STEP 3 2007

In this question, do not consider the special cases in which the denominators of any of your expressions are zero.

Express $\tan ({\theta }_1+{\theta }_2+{\theta }_3+{\theta }_4)$ in terms of $t_i$, where $t_1=\tan {\theta }_1$, etc.

Given that $\tan {\theta }_1$, $\tan {\theta }_2$, $\tan {\theta }_3$ and $\tan {\theta }_4$ are the four roots of the equation $$a{t}^4+b{t}^3+c{t}^2+dt+e=0$$(where $a\ne 0$), find an expression in terms of $a$, $b$, $c$, $d$ and $e$ for $\tan ({\theta }_1+{\theta }_2+{\theta }_3+{\theta }_4)$.

The four real numbers ${\theta }_1$, ${\theta }_2$, ${\theta }_3$ and ${\theta }_4$ lie in the range $0⩽{\theta }_i<2\pi$ and satisfy the equation $$p\cos 2\theta +\cos (\theta -\alpha )+p=0,$$ where $p$ and $\alpha$ are independent of $\theta$. Show that ${\theta }_1+{\theta }_2+{\theta }_3+{\theta }_4=n\pi$ for some integer $n$.

A sequence of numbers, $F_1$, $F_2$, $\dots$, is defined by $F_1=1$, $F_2=1$, and $$F_n=F_{n-1}+F_{n-2}\text{ for n⩾3}.$$

1. Write down the values of $F_3$, $F_4$, $\dots$ , $F_8$.
2. Prove that $F_{2k+3}^{}{F}_{2k+1}^{}-F_{2k+2}^2=-F_{2k+2}^{}{F}_{2k}^{}+F_{2k+1}^2$.
3. Prove by induction or otherwise that $F_{2n+1}^{}{F}_{2n-1}^{}-F_{2n}^2=1$ and deduce that $F_{2n}^2+1$ is divisible by $F_{2n+1}^{}.$
4. Prove that $F_{2n-1}^2+1$ is divisible by $F_{2n+1}^{}.$

A curve is given parametrically by $$\begin{array}{rl}x& =a(\cos t+\ln \tan \frac{1}{2}t),\\ y& =a\sin t,\end{array}$$ where $0<t<\frac{1}{2}\pi$ and $a$ is a positive constant. Show that $\frac{\mathrm{d}y}{\mathrm{d}x}=\tan t$ and sketch the curve.

Let $P$ be the point with parameter $t$ and let $Q$ be the point where the tangent to the curve at $P$ meets the $x$-axis. Show that $PQ=a$.

The radius of curvature, $\rho$, at $P$ is defined by $$\rho =\frac{({\dot{x}}^2+{\dot{y}}^2{)}^{\frac{3}{2}}}{|\dot{x}\ddot{y}-\dot{y}\ddot{x}|},$$ where the dots denote differentiation with respect to $t$. Show that $\rho =a\cot t$.

The point $C$ lies on the normal to the curve at $P$, a distance $\rho$ from $P$ and above the curve. Show that $CQ$ is parallel to the $y$-axis.

Let $y=\ln (x^2-1)$, where $x>1$, and let $r$ and $\theta$ be functions of $x$ determined by $r=\sqrt{x^2-1}$ and $\coth \theta =x$. Show that $$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{2\cosh \theta }{r}\text{ and }\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}=-\frac{2\cosh 2\theta }{r^2},$$ and find an expression in terms of $r$ and $\theta$ for $\frac{{\mathrm{d}}^{3}y}{\mathrm{d}{x}^3}$.

Find, with proof, a similar formula for $\frac{{\mathrm{d}}^{n}y}{\mathrm{d}{x}^n}$ in terms of $r$ and $\theta$.

The distinct points $P$, $Q$, $R$ and $S$ in the Argand diagram lie on a circle of radius $a$ centred at the origin and are represented by the complex numbers $p$, $q$, $r$ and $s$, respectively. Show that $$pq=-a^2\frac{p-q}{p^{\ast }-q^{\ast }}.$$ Deduce that, if the chords $PQ$ and $RS$ are perpendicular, then $pq+rs=0$.

The distinct points $A_1$, $A_2$, $\dots$, $A_n$ (where $n⩾3$) lie on a circle. The points {$B_1$, $B_2$, $\dots$, $B_n$} lie on the same circle and are chosen so that the chords $B_1{B}_2$, $B_2{B}_3$, $\dots$, $B_n{B}_1$ are perpendicular, respectively, to the chords $A_1{A}_2$, $A_2{A}_3$, $\dots$, $A_n{A}_1$. Show that, for $n=3$, there are only two choices of $B_1$ for which this is possible. What is the corresponding result for $n=4$? State the corresponding results for values of $n$ greater than 4.

The functions $\mathrm{s}(x)$ ($0⩽x<1$) and $\mathrm{t}(x)$ ($x⩾0$), and the real number $p$, are defined by $$\mathrm{s}(x)={\int }_0^x\frac{1}{\sqrt{1-u^2}}\mathrm{d}u,\mathrm{t}(x)={\int }_0^x\frac{1}{1+u^2}\mathrm{d}u,p=2{\int }_0^{\infty }\frac{1}{1+u^2}\mathrm{d}u.$$ For this question, do not evaluate any of the above integrals explicitly in terms of inverse trigonometric functions or the number $\pi$.

