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120

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Chapters covered

OCR (Cambridge University Press & Assessment) · United Kingdom

STEP 2 2002

2002

14 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.

Show that $\int_{\frac{1}{6} π}^{\frac{1}{4} π} \frac{1}{1 - cos 2 θ} d θ = \frac{3}{2} - \frac{1}{2} .$ By using the substitution $x = sin 2 θ$ , or otherwise, show that $\int_{3 /2}^{1} \frac{1}{1 - 1 - x ^{2}} d x = 3 - 1 - \frac{π}{6} .$ Hence evaluate the integral $\int_{1}^{2/ 3} \frac{1}{y ( y - y ^{2} - 1 ^{2} )} d y .$
Integration
Show that setting $z - z^{- 1} = w$ in the quartic equation $z^{4} + 5 z^{3} + 4 z^{2} - 5 z + 1 = 0$ results in the quadratic equation $w^{2} + 5 w + 6 = 0$ . Hence solve the above quartic equation.

Solve similarly the equation $2 z^{8} - 3 z^{7} - 12 z^{6} + 12 z^{5} + 22 z^{4} - 12 z^{3} - 12 z^{2} + 3 z + 2 = 0 .$
Algebra & Inequalities
The $n$ th Fermat number, $F_{n}$ , is defined by $F_{n} = 2^{2^{n}} + 1, n = 0, 1, 2, \dots,$ where $2^{2^{n}}$ means $2$ raised to the power $2^{n}$ . Calculate $F_{0}$ , $F_{1}$ , $F_{2}$ and $F_{3}$ . Show that, for $k = 1$ , $k = 2$ and $k = 3$ , $F_{0} F_{1} \dots F_{k - 1} = F_{k} - 2 . (*)$ Prove, by induction, or otherwise, that $(*)$ holds for all $k ⩾ 1$ . Deduce that no two Fermat numbers have a common factor greater than $1$ .

Hence show that there are infinitely many prime numbers.
Proof & Number Theory
Give a sketch to show that, if $f (x) > 0$ for $p < x < q$ , then $\int_{p}^{q} f (x) d x > 0$ .
By considering $f (x) = a x^{2} - b x + c$ show that, if $a > 0$ and $b^{2} < 4 a c$ , then $3 b < 2 a + 6 c$ .
By considering $f (x) = a sin^{2} x - b sin x + c$ show that, if $a > 0$ and $b^{2} < 4 a c$ , then $4 b < (a + 2 c) π .$
Show that, if $a > 0$ , $b^{2} < 4 a c$ and $q > p > 0$ , then $b ln (q / p) < a (\frac{1}{p} - \frac{1}{q}) + c (q - p) .$
Trigonometry
The numbers $x_{n}$ , where $n = 0$ , $1$ , $2$ , $\dots$ , satisfy $x_{n + 1} = k x_{n} (1 - x_{n}) .$
Prove that, if $0 < k < 4$ and $0 < x_{0} < 1$ , then $0 < x_{n} < 1$ for all $n$ .
Given that $x_{0} = x_{1} = x_{2} = \dots = a$ , with $a \neq = 0$ and $a \neq = 1$ , find $k$ in terms of $a$ .
Given instead that $x_{0} = x_{2} = x_{4} = \dots = a$ , with $a \neq = 0$ and $a \neq = 1$ , show that $a b^{3} - b^{2} + (1 - a) = 0$ , where $b = k (1 - a)$ . Given, in addition, that $x_{1} \neq = a$ , find the possible values of $k$ in terms of $a$ .
Sequences & Series
The lines $l_{1}$ , $l_{2}$ and $l_{3}$ lie in an inclined plane $P$ and pass through a common point $A$ . The line $l_{2}$ is a line of greatest slope in $P$ . The line $l_{1}$ is perpendicular to $l_{3}$ and makes an acute angle $α$ with $l_{2}$ . The angles between the horizontal and $l_{1}$ , $l_{2}$ and $l_{3}$ are $π /6$ , $β$ and $π /4$ , respectively. Show that $cos α sin β = \frac{1}{2}$ and find the value of $sin α sin β$ . Deduce that $β = π /3$ .

