STEP 3 2017

1. Prove that, for any positive integers $n$ and $r$, $$\frac{1}{{}^{n+r}{\mathrm{C}}_{r+1}}=\frac{r+1}{r}\left(\frac{1}{{}^{n+r-1}{\mathrm{C}}_r}-\frac{1}{{}^{n+r}{\mathrm{C}}_r}\right).$$ Hence determine $$\sum _{n=1}^{\infty }\frac{1}{{}^{n+r}{\mathrm{C}}_{r+1}},$$ and deduce that $\sum _{n=2}^{\infty }\frac{1}{{}^{n+2}{\mathrm{C}}_3}=\frac{1}{2}$.
2. Show that, for $n⩾3$, $$\frac{3!}{n^3}<\frac{1}{{}^{n+1}{\mathrm{C}}_3}\text{and}\frac{20}{{}^{n+1}{\mathrm{C}}_3}-\frac{1}{{}^{n+2}{\mathrm{C}}_5}<\frac{5!}{n^3}.$$ By summing these inequalities for $n⩾3$, show that $$\frac{115}{96}<\sum _{n=1}^{\infty }\frac{1}{n^3}<\frac{116}{96}.$$

Note: ${}^n{\mathrm{C}}_r$ is another notation for $\left(\genfrac{}{}{0ex}{}{n}{r}\right)$.

The transformation $R$ in the complex plane is a rotation (anticlockwise) by an angle $\theta$ about the point represented by the complex number $a$. The transformation $S$ in the complex plane is a rotation (anticlockwise) by an angle $\varphi$ about the point represented by the complex number $b$.

1. The point $P$ is represented by the complex number $z$. Show that the image of $P$ under $R$ is represented by $${\mathrm{e}}^{\mathrm{i}\theta }z+a(1-{\mathrm{e}}^{\mathrm{i}\theta }).$$
2. Show that the transformation $SR$ (equivalent to $R$ followed by $S$) is a rotation about the point represented by $c$, where $$c\sin \frac{1}{2}(\theta +\varphi )=a{\mathrm{e}}^{\mathrm{i}\varphi /2}\sin \frac{1}{2}\theta +b{\mathrm{e}}^{-\mathrm{i}\theta /2}\sin \frac{1}{2}\varphi ,$$ provided $\theta +\varphi \ne 2n\pi$ for any integer $n$. What is the transformation $SR$ if $\theta +\varphi =2\pi$?
3. Under what circumstances is $RS=SR$?

Let $\alpha$, $\beta$, $\gamma$ and $\delta$ be the roots of the quartic equation $$x^4+p{x}^3+q{x}^2+rx+s=0.$$You are given that, for any such equation, $\alpha \beta +\gamma \delta$, $\alpha \gamma +\beta \delta$ and $\alpha \delta +\beta \gamma$ satisfy a cubic equation of the form $$y^3+A{y}^2+(pr-4s)y+(4qs-p^{2}s-r^2)=0.$$ Determine $A$.

Now consider the quartic equation given by $p=0$, $q=3$, $r=-6$ and $s=10$.

1. Find the value of $\alpha \beta +\gamma \delta$, given that it is the largest root of the corresponding cubic equation.
2. Hence, using the values of $q$ and $s$, find the value of $(\alpha +\beta )(\gamma +\delta )$ and the value of $\alpha \beta$ given that $\alpha \beta >\gamma \delta$.
3. Using these results, and the values of $p$ and $r$, solve the quartic equation.

For any function $\mathrm{f}$ satisfying $\mathrm{f}(x)>0$, we define the geometric mean, F, by $$\mathrm{F}(y)=\text{ e}\text{ y1∫0ylnf(x)dx }(y>0).$$

1. The function f satisfies $\mathrm{f}(x)>0$ and $a$ is a positive number with $a\ne 1$. Prove that $$\mathrm{F}(y)=\text{ a}\text{ y1∫0ylogaf(x)dx }.$$
2. The functions f and g satisfy $\mathrm{f}(x)>0$ and $\mathrm{g}(x)>0$, and the function $\mathrm{h}$ is defined by $\mathrm{h}(x)=\mathrm{f}(x)\mathrm{g}(x)$. Their geometric means are F, G and H, respectively. Show that $\mathrm{H}(y)=\mathrm{F}(y)\mathrm{G}(y)$.
3. Prove that, for any positive number $b$, the geometric mean of $b^x$ is $\sqrt{b^y}$.
4. Prove that, if $\mathrm{f}(x)>0$ and the geometric mean of $\mathrm{f}(x)$ is $\sqrt{\mathrm{f}(y)}$, then $\mathrm{f}(x)=b^x$ for some positive number $b$.

The point with cartesian coordinates $(x,y)$ lies on a curve with polar equation $r=\mathrm{f}(\theta )$. Find an expression for $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $\mathrm{f}(\theta )$, ${\mathrm{f}}^{\prime }(\theta )$ and $\tan \theta$.

