Prove that, if \[ u_2 = u_1^2 - 1, \quad u_1u_3 = u_2^2 - 1, \quad u_2u_4 = u_3^2 - 1, \quad u_3u_5 = u_4^2 - 1, \dots \] then \[ u_1+u_3 = u_1u_2, \quad u_2+u_4 = u_1u_3, \quad u_3+u_5 = u_1u_4, \quad u_4+u_6 = u_1u_5, \dots. \]
\(ABCD\) is a quadrilateral of smoothly jointed rods, having the angles at \(A\) and \(B\) equal to \(60^\circ\), the angle \(ACB\) equal to \(90^\circ\), and \(AB\) parallel to \(DC\). The framework is kept in shape by a rod joining \(A\) and \(C\). It is supported at \(A\) and \(B\) with \(AB\) horizontal, having loads \(X\) and \(Y\) placed at \(C\) and \(D\). Draw a diagram giving the stresses in the rods, and shew by use of it that the stress in the rod \(AC\) is \(\frac{1}{2}(X-Y)\).
State any rules you know for determining whether a number is divisible by 2, 3, 4, 5, 8, 9, and 11. \par Find a number of three digits, not necessarily different, such that (i) all its digits are prime, (ii) all numbers that can be formed by taking two of its digits are prime, (iii) all numbers that can be formed by taking all three of its digits are prime.
Prove that a function which vanishes with \(x\), is continuous, and has a differential coefficient positive for all positive values of \(x\), is itself positive for all positive values of \(x\). \par Prove that each of the functions \[ 1-\cos x, \quad x-\sin x, \quad \tfrac{1}{2}x^2-1+\cos x, \quad \tfrac{1}{6}x^3-x+\sin x, \quad \tfrac{1}{24}x^4-\tfrac{1}{2}x^2+1-\cos x, \] is positive for all positive values of \(x\). Hence obtain expansions of \(\cos x\) and \(\sin x\) in powers of \(x\) valid for all real values of \(x\).
A 50-ton engine starts from rest with a 10-ton truck: the coupling is initially slack, and when it tightens the engine is running at 2 f.s. Each coupling hook is attached to a spring which extends \(\frac{1}{4}\) inch per ton of pull, the chain itself being inextensible. Find the maximum tension in the chain during the jerk, on the assumption that the engine is running with steam cut off during that time.
Prove that, if \(4x\) lies between \(+1\) and \(-1\), \begin{align*} (1 + \sqrt{1-4x})^4 &= 16 - 64x + 32x^2 - 16x^4 \\ &\quad - 64\left\{x^5 + \frac{7}{2!}x^6 + \frac{8\cdot9}{3!}x^7 + \frac{9\cdot10\cdot11}{4!}x^8 + \dots + \frac{(n+1)(n+2)\cdots(2n-5)}{(n-4)!}x^n + \dots \right\} \end{align*}
A circular disc rests in a vertical plane on a horizontal plane, and in contact with it in the same vertical plane there is a second equal disc which is also in contact with a peg. If all the surfaces in contact are equally rough (\(\mu\)), and the equilibrium is limiting both at the peg and between the two discs, find the position of the peg and the inclination of the line joining the centres of the discs.
Show that if \(ax+by+cz=0\) for all values of \(x, y,\) and \(z\) such that \(\alpha x + \beta y + \gamma z = 0\), then \[ \frac{a}{\alpha} = \frac{b}{\beta} = \frac{c}{\gamma}. \]
The number \(e\) may be defined
A thin uniform metal plate is moving in any manner on a smooth horizontal table; investigate the question whether it can be stopped completely by stopping one point of it, and if so find the point.