The lines $l_{1}$ and $l_{3}$ are rotated in $P$ about $A$ so that $l_{1}$ and $l_{3}$ remain perpendicular to each other. The new acute angle between $l_{1}$ and $l_{2}$ is $θ$ . The new angles which $l_{1}$ and $l_{3}$ make with the horizontal are $ϕ$ and $2 ϕ$ , respectively. Show that $tan^{2} θ = \frac{3 + 13}{2} .$
Coordinate Geometry & Conics
In 3-dimensional space, the lines $m_{1}$ and $m_{2}$ pass through the origin and have directions $i + j$ and $i + k$ , respectively. Find the directions of the two lines $m_{3}$ and $m_{4}$ that pass through the origin and make angles of $π /4$ with both $m_{1}$ and $m_{2}$ . Find also the cosine of the acute angle between $m_{3}$ and $m_{4}$ .

The points $A$ and $B$ lie on $m_{1}$ and $m_{2}$ respectively, and are each at distance $λ \sqrt2$ units from $O$ . The points $P$ and $Q$ lie on $m_{3}$ and $m_{4}$ respectively, and are each at distance $1$ unit from $O$ . If all the coordinates (with respect to axes $i$ , $j$ and $k$ ) of $A$ , $B$ , $P$ and $Q$ are non-negative, prove that:
there are only two values of $λ$ for which $A Q$ is perpendicular to $B P$ ;
there are no non-zero values of $λ$ for which $A Q$ and $B P$ intersect.
Vectors & Matrices
Find $y$ in terms of $x$ , given that: \begin{aligned} \text{for $x < 0$ }, && \frac{{\mathrm d} y}{{\mathrm d} x} = -y \text{ \ \ and \ \ } y = a \text{ when } x = -1\;; \\ \text{for $x > 0$ }, && \frac{{\mathrm d} y}{{\mathrm d} x} = y \text{ \ \ \ \ and \ \ } y = b \ \text{ when } x = 1\;. \end{aligned}

Sketch a solution curve. Determine the condition on $a$ and $b$ for the solution curve to be continuous (that is, for there to be no `jump' in the value of $y$ ) at $x = 0$ .

Solve the differential equation $\frac{d y}{d x} = ∣ e^{x} - 1 ∣ y$ given that $y = e^{e}$ when $x = 1$ and that $y$ is continuous at $x = 0$ . Write down the following limits: $\text{⟦B⟧(i)⟦/B⟧} \ \ \lim_ {x \to +\infty} y\exp(-{\mathrm e}^x)\;; \ \ \ \ \ \ \ \ \ \text{⟦B⟧(ii)⟦/B⟧} \ \ \lim_{x \to -\infty}y {\mathrm e}^{-x}\,.$
Differential Equations
A particle is projected from a point $O$ on a horizontal plane with speed $V$ and at an angle of elevation $α$ . The vertical plane in which the motion takes place is perpendicular to two vertical walls, both of height $h$ , at distances $a$ and $b$ from $O$ . Given that the particle just passes over the walls, find $tan α$ in terms of $a$ , $b$ and $h$ and show that $\frac{2 V ^{2}}{g} = \frac{ab}{h} + \frac{( a + b ) ^{2} h}{ab} .$ The heights of the walls are now increased by the same small positive amount $δ h$ . A second particle is projected so that it just passes over both walls, and the new angle and speed of projection are $α + δ α$ and $V + δ V$ , respectively. Show that $sec^{2} α δ α \approx \frac{a + b}{ab} δ h,$ and deduce that $δ α > 0$ . Show also that $δ V$ is positive if $h > ab / (a + b)$ and negative if $h < ab / (a + b)$ .
Mechanics
A competitor in a Marathon of $42 \frac{3}{8}$ km runs the first $t$ hours of the race at a constant speed of 13 km h $^{- 1}$ and the remainder at a constant speed of $14 + 2 t / T$ km h $^{- 1}$ , where $T$ hours is her time for the race. Show that the minimum possible value of $T$ over all possible values of $t$ is 3.

