ZIMSEC A Level Pure Mathematics Paper 2 November 2009

Section a

Section a, Question 1

[2 marks]Proof by mathematical induction

In a proof by induction that

5^{2n}-3^{2n}

is a multiple of 8 for every positive integer

n

, the first step is to test

n=1

. Evaluate

5^{2n}-3^{2n}

when

n=1

Answer this when you sit the paper.

[3 marks]Proof by mathematical induction

In proving by induction that

5^{2n}-3^{2n}

is a multiple of 8, you assume that

5^{2k}-3^{2k}=8m

for some integer

m

. Which of these correctly rewrites

5^{2(k+1)}-3^{2(k+1)}

so that the assumption can be used?

A $2\left(5^{2k}-3^{2k}\right)+16\cdot3^{2k}$
B $25\left(5^{2k}-3^{2k}\right)-16\cdot3^{2k}$
C $25\left(5^{2k}-3^{2k}\right)+16\cdot3^{2k}$
D $25\left(5^{2k}-3^{2k}\right)+9\cdot3^{2k}$

[2 marks]Proof by mathematical induction

A student shows that if

5^{2k}-3^{2k}

is a multiple of 8 then

5^{2(k+1)}-3^{2(k+1)}

is also a multiple of 8, and stops there. Why is the proof by induction not yet complete?

AThe implication must also be proved in the reverse direction.
BThe result must be checked for at least three consecutive values of $k$ .
CThe assumption is only valid once $k$ has been shown to be even.
DNo case has been shown true, so nothing starts the chain.

Section a, Question 2

[2 marks]De Moivre's theorem and powers of sine

Given that

z=\cos\theta+i\sin\theta

, express

\dfrac{1}{z}

in terms of

\theta

A $\sin\theta-i\cos\theta$
B $-\cos\theta-i\sin\theta$
C $\cos\theta-i\sin\theta$
D $\cos\theta+i\sin\theta$

[3 marks]De Moivre's theorem and powers of sine

Given that

z=\cos\theta+i\sin\theta

, so that

z-\dfrac{1}{z}=2i\sin\theta

and

z^{n}+z^{-n}=2\cos n\theta

, use De Moivre's theorem to express

\sin^{4}\theta

in terms of

\cos4\theta

and

\cos2\theta

A $\dfrac{1}{8}\cos4\theta-\dfrac{1}{2}\cos2\theta+\dfrac{3}{8}$
B $\dfrac{1}{8}\cos4\theta+\dfrac{1}{2}\cos2\theta+\dfrac{3}{8}$
C $\dfrac{1}{16}\cos4\theta-\dfrac{1}{4}\cos2\theta+\dfrac{3}{16}$
D $\dfrac{1}{2}\cos4\theta-\dfrac{1}{8}\cos2\theta+\dfrac{3}{8}$

[2 marks]De Moivre's theorem and powers of sine

Given that

z=\cos\theta+i\sin\theta

and

n

is a positive integer, simplify

z^{n}+z^{-n}

as a single trigonometric term.

Answer this when you sit the paper.

Section a, Question 3

[3 marks]Binomial expansion

In the expansion of

\left(1-\dfrac{x}{n}\right)^{n}

in ascending powers of

x

, the term in

x^{2}

kx^{2}

. Give

k

in terms of

n

, simplified as far as possible.

Answer this when you sit the paper.

[2 marks]Binomial expansion

In the expansion of

\left(1-\dfrac{x}{n}\right)^{n}

in ascending powers of

x

, what is the term in

x^{3}

A $-\dfrac{(n-1)(n-2)}{3n^{2}}x^{3}$
B $-\dfrac{(n-1)(n-2)}{6n^{2}}x^{3}$
C $\dfrac{(n-1)(n-2)}{6n^{2}}x^{3}$
D $-\dfrac{(n-1)(n-2)}{6n^{3}}x^{3}$

[3 marks]Binomial expansion

In the expansion of

\left(1-\dfrac{x}{n}\right)^{n}

, the term in

x^{3}

takes the value

\dfrac{7}{8}

when

x=-2

. Given that

n

is a positive integer, find

n

Answer this when you sit the paper.

Section a, Question 4

[2 marks]Differential equations and exponential growth

A population has a birth rate of 55 persons per 1 000 per year and a death rate of 14 persons per 1 000 per year, with no other changes. Write down the net rate of increase per person per year, as a fraction with denominator 1 000.

Answer this when you sit the paper.

[3 marks]Differential equations and exponential growth

A population of

P

persons at time

t

years satisfies

\dfrac{\mathrm{d}P}{\mathrm{d}t}=\dfrac{41}{1000}P

, and

P=10\,000

when

t=0

. Which is the solution of this equation?

