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Questions

Worked solutions

85%

120

Marks total

Chapters covered

OCR (Cambridge University Press & Assessment) · United Kingdom

STEP 2 2016

2016

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 curve $C_{1}$ has parametric equations $x = t^{2}$ , $y = t^{3}$ , where $- \infty < t < \infty$ . Let $O$ denote the point $(0, 0)$ . The points $P$ and $Q$ on $C_{1}$ are such that $∠ P O Q$ is a right angle. Show that the tangents to $C_{1}$ at $P$ and $Q$ intersect on the curve $C_{2}$ with equation $4 y^{2} = 3 x - 1$ .

Determine whether $C_{1}$ and $C_{2}$ meet, and sketch the two curves on the same axes.
Coordinate Geometry & Conics
Use the factor theorem to show that $a + b - c$ is a factor of $(a + b + c)^{3} - 6 (a + b + c) (a^{2} + b^{2} + c^{2}) + 8 (a^{3} + b^{3} + c^{3}) . (*)$ Hence factorise ( $*$ ) completely.
Use the result above to solve the equation $(x + 1)^{3} - 3 (x + 1) (2 x^{2} + 5) + 2 (4 x^{3} + 13) = 0 .$
By setting $d + e = c$ , or otherwise, show that $(a + b - d - e)$ is a factor of $(a + b + d + e)^{3} - 6 (a + b + d + e) (a^{2} + b^{2} + d^{2} + e^{2}) + 8 (a^{3} + b^{3} + d^{3} + e^{3})$ and factorise this expression completely. Hence solve the equation $(x + 6)^{3} - 6 (x + 6) (x^{2} + 14) + 8 (x^{3} + 36) = 0 .$
Proof & Number Theory
For each non-negative integer $n$ , the polynomial $f_{n}$ is defined by $f_{n} (x) = 1 + x + \frac{x ^{2}}{2 !} + \frac{x ^{3}}{3 !} + \dots + \frac{x ^{n}}{n !} .$
Show that $f_{n}^{'} (x) = f_{n - 1} (x)$ (for $n ⩾ 1$ ).
Show that, if $a$ is a real root of the equation $f_{n} (x) = 0, (*)$ then $a < 0$ .
Let $a$ and $b$ be distinct real roots of ( $*$ ), for $n ⩾ 2$ . Show that $f_{n}^{'} (a) f_{n}^{'} (b) > 0$ and use a sketch to deduce that $f_{n} (c) = 0$ for some number $c$ between $a$ and $b$ . Deduce that ( $*$ ) has at most one real root. How many real roots does ( $*$ ) have if $n$ is odd? How many real roots does ( $*$ ) have if $n$ is even?
Sequences & Series
Let $y = \frac{x ^{2} + x sin θ + 1}{x ^{2} + x cos θ + 1} .$
Given that $x$ is real, show that $(y cos θ - sin θ)^{2} ⩾ 4 (y - 1)^{2} .$ Deduce that $y^{2} + 1 ⩾ 4 (y - 1)^{2},$ and hence that $\frac{4 - 7}{3} ⩽ y ⩽ \frac{4 + 7}{3} .$
In the case $y = \frac{4 + 7}{3}$ , show that $y^{2} + 1 = 2 (y - 1)$ and find the corresponding values of $x$ and $tan θ$ .
Functions & Curve Sketching
In this question, the definition of $(q p)$ is taken to be $(q p) = ⎩ ⎨ ⎧ \frac{p !}{q ! ( p - q )!} 0 if p ⩾ q ⩾ 0, otherwise .$
Write down the coefficient of $x^{n}$ in the binomial expansion for $(1 - x)^{- N}$ , where $N$ is a positive integer, and write down the expansion using the $Σ$ summation notation. By considering $(1 - x)^{- 1} (1 - x)^{- N},$ where $N$ is a positive integer, show that $j = 0 \sum n (j N + j - 1) = (n N + n) .$
Show that, for any positive integers $m$ , $n$ and $r$ with $r ⩽ m + n$ , $(r m + n) = j = 0 \sum r (j m) (r - j n) .$
Show that, for any positive integers $m$ and $N$ , $j = 0 \sum n (- 1)^{j} (n - j N + m) (j m + j - 1) = (n N) .$
Combinatorics & Binomial
This question concerns solutions of the differential equation $(1 - x^{2}) (\frac{d y}{d x})^{2} + k^{2} y^{2} = k^{2} (*)$ where $k$ is a positive integer.

