ZIMSEC A Level Pure Mathematics Paper 2 November 2010

Section A

Section A, Question 1

[2 marks]Trigonometric identities

Angles

A

and

B

satisfy

A + B = 45^\circ

, and it follows that

\tan A + \tan B + \tan A\tan B = 1

. By putting

A = B = 22\tfrac12^\circ

and writing

t = \tan 22\tfrac12^\circ

, write down the resulting quadratic equation in

t

Answer this when you sit the paper.

[3 marks]Trigonometric identities

For angles with

A + B = 45^\circ

it can be shown that

\tan A + \tan B + \tan A\tan B = 1

. Use this result to find the exact value of

\tan 22\tfrac12^\circ

in its simplest surd form.

Answer this when you sit the paper.

[1 marks]Trigonometric identities

Angles

A

and

B

satisfy

A + B = 45^\circ

. Which standard result is the starting point for proving that

\tan A + \tan B + \tan A\tan B = 1

AThe identity $\sec^2 A = 1 + \tan^2 A$ , applied to each of the angles in turn.
BThe compound angle formula for $\tan(A+B)$ , together with $\tan 45^\circ = 1$ .
CThe sine rule applied to a triangle whose angles are $A$ , $B$ and $45^\circ$ .
DThe double angle formula for $\tan 2A$ , together with $\tan 45^\circ = 1$ .

Section A, Question 2

[2 marks]Mathematical induction

The statement '

7^n - 6n - 1

is divisible by 36' is to be proved by induction for

n \in \mathbb{N}

n > 1

. Evaluate

7^n - 6n - 1

for the base case

n = 2

Answer this when you sit the paper.

[3 marks]Mathematical induction

In an induction proof that

7^n - 6n - 1

is divisible by 36, assume the result for

n = k

, so that

7^k - 6k - 1 = 36p

and hence

7^k = 36p + 6k + 1

. Simplify

7^{k+1} - 6(k+1) - 1

to a single product showing the factor 36.

Answer this when you sit the paper.

[1 marks]Mathematical induction

In proving by induction that

7^n - 6n - 1

is divisible by 36, the inductive step replaces

7^k

36p + 6k + 1

. Why is that substitution the key move?

AIt establishes the base case, which is the part of an induction argument that does the real work.
BIt shows that $7^k$ is itself divisible by 36, so every multiple of it is divisible by 36 too.
CIt brings in the inductive assumption, so the $n = k+1$ expression can be written as a multiple of 36.
DIt removes the variable $k$ entirely, leaving an expression that is plainly a multiple of 36.

Section A, Question 3

[2 marks]Maclaurin series

Using the Maclaurin series for

e^x

, write down the series for

e^{-2x}

up to and including the term in

x^3

, with the terms simplified.

Answer this when you sit the paper.

[2 marks]Maclaurin series

Using the Maclaurin series for

\sin x

, write down the series for

\sin 3x

up to and including the term in

x^3

, with the terms simplified.

Answer this when you sit the paper.

[2 marks]Maclaurin series

Given that

e^{-2x} = 1 - 2x + 2x^2 - \dfrac{4x^3}{3}

and

\sin 3x = 3x - \dfrac{9x^3}{2}

up to the

x^3

terms, find the series expansion of

f(x) = e^{-2x}\sin 3x

up to the term in

x^3

Answer this when you sit the paper.

[2 marks]Maclaurin series

For

f(x) = e^{-2x}\sin 3x

, differentiate once and hence state the value of

f'(0)

Answer this when you sit the paper.

[1 marks]Maclaurin series

The expansion of

f(x) = e^{-2x}\sin 3x

up to

x^3

3x - 6x^2 + \tfrac32 x^3

, and differentiating

f(x)

directly gives

f'(0) = 3

. What does that check confirm?

AThe coefficient of $x^3$ , because the highest term of a Maclaurin series is fixed by the first derivative.
BThat the expansion has no constant term, which is the only feature a first derivative can test.
CThat two series may be multiplied term by term without any loss of accuracy in the product.
DThe coefficient of $x$ in the expansion, because the $x$ coefficient of a Maclaurin series is $f'(0)$ .

