STEP 3 2010

Let $x_{{}^{}1}$, $x_{{}^{}2}$, …, $x_n$ and $x_{n+1}$ be any fixed real numbers. The numbers $A$ and $B$ are defined by $$A=\frac{1}{n}\sum _{k=1}^n{x}_{{}^{}k},B=\frac{1}{n}\sum _{k=1}^n(x_{{}^{}k}-A{)}^2,$$ and the numbers $C$ and $D$ are defined by $$C=\frac{1}{n+1}\sum _{k=1}^{n+1}{x}_{{}^{}k},D=\frac{1}{n+1}\sum _{k=1}^{n+1}(x_{{}^{}k}-C{)}^2.$$

1. Express $C$ in terms of $A$, $x_{{}^{}n+1}$ and $n$.
2. Show that $B=\frac{1}{n}\sum _{k=1}^n{x}_{{}^{}k}^2-A^2$.
3. Express $D$ in terms of $B$, $A$, $x_{{}^{}n+1}$ and $n$. Hence show that $(n+1)D⩾nB$ for all values of $x_{{}^{}n+1}$, but that $D<B$ if and only if $$A-\sqrt{\frac{(n+1)B}{n}}<x_{{}^{}n+1}<A+\sqrt{\frac{(n+1)B}{n}}.$$

In this question, $a$ is a positive constant.

1. Express $\cosh a$ in terms of exponentials. By using partial fractions, prove that $${\int }_0^1\frac{1}{x^2+2x\cosh a+1}\mathrm{d}x=\frac{a}{2\sinh a}.$$
2. Find, expressing your answers in terms of hyperbolic functions, $${\int }_1^{\infty }\frac{1}{x^2+2x\sinh a-1}\mathrm{d}x$$ and $${\int }_0^{\infty }\frac{1}{x^4+2x^2\cosh a+1}\mathrm{d}x.$$

For any given positive integer $n$, a number $a$ (which may be complex) is said to be a primitive $n$th root of unity if $a^n=1$ and there is no integer $m$ such that $0<m<n$ and $a^m=1$. Write down the two primitive 4th roots of unity.

Let ${\mathrm{C}}_n(x)$ be the polynomial such that the roots of the equation ${\mathrm{C}}_n(x)=0$ are the primitive $n$th roots of unity, the coefficient of the highest power of $x$ is one and the equation has no repeated roots. Show that ${\mathrm{C}}_4(x)=x^2+1$.

1. Find ${\mathrm{C}}_1(x)$, ${\mathrm{C}}_2(x)$, ${\mathrm{C}}_3(x)$, ${\mathrm{C}}_5(x)$ and ${\mathrm{C}}_6(x)$, giving your answers as unfactorised polynomials.
2. Find the value of $n$ for which ${\mathrm{C}}_n(x)=x^4+1$.
3. Given that $p$ is prime, find an expression for ${\mathrm{C}}_p(x)$, giving your answer as an unfactorised polynomial.
4. Prove that there are no positive integers $q$, $r$ and $s$ such that ${\mathrm{C}}_q(x)\equiv {\mathrm{C}}_r(x){\mathrm{C}}_s(x)$.

The vertices $A$, $B$, $C$ and $D$ of a square have coordinates $(0,0)$, $(a,0)$, $(a,a)$ and $(0,a)$, respectively. The points $P$ and $Q$ have coordinates $(an,0)$ and $(0,am)$ respectively, where $0<m<n<1$. The line $CP$ produced meets $DA$ produced at $R$ and the line $CQ$ produced meets $BA$ produced at $S$. The line $PQ$ produced meets the line $RS$ produced at $T$. Show that $TA$ is perpendicular to $AC$.

Explain how, given a square of area $a^2$, a square of area $2a^2$ may be constructed using only a straight-edge.

[Note: a straight-edge is a ruler with no markings on it; no measurements (and no use of compasses) are allowed in the construction.]

The points $P$, $Q$ and $R$ lie on a sphere of unit radius centred at the origin, $O$, which is fixed. Initially, $P$ is at $P_0(1,0,0)$, $Q$ is at $Q_0(0,1,0)$ and $R$ is at $R_0(0,0,1)$.

