A uniform heavy rod of length \(2l\) rests with its ends on a fixed smooth parabola with axis vertical and vertex downwards (latus rectum \(= 4a\)). Shew that if \(l > 2a\) there are three positions of equilibrium and that the horizontal position is then unstable, but that if \(l < 2a\) the only position of equilibrium is horizontal.
Give an account of the properties of the system of confocal conics \[ \frac{x^2}{a^2+\lambda} + \frac{y^2}{b^2+\lambda} = 1. \] Include in your account a proof that the locus of the poles of a given line \(p\) in regard to the confocals is a straight line perpendicular to \(p\) and meeting it in the point where \(p\) touches a confocal; also that the envelope of the polars of a given point \(O\) in regard to the confocals is a parabola which touches the axes of the confocals and whose directrix is the line joining \(O\) to the centre of the confocals. Taking the general conic (in rectangular coordinates) \[ S \equiv ax^2+2hxy+by^2+2gx+2fy+c=0 \] whose tangential equation is \[ \Sigma \equiv Al^2+2Hlm+Bm^2+2Gnl+2Fmn+Cn^2=0 \] shew that the tangential equation of any conic confocal with \(S\) is of the form \[ \Sigma + \mu(l^2+m^2)=0; \] and hence that the conics confocal with \(S\) are given by \[ \Delta S + \mu D + \mu^2 = 0 \] where \(\Delta\) is the discriminant of \(S\) and \(D=0\) is the equation of its director circle.
If \(\alpha, \beta, \gamma\) are the roots of \(x^3 + px + q = 0\), prove that \[ \frac{\alpha^5 + \beta^5 + \gamma^5}{5} = \frac{\alpha^3 + \beta^3 + \gamma^3}{3} \frac{\alpha^2 + \beta^2 + \gamma^2}{2}. \]
On each of a system of confocal ellipses the points whose eccentric angles are \(\alpha\) and \(\beta\) are taken. Prove that the locus of the intersection of the tangents at these points is a hyperbola of eccentricity \(\sec \frac{1}{2}(\alpha+\beta)\). Prove also that the lines which join the two points on any ellipse are normals to another hyperbola of the same eccentricity.
A chain hangs freely in the form of an arc of a circle. Shew that its weight per unit length at any point varies as the square of the secant of the angle which the radius to that point makes with the vertical.
Define a determinant of any order; and give an account, with proofs as far as you think desirable, of the chief properties of determinants of order not exceeding three, including their application to the solution of simultaneous linear equations. By multiplying together the two determinants, \[ \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix}, \quad \begin{vmatrix} bc-f^2 & fg-ch & hf-bg \\ fg-ch & ca-g^2 & gh-af \\ hf-bg & gh-af & ab-h^2 \end{vmatrix} \] or otherwise, prove that the second is the square of the first.
Prove the Binomial Theorem for a positive integral index. If the binomial expansion of \((1+x)^m\), where \(m\) is a positive integer, be written \[ (1+x)^m = 1 + p_1x + p_2x^2 + \dots + p_mx^m, \] shew that \begin{align*} 1 + p_1 + p_2 + \dots + p_m &= 2^m, \\ p_1 + 2p_2 + 3p_3 + \dots + mp_m &= m2^{m-1}, \end{align*} and find the value of \[ 1+p_1^2+p_2^2+\dots+p_m^2. \]
By taking the asymptotes as axes, the equation of a rectangular hyperbola \[ x^2 - y^2 + 2hxy + 2fy + 2gx = 0 \] is reduced to \[ x'y' - k^2 = 0. \] Prove, by the use of invariants, or otherwise, that \[ k^2 = \pm \tfrac{1}{2} (-f^2 + 2fgh + g^2) (1+h^2)^{-\frac{3}{2}}. \]
A uniform cylinder rests on two fixed planes as shewn in the figure; the plane \(AB\) is smooth and the coefficient of friction between the cylinder and the plane \(AC\) is \(\mu\). A horizontal force equal to the weight of the cylinder acts at \(D\), the middle point of the highest generator of the cylinder. Shew that equilibrium is impossible unless \(\alpha\) is greater than \(\frac{\pi}{4}\), and that if \(\alpha = \tan^{-1} 2.4\) there will be equilibrium if \(\mu\) is not less than \(\frac{1}{4}\).
Give an account of methods by which the \(n\)th differential coefficient of certain functions can be found, giving illustrations. Prove that the method of partial fractions enables us to find the \(n\)th differential coefficient of any rational algebraic fraction. Find the \(n\)th differential coefficients of \[ \text{(i) } e^{ax}\cos bx, \quad \text{(ii) } (2+x)^2/(1-x^3). \] Prove that, if \(\sin^{-1}y = a+b\sin^{-1}x\), the values when \(x=0\) of the successive differential coefficients of \(y\) satisfy \[ \frac{d^{n+2}y}{dx^{n+2}} = (n^2-b^2)\frac{d^ny}{dx^n}. \]