Section A, Question 4

[1 marks]Vectors in three dimensions

The points A and B have coordinates

(1; 2; 4)

and

(3; -2; 1)

respectively. Find

\overrightarrow{AB}

Answer this when you sit the paper.

[2 marks]Vectors in three dimensions

The points A and B have coordinates

(1; 2; 4)

and

(3; -2; 1)

respectively. Find

\left|\overrightarrow{AB}\right|

, leaving your answer in surd form.

Answer this when you sit the paper.

[1 marks]Vectors in three dimensions

The points A and B have coordinates

(1; 2; 4)

and

(3; -2; 1)

respectively. Write down the vector equation of the straight line that passes through A and B.

Answer this when you sit the paper.

[3 marks]Vectors in three dimensions

Plane

p

has equation

\mathbf{r}.(2\mathbf{i} + \mathbf{j} - 3\mathbf{k}) = 28

and plane

q

has equation

\mathbf{r}.(4\mathbf{i} - 7\mathbf{j} + \mathbf{k}) = 31

. Find a direction vector of their line of intersection, in its simplest integer form.

Answer this when you sit the paper.

[3 marks]Vectors in three dimensions

Plane

p

has equation

\mathbf{r}.(2\mathbf{i} + \mathbf{j} - 3\mathbf{k}) = 28

and plane

q

has equation

\mathbf{r}.(4\mathbf{i} - 7\mathbf{j} + \mathbf{k}) = 31

. Find the angle between plane

p

and plane

q

, giving your answer correct to the nearest

0,1^\circ

Answer this when you sit the paper.

Section A, Question 5

[1 marks]Matrices and transformations

A sequence of matrices is given by

\begin{pmatrix}0&1\\1&0\end{pmatrix}^{n}

n \in \mathbb{N}

. Evaluate the second term of the sequence, that is the case

n = 2

Answer this when you sit the paper.

[2 marks]Matrices and transformations

The sequence of matrices

\begin{pmatrix}0&1\\1&0\end{pmatrix}^{n}

n \in \mathbb{N}

, has first three terms

\begin{pmatrix}0&1\\1&0\end{pmatrix}

\begin{pmatrix}1&0\\0&1\end{pmatrix}

\begin{pmatrix}0&1\\1&0\end{pmatrix}

. State the behaviour of the sequence as

n

increases.

AIts entries grow without bound, so the sequence diverges as $n$ increases without limit.
BIt converges to the zero matrix, because the repeated swapping cancels the entries out.
CIt converges to the identity matrix, because the even powers come to dominate as $n$ grows.
DIt oscillates with period 2 between $\begin{pmatrix}0&1\\1&0\end{pmatrix}$ and the identity, so it never settles.

[3 marks]Matrices and transformations

Given that

A = \begin{pmatrix}1&x&y\\-2&1&3\\-3&2&4\end{pmatrix}

, show that

|A| = 2

reduces to the equation of a straight line, and state that equation.

Answer this when you sit the paper.

[2 marks]Matrices and transformations

Describe completely the single transformation represented by the matrix

T = \begin{pmatrix}1&0\\2&1\end{pmatrix}

AA shear of factor 2 parallel to the $y$ -axis, with the $y$ -axis invariant.
BA shear of factor 2 parallel to the $x$ -axis, with the $x$ -axis invariant.
CA stretch of factor 2 parallel to the $y$ -axis, with the $x$ -axis invariant.
DA reflection in the line $y = 2x$ , which leaves every point of that line fixed.

[2 marks]Matrices and transformations

Describe completely the single transformation represented by the matrix

M = \begin{pmatrix}0&1\\-1&0\end{pmatrix}

AA reflection in the line $y = x$ , which leaves every point of that line fixed.
BA rotation of $90^\circ$ clockwise about the origin, that is $270^\circ$ anticlockwise.
CA reflection in the $x$ -axis followed by a stretch of factor 2 parallel to it.
DA rotation of $90^\circ$ anticlockwise about the origin, that is $270^\circ$ clockwise.

[3 marks]Matrices and transformations

T = \begin{pmatrix}1&0\\2&1\end{pmatrix}

and

M = \begin{pmatrix}0&1\\-1&0\end{pmatrix}

. Find the coordinates of the point whose image under

(TM)^{-1}

(3; 1)

Answer this when you sit the paper.

