Problems

A family of conics having a fixed point S as one focus and major axes of given length \(2a\) along a given line through S is inverted with respect to a circle, centre S and radius \(k\). Find the envelope of the inverse curves.

Find the equation of the normal and the centre and radius of curvature of the curve \(ay^2=x^3\) at the point \((at^2, at^3)\). \par Shew that the length of the arc of the evolute between the points corresponding to \(t=0\) and \(t=1\) is \(\dfrac{13\sqrt{13}-8}{6}a\).

Determine, in terms of \(\theta\) and the length of the latus rectum, the area of the region bounded by a parabola and a focal chord inclined at an angle \(\theta\) to the axis of the parabola. \par Find the volume swept out by this region when revolved about the directrix through an angle \(2\pi\). \par Check both results by considering the special case \(\theta=\dfrac{\pi}{2}\).

O is the centre of a rectangle ABCD. E is the mid-point of CD and F is the mid-point of AD. AB, BC are of length \(2a, 2b\) respectively. \par A uniform lamina in the shape ABCEFA is of mass M. Find the moment of inertia of the lamina about the line through its mass centre and parallel to AB.

(i) A plane figure consisting of points, straight lines and circles is inverted with respect to a circle in the plane; state (without proof) how the most important relations between the elements of the original figure are transferred to the inverse figure. \par (ii) A, B, C, D are four coplanar points, A' is the inverse of A with respect to the circle BCD and B', C', D' are determined similarly; prove, by inversion with respect to any circle whose centre is D, that the six circles BCA', CAB', ABC', ADA', BDB', CDC' have a common point. \par (iii) Expose the fallacy in the following ``argument'':

1. [(a)] ``If two circles touch, their inverses touch'' (true);
2. [(b)] ``concentric circles have double contact and circles having double contact are concentric'' (true);
3. [(c)] ``therefore concentric circles invert into concentric circles'' (untrue).

In a system of generalized homogeneous coordinates \((x,y,z)\) the condition that the lines \(lx+my+nz=0, l'x+m'y+n'z=0\) should be perpendicular is \[ ll'+mm'+2nn' + mn'+m'n+nl'+n'l=0; \] find the envelope (tangential) equation of the circular points at infinity and prove (without using any formulae for areal or trilinear coordinates) that

1. [(i)] the equation of the line at infinity is \(x+y-z=0\);
2. [(ii)] the equation of the circumcircle of the triangle of reference is \[ yz+zx-2xy=0. \]

State (without proof) the rule for expressing the product of two determinants each of the third order as a determinant of the third order. \par If \(\alpha, \beta, \gamma, \delta\) are the roots of the equation \[ ax^4+4bx^3+6cx^2+4dx+e=0, \] and \(\theta = \alpha(\beta\gamma+\alpha\delta)\), prove that \[ -a^3 \times \begin{vmatrix} 1 & \theta & \theta^2 \\ 1 & \beta+\gamma & \beta\gamma \end{vmatrix} \begin{vmatrix} 1 & \alpha+\delta & \alpha\delta \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{vmatrix}^2 = \begin{vmatrix} 2a & -4b & 0 \\ -4b & 12c-2\theta & -4d \\ 0 & -4d & 2e \end{vmatrix} \] and deduce that \(\beta\gamma+\alpha\delta, \gamma\alpha+\beta\delta, \alpha\beta+\gamma\delta\) are the roots of the equation in \(y\) \[ (ay-2c)^3 - 4(ae-4bd+3c^2)(ay-2c) + 16 \begin{vmatrix} a & b & c \\ b & c & d \\ c & d & e \end{vmatrix} = 0. \]

Prove that the arithmetic mean of \(n\) positive numbers which are not all equal exceeds their geometric mean. \par Deduce that \[ nx^{\frac{n-1}{2}} < 1+x+x^2+\dots+x^{n-1}, \] so that, if \(0

Explain in precise language what you mean by the statement that \(u_n\) tends to a limit \(l\) as \(n\) tends to infinity, and evaluate \(\lim_{n\to\infty} x^n\) when it exists, considering the special cases which may arise for various values of \(x\). \par Prove that \[ x\sin\alpha+x^2\sin 2\alpha+\dots+x^n\sin n\alpha = \frac{x\sin\alpha}{1-2x\cos\alpha+x^2} - \frac{\sin(n+1)\alpha - x\sin n\alpha}{1-2x\cos\alpha+x^2}x^{n+1}, \] and examine fully the convergence of the series as \(n\) tends to infinity.

P, Q are two polynomials in \(x\) which satisfy the identity \[ \sqrt{P^2-1} = Q\sqrt{x^2-1}. \] Prove the following results:

1. [(i)] P, Q have no common factor.
2. [(ii)] If P is of degree \(n\) in \(x\), Q is of degree \(n-1\) in \(x\).
3. [(iii)] \(\dfrac{dP}{dx}=kQ\), where \(k\) is some constant.