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Questions

Worked solutions

100%

120

Marks total

Chapters covered

OCR (Cambridge University Press & Assessment) · United Kingdom

STEP 3 2009

2009

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.

The points $S$ , $T$ , $U$ and $V$ have coordinates $(s, m s)$ , $(t, m t)$ , $(u, n u)$ and $(v, n v)$ , respectively. The lines $S V$ and $U T$ meet the line $y = 0$ at the points with coordinates $(p, 0)$ and $(q, 0)$ , respectively. Show that $p = \frac{( m - n ) s v}{m s - n v},$ and write down a similar expression for $q$ .

Given that $S$ and $T$ lie on the circle $x^{2} + (y - c)^{2} = r^{2}$ , find a quadratic equation satisfied by $s$ and by $t$ , and hence determine $s t$ and $s + t$ in terms of $m$ , $c$ and $r$ .

Given that $S$ , $T$ , $U$ and $V$ lie on the above circle, show that $p + q = 0$ .
Coordinate Geometry & Conics
Let $y = n = 0 \sum \infty a_{n} x^{n}$ , where the coefficients $a_{n}$ are independent of $x$ and are such that this series and all others in this question converge. Show that $y^{'} = n = 1 \sum \infty n a_{n} x^{n - 1},$ and write down a similar expression for $y^{''}$ . Write out explicitly each of the three series as far as the term containing $a_{3}$ .
It is given that $y$ satisfies the differential equation $x y^{''} - y^{'} + 4 x^{3} y = 0 .$ By substituting the series of part (i) into the differential equation and comparing coefficients, show that $a_{1} = 0$ . Show that, for $n ⩾ 4$ , $a_{n} = - \frac{4}{n ( n - 2 )} a_{n - 4},$ and that, if $a_{0} = 1$ and $a_{2} = 0$ , then $y = cos (x^{2})$ . Find the corresponding result when $a_{0} = 0$ and $a_{2} = 1$ .
Differential Equations
The function $f (t)$ is defined, for $t \neq = 0$ , by $f (t) = \frac{t}{e ^{t} - 1} .$
By expanding $e^{t}$ , show that $t \to 0 lim f (t) = 1$ . Find $f^{'} (t)$ and evaluate $t \to 0 lim f^{'} (t)$ .
Show that $f (t) + \frac{1}{2} t$ is an even function. [Note: A function $g (t)$ is said to be even if $g (t) \equiv g (- t)$ .]
Show with the aid of a sketch that $e^{t} (1 - t) ⩽ 1$ and deduce that $f^{'} (t) \neq = 0$ for $t \neq = 0$ .
Sketch the graph of $f (t)$ .
Functions & Curve Sketching
For any given (suitable) function $f$ , the Laplace transform of $f$ is the function $F$ defined by $F (s) = \int_{0}^{\infty} e^{- s t} f (t) d t (s > 0) .$
Show that the Laplace transform of $e^{- b t} f (t)$ , where $b > 0$ , is $F (s + b)$ .
Show that the Laplace transform of $f (a t)$ , where $a > 0$ , is $a^{- 1} F (\frac{s}{a})$ .
Show that the Laplace transform of $f^{'} (t)$ is $s F (s) - f (0)$ .
In the case $f (t) = sin t$ , show that $F (s) = \frac{1}{s ^{2} + 1}$ .
Using only these four results, find the Laplace transform of $e^{- pt} cos q t$ , where $p > 0$ and $q > 0$ .
Differential Equations
The numbers $x$ , $y$ and $z$ satisfy $x + y + z x^{2} + y^{2} + z^{2} x^{3} + y^{3} + z^{3} = 1 = 2 = 3 .$ Show that $y z + z x + x y = - \frac{1}{2} .$ Show also that $x^{2} y + x^{2} z + y^{2} z + y^{2} x + z^{2} x + z^{2} y = - 1$ , and hence that $x y z = \frac{1}{6} .$ Let $S_{n} = x^{n} + y^{n} + z^{n}$ . Use the above results to find numbers $a$ , $b$ and $c$ such that the relation $S_{n + 1} = a S_{n} + b S_{n - 1} + c S_{n - 2},$ holds for all $n$ .
Sequences & Series
Show that $e^{i β} - e^{i α} = 2 sin \frac{1}{2} (β - α)$ for $0 < α < β < 2 π$ . Hence show that $e^{i α} - e^{i β} e^{i γ} - e^{i δ} + e^{i β} - e^{i γ} e^{i α} - e^{i δ} = e^{i α} - e^{i γ} e^{i β} - e^{i δ},$ where $0 < α < β < γ < δ < 2 π$ .