1. Use the substitution $u=v^{-1}$ to show that $\mathrm{t}(x)={\int }_{1/x}^{\infty }\frac{1}{1+v^2}\mathrm{d}v$. Hence evaluate $\mathrm{t}(1/x)+\mathrm{t}(x)$ in terms of $p$ and deduce that $2\mathrm{t}(1)=\frac{1}{2}p$.
2. Let $y=\frac{u}{\sqrt{1+u^2}}$\;. Express $u$ in terms of $y$, and show that $\frac{\mathrm{d}u}{\mathrm{d}y}=\frac{1}{\sqrt{(1-y^2{)}^3}}.$ By making a substitution in the integral for $\mathrm{t}(x)$, show that $$\mathrm{t}(x)=\mathrm{s}\left(\frac{x}{\sqrt{1+x^2}}\right).$$ Deduce that $\mathrm{s}\left(\frac{1}{\sqrt{2}}\right)=\frac{1}{4}p$.
3. Let $z=\frac{u+\frac{1}{\sqrt{3}}}{1-\frac{1}{\sqrt{3}}u}$. Show that $\mathrm{t}(\frac{1}{\sqrt{3}})={\int }_{\frac{1}{\sqrt{3}}}^{\sqrt{3}}\frac{1}{1+z^2}\mathrm{d}z,$ and hence that $3\mathrm{t}(\frac{1}{\sqrt{3}})=\frac{1}{2}p$.

Two small beads, $A$ and $B$, each of mass $m$, are threaded on a smooth horizontal circular hoop of radius $a$ and centre $O$. The angle $\theta$ is the acute angle determined by $2\theta =\angle AOB$.

The beads are connected by a light straight spring. The energy stored in the spring is $$m{k}^2{a}^2(\theta -\alpha {)}^2,$$ where $k$ and $\alpha$ are constants satisfying $k>0$ and $\frac{\pi }{4}<\alpha <\frac{\pi }{2}$.

The spring is held in compression with $\theta =\beta$ and then released. Find the period of oscillations in the two cases that arise according to the value of $\beta$ and state the value of $\beta$ for which oscillations do not occur.

A particle is projected from a point on a plane that is inclined at an angle $\varphi$ to the horizontal. The position of the particle at time $t$ after it is projected is $(x,y)$, where $(0,0)$ is the point of projection, $x$ measures distance up the line of greatest slope and $y$ measures perpendicular distance from the plane. Initially, the velocity of the particle is given by $(\dot{x},\dot{y})=(V\cos \theta ,V\sin \theta )$, where $V>0$ and $\varphi +\theta <\pi /2$. Write down expressions for $x$ and $y$.

The particle bounces on the plane and returns along the same path to the point of projection. Show that $$2\tan \varphi \tan \theta =1$$ and that $$R=\frac{V^2{\cos }^2\theta }{2g\sin \varphi },$$ where $R$ is the range along the plane.

Show further that $$\frac{2V^2}{gR}=3\sin \varphi +\mathrm{cosec}\varphi$$ and deduce that the largest possible value of $R$ is $V^2/(\sqrt{3}g)$.

I choose a number from the integers $1,2,\dots$, $(2n-1)$ and the outcome is the random variable $N$. Calculate $\mathrm{E}(N)$ and $\mathrm{E}(N^2)$.

I then repeat a certain experiment $N$ times, the outcome of the $i$th experiment being the random variable $X_i$ ($1⩽i⩽N$). For each $i$, the random variable $X_i$ has mean $\mu$ and variance ${\sigma }^2$, and $X_i$ is independent of $X_j$ for $i\ne j$ and also independent of $N$. The random variable $Y$ is defined by $Y=\sum _{i=1}^N{X}_i$. Show that $\mathrm{E}(Y)=n\mu$ and that $\mathrm{Cov}(Y,N)=\frac{1}{3}n(n-1)\mu$. Find $\mathrm{Var}(Y)$ in terms of $n$, ${\sigma }^2$ and $\mu$.

A frog jumps towards a large pond. Each jump takes the frog either $1\mathrm{m}$ or $2\mathrm{m}$ nearer to the pond. The probability of a $1\mathrm{m}$ jump is $p$ and the probability of a $2\mathrm{m}$ jump is $q$, where $p+q=1$, the occurence of long and short jumps being independent.

1. Let $p_n(j)$ be the probability that the frog, starting at a point $(n-\frac{1}{2})\mathrm{m}$ away from the edge of the pond, lands in the pond for the first time on its $j$th jump. Show that $p_2(2)=p$.
2. Let $u_n$ be the expected number of jumps, starting at a point $(n-\frac{1}{2})\mathrm{m}$ away from the edge of the pond, required to land in the pond for the first time. Write down the value of $u_1$. By finding first the relevant values of $p_n(m)$, calculate $u_2$ and show that $u_3=3-2q+q^2$.
3. Given that $u_n$ can be expressed in the form $u_n=A(-q{)}^{n-1}+B+Cn$, where $A$, $B$ and $C$ are constants (independent of $n$), show that $C=(1+q{)}^{-1}$ and find $A$ and $B$ in terms of $q$. Hence show that, for large $n$, $u_n\approx \frac{n}{p+2q}$ and explain carefully why this result is to be expected.