STEP 2 2005

Find the three values of $x$ for which the derivative of $x^2{\mathrm{e}}^{-x^2}$ is zero.

Given that $a$ and $b$ are distinct positive numbers, find a polynomial $\mathrm{P}(x)$ such that the derivative of $\mathrm{P}(x){\mathrm{e}}^{-x^2}$ is zero for $x=0$, $x=\pm a$ and $x=\pm b$, but for no other values of $x$.

For any positive integer $N$, the function $\mathrm{f}(N)$ is defined by $$\mathrm{f}(N)=N(1-\frac{1}{p_1})(1-\frac{1}{p_2})\cdots (1-\frac{1}{p_k})$$ where $p_1$, $p_2$, $\dots$ , $p_k$ are the only prime numbers that are factors of $N$.

Thus $\mathrm{f}(80)=80(1-\frac{1}{2})(1-\frac{1}{5})$.

1. (a) Evaluate $\mathrm{f}(12)$ and $\mathrm{f}(180)$. (b) Show that $\mathrm{f}(N)$ is an integer for all $N$.
2. Prove, or disprove by means of a counterexample, each of the following: (a) $\mathrm{f}(m)\mathrm{f}(n)=\mathrm{f}(mn)$; (b) $\mathrm{f}(p)\mathrm{f}(q)=\mathrm{f}(pq)$ if $p$ and $q$ are distinct prime numbers; (c) $\mathrm{f}(p)\mathrm{f}(q)=\mathrm{f}(pq)$ only if $p$ and $q$ are distinct prime numbers.
3. Find a positive integer $m$ and a prime number $p$ such that $\mathrm{f}(p^m)=146410$.

Give a sketch, for $0⩽x⩽\frac{1}{2}\pi$, of the curve $$y=(\sin x-x\cos x),$$ and show that $0⩽y⩽1$.

Show that:

1. ${\int }_0^{\frac{1}{2}\pi }y\mathrm{d}x=2-\frac{\pi }{2};$
2. ${\int }_0^{\frac{1}{2}\pi }{y}^2\mathrm{d}x=\frac{{\pi }^3}{48}-\frac{\pi }{8}.$

Deduce that ${\pi }^3+18\pi <96$.

The positive numbers $a$, $b$ and $c$ satisfy $bc=a^2+1$. Prove that $$\arctan \left(\frac{1}{a+b}\right)+\arctan \left(\frac{1}{a+c}\right)=\arctan \left(\frac{1}{a}\right).$$The positive numbers $p$, $q$, $r$, $s$, $t$, $u$ and $v$ satisfy $$st=(p+q{)}^2+1,uv=(p+r{)}^2+1,qr=p^2+1.$$ Prove that $$\arctan \left(\frac{1}{p+q+s}\right)+\arctan \left(\frac{1}{p+q+t}\right)+\arctan \left(\frac{1}{p+r+u}\right)+\arctan \left(\frac{1}{p+r+v}\right)=\arctan \left(\frac{1}{p}\right).$$Hence show that $$\arctan \left(\frac{1}{13}\right)+\arctan \left(\frac{1}{21}\right)+\arctan \left(\frac{1}{82}\right)+\arctan \left(\frac{1}{187}\right)=\arctan \left(\frac{1}{7}\right).$$ [ Note that $\arctan x$ is another notation for ${\tan }^{-1}x.$]

The angle $A$ of triangle $ABC$ is a right angle and the sides $BC$, $CA$ and $AB$ are of lengths $a$, $b$ and $c$, respectively. Each side of the triangle is tangent to the circle $S_1$ which is of radius $r$. Show that $2r=b+c-a$.

