Problems

The function \(\f\) satisfies \(0\leqslant\f(t)\leqslant K\) when \(0\leqslant t\leqslant x\). Explain by means of a sketch, or otherwise, why \[0\leqslant\int_{0}^{x} \f (t)\,{\mathrm d}t \leqslant Kx.\] By considering \(\displaystyle \int_{0}^{1}\frac{t}{n(n-t)}\,{\mathrm d}t\), or otherwise, show that, if \(n>1\), \[ 0\le \ln \left( \frac n{n-1}\right) -\frac 1n \le \frac 1 {n-1} - \frac 1n \] and deduce that \[ 0\le \ln N -\sum_{n=2}^N \frac1n \le 1. \] Deduce that as \(N\to \infty\) \[ \sum_{n=1}^N \frac1n \to\infty. \] Noting that \(2^{10}=1024\), show also that if \(N<10^{30}\) then \[ \sum_{n=1}^N \frac1n <101. \]

A tortoise and a hare have a race to the vegetable patch, a distance \(X\) kilometres from the starting post, and back. The tortoise sets off immediately, at a steady \(v\) kilometers per hour. The hare goes to sleep for half an hour and then sets off at a steady speed \(V\) kilometres per hour. The hare overtakes the tortoise half a kilometre from the starting post, and continues on to the vegetable patch, where she has another half an hour's sleep before setting off for the return journey at her previous pace. One and quarter kilometres from the vegetable patch, she passes the tortoise, still plodding gallantly and steadily towards the vegetable patch. Show that \[ V= \frac{10}{4X-9} \] and find \(v\) in terms of \(X\). Find \(X\) if the hare arrives back at the starting post one and a half hours after the start of the race.

A particle is attached to a point \(P\) of an unstretched light uniform spring \(AB\) of modulus of elasticity \(\lambda\) in such a way that \(AP\) has length \(a\) and \(PB\) has length \(b\). The ends \(A\) and \(B\) of the spring are now fixed to points in a vertical line a distance \(l\) apart, The particle oscillates along this line. Show that the motion is simple harmonic. Show also that the period is the same whatever the value of \(l\) and whichever end of the string is uppermost.

The force of attraction between two stars of masses \(m_{1}\) and \(m_{2}\) a distance \(r\) apart is \(\gamma m_{1}m_{2}/r^{2}\). The Starmakers of Kryton place three stars of equal mass \(m\) at the corners of an equilateral triangle of side \(a\). Show that it is possible for each star to revolve round the centre of mass of the system with angular velocity \((3\gamma m/a^{3})^{1/2}\). Find a corresponding result if the Starmakers place a fourth star, of mass \(\lambda m\), at the centre of mass of the system.

Show Solution

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Solution: The net force on the planets will always be towards the centre of mass (by symmetry or similar arguments). Therefore it suffices to check whether we can find a speed where the planets follow uniform circular motion, ie \(F = mr \omega^2\). (But clearly this is possible, we just need to find the speed)

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\begin{align*} && F &= m r \omega^2 \\ && 2\frac{\gamma m^2}{a^2} \cos 30^{\circ} &= m \frac{a}{\sqrt{3}} \omega^2 \\ \Rightarrow && \frac{\sqrt{3}\gamma m^2}{a^2} &= \frac{ma \omega^2}{\sqrt{3}} \\ \Rightarrow && \omega^2 &= \frac{3\gamma m}{a^3} \\ \Rightarrow && \omega &= \left ( \frac{3\gamma m}{a^3}\right)^{1/2} \end{align*}

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In the second scenario, we are interested in when: \begin{align*} && F &= m r \omega^2 \\ && \underbrace{2\frac{\gamma m^2}{a^2} \cos 30^{\circ}}_{\text{to other symmetric planets}} + \underbrace{\frac{\gamma \lambda m^2}{a^2}}_{\text{central planet}} &= m \frac{a}{\sqrt{3}} \omega^2 \\ \Rightarrow && \frac{(\sqrt{3}+\lambda)\gamma m^2}{a^2} &= \frac{ma \omega^2}{\sqrt{3}} \\ \Rightarrow && \omega^2 &= \frac{(3+\sqrt{3}\lambda)\gamma m}{a^3} \\ \Rightarrow && \omega &= \left ( \frac{(3+\sqrt{3}\lambda)\gamma m}{a^3}\right)^{1/2} \end{align*}

