STEP 3 2023

The distinct points $P(2ap,a{p}^2)$ and $Q(2aq,a{q}^2)$ lie on the curve $x^2=4ay$, where $a>0$.

(i) Given that$$(p+q{)}^2=p^2{q}^2+6pq+5,(\ast )$$show that the line through $P$ and $Q$ is a tangent to the circle with centre $(0,3a)$ and radius $2a$.

(ii) Show that, for any given value of $p$ with $p^2\ne 1$, there are two distinct real values of $q$ that satisfy equation $(\ast )$. Let these values be $q_1$ and $q_2$. Find expressions, in terms of $p$, for $q_1+q_2$ and $q_1{q}_2$.

(iii) Show that, for any given value of $p$ with $p^2\ne 1$, there is a triangle with one vertex at $P$ such that all three vertices lie on the curve $x^2=4ay$ and all three sides are tangents to the circle with centre $(0,3a)$ and radius $2a$.

The polar curves $C_1$ and $C_2$ are defined for $0\le \theta \le \pi$ by$$r=k(1+\sin \theta )$$$$r=k+\cos \theta$$respectively, where $k$ is a constant greater than $1$.

(i) Sketch the curves on the same diagram. Show that if $\theta =\alpha$ at the point where the curves intersect, $\tan \alpha =\frac{1}{k}$.

(ii) The region $A$ is defined by the inequalities$$0\le \theta \le \alpha \text{and}r\le k(1+\sin \theta ).$$Show that the area of $A$ can be written as$$\frac{k^2}{4}(3\alpha -\sin \alpha \cos \alpha )+k^2(1-\cos \alpha ).$$(iii) The region $B$ is defined by the inequalities$$\alpha \le \theta \le \pi \text{and}r\le k+\cos \theta .$$Find an expression in terms of $k$ and $\alpha$ for the area of $B$.

(iv) The total area of regions $A$ and $B$ is denoted by $R$. The area of the region enclosed by $C_1$ and the lines $\theta =0$ and $\theta =\pi$ is denoted by $S$. The area of the region enclosed by $C_2$ and the lines $\theta =0$ and $\theta =\pi$ is denoted by $T$.

Show that as $k\to \infty$,$$\frac{R}{T}\to 1$$and find the limit of $\frac{R}{S}$ as $k\to \infty$.

(i) Show that, if $a$ and $b$ are complex numbers, with $b\ne 0$, and $s$ is a positive real number, then the points in the Argand diagram representing the complex numbers $a+sbi$, $a-sbi$ and $a+b$ form an isosceles triangle. Given three points which form an isosceles triangle in the Argand diagram, explain with the aid of a diagram how to determine the values of $a$, $b$ and $s$ so that the vertices of the triangle represent complex numbers $a+sbi$, $a-sbi$ and $a+b$.

(ii) Show that, if the roots of the equation $z^3+pz+q=0$, where $p$ and $q$ are complex numbers, are represented in the Argand diagram by the vertices of an isosceles triangle, then there is a non-zero real number $s$ such that$$\frac{p^3}{q^2}=\frac{27(3s^2-1{)}^3}{4(9s^2+1{)}^2}.$$(iii) Sketch the graph $y=\frac{(3x-1{)}^3}{(9x+1{)}^2}$, identifying any stationary points.

(iv) Show that if the roots of the equation $z^3+pz+q=0$ are represented in the Argand diagram by the vertices of an isosceles triangle then $\frac{p^3}{q^2}$ is a real number and $\frac{p^3}{q^2}>-\frac{27}{4}$.

Let $n$ be a positive integer. The polynomial $p$ is defined by the identity$$p(\cos \theta )\equiv \cos ((2n+1)\theta )+1.$$(i) Show that$$\cos ((2n+1)\theta )=\sum _{r=0}^n\left(\genfrac{}{}{0ex}{}{2n+1}{2r}\right){\cos }^{2n+1-2r}\theta ({\cos }^2\theta -1{)}^r.$$(ii) By considering the expansion of $(1+t{)}^{2n+1}$ for suitable values of $t$, show that the coefficient of $x^{2n+1}$ in the polynomial $p(x)$ is $2^{2n}$.

