STEP 3 1993

The curve $P$ has the parametric equations $$x=\sin \theta ,y=\cos 2\theta for-\pi /2⩽\theta ⩽\pi /2.$$ Show that $P$ is part of the parabola $y=1-2x^2$ and sketch $P$.

Show that the length of $P$ is $\surd (17)+\frac{1}{4}{\sinh }^{-1}4$.

Obtain the volume of the solid enclosed when $P$ is rotated through $2\pi$ radians about the line $y=-1$.

The curve $C$ has the equation $x^3+y^3=3xy$.

1. Show that there is no point of inflection on $C$. You may assume that the origin is not a point of inflection.
2. The part of $C$ which lies in the first quadrant is a closed loop touching the axes at the origin. By converting to polar coordinates, or otherwise, evaluate the area of this loop.

The matrices $\mathbf{A},\mathbf{B}$ and $\mathbf{M}$ are given by $$\mathbf{A}=\left(\begin{array}{ccc}a& 0& 0\\ b& c& 0\\ d& e& f\end{array}\right),\mathbf{B}=\left(\begin{array}{ccc}1& p& q\\ 0& 1& r\\ 0& 0& 1\end{array}\right),\mathbf{M}=\left(\begin{array}{ccc}1& 3& 2\\ 4& 13& 5\\ 3& 8& 7\end{array}\right),$$ where $a,b,\dots ,r$ are real numbers. Given that $\mathbf{M}=\mathbf{AB},$ show that $a=1,b=4,c=1,d=3,e=1,f=-2,p=3,q=2$ and $r=-3$ gives the unique solution for $\mathbf{A}$ and $\mathbf{B}.$ Evaluate ${\mathbf{A}}^{-1}$ and ${\mathbf{B}}^{-1},$

Hence, or otherwise, solve the simultaneous equations $$\begin{array}{rl}x+3y+2z& =7\\ 4x+13y+5z& =18\\ 3x+8y+7z& =25.\end{array}$$

Sum the following infinite series.

1. $1+\frac{1}{3}(\frac{1}{2}{)}^2+\frac{1}{5}(\frac{1}{2}{)}^4+\cdots +\frac{1}{2n+1}(\frac{1}{2}{)}^{2n}+\cdots$ .
2. $2-x-x^3+2x^4-\cdots +2x^{4k}-x^{4k+1}-x^{4k+3}+\cdots$ where $|x|<1$.
3. $\sum _{r=2}^{\infty }\frac{r{2}^{r-2}}{3^{r-1}}.$
4. $\sum _{r=2}^{\infty }\frac{2}{r(r^2-1)}$.

The set $S$ consists of ordered pairs of complex numbers $(z_1,z_2)$ and a binary operation $\circ$ on $S$ is defined by $$(z_1,z_2)\circ (w_1,w_2)=(z_1{w}_1-z_2{w}_2^{\ast },z_1{w}_2+z_2{w}_1^{\ast }).$$ Show that the operation $\circ$ is associative and determine whether it is commutative. Evaluate $(z,0)\circ (w,0)$, $(z,0)\circ (0,w)$, $(0,z)\circ (w,0)$ and $(0,z)\circ (0,w)$.

The set $S_1$ is the subset of $S$ consisting of $A$, $B$, $\dots$, $H$, where $A=(1,0)$, $B=(0,1)$, $C=(i,0)$, $D=(0,i)$, $E=(-1,0)$, $F=(0,-1)$, $G=(-i,0)$ and $H=(0,-i)$. Show that $S_1$ is closed under $\circ$ and that it has an identity element. Determine the inverse and order of each element of $S_1$. Show that $S_1$ is a group under $\circ$. [You are not required to compute the multiplication table in full.]

Show that $\{A,B,E,F\}$ is a subgroup of $S_1$ and determine whether it is isomorphic to the group generated by the $2\times 2$ matrix $\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$ under matrix multiplication.

The point in the Argand diagram representing the complex number $z$ lies on the circle with centre $K$ and radius $r$, where $K$ represents the complex number $k$. Show that $$z{z}^{\ast }-k{z}^{\ast }-k^{\ast }z+k{k}^{\ast }-r^2=0.$$The points $P$, $Q_1$ and $Q_2$ represent the complex numbers $z$, $w_1$ and $w_2$ respectively. The point $P$ lies on the circle with $OA$ as diameter, where $O$ and $A$ represent $0$ and $2i$ respectively. Given that $w_1=z/(z-1)$, find the equation of the locus $L$ of $Q_1$ in terms of $w_1$ and describe the geometrical form of $L$.

