ZIMSEC A Level Pure Mathematics Paper 2 June 2009

Section a

Section a, Question 1

[1 marks]Recurring decimals as fractions, geometric series

The recurring decimal

5,4\dot{2}\dot{7}

can be written as

5,4+0,027+0,00027+0,0000027+\ldots

. What is the common ratio of the geometric series that follows the 5,4?

A0,001
B0,01
C0,027
D0,1

[3 marks]Recurring decimals as fractions, geometric series

Given that

5,4\dot{2}\dot{7}=5,4+0,027+0,00027+0,0000027+\ldots

, express the number

5,4\dot{2}\dot{7}

in the form

\dfrac{a}{b}

, where

a

and

b

are integers.

Answer this when you sit the paper.

Section a, Question 2

[3 marks]Parametric differentiation, eliminating a parameter

A curve has parametric equations

x=a\sin^{2}\theta+\theta

and

y=a\cos^{2}\theta-\theta

. Find

\dfrac{dy}{dx}

Answer this when you sit the paper.

[2 marks]Parametric differentiation, eliminating a parameter

A curve has parametric equations

x=a\sin^{2}\theta+\theta

and

y=a\cos^{2}\theta-\theta

. By eliminating

\theta

, express the curve as a Cartesian equation.

Answer this when you sit the paper.

Section a, Question 3

[1 marks]Trigonometry: angles of a triangle and the sine rule

In a triangle ABC,

\hat{BAC}=\left(\dfrac{\pi}{2}+\theta\right)

radians and

\hat{ACB}=\left(\theta-\dfrac{\pi}{2}\right)

radians. Find

\hat{ABC}

in radians.

Answer this when you sit the paper.

[3 marks]Trigonometry: angles of a triangle and the sine rule

In a triangle ABC,

\hat{BAC}=\left(\dfrac{\pi}{2}+\theta\right)

radians and

\hat{ABC}=(\pi-2\theta)

radians. Use the sine rule to express

\dfrac{BC}{AC}

in terms of

\theta

as simply as possible.

Answer this when you sit the paper.

[2 marks]Trigonometry: angles of a triangle and the sine rule

In a triangle ABC it is known that

\dfrac{BC}{AC}=\dfrac{1}{2\sin\theta}

. Find the value of

\theta

, in radians, for which

BC=2AC

Answer this when you sit the paper.

Section a, Question 4

[2 marks]Proof by induction applied to repeated differentiation

The result

\dfrac{d^{n}(x^{m})}{dx^{n}}=\dfrac{m!}{(m-n)!}x^{m-n}

is to be proved by induction. Write down the expression the formula gives when

n=1

Answer this when you sit the paper.

[3 marks]Proof by induction applied to repeated differentiation

In proving

\dfrac{d^{n}(x^{m})}{dx^{n}}=\dfrac{m!}{(m-n)!}x^{m-n}

by induction, you assume it for

n=k

and differentiate once more. Which step completes the argument?

A $\dfrac{m!}{(m-k)!}(m-k)x^{m-k}=\dfrac{m!}{(m-k-1)!}x^{m-k}$
B $\dfrac{m!}{(m-k)!}(m-k)x^{m-k-1}=\dfrac{m!}{(m-k-1)!}x^{m-k-1}$
C $\dfrac{m!}{(m-k)!}x^{m-k}\times x=\dfrac{m!}{(m-k)!}x^{m-k+1}$
D $\dfrac{(m-k)!}{m!}(m-k)x^{m-k-1}=\dfrac{m!}{(m-k+1)!}x^{m-k-1}$

[3 marks]Proof by induction applied to repeated differentiation

Use the result

\dfrac{d^{n}(x^{m})}{dx^{n}}=\dfrac{m!}{(m-n)!}x^{m-n}

to find the third derivative of

x^{7}

Answer this when you sit the paper.

Section a, Question 5

[3 marks]Binomial and Maclaurin series expansions

Expand

\dfrac{1+ax}{1-bx}

as a series in ascending powers of

x

, up to and including the term in

x^{3}

A $1+(a+b)x+(a+b)^{2}x^{2}+(a+b)^{3}x^{3}$
B $1+ax+abx^{2}+ab^{2}x^{3}$
C $1+(a+b)x+b(a+b)x^{2}+b^{2}(a+b)x^{3}$
D $1+(a-b)x+b(a-b)x^{2}+b^{2}(a-b)x^{3}$

[2 marks]Binomial and Maclaurin series expansions

Write down the Maclaurin's theorem expansion of

e^{x}

up to and including the term in

x^{3}

Answer this when you sit the paper.