A $P=10\,000e^{0.41t}$
B $P=10\,000+0.041t^{2}-3t$
C $P=10\,000t^{0.041}$
D $P=10\,000e^{0.041t}$

[3 marks]Differential equations and exponential growth

A community of 10 000 people in 2005 grows so that its population

t

years after 2005 is

P=10\,000e^{0.041t}

. Find the predicted population in the year 2020, to the nearest whole person.

Answer this when you sit the paper.

Section a, Question 5

[3 marks]Matrix multiplication and linear systems

Given

A=\begin{pmatrix}1&0&2\\2&1&0\\3&1&1\end{pmatrix}

and

B=\begin{pmatrix}1&-1&1\\0&2&1\\1&3&0\end{pmatrix}

, find

AB

A $\begin{pmatrix}3&2&4\\5&0&2\\1&3&4\end{pmatrix}$
B $\begin{pmatrix}3&5&1\\2&0&3\\4&2&4\end{pmatrix}$
C $\begin{pmatrix}3&5&1\\2&0&3\\4&2&3\end{pmatrix}$
D $\begin{pmatrix}1&0&2\\0&2&0\\3&3&0\end{pmatrix}$

[2 marks]Matrix multiplication and linear systems

Let

M=\begin{pmatrix}3&5&1\\2&0&3\\4&2&4\end{pmatrix}

and

C=\begin{pmatrix}-6&-18&15\\4&8&-7\\4&14&-10\end{pmatrix}

. Given that

MC=\begin{pmatrix}a&0&0\\0&a&0\\0&0&a\end{pmatrix}

, find the value of

a

Answer this when you sit the paper.

[2 marks]Matrix multiplication and linear systems

3\times3

matrix

M

satisfies

MC=6I

, where

C

is a known

3\times3

matrix and

I

is the identity. What is

M^{-1}

A $\tfrac{1}{6}C$
B $6C$
C $C$
D $\tfrac{1}{6}C^{2}$

[2 marks]Matrix multiplication and linear systems

Let

M=\begin{pmatrix}3&5&1\\2&0&3\\4&2&4\end{pmatrix}

. Given that

M\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}3\\1\\2\end{pmatrix}

, find the value of

x

Answer this when you sit the paper.

Section a, Question 6

[2 marks]Vectors: lines and planes in three dimensions

The line

l

passes through the points

Q(3,1,-2)

and

R(2,7,-4)

. Which vector is a direction vector of

l

A $(1,6,2)$
B $(6,-1,-2)$
C $(-1,6,-2)$
D $(5,8,-6)$

[3 marks]Vectors: lines and planes in three dimensions

The line

l

passes through

Q(3,1,-2)

and

R(2,7,-4)

. What is the cartesian equation of

l

A $\dfrac{x-3}{2}=\dfrac{y-1}{7}=\dfrac{z+2}{-4}$
B $\dfrac{x-3}{-1}=\dfrac{y-1}{6}=\dfrac{z+2}{-2}$
C $\dfrac{x-3}{-1}=\dfrac{y-1}{6}=\dfrac{z-2}{-2}$
D $\dfrac{x+3}{-1}=\dfrac{y+1}{6}=\dfrac{z-2}{-2}$

[2 marks]Vectors: lines and planes in three dimensions

The line

l

passes through

Q(3,1,-2)

and

R(2,7,-4)

. It meets the plane

x=y

at one point. Find the value of

x

at that point, as a fraction.

Answer this when you sit the paper.

[2 marks]Vectors: lines and planes in three dimensions

The plane

P_{1}

has equation

2x-3y-z-5=0

and the plane

P_{2}

has equation

-6x+9y+3z+2=0

. Which fact shows that

P_{1}

and

P_{2}

are parallel and distinct?

A $(-6,9,3)=-3(2,-3,-1)$ but $-3(-5)=15\ne2$ .
B $(-6,9,3)\cdot(2,-3,-1)=0$ and the constants differ.
CBoth equations have three variables and no common solution point.
DThe sum of the coefficients of $P_{2}$ is $-3$ times that of $P_{1}$ .

[3 marks]Vectors: lines and planes in three dimensions

The plane

P_{3}

has equation

3x+2y-6z+10=0

. The plane

P_{4}

passes through the point

P(5,-2,4)

and is parallel to

P_{3}

. Find the equation of

P_{4}

Answer this when you sit the paper.

[3 marks]Vectors: lines and planes in three dimensions

Find the perpendicular distance of the point

P(5,-2,4)

from the plane

2x-3y-z-5=0

, correct to 3 significant figures.

Answer this when you sit the paper.

