Problems

A chain of length \(b\) is trailed on level ground behind a uniformly moving cart to which it is attached at height \(h\) above the ground. Prove that the length \(a\) of chain in contact with the ground satisfies the equation \[ a^2 - 2(b+\mu h)a + b^2 - h^2 = 0, \] where \(\mu\) is the coefficient of friction. Find which of the roots of this quadratic equation represent possible values of \(a\).

A particle of mass \(m\) moves in a straight line under a force \(mf(t)\); the motion is opposed by a resistance \(mkv\) where \(k\) is constant and \(v\) is the velocity. The function \(f(t)\) has the constant value \(F\) when \(0< t< T, 2T 0\), \[ v = u(t) + Ae^{-kt}, \] where \(A\) is an arbitrary constant and \(u(t)\) is a periodic function of \(t\), of period \(2T\). Determine the function \(u(t)\).

Find whether any of the roots of the equation \[ x^5 + 8x^4 + 6x^3 - 42x^2 - 19x - 2 = 0 \] are integers, and solve it completely.

If \(a, b, c\) are three constants, all different, show that the system of equations \begin{align*} x+y+z &= 3(a+b+c), \\ ax+by+cz &= (a+b+c)^2, \\ xyz &= ayz+bzx+cxy \end{align*} has in general only one set of unequal solutions, and find that set.

Resolve the expression \[ y = \frac{2(1-x)}{(x^2+1)^2(x+1)} \] into real partial fractions. Show that \[ \int_0^\infty y\,dx = \frac{\pi}{2}-1. \]

If \(\alpha, \beta, \gamma\) are three real angles, and if the equations \begin{align*} x\cos\alpha + y\sin\alpha &= 1, \\ x\cos\beta + y\sin\beta &= 1, \\ x\cos\gamma + y\sin\gamma &= 1 \end{align*} hold simultaneously, show that \(\cos 2\alpha, \cos 2\beta\) and \(\cos 2\gamma\) cannot all be different.

The altitudes of an obtuse-angled triangle \(ABC\) intersect at a point \(H\). Prove that the circumcircle subtends at \(H\) an angle \(\theta\) whose cosine is \[ \frac{8\cos A\cos B\cos C+1}{8\cos A\cos B\cos C-1}, \] and show that \(\theta\) is always greater than \(2\sin^{-1}\frac{1}{3}\).

Obtain an expression for \(\tan 7\theta\) in terms of \(\tan\theta\), and find the value of \[ \cot\frac{\pi}{7}\cot\frac{2\pi}{7}\cot\frac{3\pi}{7}. \] Prove that \[ \cot\frac{\pi}{7}+\cot\frac{2\pi}{7}-\cot\frac{3\pi}{7} = \sqrt{7}. \]

Find the sum of the first \(n\) terms of each of the following series

1. \(\displaystyle\frac{1}{2\cdot5\cdot8} + \frac{1}{5\cdot8\cdot11} + \frac{1}{8\cdot11\cdot14} + \dots\);
2. \(\cos^3\theta - \frac{1}{3}\cos^3 3\theta + \frac{1}{9}\cos^3 9\theta - \frac{1}{27}\cos^3 27\theta + \dots\).

A circular field of unit radius is divided in two by a straight fence. In the smaller segment a circular enclosure as large as possible is fenced off. Show that the total area of the two remaining pieces of the segment can at most be \(\psi-\sin\psi\), where \(\psi=2\tan^{-1}(4/\pi)\).