STEP 2 2011

Write down the cubes of the integers 1, 2, $\dots$ , 10.

The positive integers $x$, $y$ and $z$, where $x<y$, satisfy $$x^3+y^3=k{z}^3,\text{\hspace{1em}(*)}$$ where $k$ is a given positive integer.

1. In the case $x+y=k$, show that $$z^3=k^2-3kx+3x^2.$$ Deduce that $(4z^3-k^2)/3$ is a perfect square and that $\frac{1}{4}{k}^2⩽z^3<k^2$. Use these results to find a solution of $(\ast )$ when $k=20$.
2. By considering the case $x+y=z^2$, find two solutions of $(\ast )$ when $k=19$.

In this question, you may assume without proof that any function $\mathrm{f}$ for which ${\mathrm{f}}^{\prime }(x)⩾0$ is increasing; that is, $\mathrm{f}(x_2)⩾\mathrm{f}(x_1)$ if $x_2⩾x_1$.

1. • (a) Let $\mathrm{f}(x)=\sin x-x\cos x$. Show that $\mathrm{f}(x)$ is increasing for $0⩽x⩽\frac{1}{2}\pi$ and deduce that $\mathrm{f}(x)⩾0$ for $0⩽x⩽\frac{1}{2}\pi$.
   • (b) Given that $\frac{\mathrm{d}}{\mathrm{d}x}(\arcsin x)⩾1$ for $0⩽x<1$, show that $$\arcsin x⩾x(0⩽x<1).$$
   • (c) Let $\mathrm{g}(x)=x\mathrm{cosec}x$ for $0<x<\frac{1}{2}\pi$. Show that $\mathrm{g}$ is increasing and deduce that $$(\arcsin x)x^{-1}⩾x\mathrm{cosec}x(0<x<1).$$
2. Given that $\frac{\mathrm{d}}{\mathrm{d}x}(\arctan x)⩽1$ for $x⩾0$, show by considering the function $x^{-1}\tan x$ that $$(\tan x)(\arctan x)⩾x^2(0<x<\frac{1}{2}\pi ).$$

The points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ with respect to an origin $O$, and $O$, $A$ and $B$ are non-collinear. The point $C$, with position vector $\mathbf{c}$, is the reflection of $B$ in the line through $O$ and $A$. Show that $\mathbf{c}$ can be written in the form $$\mathbf{c}=\lambda \mathbf{a}-\mathbf{b}$$ where $\lambda =\frac{2[[B]]a.b[[/B]]}{[[B]]a.a[[/B]]}$.

The point $D$, with position vector $\mathbf{d}$, is the reflection of $C$ in the line through $O$ and $B$. Show that $\mathbf{d}$ can be written in the form $$\mathbf{d}=\mu \mathbf{b}-\lambda \mathbf{a}$$ for some scalar $\mu$ to be determined.

Given that $A$, $B$ and $D$ are collinear, find the relationship between $\lambda$ and $\mu$. In the case $\lambda =-\frac{1}{2}$, determine the cosine of $\angle AOB$ and describe the relative positions of $A$, $B$ and $D$.

For any given function $\mathrm{f}$, let $$I=\int [{\mathrm{f}}^{\prime }(x){]}^2[\mathrm{f}(x){]}^n\mathrm{d}x,\text{\hspace{1em}(*)}$$ where $n$ is a positive integer. Show that, if $\mathrm{f}(x)$ satisfies ${\mathrm{f}}^{\prime \prime }(x)=k\mathrm{f}(x){\mathrm{f}}^{\prime }(x)$ for some constant $k$, then ($\ast$) can be integrated to obtain an expression for $I$ in terms of $\mathrm{f}(x)$, ${\mathrm{f}}^{\prime }(x)$, $k$ and $n$.