Section a, Question 7

[2 marks]Complex numbers, roots of polynomials and De Moivre's theorem

Express

4\left(\sqrt{3}-i\right)

in the form

re^{i\theta}

with

r>0

. Find the value of

r

Answer this when you sit the paper.

[2 marks]Complex numbers, roots of polynomials and De Moivre's theorem

Express

4\left(\sqrt{3}-i\right)

in the form

re^{i\theta}

where

r>0

and

-\pi<\theta\le\pi

. Find

\theta

as a multiple of

\pi

Answer this when you sit the paper.

[2 marks]Complex numbers, roots of polynomials and De Moivre's theorem

The equation

x^{4}-4x^{3}-6x^{2}+20x-75=0

has

1+2i

as a root. Why must

1-2i

also be a root?

AThe constant term is negative, which forces conjugate roots.
BThe sum of the roots is 4, which is real, so the roots pair up.
CAll the coefficients are real, so complex roots occur in conjugate pairs.
DA quartic equation must have exactly two complex roots and two real roots.

[3 marks]Complex numbers, roots of polynomials and De Moivre's theorem

Express

\sin5\theta

in terms of powers of

\sin\theta

using De Moivre's theorem.

A $16\sin^{5}\theta-20\sin^{3}\theta-5\sin\theta$
B $16\sin^{5}\theta-20\sin^{3}\theta+5\sin\theta$
C $16\sin^{5}\theta+20\sin^{3}\theta-5\sin\theta$
D $5\sin^{5}\theta-20\sin^{3}\theta+16\sin\theta$

[3 marks]Complex numbers, roots of polynomials and De Moivre's theorem

Given that

x_{1}=1+2i

is a root of

x^{4}-4x^{3}-6x^{2}+20x-75=0

, find the two real roots of the equation.

Answer this when you sit the paper.

[3 marks]Complex numbers, roots of polynomials and De Moivre's theorem

Using the identity

\sin5\theta-5\sin\theta=16\sin^{5}\theta-20\sin^{3}\theta

, find the exact value of

\displaystyle\int_{\pi/6}^{\pi/2}\left(16\sin^{5}\theta-20\sin^{3}\theta\right)\mathrm{d}\theta

Answer this when you sit the paper.

Section b

Section b, Question 8

[3 marks]Motion on a rough inclined plane

A block is released from rest on a plane inclined at angle

\alpha

to the horizontal, where

\tan\alpha=\dfrac{3}{4}

, and slides down. The coefficient of friction between the block and the plane is

\dfrac{1}{4}

. Taking

g=9.81

^{-2}

, calculate the acceleration of the block down the plane, in ms

^{-2}

Answer this when you sit the paper.

[2 marks]Motion on a rough inclined plane

A block slides down a rough plane inclined at

\alpha

to the horizontal, with coefficient of friction

\mu

. Why does the acceleration down the plane not depend on the mass of the block?

AFriction on an inclined plane does not depend on the normal reaction.
BThe normal reaction is equal to the weight, whatever the angle of the plane.
CMass affects the speed reached but never the rate at which speed changes.
DWeight component and friction are both proportional to mass, so it cancels.

[2 marks]Motion on a rough inclined plane

A block of mass 3.5 kg rests on a plane inclined at angle

\alpha

to the horizontal, where

\tan\alpha=\dfrac{3}{4}

, and slides down it. The coefficient of friction is

\dfrac{1}{4}

. Taking

g=9.81

^{-2}

, calculate the friction force on the block, in newtons, to 3 significant figures.

Answer this when you sit the paper.

Section b, Question 9

[2 marks]Friction and Newton's second law

A ring of mass

m

kg slides along a rough horizontal wire. A force of 6 N acts on it at

60^{\circ}

above the wire, in the same vertical plane as the wire. Write down the normal reaction between the ring and the wire, in terms of

m

and

g

, in exact form.

Answer this when you sit the paper.

[2 marks]Friction and Newton's second law

A ring of mass

m

kg slides along a rough horizontal wire under a force of 6 N acting at

60^{\circ}

above the wire. The coefficient of friction between the ring and the wire is

\dfrac{1}{4}

and the normal reaction is

mg-3\sqrt3

N. Write down the friction force in exact form.

Answer this when you sit the paper.

[3 marks]Friction and Newton's second law

A ring of mass

m

kg accelerates at 1 ms

^{-2}

along a rough horizontal wire under a force of 6 N acting at

60^{\circ}

above the wire, in the same vertical plane as the wire. The coefficient of friction is

\dfrac{1}{4}

. Taking

g=9.81

^{-2}

, find

m

to 3 decimal places.

Answer this when you sit the paper.