Each vertex of the triangle lies on the circle $S_2$. The ratio of the area of the region between $S_1$ and the triangle to the area of $S_2$ is denoted by $R$. Show that $$\pi R=-(\pi -1)q^2+2\pi q-(\pi +1),$$ where $q=\frac{b+c}{a}$. Deduce that $$R⩽\frac{1}{\pi (\pi -1)}.$$

The position vectors, relative to an origin $O$, at time $t$ of the particles $P$ and $Q$ are $$\cos t[[B]]i[[/B]]+\sin t[[B]]j[[/B]]+0[[B]]k[[/B]]and\cos (t+\frac{1}{4}\pi )[\frac{3}{2}[[B]]i[[/B]]+\frac{3\sqrt{3}}{2}[[B]]k[[/B]]]+3\sin (t+\frac{1}{4}\pi )[[B]]j[[/B]],$$ respectively, where $0⩽t⩽2\pi$.

1. Give a geometrical description of the motion of $P$ and $Q$.
2. Let $\theta$ be the angle $POQ$ at time $t$ that satisfies $0⩽\theta ⩽\pi$. Show that $$\cos \theta =\frac{3\surd 2}{8}-\frac{1}{4}\cos (2t+\frac{1}{4}\pi ).$$
3. Show that the total time for which $\theta ⩾\frac{1}{4}\pi$ is $\frac{3}{2}\pi$.

For $x⩾0$ the curve $C$ is defined by $$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{x^3{y}^2}{(1+x^2{)}^{5/2}}$$ with $y=1$ when $x=0$. Show that $$\frac{1}{y}=\frac{2+3x^2}{3(1+x^2{)}^{3/2}}+\frac{1}{3}$$ and hence that for large positive $x$ $$y\approx 3-\frac{9}{x}.$$ Draw a sketch of $C$.

On a separate diagram draw a sketch of the two curves defined for $x⩾0$ by $$\frac{\mathrm{d}z}{\mathrm{d}x}=\frac{x^3{z}^3}{2(1+x^2{)}^{5/2}}$$ with $z=1$ at $x=0$ on one curve, and $z=-1$ at $x=0$ on the other.

Two particles, $A$ and $B$, of masses $m$ and $2m$, respectively, are placed on a line of greatest slope, $\ell$, of a rough inclined plane which makes an angle of $30^{\circ }$ with the horizontal. The coefficient of friction between $A$ and the plane is $\frac{1}{6}\sqrt{3}$ and the coefficient of friction between $B$ and the plane is $\frac{1}{3}\sqrt{3}$. The particles are at rest with $B$ higher up $\ell$ than $A$ and are connected by a light inextensible string which is taut. A force $P$ is applied to $B$.

1. Show that the least magnitude of $P$ for which the two particles move upwards along $\ell$ is $\frac{11}{8}\sqrt{3}mg$ and give, in this case, the direction in which $P$ acts.
2. Find the least magnitude of $P$ for which the particles do not slip downwards along $\ell$.

The points $A$ and $B$ are $180$ metres apart and lie on horizontal ground. A missile is launched from $A$ at speed of $100{\mathrm{m}\mathrm{s}}^{-1}$ and at an acute angle of elevation to the line $AB$ of $\arcsin \frac{3}{5}$. A time $T$ seconds later, an anti-missile missile is launched from $B$, at speed of $200{\mathrm{m}\mathrm{s}}^{-1}$ and at an acute angle of elevation to the line $BA$ of $\arcsin \frac{4}{5}$. The motion of both missiles takes place in the vertical plane containing $A$ and $B$, and the missiles collide.

Taking $g=10{\mathrm{m}\mathrm{s}}^{-2}$ and ignoring air resistance, find $T$.

[Note that $\arcsin \frac{3}{5}$ is another notation for ${\sin }^{-1}\frac{3}{5}$.]

A plane is inclined at an angle $\arctan \frac{3}{4}$ to the horizontal and a small, smooth, light pulley $P$ is fixed to the top of the plane. A string, $APB$, passes over the pulley. A particle of mass $m_1$ is attached to the string at $A$ and rests on the inclined plane with $AP$ parallel to a line of greatest slope in the plane. A particle of mass $m_2$, where $m_2>m_1$, is attached to the string at $B$ and hangs freely with $BP$ vertical. The coefficient of friction between the particle at $A$ and the plane is $\frac{1}{2}$.

