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120

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OCR (Cambridge University Press & Assessment) · United Kingdom

STEP 3 2015

2015

13 questions worth 120 marks in total. Each part is annotated: check the hint if you are stuck, or reveal the worked solution once you have committed to an approach.

Let $I_{n} = \int_{0}^{\infty} \frac{1}{( 1 + u ^{2} ) ^{n}} d u,$ where $n$ is a positive integer. Show that $I_{n} - I_{n + 1} = \frac{1}{2 n} I_{n}$ and deduce that $I_{n + 1} = \frac{( 2 n )! π}{2 ^{2 n + 1} ( n ! ) ^{2}} .$
Let $J = \int_{0}^{\infty} f ((x - x^{- 1})^{2}) d x,$ where $f$ is any function for which the integral exists. Show that $J = \int_{0}^{\infty} x^{- 2} f ((x - x^{- 1})^{2}) d x = \frac{1}{2} \int_{0}^{\infty} (1 + x^{- 2}) f ((x - x^{- 1})^{2}) d x = \int_{0}^{\infty} f (u^{2}) d u .$
Hence evaluate $\int_{0}^{\infty} \frac{x ^{2 n - 2}}{( x ^{4} - x ^{2} + 1 ) ^{n}} d x,$ where $n$ is a positive integer.
Integration
If $s_{1}$ , $s_{2}$ , $s_{3}$ , $\dots$ and $t_{1}$ , $t_{2}$ , $t_{3}$ , $\dots$ are sequences of positive numbers, we write $(s_{n}) ⩽ (t_{n})$ to mean

"there exists a positive integer $m$ such that $s_{n} ⩽ t_{n}$ whenever $n ⩾ m$ ".

Determine whether each of the following statements is true or false. In the case of a true statement, you should give a proof which includes an explicit determination of an appropriate $m$ ; in the case of a false statement, you should give a counterexample.
$(1000 n) ⩽ (n^{2})$ .
If it is not the case that $(s_{n}) ⩽ (t_{n})$ , then it is the case that $(t_{n}) ⩽ (s_{n})$ .
If $(s_{n}) ⩽ (t_{n})$ and $(t_{n}) ⩽ (u_{n})$ , then $(s_{n}) ⩽ (u_{n})$ .
$(n^{2}) ⩽ (2^{n})$ .
Sequences & Series
In this question, $r$ and $θ$ are polar coordinates with $r ⩾ 0$ and $- π < θ ⩽ π$ , and $a$ and $b$ are positive constants.

