STEP 2 1999

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

1. Use Stirling's approximation $n!\approx \sqrt{2\pi }{n}^{n+\frac{1}{2}}{\mathrm{e}}^{-n}$, which is valid for large $n$, to show that ${\log }_{10}\left({\log }_{10}{a}_1\right)\approx 102$.
2. Arrange the seven numbers $a_1$, $\dots$ , $a_7$ in ascending order of magnitude, justifying your result.

Consider the quadratic equation $$n{x}^2+2x\sqrt{p{n}^2+q}+rn+s=0,(\ast )$$ where $p>0$, $p\ne r$ and $n=1$, $2$, $3$, $\dots$ .

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

Let $${\mathrm{S}}_n(x)={\mathrm{e}}^{x^3}\frac{{\mathrm{d}}^n}{\mathrm{d}{x}^n}({\mathrm{e}}^{-x^3}).$$ Show that ${\mathrm{S}}_2(x)=9x^4-6x$ and find ${\mathrm{S}}_3(x)$.

Prove by induction on $n$ that ${\mathrm{S}}_n(x)$ 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 $\frac{\mathrm{d}{S}_n}{\mathrm{d}x}=0$ for some value $a$ of $x$, then $S_n(a)S_{n+1}(a)⩽0$.

By considering the expansions in powers of $x$ of both sides of the identity $$(1+x{)}^n(1+x{)}^n\equiv (1+x{)}^{2n},$$ show that $$\sum _{s=0}^n{\left(\genfrac{}{}{0ex}{}{n}{s}\right)}^2=\left(\genfrac{}{}{0ex}{}{2n}{n}\right),$$ where $\left(\genfrac{}{}{0ex}{}{n}{s}\right)=\frac{n!}{s!(n-s)!}$.

By considering similar identities, or otherwise, show also that:

1. if $n$ is an even integer, then $\sum _{s=0}^n{(-1)}^s{\left(\genfrac{}{}{0ex}{}{n}{s}\right)}^2=(-1{)}^{n/2}(\genfrac{}{}{0ex}{}{n}{n/2});$
2. $\sum _{t=1}^{n}2t{\left(\genfrac{}{}{0ex}{}{n}{t}\right)}^2=n\left(\genfrac{}{}{0ex}{}{2n}{n}\right).$

Show that if $\alpha$ is a solution of the equation $$5\cos x+12\sin x=7,$$ then either $$\cos \alpha =\frac{35-12\sqrt{120}}{169}$$ or $\cos \alpha$ has one other value which you should find.

Prove carefully that if $\frac{1}{2}\pi <\alpha <\pi$, then $\alpha <\frac{3}{4}\pi$.

Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ if $$y=\frac{ax+b}{cx+d}.(\ast )$$By using changes of variable of the form $(\ast )$, or otherwise, show that $${\int }_0^1\frac{1}{(x+3{)}^2}\ln \left(\frac{x+1}{x+3}\right)\mathrm{d}x=\frac{1}{6}\ln 3-\frac{1}{4}\ln 2-\frac{1}{12},$$ and evaluate the integrals $${\int }_0^1\frac{1}{(x+3{)}^2}\ln \left(\frac{x^2+3x+2}{(x+3{)}^2}\right)\mathrm{d}x\text{ and }{\int }_0^1\frac{1}{(x+3{)}^2}\ln \left(\frac{x+1}{x+2}\right)\mathrm{d}x.$$

The curve $C$ has equation $$y=\frac{x}{\sqrt{x^2-2x+a}},$$ where the square root is positive. Show that, if $a>1$, then $C$ has exactly one stationary point.

Sketch $C$ when (i) $a=2$ and (ii) $a=1$.

Prove that $$\sum _{k=0}^n\sin k\theta =\frac{\cos \frac{1}{2}\theta -\cos (n+\frac{1}{2})\theta }{2\sin \frac{1}{2}\theta }.(\ast )$$

1. Deduce that, when $n$ is large, $$\sum _{k=0}^n\sin \left(\frac{k\pi }{n}\right)\approx \frac{2n}{\pi }.$$
2. By differentiating $(\ast )$ with respect to $\theta$, or otherwise, show that, when $n$ is large, $$\sum _{k=0}^{n}k{\sin }^2\left(\frac{k\pi }{2n}\right)\approx \left(\frac{1}{4}+\frac{1}{{\pi }^2}\right)n^2.$$

[The approximations, valid for small $\theta$, $\sin \theta \approx \theta$ and $\cos \theta \approx 1-\frac{1}{2}{\theta }^2$ may be assumed.]

