STEP 2 2008

A sequence of points $(x_1,y_1)$, $(x_2,y_2)$, $\dots$ in the cartesian plane is generated by first choosing $(x_1,y_1)$ then applying the rule, for $n=1$, $2$, $\dots$, $$(x_{n+1},y_{n+1})=(x_n^2-y_n^2+a,2x_n{y}_n+b+2),$$ where $a$ and $b$ are given real constants.

1. In the case $a=1$ and $b=-1$, find the values of $(x_1,y_1)$ for which the sequence is constant.
2. Given that $(x_1,y_1)=(-1,1)$, find the values of $a$ and $b$ for which the sequence has period 2.

Let $a_n$ be the coefficient of $x^n$ in the series expansion, in ascending powers of $x$, of $$\frac{1+x}{(1-x{)}^2(1+x^2)},$$ where $|x|<1$. Show, using partial fractions, that either $a_n=n+1$ or $a_n=n+2$ according to the value of $n$.

Hence find a decimal approximation, to nine significant figures, for the fraction $\frac{11000}{8181}$.

[You are not required to justify the accuracy of your approximation.]

A curve is given by $$x^2+y^2+2axy=1,$$ where $a$ is a constant satisfying $0<a<1$. Show that the gradient of the curve at the point $P$ with coordinates $(x,y)$ is $$-\frac{x+ay}{ax+y},$$ provided $ax+y\ne 0$. Show that $\theta$, the acute angle between $OP$ and the normal to the curve at $P$, satisfies $$\tan \theta =a|y^2-x^2|.$$Show further that, if $\frac{\mathrm{d}\theta }{\mathrm{d}x}=0$ at $P$, then:

1. $a(x^2+y^2)+2xy=0$;
2. $(1+a)(x^2+y^2+2xy)=1$;
3. $\tan \theta =\frac{a}{\sqrt{1-a^2}}$.

Evaluate the integrals $${\int }_0^{\frac{1}{2}\pi }\frac{\sin 2x}{1+{\sin }^{2}x}\mathrm{d}x\text{ and }{\int }_0^{\frac{1}{2}\pi }\frac{\sin x}{1+{\sin }^{2}x}\mathrm{d}x.$$Show, using the binomial expansion, that $(1+\sqrt{2}{)}^5<99$. Show also that $\sqrt{2}>1.4$. Deduce that $2^{\sqrt{2}}>1+\sqrt{2}$. Use this result to determine which of the above integrals is greater.

A curve has the equation $y=\mathrm{f}(x)$, where $$\mathrm{f}(x)=\cos (2x+\frac{\pi }{3})+\sin (\frac{3x}{2}-\frac{\pi }{4}).$$

1. Find the period of $\mathrm{f}(x)$.
2. Determine all values of $x$ in the interval $-\pi ⩽x⩽\pi$ for which $\mathrm{f}(x)=0$. Find a value of $x$ in this interval at which the curve touches the $x$-axis without crossing it.
3. Find the value or values of $x$ in the interval $0⩽x⩽2\pi$ for which $\mathrm{f}(x)=2$.

The points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$, respectively, relative to the origin $O$. The points $A$, $B$ and $O$ are not collinear. The point $P$ lies on $AB$ between $A$ and $B$ such that $$AP:PB=(1-\lambda ):\lambda .$$ Write down the position vector of $P$ in terms of $\mathbf{a}$, $\mathbf{b}$ and $\lambda$. Given that $OP$ bisects $\angle AOB$, determine $\lambda$ in terms of $a$ and $b$, where $a=|a|$ and $b=|\mathbf{b}|$.

The point $Q$ also lies on $AB$ between $A$ and $B$, and is such that $AP=BQ$. Prove that $$O{Q}^2-O{P}^2=(b-a{)}^2.$$

In this question, use $g=10{\mathrm{m}\mathrm{s}}^{-2}$.

In cricket, a fast bowler projects a ball at $40{\mathrm{m}\mathrm{s}}^{-1}$ from a point $h\mathrm{m}$ above the ground, which is horizontal, and at an angle $\alpha$ above the horizontal. The trajectory is such that the ball will strike the stumps at ground level a horizontal distance of $20\mathrm{m}$ from the point of projection.

