STEP 3 2022

Let $C_1$ be the curve given by the parametric equations$$x=ct,y=\frac{c}{t},$$where $c>0$ and $t\ne 0$, and let $C_2$ be the circle$$(x-a{)}^2+(y-b{)}^2=r^2.$$$C_1$ and $C_2$ intersect at the four points $P_i$ ($i=1,2,3,4$), and the corresponding values of the parameter $t$ at these points are $t_i$.

(i) Show that $t_i$ are the roots of the equation$$c^2{t}^4-2ac{t}^3+(a^2+b^2-r^2)t^2-2bct+c^2=0.(\ast )$$(ii) Show that$$\sum _{i=1}^4{t}_i^2=\frac{2}{c^2}(a^2-b^2+r^2)$$and find a similar expression for $\sum _{i=1}^4\frac{1}{t_i^2}$.

(iii) Hence show that $\sum _{i=1}^{4}O{P}_i^2=4r^2$, where $O{P}_i$ denotes the distance of the point $P_i$ from the origin.

(iv) Suppose that the curves $C_1$ and $C_2$ touch at two distinct points. By considering the product of the roots of $(\ast )$, or otherwise, show that the centre of circle $C_2$ must lie on either the line $y=x$ or $y=-x$.

(i) Suppose that there are three non-zero integers $a$, $b$ and $c$ for which $a^3+2b^3+4c^3=0$. Explain why there must exist an integer $p$, with $|p|<|a|$, such that $4p^3+b^3+2c^3=0$, and show further that there must exist integers $p$, $q$ and $r$, with $|p|<|a|$, $|q|<|b|$ and $|r|<|c|$, such that $p^3+2q^3+4r^3=0$. Deduce that no such integers $a$, $b$ and $c$ can exist.

(ii) Prove that there are no non-zero integers $a$, $b$ and $c$ for which $9a^3+10b^3+6c^3=0$.

(iii) By considering the expression $(3n\pm 1{)}^2$, prove that, unless an integer is a multiple of three, its square is one more than a multiple of three. Deduce that the sum of the squares of two integers can only be a multiple of three if each of the integers is a multiple of three. Hence prove that there are no non-zero integers $a$, $b$ and $c$ for which $a^2+b^2=3c^2$.

(iv) Prove that there are no non-zero integers $a$, $b$ and $c$ for which $a^2+b^2+c^2=4abc$.

(i) The curve $C_1$ has equation$$a{x}^2+bxy+c{y}^2=1$$where $abc\ne 0$ and $a>0$.

Show that, if the curve has two stationary points, then $b^2<4ac$.

(ii) The curve $C_2$ has equation$$a{y}^3+b{x}^{2}y+cx=1$$where $abc\ne 0$ and $b>0$.

Show that the $x$-coordinates of stationary points on this curve satisfy$$4c{b}^3{x}^4-8b^3{x}^3-a{c}^3=0.$$Show that, if the curve has two stationary points, then $4a{c}^6+27b^3>0$.

(iii) Consider the simultaneous equations$$a{y}^3+b{x}^{2}y+cx=1$$$$2bxy+c=0$$$$3a{y}^2+b{x}^2=0$$where $abc\ne 0$ and $b>0$.

Show that, if these simultaneous equations have a solution, then $4a{c}^6+27b^3=0$.

You may assume that all infinite sums and products in this question converge.

(i) Prove by induction that for all positive integers $n$,$$\sinh x=2^n\cosh \left(\frac{x}{2}\right)\cosh \left(\frac{x}{4}\right)\cdots \cosh \left(\frac{x}{2^n}\right)\sinh \left(\frac{x}{2^n}\right)$$and deduce that, for $x\ne 0$,$$\frac{\sinh x}{x}=\frac{2^n}{\sinh \left(\frac{x}{2^n}\right)/\left(\frac{x}{2^n}\right)}\cdot \cosh \left(\frac{x}{2}\right)\cosh \left(\frac{x}{4}\right)\cdots \cosh \left(\frac{x}{2^n}\right).$$Wait — more precisely, deduce that for $x\ne 0$,$$\frac{\sinh x}{x}=\cosh \left(\frac{x}{2}\right)\cosh \left(\frac{x}{4}\right)\cdots \cosh \left(\frac{x}{2^n}\right)\cdot \frac{\sinh \left(\frac{x}{2^n}\right)}{\frac{x}{2^n}}.$$(ii) You are given that the Maclaurin series for $\sinh x$ is$$\sinh x=\sum _{r=0}^{\infty }\frac{x^{2r+1}}{(2r+1)!}.$$Use this result to show that, as $y$ tends to $0$, $\frac{y}{\sinh y}$ tends to $1$.

