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Questions

Worked solutions

92%

120

Marks total

Chapters covered

OCR (Cambridge University Press & Assessment) · United Kingdom

STEP 2 2013

2013

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.

Find the value of $m$ for which the line $y = m x$ touches the curve $y = ln x$ . If instead the line intersects the curve when $x = a$ and $x = b$ , where $a < b$ , show that $a^{b} = b^{a}$ . Show by means of a sketch that $a < e < b$ .
The line $y = m x + c$ , where $c > 0$ , intersects the curve $y = ln x$ when $x = p$ and $x = q$ , where $p < q$ . Show by means of a sketch, or otherwise, that $p^{q} > q^{p}$ .
Show by means of a sketch that the straight line through the points $(p, ln p)$ and $(q, ln q)$ , where $e ⩽ p < q$ , intersects the $y$ -axis at a positive value of $y$ . Which is greater, $π^{e}$ or $e^{π}$ ?
Show, using a sketch or otherwise, that if $0 < p < q$ and $\frac{ln q - ln p}{q - p} = e^{- 1}$ , then $q^{p} > p^{q}$ .
Functions & Curve Sketching
For $n ⩾ 0$ , let $I_{n} = \int_{0}^{1} x^{n} (1 - x)^{n} d x .$
For $n ⩾ 1$ , show by means of a substitution that $\int_{0}^{1} x^{n - 1} (1 - x)^{n} d x = \int_{0}^{1} x^{n} (1 - x)^{n - 1} d x$ and deduce that $2 \int_{0}^{1} x^{n - 1} (1 - x)^{n} d x = I_{n - 1} .$ Show also, for $n ⩾ 1$ , that $I_{n} = \frac{n}{n + 1} \int_{0}^{1} x^{n - 1} (1 - x)^{n + 1} d x$ and hence that $I_{n} = \frac{n}{2 ( 2 n + 1 )} I_{n - 1} .$
When $n$ is a positive integer, show that $I_{n} = \frac{( n ! ) ^{2}}{( 2 n + 1 )!} .$
Use the substitution $x = sin^{2} θ$ to show that $I_{\frac{1}{2}} = \frac{π}{8}$ , and evaluate $I_{\frac{3}{2}}$ .
Integration
Given that the cubic equation $x^{3} + 3 a x^{2} + 3 b x + c = 0$ has three distinct real roots and $c < 0$ , show with the help of sketches that either exactly one of the roots is positive or all three of the roots are positive.
Given that the equation $x^{3} + 3 a x^{2} + 3 b x + c = 0$ has three distinct real positive roots show that $a^{2} > b > 0, a < 0, c < 0 . (*)$ [Hint: Consider the turning points.]
Given that the equation $x^{3} + 3 a x^{2} + 3 b x + c = 0$ has three distinct real roots and that $ab < 0, c > 0,$ determine, with the help of sketches, the signs of the roots.
Show by means of an explicit example (giving values for $a$ , $b$ and $c$ ) that it is possible for the conditions ( $*$ ) to be satisfied even though the corresponding cubic equation has only one real root.
Functions & Curve Sketching
The line passing through the point $(a, 0)$ with gradient $b$ intersects the circle of unit radius centred at the origin at $P$ and $Q$ , and $M$ is the midpoint of the chord $P Q$ . Find the coordinates of $M$ in terms of $a$ and $b$ .
Suppose $b$ is fixed and positive. As $a$ varies, $M$ traces out a curve (the locus of $M$ ). Show that $x = - b y$ on this curve. Given that $a$ varies with $- 1 ⩽ a ⩽ 1$ , show that the locus is a line segment of length $2 b / (1 + b^{2})^{\frac{1}{2}}$ . Give a sketch showing the locus and the unit circle.
Find the locus of $M$ in the following cases, giving in each case its cartesian equation, describing it geometrically and sketching it in relation to the unit circle:
( a) $a$ is fixed with $0 < a < 1$ , and $b$ varies with $- \infty < b < \infty$ ;
( b) $ab = 1$ , and $b$ varies with $0 < b ⩽ 1$ .
Coordinate Geometry & Conics
A function $f (x)$ satisfies $f (x) = f (1 - x)$ for all $x$ . Show, by differentiating with respect to $x$ , that $f^{'} (\frac{1}{2}) = 0$ . If, in addition, $f (x) = f (\frac{1}{x})$ for all (non-zero) $x$ , show that $f^{'} (- 1) = 0$ and that $f^{'} (2) = 0$ .
The function $f$ is defined, for $x \neq = 0$ and $x \neq = 1$ , by $f (x) = \frac{( x ^{2} - x + 1 ) ^{3}}{( x ^{2} - x ) ^{2}} .$ Show that $f (x) = f (\frac{1}{x})$ and $f (x) = f (1 - x)$ . Given that it has exactly three stationary points, sketch the curve $y = f (x)$ .
Hence, or otherwise, find all the roots of the equation $f (x) = \frac{27}{4}$ and state the ranges of values of $x$ for which $f (x) > \frac{27}{4}$ . Find also all the roots of the equation $f (x) = \frac{343}{36}$ and state the ranges of values of $x$ for which $f (x) > \frac{343}{36}$ .
Functions & Curve Sketching
In this question, the following theorem may be used.