In the $Z$--universe, a star of mass $M$ suddenly blows up, and the fragments, with various initial speeds, start to move away from the centre of mass $G$ which may be regarded as a fixed point. In the subsequent motion the acceleration of each fragment is directed towards $G$. Moreover, in accordance with the laws of physics of the $Z$--universe, there are positive constants $k_1$, $k_2$ and $R$ such that when a fragment is at a distance $x$ from $G$, the magnitude of its acceleration is $k_1{x}^3$ if $x<R$ and is $k_2{x}^{-4}$ if $x⩾R$. The initial speed of a fragment is denoted by $u$.

1. For $x<R$, write down a differential equation for the speed $v$, and hence determine $v$ in terms of $u$, $k_1$ and $x$ for $x<R$.
2. Show that if $u<a$, where $2a^2=k_1{R}^4$, then the fragment does not reach a distance $R$ from $G$.
3. Show that if $u⩾b$, where $6b^2=3k_1{R}^4+4k_2/R^3,$ then from the moment of the explosion the fragment is always moving away from $G$.
4. If $a<u<b$, determine in terms of $k_2$, $b$ and $u$ the maximum distance from $G$ attained by the fragment.

$N$ particles $P_1$, $P_2$, $P_3$, $\dots$, $P_N$ with masses $m$, $qm$, $q^{2}m$, $\dots$ , $q^{N-1}m$, respectively, are at rest at distinct points along a straight line in gravity-free space. The particle $P_1$ is set in motion towards $P_2$ with velocity $V$ and in every subsequent impact the coefficient of restitution is $e$, where $0<e<1$. Show that after the first impact the velocities of $P_1$ and $P_2$ are $$\left(\frac{1-eq}{1+q}\right)V\text{ and }\left(\frac{1+e}{1+q}\right)V,$$ respectively.

Show that if $q⩽e$, then there are exactly $N-1$ impacts and that if $q=e$, then the total loss of kinetic energy after all impacts have occurred is equal to $$\frac{1}{2}me\left(1-e^{N-1}\right)V^2.$$

An automated mobile dummy target for gunnery practice is moving anti-clockwise around the circumference of a large circle of radius $R$ in a horizontal plane at a constant angular speed $\omega$. A shell is fired from $O$, the centre of this circle, with initial speed $V$ and angle of elevation $\alpha$. Show that if $V^2<gR$, then no matter what the value of $\alpha$, or what vertical plane the shell is fired in, the shell cannot hit the target.

Assume now that $V^2>gR$ and that the shell hits the target, and let $\beta$ be the angle through which the target rotates between the time at which the shell is fired and the time of impact. Show that $\beta$ satisfies the equation $$g^2{\beta }^4-4{\omega }^2{V}^2{\beta }^2+4R^2{\omega }^4=0.$$ Deduce that there are exactly two possible values of $\beta$.

Let ${\beta }_1$ and ${\beta }_2$ be the possible values of $\beta$ and let $P_1$ and $P_2$ be the corresponding points of impact. By considering the quantities $({\beta }_1^2+{\beta }_2^2)$ and ${\beta }_1^2{\beta }_2^2$, or otherwise, show that the linear distance between $P_1$ and $P_2$ is $$2R\sin \left(\frac{\omega }{g}\sqrt{V^2-Rg}\right).$$

It is known that there are three manufacturers $A,B,C,$ who can produce micro chip MB666. The probability that a randomly selected MB666 is produced by $A$ is $2p$, and the corresponding probabilities for $B$ and $C$ are $p$ and $1-3p$, respectively, where $0⩽p⩽\frac{1}{3}.$ It is also known that $70\%$ of MB666 micro chips from $A$ are sound and that the corresponding percentages for $B$ and $C$ are $80\%$ and $90\%$, respectively.

Find in terms of $p$, the conditional probability, $\mathrm{P}(A|S)$, that if a randomly selected MB666 chip is found to be sound then it came from $A$, and also the conditional probability, $\mathrm{P}(C|S)$, that if it is sound then it came from $C$.

A quality inspector took a random sample of one MB666 micro chip and found it to be sound. She then traced its place of manufacture to be $A$, and so estimated $p$ by calculating the value of $p$ that corresponds to the greatest value of $\mathrm{P}(A|S)$. A second quality inspector also a took random sample of one MB666 chip and found it to be sound. Later he traced its place of manufacture to be $C$ and so estimated $p$ by applying the procedure of his colleague to $\mathrm{P}(C|S)$.

Determine the values of the two estimates and comment briefly on the results obtained.

A stick is broken at a point, chosen at random, along its length. Find the probability that the ratio, $R$, of the length of the shorter piece to the length of the longer piece is less than $r$.

Find the probability density function for $R$, and calculate the mean and variance of $R$.

You play the following game. You throw a six-sided fair die repeatedly. You may choose to stop after any throw, except that you must stop if you throw a 1. Your score is the number obtained on your last throw. Determine the strategy that you should adopt in order to maximize your expected score, explaining your reasoning carefully.