STEP 2 2010

Let $P$ be a given point on a given curve $C$. The osculating circle to $C$ at $P$ is defined to be the circle that satisfies the following two conditions at $P$: it touches $C$; and the rate of change of its gradient is equal to the rate of change of the gradient of $C$.

Find the centre and radius of the osculating circle to the curve $y=1-x+\tan x$ at the point on the curve with $x$-coordinate $\frac{1}{4}\pi$.

Prove that $$\cos 3x=4{\cos }^{3}x-3\cos x.$$ Find and prove a similar result for $\sin 3x$ in terms of $\sin x$.

1. Let $$\mathrm{I}(\alpha )={\int }_0^{\alpha }(7\sin x-8{\sin }^{3}x)\mathrm{d}x.$$ Show that $$\mathrm{I}(\alpha )=-\frac{8}{3}{c}^3+c+\frac{5}{3},$$ where $c=\cos \alpha$. Write down one value of $c$ for which $\mathrm{I}(\alpha )=0$.
2. Useless Eustace believes that $$\int {\sin }^{n}x\mathrm{d}x=\frac{{\sin }^{n+1}x}{n+1}$$ for $n=1,2,3,\dots$. Show that Eustace would obtain the correct value of $\mathrm{I}(\beta )$, where $\cos \beta =-\frac{1}{6}$. Find all values of $\alpha$ for which he would obtain the correct value of $\mathrm{I}(\alpha )$.

The first four terms of a sequence are given by $F_0=0$, $F_1=1$, $F_2=1$ and $F_3=2$. The general term is given by $$F_n=a{\lambda }^n+b{\mu }^n,\text{\hspace{1em}(*)}$$ where $a$, $b$, $\lambda$ and $\mu$ are independent of $n$, and $a$ is positive.

1. Show that ${\lambda }^2+\lambda \mu +{\mu }^2=2$, and find the values of $\lambda$, $\mu$, $a$ and $b$.
2. Use $(\ast )$ to evaluate $F_6$.
3. Evaluate $\sum _{n=0}^{\infty }\frac{F_n}{2^{n+1}}.$

The points $A$ and $B$ have position vectors $\mathbf{i}+\mathbf{j}+\mathbf{k}$ and $5[[B]]i[[/B]]-[[B]]j[[/B]]-[[B]]k[[/B]]$, respectively, relative to the origin $O$. Find $\cos 2\alpha$, where $2\alpha$ is the angle $\angle AOB$.

1. The line $L_1$ has equation $<strong>r</strong>=\lambda (m<strong>i</strong>+n<strong>j</strong>+p<strong>k</strong>)$. Given that $L_1$ is inclined equally to $OA$ and to $OB$, determine a relationship between $m$, $n$ and $p$. Find also values of $m$, $n$ and $p$ for which $L_1$ is the angle bisector of $\angle AOB$.
2. The line $L_2$ has equation $<strong>r</strong>=\mu (u<strong>i</strong>+v<strong>j</strong>+w<strong>k</strong>)$. Given that $L_2$ is inclined at an angle $\alpha$ to $OA$, where $2\alpha =\angle AOB$, determine a relationship between $u$, $v$ and $w$. Hence describe the surface with Cartesian equation $x^2+y^2+z^2=2(yz+zx+xy)$.

Each edge of the tetrahedron $ABCD$ has unit length. The face $ABC$ is horizontal, and $P$ is the point in $ABC$ that is vertically below $D$.

1. Find the length of $PD$.
2. Show that the cosine of the angle between adjacent faces of the tetrahedron is $1/3$.
3. Find the radius of the largest sphere that can fit inside the tetrahedron.

The curves $C_1$ and $C_2$ are defined by $$y={\mathrm{e}}^{-x}(x>0)\text{ and }y={\mathrm{e}}^{-x}\sin x(x>0),$$ respectively. Sketch roughly $C_1$ and $C_2$ on the same diagram.

Let $x_n$ denote the $x$-coordinate of the $n$th point of contact between the two curves, where $0<x_1<x_2<\cdots$, and let $A_n$ denote the area of the region enclosed by the two curves between $x_n$ and $x_{n+1}$. Show that $$A_n=\frac{1}{2}({\mathrm{e}}^{2\pi }-1){\mathrm{e}}^{-(4n+1)\pi /2}$$ and hence find $\sum _{n=1}^{\infty }{A}_n$.

Two points $A$ and $B$ lie on horizontal ground. A particle $P_1$ is projected from $A$ towards $B$ at an acute angle of elevation $\alpha$ and simultaneously a particle $P_2$ is projected from $B$ towards $A$ at an acute angle of elevation $\beta$. Given that the two particles collide in the air a horizontal distance $b$ from $B$, and that the collision occurs after $P_1$ has attained its maximum height $h$, show that $$2h\cot \beta <b<4h\cot \beta$$ and $$2h\cot \alpha <a<4h\cot \alpha ,$$ where $a$ is the horizontal distance from $A$ to the point of collision.

A uniform rod $AB$ of length $4L$ and weight $W$ is inclined at an angle $\theta$ to the horizontal. Its lower end $A$ rests on a fixed support and the rod is held in equilibrium by a string attached to the rod at a point $C$ which is $3L$ from $A$. The reaction of the support on the rod acts in a direction $\alpha$ to $AC$ and the string is inclined at an angle $\beta$ to $CA$. Show that $$\cot \alpha =3\tan \theta +2\cot \beta .$$Given that $\theta =30^{\circ }$ and $\beta =45^{\circ }$, show that $\alpha =15^{\circ }$.

The continuous random variable $X$ has probability density function $\mathrm{f}(x)$, where $$\mathrm{f}(x)=\{\begin{array}{ll}a& \text{for }0⩽x<k\\ b& \text{for }k⩽x⩽1\\ 0& \text{otherwise},\end{array}$$ where $a>b>0$ and $0<k<1$. Show that $a>1$ and $b<1$.

1. Show that $$\mathrm{E}(X)=\frac{1-2b+ab}{2(a-b)}.$$
2. Show that the median, $M$, of $X$ is given by $M=\frac{1}{2a}$ if $a+b⩾2ab$ and obtain an expression for the median if $a+b⩽2ab$.
3. Show that $M<\mathrm{E}(X)$.

Rosalind wants to join the Stepney Chess Club. In order to be accepted, she must play a challenge match consisting of several games against Pardeep (the Club champion) and Quentin (the Club secretary), in which she must win at least one game against each of Pardeep and Quentin. From past experience, she knows that the probability of her winning a single game against Pardeep is $p$ and the probability of her winning a single game against Quentin is $q$, where $0<p<q<1$.

1. The challenge match consists of three games. Before the match begins, Rosalind must choose either to play Pardeep twice and Quentin once or to play Quentin twice and Pardeep once. Show that she should choose to play Pardeep twice.
2. In order to ease the entry requirements, it is decided instead that the challenge match will consist of four games. Now, before the match begins, Rosalind must choose whether to play Pardeep three times and Quentin once (strategy 1), or to play Pardeep twice and Quentin twice (strategy 2) or to play Pardeep once and Quentin three times (strategy 3). Show that, if $q-p>\frac{1}{2}$, Rosalind should choose strategy 1. If $q-p<\frac{1}{2}$, give examples of values of $p$ and $q$ to show that strategy 2 can be better or worse than strategy 1.