Section A, Question 6

[1 marks]Complex numbers

The complex number

z = a + bi

, where

a

and

b

are positive real numbers, and

w = iz

. Write down

w

in terms of

a

and

b

Answer this when you sit the paper.

[2 marks]Complex numbers

The complex number

z = a + bi

, where

a

and

b

are positive real numbers, and

w = iz = -b + ai

. Explain the geometrical relationship between

z

and

w

on an Argand diagram.

A $w$ is $z$ enlarged about the origin by scale factor $i$ , so that $|w| = |z|^2$ .
B $w$ is $z$ rotated $90^\circ$ anticlockwise about the origin, and $|w| = |z|$ .
C $w$ is $z$ rotated $90^\circ$ clockwise about the origin, and $|w| = |z|$ .
D $w$ is the reflection of $z$ in the real axis, so $w$ is the conjugate of $z$ .

[2 marks]Complex numbers

Given that

w = iz

and

z = 3 + 2i

, find

w

in the form

x + yi

Answer this when you sit the paper.

[3 marks]Complex numbers

The complex numbers

w

and

v

are defined by

w = iz

and

v = \tfrac12 z + w

. Find

v

in the form

x + yi

when

z = 3 + 2i

Answer this when you sit the paper.

[2 marks]Complex numbers

The complex number

z = a + bi

has

w = iz = -b + ai

, and

v = \tfrac12 z + w

. Write down the coordinates of the point representing

v

on an Argand diagram, in terms of

a

and

b

Answer this when you sit the paper.

[3 marks]Complex numbers

Use De Moivre's theorem to find the 4 roots of unity, giving your answers in exponential form.

Answer this when you sit the paper.

Section A, Question 7

[2 marks]Trigonometric identities and three dimensional trigonometry

Simplify

\dfrac{\cos^4\theta - \sin^4\theta}{\sin\theta\cos^3\theta + \sin^3\theta\cos\theta}

by factorising the numerator and the denominator.

Answer this when you sit the paper.

[2 marks]Trigonometric identities and three dimensional trigonometry

Express

\dfrac{\cos^2\theta - \sin^2\theta}{\sin\theta\cos\theta}

as a single trigonometric function of

2\theta

Answer this when you sit the paper.

[2 marks]Trigonometric identities and three dimensional trigonometry

A rectangular piece of cardboard ABCD is fixed against a vertical wall. The edge AB is inclined at

\theta^\circ

and the edge BC at

\alpha^\circ

to the horizontal. Why must

\theta + \alpha = 90

AThe three heights 41, 62 and 104 are in arithmetic progression, which forces a right angle at B.
BThe wall is vertical, so every edge of the cardboard makes an angle of $45^\circ$ with the ground.
CABCD is a rectangle, so AB and BC are perpendicular and their inclinations must add to $90^\circ$ .
DAB is three times BC, and the ratio of two sides always fixes their inclinations at $90^\circ$ .

[2 marks]Trigonometric identities and three dimensional trigonometry

A rectangular cardboard ABCD leans against a vertical wall with A, B and C at heights 41 cm, 62 cm and 104 cm above the ground. AB and BC are inclined at

\theta^\circ

and

\alpha^\circ

to the horizontal, with

\theta + \alpha = 90

and

6\sin\theta = \sin\alpha

. Find

\tan\theta

Answer this when you sit the paper.

[2 marks]Trigonometric identities and three dimensional trigonometry

For a rectangular cardboard leaning on a wall, the inclinations satisfy

\theta + \alpha = 90

and

6\sin\theta = \sin\alpha

. Solve for

\theta

, giving your answer in degrees correct to 2 decimal places.

Answer this when you sit the paper.

[3 marks]Trigonometric identities and three dimensional trigonometry

A rectangular cardboard ABCD leans against a vertical wall, with A at height 41 cm and B at height 62 cm above the ground. AB is inclined at

\theta^\circ

to the horizontal where

\tan\theta = \tfrac16

. Calculate the length of AB, in cm, correct to 1 decimal place.

Answer this when you sit the paper.

