STEP 3 2004

Show that $${\int }_0^a\frac{\sinh x}{2{\cosh }^{2}x-1}\mathrm{d}x=\frac{1}{2\sqrt{2}}\ln \left(\frac{\sqrt{2}\cosh a-1}{\sqrt{2}\cosh a+1}\right)+\frac{1}{2\sqrt{2}}\ln \left(\frac{\sqrt{2}+1}{\sqrt{2}-1}\right)$$ and find $${\int }_0^a\frac{\cosh x}{1+2{\sinh }^{2}x}\mathrm{d}x.$$Hence show that $${\int }_0^{\infty }\frac{\cosh x-\sinh x}{1+2{\sinh }^{2}x}\mathrm{d}x=\frac{\pi }{2\sqrt{2}}-\frac{1}{2\sqrt{2}}\ln \left(\frac{\sqrt{2}+1}{\sqrt{2}-1}\right).$$By substituting $u={\mathrm{e}}^x$ in this result, or otherwise, find $${\int }_1^{\infty }\frac{1}{1+u^4}\mathrm{d}u.$$

The equation of a curve is $y=\mathrm{f}(x)$ where $$\mathrm{f}(x)=x-4-\frac{16{\left(2x+1\right)}^2}{x^2\left(x-4\right)}.$$

1. Write down the equations of the vertical and oblique asymptotes to the curve and show that the oblique asymptote is a tangent to the curve.
2. Show that the equation $\mathrm{f}(x)=0$ has a double root.
3. Sketch the curve.

Given that ${\mathrm{f}}^{\prime \prime }(x)>0$ when $a⩽x⩽b$, explain with the aid of a sketch why $$(b-a)\mathrm{f}\left(\frac{a+b}{2}\right)<{\int }_a^b\mathrm{f}(x)\mathrm{d}x<(b-a)\frac{\mathrm{f}(a)+\mathrm{f}(b)}{2}.$$ By choosing suitable $a$, $b$ and $\mathrm{f}(x)$, show that $$\frac{4}{(2n-1{)}^2}<\frac{1}{n-1}-\frac{1}{n}<\frac{1}{2}\left(\frac{1}{n^2}+\frac{1}{(n-1{)}^2}\right),$$ where $n$ is an integer greater than 1.

Deduce that $$4\left(\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+\cdots \right)<1<\frac{1}{2}+\left(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\cdots \right).$$Show that $$\frac{1}{2}\left(\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+\frac{1}{6^2}+\cdots \right)<\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+\cdots$$ and hence show that $$\frac{3}{2}<\sum _{n=1}^{\infty }\frac{1}{n^2}<\frac{7}{4}.$$

The triangle $OAB$ is isosceles, with $OA=OB$ and angle $AOB=2\alpha$ where $0<\alpha <\frac{\pi }{2}$. The semi-circle ${\mathrm{C}}_0$ has its centre at the midpoint of the base $AB$ of the triangle, and the sides $OA$ and $OB$ of the triangle are both tangent to the semi-circle. ${\mathrm{C}}_1,{\mathrm{C}}_2,{\mathrm{C}}_3,\dots$ are circles such that ${\mathrm{C}}_n$ is tangent to ${\mathrm{C}}_{n-1}$ and to sides $OA$ and $OB$ of the triangle.

Let $r_n$ be the radius of ${\mathrm{C}}_n$. Show that $$\frac{r_{n+1}}{r_n}=\frac{1-\sin \alpha }{1+\sin \alpha }.$$Let $S$ be the total area of the semi-circle ${\mathrm{C}}_0$ and the circles ${\mathrm{C}}_1$, ${\mathrm{C}}_2$, ${\mathrm{C}}_3$, $\dots$. Show that $$S=\frac{1+{\sin }^2\alpha }{4\sin \alpha }\pi r_0^2.$$Show that there are values of $\alpha$ for which $S$ is more than four fifths of the area of triangle $OAB$.

Show that if $\cos (x-\alpha )=\cos \beta$ then either $\tan x=\tan (\alpha +\beta )$ or $\tan x=\tan (\alpha -\beta )$. By choosing suitable values of $x$, $\alpha$ and $\beta$, give an example to show that if $\tan x=\tan (\alpha +\beta )$, then $\cos (x-\alpha )$ need not equal $\cos \beta$.

Let $\omega$ be the acute angle such that $\tan \omega =\frac{4}{3}$.