Given that $w_2=z^{\ast }$, show that the locus of $Q_2$ is also $L$. Determine the positions of $P$ for which $Q_1$ coincides with $Q_2$.

The real numbers $x$ and $y$ satisfy the simultaneous equations $$\sinh (2x)=\cosh yand\sinh (2y)=2\cosh x.$$ Show that ${\sinh }^{2}y$ is a root of the equation $$4t^3+4t^2-4t-1=0$$ and demonstrate that this gives at most one valid solution for $y$. Show that the relevant value of $t$ lies between $0.7$ and $0.8$, and use an iterative process to find $t$ to 6 decimal places.

Find $y$ and hence find $x$, checking your answers and stating the final answers to four decimal places.

A square pyramid has its base vertices at the points $A$ $(a,0,0)$, $B$ $(0,a,0)$, $C$ $(-a,0,0)$ and $D$ $(0,-a,0)$, and its vertex at $E$ $(0,0,a)$. The point $P$ lies on $AE$ with $x$-coordinate $\lambda a$, where $0<\lambda <1$, and the point $Q$ lies on $CE$ with $x$-coordinate $-\mu a$, where $0<\mu <1$. The plane $BPQ$ cuts $DE$ at $R$ and the $y$-coordinate of $R$ is $-\gamma a$. Prove that $$\gamma =\frac{\lambda \mu }{\lambda +\mu -\lambda \mu }.$$Show that the quadrilateral $BPRQ$ cannot be a parallelogram.

For the real numbers $a_1$, $a_2$, $a_3$, $\dots$,

1. prove that $a_1^2+a_2^2⩾2a_1{a}_2$,
2. prove that $a_1^2+a_2^2+a_3^2⩾a_2{a}_3+a_3{a}_1+a_1{a}_2$,
3. prove that $3(a_1^2+a_2^2+a_3^2+a_4^2)⩾2(a_1{a}_2+a_1{a}_3+a_1{a}_4+a_2{a}_3+a_2{a}_4+a_3{a}_4)$,
4. state and prove a generalisation of (iii) to the case of $n$ real numbers,
5. prove that $${\left(\sum _{i=1}^n{a}_i\right)}^2⩾\frac{2n}{n-1}\sum _{i,j}{a}_i{a}_j,$$ where the latter sum is taken over all pairs $(i,j)$ with $1⩽i<j⩽n$.

The transformation $T$ of the point $P$ in the $x$,$y$ plane to the point $P^{\prime }$ is constructed as follows: Lines are drawn through $P$ parallel to the lines $y=mx$ and $y=-mx$ to cut the line $y=kx$ at $Q$ and $R$ respectively, $m$ and $k$ being given constants. $P^{\prime }$ is the fourth vertex of the parallelogram $PQ{P}^{\prime }R$.

Show that if $P$ is $(x_1,y_1)$ then $Q$ is $$\left(\frac{m{x}_1-y_1}{m-k},\frac{k(m{x}_1-y_1)}{m-k}\right).$$ Obtain the coordinates of $P^{\prime }$ in terms of $x_1$, $y_1$, $m$ and $k$, and express $T$ as a matrix transformation. Show that areas are transformed under $T$ into areas of the same magnitude.

In this question, all gravitational forces are to be neglected.

A rigid frame is constructed from 12 equal uniform rods, each of length $a$ and mass $m,$ forming the edges of a cube. Three of the edges are $OA,OB$ and $OC,$ and the vertices opposite $O,A,B$ and $C$ are $O^{\prime },A^{\prime },B^{\prime }$ and $C^{\prime }$ respectively. Forces act along the lines as follows, in the directions indicated by the order of the letters: $$\begin{array}{rlrlr}2mg\text{ along }OA,& & mg\text{ along }A{C}^{\prime },& & \sqrt{2}mg\text{ along }{O}^{\prime }A,\\ \sqrt{2}mg\text{ along }O{A}^{\prime },& & 2mg\text{ along }{C}^{\prime }B,& & mg\text{ along }{A}^{\prime }C.\end{array}$$

