ZIMSEC A Level Pure Mathematics Paper 2 June 2008

Section a

Section a, Question 1

[3 marks]Integration: area between a curve and lines, logarithms

A region R is enclosed by the curve

y=\dfrac{a}{x}

, where

a

is a positive constant, the line

y=\dfrac12

, the

y

-axis and the line

y=2

. Find the exact area of R, giving your answer in a form involving a single logarithm.

Answer this when you sit the paper.

[2 marks]Integration: area between a curve and lines, logarithms

The region enclosed by

y=\dfrac{a}{x}

y=\dfrac12

, the

y

-axis and

y=2

has exact area

a\ln 4

, where

a

is a positive integer. Find the minimum value of

a

for which this area exceeds 100.

Answer this when you sit the paper.

Section a, Question 2

[3 marks]Integration by substitution, logarithms

The substitution

t=e^{x}

is used in

\displaystyle\int_{0}^{1}\frac{e^{x}}{9-e^{2x}}dx

. Which integral in

t

does it become?

A $\displaystyle\int_{1}^{e}\frac{dt}{9-t}$
B $\displaystyle\int_{0}^{1}\frac{dt}{9-t^{2}}$
C $\displaystyle\int_{1}^{e}\frac{dt}{9-t^{2}}$
D $\displaystyle\int_{1}^{e}\frac{t\,dt}{9-t^{2}}$

[3 marks]Integration by substitution, logarithms

Evaluate

\displaystyle\int_{0}^{1}\frac{e^{x}}{9-e^{2x}}dx

, giving your answer correct to 3 significant figures.

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Section a, Question 3

[2 marks]Proof by mathematical induction, arithmetic series

The statement

-2+3+8+\ldots+(5n-7)=\dfrac{n}{2}(5n-9)

is to be proved by induction. Show the base case holds by giving the common value of both sides when

n=1

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[3 marks]Proof by mathematical induction, arithmetic series

In proving

-2+3+8+\ldots+(5n-7)=\dfrac{n}{2}(5n-9)

by induction, you assume the result for

n=k

and add the next term. Which expression is the correct next term to add?

A $5(k+1)-7=5k-2$
B $5k-7$
C $5(k+1)-9=5k-4$
D $\dfrac{k+1}{2}$

[2 marks]Proof by mathematical induction, arithmetic series

For the series whose sum is

-2+3+8+\ldots+(5n-7)=\dfrac{n}{2}(5n-9)

, find the sum of the first 20 terms.

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Section a, Question 4

[3 marks]Maxima and minima, related rates of change

A bus company will take a minimum of 50 passengers and a maximum of 70 on a trip. At exactly 50 passengers the fare is \$600 000 per passenger, and for every passenger above 50 the fare is reduced by \$10 000. Find the number of passengers that maximises the company's intake.

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[2 marks]Maxima and minima, related rates of change

A bus company charges \$600 000 per passenger for exactly 50 passengers, reducing the fare by \$10 000 for each passenger above 50. The intake is greatest at 55 passengers. Find that maximum intake in dollars.

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[3 marks]Maxima and minima, related rates of change

A heated metal cube of side

x

cm is cooled uniformly so that the side decreases at a rate of 10 cm per hour. Find the rate at which the volume is changing at the instant when the side is 15 cm long.

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[2 marks]Maxima and minima, related rates of change

A cube's side decreases at 10 cm per hour. Why is

\dfrac{dV}{dt}

not the same at every instant, even though the side shrinks at a steady rate?

ABecause the rate of change of volume of a cooling solid is always constant once the side is shrinking steadily.
BBecause $\dfrac{dV}{dt}=3x^{2}\dfrac{dx}{dt}$ depends on the current side length $x$ , which is changing.
CBecause $\dfrac{dx}{dt}$ becomes positive as the cube cools further.
DBecause the volume of a cube is not $x^{3}$ once it is cooling.

Section a, Question 5

[2 marks]Circular measure, sector area, complex numbers in polar and exponential form

Two tangents from a point O touch a circle of centre C and radius 1 unit at the points A and B. Given that

\hat{AOB}=\dfrac{\pi}{3}

radians, find

\hat{ACB}

in radians.

Answer this when you sit the paper.

[2 marks]Circular measure, sector area, complex numbers in polar and exponential form

Tangents from O touch a circle of centre C and radius 1 unit at A and B, and

\hat{AOB}=\dfrac{\pi}{3}

radians. Find the exact length of OA.

