Prove that the orbit of a projectile in vacuo is a parabola. \par If any number of particles are projected simultaneously from the same point with equal velocities in the same vertical plane, prove that at any instant the particles all lie on a circle, also that their relative directions as seen from one another remain unchanged.
Enunciate and prove the principle of conservation of linear momentum. \par Two equal particles connected by a light string rest with the string taut on a smooth horizontal plane. If one of the particles is struck a blow at right angles to the string, determine the paths of the particles.
State the principle of conservation of energy and prove it for the motion of a particle under gravity. \par Two small rings of equal mass can slide on a smooth parabolic wire with axis vertical and vertex upwards. The rings are connected by an elastic string equal in length to the latus rectum, the modulus of elasticity being equal to four times the weight of either ring. If the rings slide down the two sides of the parabola starting from rest at the vertex, shew that they will come to rest at a depth equal to the latus rectum.
Find the linear factors of \[ a(b-c)^3+b(c-a)^3+c(a-b)^3. \] If \[ x^3+y^3+z^3=mxyz \quad \text{and} \quad \frac{x}{a}+\frac{y}{b}+\frac{z}{c}=0, \quad a^2x+b^2y+c^2z=0, \] shew that \[ a^3+b^3+c^3=mabc. \]
Find the number of combinations of \(n\) letters \(r\) at a time (1) when they are all unlike, (2) when \(r\) of them are alike and the rest unlike. \par Shew that the number of combinations taken \(n\) together of \(3n\) letters, of which \(n\) are \(a\), \(n\) are \(b\), and the rest are unlike, is \(2^{n-1}(n+2)\).
Shew how to sum the series \(a_0+a_1x+\dots+a_nx^n+\dots\), whose coefficients satisfy the relation \(a_n+pa_{n-1}+qa_{n-2}=0\), \(p,q\) being given numbers. \par In the case where \(a_n-8a_{n-1}+7a_{n-2}=0\) and \(a_0=1, a_1=8\), shew that \[ a_n = \tfrac{1}{6}(7^{n+1}-1). \]
If \(p_n\) is the numerator of the \(n\)th convergent of \(a_1+\frac{1}{a_2+}\frac{1}{a_3+}\dots\), shew that \(p_n=a_np_{n-1}+p_{n-2}\). \par Prove that \[ p_nq_{n-4}-q_np_{n-4} = (-1)^{n-1}(a_na_{n-1}a_{n-2}+a_n+a_{n-2}). \]
Shew that if \(n\) can be found so that \(\frac{v_m}{u_m}\) is finite whenever \(m>n\), and the series \(u_1+u_2+\dots\) converges, then the series \(v_1+v_2+\dots\) converges. \par Examine the convergence of the series whose \(n\)th term is \(\frac{x^n}{n^2-x^{2n}}\) for all values of \(x\).
From a variable point \(P\) on a fixed line \(OX\), tangents \(PA, PB\) are drawn to a given circle; prove that \(\tan\angle XPA \cot\angle XPB\) is constant. \par Shew that all the lines for which this constant has the same value with respect to two given circles pass through one or other of two fixed points.
If \(A+B+C=\pi\), prove that