Two curves, with polar equations $r=\mathrm{f}(\theta )$ and $r=\mathrm{g}(\theta )$, meet at right angles. Show that where they meet $${\mathrm{f}}^{\prime }(\theta ){\mathrm{g}}^{\prime }(\theta )+\mathrm{f}(\theta )\mathrm{g}(\theta )=0.$$The curve $C$ has polar equation $r=\mathrm{f}(\theta )$ and passes through the point given by $r=4$, $\theta =-\frac{1}{2}\pi$. For each positive value of $a$, the curve with polar equation $r=a(1+\sin \theta )$ meets $C$ at right angles. Find $\mathrm{f}(\theta )$.

Sketch on a single diagram the three curves with polar equations $r=1+\sin \theta$, $r=4(1+\sin \theta )$ and $r=\mathrm{f}(\theta )$.

In this question, you are not permitted to use any properties of trigonometric functions or inverse trigonometric functions.

The function $\mathrm{T}$ is defined for $x>0$ by $$\mathrm{T}(x)={\int }_0^x\frac{1}{1+u^2}\mathrm{d}u,$$ and $T_{\infty }={\int }_0^{\infty }\frac{1}{1+u^2}\mathrm{d}u$ (which has a finite value).

1. By making an appropriate substitution in the integral for $\mathrm{T}(x)$, show that $$\mathrm{T}(x)={\mathrm{T}}_{\infty }-\mathrm{T}(x^{-1}).$$
2. Let $v=\frac{u+a}{1-au}$, where $a$ is a constant. Verify that, for $u\ne a^{-1}$, $$\frac{\mathrm{d}v}{\mathrm{d}u}=\frac{1+v^2}{1+u^2}.$$ Hence show that, for $a>0$ and $x<\frac{1}{a}$, $$\mathrm{T}(x)=\mathrm{T}\left(\frac{x+a}{1-ax}\right)-\mathrm{T}(a).$$ Deduce that $$\mathrm{T}(x^{-1})=2{\mathrm{T}}_{\infty }-\mathrm{T}\left(\frac{x+a}{1-ax}\right)-\mathrm{T}(a^{-1})$$ and hence that, for $b>0$ and $y>\frac{1}{b}$, $$\mathrm{T}(y)=2{\mathrm{T}}_{\infty }-\mathrm{T}\left(\frac{y+b}{by-1}\right)-\mathrm{T}(b).$$
3. Use the above results to show that $\mathrm{T}(\sqrt{3})=\frac{2}{3}{\mathrm{T}}_{\infty }$ and $\mathrm{T}(\sqrt{2}-1)=\frac{1}{4}{\mathrm{T}}_{\infty }$.

Show that the point $T$ with coordinates $$\left(\frac{a(1-t^2)}{1+t^2},\frac{2bt}{1+t^2}\right)\text{\hspace{1em}(*)}$$(where $a$ and $b$ are non-zero) lies on the ellipse $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1.$$

1. The line $L$ is the tangent to the ellipse at $T$. The point $(X,Y)$ lies on $L$, and $X^2\ne a^2$. Show that $$(a+X)b{t}^2-2aYt+b(a-X)=0.$$ Deduce that if $a^2{Y}^2>(a^2-X^2)b^2$, then there are two distinct lines through $(X,Y)$ that are tangents to the ellipse. Interpret this result geometrically. Show, by means of a sketch, that the result holds also if $X^2=a^2$.
2. The distinct points $P$ and $Q$ are given by $(\ast )$, with $t=p$ and $t=q$, respectively. The tangents to the ellipse at $P$ and $Q$ meet at the point with coordinates $(X,Y)$, where $X^2\ne a^2$. Show that $$(a+X)pq=a-X$$ and find an expression for $p+q$ in terms of $a$, $b$, $X$ and $Y$. Given that the tangents meet the $y$-axis at points $(0,y_1)$ and $(0,y_2)$, where $y_1+y_2=2b$, show that $$\frac{X^2}{a^2}+\frac{Y}{b}=1.$$

Prove that, for any numbers $a_1$, $a_2$, $\dots$, and $b_1$, $b_2$, $\dots$, and for $n⩾1$, $$\sum _{m=1}^n{a}_m(b_{m+1}-b_m)=a_{n+1}{b}_{n+1}-a_1{b}_1-\sum _{m=1}^n{b}_{m+1}(a_{m+1}-a_m).$$

1. By setting $b_m=\sin mx$, show that $$\sum _{m=1}^n\cos (m+\frac{1}{2})x=\frac{1}{2}(\sin (n+1)x-\sin x)\mathrm{cosec}\frac{1}{2}x.$$ Note: $\sin A-\sin B=2\cos \left(\frac{A+B}{2}\right)\sin \left(\frac{A-B}{2}\right)$.
2. Show that $$\sum _{m=1}^{n}m\sin mx=(p\sin (n+1)x+q\sin nx){\mathrm{cosec}}^2\frac{1}{2}x,$$ where $p$ and $q$ are to be determined in terms of $n$. Note: $2\sin A\sin B=\cos (A-B)-\cos (A+B)$; $2\cos A\sin B=\sin (A+B)-\sin (A-B)$.

