesian coordinates are \(x=at+bt^2, y=ct+dt^2\), where \(ad \ne bc\), is a parabola. Show that the chord joining the points whose parameters are \(t_1\) and \(t_2\) is \[ x\{c+d(t_1+t_2)\} - y\{a+b(t_1+t_2)\} + (bc-ad)t_1t_2 = 0. \] Show that the equation of the directrix is \[ 4bx+4dy+a^2+c^2=0. \] Obtain the equation of the axis and the coordinates of the focus. \subsubsection*{SECTION B}
A uniform plank of thickness \(2h\) is placed on top of a perfectly rough fixed circular cylinder of radius \(a\) (\(h < a\)). The axis of the cylinder is horizontal and the long edges of the plank are perpendicular to the axis of the cylinder. When the plank is rolled into the horizontal position its centre of gravity is a distance \(l\) from the vertical plane through the axis of the cylinder. Prove that there are three positions of equilibrium in which the plank rests on the cylinder, provided that \(l < l'\), where \[ l' = a \arccos(h/a)^{\frac{1}{2}} - ((a-h)h)^{\frac{1}{2}} \] and show that only the intermediate position is stable. Examine the case \(l > l'\).
A particle \(A\) of mass \(m_1\) is hung from a fixed point \(O\) by a string of length \(l\) and a particle \(B\) of mass \(m_2\) is hung from \(A\) by a string of equal length. The strings have negligible weight and are inextensible. The particles are constrained to move in a fixed vertical plane through \(O\), and the inclinations of the upper and lower strings to the vertical, \(\theta_1\) and \(\theta_2\) respectively, are so small that their squares and products may be neglected. Show that the equations of motion may be written \begin{align*} (m_1+m_2)l\ddot{\theta}_1 + m_2l\ddot{\theta}_2 + (m_1+m_2)g\theta_1 &= 0, \\ l\ddot{\theta}_1 + l\ddot{\theta}_2 + g\theta_2 &= 0. \end{align*} Deduce that \(\theta_1+\alpha\theta_2\) and \(\theta_1-\alpha\theta_2\) vary in a simple harmonic way with periods \(2\pi\{l(1+\alpha)/g\}^{\frac{1}{2}}\) and \(2\pi\{l(1-\alpha)/g\}^{\frac{1}{2}}\) respectively, where \(\alpha = (1+\frac{m_1}{m_2})^{-\frac{1}{2}}\). Show that if the system is started from rest in the vertical position by giving \(B\) a horizontal velocity \(v\), at time \(t\) later \begin{align*} \theta_1 + \alpha\theta_2 &= \alpha v \left(\frac{1+\alpha}{gl}\right)^{\frac{1}{2}} \sin\left(t\left(\frac{g}{l(1+\alpha)}\right)^{\frac{1}{2}}\right), \\ \theta_1 - \alpha\theta_2 &= -\alpha v \left(\frac{1-\alpha}{gl}\right)^{\frac{1}{2}} \sin\left(t\left(\frac{g}{l(1-\alpha)}\right)^{\frac{1}{2}}\right). \end{align*}
The mass \(m\) of a particle varies with the speed \(v\) according to the law \[ m=m_0(1-v^2/c^2)^{-\frac{1}{2}} \] where \(m_0, c\) are constants. Assuming that the force required to accelerate the particle is the rate of change of momentum, show that when the particle moves in space the force required to produce an acceleration \(\mathbf{a}\) is the resultant of a component \(\frac{m_0\mathbf{a}}{(1-v^2/c^2)^{\frac{3}{2}}}\) in the direction of the acceleration, and a component \(-\frac{m_0(v/c^2)^2 a \sin\theta}{(1-v^2/c^2)^{\frac{1}{2}}}\) which is in the plane containing the velocity and the acceleration and is perpendicular to the velocity; \(\theta\) is the angle between the acceleration and the velocity. Deduce that the equation of energy for motion under a force whose potential energy is \(V\) is \[ m_0c^2(1-v^2/c^2)^{-\frac{1}{2}} + V = E \] where \(E\) is a constant.
A point \(A\) describes a circle of radius \(a\) about the fixed centre \(O\) with constant speed \(a\omega\). A point \(B\) moves along a fixed diameter of the circle and is connected to \(A\) by a rigid rod \(AB\) whose length is \(l\) (\(l>a\)). Find the instantaneous centre of rotation \(I\) of the rod \(AB\), and show that \[ IA = l \left| \frac{\cos\phi}{\cos\theta} \right|, \] where \(\angle AOB = \theta, \angle ABO = \phi\). Prove that the component \(V\) along \(AB\) of the velocity of any point of \(AB\) satisfies \[ \frac{dV}{d\theta} = \frac{1}{4}a\omega \frac{\sin 2(\theta+\phi)}{\sin\theta\cos\phi}. \] Hence or otherwise deduce that if \(v\) is the velocity of \(B\), then \(\frac{dv}{d\theta}\) vanishes when \[ \cot\theta = \frac{1}{2} \sin(2\phi). \]