(iii) Show that the coefficient of $x^{2n-1}$ in the polynomial $p(x)$ is $-(2n+1)2^{2n-2}$.

(iv) It is given that there exists a polynomial $q$ such that$$p(x)=(x+1)[q(x){]}^2$$and the coefficient of $x^n$ in $q(x)$ is greater than $0$.

Write down the coefficient of $x^n$ in the polynomial $q(x)$ and, for $n\ge 2$, show that the coefficient of $x^{n-2}$ in the polynomial $q(x)$ is$$2^{n-2}(1-n).$$

(i) Show that if$$\frac{1}{x}+\frac{2}{y}=\frac{2}{7},$$then $(2x-7)(y-7)=49$.

By considering the factors of 49, find all the pairs of positive integers $x$ and $y$ such that$$\frac{1}{x}+\frac{2}{y}=\frac{2}{7}.$$(ii) Let $p$ and $q$ be prime numbers such that$$p^2+pq+q^2=n^2$$where $n$ is a positive integer. Show that$$(p+q+n)(p+q-n)=pq$$and hence explain why $p+q=n+1$.

Hence find the possible values of $p$ and $q$.

(iii) Let $p$ and $q$ be positive and$$p^3+q^3+3p{q}^2=n^3.$$Show that $p+q-n<p$ and $p+q-n<q$.

Show that there are no prime numbers $p$ and $q$ such that $p^3+q^3+3p{q}^2$ is the cube of an integer.

(i) By considering the Maclaurin series for $e^x$, show that for all real $x$,$${\cosh }^{2}x\ge 1+x^2.$$Hence show that the function f, defined for all real $x$ by $f(x)={\tan }^{-1}x-\tanh x$, is an increasing function. Sketch the graph $y=f(x)$.

(ii) Function g is defined for all real $x$ by $g(x)={\tan }^{-1}x-\frac{1}{2}\pi \tanh x$.

(a) Show that g has at least two stationary points.
(b) Show, by considering its derivative, that $(1+x^2)\sinh x-x\cosh x$ is non-negative for $x\ge 0$.

(c) Show that $\frac{{\cosh }^{2}x}{1+x^2}$ is an increasing function for $x\ge 0$.

(d) Hence or otherwise show that g has exactly two stationary points.
(e) Sketch the graph $y=g(x)$.

(i) Let $f$ be a continuous function defined for $0\le x\le 1$. Show that$${\int }_0^{1}f(\sqrt{x})dx=2{\int }_0^{1}xf(x)dx.$$(ii) Let $g$ be a continuous function defined for $0\le x\le 1$ such that$${\int }_0^1(g(x){)}^{2}dx={\int }_0^{1}g(\sqrt{x})dx-\frac{1}{3}.$$Show that ${\int }_0^1(g(x)-x{)}^{2}dx=0$ and explain why $g(x)=x$ for $0\le x\le 1$.

(iii) Let $h$ be a continuous function defined for $0\le x\le 1$ with derivative $h^{\prime }$ such that$${\int }_0^1(h^{\prime }(x){)}^{2}dx=2h(1)-2{\int }_0^{1}h(x)dx-\frac{1}{3}.$$Given that $h(0)=0$, find $h$.

(iv) Let $k$ be a continuous function defined for $0\le x\le 1$ and $a$ be a real number, such that$${\int }_0^1{e}^{ax}(k(x){)}^{2}dx=2{\int }_0^{1}k(x)dx+\frac{e^{-a}}{a}-\frac{1}{a^2}-\frac{1}{4}.$$Show that $a$ must be equal to $2$ and find $k$.

If$$y=\{\begin{array}{ll}{k}_1(x)& x\le b\\ k_2(x)& x\ge b\end{array}$$with $k_1(b)=k_2(b)$, then $y$ is said to be continuously differentiable at $x=b$ if $k_1^{\prime }(b)=k_2^{\prime }(b)$.