Interpret this result as a theorem about cyclic quadrilaterals.
Complex Numbers
The functions $f_{n} (x)$ are defined for $n = 0$ , $1$ , $2$ , $\dots$ , by $f_{0} (x) = \frac{1}{1 + x ^{2}} and f_{n + 1} (x) = \frac{d f _{n} ( x )}{d x} .$ Prove, for $n ⩾ 1$ , that $(1 + x^{2}) f_{n + 1} (x) + 2 (n + 1) x f_{n} (x) + n (n + 1) f_{n - 1} (x) = 0 .$
The functions $P_{n} (x)$ are defined for $n = 0$ , $1$ , $2$ , $\dots$ , by $P_{n} (x) = (1 + x^{2})^{n + 1} f_{n} (x) .$ Find expressions for $P_{0} (x)$ , $P_{1} (x)$ and $P_{2} (x)$ . Prove, for $n ⩾ 0$ , that $P_{n + 1} (x) - (1 + x^{2}) \frac{d P _{n} ( x )}{d x} + 2 (n + 1) x P_{n} (x) = 0,$ and that $P_{n} (x)$ is a polynomial of degree $n$ .
Differential Equations
Let $m$ be a positive integer and let $n$ be a non-negative integer.
Use the result $t \to \infty lim e^{- m t} t^{n} = 0$ to show that $x \to 0 lim x^{m} (ln x)^{n} = 0 .$ By writing $x^{x}$ as $e^{x l n x}$ show that $x \to 0 lim x^{x} = 1 .$
Let $I_{n} = \int_{0}^{1} x^{m} (ln x)^{n} d x$ . Show that $I_{n + 1} = - \frac{n + 1}{m + 1} I_{n}$ and hence evaluate $I_{n}$ .
Show that $\int_{0}^{1} x^{x} d x = 1 - (\frac{1}{2})^{2} + (\frac{1}{3})^{3} - (\frac{1}{4})^{4} + \dots .$
Integration
A particle is projected under gravity from a point $P$ and passes through a point $Q$ . The angles of the trajectory with the positive horizontal direction at $P$ and at $Q$ are $θ$ and $ϕ$ , respectively. The angle of elevation of $Q$ from $P$ is $α$ .
Show that $tan θ + tan ϕ = 2 tan α$ .
It is given that there is a second trajectory from $P$ to $Q$ with the same speed of projection. The angles of this trajectory with the positive horizontal direction at $P$ and at $Q$ are $θ^{'}$ and $ϕ^{'}$ , respectively. By considering a quadratic equation satisfied by $tan θ$ , show that $tan (θ + θ^{'}) = - cot α$ . Show also that $θ + θ^{'} = π + ϕ + ϕ^{'}$ .
Mechanics
A light spring is fixed at its lower end and its axis is vertical. When a certain particle $P$ rests on the top of the spring, the compression is $d$ . When, instead, $P$ is dropped onto the top of the spring from a height $h$ above it, the compression at time $t$ after $P$ hits the top of the spring is $x$ . Obtain a second-order differential equation relating $x$ and $t$ for $0 ⩽ t ⩽ T$ , where $T$ is the time at which $P$ first loses contact with the spring.

Find the solution of this equation in the form $x = A + B cos (ω t) + C sin (ω t),$ where the constants $A$ , $B$ , $C$ and $ω$ are to be given in terms of $d$ , $g$ and $h$ as appropriate.

Show that $T = d / g (2 π - 2 arctan 2 h / d) .$
Mechanics
A comet in deep space picks up mass as it travels through a large stationary dust cloud. It is subject to a gravitational force of magnitude $M f$ acting in the direction of its motion. When it entered the cloud, the comet had mass $M$ and speed $V$ . After a time $t$ , it has travelled a distance $x$ through the cloud, its mass is $M (1 + b x)$ , where $b$ is a positive constant, and its speed is $v$ .
In the case when $f = 0$ , write down an equation relating $V$ , $x$ , $v$ and $b$ . Hence find an expression for $x$ in terms of $b$ , $V$ and $t$ .
In the case when $f$ is a non-zero constant, use Newton's second law in the form $force = rate of change of momentum$ to show that $v = \frac{f t + V}{1 + b x} .$ Hence find an expression for $x$ in terms of $b$ , $V$ , $f$ and $t$ . Show that it is possible, if $b$ , $V$ and $f$ are suitably chosen, for the comet to move with constant speed. Show also that, if the comet does not move with constant speed, its speed tends to a constant as $t \to \infty$ .
Mechanics
Albert tosses a fair coin $k$ times, where $k$ is a given positive integer. The number of heads he gets is $X_{1}$ . He then tosses the coin $X_{1}$ times, getting $X_{2}$ heads. He then tosses the coin $X_{2}$ times, getting $X_{3}$ heads. The random variables $X_{4}$ , $X_{5}$ , $\dots$ are defined similarly. Write down $E (X_{1})$ . By considering $E (X_{2} X_{1} = x_{1})$ , or otherwise, show that $E (X_{2}) = \frac{1}{4} k$ . Find $i = 1 \sum \infty E (X_{i})$ .
Bertha has $k$ fair coins. She tosses the first coin until she gets a tail. The number of heads she gets before the first tail is $Y_{1}$ . She then tosses the second coin until she gets a tail and the number of heads she gets with this coin before the first tail is $Y_{2}$ . The random variables $Y_{3}$ , $Y_{4}$ , $\dots$ , $Y_{k}$ are defined similarly, and $Y = i = 1 \sum k Y_{i}$ . Obtain the probability generating function of $Y$ , and use it to find $E (Y)$ , $Var (Y)$ and $P (Y = r)$ .
Probability & Statistics
The point $P$ lies on the circumference of a circle of unit radius and centre $O$ . The angle, $θ$ , between $O P$ and the positive $x$ -axis is a random variable, uniformly distributed on the interval $0 ⩽ θ < 2 π$ . The cartesian coordinates of $P$ with respect to $O$ are $(X, Y)$ . Find the probability density function for $X$ , and calculate $Var (X)$ . Show that $X$ and $Y$ are uncorrelated and discuss briefly whether they are independent.
The points $P_{i}$ ( $i = 1$ , $2$ , $\dots$ , $n$ ) are chosen independently on the circumference of the circle, as in part (i), and have cartesian coordinates $(X_{i}, Y_{i})$ . The point $\overline{P}$ has coordinates $(\overline{X}, \overline{Y})$ , where $\overline{X} = \frac{1}{n} i = 1 \sum n X_{i}$ and $\overline{Y} = \frac{1}{n} i = 1 \sum n Y_{i}$ . Show that $\overline{X}$ and $\overline{Y}$ are uncorrelated. Show that, for large $n$ , $P (∣ \overline{X} ∣ ⩽ \frac{2}{n}) \approx 0.95$ .
Probability & Statistics