If \[ p_r = \frac{1 \cdot 4 \cdot 7 \dots (3r-2)}{3 \cdot 6 \cdot 9 \dots (3r)} \quad (r=1, 2, \dots), \] show that \[ 2(p_{2n+1} + p_1 p_{2n} + \dots + p_n p_{n+1}) = \frac{2 \cdot 5 \cdot 8 \dots (6n+2)}{3 \cdot 6 \cdot 9 \dots (6n+3)} \quad (n=0, 1, 2, \dots). \]
If \(k\) and \(l\) are positive numbers, and the sequence \((a_n)\) satisfies the recurrence relation \[ a_{n+1} = k a_n + l a_{n-1}, \] prove that \[ \lim_{n\to\infty} \frac{a_n}{\alpha^n} = \frac{a_2 - \beta a_1}{\alpha(\alpha-\beta)}, \] where \(\alpha\) is the positive root and \(\beta\) the negative root of the equation \[ x^2 - kx - l = 0. \]
If \[ I = \int_0^\pi \frac{x \cos^2 x \sin x}{\sqrt{(1+3 \cos^2 x)}}\,dx, \] show that \[ I = \frac{\pi}{2} \int_0^\pi \frac{\cos^2 x \sin x}{\sqrt{(1+3\cos^2 x)}}\,dx, \] and hence evaluate \(I\).
If \(y = \frac{\sin x}{x}\), show that \[ \frac{d^n y}{dx^n} = u_n \sin x + v_n \cos x, \] where \(u_n\) and \(v_n\) are polynomials in \(1/x\) satisfying the relations \begin{align*} x u_{2n+1} + (2n+1)u_{2n} &= 0, \\ x v_{2n+1} + (2n+1)v_{2n} &= (-1)^n. \end{align*} Prove also that \begin{align*} u_{2n+1} &= u'_{2n} - v_{2n}, \\ v_{2n+1} &= u_{2n} + v'_{2n}, \end{align*} where the dash denotes differentiation with respect to \(x\). Deduce that \(u=u_{2n}\) satisfies the equation \[ x^2 \frac{d^2u}{dx^2} + 2(2n+1)x \frac{du}{dx} + [x^2+2n(2n+1)]u = (-1)^n x. \]
Find the equation of the normal at the point \(T(ct, c/t)\) to the rectangular hyperbola \(xy=c^2\). The normals at three points \(P, Q, R\), with parameters \(p, q, r\), are concurrent. Prove that \[ qr+rp+pq+up+uq+ur=0, \] where \[ pqru = -1. \] Find the quadratic equation whose roots give the feet of the two further normals from the point of intersection of the normals at the points whose parameters have given values \(\theta, \phi\).
Prove that, if the chord joining the points \(P(ap^2, 2ap)\), \(Q(aq^2, 2aq)\) of the parabola \(y^2=4ax\) touches the circle of centre \((b,0)\) and radius \(k\), then \[ k^2(p+q)^2 = 4(apq+b+k)(apq+b-k). \] Prove that, if \(k\) satisfies the equation \[ k^2+4ak-4ab=0, \] then there are an infinite number of triangles inscribed in the parabola and circumscribed to the circle.
A long narrow hollow tube is inclined at an angle \(\alpha\) to the vertical, and a particle of mass \(m\) which lies inside the tube is attached to a point \(O\) of the tube by an elastic string of modulus \(\lambda\) and natural length \(a\). The vertical plane in which the tube lies is rotated about a vertical axis through \(O\) with constant angular velocity \(\omega\). Assuming that the particle lies below the point \(O\), find the least value \(\lambda_0\) of \(\lambda\) which will ensure that the string does not stretch indefinitely. Show that for \(\lambda > \lambda_0\) the particle can oscillate about its position of equilibrium with a period \(2\pi \sqrt{[am/(\lambda-\lambda_0)]}\).
A rod \(OA\) of length \(a\) which lies on a smooth horizontal table is made to rotate with constant angular velocity \(\omega\) about the end \(O\) which is fixed. Another rod \(AB\) of length \(b\) and of negligible weight, which also rests on the table, is clamped at \(A\) so that the angle \(OAB\) is kept constant, and a particle of mass \(m\) is attached to the end \(B\). Find the force and the couple exerted by the clamp at \(A\).
A particle of mass \(m\) is projected with velocity \(v_0\) at an inclination \(\psi_0\) to the horizontal in a medium whose resistance to the particle's motion is \(mkv^2\) at speed \(v\), where \(k\) is a constant. Prove that the horizontal component of the velocity decreases exponentially with the arc length traversed, and show that the length of the trajectory to the highest point of flight is \[ \frac{1}{2k} \log \{1+(kv_0^2/g)[\sin\psi_0+\cos^2\psi_0\log\tan(\frac{1}{2}\psi_0+\frac{1}{4}\pi)]\}. \]
A reel consists of two circular discs of radius \(a\) and negligible weight, joined coaxially to both ends of a uniform solid cylinder of mass \(m\) and radius \(b\). The discs are in contact with a rough horizontal table, and a string wound around the cylinder is pulled in a vertical direction with a tension \(T\) which is less than \(mg\). Prove that if the string is kept vertical the reel will roll along the table without slipping provided \[ T \le \frac{\mu mg(1+\frac{1}{2}b^2/a^2)}{b/a+\mu(1+\frac{1}{2}b^2/a^2)}, \] \(\mu\) being the coefficient of friction between the discs and the table.