(i) Let $f(x)=x{e}^{-x}$. Verify that, for all real $x$, $y=f(x)$ is a solution to the differential equation$$\frac{d^{2}y}{d{x}^2}+2\frac{dy}{dx}+y=0$$and that $y=0$ and $\frac{dy}{dx}=1$ when $x=0$. Show that $f^{\prime }(x)\ge 0$ for $x\le 1$.

(ii) You are given the differential equation$$\frac{d^{2}y}{d{x}^2}+2\frac{dy}{dx}+y=0$$where $y=0$ and $\frac{dy}{dx}=1$ when $x=0$. Let$$y=\{\begin{array}{ll}{g}_1(x)& x\le 1\\ g_2(x)& x\ge 1\end{array}$$be a solution of the differential equation which is continuously differentiable at $x=1$. Write down an expression for $g_1(x)$ and find an expression for $g_2(x)$.

(iii) State the geometrical relationship between the curves $y=g_1(x)$ and $y=g_2(x)$.

(iv) Prove that if $y=k(x)$ is a solution of the differential equation$$\frac{d^{2}y}{d{x}^2}+p\frac{dy}{dx}+qy=0$$in the interval $r\le x\le s$, where $p$ and $q$ are constants, then, in a suitable interval which you should state, $y=k(c-x)$ satisfies the differential equation$$\frac{d^{2}y}{d{x}^2}-p\frac{dy}{dx}+qy=0.$$(v) You are given the differential equation$$\frac{d^{2}y}{d{x}^2}+2\frac{dy}{dx}+2y=0$$where $y=0$ and $\frac{dy}{dx}=1$ when $x=0$.

Let $h(x)=e^{-x}\sin x$. Show that $h^{\prime }\left(\frac{1}{4}\pi \right)=0$.

It is given that $y=h(x)$ satisfies the differential equation in the interval $-\frac{3}{4}\pi \le x\le \frac{1}{4}\pi$ and that $h^{\prime }(x)\ge 0$ in this interval.

In a solution to the differential equation which is continuously differentiable at $\left(n+\frac{1}{4}\right)\pi$ for all $n\in \mathbb{Z}$, find $y$ in terms of $x$ in the intervals

(a) $\frac{1}{4}\pi \le x\le \frac{5}{4}\pi$,
(b) $\frac{5}{4}\pi \le x\le \frac{9}{4}\pi$.

Two particles, $A$ of mass $m$ and $B$ of mass $M$, are fixed to the ends of a light inextensible string $AB$ of length $r$ and lie on a smooth horizontal plane. The origin of coordinates and the $x$- and $y$-axes are in the plane.

Initially, $A$ is at $(0,0)$ and $B$ is at $(r,0)$. $B$ is at rest and $A$ is given an instantaneous velocity of magnitude $u$ in the positive $y$ direction.

At a time $t$ after this, $A$ has position $(x,y)$ and $B$ has position $(X,Y)$. You may assume that, in the subsequent motion, the string remains taut.

(i) Explain by means of a diagram why$$X=x+r\cos \theta ,Y=y-r\sin \theta ,$$where $\theta$ is the angle clockwise from the positive $x$-axis of the vector $\overrightarrow{AB}$.

(ii) Find expressions for $\dot{X}$, $\dot{Y}$, $\ddot{X}$ and $\ddot{Y}$ in terms of $\ddot{x}$, $\ddot{y}$, $\dot{x}$, $\dot{y}$, $r$, $\ddot{\theta }$, $\dot{\theta }$ and $\theta$, as appropriate.

Assume that the tension $T$ in the string is the only force acting on either particle.

(iii) Show that$$\ddot{x}\sin \theta +\ddot{y}\cos \theta =0,\ddot{X}\sin \theta +\ddot{Y}\cos \theta =0,$$and hence that $\theta =\frac{ut}{r}$.

(iv) Show that$$m\ddot{x}+M\ddot{X}=0,m\ddot{y}+M\ddot{Y}=0,$$and find $my+MY$ in terms of $t$ and $m$, $M$, $u$, $r$ as appropriate.

