An engine and tender contain a quantity of fuel which is steadily consumed at the uniform rate of \(\mu\) units of mass per unit time. The engine thereby always does \(Q\) units of work per unit time and the resistance to its motion is \(k\) times the velocity. Assuming that the engine is moving on a level track, and denoting by \(M_0, v_0\) respectively the total mass and velocity at time \(t=0\), show that the velocity \(v\) at time \(t\) is given by the relation \[ Q - kv^3 = (Q-kv_0^3)(1-\mu t/M_0)^{2k/\mu}. \]
If two triangles are in perspective from a point, prove that the three points of intersection of pairs of corresponding sides lie in a straight line (the axis of perspective). Show that, if three triangles are in perspective, the axes of perspective of the three pairs of triangles are concurrent.
The coordinates of a point on a curve are \((at+bt^2, ct+dt^2)\), where \(t\) is a parameter. Prove that the equation of the chord joining the points \(t_1\) and \(t_2\) is \[ \begin{vmatrix} x & y & t_1t_2 \\ a & c & t_1+t_2 \\ b & d & -1 \end{vmatrix} = 0. \] If the tangents at the points \(t_1\) and \(t_2\) are at right angles, show that the chord passes through a fixed point, and find its coordinates.
A, B, C and D are the points of intersection of two conics S and S'. A variable line through A meets S in X, S' in Y and BC in Z. Prove that the locus of P, the harmonic conjugate of Z with respect to X and Y, is a conic passing through A, B, C and D.
In an Argand diagram a quadrilateral (which may be crossed) has its vertices at the points \(ab, aB, AB\) and \(Ab\) taken in that order, where \(a, b, A\) and \(B\) are any non-zero complex numbers. Prove that the origin \(z=0\) cannot be inside the quadrilateral (or, in the case of a crossed quadrilateral, inside either of the triangles formed by the sides of the quadrilateral).
If \(\Delta_n\) denotes the determinant \[ \begin{vmatrix} \lambda & 1 & 0 & 0 & \cdots \\ 1 & \lambda & 1 & 0 & \cdots \\ 0 & 1 & \lambda & 1 & \cdots \\ 0 & 0 & 1 & \lambda & \cdots \\ \vdots & \vdots & \vdots & \vdots & \ddots \end{vmatrix} \] with \(n\) rows and columns (where elements \(a_{rr}\) in the main diagonal all have the value \(\lambda\), elements \(a_{r,r+1}\) and \(a_{r+1,r}\) all have the value 1, and the rest vanish) prove that \[ \Delta_n = \lambda \Delta_{n-1} - \Delta_{n-2}. \] Deduce that, if \(\lambda=2\cos\theta\), the value of the determinant \(\Delta_n\) is \(\sin(n+1)\theta \operatorname{cosec}\theta\).
Assume that, if a function of \(x\) vanishes for two values of \(x\), its derivative vanishes for an intermediate value of \(x\). If \[ \phi(x) = \int_x^b f(t) dt - \tfrac{1}{2}(b-x)\{f(x)+f(b)\} + \tfrac{1}{12}(b-x)^3 R, \] where the constant \(R\) is so chosen that \(\phi(a)\) vanishes, show that \(\phi'(x)\) vanishes for \(x=\alpha\) and \(x=b\), where \(a< \alpha< b\). Deduce that \[ \int_a^b f(t) dt = \tfrac{1}{2}(b-a)\{f(a)+f(b)\} - \tfrac{1}{12}(b-a)^3 f''(\beta), \] where \(a< \beta < b\). Hence show that the difference between \(\int_a^{a+nh} f(t) dt\) and \[ \tfrac{1}{2}h\{f(a)+f(a+nh)\} + h\{f(a+h)+f(a+2h)+ \dots + f(a+\overline{n-1}h)\} \] is less than \(\frac{1}{12}nh^3M\), where \(M\) is the greatest value of \(|f''(t)|\) in \(a< t< a+nh\).
By the use of Maclaurin's theorem, or otherwise, prove that \[ \sin x \sinh x = \frac{2x^2}{2!} - \frac{2^3x^6}{6!} + \frac{2^5x^{10}}{10!} - \dots. \]
A function \(z=f_m(x)\) is defined as the solution of the differential equation \[ \frac{dz}{dx} = m \frac{z}{x} \] (where \(m\) is constant) such that \(z=1\) when \(x=1\). Without solving the differential equation explicitly prove that \begin{align*} f_m(x) f_n(x) &= f_{m+n}(x), \\ f_m(x) f_m(y) &= f_m(xy). \end{align*} Deduce the values of \(f_m(1)\) and \(f_0(x)\), and prove that \[ f_{-m}(x) = f_m\left(\frac{1}{x}\right) = [f_m(x)]^{-1}. \]
Establish the equations \[ x=c\sinh^{-1}\frac{s}{c}, \quad y=\sqrt{(s^2+c^2)}, \quad T=wy \] for a uniform flexible chain hanging under gravity. If \(O\) is the lowest point of the chain (\(s=0\)) and \(P\) is any other point, verify from these equations that the moment about \(O\) of the weight of the portion \(OP\) is equal to the moment of the tension at \(P\).