1. Verify your result in the case $\mathrm{f}(x)=\tan x$. Hence find $$\int \frac{{\sin }^{4}x}{{\cos }^{8}x}\mathrm{d}x.$$
2. Find $$\int {\sec }^{2}x(\sec x+\tan x{)}^6\mathrm{d}x.$$

The two sequences $a_0$, $a_1$, $a_2$, $\dots$ and $b_0$, $b_1$, $b_2$, $\dots$ have general terms $$a_n={\lambda }^n+{\mu }^n\text{ and }{b}_n={\lambda }^n-{\mu }^n,$$ respectively, where $\lambda =1+\sqrt{2}$ and $\mu =1-\sqrt{2}$.

1. Show that $\sum _{r=0}^n{b}_r=-\sqrt{2}+\frac{1}{\sqrt{2}}{a}_{{}^{}n+1}$, and give a corresponding result for $\sum _{r=0}^n{a}_r$.
2. Show that, if $n$ is odd, $$\sum _{m=0}^{2n}\left(\sum _{r=0}^m{a}_{{}^{}r}\right)=\frac{1}{2}{b}_{n+1}^2,$$ and give a corresponding result when $n$ is even.
3. Show that, if $n$ is even, $${\left(\sum _{r=0}^n{a}_r\right)}^2-\sum _{r=0}^n{a}_{{}^{}2r+1}=2,$$ and give a corresponding result when $n$ is odd.

The end $A$ of an inextensible string $AB$ of length $\pi$ is attached to a point on the circumference of a fixed circle of unit radius and centre $O$. Initially the string is straight and tangent to the circle. The string is then wrapped round the circle until the end $B$ comes into contact with the circle. The string remains taut during the motion, so that a section of the string is in contact with the circumference and the remaining section is straight.

Taking $O$ to be the origin of cartesian coordinates with $A$ at $(-1,0)$ and $B$ initially at $(-1,\pi )$, show that the curve described by $B$ is given parametrically by $$x=\cos t+t\sin t,y=\sin t-t\cos t,$$ where $t$ is the angle shown in the diagram.

Find the value, $t_0$, of $t$ for which $x$ takes its maximum value on the curve, and sketch the curve.

Use the area integral $\int y\frac{\mathrm{d}x}{\mathrm{d}t}\mathrm{d}t$ to find the area between the curve and the $x$ axis for {$\pi ⩾t⩾t_0$}.

Find the area swept out by the string (that is, the area between the curve described by $B$ and the semicircle shown in the diagram).

Two particles, $A$ of mass $2m$ and $B$ of mass $m$, are moving towards each other in a straight line on a smooth horizontal plane, with speeds $2u$ and $u$ respectively. They collide directly. Given that the coefficient of restitution between the particles is $e$, where $0<e⩽1$, determine the speeds of the particles after the collision.

After the collision, $B$ collides directly with a smooth vertical wall, rebounding and then colliding directly with $A$ for a second time. The coefficient of restitution between $B$ and the wall is $f$, where $0<f⩽1$. Show that the velocity of $B$ after its second collision with $A$ is $$\frac{2}{3}(1-e^2)u-\frac{1}{3}(1-4e^2)fu$$ towards the wall and that $B$ moves towards (not away from) the wall for all values of $e$ and $f$.

A particle is projected from a point on a horizontal plane, at speed $u$ and at an angle $\theta$ above the horizontal. Let $H$ be the maximum height of the particle above the plane. Derive an expression for $H$ in terms of $u$, $g$ and $\theta$.

A particle $P$ is projected from a point $O$ on a smooth horizontal plane, at speed $u$ and at an angle $\theta$ above the horizontal. At the same instant, a second particle $R$ is projected horizontally from $O$ in such a way that $R$ is vertically below $P$ in the ensuing motion. A light inextensible string of length $\frac{1}{2}H$ connects $P$ and $R$. Show that the time that elapses before the string becomes taut is $$(\sqrt{2}-1)\sqrt{H/g}.$$When the string becomes taut, $R$ leaves the plane, the string remaining taut. Given that $P$ and $R$ have equal masses, determine the total horizontal distance, $D$, travelled by $R$ from the moment its motion begins to the moment it lands on the plane again, giving your answer in terms of $u$, $g$ and $\theta$.

Given that $D=H$, find the value of $\tan \theta$.