Let $L$ be a fixed line and let $A$ be a fixed point not lying on $L$ . Then the locus of points that are a fixed distance (call it $d$ ) from $L$ measured along lines through $A$ is called a conchoid of Nicomedes.
Show that if $∣ r - a sec θ ∣ = b, (*)$ where $a > b$ , then $sec θ > 0$ . Show that all points with coordinates satisfying ( $*$ ) lie on a certain conchoid of Nicomedes (you should identify $L$ , $d$ and $A$ ). Sketch the locus of these points.
In the case $a < b$ , sketch the curve (including the loop for which $sec θ < 0$ ) given by $∣ r - a sec θ ∣ = b .$ Find the area of the loop in the case $a = 1$ and $b = 2$ . [Note: $\int sec θ d θ = ln ∣ sec θ + tan θ ∣ + C .$ ]
Integration
If $a$ , $b$ and $c$ are all real, show that the equation $z^{3} + a z^{2} + b z + c = 0 (*)$ has at least one real root.
Let $S_{1} = z_{1} + z_{2} + z_{3}, S_{2} = z_{1}^{2} + z_{2}^{2} + z_{3}^{2}, S_{3} = z_{1}^{3} + z_{2}^{3} + z_{3}^{3},$ where $z_{1}$ , $z_{2}$ and $z_{3}$ are the roots of the equation $(*)$ . Express $a$ and $b$ in terms of $S_{1}$ and $S_{2}$ , and show that $6 c = - S_{1}^{3} + 3 S_{1} S_{2} - 2 S_{3} .$
The six real numbers $r_{k}$ and $θ_{k}$ ( $k = 1, 2, 3$ ), where $r_{k} > 0$ and $- π < θ_{k} < π$ , satisfy $k = 1 \sum 3 r_{k} sin (θ_{k}) = 0, k = 1 \sum 3 r_{k}^{2} sin (2 θ_{k}) = 0, k = 1 \sum 3 r_{k}^{3} sin (3 θ_{k}) = 0 .$ Show that $θ_{k} = 0$ for at least one value of $k$ . Show further that if $θ_{1} = 0$ then $θ_{2} = - θ_{3}$ .
Trigonometry
In the following argument to show that $2$ is irrational, give proofs appropriate for steps 3, 5 and 6. 1. Assume that $2$ is rational. 2. Define the set $S$ to be the set of positive integers with the following property: $n$ is in $S$ if and only if $n 2$ is an integer. 3. Show that the set $S$ contains at least one positive integer. 4. Define the integer $k$ to be the smallest positive integer in $S$ . 5. Show that $(2 - 1) k$ is in $S$ . 6. Show that steps 4 and 5 are contradictory and hence that $2$ is irrational.
Prove that $2^{\frac{1}{3}}$ is rational if and only if $2^{\frac{2}{3}}$ is rational. Use an argument similar to that of part (i) to prove that $2^{\frac{1}{3}}$ and $2^{\frac{2}{3}}$ are irrational.
Proof & Number Theory
Let $w$ and $z$ be complex numbers, and let $u = w + z$ and $v = w^{2} + z^{2}$ . Prove that $w$ and $z$ are real if and only if $u$ and $v$ are real and $u^{2} ⩽ 2 v$ .
The complex numbers $u$ , $w$ and $z$ satisfy the equations $w + z - u w^{2} + z^{2} - u^{2} w^{3} + z^{3} - λ u = 0 = - \frac{2}{3} = - λ$ where $λ$ is a positive real number. Show that for all values of $λ$ except one (which you should find) there are three possible values of $u$ , all real. Are $w$ and $z$ necessarily real? Give a proof or counterexample.
Complex Numbers
An operator $D$ is defined, for any function $f$ , by $D f (x) = x \frac{d f ( x )}{d x} .$ The notation $D^{n}$ means that $D$ is applied $n$ times; for example $D^{2} f (x) = x \frac{d}{d x} (x \frac{d f ( x )}{d x}) .$ Show that, for any constant $a$ , $D^{2} x^{a} = a^{2} x^{a}$ .
Show that if $P (x)$ is a polynomial of degree $r$ (where $r ⩾ 1$ ) then, for any positive integer $n$ , $D^{n} P (x)$ is also a polynomial of degree $r$ .
Show that if $n$ and $m$ are positive integers with $n < m$ , then $D^{n} (1 - x)^{m}$ is divisible by $(1 - x)^{m - n}$ .
Deduce that, if $m$ and $n$ are positive integers with $n < m$ , then $r = 0 \sum m (- 1)^{r} (r m) r^{n} = 0 .$
Proof & Number Theory
Show that under the changes of variable $x = r cos θ$ and $y = r sin θ$ , where $r$ is a function of $θ$ with $r > 0$ , the differential equation $(y + x) \frac{d y}{d x} = y - x$ becomes $\frac{d r}{d θ} + r = 0 .$ Sketch a solution in the $x$ - $y$ plane.
Show that the solutions of $(y + x - x (x^{2} + y^{2})) \frac{d y}{d x} = y - x - y (x^{2} + y^{2})$ can be written in the form $r^{2} = \frac{1}{1 + A e ^{2 θ}}$ and sketch the different forms of solution that arise according to the value of $A$ .
Differential Equations
A particle $P$ of mass $m$ moves on a smooth fixed straight horizontal rail and is attached to a fixed peg $Q$ by a light elastic string of natural length $a$ and modulus $λ$ . The peg $Q$ is a distance $a$ from the rail. Initially $P$ is at rest with $P Q = a$ .

An impulse imparts to $P$ a speed $v$ along the rail. Let $x$ be the displacement at time $t$ of $P$ from its initial position. Obtain the equation $\overset{x}{˙}^{2} = v^{2} - k^{2} (x^{2} + a^{2} - a)^{2}$ where $k^{2} = λ / (ma)$ , $k > 0$ and the dot denotes differentiation with respect to $t$ .

Find, in terms of $k$ , $a$ and $v$ , the greatest value, $x_{0}$ , attained by $x$ . Find also the acceleration of $P$ at $x = x_{0}$ .

Obtain, in the form of an integral, an expression for the period of the motion. Show that in the case $v ≪ k a$ (that is, $v$ is much less than $k a$ ), this is approximately $\frac{32 a}{k v} \int_{0}^{1} \frac{1}{1 - u ^{4}} d u .$
Mechanics
A light rod of length $2 a$ has a particle of mass $m$ attached to each end and it moves in a vertical plane. The midpoint of the rod has coordinates $(x, y)$ , where the $x$ -axis is horizontal (within the plane of motion) and $y$ is the height above a horizontal table. Initially, the rod is vertical, and at time $t$ later it is inclined at an angle $θ$ to the vertical.