Two particles $A$ and $B$ of masses $m$ and $2m$, respectively, are connected by a light spring of natural length $a$ and modulus of elasticity $\lambda$. They are placed on a smooth horizontal table with $AB$ perpendicular to the edge of the table, and $A$ is held on the edge of the table. Initially the spring is at its natural length.

Particle $A$ is released. At a time $t$ later, particle $A$ has dropped a distance $y$ and particle $B$ has moved a distance $x$ from its initial position (where $x<a$). Show that $y+2x=\frac{1}{2}g{t}^2$.

The value of $\lambda$ is such that particle $B$ reaches the edge of the table at a time $T$ given by $T=\sqrt{6a/g}$. By considering the total energy of the system (without solving any differential equations), show that the speed of particle $B$ at this time is $\sqrt{2ag/3}$.

A uniform rod $PQ$ of mass $m$ and length $3a$ is freely hinged at $P$.

The rod is held horizontally and a particle of mass $m$ is placed on top of the rod at a distance $\ell$ from $P$, where $\ell <2a$. The coefficient of friction between the rod and the particle is $\mu$.

The rod is then released. Show that, while the particle does not slip along the rod, $$(3a^2+{\ell }^2){\dot{\theta }}^2=g(3a+2\ell )\sin \theta ,$$ where $\theta$ is the angle through which the rod has turned, and the dot denotes the time derivative.

Hence, or otherwise, find an expression for $\ddot{\theta }$ and show that the normal reaction of the rod on the particle is non-zero when $\theta$ is acute.

Show further that, when the particle is on the point of slipping, $$\tan \theta =\frac{\mu a(2a-\ell )}{2({\ell }^2+a\ell +a^2)}.$$What happens at the moment the rod is released if, instead, $\ell >2a$?

A railway truck, initially at rest, can move forwards without friction on a long straight horizontal track. On the truck, $n$ guns are mounted parallel to the track and facing backwards, where $n>1$. Each of the guns is loaded with a single projectile of mass $m$. The mass of the truck and guns (but not including the projectiles) is $M$.

When a gun is fired, the projectile leaves its muzzle horizontally with a speed $v-V$ relative to the ground, where $V$ is the speed of the truck immediately before the gun is fired.

1. All $n$ guns are fired simultaneously. Find the speed, $u$, with which the truck moves, and show that the kinetic energy, $K$, which is gained by the system (truck, guns and projectiles) is given by $$K=\frac{1}{2}nm{v}^2\left(1+\frac{nm}{M}\right).$$
2. Instead, the guns are fired one at a time. Let $u_r$ be the speed of the truck when $r$ guns have been fired, so that $u_0=0$. Show that, for $1⩽r⩽n$, $$u_r-u_{r-1}=\frac{mv}{M+(n-r)m}\text{\hspace{1em}(*)}$$ and hence that $u_n<u$.
3. Let $K_r$ be the total kinetic energy of the system when $r$ guns have been fired (one at a time), so that $K_0=0$. Using $(\ast )$, show that, for $1⩽r⩽n$, $$K_r-K_{r-1}=\frac{1}{2}m{v}^2+\frac{1}{2}mv(u_r-u_{r-1})$$ and hence show that $$K_n=\frac{1}{2}nm{v}^2+\frac{1}{2}mv{u}_n.$$ Deduce that $K_n<K$.

The discrete random variables $X$ and $Y$ can each take the values $1$, $\dots$, $n$ (where $n⩾2$). Their joint probability distribution is given by $$\mathrm{P}(X=x,Y=y)=k(x+y),$$ where $k$ is a constant.

1. Show that $$\mathrm{P}(X=x)=\frac{n+1+2x}{2n(n+1)}.$$ Hence determine whether $X$ and $Y$ are independent.
2. Show that the covariance of $X$ and $Y$ is negative.

The random variable $X$ has mean $\mu$ and variance ${\sigma }^2$, and the function $\mathrm{V}$ is defined, for $-\infty <x<\infty$, by $$\mathrm{V}(x)=\mathrm{E}((X-x{)}^2).$$ Express $\mathrm{V}(x)$ in terms of $x$, $\mu$ and $\sigma$.

The random variable $Y$ is defined by $Y=\mathrm{V}(X)$. Show that $$\mathrm{E}(Y)=2{\sigma }^2.\text{\hspace{1em}(*)}$$Now suppose that $X$ is uniformly distributed on the interval $0⩽x⩽1$. Find $\mathrm{V}(x)$. Find also the probability density function of $Y$ and use it to verify that $(\ast )$ holds in this case.