STEP 2 2020

(i) Use the substitution $x=\frac{1}{1-u}$, where $0<u<1$, to find in terms of $x$ the integral$$\int \frac{1}{x^2(x-1{)}^{\frac{3}{2}}}dx(\text{where }x>1).$$(ii) Find in terms of $x$ the integral$$\int \frac{1}{(x-2{)}^{\frac{3}{2}}(x+1{)}^{\frac{1}{2}}}dx(\text{where }x>2).$$(iii) Show that$${\int }_2^{\infty }\frac{1}{(x-1)(x-2{)}^{\frac{1}{2}}(3x-2{)}^{\frac{1}{2}}}dx=\frac{1}{3}\pi .$$

The curves $C_1$ and $C_2$ both satisfy the differential equation$$\frac{dy}{dx}=\frac{kxy-y}{x-kxy},$$where $k=\ln 2$.

All points on $C_1$ have positive $x$ and $y$ co-ordinates and $C_1$ passes through $(1,1)$. All points on $C_2$ have negative $x$ and $y$ co-ordinates and $C_2$ passes through $(-1,-1)$.

(i) Show that the equation of $C_1$ can be written as $(x-y{)}^2=(x+y{)}^2-2^{x+y}$. Determine a similar result for curve $C_2$. Hence show that $y=x$ is a line of symmetry of each curve.

(ii) Sketch on the same axes the curves $y=x^2$ and $y=2^x$, for $x\ge 0$. Hence show that $C_1$ lies between the lines $x+y=2$ and $x+y=4$. Sketch curve $C_1$.

(iii) Sketch curve $C_2$.

A sequence $u_1,u_2,\dots ,u_n$ of positive real numbers is said to be unimodal if there is a value $k$ such that$$u_1\le u_2\le \cdots \le u_k$$and$$u_k\ge u_{k+1}\ge \cdots \ge u_n.$$So the sequences $1,2,3,2,1$; $1,2,3,4,5$; $1,1,3,3,2$ and $2,2,2,2,2$ are all unimodal, but $1,2,1,3,1$ is not.

A sequence $u_1,u_2,\dots ,u_n$ of positive real numbers is said to have property L if $u_{r-1}{u}_{r+1}\le u_r^2$ for all $r$ with $2\le r\le n-1$.

(i) Show that, in any sequence of positive real numbers with property L,$$u_{r-1}\ge u_r⟹u_r\ge u_{r+1}.$$Prove that any sequence of positive real numbers with property L is unimodal.

(ii) A sequence $u_1,u_2,\dots ,u_n$ of real numbers satisfies $u_r=2\alpha u_{r-1}-{\alpha }^2{u}_{r-2}$ for $3\le r\le n$, where $\alpha$ is a positive real constant. Prove that, for $2\le r\le n$,$$u_r-\alpha u_{r-1}={\alpha }^{r-2}(u_2-\alpha u_1)$$and, for $2\le r\le n-1$,$$u_r^2-u_{r-1}{u}_{r+1}=(u_r-\alpha u_{r-1}{)}^2.$$Hence show that the sequence consists of positive terms and is unimodal, provided $u_2>\alpha u_1>0$.

In the case $u_1=1$ and $u_2=2$, prove by induction that$$u_r=(2-r){\alpha }^{r-1}+2(r-1){\alpha }^{r-2}.$$Let $\alpha =1-\frac{1}{N}$, where $N$ is an integer with $2\le N\le n$.

In the case $u_1=1$ and $u_2=2$, prove that $u_r$ is largest when $r=N$.

(i) Given that $a$, $b$ and $c$ are the lengths of the sides of a triangle, explain why$$c<a+b,a<b+c\text{and}b<a+c.$$(ii) Use a diagram to show that the converse of the result in part (i) also holds: if $a$, $b$ and $c$ are positive numbers such that $c<a+b$, $a<b+c$ and $b<c+a$ then it is possible to construct a triangle with sides of length $a$, $b$ and $c$.

(iii) When $a$, $b$ and $c$ are the lengths of the sides of a triangle, determine in each case whether the following sets of three lengths can

- always
- sometimes but not always
- never

form the sides of a triangle. Prove your claims.

(A) $a+1$, $b+1$, $c+1$.

(B) $\frac{a}{b}$, $\frac{b}{c}$, $\frac{c}{a}$.

(C) $|a-b|$, $|b-c|$, $|c-a|$.

(D) $a^2+bc$, $b^2+ca$, $c^2+ab$.