[2 marks]Trigonometric identities and three dimensional trigonometry

A rectangular piece of cardboard ABCD is fixed against a vertical wall. The heights of A, B and C above the ground are 41 cm, 62 cm and 104 cm. Find the height of D above the horizontal, in cm.

Answer this when you sit the paper.

Section B

Section B, Question 8

[2 marks]Projectiles

A stone is projected from a point A with a speed of

20\,\mathrm{ms}^{-1}

30^\circ

to the horizontal, towards a bird 30 m away horizontally. Find the time taken for the stone to travel the 30 m horizontally.

Answer this when you sit the paper.

[3 marks]Projectiles

A stone is projected from A at

20\,\mathrm{ms}^{-1}

30^\circ

to the horizontal towards a bird standing 2,5 m above level ground on a pole 30 m away. Taking

g = 9,81\,\mathrm{ms}^{-2}

and reaching the pole after 1,73 s, find how far vertically above or below the bird the stone passes.

Answer this when you sit the paper.

Section B, Question 9

[2 marks]Friction on an inclined plane

A block of weight 20 N rests on a rough plane inclined at

\beta

where

\sin\beta = \tfrac45

, acted on by a horizontal force of

P

N pressing towards the plane. Resolving perpendicular to the plane, express the normal reaction

R

in terms of

P

Answer this when you sit the paper.

[2 marks]Friction on an inclined plane

A block of weight 20 N is on the point of slipping down a rough plane inclined at

\beta

where

\sin\beta = \tfrac45

, held by a horizontal force of

P

N and by limiting friction with

\mu = \tfrac14

. Resolving along the plane, express the normal reaction

R

in terms of

P

Answer this when you sit the paper.

[2 marks]Friction on an inclined plane

For a block on an inclined plane the two resolved equations give

R = 64 - \dfrac{12P}{5}

and

R = \dfrac{4P}{5} + 12

. Solve for the horizontal force

P

, in newtons.

Answer this when you sit the paper.

Section B, Question 10

[2 marks]Connected particles

Masses of 2 kg and 5 kg hang on either side of a smooth fixed pulley, joined by a light inextensible string, and are released from rest. Find the acceleration of the particles in terms of

g

Answer this when you sit the paper.

[2 marks]Connected particles

Masses of 2 kg and 5 kg hang either side of a smooth fixed pulley and are released from rest, each 2,5 m above the ground, with acceleration

\dfrac{3g}{7}

. Taking

g = 9,81\,\mathrm{ms}^{-2}

, find the speed of the 5 kg mass as it hits the ground.

Answer this when you sit the paper.

[3 marks]Connected particles

Masses of 2 kg and 5 kg hang either side of a smooth fixed pulley, released from rest 2,5 m above the ground. The 5 kg mass lands at

4,58\,\mathrm{ms}^{-1}

and does not rebound. Taking

g = 9,81\,\mathrm{ms}^{-2}

, find the greatest height above the ground reached by the 2 kg mass, assuming it does not reach the pulley.

Answer this when you sit the paper.

Section B, Question 11

[2 marks]Kinematics of a particle

A particle starts from rest and moves in a straight line with acceleration

a = 3\,\mathrm{ms}^{-2}

for

0 \le t \le 2

and

a = -3\,\mathrm{ms}^{-2}

for

2 < t \le 8

. Find its velocity when

t = 2

seconds.

Answer this when you sit the paper.

[2 marks]Kinematics of a particle

A particle starts from rest and moves in a straight line with acceleration

a = 3\,\mathrm{ms}^{-2}

for

0 \le t \le 2

and

a = -3\,\mathrm{ms}^{-2}

for

2 < t \le 8

. Find the total distance it travels in the first 8 seconds.

Answer this when you sit the paper.

Section C

Section C, Question 12

[2 marks]Binomial and normal distributions

In a large batch of bolts the probability that a randomly chosen bolt is defective is

\tfrac16

, and a random sample of 42 bolts is taken. State the distribution of the number of defective bolts and the normal distribution that approximates it.

A $X\sim B\left(42;\tfrac16\right)$ , approximated by $Y\sim N\left(7;\tfrac{35}{6}\right)$ .
B $X\sim B\left(42;\tfrac16\right)$ , approximated by $Y\sim N\left(7;\,7\right)$ .
C $X\sim B\left(42;\tfrac56\right)$ , approximated by $Y\sim N\left(35;\tfrac{35}{6}\right)$ .
D $X\sim Po\left(7\right)$ , approximated by $Y\sim N\left(\tfrac16;\tfrac{35}{6}\right)$ .