Section b, Question 10

[3 marks]Projectile motion

A particle is projected from a point O on the ground with speed

V

^{-1}

60^{\circ}

to the horizontal. Taking O as the origin, it passes through the point

A

with coordinates

\left(\sqrt3,\,2\right)

metres. Express

V^{2}

in terms of

g

Answer this when you sit the paper.

[2 marks]Projectile motion

A particle is projected from a point O on the ground at

60^{\circ}

to the horizontal with speed

V

^{-1}

, where

V^{2}=6g

. Find the greatest height, in metres, that the particle reaches above O.

Answer this when you sit the paper.

[2 marks]Projectile motion

A particle is projected from a point O on the ground at

60^{\circ}

to the horizontal with speed

V

^{-1}

, where

V^{2}=6g

. Find the horizontal distance from O, in metres, at which the particle reaches its greatest height, correct to 3 significant figures.

Answer this when you sit the paper.

[3 marks]Projectile motion

A particle projected from O passes through

A\left(\sqrt3,\,2\right)

on its way up and reaches its greatest height at

B\left(\dfrac{3\sqrt3}{2},\,\dfrac{9}{4}\right)

, distances in metres. Find the angle, in degrees, that AB makes with the horizontal, correct to 1 decimal place.

Answer this when you sit the paper.

Section c

Section c, Question 11

[2 marks]Binomial probability

In the long run 40% of the people who inquire about a room at a motel actually book one. The owners want to be 99% sure of getting at least one booking from

n

inquiries. Which inequality expresses that requirement?

A $0.4^{n}\ge0.99$
B $1-0.4^{n}\ge0.99$
C $0.6^{n}\ge0.99$
D $1-0.6^{n}\ge0.99$

[3 marks]Binomial probability

In the long run 40% of the people who inquire about a room at a motel actually book one, independently of each other. How many inquiries must the owners answer to be 99% sure of at least one booking?

Answer this when you sit the paper.

Section c, Question 12

[2 marks]Data presentation and the normal distribution

Give one advantage of using a stem and leaf diagram rather than a grouped frequency table to analyse a set of data.

AEvery original data value is kept and shown in rank order.
BIt can display data that has no numerical order at all.
CIt removes outliers from the data automatically.
DIt shows the mean and the standard deviation without any calculation.

[1 marks]Data presentation and the normal distribution

In which of these situations is the mode the most appropriate measure of central tendency?

AA shop deciding which shoe size to stock the most of.
BA teacher reporting the average score of a class in a test.
CAn economist reporting the typical household income in a country.
DA farmer working out the average mass of a batch of harvested maize.

[3 marks]Data presentation and the normal distribution

The masses of a group of adult males are normally distributed with mean 65 kg and standard deviation 10 kg. A male is considered overweight if he is in the top 5% of the group by mass. Find the least mass, in kg, to be considered overweight, correct to 1 decimal place.

Answer this when you sit the paper.

Section c, Question 13

[3 marks]Probability with selections without replacement

Fifteen girls and ten boys each want one of three tickets. The three people to receive a ticket are chosen at random from the 25. Find the probability that exactly 2 boys receive a ticket, correct to 3 significant figures.

Answer this when you sit the paper.

[3 marks]Probability with selections without replacement

Fifteen girls and ten boys each want one of three tickets. The three people to receive a ticket are chosen at random from the 25. Find the probability that at least 2 girls receive a ticket, correct to 3 significant figures.

Answer this when you sit the paper.

Section c, Question 14

[2 marks]Continuous random variables and conditional probability

The duration

X

minutes of a telephone call has probability density function

f(x)=x^{-2}

for

x\ge1

and

f(x)=0

otherwise. Find

P(X>5)

Answer this when you sit the paper.

[2 marks]Continuous random variables and conditional probability

The duration

X

minutes of a telephone call has probability density function

f(x)=x^{-2}

for

x\ge1

. A call has already lasted 5 minutes. Which expression gives the probability that its total duration is less than 7 minutes?

A $\dfrac{P(X>5)}{P(5<X<7)}$
B $\dfrac{P(5<X<7)}{P(X>5)}$
C $\dfrac{P(X<7)}{P(X>5)}$
D $P(5<X<7)\times P(X>5)$

[3 marks]Continuous random variables and conditional probability

The duration

X

minutes of a telephone call has probability density function

f(x)=x^{-2}

for

x\ge1

and

f(x)=0

otherwise. Given that a call has already lasted 5 minutes, find the probability that its total duration will be less than 7 minutes.

Answer this when you sit the paper.

Pure Mathematics Paper 2 November 2009

Sit this paper online

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 c

Section c, Question 11

Section c, Question 12

Section c, Question 13

Section c, Question 14

More sittings of this paper

The answers, and why they are the answers

Sit this paper online