1. Determine, in terms of $h$, the two possible values of $\tan \alpha$. Explain which of these two values is the more appropriate one, and deduce that the ball hits the stumps after approximately half a second.
2. State the range of values of $h$ for which the bowler projects the ball below the horizontal.
3. In the case $h=2.5$, give an approximate value in degrees, correct to two significant figures, for $\alpha$. You need not justify the accuracy of your approximation.

[You may use the small-angle approximations $\cos \theta \approx 1$ and $\sin \theta \approx \theta$.]

The lengths of the sides of a rectangular billiards table $ABCD$ are given by $AB=DC=a$ and $AD=BC=2b$. There are small pockets at the midpoints $M$ and $N$ of the sides $AD$ and $BC$, respectively. The sides of the table may be taken as smooth vertical walls.

A small ball is projected along the table from the corner $A$. It strikes the side $BC$ at $X$, then the side $DC$ at $Y$ and then goes directly into the pocket at $M$. The angles $BAX$, $CXY$ and $DYM$ are $\alpha$, $\beta$ and $\gamma$ respectively. On each stage of its path, the ball moves with constant speed in a straight line, the speeds being $u$, $v$ and $w$ respectively. The coefficient of restitution between the ball and the sides is $e$, where $e>0$.

1. Show that $\tan \alpha \tan \beta =e$ and find $\gamma$ in terms of $\alpha$.
2. Show that $\tan \alpha =\frac{(1+2e)b}{(1+e)a}$ and deduce that the shot is possible whatever the value of $e$.
3. Find an expression in terms of $e$ for the fraction of the kinetic energy of the ball that is lost during the motion.

A wedge of mass $km$ has the shape (in cross-section) of a right-angled triangle. It stands on a smooth horizontal surface with one face vertical. The inclined face makes an angle $\theta$ with the horizontal surface. A particle $P$, of mass $m$, is placed on the inclined face and released from rest. The horizontal face of the wedge is smooth, but the inclined face is rough and the coefficient of friction between $P$ and this face is $\mu$.

1. When $P$ is released, it slides down the inclined plane at an acceleration $a$ relative to the wedge. Show that the acceleration of the wedge is $$\frac{a\cos \theta }{k+1}.$$ To a stationary observer, $P$ appears to descend along a straight line inclined at an angle $45^{\circ }$ to the horizontal. Show that $$\tan \theta =\frac{k}{k+1}.$$ In the case $k=3$, find an expression for $a$ in terms of $g$ and $\mu$.
2. What happens when $P$ is released if $\tan \theta ⩽\mu$?

In the High Court of Farnia, the outcome of each case is determined by three judges: the ass, the beaver and the centaur. Each judge decides its verdict independently. Being simple creatures, they make their decisions entirely at random. Past verdicts show that the ass gives a guilty verdict with probability $p$, the beaver gives a guilty verdict with probability $p/3$ and the centaur gives a guilty verdict with probability $p^2$.

Let $X$ be the number of guilty verdicts given by the three judges in a case. Given that $\mathrm{E}(X)=4/3$, find the value of $p$.

The probability that a defendant brought to trial is guilty is $t$. The King pronounces that the defendant is guilty if at least two of the judges give a guilty verdict; otherwise, he pronounces the defendant not guilty. Find the value of $t$ such that the probability that the King pronounces correctly is $1/2$.

Bag $P$ and bag $Q$ each contain $n$ counters, where $n⩾2$. The counters are identical in shape and size, but coloured either black or white. First, $k$ counters ($0⩽k⩽n$) are drawn at random from bag $P$ and placed in bag $Q$. Then, $k$ counters are drawn at random from bag $Q$ and placed in bag $P$.

1. If initially $n-1$ counters in bag $P$ are white and one is black, and all $n$ counters in bag $Q$ are white, find the probability in terms of $n$ and $k$ that the black counter ends up in bag $P$. Find the value or values of $k$ for which this probability is maximised.
2. If initially $n-1$ counters in bag $P$ are white and one is black, and $n-1$ counters in bag $Q$ are white and one is black, find the probability in terms of $n$ and $k$ that the black counters end up in the same bag. Find the value or values of $k$ for which this probability is maximised.