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Questions

Worked solutions

92%

120

Marks total

Chapters covered

OCR (Cambridge University Press & Assessment) · United Kingdom

STEP 2 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.

By use of calculus, show that $x - ln (1 + x)$ is positive for all positive $x$ . Use this result to show that $k = 1 \sum n \frac{1}{k} > ln (n + 1) .$
By considering $x + ln (1 - x)$ , show that $k = 1 \sum \infty \frac{1}{k ^{2}} < 1 + ln 2 .$
Sequences & Series
In the triangle $A B C$ , angle $B A C = α$ and angle $C B A = 2 α$ , where $2 α$ is acute, and $B C = x$ . Show that $A B = (3 - 4 sin^{2} α) x$ .

The point $D$ is the midpoint of $A B$ and the point $E$ is the foot of the perpendicular from $C$ to $A B$ . Find an expression for $D E$ in terms of $x$ .

The point $F$ lies on the perpendicular bisector of $A B$ and is a distance $x$ from $C$ . The points $F$ and $B$ lie on the same side of the line through $A$ and $C$ . Show that the line $F C$ trisects the angle $A C B$ .
Trigonometry
Three rods have lengths $a$ , $b$ and $c$ , where $a < b < c$ . The three rods can be made into a triangle (possibly of zero area) if $a + b ⩾ c$ .

Let $T_{n}$ be the number of triangles that can be made with three rods chosen from $n$ rods of lengths $1$ , $2$ , $3$ , $\dots$ , $n$ (where $n ⩾ 3$ ). Show that $T_{8} - T_{7} = 2 + 4 + 6$ and evaluate $T_{8} - T_{6}$ . Write down expressions for $T_{2 m} - T_{2 m - 1}$ and $T_{2 m} - T_{2 m - 2}$ .

