Two equal heavy beads \(A, B\) each of mass \(m\) move on a smooth horizontal wire in the form of a circle of radius \(a\) and centre \(O\). They are joined by a light spring of natural length \(2a\sin\alpha\) and modulus of elasticity \(\lambda\). If the angle \(AOB\) is denoted by \(2\theta\) show that during the motion \[ ma\sin\alpha\,\dot{\theta}^2 + \lambda(\sin\theta-\sin\alpha)^2 \] remains constant. If when the spring is at its greatest compression \(\theta=\beta\), show that maximum extension occurs when \(\sin\theta = 2\sin\alpha - \sin\beta\). What happens if \(2\sin\alpha-\sin\beta>1\)?
Prove that the sum of the roots of the equation \[ \begin{vmatrix} x & h & g \\ h & x & f \\ g & f & x \end{vmatrix} = 0 \] is zero, and that the sum of the squares of the roots is \[ 2 (f^2+g^2+h^2). \] Taking \(f, g, h\) to be real, and assuming that the roots are then all real, prove that no root exceeds \[ 2\sqrt{\tfrac{1}{3}(f^2+g^2+h^2)} \] in absolute value. In what circumstances (if any) can a root be equal to this in absolute value?
If \[ f(\theta) = \sum_{r=1}^n a_r \sin (2r-1)\theta, \] where \(a_1 > a_2 > \dots > a_n > 0\), show by considering \(f(\theta)\sin\theta\), or otherwise, that \[ f(\theta) > 0 \quad (0 < \theta < \pi). \]
A conic \(S\) and three points \(A, B, C\) are given in a plane. A variable point \(P\) is taken on \(S\), the line \(PC\) meets \(S\) again in \(Q\), the line \(QA\) meets \(S\) again in \(R\), and the line \(RB\) meets \(S\) again in \(P'\). By considering the relationship thus set up between \(P\) and \(P'\), or otherwise, prove that in general two triangles \(PQR\) (real or imaginary) can be inscribed in \(S\) so that \(QR, RP, PQ\) pass through \(A, B, C\), respectively. Give a construction for these two triangles, using only straight lines joining known points, and intersections of known lines with one another or with \(S\). Use as few lines as you can.
A conic \(U\) passes through two points \(X, Y\). Show that, by taking \(X, Y\) as two vertices of a triangle of reference \(XYZ\), we can in general write the equation of \(U\) in the form \[ xy=zu, \] where \(u\) is a homogeneous linear function of \(x, y, z\). Hence, or otherwise, prove that, if three (non-degenerate) conics have two points \(X, Y\) in common, the three common chords, not passing through \(X\) or \(Y\), of the three conics taken in pairs are concurrent.
The polynomial \(P(x)\) is defined, for a given positive integer \(n\), by \[ P(x) = \frac{d^n y}{dx^n}, \] where \(y=(x^2-1)^n\). Find the values of \(P(0)\), \(P(1)\), \(P(-1)\). Prove that \[ (x^2-1)P''(x) + 2xP'(x) - n(n+1)P(x) = 0. \]
A function \(f(x, t)\) satisfies the equation \[ k \frac{\partial^2 f}{\partial x^2} = \frac{\partial f}{\partial t}. \quad \text{(I)} \] On transforming the independent variables from \(x, t\) to \(\xi, \tau\), where \[ \xi = \frac{x}{\sqrt(kt)} \quad \text{and} \quad \tau=t, \] the function \(f(x, t)\) is transformed into \(\phi(\xi, \tau)\). Show that \[ \frac{\partial^2\phi}{\partial\xi^2} + \frac{\xi}{2} \frac{\partial\phi}{\partial\xi} = \tau \frac{\partial\phi}{\partial\tau}. \quad \text{(II)} \] Find in the form of an integral the most general solution of (I) of the form \(F\left(\dfrac{x}{\sqrt(kt)}\right)\).
Obtain a reduction formula for \[ u_n = \int_0^{\pi/2} \sin^n x \, dx. \] Prove that, for any positive integer \(n\), \[ n u_n u_{n-1} = \tfrac{1}{2}\pi, \] \[ 0 < u_n < u_{n-1}. \] Hence, or otherwise, prove that \[ n u_n^2 \to \tfrac{1}{2}\pi \quad \text{as} \quad n \to \infty. \]
Five equal straight rods \(AB, BC, CD, DE, EA\), each of weight \(W\), are smoothly hinged together at \(A, B, C, D, E\). The rods are suspended from \(A\), and are kept in the form of a regular pentagon by two light strings \(AC, AD\). Show that the tension in each string is about \(1\cdot902W\).
A stream of particles impinges on a plane surface \(S\). Before impact the stream contains a mass \(\rho\) of particles per unit volume, and the particles are all moving in the same direction with constant velocity \(V\) at an angle \(\alpha\) with the normal to \(S\). The impact is frictionless, and the coefficient of restitution is \(e\). Calculate (i) the force \(p\) per unit area of \(S\) exerted by the stream on \(S\), and (ii) the loss \(\tau\) of kinetic energy per unit volume of the impinging stream caused by the impact. Verify that \[ \tau = \tfrac{1}{2}(1-e)p. \]