ZIMSEC O Level Mathematics Paper 2 November 2008

Question 101

[2 marks]Fractions, Algebra & Mensuration

Express

3\frac{2}{5} - 2\frac{13}{20}

as a single fraction in its lowest terms.

Answer this when you sit the paper.

Question 102

[2 marks]Fractions, Algebra & Mensuration

Remove the brackets and simplify

3(a + 2c) - 4(2a - c)

Answer this when you sit the paper.

Question 103

[3 marks]Fractions, Algebra & Mensuration

Solve the equation

\frac{4x - 5}{7} = 1\frac{3}{4}

Answer this when you sit the paper.

Question 104

[3 marks]Fractions, Algebra & Mensuration

Find the number of circular rings, each of diameter 6,3 cm, that can be made from a wire 19,8 m long. Take

\pi = \frac{22}{7}

Answer this when you sit the paper.

Question 201

[2 marks]Factorisation, Vectors & Linear Equations

Factorise completely

2x^2 + ax - 2bx - ab

Answer this when you sit the paper.

Question 202

[2 marks]Factorisation, Vectors & Linear Equations

Factorise completely

3 - 12y^2

Answer this when you sit the paper.

Question 203

[2 marks]Factorisation, Vectors & Linear Equations

P is the point

(4; 8)

and R is the point

(-4; -2)

on the Cartesian plane. Write

\overrightarrow{PR}

as a column vector, giving the top entry first.

Answer this when you sit the paper.

Question 204

[2 marks]Factorisation, Vectors & Linear Equations

P is the point

(4; 8)

and R is the point

(-4; -2)

on the Cartesian plane. Find

\left|\overrightarrow{PR}\right|

, correct to 3 significant figures.

Answer this when you sit the paper.

Question 205

[2 marks]Factorisation, Vectors & Linear Equations

Two cyclists, Alice and John, start at the same time from two villages 27 km apart and ride towards each other. Alice rides at

x

km/h, John rides at

2x

km/h, and they meet after

\frac{3}{4}

hour. Find John's speed, in km/h.

Answer this when you sit the paper.

Question 301

[2 marks]Standard Form, Consumer Arithmetic & Inequalities

Write

1,496 \times 10^8

in ordinary form.

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Question 302

[2 marks]Standard Form, Consumer Arithmetic & Inequalities

The Sun, the Earth and Mars lie in a straight line with the Earth between the other two. The Earth is

1,496 \times 10^8

km from the Sun and Mars is

2,279 \times 10^8

km from the Sun. Find, in standard form, the distance of Mars from the Earth.

Answer this when you sit the paper.

Question 303

[3 marks]Standard Form, Consumer Arithmetic & Inequalities

A paint manufacturer mixed 27 litres of white paint with 9 litres of red paint to make 36 litres of pink paint. One litre of the white paint cost

\$36\,800

and the average cost of the pink paint was

\$33\,575

per litre. Calculate the cost, in dollars, of one litre of the red paint.

Answer this when you sit the paper.

Question 304

[3 marks]Standard Form, Consumer Arithmetic & Inequalities

Solve the simultaneous inequalities

2x - 6 < 5x + 3 \le 3x + 11

, giving the answer in the form

a < x \le b

where

a

and

b

are integers.

A $-3 \le x < 4$ , keeping the inclusive sign on the left end and the strict sign on the right end
B $-9 < x \le 8$ , stopping at $-9 < 3x$ and $2x \le 8$ without dividing either side by the coefficient
C $-3 < x \le 4$ , since $-9 < 3x$ gives $x > -3$ and $2x \le 8$ gives $x \le 4$
D $3 < x \le 4$ , moving the $-6$ across the first inequality as $+6$ instead of subtracting it

Question 401

[2 marks]Similar Triangles & Circle Geometry

ABCD is a quadrilateral in which AD is parallel to BC. The diagonals AC and BD cut at X, with BX : XD

= 3 : 2

. Triangle ABX has an area of 9 cm

^2

. Calculate the area, in cm

^2

, of triangle ADX.

Answer this when you sit the paper.

Question 402

[2 marks]Similar Triangles & Circle Geometry

ABCD is a quadrilateral in which AD is parallel to BC, and the diagonals AC and BD cut at X. Which triangle is similar to triangle BCX, with its vertices named in the matching order?

ATriangle DAX, because angle XBC equals angle XDA and angle XCB equals angle XAD
BTriangle ADX, because the two triangles share the vertex X and each contains a diagonal
CTriangle ABX, because it stands on the same line BD and has the same height from A
DTriangle CDX, because CD is the remaining side and X lies on both diagonals of ABCD

Question 403

[3 marks]Similar Triangles & Circle Geometry

ABCD is a quadrilateral in which AD is parallel to BC. The diagonals AC and BD cut at X, with BX : XD

= 3 : 2

. Triangle ABX has an area of 9 cm

^2

and triangle ADX has an area of 6 cm

^2

. Calculate the area, in cm

^2

, of triangle BCX.

Answer this when you sit the paper.

Question 404

[2 marks]Similar Triangles & Circle Geometry

Two circles cut each other at A and B. AP is a tangent at A to the circle through A, B and Q, and AQ is a tangent at A to the circle through A, B, P and C. Given that

A\hat{P}B = 30^\circ

and

A\hat{Q}B = 50^\circ

, calculate

B\hat{A}P

, in degrees.

