STEP 2 2003

Consider the equations $$\begin{array}{rlrl}ax-& y-z& & =3,\\ 2ax-& y-3z& & =7,\\ 3ax-& y-5z& & =b,\end{array}$$ where $a$ and $b$ are given constants.

1. In the case $a=0$, show that the equations have a solution if and only if $b=11$.
2. In the case $a\ne 0$ and $b=11$ show that the equations have a solution with $z=\lambda$ for any given number $\lambda$.
3. In the case $a=2$ and $b=11$ find the solution for which $x^2+y^2+z^2$ is least.
4. Find a value for $a$ for which there is a solution such that $x>10^6$ and $y^2+z^2<1$.

Write down a value of $\theta$ in the interval $\frac{1}{4}\pi <\theta <\frac{1}{2}\pi$ that satisfies the equation $$4\cos \theta +2\sqrt{3}\sin \theta =5.$$ Hence, or otherwise, show that $$\pi =3\arccos (5/\sqrt{2}8)+3\arctan (\sqrt{3}/2).$$Show that $$\pi =4\arcsin (7\sqrt{2}/10)-4\arctan (3/4).$$

Prove that the cube root of any irrational number is an irrational number.

Let $u_n=5^{1/(3^n)}$. Given that $\sqrt[3]{5}$ is an irrational number, prove by induction that $u_n$ is an irrational number for every positive integer $n$.

Hence, or otherwise, give an example of an infinite sequence of irrational numbers which converges to a given integer $m$.

[An irrational number is a number that cannot be expressed as the ratio of two integers.]

The line $y=d$, where $d>0$, intersects the circle $x^2+y^2=R^2$ at $G$ and $H$. Show that the area of the minor segment $GH$ is equal to $$R^2\arccos \left(\frac{d}{R}\right)-d\sqrt{R^2-d^2}.\text{\hspace{1em}(∗)}$$In the following cases, the given line intersects the given circle. Determine how, in each case, the expression $(\ast )$ should be modified to give the area of the minor segment.

1. Line: $y=c$; circle: $(x-a{)}^2+(y-b{)}^2=R^2$.
2. Line: $y=mx+c$; circle: $x^2+y^2=R^2$.
3. Line: $y=mx+c$; circle: $(x-a{)}^2+(y-b{)}^2=R^2$.

The position vectors of the points $A$, $B$ and $P$ with respect to an origin $O$ are $a[[B]]i[[/B]]$, $b[[B]]j[[/B]]$ and $l[[B]]i[[/B]]+m[[B]]j[[/B]]+n[[B]]k[[/B]]$, respectively, where $a$, $b$, and $n$ are all non-zero. The points $E$, $F$, $G$ and $H$ are the midpoints of $OA$, $BP$, $OB$ and $AP$, respectively. Show that the lines $EF$ and $GH$ intersect.

Let $D$ be the point with position vector $d[[B]]k[[/B]]$, where $d$ is non-zero, and let $S$ be the point of intersection of $EF$ and $GH.$ The point $T$ is such that the mid-point of $DT$ is $S$. Find the position vector of $T$ and hence find $d$ in terms of $n$ if $T$ lies in the plane $OAB$.

The function $\mathrm{f}$ is defined by $$\mathrm{f}(x)=|x-1|,$$ where the domain is $[[B]]R[[/B]]$, the set of all real numbers. The function ${\mathrm{g}}_n={\mathrm{f}}^n$, with domain $[[B]]R[[/B]]$, so for example ${\mathrm{g}}_3(x)=\mathrm{f}(\mathrm{f}(\mathrm{f}(x)))$. In separate diagrams, sketch graphs of ${\mathrm{g}}_1$, ${\mathrm{g}}_2$, ${\mathrm{g}}_3$ and ${\mathrm{g}}_4$.

The function $\mathrm{h}$ is defined by $$\mathrm{h}(x)=\left|\sin \frac{\pi x}{2}\right|,$$ where the domain is $[[B]]R[[/B]]$. Show that if $n$ is even, $${\int }_0^n(\mathrm{h}(x)-{\mathrm{g}}_n(x))\mathrm{d}x=\frac{2n}{\pi }-\frac{n}{2}.$$

Show that, if $n>0$, then $${\int }_{e^{1/n}}^{\infty }\frac{\ln x}{x^{n+1}}\mathrm{d}x=\frac{2}{n^2\mathrm{e}}.$$ You may assume that $\frac{\ln x}{x}\to 0$ as $x\to \infty$.

Explain why, if $1<a<b$, then $${\int }_b^{\infty }\frac{\ln x}{x^{n+1}}\mathrm{d}x<{\int }_a^{\infty }\frac{\ln x}{x^{n+1}}\mathrm{d}x.$$Deduce that $$\sum _{n=1}^N\frac{1}{n^2}<\frac{\mathrm{e}}{2}{\int }_{{\mathrm{e}}^{1/N}}^{\infty }\left(\frac{1-x^{-N}}{x^2-x}\right)\ln x\mathrm{d}x,$$ where $N$ is any integer greater than $1$.

It is given that $y$ satisfies $$\frac{\mathrm{d}y}{\mathrm{d}t}+k\left(\frac{t^2-3t+2}{t+1}\right)y=0,$$ where $k$ is a constant, and $y=A$ when $t=0$, where $A$ is a positive constant. Find $y$ in terms of $t$, $k$ and $A$.