[2 marks]Binomial and Maclaurin series expansions

Given that

\dfrac{1+ax}{1-bx}=1+(a+b)x+b(a+b)x^{2}+b^{2}(a+b)x^{3}+\ldots

and

e^{x}=1+x+\dfrac{x^{2}}{2}+\dfrac{x^{3}}{6}+\ldots

, determine the values of

a

and

b

for which

\dfrac{1+ax}{1-bx}-e^{x}=cx^{3}

Answer this when you sit the paper.

[2 marks]Binomial and Maclaurin series expansions

With

a=b=\dfrac12

, the expansion of

\dfrac{1+ax}{1-bx}

differs from that of

e^{x}

only in the term in

x^{3}

, so that

\dfrac{1+ax}{1-bx}-e^{x}=cx^{3}

. State the value of

c

Answer this when you sit the paper.

Section a, Question 6

[2 marks]Inverse of a 3 by 3 matrix, simultaneous equations

Given

\mathbf{A}=\begin{pmatrix}-1&3&1\\2&5&0\\3&1&-2\end{pmatrix}

, find

\det\mathbf{A}

Answer this when you sit the paper.

[2 marks]Inverse of a 3 by 3 matrix, simultaneous equations

For the matrix

\mathbf{A}=\begin{pmatrix}-1&3&1\\2&5&0\\3&1&-2\end{pmatrix}

, find the cofactor of the top-left element

-1

Answer this when you sit the paper.

[3 marks]Inverse of a 3 by 3 matrix, simultaneous equations

Given

\mathbf{A}=\begin{pmatrix}-1&3&1\\2&5&0\\3&1&-2\end{pmatrix}

with

\det\mathbf{A}=9

, which matrix is

\mathbf{A}^{-1}

A $\dfrac19\begin{pmatrix}-10&7&-5\\4&-1&2\\-13&10&-11\end{pmatrix}$
B $\dfrac19\begin{pmatrix}10&-7&5\\-4&1&-2\\13&-10&11\end{pmatrix}$
C $\dfrac19\begin{pmatrix}-1&3&1\\2&5&0\\3&1&-2\end{pmatrix}$
D $\dfrac19\begin{pmatrix}-10&4&-13\\7&-1&10\\-5&2&-11\end{pmatrix}$

[3 marks]Inverse of a 3 by 3 matrix, simultaneous equations

Using

\mathbf{A}^{-1}=\dfrac19\begin{pmatrix}-10&7&-5\\4&-1&2\\-13&10&-11\end{pmatrix}

, solve the equations

-x+3y+z=1

2x+5y=3

3x+y-2z=3

Answer this when you sit the paper.

Section a, Question 7

[2 marks]Complex numbers: roots of a quartic, exponential form, loci

Express the complex number

e^{i\frac{5\pi}{6}}

in the form

a+ib

, where

a

and

b

are real.

Answer this when you sit the paper.

[2 marks]Complex numbers: roots of a quartic, exponential form, loci

The polynomial

f(z)=z^{4}+\sqrt3z^{3}+2z^{2}+\sqrt3z+1

has real coefficients and

e^{i\frac{5\pi}{6}}

as a root. Why must

e^{-i\frac{5\pi}{6}}

also be a root?

ABecause the roots of any polynomial with irrational coefficients are always symmetric about the imaginary axis.
BBecause the non-real roots of a polynomial with real coefficients occur in conjugate pairs.
CBecause the coefficients of $f(z)$ include $\sqrt3$ , which is irrational.
DBecause every quartic must have four roots of equal modulus.

[3 marks]Complex numbers: roots of a quartic, exponential form, loci

Given that

e^{i\frac{5\pi}{6}}

and its conjugate are roots of

f(z)=z^{4}+\sqrt3z^{3}+2z^{2}+\sqrt3z+1

, find the two quadratic factors of

f(z)

Answer this when you sit the paper.

[3 marks]Complex numbers: roots of a quartic, exponential form, loci

Solve

f(z)=z^{4}+\sqrt3z^{3}+2z^{2}+\sqrt3z+1=0

completely, given that

f(z)=(z^{2}+\sqrt3z+1)(z^{2}+1)

. Give the roots in the form

e^{i\theta}

Answer this when you sit the paper.

[2 marks]Complex numbers: roots of a quartic, exponential form, loci

Describe fully the locus of points in the Argand diagram for which

|z-1+i|=2

Answer this when you sit the paper.

[2 marks]Complex numbers: roots of a quartic, exponential form, loci

Which region of the Argand diagram is defined by

-\dfrac{\pi}{3}\le\arg(z)\le\dfrac{\pi}{3}

AThe infinite wedge with vertex at the origin between the half-lines at $\dfrac{\pi}{3}$ above and $\dfrac{\pi}{3}$ below the negative real axis, with both boundary half-lines included.
BA disc of radius $\dfrac{\pi}{3}$ centred at the origin.
CThe half-plane to the right of the imaginary axis.
DThe infinite wedge with vertex at the origin between the half-lines at $\dfrac{\pi}{3}$ above and $\dfrac{\pi}{3}$ below the positive real axis.