Deduce that, for $x\ne 0$,$$\frac{\sinh x}{x}=\cosh \left(\frac{x}{2}\right)\cosh \left(\frac{x}{4}\right)\cdots \cosh \left(\frac{x}{2^n}\right)\cdots$$(iii) Let $x=\ln 2$. Evaluate $\cosh \left(\frac{x}{2}\right)$ and show that$$\cosh \left(\frac{x}{4}\right)=\frac{1+2^{1/2}}{2\times 2^{1/4}}.$$Use part (ii) to show that$$\frac{1}{\ln 2}=\frac{1+2^{1/2}}{2}\times \frac{1+2^{1/4}}{2}\times \frac{1+2^{1/8}}{2}\cdots$$(iv) Show that$$\frac{2}{\pi }=\frac{\sqrt{2}}{2}\times \frac{\sqrt{2+\sqrt{2}}}{2}\times \frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2}\cdots$$

(i) Show that$${\int }_{-a}^a\frac{1}{1+e^x}dx=a\text{for all }a\ge 0.$$(ii) Explain why, if $g$ is a continuous function and$${\int }_0^{a}g(x)dx=0\text{for all }a\ge 0,$$then $g(x)=0$ for all $x\ge 0$.

Let $f$ be a continuous function with $f(x)\ge 0$ for all $x$. Show that$${\int }_{-a}^a\frac{1}{1+f(x)}dx=a\text{for all }a\ge 0$$if and only if$$\frac{1}{1+f(x)}+\frac{1}{1+f(-x)}-1=0\text{for all }x\ge 0,$$and hence if and only if $f(x)f(-x)=1$ for all $x$.

(iii) Let $f$ be a continuous function such that, for all $x$, $f(x)\ge 0$ and $f(x)f(-x)=1$. Show that, if $h$ is a continuous function with $h(x)=h(-x)$ for all $x$, then$${\int }_{-a}^a\frac{h(x)}{1+f(x)}dx={\int }_0^{a}h(x)dx.$$(iv) Hence find the exact value of$${\int }_{-\frac{1}{2}\pi }^{\frac{1}{2}\pi }\frac{e^{-x}\cos x}{\cosh x}dx.$$

(i) Show that when $\alpha$ is small, $\cos (\theta +\alpha )-\cos \theta \approx -\alpha \sin \theta -\frac{1}{2}{\alpha }^2\cos \theta$.

Find the limit as $\alpha \to 0$ of$$\frac{\sin (\theta +\alpha )-\sin \theta }{\cos (\theta +\alpha )-\cos \theta }(\ast )$$in the case $\sin \theta \ne 0$.

In the case $\sin \theta =0$, what happens to the value of expression $(\ast )$ when $\alpha \to 0$?

(ii) A circle $C_1$ of radius $a$ rolls without slipping in an anti-clockwise direction on a fixed circle $C_2$ with centre at the origin $O$ and radius $(n-1)a$, where $n$ is an integer greater than $2$. The point $P$ is fixed on $C_1$. Initially the centre of $C_1$ is at $(na,0)$ and $P$ is at $((n+1)a,0)$.

(a) Let $Q$ be the point of contact of $C_1$ and $C_2$ at any time in the rolling motion. Show that when $OQ$ makes an angle $\theta$, measured anticlockwise, with the positive $x$-axis, the $x$-coordinate of $P$ is $x(\theta )=a(n\cos \theta +\cos n\theta )$, and find the corresponding expression for the $y$-coordinate, $y(\theta )$, of $P$.

