Sketch the curve whose polar equation is \(r^2(\sec n\theta+\tan n\theta)=a^2\), where \(n\) is a positive integer and \(a\) is a constant. In the case \(n=1\) shew that the only real point at which the circle of curvature passes through the pole is given by \(\theta=\tan^{-1}\sqrt{1+\sqrt{\frac{28}{3}}}\).
Explain how to find the intrinsic \((s, \psi)\) form of the equation of a plane curve whose pedal \((p,r)\) equation is known. Shew that the \((s, \psi)\) equation of the curve \(p^3=ar^2\) is \(s=3a\tan\frac{\psi}{2}+a\tan^3\frac{\psi}{2}\), where \(s\) and \(\psi\) are measured from the apse.
Prove that the centre of mass of a uniform lamina bounded by part of the parabola \(y^2=2lx\) and a focal chord of the parabola always lies on the parabola \[ y^2 = \frac{5l}{4}\left(x-\frac{3l}{10}\right) \] whatever the inclination of the focal chord to the axis of the parabola.
On two fixed straight lines, \(p, p'\), fixed points \(A, B, C, A', B', C'\) are taken. Variable points \(P, P'\) are taken on \(p, p'\), respectively, such that the cross ratios \((ABCP), (A'B'C'P')\) are equal. Prove that the line \(PP'\) envelops a conic, and discuss what happens if \(AA', BB', CC'\) meet in a point. A variable conic is drawn through four fixed points, \(A, B, C, D\). A fixed line through \(D\) cuts the conic again in \(P\). Prove that the tangent to the conic at \(P\) envelops a fixed conic inscribed in the triangle \(ABC\), and touching the fixed line at \(D\).
(i) Shew that with a suitable choice of the triangle of reference, the equations of any two coplanar conics may be taken in the form \[ ax^2+by^2+cz^2=0 \] \[ a'x^2+b'y^2+c'z^2=0. \] Deduce that if two conics have four real common tangents, their points of intersection are either all real or all imaginary. Hence or otherwise, shew that the four conics which can be drawn through two given points to touch the sides of a given triangle are either all real, or all imaginary. (ii) A conic touches the sides \(QR, RP, PQ\) of a triangle in the points \(A, B, C\). From any point \(O\) the lines \(OP, OQ, OR\) are drawn to meet the lines \(QR, RP, PQ\), in points \(X, Y, Z\). From \(X, Y, Z\) further tangents are drawn to the conic, meeting each other in points \(A', B', C'\). Prove that the lines \(AA', BB', CC'\) meet in the point \(O\).
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 \(\sin^3\beta=\sin\alpha\).