Section a, Question 8

[3 marks]Vectors: intersecting lines, line and plane intersections, area of a triangle

The lines

\ell_1

and

\ell_2

have equations

\mathbf{r}=(0,1,1)+\mu(1,3,2)

and

\mathbf{r}=(1,2,3)+\lambda(1,1,2)

. Find the coordinates of the point A where they intersect.

Answer this when you sit the paper.

[3 marks]Vectors: intersecting lines, line and plane intersections, area of a triangle

The line

\ell_1

has equation

\mathbf{r}=(0,1,1)+\mu(1,3,2)

and the plane

\pi

has equation

\mathbf{r}\cdot(1,2,-3)=2

. Find the coordinates of the point B where

\ell_1

meets

\pi

Answer this when you sit the paper.

[3 marks]Vectors: intersecting lines, line and plane intersections, area of a triangle

The line

\ell_2

has equation

\mathbf{r}=(1,2,3)+\lambda(1,1,2)

and the plane

\pi

has equation

\mathbf{r}\cdot(1,2,-3)=2

. Find the coordinates of the point C where

\ell_2

meets

\pi

Answer this when you sit the paper.

[3 marks]Vectors: intersecting lines, line and plane intersections, area of a triangle

The lines

\ell_1

and

\ell_2

have direction vectors

(1,3,2)

and

(1,1,2)

. Find

\cos\theta

, where

\theta

is the acute angle between them.

Answer this when you sit the paper.

[2 marks]Vectors: intersecting lines, line and plane intersections, area of a triangle

The acute angle

\theta

between two lines satisfies

\cos\theta=\dfrac{4}{\sqrt{21}}

. Find

\sin\theta

in surd form.

Answer this when you sit the paper.

[2 marks]Vectors: intersecting lines, line and plane intersections, area of a triangle

Points

A(0,1,1)

B(3,10,7)

and

C(-1,0,-1)

are such that AB lies along a line

\ell_1

and AC along a line

\ell_2

, and the acute angle between the lines has

\sin\theta=\sqrt{\dfrac{5}{21}}

. Find the exact area of triangle ABC.

Answer this when you sit the paper.

Section b

Section b, Question 9

[3 marks]Equilibrium of coplanar forces

Two forces P and Q, each of magnitude 80 N, act on an object and are inclined at

60^\circ

to each other. Find the magnitude of the single force required to keep the object in equilibrium.

Answer this when you sit the paper.

Section b, Question 10

[2 marks]Kinematics: uniform acceleration in a straight line

A cheetah starts from rest at A and accelerates at 3 ms

^{-2}

in a straight line, catching an antelope at C where AC = 54 m. Find the time taken by the cheetah to catch the antelope.

Answer this when you sit the paper.

[2 marks]Kinematics: uniform acceleration in a straight line

A cheetah at A and an antelope at B, where AB = 24 m, both start from rest at the same instant. The antelope accelerates uniformly away in the direction AB and is caught 6 seconds later at C on AB produced, where AC = 54 m. Find the acceleration of the antelope.

Answer this when you sit the paper.

Section b, Question 11

[2 marks]Connected particles on an inclined plane with friction

A 1 kg mass lies on a rough plane inclined at

\theta

, where

\sin\theta=\dfrac35

, and the coefficient of friction is

\dfrac14

. Find the total force, in terms of

g

, resisting motion of the mass up the plane.

Answer this when you sit the paper.

[2 marks]Connected particles on an inclined plane with friction

A 1 kg mass on a rough plane inclined at

\theta

(where

\sin\theta=\dfrac35

, coefficient of friction

\dfrac14

) is joined by a string over a smooth pulley to a 4 kg mass hanging freely. Why does the 1 kg mass slide up the plane?

ABecause the 4 kg weight $4g$ exceeds the total resistance of $0{,}8g$ .
BBecause $\sin\theta>\cos\theta$ for this angle.
CBecause friction always acts up a plane when a string is attached.
DBecause the 1 kg mass is lighter than the 4 kg mass, whatever the angle.

[2 marks]Connected particles on an inclined plane with friction

A 1 kg mass on a rough incline (total resistance

0{,}8g

N) is connected over a smooth pulley to a 4 kg mass hanging freely. Taking

g=9{,}81

^{-2}

, find the acceleration of the system.

Answer this when you sit the paper.

[2 marks]Connected particles on an inclined plane with friction

A 4 kg mass hanging 2,5 m above the floor accelerates downwards from rest at

6{,}28

^{-2}

. Find the velocity with which it hits the floor.

Answer this when you sit the paper.

