STEP 2 1992

Find the limit, as $n\to \infty ,$ of each of the following. You should explain your reasoning briefly. $$\begin{array}{rlrlrlrl}(\mathbf{i})& \frac{n}{n+1},& & (\mathbf{ii})& \frac{5n+1}{n^2-3n+4},& & (\mathbf{iii})& \frac{\sin n}{n},\\ & & & & & & & \\ (\mathbf{iv})& \frac{\sin (1/n)}{(1/n)},& & (\mathbf{v})& (\arctan n{)}^{-1},& & (\mathbf{vi})& \frac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n+2}-\sqrt{n}}.\end{array}$$

Suppose that $y$ satisfies the differential equation $$y=x\frac{\mathrm{d}y}{\mathrm{d}x}-\cosh \left(\frac{\mathrm{d}y}{\mathrm{d}x}\right).\text{\hspace{1em}(*)}$$ By differentiating both sides of $(\ast )$ with respect to $x$, show that either $$\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}=0\text{ or }x-\sinh \left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)=0.$$ Find the general solutions of each of these two equations. Determine the solutions of $(\ast )$.

In the figure, the large circle with centre $O$ has radius $4$ and the small circle with centre $P$ has radius $1$. The small circle rolls around the inside of the larger one. When $P$ was on the line $OA$ (before the small circle began to roll), the point $B$ was in contact with the point $A$ on the large circle.

Sketch the curve $C$ traced by $B$ as the circle rolls. Show that if we take $O$ to be the origin of cartesian coordinates and the line $OA$ to be the $x$-axis (so that $A$ is the point $(4,0)$) then $B$ is the point $$(3\cos \varphi +\cos 3\varphi ,3\sin \varphi -\sin 3\varphi ).$$ It is given that the area of the region enclosed by the curve $C$ is $${\int }_0^{2\pi }x\frac{\mathrm{d}y}{\mathrm{d}\varphi }\mathrm{d}\varphi ,$$ where $B$ is the point $(x,y).$ Calculate this area.

$◊$ is an operation which take polynomials in $x$ to polynomials in $x$; that is, given a polynomial $\mathrm{h}(x)$ there is another polynomial called $◊\mathrm{h}(x)$. It is given that, if $\mathrm{f}(x)$ and $\mathrm{g}(x)$ are any two polynomials in $x$, the following are always true:

• (i) $◊(\mathrm{f}(x)\mathrm{g}(x))=\mathrm{g}(x)◊\mathrm{f}(x)+\mathrm{f}(x)◊\mathrm{g}(x),$
• (ii) $◊(\mathrm{f}(x)+\mathrm{g}(x))=◊\mathrm{f}(x)+◊\mathrm{g}(x),$
• (iii) $◊x=1$
• (iv) if $\lambda$ is a constant then $◊(\lambda \mathrm{f}(x))=\lambda ◊\mathrm{f}(x).$

Show that, if $\mathrm{f}(x)$ is a constant (i.e., a polynomial of degree zero), then $◊\mathrm{f}(x)=0.$

Calculate $◊x^2$ and $◊x^3.$ Prove that $◊\mathrm{h}(x)=\frac{\mathrm{d}}{\mathrm{d}x}(\mathrm{h}(x))$ for any polynomial $\mathrm{h}(x)$.

Explain what is meant by the order of an element $g$ of a group $G$.

The set $S$ consists of all $2\times 2$ matrices whose determinant is $1$. Find the inverse of the element $\mathbf{A}$ of $S$, where $$\mathbf{A}=\left(\begin{array}{cc}w& x\\ y& x\end{array}\right).$$ Show that $S$ is a group under matrix multiplication (you may assume that matrix multiplication is associative). For which elements $\mathbf{A}$ is ${\mathbf{A}}^{-1}=\mathbf{A}$? Which element or elements have order 2? Show that the element $\mathbf{A}$ of $S$ has order 3 if, and only if, $w+z+1=0.$ Write down one such element.