Three non-collinear points $A$, $B$ and $C$ lie in a horizontal ceiling. A particle $P$ of weight $W$ is suspended from this ceiling by means of three light inextensible strings $AP$, $BP$ and $CP$, as shown in the diagram. The point $O$ lies vertically above $P$ in the ceiling.

The angles $AOB$ and $AOC$ are $90^{\circ }+\theta$ and $90^{\circ }+\varphi$, respectively, where $\theta$ and $\varphi$ are acute angles such that $\tan \theta =\sqrt{2}$ and $\tan \varphi =\frac{1}{4}\sqrt{2}$.

The strings $AP$, $BP$ and $CP$ make angles $30^{\circ }$, $90^{\circ }-\theta$ and $60^{\circ }$, respectively, with the vertical, and the tensions in these strings have magnitudes $T$, $U$ and $V$ respectively.

1. Show that the unit vector in the direction $PB$ can be written in the form $$-\frac{1}{3}<strong>i</strong>-\frac{\sqrt{2}}{3}<strong>j</strong>+\frac{\sqrt{2}}{\sqrt{3}}<strong>k</strong>,$$ where $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are the usual mutually perpendicular unit vectors with $\mathbf{j}$ parallel to $OA$ and $\mathbf{k}$ vertically upwards.
2. Find expressions in vector form for the forces acting on $P$.
3. Show that $U=\sqrt{6}V$ and find $T$, $U$ and $V$ in terms of $W$.

Xavier and Younis are playing a match. The match consists of a series of games and each game consists of three points.

Xavier has probability $p$ and Younis has probability $1-p$ of winning the first point of any game. In the second and third points of each game, the player who won the previous point has probability $p$ and the player who lost the previous point has probability $1-p$ of winning the point. If a player wins two consecutive points in a single game, the match ends and that player has won; otherwise the match continues with another game.

1. Let $w$ be the probability that Younis wins the match. Show that, for $p\ne 0$, $$w=\frac{1-p^2}{2-p}.$$ Show that $w>\frac{1}{2}$ if $p<\frac{1}{2}$, and $w<\frac{1}{2}$ if $p>\frac{1}{2}$. Does $w$ increase whenever $p$ decreases?
2. If Xavier wins the match, Younis gives him $\pounds 1$; if Younis wins the match, Xavier gives him $\pounds k$. Find the value of $k$ for which the game is `fair' in the case when $p=\frac{2}{3}$.
3. What happens when $p=0$?

What property of a distribution is measured by its skewness?

1. One measure of skewness, $\gamma$, is given by $$\gamma =\frac{\mathrm{E}((X-\mu {)}^3)}{{\sigma }^3},$$ where $\mu$ and ${\sigma }^2$ are the mean and variance of the random variable $X$. Show that $$\gamma =\frac{\mathrm{E}(X^3)-3\mu {\sigma }^2-{\mu }^3}{{\sigma }^3}.$$ The continuous random variable $X$ has probability density function $\mathrm{f}$ where $$\mathrm{f}(x)=\{\begin{array}{ll}2x& \text{for }0⩽x⩽1,\\ 0& \text{otherwise}.\end{array}$$ Show that for this distribution $\gamma =-\frac{2\sqrt{2}}{5}$.
2. The decile skewness, $D$, of a distribution is defined by $$D=\frac{{\mathrm{F}}^{-1}(\frac{9}{10})-2{\mathrm{F}}^{-1}(\frac{1}{2})+{\mathrm{F}}^{-1}(\frac{1}{10})}{{\mathrm{F}}^{-1}(\frac{9}{10})-{\mathrm{F}}^{-1}(\frac{1}{10})},$$ where ${\mathrm{F}}^{-1}$ is the inverse of the cumulative distribution function. Show that, for the above distribution, $D=2-\sqrt{5}.$ The Pearson skewness, $P$, of a distribution is defined by $$P=\frac{3(\mu -M)}{\sigma },$$ where $M$ is the median. Find $P$ for the above distribution and show that $D>P>\gamma$.