Answer this when you sit the paper.

[3 marks]Circular measure, sector area, complex numbers in polar and exponential form

Tangents from O touch a circle of centre C and radius 1 unit at A and B, with

\hat{AOB}=\dfrac{\pi}{3}

radians,

\hat{ACB}=\dfrac{2\pi}{3}

radians and

OA=\sqrt3

. Find the exact area of the region bounded by OA, OB and the arc AB.

Answer this when you sit the paper.

[2 marks]Circular measure, sector area, complex numbers in polar and exponential form

On an Argand diagram with origin O, the point B lies on the positive real axis and OA makes an angle of

\dfrac{\pi}{3}

with it, with

OA=\sqrt3

. Write down the complex number representing A in the form

a+ib

Answer this when you sit the paper.

[2 marks]Circular measure, sector area, complex numbers in polar and exponential form

On an Argand diagram, the point C is the centre of a circle of radius 1 unit to which OA and OB are tangents, where

\hat{AOB}=\dfrac{\pi}{3}

radians and OB lies along the positive real axis. The complex number

Z_1

represents C. Find

Z_1

in the form

r(\cos\theta+i\sin\theta)

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[2 marks]Circular measure, sector area, complex numbers in polar and exponential form

Given

Z_1=2\left(\cos\dfrac{\pi}{6}+i\sin\dfrac{\pi}{6}\right)

, write down

Z_1^{2}

in the form

Re^{i\alpha}

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Section a, Question 6

[3 marks]Matrix multiplication, singular matrices, matrix inverse

Given

P=\begin{pmatrix}k&1&0&0\\0&1&1&0\\0&1&0&1\end{pmatrix}

and

Q=\begin{pmatrix}1&1&k\\-1&0&0\\0&1&1\\0&0&1\end{pmatrix}

, which matrix is

M=PQ

A $\begin{pmatrix}k+1&k&k^{2}\\1&1&1\\-1&0&1\end{pmatrix}$
B $\begin{pmatrix}k-1&k&k^{2}\\-1&1&1\\-1&1&0\end{pmatrix}$
C $\begin{pmatrix}k-1&1&k\\-1&1&1\\-1&0&1\end{pmatrix}$
D $\begin{pmatrix}k-1&k&k^{2}\\-1&1&1\\-1&0&1\end{pmatrix}$

[3 marks]Matrix multiplication, singular matrices, matrix inverse

The matrix

M=\begin{pmatrix}k-1&k&k^{2}\\-1&1&1\\-1&0&1\end{pmatrix}

. Find

\det M

in terms of

k

Answer this when you sit the paper.

[3 marks]Matrix multiplication, singular matrices, matrix inverse

The matrix

M=\begin{pmatrix}k-1&k&k^{2}\\-1&1&1\\-1&0&1\end{pmatrix}

is singular and

k>0

. Find the exact value of

k

Answer this when you sit the paper.

[2 marks]Matrix multiplication, singular matrices, matrix inverse

The matrix

M=\begin{pmatrix}k-1&k&k^{2}\\-1&1&1\\-1&0&1\end{pmatrix}

has

\det M=k^{2}+k-1

. Find

\det M

when

k=2

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[3 marks]Matrix multiplication, singular matrices, matrix inverse

When

k=2

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

and

\det M=5

. Which matrix is

M^{-1}

A $\dfrac15\begin{pmatrix}1&-2&-2\\0&5&5\\1&2&3\end{pmatrix}$
B $\dfrac15\begin{pmatrix}1&-2&-2\\0&5&-5\\1&-2&3\end{pmatrix}$
C $\dfrac15\begin{pmatrix}1&2&4\\-1&1&1\\-1&0&1\end{pmatrix}$
D $\dfrac15\begin{pmatrix}1&0&1\\-2&5&-2\\-2&-5&3\end{pmatrix}$

Section a, Question 7

[3 marks]Vectors in three dimensions, equation of a plane

A deep freezer OABCDEFG has a square base of side 2 m and height 1 m. With O as origin and the

x

y

and

z

axes along OA, OC and OE, the vertices are

A(2,0,0)

and

D(0,2,1)

. Find the unit vector in the direction of

\overrightarrow{AD}

Answer this when you sit the paper.