Let $u_{1}$ , $u_{2}$ , $\dots$ be a sequence of (real) numbers. If the sequence is bounded above (that is, $u_{n} ⩽ b$ for all $n$ , where $b$ is some fixed number) and increasing (that is, $u_{n} ⩾ u_{n - 1}$ for all $n$ ), then the sequence tends to a limit (that is, converges).

The sequence $u_{1}$ , $u_{2}$ , $\dots$ is defined by $u_{1} = 1$ and $u_{n + 1} = 1 + \frac{1}{u _{n}} (n ⩾ 1) . (*)$
Show that, for $n ⩾ 3$ , $u_{n + 2} - u_{n} = \frac{u _{n} - u _{n - 2}}{( 1 + u _{n} ) ( 1 + u _{n - 2} )} .$
Prove, by induction or otherwise, that $1 ⩽ u_{n} ⩽ 2$ for all $n$ .
Show that the sequence $u_{1}$ , $u_{3}$ , $u_{5}$ , $\dots$ tends to a limit, and that the sequence $u_{2}$ , $u_{4}$ , $u_{6}$ , $\dots$ tends to a limit. Find these limits and deduce that the sequence $u_{1}$ , $u_{2}$ , $u_{3}$ , $\dots$ tends to a limit. Would this conclusion change if the sequence were defined by $(*)$ and $u_{1} = 3$ ?
Sequences & Series
Write down a solution of the equation $x^{2} - 2 y^{2} = 1, (*)$ for which $x$ and $y$ are non-negative integers. Show that, if $x = p$ , $y = q$ is a solution of ( $*$ ), then so also is $x = 3 p + 4 q$ , $y = 2 p + 3 q$ . Hence find two solutions of $(*)$ for which $x$ is a positive odd integer and $y$ is a positive even integer.
Show that, if $x$ is an odd integer and $y$ is an even integer, $(*)$ can be written in the form $n^{2} = \frac{1}{2} m (m + 1),$ where $m$ and $n$ are integers.
The positive integers $a$ , $b$ and $c$ satisfy $b^{3} = c^{4} - a^{2},$ where $b$ is a prime number. Express $a$ and $c^{2}$ in terms of $b$ in the two cases that arise. Find a solution of $a^{2} + b^{3} = c^{4}$ , where $a$ , $b$ and $c$ are positive integers but $b$ is not prime.
Proof & Number Theory
The function $f$ satisfies $f (x) > 0$ for $x ⩾ 0$ and is strictly decreasing (which means that $f (b) < f (a)$ for $b > a$ ).
For $t ⩾ 0$ , let $A_{0} (t)$ be the area of the largest rectangle with sides parallel to the coordinate axes that can fit in the region bounded by the curve $y = f (x)$ , the $y$ -axis and the line $y = f (t)$ . Show that $A_{0} (t)$ can be written in the form $A_{0} (t) = x_{0} (f (x_{0}) - f (t)),$ where $x_{0}$ satisfies $x_{0} f^{'} (x_{0}) + f (x_{0}) = f (t)$ .
The function g is defined, for $t > 0$ , by $g (t) = \frac{1}{t} \int_{0}^{t} f (x) d x .$ Show that $t g^{'} (t) = f (t) - g (t)$ . Making use of a sketch show that, for $t > 0$ , $\int_{0}^{t} (f (x) - f (t)) d x > A_{0} (t)$ and deduce that $- t^{2} g^{'} (t) > A_{0} (t)$ .
In the case $f (x) = \frac{1}{1 + x}$ , use the above to establish the inequality $ln 1 + t > 1 - \frac{1}{1 + t},$ for $t > 0$ .
Functions & Curve Sketching
The diagram shows three identical discs in equilibrium in a vertical plane. Two discs rest, not in contact with each other, on a horizontal surface and the third disc rests on the other two. The angle at the upper vertex of the triangle joining the centres of the discs is $2 θ$ .

