STEP 2 2023

(i) Show that making the substitution $x=\frac{1}{t}$ in the integral$${\int }_a^b\frac{1}{(1+x^2{)}^{\frac{3}{2}}}dx,$$where $b>a>0$, gives the integral$${\int }_{a^{-1}}^{b^{-1}}\frac{-t}{(1+t^2{)}^{\frac{3}{2}}}dt.$$(ii) Evaluate:
(a) ${\int }_{\frac{1}{2}}^2\frac{1}{(1+x^2{)}^{\frac{3}{2}}}dx$;

(b) ${\int }_{-2}^2\frac{1}{(1+x^2{)}^{\frac{3}{2}}}dx$.

(iii) (a) Show that$${\int }_{\frac{1}{2}}^2\frac{1}{(1+x^2{)}^2}dx={\int }_{\frac{1}{2}}^2\frac{x^2}{(1+x^2{)}^2}dx=\frac{1}{2}{\int }_{\frac{1}{2}}^2\frac{1}{1+x^2}dx,$$and hence evaluate ${\int }_{\frac{1}{2}}^2\frac{1}{(1+x^2{)}^2}dx$.

(b) Evaluate$${\int }_{\frac{1}{2}}^2\frac{1-x}{x(1+x^2{)}^{\frac{1}{2}}}dx.$$

(i) The real numbers $x$, $y$ and $z$ satisfy the equations$$y=\frac{2x}{1-x^2},z=\frac{2y}{1-y^2},x=\frac{2z}{1-z^2}.$$Let $x=\tan \alpha$. Deduce that $y=\tan 2\alpha$ and show that $\tan \alpha =\tan 8\alpha$.

Find all solutions of the equations, giving each value of $x$, $y$ and $z$ in the form $\tan \theta$ where $-\frac{1}{2}\pi <\theta <\frac{1}{2}\pi$.

(ii) Determine the number of real solutions of the simultaneous equations$$y=\frac{3x-x^3}{1-3x^2},z=\frac{3y-y^3}{1-3y^2},x=\frac{3z-z^3}{1-3z^2}.$$(iii) Consider the simultaneous equations$$y=2x^2-1,z=2y^2-1,x=2z^2-1.$$(a) Determine the number of real solutions of these simultaneous equations with $|x|\le 1$, $|y|\le 1$, $|z|\le 1$.

(b) By finding the degree of a single polynomial equation which is satisfied by $x$, show that all solutions of these simultaneous equations have $|x|\le 1$, $|y|\le 1$, $|z|\le 1$.

Let $p(x)$ be a polynomial of degree $n$ with $p(x)>0$ for all $x$ and let$$q(x)=\sum _{k=0}^n{p}^{(k)}(x),$$where $p^{(k)}(x)\equiv \frac{d^{k}p(x)}{d{x}^k}$ for $k\ge 1$ and $p^{(0)}(x)\equiv p(x)$.

(i)

(a) Explain why $n$ must be even and show that $q(x)$ takes positive values for some values of $x$.

(b) Show that $q^{\prime }(x)=q(x)-p(x)$.

(ii) In this part you will be asked to show the same result in three different ways.
(a) Show that the curves $y=p(x)$ and $y=q(x)$ meet at every stationary point of $y=q(x)$. Hence show that $q(x)>0$ for all $x$.

(b) Show that $e^{-x}q(x)$ is a decreasing function. Hence show that $q(x)>0$ for all $x$.

(c) Show that$${\int }_0^{\infty }p(x+t)e^{-t}dt=p(x)+{\int }_0^{\infty }{p}^{(1)}(x+t)e^{-t}dt.$$Show further that$${\int }_0^{\infty }p(x+t)e^{-t}dt=q(x).$$Hence show that $q(x)>0$ for all $x$.

(i) Show that, if ${\left(x-\sqrt{2}\right)}^2=3$, then $x^4-10x^2+1=0$.

Deduce that, if $f(x)=x^4-10x^2+1$, then $f\left(\sqrt{2}+\sqrt{3}\right)=0$.

(ii) Find a polynomial $g$ of degree $8$ with integer coefficients such that $g\left(\sqrt{2}+\sqrt{3}+\sqrt{5}\right)=0$. Write your answer in a form without brackets.

(iii) Let $a$, $b$ and $c$ be the three roots of $t^3-3t+1=0$.

Find a polynomial $h$ of degree $6$ with integer coefficients such that $h\left(a+\sqrt{2}\right)=0$, $h\left(b+\sqrt{2}\right)=0$ and $h\left(c+\sqrt{2}\right)=0$. Write your answer in a form without brackets.

(iv) Find a polynomial $k$ with integer coefficients such that $k\left(\sqrt[3]{2}+\sqrt[3]{3}\right)=0$. Write your answer in a form without brackets.

(i) The sequence $x_n$ for $n=0,1,2,\dots$ is defined by $x_0=1$ and by$$x_{n+1}=\frac{x_n+2}{x_n+1}$$for $n\ge 0$.

(a) Explain briefly why $x_n\ge 1$ for all $n$.