The system is released from rest with the string taut. Show that the acceleration of the particles is $\frac{m_2-m_1}{m_2+m_1}g$.

At a time $T$ after release, the string breaks. Given that the particle at $A$ does not reach the pulley at any point in its motion, find an expression in terms of $T$ for the time after release at which the particle at $A$ reaches its maximum height. It is found that, regardless of when the string broke, this time is equal to the time taken by the particle at $A$ to descend from its point of maximum height to the point at which it was released. Find the ratio $m_1:m_2$.

[Note that $\arctan \frac{3}{4}$ is another notation for ${\tan }^{-1}\frac{3}{4}$.]

The twins Anna and Bella share a computer and never sign their e-mails. When I e-mail them, only the twin currently online responds. The probability that it is Anna who is online is $p$ and she answers each question I ask her truthfully with probability $a$, independently of all her other answers, even if a question is repeated. The probability that it is Bella who is online is $q$, where $q=1-p$, and she answers each question truthfully with probability $b$, independently of all her other answers, even if a question is repeated.

1. I send the twins the e-mail: `Toss a fair coin and answer the following question. Did the coin come down heads?'. I receive the answer `yes'. Show that the probability that the coin did come down heads is $\frac{1}{2}$ if and only if $2(ap+bq)=1$.
2. I send the twins the e-mail: `Toss a fair coin and answer the following question. Did the coin come down heads?'. I receive the answer `yes'. I then send the e-mail: `Did the coin come down heads?' and I receive the answer `no'. Show that the probability (taking into account these answers) that the coin did come down heads is $\frac{1}{2}$.
3. I send the twins the e-mail: `Toss a fair coin and answer the following question. Did the coin come down heads?'. I receive the answer `yes'. I then send the e-mail: `Did the coin come down heads?' and I receive the answer `yes'. Show that, if $2(ap+bq)=1$, the probability (taking into account these answers) that the coin did come down heads is $\frac{1}{2}$.

The number of printing errors on any page of a large book of $N$ pages is modelled by a Poisson variate with parameter $\lambda$ and is statistically independent of the number of printing errors on any other page. The number of pages in a random sample of $n$ pages (where $n$ is much smaller than $N$ and $n⩾2$) which contain fewer than two errors is denoted by $Y$. Show that $\mathrm{P}(Y=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k{q}^{n-k}$ where $p=(1+\lambda )e^{-\lambda }$ and $q=1-p$.

Show also that, if $\lambda$ is sufficiently small,

1. $q\approx \frac{1}{2}{\lambda }^2$;
2. the largest value of $n$ for which $\mathrm{P}(Y=n)⩾1-\lambda$ is approximately $2/\lambda$;
3. $\mathrm{P}(Y>1|Y>0)\approx 1-n({\lambda }^2/2{)}^{n-1}.$

The probability density function $\mathrm{f}(x)$ of the random variable $X$ is given by $$\mathrm{f}(x)=k\left[\varphi (x)+\lambda \mathrm{g}(x)\right],$$ where $\varphi (x)$ is the probability density function of a normal variate with mean 0 and variance 1, $\lambda$ is a positive constant, and $\mathrm{g}(x)$ is a probability density function defined by $$\mathrm{g}(x)=\{\begin{array}{ll}1/\lambda & \text{for 0⩽x⩽λ};\\ 0& \text{otherwise}.\end{array}$$ Find $\mu$, the mean of $X$, in terms of $\lambda$, and prove that $\sigma$, the standard deviation of $X$, satisfies. $${\sigma }^2=\frac{{\lambda }^4+4{\lambda }^3+12\lambda +12}{12(1+\lambda {)}^2}.$$In the case $\lambda =2$:

1. draw a sketch of the curve $y=\mathrm{f}(x)$;
2. express the cumulative distribution function of $X$ in terms of $\Phi (x)$, the cumulative distribution function corresponding to $\varphi (x)$;
3. evaluate $\mathrm{P}(0<X<\mu +2\sigma )$, given that $\Phi (\frac{2}{3}+\frac{2}{3}\surd 7)=0.9921$.