Problems

State the rule for expanding a determinant of order \(n\), and find in the form of a determinant the eliminant of the equations \begin{align*} a_{11}x_1 + a_{12}x_2 + \dots + a_{1n}x_n &= 0 \\ a_{21}x_1 + a_{22}x_2 + \dots + a_{2n}x_n &= 0 \\ \vdots \qquad & \vdots \\ a_{n1}x_1 + a_{n2}x_2 + \dots + a_{nn}x_n &= 0. \end{align*} Prove that if \(u_n\) denote the determinant of \(n\)th order \[ \begin{vmatrix} a & 1 & 0 & 0 & \dots \\ 1 & a & 1 & 0 & \dots \\ 0 & 1 & a & 1 & \dots \\ 0 & 0 & 1 & a & \dots \\ \vdots & & & & \ddots \end{vmatrix} \] then \[ u_{n+1}-au_n+u_{n-1}=0. \] Hence prove that \[ u_n = \frac{p^{n+1}-q^{n+1}}{p-q}, \] where \(p, q\) are the roots of \[ x^2-ax+1=0. \]

Prove that, if \[ -1 < x < 1, \] then \(x^n n^s\) tends to zero as the positive integer \(n\) tends to infinity. Sum to \(n\) terms the series whose \(r\)th term is \(rx^r\), and hence find for what values of \(x\) the infinite series is convergent, and find its sum. Sum to infinity, the series whose \(r\)th term is \(r x^r \cosh r\theta\), stating for what values of \(x, \theta\) the series is convergent.

A function \(f(x)\) may be expanded by Taylor's theorem in the neighbourhood of the point \(x=x_0\). Find necessary and sufficient conditions that \(f(x)\) shall be a minimum at the point \(x=x_0\). Assuming that a polynomial \(\phi(x,y)\) may be expanded in the form \[ \phi(x,y) = A + B_1(x-x_0) + B_2(y-y_0) + C_{11}(x-x_0)^2 + 2C_{12}(x-x_0)(y-y_0) + C_{22}(y-y_0)^2 + \dots, \] find values for the constants \(A, B_1, B_2, C_{11}, C_{12}, C_{22}\) in terms of the partial differential coefficients of \(\phi\) at the point \((x_0, y_0)\). Hence find sufficient conditions that \(\phi\) shall be (1) stationary, (2) a minimum, at the point \((x_0, y_0)\), assuming that the constants \(C_{11}, C_{12}, C_{22}\) do not all vanish. Hence prove that the point \(P\) within a triangle such that the sum of the squares of the distances from \(P\) to the vertices is a minimum is the centroid. Shew also that if \(P\) is restricted to lie on a given circle, it will lie on the line joining the centre of the circle to the centroid.

Prove that coplanar couples of equal moment acting on a rigid body are equivalent. A system of forces in one plane acts on a rigid body, and the moments of the system about three non-collinear points of the plane are equal. Prove that the system has the same moment about all points of the plane. Forces \(A, B, C, D, E, F\) act in order in the sides of a regular hexagon. Shew that necessary and sufficient conditions for equilibrium are \[ A-D = C-F = E-B, \] \[ A+B+C+D+E+F = 0. \] What conclusion follows if the first two conditions are satisfied and not the third?

Determine the potential energy of a stretched string. A uniform elastic ring rests horizontally on a smooth sphere of radius \(a\); the natural length of the ring is \(2\pi a \sin\alpha\), and the tension needed to double its length is \(k\) times its weight. Shew that there is equilibrium when the plane of the ring is at a height \(a \cos\theta\) above the centre of the sphere, where \(\theta\) satisfies the equation \[ \tan\theta + 2\pi k = \frac{2\pi k}{\sin\alpha}\sin\theta. \] By considering the graphs of \((\tan\theta+2\pi k)\) and of \(\left(\frac{2\pi k}{\sin\alpha}\sin\theta\right)\) in the range \(0\) to \(\frac{1}{2}\pi\), or otherwise, shew that no such position of equilibrium exists if \(k\) is less than \(\tan^3\beta/2\pi\), where \(\beta\) is the acute angle given by the equation \(\sin^3\beta=\sin\alpha\).

Six uniform heavy rods \(AB, BC, CD, DE, EF, FG\), each of length \(2a\) and weight \(W\), are freely jointed together to form a chain. \(C\) is joined to \(E\) by a light string of length \(\sqrt{2}a\), and \(B\) is joined to \(F\) by a light string of length \(4a/\sqrt{3}\). The ends \(A, G\) are freely hinged to small supports which are held together and slowly moved apart at the same level. Shew that the string \(CE\) is the first to tighten, and that this happens when the length \(AG\) is approximately equal to \(3.96a\).

Each of three particles \(A, B, C\) has a mass \(m\), and \(A\) is joined to \(B\), and \(B\) to \(C\) by similar light springs of natural length \(a\). The particles move in a straight line under no forces save the tensions of the springs. Shew that if the lengths of \(AB, BC\) respectively at time \(t\) are denoted by \(a+x, a+y\) respectively, then \[ \frac{d^2u}{dt^2} + n^2u = 0, \quad \frac{d^2v}{dt^2} + 3n^2v = 0, \] where \(u=x+y, v=x-y\), and \(amn^2\) is the tension required to double the length of either spring. Hence determine the length of \(AB\) at any time if the system, originally at rest with the springs unstretched, is set in motion by an impulse \(I\) on the particle \(C\) in the direction \(AC\).

A point moves in a circle of radius \(a\). If the radius through the point at time \(t\) makes an angle \(\theta\) with a fixed radius, shew that the acceleration of the point has components \(a\ddot{\theta}\) tangentially and \(a\dot{\theta}^2\) towards the centre of the circle. A light rod \(PQ\), of length \(b\), has a massive particle attached at \(Q\). The rod rests on a smooth table when the end \(P\) is seized and moved off in a horizontal circle of radius \(a\) with constant velocity \(a\omega\), the initial position of the rod being outside the circle and in line with the centre \(O\). In the subsequent motion the angle which \(PQ\) makes with \(OP\) produced is denoted by \(\phi\); shew that \[ b\dot{\phi}^2 = (a^2+b^2+2ab\cos\phi)\omega^2. \] Shew further that if \(a=b\), then \(\phi \to \pi\) as \(t\to\infty\).

Show that angles in the same segment of a circle are equal. A rod \(PQ\) slides with its ends \(P, Q\) on the two straight arms of a bent rod. At each position of \(P\) and \(Q\) lines \(PR, QR\) are drawn perpendicular respectively to the arms on which \(P\) and \(Q\) move. Show that, when the bent rod is fixed and \(PQ\) moves, the locus of \(R\) is a circle, and that, when \(PQ\) is fixed and the bent rod is moved, the locus of \(R\) is again a circle, of radius half the former circle and touching it at \(R\).

Prove that the number of combinations of \(n\) things \(r\) at a time is \(n!/\{r!(n-r)!\}\). A pack of cards is dealt (in the usual way) to four players. One player has just 5 cards of a particular suit; prove that the chance that his partner has the remaining 8 cards of that suit is \(1/(4.17.19.37)\).