STEP 3 2019

The coordinates of a particle at time $t$ are $x$ and $y$. For $t\ge 0$, they satisfy the pair of coupled differential equations$$\dot{x}=-x-ky$$$$\dot{y}=x-y$$where $k$ is a constant. When $t=0$, $x=1$ and $y=0$.

(i) Let $k=1$. Find $x$ and $y$ in terms of $t$ and sketch $y$ as a function of $t$.

Sketch the path of the particle in the $x$-$y$ plane, giving the coordinates of the point at which $y$ is greatest and the coordinates of the point at which $x$ is least.

(ii) Instead, let $k=0$. Find $x$ and $y$ in terms of $t$ and sketch the path of the particle in the $x$-$y$ plane.

The definition of the derivative $f^{\prime }$ of a (differentiable) function $f$ is$$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}.(\ast )$$(i) The function $f$ has derivative $f^{\prime }$ and satisfies$$f(x+y)=f(x)f(y)$$for all $x$ and $y$, and $f^{\prime }(0)=k$ where $k\ne 0$. Show that $f(0)=1$.

Using $(\ast )$, show that $f^{\prime }(x)=kf(x)$ and find $f(x)$ in terms of $x$ and $k$.

(ii) The function $g$ has derivative $g^{\prime }$ and satisfies$$g(x+y)=\frac{g(x)+g(y)}{1+g(x)g(y)}$$for all $x$ and $y$, $|g(x)|<1$ for all $x$, and $g^{\prime }(0)=k$ where $k\ne 0$.

Find $g^{\prime }(x)$ in terms of $g(x)$ and $k$, and hence find $g(x)$ in terms of $x$ and $k$.

The matrix $A$ is given by$$A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right).$$(i) You are given that the transformation represented by $A$ has a line $L_1$ of invariant points (so that each point on $L_1$ is transformed to itself). Let $(x,y)$ be a point on $L_1$. Show that$$(a-1)(d-1)-bc\cdot \frac{x}{y}=0.$$Show further that $(a-1)(d-1)=bc$.

What can be said about $A$ if $L_1$ does not pass through the origin?

(ii) By considering the cases $b\ne 0$ and $b=0$ separately, show that if $(a-1)(d-1)=bc$ then the transformation represented by $A$ has a line of invariant points. You should identify the line in the different cases that arise.

(iii) You are given instead that the transformation represented by $A$ has an invariant line $L_2$ (so that each point on $L_2$ is transformed to a point on $L_2$) and that $L_2$ does not pass through the origin. If $L_2$ has the form $y=mx+k$, show that $(a-1)(d-1)=bc$.

The $n$th degree polynomial $P(x)$ is said to be reflexive if:

(a) $P(x)$ is of the form $x^n-a_1{x}^{n-1}+a_2{x}^{n-2}-\cdots +(-1{)}^n{a}_n$ where $n\ge 1$;
(b) $a_1,a_2,\dots ,a_n$ are real;
(c) the $n$ (not necessarily distinct) roots of the equation $P(x)=0$ are $a_1,a_2,\dots ,a_n$.

(i) Find all reflexive polynomials of degree less than or equal to $3$.

(ii) For a reflexive polynomial with $n>3$, show that$$2a_2=-a_2^2-a_3^2-\cdots -a_n^2.$$Deduce that, if all the coefficients of a reflexive polynomial of degree $n$ are integers and $a_n\ne 0$, then $n\le 3$.

(iii) Determine all reflexive polynomials with integer coefficients.

(i) Let$$f(x)=\frac{x}{\sqrt{x^2+p}},$$where $p$ is a non-zero constant. Sketch the curve $y=f(x)$ for $x\ge 0$ in the case $p>0$.