[2 marks]Vectors in three dimensions, equation of a plane

A deep freezer has a square base of side 2 m and is 1 m high. With O as origin and the

x

y

and

z

axes along OA, OC and OE, write down the position vectors of D (the corner above C) and E (the corner above O).

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[2 marks]Vectors in three dimensions, equation of a plane

A freezer lid hinged along ED swings the point

F(2,0,1)

F^{1}

with position vector

\mathbf{r}=(2\cos\theta)\mathbf{i}+(1+2\sin\theta)\mathbf{k}

. Find the position vector of

F^{1}

when

\theta=30^\circ

Answer this when you sit the paper.

[3 marks]Vectors in three dimensions, equation of a plane

A plane contains

E(0,0,1)

D(0,2,1)

and

F^{1}\big(2\cos\theta,\,0,\,1+2\sin\theta\big)

. Which vector is normal to that plane?

A $\sin\theta\,\mathbf{i}-\cos\theta\,\mathbf{k}$
B $\sin\theta\,\mathbf{i}+\mathbf{j}-\cos\theta\,\mathbf{k}$
C $2\cos\theta\,\mathbf{i}+2\sin\theta\,\mathbf{k}$
D $\cos\theta\,\mathbf{i}-\sin\theta\,\mathbf{k}$

[2 marks]Vectors in three dimensions, equation of a plane

A plane has vector equation

\mathbf{r}\cdot(\sin\theta\,\mathbf{i}-\cos\theta\,\mathbf{k})=-\cos\theta

. Find its Cartesian equation when

\theta=60^\circ

Answer this when you sit the paper.

[2 marks]Vectors in three dimensions, equation of a plane

A freezer OABCDEFG has a square base of side 2 m and is 1 m high, with O the origin and the

x

y

and

z

axes along OA, OC and OE. Write down the coordinates of the corner G, which is vertically above B.

Answer this when you sit the paper.

Section b

Section b, Question 8

[2 marks]Friction and limiting equilibrium

A crate of weight 20 N rests on a rough horizontal floor. A string inclined at

60^\circ

to the horizontal pulls it with a force of 4 N and the crate is in limiting equilibrium. Find the exact normal reaction between the crate and the floor.

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[3 marks]Friction and limiting equilibrium

A crate of weight 20 N is in limiting equilibrium on a rough horizontal floor while a string at

60^\circ

to the horizontal pulls it with a force of 4 N. The normal reaction is

20-2\sqrt3

N. Find the coefficient of friction between the crate and the floor.

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Section b, Question 9

[2 marks]Kinematics: velocity-time and displacement-time graphs

A car travelling at 40 ms

^{-1}

decelerates uniformly to 25 ms

^{-1}

over 30 seconds. Find its deceleration.

Answer this when you sit the paper.

[2 marks]Kinematics: velocity-time and displacement-time graphs

A car travelling at 40 ms

^{-1}

decelerates uniformly to 25 ms

^{-1}

over 30 seconds. Find the distance it covers during this deceleration.

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[2 marks]Kinematics: velocity-time and displacement-time graphs

A car decelerates uniformly from 40 ms

^{-1}

to 25 ms

^{-1}

over 30 seconds, covering 975 m, then travels at 25 ms

^{-1}

for a further 500 m. Find the total time for the whole journey.

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Section b, Question 10

[2 marks]Projectile motion

A stone is thrown from the top of a vertical wall with speed V ms

^{-1}

at an angle of depression

\theta

. It hits the ground 0,5 seconds later at a point 1,6 metres from the foot of the wall. Find the horizontal component

V\cos\theta

of the initial velocity.

Answer this when you sit the paper.

[2 marks]Projectile motion

A stone thrown from the top of a 3 m wall at an angle of depression

\theta

hits the ground 0,5 s later, 1,6 m from the foot of the wall. Taking

g=9{,}81

^{-2}

and air resistance as negligible, find

\theta

to the nearest degree.

Answer this when you sit the paper.

[2 marks]Projectile motion

For a stone thrown from a wall, the horizontal component of the initial velocity is

V\cos\theta=3{,}2

^{-1}

and the vertical component is

V\sin\theta=3{,}5475

^{-1}

. Find V, correct to 2 significant figures.

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Section b, Question 11

[1 marks]Connected particles, friction, force on a pulley

A toy car P of mass 4 kg rests on a rough horizontal plane where the coefficient of friction is

\dfrac12

. Taking

g

as the acceleration due to gravity, find the frictional force acting on P once it starts to move, in terms of

g

Answer this when you sit the paper.