The weight of each disc is $W$ . The coefficient of friction between a disc and the horizontal surface is $μ$ and the coefficient of friction between the discs is also $μ$ .
Show that the normal reaction between the horizontal surface and a disc in contact with the surface is $\frac{3}{2} W$ .
Find the normal reaction between two discs in contact and show that the magnitude of the frictional force between two discs in contact is $\frac{W sin θ}{2 ( 1 + cos θ )}$ .
Show that if $μ < 2 - \sqrt3$ there is no value of $θ$ for which equilibrium is possible.
Mechanics
A particle is projected at an angle of elevation $α$ (where $α > 0$ ) from a point $A$ on horizontal ground. At a general point in its trajectory the angle of elevation of the particle from $A$ is $θ$ and its direction of motion is at an angle $ϕ$ above the horizontal (with $ϕ ⩾ 0$ for the first half of the trajectory and $ϕ ⩽ 0$ for the second half).

Let $B$ denote the point on the trajectory at which $θ = \frac{1}{2} α$ and let $C$ denote the point on the trajectory at which $ϕ = - \frac{1}{2} α$ .
Show that, at a general point on the trajectory, $2 tan θ = tan α + tan ϕ$ .
Show that, if $B$ and $C$ are the same point, then $α = 6 0^{\circ}$ .
Given that $α < 6 0^{\circ}$ , determine whether the particle reaches the point $B$ first or the point $C$ first.
Mechanics
Three identical particles lie, not touching one another, in a straight line on a smooth horizontal surface. One particle is projected with speed $u$ directly towards the other two which are at rest. The coefficient of restitution in all collisions is $e$ , where $0 < e < 1$ .
Show that, after the second collision, the speeds of the particles are $\frac{1}{2} u (1 - e)$ , $\frac{1}{4} u (1 - e^{2})$ and $\frac{1}{4} u (1 + e)^{2}$ . Deduce that there will be a third collision whatever the value of $e$ .
Show that there will be a fourth collision if and only if $e$ is less than a particular value which you should determine.
Mechanics
The random variable $U$ has a Poisson distribution with parameter $λ$ . The random variables $X$ and $Y$ are defined as follows. \begin{align*} X&= \begin{cases} U & \text{ if $U$ is 1, 3, 5, 7, $\dots$ } \\ 0 & \text{ otherwise} \end{cases} \\ Y&= \begin{cases} U & \text{ if $U$ is 2, 4, 6, 8, $\dots$ } \\ 0 & \text{ otherwise} \end{cases} \end{align*}
Find $E (X)$ and $E (Y)$ in terms of $λ$ , $α$ and $β$ , where $α = 1 + \frac{λ ^{2}}{2 !} + \frac{λ ^{4}}{4 !} + \dots and β = \frac{λ}{1 !} + \frac{λ ^{3}}{3 !} + \frac{λ ^{5}}{5 !} + \dots .$
Show that $Var (X) = \frac{λ α + λ ^{2} β}{α + β} - \frac{λ ^{2} α ^{2}}{( α + β ) ^{2}}$ and obtain the corresponding expression for $Var (Y)$ . Are there any non-zero values of $λ$ for which $Var (X) + Var (Y) = Var (X + Y)$ ?
Probability & Statistics
A biased coin has probability $p$ of showing a head and probability $q$ of showing a tail, where $p \neq = 0$ , $q \neq = 0$ and $p \neq = q$ . When the coin is tossed repeatedly, runs occur. A straight run of length $n$ is a sequence of $n$ consecutive heads or $n$ consecutive tails. An alternating run of length $n$ is a sequence of length $n$ alternating between heads and tails. An alternating run can start with either a head or a tail.

Let $S$ be the length of the longest straight run beginning with the first toss and let $A$ be the length of the longest alternating run beginning with the first toss.
Explain why $P (A = 1) = p^{2} + q^{2}$ and find $P (S = 1)$ . Show that $P (S = 1) < P (A = 1)$ .
Show that $P (S = 2) = P (A = 2)$ and determine the relationship between $P (S = 3)$ and $P (A = 3)$ .
Show that, for $n > 1$ , $P (S = 2 n) > P (A = 2 n)$ and determine the corresponding relationship between $P (S = 2 n + 1)$ and $P (A = 2 n + 1)$ . [You are advised not to use $p + q = 1$ in this part.]
Probability & Statistics