STEP 3 2016

Let $$I_n={\int }_{-\infty }^{\infty }\frac{1}{(x^2+2ax+b{)}^n}\mathrm{d}x,$$ where $a$ and $b$ are constants with $b>a^2$, and $n$ is a positive integer.

1. By using the substitution $x+a=\sqrt{b-a^2}\tan u$, or otherwise, show that $$I_1=\frac{\pi }{\sqrt{b-a^2}}.$$
2. Show that $2n(b-a^2)I_{n+1}=(2n-1)I_n$.
3. Hence prove by induction that $$I_n=\frac{\pi }{2^{2n-2}(b-a^2{)}^{n-\frac{1}{2}}}\left(\genfrac{}{}{0ex}{}{2n-2}{n-1}\right).$$

The distinct points $P(a{p}^2,2ap)$, $Q(a{q}^2,2aq)$ and $R(a{r}^2,2ar)$ lie on the parabola $y^2=4ax$, where $a>0$. The points are such that the normal to the parabola at $Q$ and the normal to the parabola at $R$ both pass through $P$.

1. Show that $q^2+qp+2=0$.
2. Show that $QR$ passes through a certain point that is independent of the choice of $P$.
3. Let $T$ be the point of intersection of $OP$ and $QR$, where $O$ is the coordinate origin. Show that $T$ lies on a line that is independent of the choice of $P$. Show further that the distance from the $x$-axis to $T$ is less than $\frac{a}{\sqrt{2}}$.

Show, by finding $R$ and $\gamma$, that $A\sinh x+B\cosh x$ can be written in the form $R\cosh (x+\gamma )$ if $B>A>0$. Determine the corresponding forms in the other cases that arise, for $A>0$, according to the value of $B$.

Two curves have equations $y=\mathrm{sech}x$ and $y=a\tanh x+b$, where $a>0$.

1. In the case $b>a$, show that if the curves intersect then the $x$-coordinates of the points of intersection can be written in the form $$\pm \mathrm{arcosh}\left(\frac{1}{\sqrt{b^2-a^2}}\right)-\mathrm{artanh}\frac{a}{b}.$$
2. Find the corresponding result in the case $a>b>0$.
3. Find necessary and sufficient conditions on $a$ and $b$ for the curves to intersect at two distinct points.
4. Find necessary and sufficient conditions on $a$ and $b$ for the curves to touch and, given that they touch, express the $y$-coordinate of the point of contact in terms of $a$.

Let $\omega ={\mathrm{e}}^{2\pi \mathrm{i}/n}$, where $n$ is a positive integer. Show that, for any complex number $z$, $$(z-1)(z-\omega )\cdots (z-{\omega }^{n-1})=z^n-1.$$The points $X_0$, $X_1$, …, $X_{n-1}$ lie on a circle with centre $O$ and radius 1, and are the vertices of a regular polygon.

1. The point $P$ is equidistant from $X_0$ and $X_1$. Show that, if $n$ is even, $$|P{X}_0|\times |P{X}_1|\times \cdots \times |P{X}_{n-1}|=|OP{|}^n+1,$$ where $|P{X}_k|$ denotes the distance from $P$ to $X_k$. Give the corresponding result when $n$ is odd. (There are two cases to consider.)
2. Show that $$|X_0{X}_1|\times |X_0{X}_2|\times \cdots \times |X_0{X}_{n-1}|=n.$$

Three pegs $P$, $Q$ and $R$ are fixed on a smooth horizontal table in such a way that they form the vertices of an equilateral triangle of side $2a$. A particle $X$ of mass $m$ lies on the table. It is attached to the pegs by three springs, $PX$, $QX$ and $RX$, each of modulus of elasticity $\lambda$ and natural length $l$, where $l<\frac{2}{\sqrt{3}}a$. Initially the particle is in equilibrium. Show that the extension in each spring is $\frac{2}{\sqrt{3}}a-l$.