Show that the velocity of one particle can be written in the form $(\overset{x}{˙} + a \dot{θ} cos θ \overset{y}{˙} - a \dot{θ} sin θ)$ and that $m (\overset{x}{¨} + a \ddot{θ} cos θ - a \dot{θ}^{2} sin θ \overset{y}{¨} - a \ddot{θ} sin θ - a \dot{θ}^{2} cos θ) = - T (sin θ cos θ) - m g (01)$ where the dots denote differentiation with respect to time $t$ and $T$ is the tension in the rod. Obtain the corresponding equations for the other particle.

Deduce that $\overset{x}{¨} = 0$ , $\overset{y}{¨} = - g$ and $\ddot{θ} = 0$ .

Initially, the midpoint of the rod is a height $h$ above the table, the velocity of the higher particle is $(u v)$ , and the velocity of the lower particle is $(0 v)$ . Given that the two particles hit the table for the first time simultaneously, when the rod has rotated by $\frac{1}{2} π$ , show that $2 h u^{2} = π^{2} a^{2} g - 2 π uv a .$
Mechanics
A horizontal disc of radius $r$ rotates about a vertical axis through its centre with angular speed $ω$ . One end of a light rod is fixed by a smooth hinge to the edge of the disc so that it can rotate freely in a vertical plane through the centre of the disc. A particle $P$ of mass $m$ is attached to the rod at a distance $d$ from the hinge. The rod makes a constant angle $α$ with the upward vertical, as shown in the diagram, and $d sin α < r$ . By considering moments about the hinge for the (light) rod, show that the force exerted on the rod by $P$ is parallel to the rod. Show also that $r cot α = a + d cos α,$ where $a = \frac{g}{ω ^{2}}$ . State clearly the direction of the force exerted by the hinge on the rod, and find an expression for its magnitude in terms of $m$ , $g$ and $α$ .
The disc and rod rotate as in part (i), but two particles (instead of $P$ ) are attached to the rod. The masses of the particles are $m_{1}$ and $m_{2}$ and they are attached to the rod at distances $d_{1}$ and $d_{2}$ from the hinge, respectively. The rod makes a constant angle $β$ with the upward vertical and $d_{1} sin β < d_{2} sin β < r$ . Show that $β$ satisfies an equation of the form $r cot β = a + b cos β,$ where $b$ should be expressed in terms of $d_{1}$ , $d_{2}$ , $m_{1}$ and $m_{2}$ .
Mechanics
A 6-sided fair die has the numbers 1, 2, 3, 4, 5, 6 on its faces. The die is thrown $n$ times, the outcome (the number on the top face) of each throw being independent of the outcome of any other throw. The random variable $S_{n}$ is the sum of the outcomes.
The random variable $R_{n}$ is the remainder when $S_{n}$ is divided by 6. Write down the probability generating function, $G (x)$ , of $R_{1}$ and show that the probability generating function of $R_{2}$ is also $G (x)$ . Use a generating function to find the probability that $S_{n}$ is divisible by 6.
The random variable $T_{n}$ is the remainder when $S_{n}$ is divided by 5. Write down the probability generating function, $G_{1} (x)$ , of $T_{1}$ and show that $G_{2} (x)$ , the probability generating function of $T_{2}$ , is given by $G_{2} (x) = \frac{1}{36} (x^{2} + 7 y)$ where $y = 1 + x + x^{2} + x^{3} + x^{4}$ . Obtain the probability generating function of $T_{n}$ and hence show that the probability that $S_{n}$ is divisible by $5$ is $\frac{1}{5} (1 - \frac{1}{6 ^{n}})$ if $n$ is not divisible by 5. What is the corresponding probability if $n$ is divisible by 5?
Probability & Statistics
Each of the two independent random variables $X$ and $Y$ is uniformly distributed on the interval $[0, 1]$ .
By considering the lines $x + y =$ $constant$ in the $x$ - $y$ plane, find the cumulative distribution function of $X + Y$ .
Hence show that the probability density function $f$ of $(X + Y)^{- 1}$ is given by $f (t) = ⎩ ⎨ ⎧ 2 t^{- 2} - t^{- 3} t^{- 3} 0 for \frac{1}{2} ⩽ t ⩽ 1 for 1 ⩽ t < \infty otherwise .$ Evaluate $E (\frac{1}{X + Y})$ .
Find the cumulative distribution function of $Y / X$ and use this result to find the probability density function of $\frac{X}{X + Y}$ . Write down $E (\frac{X}{X + Y})$ and verify your result by integration.
Probability & Statistics