Prove by induction that $T_{2 m} = \frac{1}{6} m (m - 1) (4 m + 1)$ , and find the corresponding result for an odd number of rods.
Proof & Number Theory
The continuous function $f$ is defined by $tan f (x) = x (- \infty < x < \infty)$ and $f (0) = π$ . Sketch the curve $y = f (x)$ .
The continuous function $g$ is defined by $tan g (x) = \frac{x}{1 + x ^{2}} (- \infty < x < \infty)$ and $g (0) = π$ . Sketch the curves $y = \frac{x}{1 + x ^{2}}$ and $y = g (x)$ .
The continuous function $h$ is defined by $h (0) = π$ and $tan h (x) = \frac{x}{1 - x ^{2}} (x \neq = \pm 1) .$ (The values of $h (x)$ at $x = \pm 1$ are such that $h (x)$ is continuous at these points.) Sketch the curves $y = \frac{x}{1 - x ^{2}}$ and $y = h (x)$ .
Functions & Curve Sketching
In this question, the $arctan$ function satisfies $0 ⩽ arctan x < \frac{1}{2} π$ for $x ⩾ 0$ .
Let $S_{n} = m = 1 \sum n arctan (\frac{1}{2 m ^{2}}),$ for $n = 1$ , 2, 3, $\dots$ . Prove by induction that $tan S_{n} = \frac{n}{n + 1} .$ Prove also that $S_{n} = arctan \frac{n}{n + 1} .$
In a triangle $A B C$ , the lengths of the sides $A B$ and $B C$ are $4 n^{2}$ and $4 n^{4} - 1$ , respectively, and the angle at $B$ is a right angle. Let $∠ B C A = 2 α_{n}$ . Show that $n = 1 \sum \infty α_{n} = \frac{1}{4} π .$
Sequences & Series
Show that $sec^{2} (\frac{1}{4} π - \frac{1}{2} x) = \frac{2}{1 + sin x} .$ Hence integrate $\frac{1}{1 + sin x}$ with respect to $x$ .
By means of the substitution $y = π - x$ , show that $\int_{0}^{π} x f (sin x) d x = \frac{π}{2} \int_{0}^{π} f (sin x) d x,$ where $f$ is any function for which these integrals exist. Hence evaluate $\int_{0}^{π} \frac{x}{1 + sin x} d x .$
Evaluate $\int_{0}^{π} \frac{2 x ^{3} - 3 π x ^{2}}{( 1 + sin x ) ^{2}} d x .$
Integration
A circle $C$ is said to be bisected by a curve $X$ if $X$ meets $C$ in exactly two points and these points are diametrically opposite each other on $C$ .
Let $C$ be the circle of radius $a$ in the $x$ - $y$ plane with centre at the origin. Show, by giving its equation, that it is possible to find a circle of given radius $r$ that bisects $C$ provided $r > a$ . Show that no circle of radius $r$ bisects $C$ if $r ⩽ a$ .
Let $C_{1}$ and $C_{2}$ be circles with centres at $(- d, 0)$ and $(d, 0)$ and radii $a_{1}$ and $a_{2}$ , respectively, where $d > a_{1}$ and $d > a_{2}$ . Let $D$ be a circle of radius $r$ that bisects both $C_{1}$ and $C_{2}$ . Show that the $x$ -coordinate of the centre of $D$ is $\frac{a _{2}^{2} - a _{1}^{2}}{4 d}$ . Obtain an expression in terms of $d$ , $r$ , $a_{1}$ and $a_{2}$ for the $y$ -coordinate of the centre of $D$ , and deduce that $r$ must satisfy $16 r^{2} d^{2} ⩾ (4 d^{2} + (a_{2} - a_{1})^{2}) (4 d^{2} + (a_{2} + a_{1})^{2}) .$
Coordinate Geometry & Conics
The diagram above shows two non-overlapping circles $C_{1}$ and $C_{2}$ of different sizes. The lines $L$ and $L^{'}$ are the two common tangents to $C_{1}$ and $C_{2}$ such that the two circles lie on the same side of each of the tangents. The lines $L$ and $L^{'}$ intersect at the point $P$ which is called the focus of $C_{1}$ and $C_{2}$ .
Let $< s t r o n g > x < / s t r o n g >_{1}$ and $< s t r o n g > x < / s t r o n g >_{2}$ be the position vectors of the centres of $C_{1}$ and $C_{2}$ , respectively. Show that the position vector of $P$ is $\frac{r _{1} < s t r o n g > x < / s t r o n g > _{2} - r _{2} < s t r o n g > x < / s t r o n g > _{1}}{r _{1} - r _{2}},$ where $r_{1}$ and $r_{2}$ are the radii of $C_{1}$ and $C_{2}$ , respectively.
The circle $C_{3}$ does not overlap either $C_{1}$ or $C_{2}$ and its radius, $r_{3}$ , satisfies $r_{1} \neq = r_{3} \neq = r_{2}$ . The focus of $C_{1}$ and $C_{3}$ is $Q$ , and the focus of $C_{2}$ and $C_{3}$ is $R$ . Show that $P$ , $Q$ and $R$ lie on the same straight line.
Find a condition on $r_{1}$ , $r_{2}$ and $r_{3}$ for $Q$ to lie half-way between $P$ and $R$ .
Vectors & Matrices
An equilateral triangle $A B C$ is made of three light rods each of length $a$ . It is free to rotate in a vertical plane about a horizontal axis through $A$ . Particles of mass $3 m$ and $5 m$ are attached to $B$ and $C$ respectively. Initially, the system hangs in equilibrium with $B C$ below $A$ .
Show that, initially, the angle $θ$ that $B C$ makes with the horizontal is given by $sin θ = \frac{1}{7}$ .
The triangle receives an impulse that imparts a speed $v$ to the particle $B$ . Find the minimum speed $v_{0}$ such that the system will perform complete rotations if $v > v_{0}$ .
Mechanics
A particle of mass $m$ is pulled along the floor of a room in a straight line by a light string which is pulled at constant speed $V$ through a hole in the ceiling. The floor is smooth and horizontal, and the height of the room is $h$ . Find, in terms of $V$ and $θ$ , the speed of the particle when the string makes an angle of $θ$ with the vertical (and the particle is still in contact with the floor). Find also the acceleration, in terms of $V$ , $h$ and $θ$ .