(b) Find the values of $\theta$ for which the distance $OP$ is $(n-1)a$.

(c) Let ${\theta }_0=\frac{1}{n-1}\pi$. Find the limit as $\alpha \to 0$ of$$\frac{y({\theta }_0+\alpha )-y({\theta }_0)}{x({\theta }_0+\alpha )-x({\theta }_0)}.$$Hence show that, at the point $(x({\theta }_0),y({\theta }_0))$, the tangent to the curve traced out by $P$ is parallel to $OP$.

Let $\mathbf{n}$ be a vector of unit length in three dimensions. For each vector $\mathbf{r}$, $f(\mathbf{r})$ is defined by $f(\mathbf{r})=\mathbf{n}\times \mathbf{r}$.

(i) Given that $\mathbf{n}=\left(\begin{array}{c}a\\ b\\ c\end{array}\right)$ and $\mathbf{r}=\left(\begin{array}{c}x\\ y\\ z\end{array}\right)$, show that the $x$-component of $f(f(\mathbf{r}))$ is $-x(b^2+c^2)+aby+acz$. Show further that$$f(f(\mathbf{r}))=(\mathbf{n}\cdot \mathbf{r})\mathbf{n}-\mathbf{r}.$$Explain, by means of a diagram, how $f(f(\mathbf{r}))$ is related to $\mathbf{n}$ and $\mathbf{r}$.

(ii) Let $R$ be the point with position vector $\mathbf{r}$ and $P$ be the point with position vector $g(\mathbf{r})$, where $g$ is defined by$$g(\mathbf{s})=\mathbf{s}+\sin \theta f(\mathbf{s})+(1-\cos \theta )f(f(\mathbf{s})).$$By considering $g(\mathbf{n})$ and $g(\mathbf{r})$ when $\mathbf{r}$ is perpendicular to $\mathbf{n}$, state, with justification, the geometric transformation which maps $R$ onto $P$.

(iii) Let $R$ be the point with position vector $\mathbf{r}$ and $Q$ be the point with position vector $h(\mathbf{r})$, where $h$ is defined by$$h(\mathbf{s})=-\mathbf{s}-2f(f(\mathbf{s})).$$State, with justification, the geometric transformation which maps $R$ onto $Q$.

(i) Use De Moivre's theorem to prove that for any positive integer $k>1$,$$\sin (k\theta )={\sin }^k\theta \left[k-\left(\genfrac{}{}{0ex}{}{k}{3}\right)({\sec }^2\theta -1)+\left(\genfrac{}{}{0ex}{}{k}{5}\right)({\sec }^2\theta -1{)}^2-\cdots \right]$$and find a similar expression for $\cos (k\theta )$.

(ii) Let $\theta ={\cos }^{-1}\left(\frac{1}{a}\right)$, where $\theta$ is measured in degrees, and $a$ is an odd integer greater than 1. Suppose that there is a positive integer $k$ such that $\sin (k\theta )=0$ and $\sin (m\theta )\ne 0$ for all integers $m$ with $0<m<k$. Show that it would be necessary to have $k$ even and $\cos (\frac{1}{2}k\theta )=0$. Deduce that $\theta$ is irrational.

(iii) Show that if $\phi ={\cot }^{-1}\left(\frac{1}{b}\right)$, where $\phi$ is measured in degrees, and $b$ is an even integer greater than 1, then $\phi$ is irrational.

(i) Two particles $A$ and $B$, of masses $m$ and $km$ respectively, lie at rest on a smooth horizontal surface. The coefficient of restitution between the particles is $e$, where $0<e<1$. Particle $A$ is then projected directly towards particle $B$ with speed $u$.

Let $v_1$ and $v_2$ be the velocities of particles $A$ and $B$, respectively, after the collision, in the direction of the initial velocity of $A$.

Show that $v_1=\alpha u$ and $v_2=\beta u$, where $\alpha =\frac{1-ke}{k+1}$ and $\beta =\frac{1+e}{k+1}$.

Particle $B$ strikes a vertical wall which is perpendicular to its direction of motion and a distance $D$ from the point of collision with $A$, and rebounds. The coefficient of restitution between particle $B$ and the wall is also $e$.

