Problems

A uniform rod \(BC\) is suspended from a fixed point \(A\) by stretched springs \(AB\), \(AC\). The springs are of different lengths but the ratio of tension to extension is the same constant \(k\) for each. The rod is not hanging vertically. Show that the ratio of sums of the stretched springs is equal to the ratio of the lengths of the unstretched springs.

A tennis player serves from height \(H\) above the ground, hitting the ball with speed \(v\) at an angle \(\alpha\) below the horizontal. The ball just clears the net of height \(h\) at distance \(a\) from the server and hits the ground a further distance \(b\) beyond the net. Show that \[v^2 = \frac{g(a+b)^2 (1 + \tan^2 \alpha)}{2[H-(a+b)\tan \alpha]}\] with \[\tan \alpha = \frac{(2a+b)H}{a(a+b)} - \frac{(a+b)h}{ab}\] What restriction must be imposed on \(H\) in terms of \(a\), \(b\) and \(h\) for such a serve to be possible?

An ellipse is given by \(x = a\cos\theta, y = b\sin\theta\), where \(a\) and \(b\) are positive.

1. [(i)] Find the equation of the normal at \(\theta\).
2. [(ii)] Show that all the lines joining the feet of the perpendiculars from any point \(P\) on the ellipse to the axes are normal to a second ellipse.
3. [(iii)] Show in a sketch the relationship of the second ellipse to the first.

For the equation \[2x \frac{d^2y}{dx^2} + \frac{dy}{dx} + \frac{1}{2}y = 0, \quad x > 0,\] look for a solution in the form \[y = \sum_{r=0}^{\infty} a_r x^{r+\frac{1}{2}},\] and find a relation between successive coefficients in the sequence \(\{a_r\}\). Hence express this solution in terms of simple functions.

1. [(i)] Prove that \(n^5 - n\) is divisible by 30 for every integer \(n\).
2. [(ii)] Suppose that \(m_1\) is a positive integer divisible by 11. Prove that the integer \(m_2\) obtained by reversing the digits of \(m_1\) is also divisible by 11.

\(C\) is a circle of radius \(r\). Determine the length \(l\) of the side of a regular \(n\)-sided polygon inscribed in \(C\). Suppose that \(P_1\) and \(P_2\) are two \(n\)-sided polygons inscribed in \(C\), and that the lengths of the sides of \(P_1\) are the same as the lengths of the sides of \(P_2\), perhaps in a different order. Deduce that \(P_1\) and \(P_2\) have the same area. Show that, if \(P_1\) is not regular, then a polygon \(P_3\) with \(n\) sides can be inscribed in \(C\) in such a way that \(P_3\) has greater area than \(P_1\), and \(P_3\) has more edges of length \(l\) than \(P_1\) has. Hence prove that of all \(n\)-sided polygons which can be inscribed in \(C\), a regular polygon has the largest area.

Let \(m\) and \(n\) be integers with \(0 \leq m \leq n\). The function \(f_{n,m}(x)\), defined for \(|x| \neq 1\), is given by \[f_{n,m}(x) = \begin{cases} \frac{(x^n-1)(x^{n-1}-1)\ldots(x^{n-m+1}-1)}{(x^m-1)(x^{m-1}-1)\ldots(x-1)} & \text{if } m > 0, \\ 1 & \text{if } m = 0\end{cases}\] Prove that for \(0 < m < n\), \[f_{n,m}(x) = f_{n-1, m-1}(x) + x^m f_{n-1, m}(x).\] Show that \(f_{n,m}(x)\) can be expressed as a polynomial of degree \(m(n-m)\) for \(|x| \neq 1\), and that the value of this polynomial when \(x = 1\) is equal to the binomial coefficient \(\displaystyle \binom{n}{m}\).

Which of the following assertions hold for each positive integer \(n\)? Justify your answer with proofs or counter-examples as appropriate.

1. [(i)] If \(G\) is a group, the equation \(x^n = e\) has at least one solution in \(G\). [\(e\) denotes the identity element of \(G\).]
2. [(ii)] If \(G\) is a group, the equation \(x^n = e\) has at most \(n\) solutions in \(G\).
3. [(iii)] There exists a group with exactly \(n\) elements.
4. [(iv)] There exist infinitely many groups with exactly \(n\) elements, no two of which are isomorphic.

If \[y = \sin^{-1}x\] show that \[(1-x^2)y'' = xy',\] and hence using Leibniz' Theorem evaluate \(y^{(n)}(0)\). Write down the MacLaurin series for \(\sin^{-1}x\). By considering the series expansions of the two functions term by term, show that \[\sin^{-1}x < \frac{x}{1-x^2} \quad \text{for } 0 < x < 1.\]

On a tropical island, there are only two species of animal. Both species feed on the abundant supplies of vegetation and species \(A\) also feeds on species \(B\), which reproduce at a higher rate than species \(A\). The numbers of individuals, \(N_A\) and \(N_B\), can be regarded as continuous functions of time, satisfying the differential equations \[\frac{dN_A}{dt} = 2N_A + N_B,\] \[\frac{dN_B}{dt} = 4N_B - 2N_A.\] At \(t = 0\), there are \(N_0\) animals in species \(A\) and \(4N_0\) in species \(B\). By obtaining a second order equation, or otherwise, find the numbers in each species for \(0 \leq t < \alpha\), where \(\tan \alpha = -2\) and \(\frac{\pi}{2} < \alpha < \pi\). What do you expect to happen to the species for \(t > \alpha\) ?