STEP Mathematics 2

The function Min is defined as$$\mathrm{Min}(a,b)=\{\begin{array}{ll}a& \text{if }a\le b,\\ b& \text{if }a>b.\end{array}$$(i) Sketch the graph $y=\mathrm{Min}(x^2,2x)$.

(ii) Solve the equation $2\mathrm{Min}(x^2,2x)=5x-3$.

(iii) Solve the equation $\mathrm{Min}(x^2,2x)+\mathrm{Min}(x^3,4x)=mx$ in the cases $m=2$ and $m=6$.

(iv) Show that $(1,-3)$ is a local maximum point on the curve $y=2\mathrm{Min}(x^2,x^3)-5x$ and find the other three local maxima and minima on this curve. Sketch the curve.

(i)

(a) Show that if the complex number $z$ satisfies the equation$$z^2+|z+b|=a,$$where $a$ and $b$ are real numbers, then $z$ must be either purely real or purely imaginary.

(b) Show that the equation$$z^2+|z+25|=72$$has no purely imaginary roots.

(c) Show that the equation$$z^2+|z+27|=52$$has no purely real roots.

(d) Show that, when $21<b<34$, the equation$$z^2+|z+b|=12$$will have at least one purely imaginary root and at least one purely real root.

(ii) Solve the equation$$z^3+|z+2{|}^2=4.$$

(i) Sketch a graph of $y=\frac{\ln x}{x}$ for $x>0$.

(ii) Use your graph to show the following.
(a) $3^{\pi }>{\pi }^3$

(b) $4^{\sqrt{5}}>5^{\sqrt{4}}$

(iii) Given that $1<x<2$, decide, with justification, which is the larger of $x^{x+2}$ or $(x+2{)}^x$.

(iv) Show that the inequalities $9^{\sqrt{2}}>(\sqrt{2}{)}^9$ and $3^{2\sqrt{2}}>2^{\frac{9}{2}}$ are equivalent. Given that $e^2<8$, decide, with justification, which is the larger of $9^{\sqrt{2}}$ and $(\sqrt{2}{)}^9$.

(v) Decide, with justification, which is the larger of $8^{\sqrt[3]{3}}$ and $3^{\sqrt[3]{8}}$.

Let $\lfloor x\rfloor$ denote the largest integer that satisfies $\lfloor x\rfloor \le x$. For example, if $x=-4.2$, then $\lfloor x\rfloor =-5$.

(i) Show that, if $n$ is an integer, then $\lfloor x+n\rfloor =\lfloor x\rfloor +n$.

(ii) Let $n$ be a positive integer and define function $f_n$ by$$f_n(x)=\lfloor x\rfloor +\lfloor x+\frac{1}{n}\rfloor +\lfloor x+\frac{2}{n}\rfloor +\cdots +\lfloor x+\frac{n-1}{n}\rfloor -\lfloor nx\rfloor .$$(a) Show that $f_n\left(x+\frac{1}{n}\right)=f_n(x)$.

(b) Evaluate $f_n(t)$ for $0\le t<\frac{1}{n}$.

(c) Hence show that $f_n(x)\equiv 0$.

(iii) (a) Show that $\lfloor \frac{x}{2}\rfloor +\lfloor \frac{x+1}{2}\rfloor =\lfloor x\rfloor$.

(b) Hence, or otherwise, simplify$$\lfloor \frac{x+1}{2}\rfloor +\lfloor \frac{x+2}{2^2}\rfloor +\cdots +\lfloor \frac{x+2^k}{2^{k+1}}\rfloor +\cdots$$

You need not consider the convergence of the improper integrals in this question.

(i) Use the substitution $x=u^{-1}$ to show that$${\int }_0^{\infty }\frac{\sqrt{x}-1}{x(x^3+1)}dx=0.$$(ii) Use the substitution $x=u^{-2}$ to show that$${\int }_0^{\infty }\frac{1}{\sqrt{x^3+1}}dx=2{\int }_0^{\infty }\frac{1}{\sqrt{x^6+1}}dx.$$(iii) Find, in terms of $p$ and $s$, a value of $r$ for which$${\int }_0^{\infty }\frac{x^r-1}{x^s(x^p+1)}dx=0,$$given that $p$ and $s$ are fixed values for which the required integrals converge.

(iv) Show that, for any positive value of $k$, it is possible to find values of $p$ and $q$ for which$${\int }_0^{\infty }\frac{1}{\sqrt{x^p+1}}dx=k{\int }_0^{\infty }\frac{1}{\sqrt{x^q+1}}dx.$$

(i) The circle $x^2+(y-a{)}^2=r^2$ touches the parabola $2ky=x^2$, where $k>0$, tangentially at two points. Show that $r^2=k(2a-k)$.