Sketch the graphs of $y=\sec x$ and $y=\ln (2\sec x)$ for $0⩽x⩽\frac{1}{2}\pi$. Show graphically that the equation $$kx=\ln (2\sec x)$$ has no solution with $0⩽x<\frac{1}{2}\pi$ if $k$ is a small positive number but two solutions if $k$ is large.

Explain why there is a number $k_0$ such that $$k_{0}x=\ln (2\sec x)$$ has exactly one solution with $0⩽x<\frac{1}{2}\pi$. Let $x_0$ be this solution, so that $0⩽x_0<\frac{1}{2}\pi$ and $k_0{x}_0=\ln (2\sec x)$. Show that $$x_0=\cot x_0\ln (2\sec x_0).$$ Use any appropriate method to find $x_0$ correct to two decimal places. Hence find an approximate value for $k_0$.

The cubic equation $$x^3-p{x}^2+qx-r=0$$ has roots $a,b$ and $c$. Express $p,q$ and $r$ in terms of $a,b$ and $c$.

1. If $p=0$ and two of the roots are equal to each other, show that $$4q^3+27r^2=0.$$
2. Show that, if two of the roots of the original equation are equal to each other, then $$4{\left(q-\frac{p^2}{3}\right)}^3+27{\left(\frac{2p^3}{27}-\frac{pq}{3}+r\right)}^2=0.$$

Calculate the following integrals

1. $\int \frac{x}{(x-1)(x^2-1)}\mathrm{d}x$;
2. $\int \frac{1}{3\cos x+4\sin x}\mathrm{d}x$;
3. $\int \frac{1}{\sinh x}\mathrm{d}x.$

Let $\mathbf{a},\mathbf{b}$ and $\mathbf{c}$ be the position vectors of points $A,B$ and $C$ in three-dimensional space. Suppose that $A,B,C$ and the origin $O$ are not all in the same plane. Describe the locus of the point whose position vector $\mathbf{r}$ is given by $$\mathbf{r}=(1-\lambda -\mu )\mathbf{a}+\lambda \mathbf{b}+\mu \mathbf{c},$$ where $\lambda$ and $\mu$ are scalar parameters. By writing this equation in the form $\mathbf{r}\cdot \mathbf{n}=p$ for a suitable vector $\mathbf{n}$ and scalar $p$, show that $$-(\lambda +\mu )\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})+\lambda \mathbf{b}\cdot (\mathbf{c}\times \mathbf{a})+\mu \mathbf{c}\cdot (\mathbf{a}\times \mathbf{b})=0$$ for all scalars $\lambda ,\mu .$

Deduce that $$\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})=\mathbf{b}\cdot (\mathbf{c}\times \mathbf{a})=\mathbf{c}\cdot (\mathbf{a}\times \mathbf{b}).$$ Say briefly what happens if $A,B,C$ and $O$ are all in the same plane.

Let $\alpha$ be a fixed angle, $0<x⩽\frac{1}{2}\pi .$ In each of the following cases, sketch the locus of $z$ in the Argand diagram (the complex plane):

1. $\arg \left(\frac{z-1}{z}\right)=\alpha ,$
2. $\arg \left(\frac{z-1}{z}\right)=\alpha -\pi ,$
3. $\left|\frac{z-1}{z}\right|=1.$

Let $z_1,z_2,z_3$ and $z_4$ be four points lying (in that order) on a circle in the Argand diagram. If $$w=\frac{(z_1-z_2)(z_3-z_4)}{(z_4-z_1)(z_2-z_3)}$$ show, by considering $\arg w$, that $w$ is real.