(ii) Let$$I=\int \frac{1}{\sqrt{(b^2-y^2)(c^2-y^2)}}dy,$$where $b$ and $c$ are positive constants. Use the substitution $y=\frac{cx}{\sqrt{x^2+p}}$, where $p$ is a suitably chosen constant, to show that$$I=\int \frac{1}{b^2+(b^2-c^2)x^2}dx.$$Evaluate$${\int }_1^{\sqrt{2}}\frac{1}{\sqrt{(3-y^2)(2-y^2)}}dy.$$$$\left[\text{Note: }\int \frac{1}{a^2+x^2}dx=\frac{1}{a}{\tan }^{-1}\frac{x}{a}+\text{constant.}\right]$$Hence evaluate$${\int }_{1/\sqrt{2}}^1\frac{y}{\sqrt{(3y^2-1)(2y^2-1)}}dy.$$(iii) By means of a suitable substitution, evaluate$${\int }_{1/\sqrt{2}}^1\frac{1}{\sqrt{(3y^2-1)(2y^2-1)}}dy.$$

6 The point $P$ in the Argand diagram is represented by the complex number $z$, which satisfies$$z{z}^{\ast }-a{z}^{\ast }-a^{\ast }z+a{a}^{\ast }-r^2=0.$$Here, $r$ is a positive real number and $r^2\ne a^{\ast }a$. By writing $|z-a{|}^2$ as $(z-a)(z-a{)}^{\ast }$, show that the locus of $P$ is a circle, $C$, the radius and the centre of which you should give.

(i) The point $Q$ is represented by $\omega$, and is related to $P$ by $\omega =\frac{1}{z}$. Let $C^{\prime }$ be the locus of $Q$. Show that $C^{\prime }$ is also a circle, and give its radius and centre.

If $C$ and $C^{\prime }$ are the same circle, show that$$(|a{|}^2-r^2{)}^2=1$$and that either $a$ is real or $a$ is imaginary. Give sketches to indicate the position of $C$ in these two cases.

(ii) Suppose instead that the point $Q$ is represented by $\omega$, where $\omega =\frac{1}{z^{\ast }}$. If the locus of $Q$ is $C$, is it the case that either $a$ is real or $a$ is imaginary?

The Devil's Curve is given by$$y^2(y^2-b^2)=x^2(x^2-a^2),$$where $a$ and $b$ are positive constants.

(i) In the case $a=b$, sketch the Devil's Curve.

(ii) Now consider the case $a=2$ and $b=\sqrt{5}$, and $x\ge 0$, $y\ge 0$.

(a) Show by considering a quadratic equation in $x^2$ that either $0\le y\le 1$ or $y\ge 2$.

(b) Describe the curve very close to and very far from the origin.
(c) Find the points at which the tangent to the curve is parallel to the $x$-axis and the point at which the tangent to the curve is parallel to the $y$-axis.

Sketch the Devil's Curve in this case.

(iii) Sketch the Devil's Curve in the case $a=2$ and $b=\sqrt{5}$ again, but with $-\infty <x<\infty$ and $-\infty <y<\infty$.

A pyramid has a horizontal rectangular base $ABCD$ and its vertex $V$ is vertically above the centre of the base. The acute angle between the face $AVB$ and the base is $\alpha$, the acute angle between the face $BVC$ and the base is $\beta$ and the obtuse angle between the faces $AVB$ and $BVC$ is $\pi -\theta$.

(i) The edges $AB$ and $BC$ are parallel to the unit vectors $\mathbf{i}$ and $\mathbf{j}$, respectively, and the unit vector $\mathbf{k}$ is vertical. Find a unit vector that is perpendicular to the face $AVB$.

Show that$$\cos \theta =\cos \alpha \cos \beta .$$(ii) The edge $BV$ makes an angle $\phi$ with the base. Show that$${\cot }^2\phi ={\cot }^2\alpha +{\cot }^2\beta .$$Show also that$${\cos }^2\phi =\frac{{\cos }^2\alpha +{\cos }^2\beta -2{\cos }^2\theta }{1-{\cos }^2\theta }\ge \frac{2\cos \theta -2{\cos }^2\theta }{1-{\cos }^2\theta }$$and deduce that $\phi <\theta$.

In this question, $\mathbf{i}$ and $\mathbf{j}$ are perpendicular unit vectors and $\mathbf{j}$ is vertically upwards. A smooth hemisphere of mass $M$ and radius $a$ rests on a smooth horizontal table with its plane face in contact with the table. The point $A$ is at the top of the hemisphere and the point $O$ is at the centre of its plane face. Initially, a particle $P$ of mass $m$ rests at $A$. It is then given a small displacement in the positive $\mathbf{i}$ direction. At a later time $t$, when the particle is still in contact with the hemisphere, the hemisphere has been displaced by $-s\mathbf{i}$ and $\angle AOP=\theta$.