(b) Show that $x_{n+1}^2-2$ and $x_n^2-2$ have opposite sign, and that$$\left|x_{n+1}^2-2\right|\le \frac{1}{4}\left|x_n^2-2\right|.$$(c) Show that$$2-10^{-6}\le x_{10}^2\le 2.$$(ii) The sequence $y_n$ for $n=0,1,2,\dots$ is defined by $y_0=1$ and by$$y_{n+1}=\frac{y_n^2+2}{2y_n}$$for $n\ge 0$.

(a) Show that, for $n\ge 0$,$$y_{n+1}-\sqrt{2}=\frac{{\left(y_n-\sqrt{2}\right)}^2}{2y_n}$$and deduce that $y_n\ge \sqrt{2}$ for $n\ge 0$ (except possibly $n=0$; deduce $y_n\ge 1$ for $n\ge 0$).

(b) Show that$$y_n-\sqrt{2}\le 2{\left(\frac{\sqrt{2}-1}{2}\right)}^{2^n}$$for $n\ge 1$.

(c) Using the fact that $\sqrt{2}-1<\frac{1}{2}$, or otherwise, show that$$\sqrt{2}\le y_{10}\le \sqrt{2}+10^{-600}.$$

The sequence $F_n$, for $n=0,1,2,\dots$, is defined by $F_0=0$, $F_1=1$ and by $F_{n+2}=F_{n+1}+F_n$ for $n\ge 0$.

Prove by induction that, for all positive integers $n$,$$\left(\begin{array}{cc}{F}_{n+1}& F_n\\ F_n& F_{n-1}\end{array}\right)=Q^n,$$where the matrix $Q$ is given by$$Q=\left(\begin{array}{cc}1& 1\\ 1& 0\end{array}\right).$$(i) By considering the matrix $Q^n$, show that $F_{n+1}{F}_{n-1}-F_n^2=(-1{)}^n$ for all positive integers $n$.

(ii) By considering the matrix $Q^{m+n}$, show that $F_{m+n}=F_{m+1}{F}_n+F_m{F}_{n-1}$ for all positive integers $m$ and $n$.

(iii) Show that $Q^2=I+Q$.

In the following parts, you may use without proof the Binomial Theorem for matrices:$$(I+A{)}^n=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})A^k.$$(a) Show that, for all positive integers $n$,$$F_{2n}=\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)F_k.$$(b) Show that, for all positive integers $n$,$$F_{3n}=\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)2^k{F}_k$$and also that$$F_{3n}=\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)F_{n+k}.$$(c) Show that, for all positive integers $n$,$$\sum _{k=0}^n(-1{)}^{n+k}(\genfrac{}{}{0ex}{}{n}{k})F_{n+k}=0.$$

(i) The complex numbers $z$ and $w$ have real and imaginary parts given by $z=a+ib$ and $w=c+id$. Prove that $|zw|=|z||w|$.

(ii) By considering the complex numbers $2+i$ and $10+11i$, find positive integers $h$ and $k$ such that $h^2+k^2=5\times 221$.

(iii) Find positive integers $m$ and $n$ such that $m^2+n^2=8045$.

(iv) You are given that $102^2+201^2=50805$. Find positive integers $p$ and $q$ such that $p^2+q^2=36\times 50805$.

(v) Find three distinct pairs of positive integers $r$ and $s$ such that $r^2+s^2=25\times 1002082$ and $r<s$.

(vi) You are given that $109\times 9193=1002037$. Find positive integers $t$ and $u$ such that $t^2+u^2=9193$.

A tetrahedron is called isosceles if each pair of edges which do not share a vertex have equal length.

(i) Prove that a tetrahedron is isosceles if and only if all four faces have the same perimeter.

Let $OABC$ be an isosceles tetrahedron and let $\overrightarrow{OA}=\mathbf{a}$, $\overrightarrow{OB}=\mathbf{b}$ and $\overrightarrow{OC}=\mathbf{c}$.

(ii) By considering the lengths of $OA$ and $BC$, show that$$2\mathbf{b}\cdot \mathbf{c}=|\mathbf{b}{|}^2+|\mathbf{c}{|}^2-|\mathbf{a}{|}^2.$$Show that$$\mathbf{a}\cdot (\mathbf{b}+\mathbf{c})=|\mathbf{a}{|}^2.$$(iii) Let $G$ be the centroid of the tetrahedron, defined by $\overrightarrow{OG}=\frac{1}{4}(\mathbf{a}+\mathbf{b}+\mathbf{c})$. Show that $G$ is equidistant from all four vertices of the tetrahedron.

(iv) By considering the length of the vector $\mathbf{a}-\mathbf{b}-\mathbf{c}$, or otherwise, show that, in an isosceles tetrahedron, none of the angles between pairs of edges which share a vertex can be obtuse. Can any of them be right angles?

A truck of mass $M$ is connected by a light, rigid tow-bar, which is parallel to the ground, to a trailer of mass $kM$. A constant driving force $D$ which is parallel to the ground acts on the truck, and the only resistance to motion is a frictional force acting on the trailer, with coefficient of friction $\mu$.