Section b, Question 12

[2 marks]Projectile motion from a raised point

A particle is projected at

35^\circ

above the horizontal with speed 25 ms

^{-1}

. Find the vertical component of its initial velocity.

Answer this when you sit the paper.

[2 marks]Projectile motion from a raised point

A particle is projected at

35^\circ

above the horizontal with speed 25 ms

^{-1}

from a point O which is 5 m above level ground. Taking

g=9{,}81

^{-2}

, find the height above the ground of H, the highest point of its path.

Answer this when you sit the paper.

[2 marks]Projectile motion from a raised point

A projectile reaches its highest point H at

15{,}48

m above level ground, then falls freely to the ground at B. Taking

g=9{,}81

^{-2}

, find the time taken to travel from H to B.

Answer this when you sit the paper.

[3 marks]Projectile motion from a raised point

A particle projected at

35^\circ

with speed 25 ms

^{-1}

from a point O 5 m above level ground takes

1{,}46

s to reach its highest point and a further

1{,}78

s to fall to the ground at B. Find the horizontal distance between O and B.

Answer this when you sit the paper.

Section c

Section c, Question 13

[2 marks]Probability: exhaustive outcomes and independent selections

An experiment has only two possible outcomes. The first occurs with probability

p

and the second with probability

p^{2}

. Find the value of

p

correct to 3 decimal places.

Answer this when you sit the paper.

[3 marks]Probability: exhaustive outcomes and independent selections

In a city the probability that a person favours capital punishment is 0,55 and that a person is against it is 0,45. Two persons are selected at random. Find the probability that at least one of them favours capital punishment.

Answer this when you sit the paper.

Section c, Question 14

[2 marks]Continuous random variables: probability density function and expectation

Show that

f(x)=\dfrac{1}{28}(6x-4)

for

2\le x\le4

is a probability density function by evaluating

\displaystyle\int_{2}^{4}\frac{1}{28}(6x-4)\,dx

Answer this when you sit the paper.

[3 marks]Continuous random variables: probability density function and expectation

The continuous random variable X has probability density function

f(x)=\dfrac{1}{28}(6x-4)

for

2\le x\le4

and zero elsewhere. Find

E(X)

Answer this when you sit the paper.

Section c, Question 15

[2 marks]Cumulative frequency curves, quartiles and interquartile range

The masses of 200 students, measured to the nearest kg, are grouped as 46-50 (20 students), 51-55 (60), 56-60 (56), 61-65 (35), 66-70 (19) and 71-75 (10). Find the cumulative frequency up to the upper class boundary 60,5 kg.

Answer this when you sit the paper.

[3 marks]Cumulative frequency curves, quartiles and interquartile range

The masses of 200 students, to the nearest kg, are grouped as 46-50 (20), 51-55 (60), 56-60 (56), 61-65 (35), 66-70 (19) and 71-75 (10). Estimate the lower quartile of the distribution.

Answer this when you sit the paper.

[3 marks]Cumulative frequency curves, quartiles and interquartile range

The masses of 200 students, to the nearest kg, are grouped as 46-50 (20), 51-55 (60), 56-60 (56), 61-65 (35), 66-70 (19) and 71-75 (10), giving

Q_1=53{,}0

kg. Estimate the interquartile range.

Answer this when you sit the paper.

[1 marks]Cumulative frequency curves, quartiles and interquartile range

The masses of 200 students, to the nearest kg, are grouped as 46-50 (20), 51-55 (60), 56-60 (56), 61-65 (35), 66-70 (19) and 71-75 (10). Estimate the percentage of students with a mass of 54 kg or less.

Answer this when you sit the paper.

Section c, Question 16

[3 marks]Normal distribution and the binomial distribution

The heights of flowers in a bed are normally distributed with mean 21,1 cm and standard deviation 4,0 cm. Find the probability that a flower has a height greater than 25 cm.

Answer this when you sit the paper.

[2 marks]Normal distribution and the binomial distribution

The heights of flowers in a bed are normally distributed with

P(\text{height}>25\text{ cm})=0{,}1648

. Eight flowers are picked at random. State the distribution of the number of these flowers whose heights are less than 25 cm.

Answer this when you sit the paper.

[3 marks]Normal distribution and the binomial distribution

The number of flowers under 25 cm among 8 picked at random follows

B(8;\,0{,}8352)

. Find the probability that fewer than three of them have heights less than 25 cm, giving your answer to 3 significant figures.

Answer this when you sit the paper.

Pure Mathematics Paper 2 June 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 a, Question 8

Section b

Section b, Question 9

Section b, Question 10

Section b, Question 11

Section b, Question 12

Section c

Section c, Question 13

Section c, Question 14

Section c, Question 15

Section c, Question 16

More sittings of this paper

The answers, and why they are the answers

Sit this paper online