(iv) Let $f$ be a function defined on the positive real numbers and such that, whenever $x>y>0$,$$f(x)>f(y)>0\text{but}\frac{f(x)}{x}<\frac{f(y)}{y}.$$Show that, whenever $a$, $b$ and $c$ are the lengths of the sides of a triangle, then $f(a)$, $f(b)$ and $f(c)$ can also be the lengths of the sides of a triangle.

If $x$ is a positive integer, the value of the function $d(x)$ is the sum of the digits of $x$ in base 10. For example, $d(249)=2+4+9=15$.

An $n$-digit positive integer $x$ is written in the form $\sum _{r=0}^{n-1}{a}_r\times 10^r$, where $0\le a_r\le 9$ for all $0\le r\le n-1$ and $a_{n-1}>0$.

(i) Prove that $x-d(x)$ is non-negative and divisible by $9$.

(ii) Prove that $x-44d(x)$ is a multiple of $9$ if and only if $x$ is a multiple of $9$. Suppose that $x=44d(x)$. Show that if $x$ has $n$ digits, then $x\le 396n$ and $x\ge 10^{n-1}$, and hence that $n\le 4$. Find a value of $x$ for which $x=44d(x)$. Show that there are no further values of $x$ satisfying this equation.

(iii) Find a value of $x$ for which $x=107d(d(x))$. Show that there are no further values of $x$ satisfying this equation.

A $2\times 2$ matrix $M$ is real if it can be written as $M=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$, where $a$, $b$, $c$ and $d$ are real.

In this case, the trace of matrix $M$ is defined to be $\mathrm{tr}(M)=a+d$ and $\det (M)$ is the determinant of matrix $M$. In this question, $M$ is a real $2\times 2$ matrix.

(i) Prove that$$\mathrm{tr}(M^2)=\mathrm{tr}(M{)}^2-2\det (M).$$(ii) Prove that$$M^2=I\text{ but }M\ne \pm I⟺\mathrm{tr}(M)=0\text{ and }\det (M)=-1,$$and that$$M^2=-I⟺\mathrm{tr}(M)=0\text{ and }\det (M)=1.$$(iii) Use part (ii) to prove that$$M^4=I⟺M^2=\pm I.$$Find a necessary and sufficient condition on $\det (M)$ and $\mathrm{tr}(M)$ so that $M^4=-I$.

(iv) Give an example of a matrix $M$ for which $M^8=I$, but which does not represent a rotation or reflection. [Note that the matrices $\pm I$ are both rotations.]

7 In this question, $w=\frac{2}{z-2}$.

(i) Let $z$ be the complex number $3+ti$, where $t\in \mathbb{R}$. Show that $|w-1|$ is independent of $t$. Hence show that, if $z$ is a complex number on the line $\mathrm{Re}(z)=3$ in the Argand diagram, then $w$ lies on a circle in the Argand diagram with centre $1$.

Let $V$ be the line $\mathrm{Re}(z)=p$, where $p$ is a real constant not equal to $2$. Show that, if $z$ lies on $V$, then $w$ lies on a circle whose centre and radius you should give in terms of $p$. For which $z$ on $V$ is $\mathrm{Im}(w)>0$?

(ii) Let $H$ be the line $\mathrm{Im}(z)=q$, where $q$ is a non-zero real constant. Show that, if $z$ lies on $H$, then $w$ lies on a circle whose centre and radius you should give in terms of $q$. For which $z$ on $H$ is $\mathrm{Re}(w)>0$?

In this question, $f(x)$ is a quartic polynomial where the coefficient of $x^4$ is equal to 1, and which has four real roots, $0$, $a$, $b$ and $c$, where $0<a<b<c$.

$F(x)$ is defined by $F(x)={\int }_0^{x}f(t)dt$.

The area enclosed by the curve $y=f(x)$ and the $x$-axis between $0$ and $a$ is equal to that between $b$ and $c$, and half that between $a$ and $b$.

(i) Sketch the curve $y=F(x)$, showing the $x$-coordinates of its turning points. Explain why $F(x)$ must have the form $F(x)=\frac{1}{5}{x}^2(x-c{)}^2(x-h)$, where $0<h<c$. Find, in factorised form, an expression for $F(x)+F(c-x)$ in terms of $c$, $h$ and $x$.

(ii) If $0\le x\le c$, explain why $F(b)+F(x)\ge 0$ and why $F(b)+F(x)>0$ if $x\ne a$. Hence show that $c-b=a$ or $c>2h$. By considering also $F(a)+F(x)$, show that $c=a+b$ and that $c=2h$.