The speed of another competitor decreases linearly with respect to time from 16 km h $^{- 1}$ at the start of the race. If both of these competitors have a run time of 3 hours, find the maximum distance between them at any stage of the race.
Mechanics
A rigid straight beam $A B$ has length $l$ and weight $W$ . Its weight per unit length at a distance $x$ from $B$ is $α W l^{- 1} (x / l)^{α - 1}$ , where $α$ is a positive constant. Show that the centre of mass of the beam is at a distance $α l / (α + 1)$ from $B$ .

The beam is placed with the end $A$ on a rough horizontal floor and the end $B$ resting against a rough vertical wall. The beam is in a vertical plane at right angles to the plane of the wall and makes an angle of $θ$ with the floor. The coefficient of friction between the floor and the beam is $μ$ and the coefficient of friction between the wall and the beam is also $μ$ . Show that, if the equilibrium is limiting at both $A$ and $B$ , then $tan θ = \frac{1 - α μ ^{2}}{( 1 + α ) μ} .$ Given that $α = 3/2$ and given also that the beam slides for any $θ < π /4$ find the greatest possible value of $μ$ .
Mechanics
On $K$ consecutive days each of $L$ identical coins is thrown $M$ times. For each coin, the probability of throwing a head in any one throw is $p$ (where $0 < p < 1$ ). Show that the probability that on exactly $k$ of these days more than $l$ of the coins will each produce fewer than $m$ heads can be approximated by $(k K) q^{k} (1 - q)^{K - k},$ where $q = Φ (\frac{2 h - 2 l - 1}{2 h}), h = L Φ (\frac{2 m - 1 - 2 M p}{2 M p ( 1 - p )})$ and $Φ (.)$ is the cumulative distribution function of a standard normal variate.

Would you expect this approximation to be accurate in the case $K = 7$ , $k = 2$ , $L = 500$ , $l = 4$ , $M = 100$ , $m = 48$ and $p = 0.6$ ?
Probability & Statistics
Let $F (x)$ be the cumulative distribution function of a random variable $X$ , which satisfies $F (a) = 0$ and $F (b) = 1$ , where $a > 0$ . Let $G (y) = \frac{F ( y )}{2 - F ( y )} .$ Show that $G (a) = 0$ , $G (b) = 1$ and that $G^{'} (y) ⩾ 0$ . Show also that $\frac{1}{2} ⩽ \frac{2}{( 2 - F ( y ) ) ^{2}} ⩽ 2 .$ The random variable $Y$ has cumulative distribution function $G (y)$ . Show that $\frac{1}{2} E (X) ⩽ E (Y) ⩽ 2 E (X),$ and that $Var (Y) ⩽ 2 Var (X) + \frac{7}{4} (E (X))^{2} .$
Probability & Statistics
A densely populated circular island is divided into $N$ concentric regions $R_{1}$ , $R_{2}$ , $\dots$ , $R_{N}$ , such that the inner and outer radii of $R_{n}$ are $n - 1$ km and $n$ km, respectively. The average number of road accidents that occur in any one day in $R_{n}$ is $2 - n / N$ , independently of the number of accidents in any other region.

Each day an observer selects a region at random, with a probability that is proportional to the area of the region, and records the number of road accidents, $X$ , that occur in it. Show that, in the long term, the average number of recorded accidents per day will be $2 - \frac{1}{6} (1 + \frac{1}{N}) (4 - \frac{1}{N}) .$ [Note: $n = 1 \sum N n^{2} = \frac{1}{6} N (N + 1) (2 N + 1)$ .]

Show also that $P (X = k) = \frac{e ^{- 2} N ^{- k - 2}}{k !} n = 1 \sum N (2 n - 1) (2 N - n)^{k} e^{n / N} .$ Suppose now that $N = 3$ and that, on a particular day, two accidents were recorded. Show that the probability that $R_{2}$ had been selected is $\frac{48}{48 + 45 e ^{1/3} + 25 e ^{- 1/3}} .$
Probability & Statistics