[3 marks]Binomial and normal distributions

In a large batch of bolts the probability that a randomly chosen bolt is defective is

\tfrac16

. A random sample of 42 bolts is taken. Find the probability that more than two bolts are defective, correct to 3 decimal places.

Answer this when you sit the paper.

Section C, Question 13

[1 marks]Discrete random variables

A discrete random variable X has

P(X = x) = \dfrac{x-1}{16}

for

x = 2, 3, 4, 5

and

P(X = x) = \dfrac{9-x}{16}

for

x = 6, 7, 8

. Find

P(X = 4)

Answer this when you sit the paper.

[2 marks]Discrete random variables

A discrete random variable X takes the values 2, 3, 4, 5, 6, 7, 8 with probabilities

\tfrac1{16}, \tfrac2{16}, \tfrac3{16}, \tfrac4{16}, \tfrac3{16}, \tfrac2{16}, \tfrac1{16}

respectively. Calculate

E(X)

Answer this when you sit the paper.

[1 marks]Discrete random variables

A discrete random variable X takes the values 2, 3, 4, 5, 6, 7, 8 with probabilities

\tfrac1{16}, \tfrac2{16}, \tfrac3{16}, \tfrac4{16}, \tfrac3{16}, \tfrac2{16}, \tfrac1{16}

respectively. State the nature of the distribution.

ASymmetrical about $x = 5$ , which is why the mean is exactly 5.
BUniform, since every one of the seven probabilities is a number of sixteenths.
CPositively skewed, with a long tail running towards the larger values of $x$ .
DNegatively skewed, with a long tail running towards the smaller values of $x$ .

Section C, Question 14

[3 marks]Continuous random variables

A continuous random variable X has probability density function

f(x) = kx

for

0 \le x \le 3

f(x) = 3k(4-x)

for

3 \le x \le 4

and

f(x) = 0

otherwise. Find the value of

k

Answer this when you sit the paper.

[1 marks]Continuous random variables

A continuous random variable X has probability density function

f(x) = \tfrac16 x

for

0 \le x \le 3

f(x) = \tfrac12(4-x)

for

3 \le x \le 4

and

f(x) = 0

otherwise. State the greatest value taken by

f(x)

, and the value of

x

at which it occurs.

Answer this when you sit the paper.

[3 marks]Continuous random variables

A continuous random variable X has probability density function

f(x) = \tfrac16 x

for

0 \le x \le 3

f(x) = \tfrac12(4-x)

for

3 \le x \le 4

and

f(x) = 0

otherwise. Find the probability that

x > 2

Answer this when you sit the paper.

Section C, Question 15

[1 marks]Probability

Three players A, B and C throw a fair cubical die in that order, and the first to throw a 6 wins. Find the probability that A wins on his first throw.

Answer this when you sit the paper.

[2 marks]Probability

Three players A, B and C throw a fair cubical die in that order, and the first to throw a 6 wins, the game continuing until someone does. Find the probability that A wins on his second throw.

Answer this when you sit the paper.

[3 marks]Probability

Three players A, B and C throw a fair cubical die in that order, and the first to throw a 6 wins, the game continuing indefinitely until one of them does. Find the probability that A wins the game.

Answer this when you sit the paper.

[2 marks]Probability

Three players A, B and C throw a fair cubical die in that order, and the first to throw a 6 wins. The probability that A wins is

\tfrac{36}{91}

and the probability that B wins is

\tfrac{30}{91}

. Find the probability that C wins.

Answer this when you sit the paper.

Pure Mathematics Paper 2 November 2010

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The questions

Section A

Section A, Question 1

Section A, Question 2

Section A, Question 3

Section A, Question 4

Section A, Question 5

Section A, Question 6

Section A, Question 7

Section B

Section B, Question 8

Section B, Question 9

Section B, Question 10

Section B, Question 11

Section C

Section C, Question 12

Section C, Question 13

Section C, Question 14

Section C, Question 15

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The answers, and why they are the answers

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