1. For $0⩽x⩽2\pi$, solve the equation $$\cos x-7\sin x=5$$ giving both solutions in terms of $\omega$.
2. For $0⩽x⩽2\pi$, solve the equation $$2\cos x+11\sin x=10$$ showing that one solution is twice the other and giving both in terms of $\omega$.

Given a sequence $w_0$, $w_1$, $w_2$, $\dots$, the sequence $F_1$, $F_2$, $\dots$ is defined by $$F_n=w_n^2+w_{n-1}^2-4w_n{w}_{n-1}.$$ Show that $F_n-F_{n-1}=\left(w_n-w_{n-2}\right)\left(w_n+w_{n-2}-4w_{n-1}\right)$ for $n⩾2$.

1. The sequence $u_0$, $u_1$, $u_2$, $\dots$ has $u_0=1$, and $u_1=2$ and satisfies $$u_n=4u_{n-1}-u_{n-2}(n⩾2).$$ Prove that $u_n^2+u_{n-1}^2=4u_n{u}_{n-1}-3$ for $n⩾1$.
2. A sequence $v_0$, $v_1$, $v_2$, $\dots$ has $v_0=1$ and satisfies $$v_n^2+v_{n-1}^2=4v_n{v}_{n-1}-3(n⩾1).\text{\hspace{1em}(*)}$$ (a) Find $v_1$ and prove that, for each $n⩾2$, either $v_n=4v_{n-1}-v_{n-2}$ or $v_n=v_{n-2}$. (b) Show that the sequence, with period 2, defined by $$v_n=\{\begin{array}{ll}1& \text{for n even}\\ 2& \text{for n odd}\end{array}$$ satisfies $(\ast )$. (c) Find a sequence $v_n$ with period 4 which has $v_0=1$, and satisfies $(\ast )$.

For $n=1$, $2$, $3$, $\dots$, let $$I_n={\int }_0^1\frac{t^{n-1}}{{\left(t+1\right)}^n}\mathrm{d}t.$$ By considering the greatest value taken by $\frac{t}{t+1}$ for $0⩽t⩽1$ show that $I_{n+1}<\frac{1}{2}{I}_n$.

Show also that $I_{n+1}=-\frac{1}{n{2}^n}+I_n$.

Deduce that $I_n<\frac{1}{n{2}^{n-1}}$.

Prove that $$\ln 2=\sum _{r=1}^n\frac{1}{r{2}^r}+I_{n+1}$$ and hence show that $\frac{2}{3}<\ln 2<\frac{17}{24}$.

Show that if $$\frac{\mathrm{d}y}{\mathrm{d}x}=\mathrm{f}(x)y+\frac{\mathrm{g}(x)}{y}$$ then the substitution $u=y^2$ gives a linear differential equation for $u(x)$.

Hence or otherwise solve the differential equation $$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{y}{x}-\frac{1}{y}.$$Determine the solution curves of this equation which pass through $(1,1)$, $(2,2)$ and $(4,4)$ and sketch graphs of all three curves on the same axes.

A circular hoop of radius $a$ is free to rotate about a fixed horizontal axis passing through a point $P$ on its circumference. The plane of the hoop is perpendicular to this axis. The hoop hangs in equilibrium with its centre, $O$, vertically below $P$. The point $A$ on the hoop is vertically below $O$, so that $POA$ is a diameter of the hoop.

A mouse $M$ runs at constant speed $u$ round the rough inner surface of the lower part of the hoop. Show that the mouse can choose its speed so that the hoop remains in equilibrium with diameter $POA$ vertical.

Describe what happens to the hoop when the mouse passes the point at which angle $AOM=2\arctan \mu$, where $\mu$ is the coefficient of friction between mouse and hoop.

A particle $P$ of mass $m$ is attached to points $A$ and $B$, where $A$ is a distance $9a$ vertically above $B$, by elastic strings, each of which has modulus of elasticity $6mg$. The string $AP$ has natural length $6a$ and the string $BP$ has natural length $2a$. Let $x$ be the distance $AP$.

The system is released from rest with $P$ on the vertical line $AB$ and $x=6a$. Show that the acceleration $\ddot{x}$ of $P$ is $\frac{4g}{a}(7a-x)$ for $6a<x<7a$ and $\frac{g}{a}(7a-x)$ for $7a<x<9a$.