(iii) Find an expression for $f(x)$ in terms of $c$ and $x$ only. Show that the points of inflection on $y=f(x)$ lie on the $x$-axis.

Point A is a distance $h$ above ground level and point N is directly below A at ground level. Point B is also at ground level, a distance $d$ horizontally from N. The angle of elevation of A from B is $\beta$. A particle is projected horizontally from A, with initial speed $V$. A second particle is projected from B with speed $U$ at an acute angle $\theta$ above the horizontal. The horizontal components of the velocities of the two particles are in opposite directions. The two particles are projected simultaneously, in the vertical plane through A, N and B.

Given that the two particles collide, show that$$d\sin \theta -h\cos \theta =\frac{Vh}{U}$$and also that

(i) $\theta >\beta$;

(ii) $U\sin \theta \ge \sqrt{\frac{gh}{2}}$;

(iii) $\frac{U}{V}>\sin \beta$.

Show that the particles collide at a height greater than $\frac{1}{2}h$ if and only if the particle projected from B is moving upwards at the time of collision.

A particle P of mass $m$ moves freely and without friction on a wire circle of radius $a$, whose axis is horizontal. The highest point of the circle is H, the lowest point of the circle is L and angle $PHL=\theta$. A light spring of modulus of elasticity $\lambda$ is attached to P and to H. The natural length of the spring is $l$, which is less than the diameter of the circle.

(i) Show that, if there is an equilibrium position of the particle at $\theta =\alpha$, where $\alpha >0$, then$$\cos \alpha =\frac{\lambda l}{2(a\lambda -mgl)}.$$Show also that there will only be such an equilibrium position if$$\lambda >\frac{2mgl}{2a-l}.$$When the particle is at the lowest point L of the circular wire, it has speed $u$.

(ii) Show that, if the particle comes to rest before reaching H, it does so when $\theta =\beta$, where $\cos \beta$ satisfies$${\left(\cos \alpha -\cos \beta \right)}^2=(1-\cos \alpha {)}^2+\frac{m{u}^2}{2a\lambda }\cos \alpha ,$$where $\cos \alpha =\frac{\lambda l}{2(a\lambda -mgl)}$.

Show also that this will only occur if$$u^2<\frac{2a\lambda }{m}(2-\sec \alpha ).$$

A coin is tossed repeatedly. The probability that a head appears is $p$ and the probability that a tail appears is $q=1-p$.

(i) A and B play a game. The game ends if two successive heads appear, in which case A wins, or if two successive tails appear, in which case B wins.

Show that the probability that the game never ends is $0$.

Given that the first toss is a head, show that the probability that A wins is$$\frac{p}{1-pq}.$$Find and simplify an expression for the probability that A wins.

(ii) A and B play another game. The game ends if three successive heads appear, in which case A wins, or if three successive tails appear, in which case B wins.

Show that$$P(\text{A wins}\mid \text{the first toss is a head})=p^2+(q+pq)P(\text{A wins}\mid \text{the first toss is a tail})$$and give a similar result for $P(\text{A wins}\mid \text{the first toss is a tail})$.

Show that$$P(\text{A wins})=\frac{p^2(1-q^3)}{1-(1-p^2)(1-q^2)}.$$(iii) A and B play a third game. The game ends if $a$ successive heads appear, in which case A wins, or if $b$ successive tails appear, in which case B wins, where $a$ and $b$ are integers greater than $1$.

Find the probability that A wins this game.

Verify that your result agrees with part (i) when $a=b=2$.

The score shown on a biased $n$-sided die is represented by the random variable $X$ which has distribution$$P(X=i)=\frac{1}{n}+{\epsilon }_i\text{for }i=1,2,\dots ,n,$$where not all the ${\epsilon }_i$ are equal to $0$.

(i) Find the probability that, when the die is rolled twice, the same score is shown on both rolls. Hence determine whether it is more likely for a fair die or a biased die to show the same score on two successive rolls.
(ii) Use part (i) to prove that, for any set of $n$ positive numbers $x_i$ ($i=1,2,\dots ,n$),$$\sum _{i=2}^n\sum _{j=1}^{i-1}{x}_i{x}_j\le \frac{n-1}{2n}{\left(\sum _{i=1}^n{x}_i\right)}^2.$$(iii) Determine, with justification, whether it is more likely for a fair die or a biased die to show the same score on three successive rolls.