(i) Let $\mathbf{r}$ be the position vector of the particle at time $t$ with respect to the initial position of $O$. Write down an expression for $\mathbf{r}$ in terms of $a$, $\theta$ and $s$ and show that$$\dot{\mathbf{r}}=(a\dot{\theta }\cos \theta -\dot{s})\mathbf{i}-a\dot{\theta }\sin \theta \mathbf{j}.$$Show also that$$\dot{s}=(1-k)a\dot{\theta }\cos \theta ,$$where $k=\frac{M}{m+M}$, and deduce that$$\dot{\mathbf{r}}=a\dot{\theta }(k\cos \theta \mathbf{i}-\sin \theta \mathbf{j}).$$(ii) Show that$$a{\dot{\theta }}^2(k{\cos }^2\theta +{\sin }^2\theta )=2g(1-\cos \theta ).$$(iii) At time $T$, when $\theta =\alpha$, the particle leaves the hemisphere. By considering the component of $\ddot{\mathbf{r}}$ parallel to the vector $\sin \theta \mathbf{i}+k\cos \theta \mathbf{j}$, or otherwise, show that at time $T$$$a{\dot{\theta }}^2=g\cos \alpha .$$Find a cubic equation for $\cos \alpha$ and deduce that $\cos \alpha >\frac{2}{3}$.

Two identical smooth spheres P and Q can move on a smooth horizontal table. Initially, P moves with speed $u$ and Q is at rest. Then P collides with Q. The direction of travel of P before the collision makes an acute angle $\alpha$ with the line joining the centres of P and Q at the moment of the collision. The coefficient of restitution between P and Q is $e$ where $e<1$.

As a result of the collision, P has speed $v$ and Q has speed $w$, and P is deflected through an angle $\theta$.

(i) Show that$$u\sin \alpha =v\sin (\alpha +\theta )$$and find an expression for $w$ in terms of $v$, $\theta$ and $\alpha$.

(ii) Show further that$$\sin \theta =\cos (\theta +\alpha )\sin \alpha +e\sin (\theta +\alpha )\cos \alpha$$and find an expression for $\tan \theta$ in terms of $\tan \alpha$ and $e$.

Find, in terms of $e$, the maximum value of $\tan \theta$ as $\alpha$ varies.

The number of customers arriving at a builders' merchants each day follows a Poisson distribution with mean $\lambda$. Each customer is offered some free sand. The probability of any given customer taking the free sand is $p$.

(i) Show that the number of customers each day who take sand follows a Poisson distribution with mean $p\lambda$.

(ii) The merchant has a mass $S$ of sand at the beginning of the day. Each customer who takes the free sand gets a proportion $k$ of the remaining sand, where $0\le k<1$. Show that by the end of the day the expected mass of sand taken is$$(1-e^{-kp\lambda })S.$$(iii) At the beginning of the day, the merchant's bag of sand contains a large number of grains, exactly one of which is made from solid gold. At the end of the day, the merchant's assistant takes a proportion $k$ of the remaining sand. Find the probability that the assistant takes the golden grain. Comment on the case $k=0$ and on the limit $k\to 1$.

In the case $p\lambda >1$, find the value of $k$ which maximises the probability that the assistant takes the golden grain.

The set $S$ is the set of all integers from $1$ to $n$. The set $T$ is the set of all distinct subsets of $S$, including the empty set $\varnothing$ and $S$ itself. Show that $T$ contains exactly $2^n$ sets.

The sets $A_1,A_2,\dots ,A_m$, which are not necessarily distinct, are chosen randomly and independently from $T$, and for each $k$ ($1\le k\le m$), the set $A_k$ is equally likely to be any of the sets in $T$.

(i) Write down the value of $P(1\in A_1)$.

(ii) By considering each integer separately, show that $P(A_1\cap A_2=\varnothing )={\left(\frac{3}{4}\right)}^n$.

Find $P(A_1\cap A_2\cap A_3=\varnothing )$ and $P(A_1\cap A_2\cap \cdots \cap A_m=\varnothing )$.

(iii) Find $P(A_1\subseteq A_2)$, $P(A_1\subseteq A_2\subseteq A_3)$ and $P(A_1\subseteq A_2\subseteq \cdots \subseteq A_m)$.