Show that, if $A$ and $B$ collide for a second time at a point $\frac{1}{2}D$ from the wall, then$$k=\frac{1+e-e^2}{e(2e+1)}.$$(ii) Three particles $A$, $B$ and $C$, of masses $m$, $km$ and $k^{2}m$ respectively, lie at rest on a smooth horizontal surface in a straight line, with $B$ between $A$ and $C$. A vertical wall is perpendicular to this line and lies on the side of $C$ away from $A$ and $B$. The distance between $B$ and $C$ is equal to $d$ and the distance between $C$ and the wall is equal to $3d$. The coefficient of restitution between each pair of particles, and between particle $C$ and the wall, is $e$, where $0<e<1$. Particle $A$ is then projected directly towards particle $B$ with speed $u$.

Show that, if all three particles collide simultaneously at a point $\frac{3}{2}d$ from the wall, then $e=\frac{1}{2}$.

Two light elastic springs each have natural length $a$. One end of each spring is attached to a particle $P$ of weight $W$. The other ends of the springs are attached to the endpoints, $B$ and $C$, of a fixed horizontal bar $BC$ of length $2a$. The moduli of elasticity of the springs $PB$ and $PC$ are $s_{1}W$ and $s_{2}W$ respectively; these values are such that the particle $P$ hangs in equilibrium with angle $BPC$ equal to $90^{\circ }$.

(i) Let angle $PBC=\theta$. Show that$$s_1=\frac{\sin \theta }{2\cos \theta -1}$$and find $s_2$ in terms of $\theta$.

(ii) Take the zero level of gravitational potential energy to be the horizontal bar $BC$ and let the total potential energy of the system be $-paW$. Show that $p$ satisfies$$\frac{1}{2}\sqrt{2}\ge p>\frac{1}{4}(1+\sqrt{3})$$and hence that $p=0.7$, correct to one significant figure.

A fair coin is tossed $N$ times and the random variable $X$ records the number of heads. The mean deviation, $\delta$, of $X$ is defined by$$\delta =E|X-\mu |$$where $\mu$ is the mean of $X$.

(i) Let $N=2n$ where $n$ is a positive integer.

(a) Show that $P(X\le n-1)=\frac{1}{2}\left(1-P(X=n)\right)$.

(b) Show that$$\delta =\sum _{r=0}^{n-1}(n-r)\left(\genfrac{}{}{0ex}{}{2n}{r}\right)\frac{1}{2^{2n-1}}.$$(c) Show that for $r>0$,$$r\left(\genfrac{}{}{0ex}{}{2n}{r}\right)=2n\left(\genfrac{}{}{0ex}{}{2n-1}{r-1}\right).$$Hence show that$$\delta =\frac{n}{2^{2n-1}}\left(\genfrac{}{}{0ex}{}{2n}{n}\right).$$(ii) Find a similar expression for $\delta$ in the case $N=2n+1$.

(i) The point $A$ lies on the circumference of a circle of radius $a$ and centre $O$. The point $B$ is chosen at random on the circumference, so that the angle $AOB$ has a uniform distribution on $[0,2\pi ]$. Find the expected length of the chord $AB$.

(ii) The point $C$ is chosen at random in the interior of a circle of radius $a$ and centre $O$, so that the probability that it lies in any given region is proportional to the area of the region. The random variable $R$ is defined as the distance between $C$ and $O$.

Find the probability density function of $R$.

Obtain a formula in terms of $a$, $R$ and $t$ for the length of a chord through $C$ that makes an acute angle of $t$ with $OC$.

Show that as $C$ varies (with $t$ fixed), the expected length $L(t)$ of such chords is given by$$L(t)=\frac{4a(1-{\cos }^{3}t)}{3{\sin }^{2}t}.$$Show further that$$L(t)=\frac{4a}{3}\left(\cos t+\frac{1}{2}{\sec }^2\left(\frac{1}{2}t\right)\right).$$(iii) The random variable $T$ is uniformly distributed on $[0,\frac{1}{2}\pi ]$. Find the expected value of $L(T)$.