I am standing next to an ice-cream van at a distance $d$ from the top of a vertical cliff of height $h$. It is not safe for me to go any nearer to the top of the cliff. My niece Padma is on the broad level beach at the foot of the cliff. I have just discovered that I have left my wallet with her, so I cannot buy her an ice-cream unless she can throw the wallet up to me. She can throw it at speed $V$, at any angle she chooses and from anywhere on the beach. Air resistance is negligible; so is Padma's height compared to that of the cliff. Show that she can throw the wallet to me if and only if $$V^2⩾g(2h+d).$$

In the figure, $W_1$ and $W_2$ are wheels, both of radius $r$. Their centres $C_1$ and $C_2$ are fixed at the same height, a distance $d$ apart, and each wheel is free to rotate, without friction, about its centre. Both wheels are in the same vertical plane. Particles of mass $m$ are suspended from $W_1$ and $W_2$ as shown, by light inextensible strings would round the wheels. A light elastic string of natural length $d$ and modulus elasticity $\lambda$ is fixed to the rims of the wheels at the points $P_1$ and $P_2.$ The lines joining $C_1$ to $P_1$ and $C_2$ to $P_2$ both make an angle $\theta$ with the vertical. The system is in equilibrium.

Show that $$\sin 2\theta =\frac{mgd}{\lambda r}.$$ For what value or values of $\lambda$ (in terms of $m,d,r$ and $g$) are there

• (i) no equilibrium positions,
• (ii) just one equilibrium position,
• (iii) exactly two equilibrium positions,
• (iv) more than two equilibrium positions?

Two particles $P_1$ and $P_2$, each of mass $m$, are joined by a light smooth inextensible string of length $\ell .$ $P_1$ lies on a table top a distance $d$ from the edge, and $P_2$ hangs over the edge of the table and is suspended a distance $b$ above the ground. The coefficient of friction between $P_1$ and the table top is $\mu ,$ and $\mu <1$. The system is released from rest. Show that $P_1$ will fall off the edge of the table if and only if $$\mu <\frac{b}{2d-b}.$$ Suppose that $\mu >b/(2d-b)$ , so that $P_1$ comes to rest on the table, and that the coefficient of restitution between $P_2$ and the floor is $e$. Show that, if $e>1/(2\mu ),$ then $P_1$ comes to rest before $P_2$ bounces a second time.

In the diagram $P_1$ and $P_2$ are smooth light pulleys fixed at the same height, and $P_3$ is a third smooth light pulley, freely suspended. A smooth light inextensible string runs over $P_1,$ under $P_3$ and over $P_2,$ as shown: the parts of the string not in contact with any pulley are vertical. A particle of mass $m_3$ is attached to $P_3.$ There is a particle of mass $m_1$ attached to the end of the string below $P_1$ and a particle of mass $m_2$ attached to the other end, below $P_2.$ The system is released from rest. Find the tension in the string, and show that the pulley $P_3$ will remain at rest if $$4m_1{m}_2=m_3(m_1+m_2).$$

A point moves in unit steps on the $x$-axis starting from the origin. At each step the point is equally likely to move in the positive or negative direction. The probability that after $s$ steps it is at one of the points $x=2,x=3,x=4$ or $x=5$ is $\mathrm{P}(s).$ Show that $\mathrm{P}(5)=\frac{3}{16},$ $\mathrm{P}(6)=\frac{21}{64}$ and $$\mathrm{P}(2k)=\left(\genfrac{}{}{0ex}{}{2k+1}{k-1}\right){\left(\frac{1}{2}\right)}^{2k}$$ where $k$ is a positive integer. Find a similar expression for $\mathrm{P}(2k+1).$

Determine the values of $s$ for which $\mathrm{P}(s)$ has its greatest value.

A taxi driver keeps a packet of toffees and a packet of mints in her taxi. From time to time she takes either a toffee (with probability $p$) or mint (with probability $q=1-p$). At the beginning of the week she has $n$ toffees and $m$ mints in the packets. On the $N$th occasion that she reaches for a sweet, she discovers (for the first time) that she has run out of that kind of sweet. What is the probability that she was reaching for a toffee?