Show that $y$ has two stationary values whose ratio is $(3/2{)}^{6k}{\mathrm{e}}^{-5k/2}.$

Describe the behaviour of $y$ as $t\to +\infty$ for the case where $k>0$ and for the case where $k<0.$

In separate diagrams, sketch the graph of $y$ for $t>0$ for each of these cases.

$AB$ is a uniform rod of weight $W$. The point $C$ on $AB$ is such that $AC>CB$. The rod is in contact with a rough horizontal floor at $A$ and with a cylinder at $C$. The cylinder is fixed to the floor with its axis horizontal. The rod makes an angle $\alpha$ with the horizontal and lies in a vertical plane perpendicular to the axis of the cylinder. The coefficient of friction between the rod and the floor is $\tan {\lambda }_1$ and the coefficient of friction between the rod and the cylinder is $\tan {\lambda }_2$.

Show that if friction is limiting both at $A$ and at $C$, and $\alpha \ne {\lambda }_2-{\lambda }_1$, then the frictional force acting on the rod at $A$ has magnitude $$\frac{W\sin {\lambda }_1\sin (\alpha -{\lambda }_2)}{\sin (\alpha +{\lambda }_1-{\lambda }_2)}.$$

A bead $B$ of mass $m$ can slide along a rough horizontal wire. A light inextensible string of length $2\ell$ has one end attached to a fixed point $A$ of the wire and the other to $B$. A particle $P$ of mass $3m$ is attached to the mid-point of the string and $B$ is held at a distance $\ell$ from $A$. The bead is released from rest.

Let $a_1$ and $a_2$ be the magnitudes of the horizontal and vertical components of the initial acceleration of $P$. Show by considering the motion of $P$ relative to $A$, or otherwise, that $a_1=\sqrt{3}{a}_2$. Show also that the magnitude of the initial acceleration of $B$ is $2a_1$.

Given that the frictional force opposing the motion of $B$ is equal to $(\sqrt{3}/6)R$, where $R$ is the normal reaction between $B$ and the wire, show that the magnitude of the initial acceleration of $P$ is $g/18$.

A particle $P_1$ is projected with speed $V$ at an angle of elevation $\alpha (>45^{\circ }),$ from a point in a horizontal plane. Find $T_1$, the flight time of $P_1$, in terms of $\alpha ,Vandg$. Show that the time after projection at which the direction of motion of $P_1$ first makes an angle of $45^{\circ }$ with the horizontal is $\frac{1}{2}(1-\cot \alpha )T_1$.

A particle $P_2$ is projected under the same conditions. When the direction of the motion of $P_2$ first makes an angle of $45^{\circ }$ with the horizontal, the speed of $P_2$ is instantaneously doubled. If $T_2$ is the total flight time of $P_2$, show that $$\frac{2T_2}{T_1}=1+\cot \alpha +\sqrt{1+3{\cot }^2\alpha }.$$

The life of a certain species of elementary particles can be described as follows. Each particle has a life time of $T$ seconds, after which it disintegrates into $X$ particles of the same species, where $X$ is a random variable with binomial distribution $\mathrm{B}(2,p)$. A population of these particles starts with the creation of a single such particle at $t=0$. Let $X_n$ be the number of particles in existence in the time interval $nT<t<(n+1)T$, where $n=1$, $2$, $\dots$.

Show that $\mathrm{P}(X_1=2\text{ and }{X}_2=2)=6p^4{q}^2$, where $q=1-p$. Find the possible values of $p$ if it is known that $\mathrm{P}(X_1=2|X_2=2)=9/25$.

Explain briefly why $\mathrm{E}(X_n)=2p\mathrm{E}(X_{n-1})$ and hence determine $\mathrm{E}(X_n)$ in terms of $p$. Show that for one of the values of $p$ found above $\underset{n\to \infty }{\lim }\mathrm{E}(X_n)=0$ and that for the other $\underset{n\to \infty }{\lim }\mathrm{E}(X_n)=+\infty$.

The random variable $X$ takes the values $k=1$, $2$, $3$, $\dots$, and has probability distribution $$\mathrm{P}(X=k)=A\frac{{\lambda }^k{\mathrm{e}}^{-\lambda }}{k!},$$ where $\lambda$ is a positive constant. Show that $A=(1-{\mathrm{e}}^{-\lambda }{)}^{-1}$. Find the mean $\mu$ in terms of $\lambda$ and show that $$\mathrm{Var}(X)=\mu (1-\mu +\lambda ).$$ Deduce that $\lambda <\mu <1+\lambda$.

Use a normal approximation to find the value of $P(X=\lambda )$ in the case where $\lambda =100$, giving your answer to 2 decimal places.

The probability of throwing a 6 with a biased die is $p$. It is known that $p$ is equal to one or other of the numbers $A$ and $B$ where $0<A<B<1$. Accordingly the following statistical test of the hypothesis $H_0:p=B$ against the alternative hypothesis $H_1:p=A$ is performed.

The die is thrown repeatedly until a 6 is obtained. Then if $X$ is the total number of throws, $H_0$ is accepted if $X⩽M$, where $M$ is a given positive integer; otherwise $H_1$ is accepted. Let $\alpha$ be the probability that $H_1$ is accepted if $H_0$ is true, and let $\beta$ be the probability that $H_0$ is accepted if $H_1$ is true.

Show that $\beta =1-{\alpha }^K,$ where $K$ is independent of $M$ and is to be determined in terms of $A$ and $B$. Sketch the graph of $\beta$ against $\alpha$.