Show further that if $r^2=k(2a-k)$ and $a>k>0$, then the circle $x^2+(y-a{)}^2=r^2$ touches the parabola $2ky=x^2$ tangentially at two points.

(ii) The lines $y=c\pm x$ are tangents to the circle $x^2+(y-a{)}^2=r^2$. Find $r^2$, and the coordinates of the points of contact, in terms of $a$ and $c$.

(iii) $C_1$ and $C_2$ are circles with equations $x^2+(y-a_1{)}^2=r_1^2$ and $x^2+(y-a_2{)}^2=r_2^2$ respectively, where $a_1\ne a_2$ and $r_1\ne r_2$.

Each circle touches the parabola $2ky=x^2$ tangentially at two points and the lines $y=c\pm x$ are tangents to both circles.

(a) Show that $a_1+a_2=2c+4k$ and that $a_1^2+a_2^2=2c^2+16kc+12k^2$.

(b) The circle $x^2+(y-d{)}^2=p^2$ passes through the four points of tangency of the lines $y=c\pm x$ to the two circles, $C_1$ and $C_2$. Find $d$ and $p^2$ in terms of $k$ and $c$.

(c) Show that the circle $x^2+(y-d{)}^2=p^2$ also touches the parabola $2ky=x^2$ tangentially at two points.

The differential equation$$\frac{d^{2}x}{d{t}^2}=2x\frac{dx}{dt}$$describes the motion of a particle with position $x(t)$ at time $t$. At $t=0$, $x=a$, where $a>0$.

(i) Solve the differential equation in the case where $\frac{dx}{dt}=a^2$ when $t=0$. What happens to the particle as $t$ increases from $0$?

(ii) Solve the differential equation in the case where $\frac{dx}{dt}=a^2+p^2$ when $t=0$, where $p>0$. What happens to the particle as $t$ increases from $0$?

(iii) Solve the differential equation in the case where $\frac{dx}{dt}=a^2-q^2$ when $t=0$, where $q>0$. What happens to the particle as $t$ increases from $0$? Give conditions on $a$ and $q$ for the different cases which arise.

If we split a set $S$ of integers into two subsets $A$ and $B$ whose intersection is empty and whose union is the whole of $S$, and such that

- the sum of the elements of $A$ is equal to the sum of the elements of $B$
- and the sum of the squares of the elements of $A$ is equal to the sum of the squares of the elements of $B$,

then we say that we have found a balanced partition of $S$ into two subsets.

(i) Find a balanced partition of the set $\{1,2,3,4,5,6,7,8\}$ into two subsets $A$ and $B$, each of size 4.

(ii) Given that $a_1,a_2,\dots ,a_m$ and $b_1,b_2,\dots ,b_m$ are sequences with$$\sum _{k=1}^m{a}_k=\sum _{k=1}^m{b}_k\text{and}\sum _{k=1}^m{a}_k^2=\sum _{k=1}^m{b}_k^2,$$show that$$\sum _{k=1}^m{a}_k^3+\sum _{k=1}^m(c+b_k{)}^3=\sum _{k=1}^m{b}_k^3+\sum _{k=1}^m(c+a_k{)}^3$$for any real number $c$.

(iii) Find, with justification, a balanced partition of the set $\{1,2,3,\dots ,16\}$ into two subsets $A$ and $B$, each of size 8, which also has the property that

- the sum of the cubes of the elements of $A$ is equal to the sum of the cubes of the elements of $B$.

(iv) You are given that the sets $A=\{1,3,4,5,9,11\}$ and $B=\{2,6,7,8,10\}$ form a balanced partition of the set $\{1,2,3,\dots ,11\}$.

Let $S=\{n^2,(n+1{)}^2,(n+2{)}^2,\dots ,(n+11{)}^2\}$, where $n$ is any positive integer. Find, with justification, two subsets $C$ and $D$ of $S$ whose intersection is empty and whose union is the whole of $S$, and such that

- the sum of the elements of $C$ is equal to the sum of the elements of $D$.

Points A and B are at the same height and a distance $2r$ apart. Two small, spherical particles of equal mass, P and Q, are suspended from A and B, respectively, by light inextensible strings of length $r$. Each particle individually may move freely around and inside a circle centred at the point of suspension.

The particles are projected simultaneously from points which are a distance $r$ vertically below their points of suspension, directly towards each other and each with speed $u$. When the particles collide, the coefficient of restitution in the collision is $e$.

(i) Show that, immediately after the collision, the horizontal component of each particle's velocity has magnitude $\frac{1}{2}ev\sqrt{2}$, where $v=u-\sqrt{gr(\sqrt{2}-2)}$, and write down the vertical component in terms of $v$.