1. Prove that if \(x>0\) then \(x+x^{-1}\ge2.\;\) I have a pair of six-faced dice, each with faces numbered from 1 to 6. The probability of throwing \(i\) with the first die is \(q_{i}\) and the probability of throwing \(j\) with the second die is \(r_{j}\) (\(1\le i,j \le 6\)). The two dice are thrown independently and the sum noted. By considering the probabilities of throwing 2, 12 and 7, show the sums \(2, 3, \dots, 12\) are not equally likely.
2. The first die described above is thrown twice and the two numbers on the die noted. Is it possible to find values of \(q_{j}\) so that the probability that the numbers are the same is less than \(1/36\)?

Bar magnets are placed randomly end-to-end in a straight line. If adjacent magnets have ends of opposite polarities facing each other, they join together to form a single unit. If they have ends of the same polarity facing each other, they stand apart. Find the expectation and variance of the number of separate units in terms of the total number \(N\) of magnets.

When I throw a dart at a target, the probability that it lands a distance \(X\) from the centre is a random variable with density function \[ \mathrm{f}(x)=\begin{cases} 2x & \text{ if }0\leqslant x\leqslant1;\\ 0 & \text{ otherwise.} \end{cases} \] I score points according to the position of the dart as follows: %

1. Show that my expected score from one dart is 15/8.
2. I play a game with the following rules. I start off with a total score 0, and each time~I throw a dart my score on that throw is added to my total. Then: \newline \hspace*{10mm} if my new total is greater than 3, I have lost and the game ends; \newline \hspace*{10mm} if my new total is 3, I have won and the game ends; \newline \hspace*{10mm} if my new total is less than 3, I throw again. Show that, if I have won such a game, the probability that I threw the dart three times is 343/2231.

Let \(x=10^{100}\), \(y=10^{x}\), \(z=10^{y}\), and let $$ a_1=x!, \quad a_2=x^y,\quad a_3=y^x,\quad a_4=z^x,\quad a_5=\e^{xyz},\quad a_6=z^{1/y},\quad a_7 = y^{z/x}. $$

1. Use Stirling's approximation \(n! \approx \sqrt{2 \pi}\, {n^{n+{1\over2}}\e^{-n}}\), which is valid for large \(n\), to show that \(\log_{10}\left(\log_{10} a_1 \right) \approx 102\).
2. Arrange the seven numbers \(a_1\), \(\ldots\) , \(a_7\) in ascending order of magnitude, justifying your result.

Consider the quadratic equation $$ nx^2+2x \sqrt{pn^2+q} + rn + s = 0, \tag{*} $$ where \(p>0\), \(p\neq r\) and \(n=1\), \(2\), \(3\), \(\ldots\) .

1. For the case where \(p=3\), \(q=50\), \(r=2\), \(s=15\), find the set of values of \(n\) for which equation \((*)\) has no real roots.
2. Prove that if \(p < r\) and \(4q(p-r) > s^2\), then \((*)\) has no real roots for any value of \(n\).
3. If \(n=1\), \(p-r=1\) and \(q={s^2}/8\), show that \((*)\) has real roots if, and only if, \(s \le 4-2\sqrt{2}\ \) or \(\ s \ge 4+2\sqrt{2}\).

Let $$ {\rm S}_n(x)=\mathrm{e}^{x^3}{{\d^n}\over{\d x^n}}{(\mathrm{e}^{-x^3})}. $$ Show that \({\rm S}_2(x)=9x^4-6x\) and find \({\rm S}_3(x)\). Prove by induction on \(n\) that \({\rm S}_n(x)\) is a polynomial. By means of your induction argument, determine the order of this polynomial and the coefficient of the highest power of~\(x\). Show also that if \ \(\displaystyle \frac{\d S_n}{\d x}=0\) \ for some value \(a\) of \(x\), then \( \ S_n(a)S_{n+1}(a)\le0\).