[2 marks]Connected particles, friction, force on a pulley

A toy car P of mass 4 kg on a rough horizontal plane (coefficient of friction

\dfrac12

) is joined by a light inextensible string over a smooth pulley to a toy car Q of mass 8 kg held on a smooth plane inclined at

30^\circ

. When Q is released, find the acceleration of the cars in terms of

g

Answer this when you sit the paper.

[2 marks]Connected particles, friction, force on a pulley

A 4 kg toy car P on a rough horizontal plane (coefficient of friction

\dfrac12

) is joined over a smooth pulley to an 8 kg toy car Q on a smooth

30^\circ

incline. The system accelerates at

\dfrac{g}{6}

. Find the tension in the string in terms of

g

Answer this when you sit the paper.

[2 marks]Connected particles, friction, force on a pulley

A string under tension

\dfrac{8g}{3}

N passes over a smooth pulley X, leaving it horizontally on one side and down a

30^\circ

incline on the other. Find the magnitude of the force exerted by X on the string, in the form

kg\cos\alpha

Answer this when you sit the paper.

Section c

Section c, Question 12

[1 marks]Stem and leaf diagrams, median, mode and mean

The masses in grammes of 24 sweets are 0,72 0,73 0,79 0,80 0,88 0,91 0,91 0,94 0,98 0,99 1,01 1,03 1,06 1,08 1,13 1,13 1,13 1,19 1,21 1,22 1,33 1,39 1,44 1,45. Find the median.

Answer this when you sit the paper.

[1 marks]Stem and leaf diagrams, median, mode and mean

The masses in grammes of 24 sweets are 0,72 0,73 0,79 0,80 0,88 0,91 0,91 0,94 0,98 0,99 1,01 1,03 1,06 1,08 1,13 1,13 1,13 1,19 1,21 1,22 1,33 1,39 1,44 1,45. Find the mode.

Answer this when you sit the paper.

[2 marks]Stem and leaf diagrams, median, mode and mean

A sweet of mass more than 1,2 g is classified as large. In a bag of 24 sweets the masses in grammes are 0,72 0,73 0,79 0,80 0,88 0,91 0,91 0,94 0,98 0,99 1,01 1,03 1,06 1,08 1,13 1,13 1,13 1,19 1,21 1,22 1,33 1,39 1,44 1,45. Calculate the mean mass of the large sweets.

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Section c, Question 13

[2 marks]Probability: binomial and independent events

A fair die is tossed three times. Find the probability that exactly one six is obtained.

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[3 marks]Probability: binomial and independent events

A fair die is tossed three times. Find the probability that the first score is even, the second is odd and the third is either a one or a two.

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Section c, Question 14

[3 marks]The normal distribution applied to a manufacturing problem

A machine fills pies whose weights are normally distributed with standard deviation 0,8 g. Exactly 0,3\% of pies weigh less than 80 g. Find the mean weight of a pie.

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[2 marks]The normal distribution applied to a manufacturing problem

Pie weights are normally distributed with mean 82,2 g and standard deviation 0,8 g. Find the probability that a pie weighs more than 83 g.

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[2 marks]The normal distribution applied to a manufacturing problem

A firm makes 500 000 pies a week, and 0,1587 of them exceed 83 g and so need extra packaging costing \$10 per pie. Find the firm's weekly cost of extra packaging.

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Section c, Question 15

[1 marks]Discrete random variables: expectation and variance

A discrete random variable X takes the values 0, 1 and 2 only, with

E(X)=\dfrac43

and

\mathrm{Var}(X)=\dfrac59

. Find

E(X^{2})

Answer this when you sit the paper.

[3 marks]Discrete random variables: expectation and variance

A discrete random variable X takes the values 0, 1 and 2 only, with probabilities

P_0

P_1

and

P_2

. Given

E(X)=\dfrac43

and

E(X^{2})=\dfrac73

, find

P_2

Answer this when you sit the paper.

[2 marks]Discrete random variables: expectation and variance

A discrete random variable X takes the values 0, 1 and 2 only, with

E(X)=\dfrac43

and

P_2=\dfrac12

. Find

P_1

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[2 marks]Discrete random variables: expectation and variance

A discrete random variable X takes the values 0, 1 and 2 only, with

P_1=\dfrac13

and

P_2=\dfrac12

. Find

P_0

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Pure Mathematics Paper 2 June 2008

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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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