STEP 3 2013

Given that $t=\tan \frac{1}{2}x$, show that $\frac{\mathrm{d}t}{\mathrm{d}x}=\frac{1}{2}(1+t^2)$ and $\sin x=\frac{2t}{1+t^2}$.

Hence show that $${\int }_0^{\frac{1}{2}\pi }\frac{1}{1+a\sin x}\mathrm{d}x=\frac{2}{\sqrt{1-a^2}}\arctan \frac{\sqrt{1-a}}{\sqrt{1+a}}(0<a<1).$$Let $$I_n={\int }_0^{\frac{1}{2}\pi }\frac{{\sin }^{n}x}{2+\sin x}\mathrm{d}x(n⩾0).$$ By considering $I_{n+1}+2I_n$, or otherwise, evaluate $I_3$.

In this question, you may ignore questions of convergence.

Let $y=\frac{\arcsin x}{\sqrt{1-x^2}}$. Show that $$(1-x^2)\frac{\mathrm{d}y}{\mathrm{d}x}-xy-1=0$$ and prove that, for any positive integer $n$, $$(1-x^2)\frac{{\mathrm{d}}^{n+2}y}{\mathrm{d}{x}^{n+2}}-(2n+3)x\frac{{\mathrm{d}}^{n+1}y}{\mathrm{d}{x}^{n+1}}-(n+1{)}^2\frac{{\mathrm{d}}^{n}y}{\mathrm{d}{x}^n}=0.$$ Hence obtain the Maclaurin series for $\frac{\arcsin x}{\sqrt{1-x^2}}$, giving the general term for odd and for even powers of $x$.

Evaluate the infinite sum $$1+\frac{1}{3!}+\frac{2^2}{5!}+\frac{2^2\times 3^2}{7!}+\cdots +\frac{2^2\times 3^2\times \cdots \times n^2}{(2n+1)!}+\cdots .$$

The four vertices $P_i$ ($i=1,2,3,4$) of a regular tetrahedron lie on the surface of a sphere with centre at $O$ and of radius 1. The position vector of $P_i$ with respect to $O$ is $[[B]]p[[/B{]]}_i$ ($i=1,2,3,4$). Use the fact that $[[B]]p[[/B{]]}_1+[[B]]p[[/B{]]}_2+[[B]]p[[/B{]]}_3+[[B]]p[[/B{]]}_4=[[B]]0[[/B]]$ to show that $[[B]]p[[/B{]]}_i.[[B]]p[[/B{]]}_j=-\frac{1}{3}$ for $i\ne j$.

Let $X$ be any point on the surface of the sphere, and let $X{P}_i$ denote the length of the line joining $X$ and $P_i$ ($i=1,2,3,4$).

1. By writing $(X{P}_i{)}^2$ as $(<strong>p</strong{>}_i-<strong>x)</strong>.(<strong>p</strong{>}_i-<strong>x</strong>)$, where $<strong>x</strong>$ is the position vector of $X$ with respect to $O$, show that $$\sum _{i=1}^4(X{P}_i{)}^2=8.$$
2. Given that $P_1$ has coordinates $(0,0,1)$ and that the coordinates of $P_2$ are of the form $(a,0,b)$, where $a>0$, show that $a=2\sqrt{2}/3$ and $b=-1/3$, and find the coordinates of $P_3$ and $P_4$.
3. Show that $$\sum _{i=1}^4(X{P}_i{)}^4=4\sum _{i=1}^4(1-<strong>x</strong>.<strong>p</strong{>}_i{)}^2.$$ By letting the coordinates of $X$ be $(x,y,z)$, show further that $\sum _{i=1}^4(X{P}_i{)}^4$ is independent of the position of $X$.

Show that $(z-{\mathrm{e}}^{\mathrm{i}\theta })(z-{\mathrm{e}}^{-\mathrm{i}\theta })=z^2-2z\cos \theta +1$.

Write down the $(2n)$th roots of $-1$ in the form ${\mathrm{e}}^{\mathrm{i}\theta }$, where $-\pi <\theta ⩽\pi$, and deduce that $$z^{2n}+1=\prod _{k=1}^n\left(z^2-2z\cos \left(\frac{(2k-1)\pi }{2n}\right)+1\right).$$ Here, $n$ is a positive integer, and the $\prod$ notation denotes the product.