1. The sphere is then rotated about the $z$-axis, so that the line $OP$ turns directly towards the positive $y$-axis through an angle $\varphi$. The position of $P$ after this rotation is denoted by $P_1$. Write down the coordinates of $P_1$.
2. The sphere is now rotated about the line in the $x$-$y$ plane perpendicular to $O{P}_1$, so that the line $OP$ turns directly towards the positive $z$-axis through an angle $\lambda$. The position of $P$ after this rotation is denoted by $P_2$. Find the coordinates of $P_2$. Find also the coordinates of the points $Q_2$ and $R_2$, which are the positions of $Q$ and $R$ after the two rotations.
3. The sphere is now rotated for a third time, so that $P$ returns from $P_2$ to its original position $P_0$. During the rotation, $P$ remains in the plane containing $P_0$, $P_2$ and $O$. Show that the angle of this rotation, $\theta$, satisfies $$\cos \theta =\cos \varphi \cos \lambda ,$$ and find a vector in the direction of the axis about which this rotation takes place.

Given that $y=\cos (m\arcsin x)$, for $|x|<1$, prove that $$(1-x^2)\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}-x\frac{\mathrm{d}y}{\mathrm{d}x}+m^{2}y=0.$$Obtain a similar equation relating $\frac{{\mathrm{d}}^{3}y}{\mathrm{d}{x}^3}$,\; $\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}$ and $\frac{\mathrm{d}y}{\mathrm{d}x}$, and a similar equation relating $\frac{{\mathrm{d}}^{4}y}{\mathrm{d}{x}^4}$, $\frac{{\mathrm{d}}^{3}y}{\mathrm{d}{x}^3}$ and $\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}$.

Conjecture and prove a relation between $\frac{{\mathrm{d}}^{n+2}y}{\mathrm{d}{x}^{n+2}}$, $\frac{{\mathrm{d}}^{n+1}y}{\mathrm{d}{x}^{n+1}}$ and $\frac{{\mathrm{d}}^{n}y}{\mathrm{d}{x}^n}$.

Obtain the first three non-zero terms of the Maclaurin series for $y$. Show that, if $m$ is an even integer, $\cos m\theta$ may be written as a polynomial in $\sin \theta$ beginning $$1-\frac{m^2{\sin }^2\theta }{2!}+\frac{m^2(m^2-2^2){\sin }^4\theta }{4!}-\cdots .\text{\hspace{1em}(∣θ∣<21π)}$$ State the degree of the polynomial.

Given that $\mathrm{P}(x)=\mathrm{Q}(x){\mathrm{R}}^{\prime }(x)-{\mathrm{Q}}^{\prime }(x)\mathrm{R}(x)$, write down an expression for $$\int \frac{\mathrm{P}(x)}{(\mathrm{Q}(x){)}^2}\mathrm{d}x.$$

1. By choosing the function $\mathrm{R}(x)$ to be of the form $a+bx+c{x}^2$, find $$\int \frac{5x^2-4x-3}{(1+2x+3x^2{)}^2}\mathrm{d}x.$$ Show that the choice of $\mathrm{R}(x)$ is not unique and, by comparing the two functions $\mathrm{R}(x)$ corresponding to two different values of $a$, explain how the different choices are related.
2. Find the general solution of $$(1+\cos x+2\sin x)\frac{\mathrm{d}y}{\mathrm{d}x}+(\sin x-2\cos x)y=5-3\cos x+4\sin x.$$

The diagram shows two particles, $P$ and $Q$, connected by a light inextensible string which passes over a smooth block fixed to a horizontal table. The cross-section of the block is a quarter circle with centre $O$, which is at the edge of the table, and radius $a$. The angle between $OP$ and the table is $\theta$. The masses of $P$ and $Q$ are $m$ and $M$, respectively, where $m<M$.

Initially, $P$ is held at rest on the table and in contact with the block, $Q$ is vertically above $O$, and the string is taut. Then $P$ is released. Given that, in the subsequent motion, $P$ remains in contact with the block as $\theta$ increases from $0$ to $\frac{1}{2}\pi$, find an expression, in terms of $m$, $M$, $\theta$ and $g$, for the normal reaction of the block on $P$ and show that $$\frac{m}{M}⩾\frac{\pi -1}{3}.$$

A small bead $B$, of mass $m$, slides without friction on a fixed horizontal ring of radius $a$. The centre of the ring is at $O$. The bead is attached by a light elastic string to a fixed point $P$ in the plane of the ring such that $OP=b$, where $b>a$. The natural length of the elastic string is $c$, where $c<b-a$, and its modulus of elasticity is $\lambda$. Show that the equation of motion of the bead is $$ma\ddot{\varphi }=-\lambda \left(\frac{a\sin \varphi }{c\sin \theta }-1\right)\sin (\theta +\varphi ),$$ where $\theta =\angle BPO$ and $\varphi =\angle BOP$.