(ii) Show that the strings will become taut again at a time $t$ after the collision, where $t$ is a non-zero root of the equation$$(r-evt{)}^2+{\left(-r+vt-\frac{1}{2}\sqrt{2}g{t}^2\right)}^2=2r^2.$$(iii) Show that, in terms of the dimensionless variables$$z=\frac{vt}{r}\text{and}c=\frac{\sqrt{2}{v}^2}{rg},$$this equation becomes$$-2{\left(\frac{z}{c}\right)}^3+{\left(\frac{z}{c}\right)}^2\left(2+1+e^2\right)-(1+e)\frac{2z}{c}-(1+e)=0.$$(iv) Show that, if this equation has three equal non-zero roots, $e=\frac{1}{3}$ and $v^2=\frac{9}{2}\sqrt{2}rg$. Explain briefly why, in this case, no energy is lost when the string becomes taut.

(v) In the case described in (iv), the particles have speed $U$ when they again reach the points of their motion vertically below their points of suspension. Find $U^2$ in terms of $r$ and $g$.

The lower end of a rigid uniform rod of mass $m$ and length $a$ rests at point M on rough horizontal ground. Each of two elastic strings, of natural length $\ell$ and modulus of elasticity $\lambda$, is attached at one end to the top of the rod. Their lower ends are attached to points A and B on the ground, which are a distance $2a$ apart. M is the midpoint of AB.

P is the point at the top of the rod and lies in the vertical plane through AMB. Suppose that the rod is in equilibrium with angle $PMB=2\theta$, where $\theta <45^{\circ }$ and $\ell$ is such that both strings are in tension.

(i) Show that angle $APB$ is a right angle.

Show that the force exerted on the rod by the elastic strings can be written as the sum of
- a force of magnitude $\frac{2a\lambda }{\ell }$ parallel to the rod,
- and a force of magnitude $\sqrt{2}\lambda$ acting along the bisector of angle $APB$.

(ii) By taking moments about point M, or otherwise, show that$$\cos \theta +\sin \theta =\frac{2\lambda }{mg}.$$Deduce that it is necessary that $\frac{1}{2}mg<\lambda <\frac{1}{2}\sqrt{2}mg$.

(iii) N and F are the magnitudes of the normal and frictional forces, respectively, exerted on the rod by the ground at M.

Show, by taking moments about an appropriate point, or otherwise, that$$N-F\tan 2\theta =\frac{1}{2}mg.$$

(i) By considering the sum of a geometric series, or otherwise, show that$$\sum _{r=1}^{\infty }r{x}^{r-1}=\frac{1}{(1-x{)}^2}\text{for }|x|<1.$$(ii) Ali plays a game with a fair $2k$-sided die. He rolls the die until the first $2k$ appears. Ali wins if all the numbers he rolls are even.

(a) Find the probability that Ali wins the game.

If Ali wins the game, he earns £$1$ for each roll, including the final one. If he loses, he earns nothing.

(b) Find Ali's expected earnings from playing the game.
(iii) Find a simplified expression for$$1+2\left(\genfrac{}{}{0ex}{}{n}{1}\right)x+3\left(\genfrac{}{}{0ex}{}{n}{2}\right)x^2+\cdots +(n+1)x^n,$$where $n$ is a positive integer.

(iv) Zen plays a different game with a fair $2k$-sided die. She rolls the die until the first $2k$ appears, and wins if the numbers rolled are strictly increasing in size. For example, if $k=3$, she wins if she rolls $2,6$ or $1,4,5,6$, but not if she rolls $1,4,2,6$ or $1,3,3,6$. If Zen wins the game, she earns £$1$ for each roll, including the final one. If she loses, she earns nothing. Find Zen's expected earnings from playing the game.

(v) Using the approximation$${\left(1+\frac{1}{n}\right)}^n\approx e\text{for large }n,$$show that, when $k$ is large, Zen's expected earnings are a little over $35\%$ more than Ali's expected earnings.

Let $X$ be a Poisson random variable with mean $\lambda$ and let $p_r=P(X=r)$, for $r=0,1,2,\dots$

Neither $\lambda$ nor $\lambda +\frac{1}{2}+\sqrt{\lambda +\frac{1}{4}}$ is an integer.

(i) Show, by considering the sequence $d_r\equiv p_r-p_{r-1}$ for $r=1,2,\dots$, that there is a unique integer $m$ such that $P(X=r)\le P(X=m)$ for all $r=0,1,2,\dots$, and that $\lambda -1<m<\lambda$.

(ii) Show that the minimum value of $d_r$ occurs at $r=k$, where $k$ is such that$$k<\lambda +\frac{1}{2}+\sqrt{\lambda +\frac{1}{4}}<k+1.$$(iii) Show that the condition for the maximum value of $d_r$ to occur at $r=1$ is$$1<\lambda <\sqrt{2}+2.$$(iv) In the case $\lambda =3.36$, sketch a graph of $p_r$ against $r$ for $r=0,1,2,\dots ,6,7$.