The particle is then pulled a small distance directly towards $P$ and released. Show that the tension $T$ in the spring $RX$ is given by $$T=\frac{\lambda }{l}\left(\sqrt{\frac{4a^2}{3}+\frac{2ax}{\sqrt{3}}+x^2}-l\right),$$ where $x$ is the displacement of $X$ from its equilibrium position.

Show further that the particle performs approximate simple harmonic motion with period $$2\pi \sqrt{\frac{4mla}{3(4a-\sqrt{3}l)\lambda }}.$$

A smooth plane is inclined at an angle $\alpha$ to the horizontal. A particle $P$ of mass $m$ is attached to a fixed point $A$ above the plane by a light inextensible string of length $a$. The particle rests in equilibrium on the plane, and the string makes an angle $\beta$ with the plane.

The particle is given a horizontal impulse parallel to the plane so that it has an initial speed of $u$. Show that the particle will not immediately leave the plane if $ag\cos (\alpha +\beta )>u^2\tan \beta$.

Show further that a necessary condition for the particle to perform a complete circle whilst in contact with the plane is $6\tan \alpha \tan \beta <1$.

A car of mass $m$ travels along a straight horizontal road with its engine working at a constant rate $P$. The resistance to its motion is such that the acceleration of the car is zero when it is moving with speed $4U$.

1. Given that the resistance is proportional to the car's speed, show that the distance $X_1$ travelled by the car while it accelerates from speed $U$ to speed $2U$, is given by $$\lambda X_1=2\ln \frac{9}{5}-1,$$ where $\lambda =P/(16m{U}^3)$.
2. Given instead that the resistance is proportional to the square of the car's speed, show that the distance $X_2$ travelled by the car while it accelerates from speed $U$ to speed $2U$ is given by $$\lambda X_2=\frac{4}{3}\ln \frac{9}{8}.$$
3. Given that $3.17<\ln 24<3.18$ and $1.60<\ln 5<1.61$, determine which is the larger of $X_1$ and $X_2$.

Let $X$ be a random variable with mean $\mu$ and standard deviation $\sigma$. Chebyshev's inequality, which you may use without proof, is $$\mathrm{P}\left(|X-\mu |>k\sigma \right)⩽\frac{1}{k^2},$$ where $k$ is any positive number.

1. The probability of a biased coin landing heads up is $0.2$. It is thrown $100n$ times, where $n$ is an integer greater than 1. Let $\alpha$ be the probability that the coin lands heads up $N$ times, where $16n⩽N⩽24n$. Use Chebyshev's inequality to show that $$\alpha ⩾1-\frac{1}{n}.$$
2. Use Chebyshev's inequality to show that $$1+n+\frac{n^2}{2!}+\cdots +\frac{n^{2n}}{(2n)!}⩾\left(1-\frac{1}{n}\right){\mathrm{e}}^n.$$

Given a random variable $X$ with mean $\mu$ and standard deviation $\sigma$, we define the kurtosis, $\kappa$, of $X$ by $$\kappa =\frac{\mathrm{E}((X-\mu {)}^4)}{{\sigma }^4}-3.$$ Show that the random variable $X-a$, where $a$ is a constant, has the same kurtosis as $X$.

1. Show by integration that a random variable which is Normally distributed with mean 0 has kurtosis 0.
2. Let $Y_1$, $Y_2$, $\dots$, $Y_n$ be $n$ independent, identically distributed, random variables with mean 0, and let $T=\sum _{r=1}^n{Y}_r$. Show that $$\mathrm{E}(T^4)=\sum _{r=1}^n\mathrm{E}(Y_r^4)+6\sum _{r=1}^{n-1}\sum _{s=r+1}^n\mathrm{E}(Y_s^2)\mathrm{E}(Y_r^2).$$
3. Let $X_1$, $X_2$, $\dots$, $X_n$ be $n$ independent, identically distributed, random variables each with kurtosis $\kappa$. Show that the kurtosis of their sum is $\frac{\kappa }{n}$.