For each value of $k$ , let $y_{k} (x)$ be the solution of $(*)$ that satisfies $y_{k} (1) = 1$ ; you may assume that there is only one such solution for each value of $k$ .
Write down the differential equation satisfied by $y_{1} (x)$ and verify that $y_{1} (x) = x$ .
Write down the differential equation satisfied by $y_{2} (x)$ and verify that $y_{2} (x) = 2 x^{2} - 1$ .
Let $z (x) = 2 (y_{n} (x))^{2} - 1$ . Show that $(1 - x^{2}) (\frac{d z}{d x})^{2} + 4 n^{2} z^{2} = 4 n^{2}$ and hence obtain an expression for $y_{2 n} (x)$ in terms of $y_{n} (x)$ .
Let $v (x) = y_{n} (y_{m} (x))$ . Show that $v (x) = y_{mn} (x)$ .
Differential Equations
Show that $\int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x, (*)$ where f is any function for which the integrals exist.
Use ( $*$ ) to evaluate $\int_{0}^{\frac{1}{2} π} \frac{sin x}{cos x + sin x} d x .$
Evaluate $\int_{0}^{\frac{1}{4} π} \frac{sin x}{cos x + sin x} d x .$
Evaluate $\int_{0}^{\frac{1}{4} π} ln (1 + tan x) d x .$
Evaluate $\int_{0}^{\frac{1}{4} π} \frac{x}{cos x ( cos x + sin x )} d x .$
Integration
Evaluate the integral $(m > \frac{1}{2}) . \int_{m - \frac{1}{2}}^{\infty} \frac{1}{x ^{2}} d x (m > \frac{1}{2}) .$ Show by means of a sketch that $r = m \sum n \frac{1}{r ^{2}} \approx \int_{m - \frac{1}{2}}^{n + \frac{1}{2}} \frac{1}{x ^{2}} d x, (*)$ where $m$ and $n$ are positive integers with $m < n$ .
You are given that the infinite series $r = 1 \sum \infty \frac{1}{r ^{2}}$ converges to a value denoted by $E$ . Use $(*)$ to obtain the following approximations for $E$ : $E \approx 2; E \approx \frac{5}{3}; E \approx \frac{33}{20} .$
Show that, when $r$ is large, the error in approximating $\frac{1}{r ^{2}}$ by $\int_{r - \frac{1}{2}}^{r + \frac{1}{2}} \frac{1}{x ^{2}} d x$ is approximately $\frac{1}{4 r ^{4}}$ . Given that $E \approx 1.645$ , show that $r = 1 \sum \infty \frac{1}{r ^{4}} \approx 1.08$ .
Integration
A small bullet of mass $m$ is fired into a block of wood of mass $M$ which is at rest. The speed of the bullet on entering the block is $u$ . Its trajectory within the block is a horizontal straight line and the resistance to the bullet's motion is $R$ , which is constant.
The block is fixed. The bullet travels a distance $a$ inside the block before coming to rest. Find an expression for $a$ in terms of $m$ , $u$ and $R$ .
Instead, the block is free to move on a smooth horizontal table. The bullet travels a distance $b$ inside the block before coming to rest relative to the block, at which time the block has moved a distance $c$ on the table. Find expressions for $b$ and $c$ in terms of $M$ , $m$ and $a$ .
Mechanics
A thin uniform wire is bent into the shape of an isosceles triangle $A B C$ , where $A B$ and $A C$ are of equal length and the angle at $A$ is $2 θ$ .

The triangle $A B C$ hangs on a small rough horizontal peg with the side $B C$ resting on the peg. The coefficient of friction between the wire and the peg is $μ$ . The plane containing $A B C$ is vertical. Show that the triangle can rest in equilibrium with the peg in contact with any point on $B C$ provided $μ ⩾ 2 tan θ (1 + sin θ) .$
Mechanics
Two particles move on a smooth horizontal surface. The positions, in Cartesian coordinates, of the particles at time $t$ are $(a + u t cos α, u t sin α)$ and $(v t cos β, b + v t sin β)$ , where $a$ , $b$ , $u$ and $v$ are positive constants, $α$ and $β$ are constant acute angles, and $t ⩾ 0$ . Given that the two particles collide, show that $u sin (θ + α) = v sin (θ + β),$ where $θ$ is the acute angle satisfying $tan θ = \frac{b}{a}$ .
A gun is placed on the top of a vertical tower of height $b$ which stands on horizontal ground. The gun fires a bullet with speed $v$ and (acute) angle of elevation $β$ . Simultaneously, a target is projected from a point on the ground a horizontal distance $a$ from the foot of the tower. The target is projected with speed $u$ and (acute) angle of elevation $α$ , in a direction directly away from the tower. Given that the target is hit before it reaches the ground, show that $2 u sin α (u sin α - v sin β) > b g .$ Explain, with reference to part (i), why the target can only be hit if $α > β$ .
Mechanics
Starting with the result $P (A \cup B) = P (A) + P (B) - P (A \cap B)$ , prove that $P (A \cup B \cup C) = P (A) + P (B) + P (C) - P (A \cap B) - P (B \cap C) - P (C \cap A) + P (A \cap B \cap C) .$ Write down, without proof, the corresponding result for four events $A$ , $B$ , $C$ and $D$ .

A pack of $n$ cards, numbered $1$ , $2$ , … , $n$ , is shuffled and laid out in a row. The result of the shuffle is that each card is equally likely to be in any position in the row. Let $E_{i}$ be the event that the card bearing the number $i$ is in the $i$ th position in the row. Write down the following probabilities:
$P (E_{i})$ ;
$P (E_{i} \cap E_{j})$ , where $i \neq = j$ ;
$P (E_{i} \cap E_{j} \cap E_{k})$ , where $i \neq = j$ , $j \neq = k$ and $k \neq = i$ .
Hence show that the probability that at least one card is in the same position as the number it bears is $1 - \frac{1}{2 !} + \frac{1}{3 !} - \dots + (- 1)^{n + 1} \frac{1}{n !} .$ Find the probability that exactly one card is in the same position as the number it bears.
Probability & Statistics
The random variable $X$ has a binomial distribution with parameters $n$ and $p$ , where $n = 16$ and $p = \frac{1}{2}$ . Show, using an approximation in terms of the standard normal density function $\frac{1}{2 π} e^{- \frac{1}{2} x^{2}}$ , that $P (X = 8) \approx \frac{1}{2 2 π} .$
By considering a binomial distribution with parameters $2 n$ and $\frac{1}{2}$ , show that $(2 n)! \approx \frac{2 ^{2 n} ( n ! ) ^{2}}{nπ} .$
By considering a Poisson distribution with parameter $n$ , show that $n! \approx 2 π n e^{- n} n^{n} .$
Probability & Statistics