- When the truck pulls the trailer up a slope which makes an angle $\alpha$ to the horizontal, the acceleration is $a_1$ and there is a tension $T_1$ in the tow-bar.
- When the truck pulls the trailer on horizontal ground, the acceleration is $a_2$ and there is a tension $T_2$ in the tow-bar.
- When the truck pulls the trailer down a slope which makes an angle $\alpha$ to the horizontal, the acceleration is $a_3$ and there is a tension $T_3$ in the tow-bar.

All accelerations are taken to be positive when in the direction of motion of the truck.

(i) Show that $T_1=T_3$ and that $M(a_1+a_3-2a_2)=2(T_2-T_1)$.

(ii) It is given that $\mu <1$.

(a) Show that$$a_2<\frac{1}{2}(a_1+a_3)<a_3.$$(b) Show further that$$a_1<a_2.$$

In this question, the $x$- and $y$-axes are horizontal and the $z$-axis is vertically upwards.

(i) A particle $P_{\alpha }$ is projected from the origin with speed $u$ at an acute angle $\alpha$ above the positive $x$-axis.

The curve $E$ is given by $z=A-B{x}^2$ and $y=0$. If $E$ and the trajectory of $P_{\alpha }$ touch exactly once, show that$$u^2-2gA=u^2\left(1-4AB{\cos }^2\alpha \right).$$$E$ and the trajectory of $P_{\alpha }$ touch exactly once for all $\alpha$ with $0<\alpha <\frac{1}{2}\pi$. Write down the values of $A$ and $B$ in terms of $u$ and $g$.

An explosion takes place at the origin and results in a large number of particles being simultaneously projected with speed $u$ in different directions. You may assume that all the particles move freely under gravity for $t\ge 0$.

(ii) Describe the set of points which can be hit by particles from the explosion, explaining your answer.
(iii) Show that, at a time $t$ after the explosion, the particles lie on a sphere whose centre and radius you should find.

(iv) Another particle $Q$ is projected horizontally from the point $(0,0,A)$ with speed $u$ in the positive $x$ direction. Show that, at all times, $Q$ lies on the curve $E$.

(v) Show that for particles $Q$ and $P_{\alpha }$ to collide, $Q$ must be projected a time$$\frac{u(1-\cos \alpha )}{g\sin \alpha }$$after the explosion.

(i) $X_1$ and $X_2$ are both random variables which take values $x_1,x_2,\dots ,x_n$, with probabilities $a_1,a_2,\dots ,a_n$ and $b_1,b_2,\dots ,b_n$ respectively. The value of random variable $Y$ is defined to be that of $X_1$ with probability $p$ and that of $X_2$ with probability $q=1-p$.

If $X_1$ has mean ${\mu }_1$ and variance ${\sigma }_1^2$, and $X_2$ has mean ${\mu }_2$ and variance ${\sigma }_2^2$, find the mean of $Y$ and show that the variance of $Y$ is $p{\sigma }_1^2+q{\sigma }_2^2+pq({\mu }_1-{\mu }_2{)}^2$.

(ii) To find the value of random variable $B$, a fair coin is tossed and a fair six-sided die is rolled. If the coin shows heads, then $B=1$ if the die shows a six and $B=0$ otherwise; if the coin shows tails, then $B=1$ if the die does not show a six and $B=0$ if it does. The value of $Z_1$ is the sum of $n$ independent values of $B$, where $n$ is large.

Show that $Z_1$ is a Binomial random variable with probability of success $\frac{1}{2}$.

Using a Normal approximation, show that the probability that $Z_1$ is within 10% of its mean tends to 1 as $n\to \infty$.

(iii) To find the value of random variable $Z_2$, a fair coin is tossed and $n$ fair six-sided dice are rolled, where $n$ is large. If the coin shows heads, then the value of $Z_2$ is the number of dice showing a six; if the coin shows tails, then the value of $Z_2$ is the number of dice not showing a six.

Use part (i) to write down the mean and variance of $Z_2$.

Explain why a Normal distribution with this mean and variance will not be a good approximation to the distribution of $Z_2$.

Show that the probability that $Z_2$ is within 10% of its mean tends to 0 as $n\to \infty$.

Each of the independent random variables $X_1,X_2,\dots ,X_n$ has the probability density function $f(x)=\frac{1}{2}\sin x$ for $0\le x\le \pi$ (and zero otherwise). Let $Y$ be the random variable whose value is the maximum of the values of $X_1,X_2,\dots ,X_n$.

(i) Explain why $P(Y\le t)=(P(X_1\le t){)}^n$ and hence, or otherwise, find the probability density function of $Y$.

Let $m(n)$ be the median of $Y$ and $\mu (n)$ be the mean of $Y$.

(ii) Find an expression for $m(n)$ in terms of $n$. How does $m(n)$ change as $n$ increases?

(iii) Show that$$\mu (n)=\pi -n{\int }_0^{\pi }(1-\cos x{)}^{n}dx.$$(a) Show that $\mu (n)$ increases with $n$.

(b) Show that $\mu (2)<m(2)$.