Find the time taken for the particle to reach $B$.

Particles $P$, of mass $2$, and $Q$, of mass $1$, move along a line. Their distances from a fixed point are $x_1$ and $x_2$, respectively where $x_2>x_1$. Each particle is subject to a repulsive force from the other of magnitude $\frac{2}{z^3}$, where $z=x_2-x_1$.

Initially, $x_1=0$, $x_2=1$, $Q$ is at rest and $P$ moves towards $Q$ with speed 1. Show that $z$ obeys the equation $\frac{{\mathrm{d}}^{2}z}{\mathrm{d}{t}^2}=\frac{3}{z^3}$.

By first writing $\frac{{\mathrm{d}}^{2}z}{\mathrm{d}{t}^2}=v\frac{\mathrm{d}v}{\mathrm{d}z}$, where $v=\frac{\mathrm{d}z}{\mathrm{d}t}$, show that $z=\sqrt{4t^2-2t+1}$.

By considering the equation satisfied by $2x_1+x_2$, find $x_1$ and $x_2$ in terms of $t$.

A team of $m$ players, numbered from $1$ to $m$, puts on a set of a $m$ shirts, similarly numbered from $1$ to $m$. The players change in a hurry, so that the shirts are assigned to them randomly, one to each player.

Let $C_i$ be the random variable that takes the value $1$ if player $i$ is wearing shirt $i$, and 0 otherwise. Show that $\mathrm{E}\left(C_1\right)=\frac{1}{m}$ and find $\mathrm{Var}\left(C_1\right)$ and $\mathrm{Cov}\left(C_1,C_2\right)$.

Let $N=C_1+C_2+\cdots +C_m$ be the random variable whose value is the number of players who are wearing the correct shirt. Show that $\mathrm{E}\left(N\right)=\mathrm{Var}\left(N\right)=1$.

Explain why a Normal approximation to $N$ is not likely to be appropriate for any $m$, but that a Poisson approximation might be reasonable.

In the case $m=4$, find, by listing equally likely possibilities or otherwise, the probability that no player is wearing the correct shirt and verify that an appropriate Poisson approximation to $N$ gives this probability with a relative error of about $2\%$. [Use $\mathrm{e}\approx 2\frac{72}{100}$.]

A men's endurance competition has an unlimited number of rounds. In each round, a competitor has, independently, a probability $p$ of making it through the round; otherwise, he fails the round. Once a competitor fails a round, he drops out of the competition; before he drops out, he takes part in every round. The grand prize is awarded to any competitor who makes it through a round which all the other remaining competitors fail; if all the remaining competitors fail at the same round the grand prize is not awarded.

If the competition begins with three competitors, find the probability that:

1. all three drop out in the same round;
2. two of them drop out in round $r$ (with $r⩾2$) and the third in an earlier round;
3. the grand prize is awarded.

In this question, $\Phi (z)$ is the cumulative distribution function of a standard normal random variable.

A random variable is known to have a Normal distribution with mean $\mu$ and standard deviation either ${\sigma }_0$ or ${\sigma }_1$, where ${\sigma }_0<{\sigma }_1$. The mean, $\overline{X}$, of a random sample of $n$ values of $X$ is to be used to test the hypothesis ${\mathrm{H}}_0:\sigma ={\sigma }_0$ against the alternative ${\mathrm{H}}_1:\sigma ={\sigma }_1$.

Explain carefully why it is appropriate to use a two sided test of the form: accept ${\mathrm{H}}_0$ if $\mu -c<\overline{X}<\mu +c$, otherwise accept ${\mathrm{H}}_1$.

Given that the probability of accepting ${\mathrm{H}}_1$ when ${\mathrm{H}}_0$ is true is $\alpha$, determine $c$ in terms of $n$, ${\sigma }_0$ and $z_{\alpha }$, where $z_{\alpha }$ is defined by $\Phi (z_{\alpha })=1-\frac{1}{2}\alpha$.

The probability of accepting ${\mathrm{H}}_0$ when ${\mathrm{H}}_1$ is true is denoted by $\beta$. Show that $\beta$ is independent of $n$.

Given that $\Phi (1.960)\approx 0.975$ and that $\Phi (0.063)\approx 0.525$, determine, approximately, the minimum value of $\frac{{\sigma }_1}{{\sigma }_0}$ if $\alpha$ and $\beta$ are both to be less than $0.05$.