A hollow cylinder of radius \(a\) rolls without slipping on the inside of a cylinder of radius \(b(b > a)\). The axes are always horizontal. If \(\theta\) is the angle between the vertical and the line-of-centres of the cylinders (in a plane perpendicular to the axes), obtain the equation of motion \[\ddot{\theta} = -\omega^2\sin\theta,\] where \(\omega^2(b-a) = g\). If the coefficient of limiting friction is \(\mu\), show that two classes of motion are possible: (i) where \(\dot{\theta}^2 \leq \omega^2[1-(1+4\mu^2)^{-\frac{1}{2}}]\), and \(\theta\) oscillates about zero; (ii) where \(\dot{\theta}^2 \geq \omega^2[(1+16\mu^2)^{\frac{1}{2}}/2\mu-1]\), and \(\theta\) increases or decreases monotonically.
A particle of unit mass orbits the sun under an inverse square law of gravity. Interplanetary gas imposes a resistive force which is \(-k\) times the velocity, in magnitude and direction. Use the equation of motion in polar coordinates to show that the angular momentum decreases exponentially with time. If the resistive force is neglected show that the particle can move in a circular orbit, say with angular frequency \(\omega\). If \(k \ll \omega\), so that \(k^2\) can be neglected in comparison with \(\omega^2\), show that the radius of the orbit decreases by a fraction \(4\pi k/\omega\) per revolution, and that the tangential velocity increases by a fraction \(2\pi k/\omega\). Comment on the fact that as a result of the resistive force the velocity actually increases.
Show that \((\mathbf{l} \wedge \mathbf{m}).\mathbf{n} = (\mathbf{n} \wedge \mathbf{l}).\mathbf{m} = (\mathbf{m} \wedge \mathbf{n}).\mathbf{l}\). Hence, or otherwise, show that \[|\mathbf{l} \wedge \mathbf{m}|^2 = |\mathbf{l}|^2|\mathbf{m}|^2-(\mathbf{l}.\mathbf{m})^2.\] If the point \(P\) has position vector \(\mathbf{r}\) given by \[\mathbf{r} = \mathbf{a} + s\mathbf{u}\] show that \(P\) lies on a line if \(s\) is allowed to vary, and explain the geometrical significance of \(\mathbf{a}\) and \(\mathbf{u}\). Suppose two lines are given by equations \[\mathbf{r}_i = \mathbf{a}_i+s_i\mathbf{u}_i, \quad i = 1, 2.\] By considering \(|(\mathbf{r}_1-\mathbf{r}_2) \wedge (\mathbf{u}_1 \wedge \mathbf{u}_2)|^2\), determine necessary and sufficient conditions for the lines to meet, and if they do not meet, find the shortest distance between them in the two cases \(\mathbf{u}_1 \wedge \mathbf{u}_2 = \mathbf{0}\) and \(\mathbf{u}_1 \wedge \mathbf{u}_2 \neq \mathbf{0}\).
Prove that \(\displaystyle \binom{n}{r} = \binom{n-1}{r} + \binom{n-1}{r-1}.\) Hence prove that for \(n > r\) \(\displaystyle \binom{n}{r} = \sum_{i=0}^r \binom{n-i-1}{r-i}.\) [The binomial coefficient \(\displaystyle \binom{n}{r}\) is defined by \(\displaystyle \binom{n}{r} = \frac{n!}{r!(n-r)!}\).]
Show that the sum of the first \(n\) odd positive integers is a perfect square. The odd positive integers are arranged in blocks with \(n\) integers in the \(n\)th block and \(a_n\) denotes the sum of the numbers in the \(n\)th block, so that \(a_1 = 1\), \(a_2 = 3+5\), \(a_3 = 7+9+11\), etc. Find an expression for \(a_n\) and deduce that \(\displaystyle \sum_{r=1}^n r^3 = \frac{1}{4}n^2(n+1)^2\). Show that \(\displaystyle \sum_{r=1}^n r^5 = \frac{1}{12}n^2(n+1)^2(2n^2+2n-1)\) by considering the situation in which the \(n\)th block has \(n^2\) integers.
Show that a number is divisible by 3 if and only if the sum of its digits is divisible by 3. All possible numbers between 1,000 and 10,000 are formed from the digits 0, 1, 2, 3, 5 and 7, no digit being repeated in any one number. What proportion of these numbers is divisible by 3 and what proportion by 6?
Find necessary and sufficient conditions on the coefficients of the quartic equation \[x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0 = 0\] which ensure that whenever \(z\) is a root so is \(1/z\). Hence show that the roots of a quartic equation of this type may be found by solving several appropriate quadratic equations.
Let \[a_n = \frac{1}{2\sqrt{2}}\{(1+\sqrt{2})^n - (1-\sqrt{2})^n\}.\] Establish a linear relationship between \(a_n\), \(a_{n+1}\) and \(a_{n+2}\), and deduce that \(a_n\) is an integer for all positive integers \(n\). Show also that the greatest integer less than or equal to \((1+\sqrt{2})^n/\sqrt{2}\) is always even.
Let \(z = \cos\theta + i\sin\theta\) (\(\theta \neq \pi\)) and \(w = (z-1)(z+1)^{-1}\). Show that \(w\) is purely imaginary, and hence show that the angle in a semi-circle is a right angle.
A matrix \(B\) satisfies \(B^2 = B\) and is known to be of the following form: \[B = \begin{pmatrix} a & 0 & a \\ -b & b & -a \\ -b & 0 & -b \end{pmatrix},\] where \(a\) and \(b\) are non-zero real numbers. Find the matrix \(B\). Find a non-zero column matrix \(Z\) such that \(BZ = 0\), and determine the condition for a column matrix \(X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}\) to satisfy \(BX = X\). Hence, by defining its columns suitably, find an invertible matrix \(P\) such that \[BP = P\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix}.\]