Prove that \[ \sin\theta \sum_{r=1}^n \sin(2r-1)\theta = \sin^2 n\theta. \] Hence, or otherwise, prove that \[ \sin^3\theta \sum_{r=1}^n (2r-1)^2 \sin(2r-1)\theta \] \[ = (4n^2+1)\sin^2\theta \sin^2 n\theta - 4n \sin\theta \sin n\theta (1-\cos\theta \cos n\theta) - 2(\sin n\theta - n\sin\theta)^2. \] By considering the case when \(\theta\) is very small, deduce the sum to \(n\) terms of the series \(1^3+3^3+5^3+\dots\).
Show that every sequence of numbers \(s_n\) (\(n=0, 1, 2, \dots\)) which satisfies the recurrence relation \[ s_{n+2} - (\alpha+\beta)s_{n+1} + \alpha\beta s_n = 0 \quad (\alpha \ne \beta) \] is of the form \(a\alpha^n + b\beta^n\), where \(a\) and \(b\) are arbitrary constants independent of \(n\). Find \(u_n, v_n\) if \begin{align*} u_{n+2} - 8u_{n+1} - 9v_n &= 0 \\ v_{n+2} - 8v_{n+1} - 9u_n &= 0 \end{align*} (\(n=0, 1, 2, \dots\)), and \(u_0=v_0=0, u_1 = -5+\sqrt{7}, v_1 = -5-\sqrt{7}\).
The nine numbers \(l_i, m_i, n_i\) (\(i=1,2,3\)) satisfy the six relations \begin{align*} l_i l_j + m_i m_j + n_i n_j &= 0, \quad i \ne j, \\ l_i^2 + m_i^2 + n_i^2 &= 1. \end{align*} If \[ \Delta = \begin{vmatrix} l_1 & m_1 & n_1 \\ l_2 & m_2 & n_2 \\ l_3 & m_3 & n_3 \end{vmatrix}, \] show that \(\Delta l_1 = m_2 n_3 - m_3 n_2\) and that \(\Delta^2 l_1^2 = l_2^2 - l_3^2\). Hence, or otherwise, show that \(\Delta^2=1\). Establish the equivalence of the two sets of equations \[ \left\{ \begin{aligned} X &= l_1 x + m_1 y + n_1 z \\ Y &= l_2 x + m_2 y + n_2 z \\ Z &= l_3 x + m_3 y + n_3 z \end{aligned} \right. \quad \text{and} \quad \left\{ \begin{aligned} x &= l_1 X + l_2 Y + l_3 Z \\ y &= m_1 X + m_2 Y + m_3 Z \\ z &= n_1 X + n_2 Y + n_3 Z. \end{aligned} \right. \]
If \[ I_{m,n} = \int_0^1 t^n (1-t)^m \, dt \quad (m > -1, n > -1) \] show that \[ (m+1)I_{m,n+1} = (n+1)I_{m+1,n}. \] Explain why the restrictions on the values of \(m\) and \(n\) are necessary. If \(n\) is a positive integer, show that
The four lines \(BCP, CAQ, ABR, PQR\) have equations \[ u_i = l_i x + m_i y + 1 = 0 \] for \(i=1, 2, 3, 4\) respectively, referred to rectangular Cartesian coordinates. Obtain the equation of the circle on \(AP\) as diameter in the form \[ (l_3 u_2 - l_2 u_3)(l_4 u_1 - l_1 u_4) + (m_3 u_2 - m_2 u_3)(m_4 u_1 - m_1 u_4) = 0. \] Deduce that the circles on \(AP, BQ, CR\) as diameters are coaxal. Hence show that the middle points of the diagonals of a complete quadrilateral are collinear.
Consider the two propositions:
A cylinder, of arbitrary cross-section, lies in equilibrium on a fixed perfectly rough horizontal plane, being in contact with the plane along a generator of the cylinder. Show that the equilibrium is stable or unstable according as \(h\) is less or greater (respectively) than \(\rho\), where \(h\) is the height of the centre of gravity of the cylinder above the plane and \(\rho\) the radius of curvature of a normal cross-section of the cylinder at a point where it touches the plane. Show that, if the cross-section of the cylinder is a parabola of latus rectum \(4a\), and its centre of gravity lies on the axis of one of its cross-sections and at a distance \(b\) from the vertex, where \(b>2a\), then there is a position of stable equilibrium in which the axis of the cross-section makes an angle \(\tan^{-1}\left(\frac{a}{b-2a}\right)^{\frac{1}{2}}\) with the horizontal.
A light uniform elastic string of natural length \(8l\) and modulus \(\lambda\) has its ends fixed at two points \(A\) and \(D\) at a distance \(8a\) apart. Two particles, each of mass \(m\), are attached to the string at \(B\) and \(C\) respectively, so that in equilibrium (gravity being neglected throughout) \(AB=CD=3a, BC=2a\). The system is then set in motion by giving the particle at \(B\) a velocity \(u\) towards \(C\). Denoting by \(x\) and \(y\) the displacements of the particles from their equilibrium positions, measured in the same sense, show that, if all parts of the string remain taut, then \(x+y\) and \(x-y\) vary in simple harmonic manner with periods \(2\pi/p\) and \(\pi/p\) respectively, where \(p^2=\lambda/3ml\). Find the largest value of \(u\) such that all parts of the string remain taut.
A raindrop is of mass \(m_0\) and at rest at time \(t=0\). It then falls through a cloud which is at rest, and, while it is falling, water condenses on the drop so that its mass increases at a constant rate \(c\). When the mass of the drop is \(m\) and its velocity \(v\) the frictional resistance to its motion is \(mkv\) where \(k\) is a constant. Obtain a differential equation governing the variation of \(v\) with the time \(t\), and hence express \(v\) explicitly in terms of \(t\) and the given constants.
A hollow uniform circular cylinder of mass \(M\) is free to roll on a perfectly rough horizontal plane. Initially the cylinder is at rest, and a smooth particle of mass \(m\) is gently placed at the mid-point of the highest generator of the cylinder, and slightly disturbed so that it moves in a plane perpendicular to the axis of the cylinder. Denoting by \(\theta\) the inclination to the vertical of the plane through the particle and the axis of the cylinder at any instant (measured so that initially \(\theta=0\)), and by \(\omega\) the angular velocity of the cylinder (measured in the same sense as \(\dot{\theta}\)) show from the equations of motion that \[ (2M+m)\omega + m\dot{\theta}\cos\theta=0 \] as long as the particle remains in contact with the cylinder. Hence, or otherwise, show that the reaction between the particle and the cylinder at any instant is \[ \frac{2Mmg}{(2M+m\sin^2\theta)^2} \{2M(3\cos\theta-2) - m(1-\cos\theta)^2(2+\cos\theta)\}. \]