If \(\alpha, \beta, \gamma, \delta\) are roots of the equation \[x^4+qx^2+rx+s=0,\] prove that \[ \Sigma \alpha^2 = -2q, \quad \Sigma \alpha^2\beta^2 = q^2+2s, \quad \Sigma \alpha^2\beta^2\gamma^2 = r^2-2qs. \] Show that the equation with roots \(\alpha\beta+\gamma\delta, \alpha\gamma+\beta\delta, \alpha\delta+\beta\gamma\) is \[ x^3 - qx^2 - 4sx - r^2 + 4qs = 0, \] and find the equation with roots \((\alpha\beta-\gamma\delta)^2, (\alpha\gamma-\beta\delta)^2, (\alpha\delta-\beta\gamma)^2\). Hence or otherwise show that, if \(\alpha, \beta, \gamma, \delta\) are four numbers whose sum is zero, a necessary and sufficient condition that the product of one pair is equal to the product of the other pair is \(\Sigma \alpha\beta\gamma=0\). Mark in an Argand diagram the points representing four such numbers.
State and prove Leibniz' theorem concerning the \(n\)th derivative of a product \(u(x)v(x)\). If \(y=y_n(x)=x^n e^{-x}\), show that \(xy'=(n-x)y\) and deduce that \[ xy^{(n+2)}+(x+1)y^{(n+1)}+(n+1)y^{(n)}=0. \] If \[ L_n(x) = e^x \frac{d^n}{dx^n}(x^n e^{-x}) \] show that \(L_n(x)\) is a polynomial of degree \(n\), and prove that it satisfies the differential equation \[ xL_n''(x)+(1-x)L_n'(x)+nL_n(x)=0. \] Prove also that \[ L_{n+2}(x)+(x-2n-3)L_{n+1}(x)+(n+1)^2 L_n(x)=0. \]
Prove that the infinite series \(\sum \frac{z^n}{n!}\) is convergent for all values of \(z\), real or complex. State any general theorems on series used in the proof. Sum the series \[ \sum_{n=1}^{\infty} \frac{n^3 z^{n-1}}{(n-1)!} \] Hence or otherwise sum the infinite series \[ 1 - \frac{3^3x^2}{2!} + \frac{5^3x^4}{4!} - \frac{7^3x^6}{6!} + \dots. \]
If \(\phi(u,v) = \phi(x,t)\), where \(u\) and \(v\) are functions of \(x\) and \(t\), show that \[ \frac{\partial \phi}{\partial x} = \frac{\partial f}{\partial u}\frac{\partial u}{\partial x} + \frac{\partial f}{\partial v}\frac{\partial v}{\partial x}, \] and obtain corresponding formulae for \(\frac{\partial^2\phi}{\partial x^2}\) and \(\frac{\partial^2\phi}{\partial t^2}\). In particular consider the transformation \(u=x-ct, v=x+ct\) where \(c\) is constant, and show that the general solution of the equation \[ \frac{\partial^2\phi}{\partial t^2} - c^2 \frac{\partial^2\phi}{\partial x^2} = 0 \] is \(\phi = F(x-ct)+G(x+ct)\) where \(F\) and \(G\) are arbitrary functions. Find the solution such that, when \(t=0\), \(\phi=0\) and \(\frac{\partial\phi}{\partial t} = a \sin kx\) for all values of \(x\).
If \[ f(x) = \frac{d^n}{dx^n}(x^2-1)^n \] and \(p(x)\) is any polynomial of degree less than \(n\), prove that \[ \int_{-1}^1 f(x)p(x)dx=0, \] and hence or otherwise show that \(f(x)\) vanishes for exactly \(n\) values of \(x\) between \(+1\) and \(-1\). If \(F(x)\) is polynomial of degree \(n\) such that \[ \int_{-1}^1 F(x)p(x)dx=0 \] for every polynomial \(p(x)\) of degree less than \(n\), show that \(F(x)\) is a constant multiple of \(f(x)\).
(i) Two coplanar triangles \(PQR\) and \(P'Q'R'\) are in perspective. \(L\) is the point of intersection of \(QR'\) and \(Q'R\); \(M\) and \(N\) are similarly defined. Prove that \(MN, QR\) and \(Q'R'\) are concurrent, and deduce that the triangles \(PQR\) and \(LMN\) are in perspective. (ii) Two coplanar triangles \(PQR\) and \(P'Q'R'\) are in perspective. The triangles \(PQR\) and \(Q'R'P'\) also are in perspective. Prove that the triangles \(PQR\) and \(R'P'Q'\) are in perspective.
State the projective form of the theorem that the locus of the centre of a variable conic through four fixed points is a conic, and prove the theorem in either its projective or its metrical form. Prove that the locus of the centre of a variable conic through four fixed concyclic points is a rectangular hyperbola. Prove also that there are exactly two parabolas through four concyclic points and that their axes are at right angles.
A particle, whose co-ordinates referred to rectangular axes are \((x,y)\), can move in a plane under a force whose components parallel to these axes are \((X,Y)\) where \(X\) and \(Y\) are functions of \(x\) and \(y\). Prove that the expression \(Xdx+Ydy\) for the work done by the force when the particle moves from \((x,y)\) to \((x+dx,y+dy)\) is independent of the choice of axes. Show that the total work done when the particle moves round a closed path is zero for every closed path if, and only if, \(Xdx+Ydy\) is a perfect (exact) differential of a function \(U(x,y)\). Show also that in this case the work done when the particle moves from \((x_1,y_1)\) to \((x_2,y_2)\) is \[ U(x_2,y_2)-U(x_1,y_1) \] for all paths joining these points.
Five equal uniform rods are smoothly jointed at their ends to form a closed pentagon \(ABCDEA\). The rod \(AE\) is fixed horizontally and the rest of the system hangs symmetrically. Prove that if \(AB\) and \(BC\) are inclined to the vertical at angles \(\theta\) and \(\phi\) respectively, \[ 3 \tan \theta = \tan \phi. \] Show that \(\sin\theta\) must lie between 0 and \(\frac{1}{2}\) and satisfy the equation \[ (1-2x)^2 = \frac{36x^2}{8x^2+1}. \] Prove that this equation has exactly one root in this range.
Two particles, of masses \(m\) and \(3m\), are joined by a light inextensible string of length \(4a\). The system can move freely in a smooth horizontal plane. Initially the string is straight, the heavier particle is stationary, and the lighter particle moves with velocity \(v\) at right angles to the string. Describe the subsequent motion, and sketch the paths of the particles. Calculate the maximum speed of the heavier particle in the subsequent motion and the tension in the string.