1. By substituting $z=\mathrm{i}$ show that, when $n$ is even, $$\cos \left(\frac{\pi }{2n}\right)\cos \left(\frac{3\pi }{2n}\right)\cos \left(\frac{5\pi }{2n}\right)\cdots \cos \left(\frac{(2n-1)\pi }{2n}\right)={(-1)}^{\frac{1}{2}n}{2}^{1-n}.$$
2. Show that, when $n$ is odd, $${\cos }^2\left(\frac{\pi }{2n}\right){\cos }^2\left(\frac{3\pi }{2n}\right){\cos }^2\left(\frac{5\pi }{2n}\right)\cdots {\cos }^2\left(\frac{(n-2)\pi }{2n}\right)=n{2}^{1-n}.$$ You may use without proof the fact that $1+z^{2n}=(1+z^2)(1-z^2+z^4-\cdots +z^{2n-2})$ when $n$ is odd.

In this question, you may assume that, if $a$, $b$ and $c$ are positive integers such that $a$ and $b$ are coprime and $a$ divides $bc$, then $a$ divides $c$. (Two positive integers are said to be coprime if their highest common factor is 1.)

1. Suppose that there are positive integers $p$, $q$, $n$ and $N$ such that $p$ and $q$ are coprime and $q^{n}N=p^n$. Show that $N=k{p}^n$ for some positive integer $k$ and deduce the value of $q$. Hence prove that, for any positive integers $n$ and $N$, $\sqrt[n]{N}$ is either a positive integer or irrational.
2. Suppose that there are positive integers $a$, $b$, $c$ and $d$ such that $a$ and $b$ are coprime and $c$ and $d$ are coprime, and $a^a{d}^b=b^a{c}^b$. Prove that $d^b=b^a$ and deduce that, if $p$ is a prime factor of $d$, then $p$ is also a prime factor of $b$. If $p^m$ and $p^n$ are the highest powers of the prime number $p$ that divide $d$ and $b$, respectively, express $b$ in terms of $a$, $m$ and $n$ and hence show that $p^n⩽n$. Deduce the value of $b$. (You may assume that if $x>0$ and $y⩾2$ then $y^x>x$.) Hence prove that, if $r$ is a positive rational number such that $r^r$ is rational, then $r$ is a positive integer.

Let $z$ and $w$ be complex numbers. Use a diagram to show that $|z-w|⩽|z|+|w|.$

For any complex numbers $z$ and $w$, $E$ is defined by $$E=z{w}^{\ast }+z^{\ast }w+2|zw|.$$

1. Show that $|z-w{|}^2={\left(|z|+|w|\right)}^2-E$, and deduce that $E$ is real and non-negative.
2. Show that $|1-z{w}^{\ast }{|}^2={\left(1+|zw|\right)}^2-E$.

Hence show that, if both $|z|>1$ and $|w|>1$, then $$\frac{|z-w|}{|1-z{w}^{\ast }|}⩽\frac{|z|+|w|}{1+|zw|}.$$ Does this inequality also hold if both $|z|<1$ and $|w|<1$?

Evaluate $\sum _{r=0}^{n-1}{\mathrm{e}}^{2\mathrm{i}(\alpha +r\pi /n)}$ where $\alpha$ is a fixed angle and $n⩾2$.

The fixed point $O$ is a distance $d$ from a fixed line $D$. For any point $P$, let $s$ be the distance from $P$ to $D$ and let $r$ be the distance from $P$ to $O$. Write down an expression for $s$ in terms of $d$, $r$ and the angle $\theta$, where $\theta$ is as shown in the diagram below.

The curve $E$ shown in the diagram is such that, for any point $P$ on $E$, the relation $r=ks$ holds, where $k$ is a fixed number with $0<k<1$.