Given that $\theta$ and $\varphi$ are small, show that $a(\theta +\varphi )\approx b\theta$. Hence find the period of small oscillations about the equilibrium position $\theta =\varphi =0$.

A bullet of mass $m$ is fired horizontally with speed $u$ into a wooden block of mass $M$ at rest on a horizontal surface. The coefficient of friction between the block and the surface is $\mu$. While the bullet is moving through the block, it experiences a constant force of resistance to its motion of magnitude $R$, where $R>(M+m)\mu g$. The bullet moves horizontally in the block and does not emerge from the other side of the block.

1. Show that the magnitude, $a$, of the deceleration of the bullet relative to the block while the bullet is moving through the block is given by $$a=\frac{R}{m}+\frac{R-(M+m)\mu g}{M}.$$
2. Show that the common speed, $v$, of the block and bullet when the bullet stops moving through the block satisfies $$av=\frac{Ru-(M+m)\mu gu}{M}.$$
3. Obtain an expression, in terms of $u$, $v$ and $a$, for the distance moved by the block while the bullet is moving through the block.
4. Show that the total distance moved by the block is $$\frac{muv}{2(M+m)\mu g}.$$

Describe briefly what happens if $R<(M+m)\mu g$.

The infinite series $S$ is given by $$S=1+(1+d)r+(1+2d)r^2+\cdots +(1+nd)r^n+\cdots ,$$ for $|r|<1$. By considering $S-rS$, or otherwise, prove that $$S=\frac{1}{1-r}+\frac{rd}{(1-r{)}^2}.$$Arthur and Boadicea shoot arrows at a target. The probability that an arrow shot by Arthur hits the target is $a$; the probability that an arrow shot by Boadicea hits the target is $b$. Each shot is independent of all others. Prove that the expected number of shots it takes Arthur to hit the target is $1/a$.

Arthur and Boadicea now have a contest. They take alternate shots, with Arthur going first. The winner is the one who hits the target first. The probability that Arthur wins the contest is $\alpha$ and the probability that Boadicea wins is $\beta$. Show that $$\alpha =\frac{a}{1-a^{\prime }{b}^{\prime }},$$ where $a^{\prime }=1-a$ and $b^{\prime }=1-b$, and find $\beta$.

Show that the expected number of shots in the contest is $\frac{\alpha }{a}+\frac{\beta }{b}.$

In this question, $\mathrm{Corr}(U,V)$ denotes the product moment correlation coefficient between the random variables $U$ and $V$, defined by $$\mathrm{Corr}(U,V)\equiv \frac{\mathrm{Cov}(U,V)}{\sqrt{\mathrm{Var}(U)\mathrm{Var}(V)}}.$$The independent random variables $Z_1$, $Z_2$ and $Z_3$ each have expectation 0 and variance 1. What is the value of $\mathrm{Corr}(Z_1,Z_2)$?

Let $Y_1=Z_1$ and let $$Y_2={\rho }_{12}^{}{Z}_1+(1-{{\rho }_{12}^2{)}_{}}^{\frac{1}{2}}{Z}_2,$$ where ${\rho }_{12}^{}$ is a given constant with $-1<{\rho }_{12}^{}<1$. Find $\mathrm{E}(Y_2)$, $\mathrm{Var}(Y_2)$ and $\mathrm{Corr}(Y_1,Y_2)$.

Now let $Y_3=a{Z}_1+b{Z}_2+c{Z}_3$, where $a$, $b$ and $c$ are real constants and $c⩾0$. Given that $\mathrm{E}(Y_3)=0$, $\mathrm{Var}(Y_3)=1$, $\mathrm{Corr}(Y_1,Y_3)={\rho }_{13}^{}$ and $\mathrm{Corr}(Y_2,Y_3)={\rho }_{23}^{}$, express $a$, $b$ and $c$ in terms of ${\rho }_{23}^{}$, ${\rho }_{13}^{}$ and ${\rho }_{12}^{}$.

Given constants ${\mu }_i$ and ${\sigma }_i$, for $i=1$, $2$ and $3$, give expressions in terms of the $Y_i$ for random variables $X_i$ such that $\mathrm{E}(X_i)={\mu }_i$, $\mathrm{Var}(X_i)={\sigma }_i^2$ and $\mathrm{Corr}(X_i,X_j)={\rho }_{ij}$.