(v) Show that$$y=\frac{1}{m+M}\left(mut+Mr\sin \frac{ut}{r}\right).$$(vi) Show that, if $M>m$, then the $y$ component of the velocity of particle $A$ will be negative at some time in the subsequent motion.

A thin uniform beam AB has mass $3m$ and length $2h$. End A rests on rough horizontal ground and the beam makes an angle of $2\beta$ to the vertical, supported by a light inextensible string attached to end B. The coefficient of friction between the beam and the ground at A is $\mu$.

The string passes over a small frictionless pulley fixed to a point C which is a distance $2h$ vertically above A. A particle of mass $km$, where $k<3$, is attached to the other end of the string and hangs freely.

(i) Given that the system is in equilibrium, find an expression for $k$ in terms of $\beta$ and show that$$k^2\le \frac{9{\mu }^2}{{\mu }^2+1}.$$(ii) A particle of mass $m$ is now fixed to the beam at a distance $xh$ from A, where $0\le x\le 2$. Given that $k=2$, and that the system is in equilibrium, show that$$\frac{F^2}{N^2}=\frac{x^2+6x+5}{4(x+2{)}^2},$$where $F$ is the frictional force and $N$ is the normal reaction on the beam at A.

By considering $\frac{1}{3}-\frac{F^2}{N^2}$, or otherwise, find the minimum value of $\mu$ for which the beam can be in equilibrium whatever the value of $x$.

Show that$$\sum _{k=1}^{\infty }\frac{k+1}{k!}{x}^k=(x+1)e^x-1.$$In the remainder of this question, $n$ is a fixed positive integer.

(i) Random variable $Y$ has a Poisson distribution with mean $n$. One observation of $Y$ is taken. Random variable $D$ is defined as follows. If the observed value of $Y$ is zero then $D=0$. If the observed value of $Y$ is $k$, where $k\ge 1$, then a fair $k$-sided die (with sides numbered $1$ to $k$) is rolled once and $D$ is the number shown on the die.

(a) Write down $P(D=0)$.

(b) Show, from the definition of the expectation of a random variable, that$$E(D)=\sum _{d=1}^{\infty }\left[\sum _{k=d}^{\infty }\frac{1}{k}\cdot \frac{n^k}{k!}\cdot e^{-n}\right].$$Show further that$$E(D)=\sum _{k=1}^{\infty }\frac{1}{k}\cdot \frac{n^k}{k!}\cdot e^{-n}\sum _{d=1}^{k}d.$$(c) Show that $E(D)=\frac{1}{2}(n+1-e^{-n})$.

(ii) Random variables $X_1,X_2,\dots ,X_n$ all have Poisson distributions. For each $k\in \{1,2,\dots ,n\}$, the mean of $X_k$ is $k$. A fair $n$-sided die, with sides numbered $1$ to $n$, is rolled. When $k$ is the number shown, one observation of $X_k$ is recorded. Let $Z$ be the number recorded.

(a) Find $P(Z=0)$.

(b) Show that $E(Z)>E(D)$.

A drawer contains $n$ pairs of socks. The two socks in each pair are indistinguishable, but each pair of socks is a different colour from all the others. A set of $2k$ socks, where $k$ is an integer with $2k\le n$, is selected at random from this drawer: that is, every possible set of $2k$ socks is equally likely to be selected.

(i) Find the probability that, among the socks selected, there is no pair of socks.
(ii) Let $X_{n,k}$ be the random variable whose value is the number of pairs of socks found amongst those selected. Show that$$P(X_{n,k}=r)=\frac{\left(\genfrac{}{}{0ex}{}{n}{r}\right)\left(\genfrac{}{}{0ex}{}{n-r}{2(k-r)}\right)2^{2(k-r)}}{\left(\genfrac{}{}{0ex}{}{2n}{2k}\right)}$$for $0\le r\le k$.

(iii) Show that$$rP(X_{n,k}=r)=\frac{k(2k-1)}{2n-1}P(X_{n-1,k-1}=r-1),$$for $1\le r\le k$, and hence find $E(X_{n,k})$.