Each of the $n$ lines $L_1$, $\dots$, $L_n$ passes through $O$ and the angle between adjacent lines is $\frac{\pi }{n}$. The line $L_j$ ($j=1$, $\dots$, $n$) intersects $E$ in two points forming a chord of length $l_j$. Show that, for $n⩾2$, $$\sum _{j=1}^n\frac{1}{l_j}=\frac{(2-k^2)n}{4kd}.$$

A sphere of radius $R$ and uniform density ${\rho }_{\text{s}}$ is floating in a large tank of liquid of uniform density $\rho$. Given that the centre of the sphere is a distance $x$ above the level of the liquid, where $x<R$, show that the volume of liquid displaced is $$\frac{\pi }{3}(2R^3-3R^{2}x+x^3).$$The sphere is acted upon by two forces only: its weight and an upward force equal in magnitude to the weight of the liquid it has displaced. Show that $$4R^3{\rho }_{\text{s}}(g+\ddot{x})=(2R^3-3R^{2}x+x^3)\rho g.$$ Given that the sphere is in equilibrium when $x=\frac{1}{2}R$, find ${\rho }_{\text{s}}$ in terms of $\rho$. Find, in terms of $R$ and $g$, the period of small oscillations about this equilibrium position.

A uniform rod $AB$ has mass $M$ and length $2a$. The point $P$ lies on the rod a distance $a-x$ from $A$. Show that the moment of inertia of the rod about an axis through $P$ and perpendicular to the rod is $$\frac{1}{3}M(a^2+3x^2).$$The rod is free to rotate, in a horizontal plane, about a fixed vertical axis through $P$. Initially the rod is at rest. The end $B$ is struck by a particle of mass $m$ moving horizontally with speed $u$ in a direction perpendicular to the rod. The coefficient of restitution between the rod and the particle is $e$. Show that the angular velocity of the rod immediately after impact is $$\frac{3mu(1+e)(a+x)}{M(a^2+3x^2)+3m(a+x{)}^2}.$$In the case $m=2M$, find the value of $x$ for which the angular velocity is greatest and show that this angular velocity is $u(1+e)/a$.

An equilateral triangle, comprising three light rods each of length $\sqrt{3}a$, has a particle of mass $m$ attached to each of its vertices. The triangle is suspended horizontally from a point vertically above its centre by three identical springs, so that the springs and rods form a tetrahedron. Each spring has natural length $a$ and modulus of elasticity $kmg$, and is light. Show that when the springs make an angle $\theta$ with the horizontal the tension in each spring is $$\frac{kmg(1-\cos \theta )}{\cos \theta }.$$ Given that the triangle is in equilibrium when $\theta =\frac{1}{6}\pi$, show that $k=4\sqrt{3}+6$.

The triangle is released from rest from the position at which $\theta =\frac{1}{3}\pi$. Show that when it passes through the equilibrium position its speed $V$ satisfies $$V^2=\frac{4ag}{3}(6+\sqrt{3}).$$

A list consists only of letters $A$ and $B$ arranged in a row. In the list, there are $a$ letter $A$s and $b$ letter $B$s, where $a⩾2$ and $b⩾2$, and $a+b=n$. Each possible ordering of the letters is equally probable. The random variable $X_1$ is defined by $$X_1=\{\begin{array}{ll}1& \text{if the first letter in the row is A};\\ 0& \text{otherwise.}\end{array}$$ The random variables $X_k$ ($2⩽k⩽n$) are defined by $$X_k=\{\begin{array}{ll}1& \text{if the (k−1)th letter is B and the kth is A};\\ 0& \text{otherwise.}\end{array}$$ The random variable $S$ is defined by $S=\sum _{i=1}^n{X}_i$.

1. Find expressions for $\mathrm{E}(X_i)$, distinguishing between the cases $i=1$ and $i\ne 1$, and show that $\mathrm{E}(S)=\frac{a(b+1)}{n}$.
2. Show that:
   1. (a) for $j⩾3$, $\mathrm{E}(X_1{X}_j)=\frac{a(a-1)b}{n(n-1)(n-2)}$;
   2. (b) $\sum _{i=2}^{n-2}(\sum _{j=i+2}^n\mathrm{E}(X_i{X}_j))=\frac{a(a-1)b(b-1)}{2n(n-1)}$;
   3. (c) $\mathrm{Var}(S)=\frac{a(a-1)b(b+1)}{n^2(n-1)}$.