1. The frame is freely pivoted at $O$. Show that the direction of the line about which it will start to rotate is $\left(\begin{array}{c}1\\ 1\\ 2\end{array}\right)$ with respect to axes along $OA$, $OB$ and $OC$ respectively.
2. Show that the moment of inertia of the rod $OA$ about the axis $O{O}^{\prime }$ is $2m{a}^2/9$ and about a parallel axis through its mid-point is $m{a}^2/18$. Hence find the moment of inertia of $B^{\prime }C$ about $O{O}^{\prime }$ and show that the moment of inertia of the frame about $O{O}^{\prime }$ is $14m{a}^2/3$. If the frame is freely pivoted about the line $O{O}^{\prime }$ and the forces continue to act along the specified lines, find the initial angular acceleration of the frame.

$ABCD$ is a horizontal line with $AB=CD=a$ and $BC=6a$. There are fixed smooth pegs at $B$ and $C$. A uniform string of natural length $2a$ and modulus of elasticity $kmg$ is stretched from $A$ to $D$, passing over the pegs at $B$ and $C$. A particle of mass $m$ is attached to the midpoint $P$ of the string. When the system is in equilibrium, $P$ is a distance $a/4$ below $BC$. Evaluate $k$.

The particle is pulled down to a point $Q$, which is at a distance $pa$ below the mid-point of $BC$, and is released from rest. $P$ rises to a point $R$, which is at a distance $3a$ above $BC$. Show that $2p^2-p-17=0$.

Show also that the tension in the strings is less when the particle is at $R$ than when the particle is at $Q$.

A uniform circular disc with radius $a$, mass $4m$ and centre $O$ is freely mounted on a fixed horizontal axis which is perpendicular to its plane and passes through $O$. A uniform heavy chain $PS$ of length $(4+\pi )a$, mass $(4+\pi )m$ and negligible thickness is hung over the rim of the disc as shown in the diagram: $Q$ and $R$ are the points of the chain at the same level as $O$. The contact between the chain and the rim of the disc is sufficiently rough to prevent slipping. Initially, the system is at rest with $PQ=RS=2a$. A particle of mass $m$ is attached to the chain at $P$ and the system is released. By considering the energy of the system, show that when $P$ has descended a distance $x$, its speed $v$ is given by $$(\pi +7)a{v}^2=2g(x^2+ax).$$By considering the part $PQ$ of the chain as a body of variable mass, show that when $S$ reaches $R$ the tension in the chain at $Q$ is $$\frac{5\pi -2}{\pi +7}mg.$$

A particle rests at a point $A$ on a horizontal table and is joined to a point $O$ on the table by a taut inextensible string of length $c$. The particle is projected vertically upwards at a speed $64\surd (6gc)$. It next strikes the table at a point $B$ and rebounds. The coefficient of restitution for any impact between the particle and the table is $\frac{1}{2}$. After rebounding at $B$, the particle will rebound alternately at $A$ and $B$ until the string becomes slack. Show that when the string becomes slack the particle is at height $c/2$ above the table.

Determine whether the first rebound between $A$ and $B$ is nearer to $A$ or to $B$.

The probability of throwing a head with a certain coin is $p$ and the probability of throwing a tail is $q=1-p$. The coin is thrown until at least two heads and at least two tails have been thrown; this happens when the coin has been thrown $N$ times. Write down an expression for the probability that $N=n$.

Show that the expectation of $N$ is $$2(\frac{1}{pq}-1-pq).$$

The time taken for me to set an acceptable examination question it $T$ hours. The distribution of $T$ is a truncated normal distribution with probability density $\mathrm{f}$ where$$\mathrm{f}(t)=\{\begin{array}{ll}\frac{1}{k\sigma \sqrt{2\pi }}\exp \left(-\frac{1}{2}{\left(\frac{t-\sigma }{\sigma }\right)}^2\right)& \text{ for }t⩾0\\ 0& \text{ for }t<0.\end{array}$$ Sketch the graph of $\mathrm{f}(t)$. Show that $k$ is approximately $0.841$ and obtain the mean of $T$ as a multiple of $\sigma$.

Over a period of years, I find that the mean setting time is 3 hours.

1. Find the approximate probability that none of the 16 questions on next year's paper will take more than 4 hours to set.
2. Given that a particular question is unsatisfactory after 2 hours work, find the probability that it will still be unacceptable after a further 2 hours work.