Two particles, masses \(M\) and \(m\) (\(M>m\)), are attached to the ends of a string, length \(2l\), which passes over a smooth peg at a height \(l\) above a smooth plane inclined at an angle \(\alpha\) to the vertical. The particles are initially held at rest on the plane at the point vertically below the peg, \(M\) being below \(m\). Prove that, if the particles are released, \(m\) will oscillate through a vertical distance \(2M(M-m)l/(m^2 \sec^2\alpha - M^2)\), provided that \(\tan^2\alpha\) is greater than \((3M+m)(M-m)/m^2\).
Find the remainder when a polynomial \(f(x)\) is divided by (i) \(x-a\), (ii) \((x-a)(x-b)\). Find all the factors of \((y-z)^5+(z-x)^5+(x-y)^5\). Prove that \((y-z)^7+(z-x)^7+(x-y)^7\) is divisible by \((x^2+y^2+z^2-yz-zx-xy)^2\).
Having given that a quadratic function of \(x\) assumes the values \(V_1, V_2, V_3\) for the values \(x=a, x=b, x=c\) prove that the function must be \[ V_1 \frac{(x-b)(x-c)}{(a-b)(a-c)} + V_2 \frac{(x-c)(x-a)}{(b-c)(b-a)} + V_3 \frac{(x-a)(x-b)}{(c-a)(c-b)}. \] A variable quantity which can be represented by a quadratic function of the time assumes the values 144, 15.6, 18 at 10 a.m., 1 p.m. and 2 p.m. on a certain day. Find the value at noon, and find at what time the quantity assumes its least value.
Find the number of homogeneous products of \(n\) dimensions formed from \(r\) letters \(a,b,c,\dots,k\); and shew that the sum of such products is equal to \(\sum \frac{a^{n+r-1}}{(a-b)(a-c)\dots(a-k)}\).
Explain briefly the theory of recurring series, shewing that if \(2r\) terms of the series are given it can in general be continued as a recurring series of the \(r\)th order in one way only. Find the \((n+1)\)th term of the recurring series \[ -2+2x+14x^2+50x^3+\dots. \]
Explain the method of proving theorems by mathematical induction. Shew that the series \[ \frac{1}{u_0} + \frac{x}{u_0 u_1} + \frac{x^2}{u_0 u_1 u_2} + \dots + \frac{x^n}{u_0 u_1 \dots u_n} \] is identically equal to a continued fraction with \(n+1\) components, in which the first is \(\frac{1}{u_0}\), the second is \(-\frac{u_0^2 x}{u_0 x+u_1}\) and the rth is \(-\frac{u_{r-1}^2 x}{u_{r-1}x+u_r}\). Hence, or otherwise, prove that \[ \frac{1}{1-} \frac{1}{3-} \frac{4}{5-} \dots \frac{n^2}{-2n+1} = 1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{n+1}. \]
Make rough drawings of the curves (i) \(y = \dfrac{x^2}{1+x^2}\); (ii) \(y = \dfrac{1-x+x^2}{1+x+x^2}\); (iii) \(y = \lim_{n\to\infty} \dfrac{x^{2n}\tan\frac{\pi x}{2} + x}{x^{2n}+1}\).
Differentiate (i) \(\dfrac{(1+x^2)^{\frac{1}{2}}+(1-x^2)^{\frac{1}{2}}}{(1+x^2)^{\frac{1}{2}}-(1-x^2)^{\frac{1}{2}}}\); (ii) \(\sin^{-1}\{\log(x^2-1)\}\). Prove that, if \[ V(u,v,w) = \begin{vmatrix} u, & v, & w \\ u', & v', & w' \\ u'', & v'', & w'' \end{vmatrix}, \] where dashes denote differentiation with regard to \(x\), then \[ V(u,v,w) = w^3 V\left(u/w, v/w, 1\right). \]
Find a formula for the radius of curvature at a point on a curve \(\phi(x,y)=0\). Prove that the equation of the evolute of the hypocycloid \(x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}\) is \[ (x+y)^{\frac{2}{3}}+(x-y)^{\frac{2}{3}} = 2a^{\frac{2}{3}}. \]
Prove the following results: \[ \int_0^\pi \frac{dx}{a+b\cos x} = \frac{\pi}{\sqrt{(a^2-b^2)}} \quad (a^2>b^2 \text{ and } a>0). \] \[ \int_{-1}^1 \frac{\sin \alpha dx}{1-2x\cos\alpha+x^2} = \frac{1}{2}\pi \quad (0<\alpha<\pi) \] \[ = -\frac{1}{2}\pi \quad (\pi<\alpha<2\pi). \] \[ \int_0^1 x^{2n-1}\log(1+x)dx = \frac{1}{2n}\left\{ \frac{1}{1\cdot2}+\frac{1}{3\cdot4}+\dots+\frac{1}{(2n-1)2n} \right\}. \]