Find the tension in the string and hence show that the particle will leave the floor when $tan^{4} θ = \frac{V ^{2}}{g h} .$
Mechanics
Three particles, $A$ , $B$ and $C$ , each of mass $m$ , lie on a smooth horizontal table. Particles $A$ and $C$ are attached to the two ends of a light inextensible string of length $2 a$ and particle $B$ is attached to the midpoint of the string. Initially, $A$ , $B$ and $C$ are at rest at points $(0, a)$ , $(0, 0)$ and $(0, - a)$ , respectively.

An impulse is delivered to $B$ , imparting to it a speed $u$ in the positive $x$ direction. The string remains taut throughout the subsequent motion.
At time $t$ , the angle between the $x$ -axis and the string joining $A$ and $B$ is $θ$ , as shown in the diagram, and $B$ is at $(x, 0)$ . Write down the coordinates of $A$ in terms of $x, a$ and $θ$ . Given that the velocity of $B$ is $(v, 0)$ , show that the velocity of $A$ is $(\overset{x}{˙} + a sin θ \dot{θ}, a cos θ \dot{θ})$ , where the dot denotes differentiation with respect to time.
Show that, before particles $A$ and $C$ first collide, $3 \overset{x}{˙} + 2 a \dot{θ} sin θ = v and \dot{θ}^{2} = \frac{v ^{2}}{a ^{2} ( 3 - 2 sin ^{2} θ )} .$
When $A$ and $C$ collide, the collision is elastic (no energy is lost). At what value of $θ$ does the second collision between particles $A$ and $C$ occur? (You should justify your answer.)
When $v = 0$ , what are the possible values of $θ$ ? Is $v = 0$ whenever $θ$ takes these values?
Mechanics
Four players $A$ , $B$ , $C$ and $D$ play a coin-tossing game with a fair coin. Each player chooses a sequence of heads and tails, as follows:

Player A: HHT; Player B: THH; Player C: TTH; Player D: HTT. The coin is then tossed until one of these sequences occurs, in which case the corresponding player is the winner.
Show that, if only $A$ and $B$ play, then $A$ has a probability of $\frac{1}{4}$ of winning.
If all four players play together, find the probabilities of each one winning.
Only $B$ and $C$ play. What is the probability of $C$ winning if the first two tosses are TT? Let the probabilities of $C$ winning if the first two tosses are HT, TH and HH be $p$ , $q$ and $r$ , respectively. Show that $p = \frac{1}{2} + \frac{1}{2} q$ . Find the probability that $C$ wins.
Probability & Statistics
The maximum height $X$ of flood water each year on a certain river is a random variable with probability density function $f$ given by $f (x) = {λ e^{- λ x} 0 for x ⩾ 0, otherwise,$ where $λ$ is a positive constant.

It costs $k y$ pounds each year to prepare for flood water of height $y$ or less, where $k$ is a positive constant and $y ⩾ 0$ . If $X ⩽ y$ no further costs are incurred but if $X > y$ the additional cost of flood damage is $a (X - y)$ pounds where $a$ is a positive constant.
Let $C$ be the total cost of dealing with the floods in the year. Show that the expectation of $C$ is given by $E (C) = k y + \frac{a}{λ} e^{- λ y} .$ How should $y$ be chosen in order to minimise $E (C)$ , in the different cases that arise according to the value of $a / k$ ?
Find the variance of $C$ , and show that the more that is spent on preparing for flood water in advance the smaller this variance.
Probability & Statistics