Answer this when you sit the paper.

Question 405

[2 marks]Similar Triangles & Circle Geometry

Two circles cut each other at A and B. AP is a tangent at A to the circle through A, B and Q, and AQ is a tangent at A to the circle through A, B, P and C. Given that

A\hat{P}B = 30^\circ

and

A\hat{Q}B = 50^\circ

, calculate

B\hat{A}Q

, in degrees.

Answer this when you sit the paper.

Question 501

[2 marks]Algebraic Fractions, Change of Subject & Matrices

Express

n + \frac{2n}{6n + 5}

as a single fraction in its simplest form.

Answer this when you sit the paper.

Question 502

[3 marks]Algebraic Fractions, Change of Subject & Matrices

Make

m

the subject of the formula

a = \frac{m - 5}{3m - 2}

Answer this when you sit the paper.

Question 503

[2 marks]Algebraic Fractions, Change of Subject & Matrices

Given that

\mathbf{A} = \begin{pmatrix} 3 & 5 \\ -2 & 7 \end{pmatrix}

, find

\mathbf{A}^2

, giving the four entries in reading order.

Answer this when you sit the paper.

Question 504

[3 marks]Algebraic Fractions, Change of Subject & Matrices

\mathbf{B} = \begin{pmatrix} 5 & y \\ y & 3 \end{pmatrix}

. Find the two possible values of

y

, given that the determinant of

\mathbf{B}

5y + 1

Answer this when you sit the paper.

Question 601

[2 marks]Constructions & Loci

A circle has centre O and radius 3,5 cm, and P is a point with OP = 9 cm. Calculate the length, in cm, of a tangent drawn from P to touch this circle, correct to 3 significant figures.

Answer this when you sit the paper.

Question 602

[2 marks]Constructions & Loci

O and P are two fixed points 9 cm apart. What is the locus of the points that are equidistant from O and from P?

AThe arc of radius 9 cm drawn from O, which passes through P and so keeps the distance fixed
BThe perpendicular bisector of OP, the straight line at right angles to OP through its midpoint
CThe circle with OP as its diameter, since every point on it is 4,5 cm from the midpoint
DThe circle of radius 9 cm centred on the midpoint of OP, since O and P would both lie on that circle

Question 603

[2 marks]Constructions & Loci

A circle is drawn with OP as its diameter. It cuts the circle of centre O and radius 3,5 cm at the point R. Why is PR a tangent to the circle of centre O?

ABecause OR and PR are both radii of the circle drawn on OP as diameter, so they are equal in length
BBecause $O\hat{R}P$ is an angle in a semicircle and so is $90^\circ$ , and OR is a radius at R
CBecause R lies on the perpendicular bisector of OP, which cuts through both of the circles at R
DBecause OP measures 9 cm and OR measures 3,5 cm, so PR works out at 8,29 cm by Pythagoras

Question 701

[2 marks]Statistics & Probability

In a grouped frequency distribution, the class

35 < m \le 45

has a frequency density of 0,5. Find the frequency of that class.

Answer this when you sit the paper.

Question 702

[2 marks]Statistics & Probability

In a grouped frequency distribution, the class

55 < m \le 60

has a frequency of 8. Find the frequency density of that class.

Answer this when you sit the paper.

Question 703

[3 marks]Statistics & Probability

The masses

m

kg of a group of students are grouped as follows:

45 < m \le 50

holds 11 students,

50 < m \le 55

holds 13,

55 < m \le 60

holds 8 and

60 < m \le 70

holds 3. Calculate an estimate of the mean mass, in kg, of these students, correct to 3 significant figures.

Answer this when you sit the paper.

Question 704

[3 marks]Statistics & Probability

A group of 40 students is grouped by mass

m

kg: 5 have

35 < m \le 45

, 11 have

45 < m \le 50

, 13 have

50 < m \le 55

, 8 have

55 < m \le 60

and 3 have

60 < m \le 70

. Two students are chosen at random from the whole group. Find the probability that each of them has a mass greater than 50 kg.

Answer this when you sit the paper.

Question 801

[3 marks]Trigonometry & Mensuration

In triangle ABC, AB = 5 m, AC = 8 m and

A\hat{C}B = 36^\circ

. Given that

A\hat{B}C

is obtuse, calculate

A\hat{B}C

, in degrees correct to one decimal place.

Answer this when you sit the paper.

Question 802

[3 marks]Trigonometry & Mensuration

A vertical beam AB, 5 m long, stands on level ground at A. A straight beam AC, 8 m long, runs from A up to C, with

A\hat{C}B = 36^\circ

and

A\hat{B}C

obtuse. A load D hangs on a string 3,6 m vertically below C. Calculate the height, in metres, of D above the ground, correct to 3 significant figures.

Answer this when you sit the paper.

Question 803

[3 marks]Trigonometry & Mensuration

Water in a vertical cylindrical container stands 9 cm deep and has a volume of 512 cm

^3

. A metal solid of volume 217 cm

^3

is lowered into the container until it is completely immersed. Calculate the rise in the water level, correct to the nearest millimetre.

Answer this when you sit the paper.

Question 804

[2 marks]Trigonometry & Mensuration

Water in a vertical cylindrical container stands 9 cm deep and that water has a volume of 512 cm

^3

. What is the area of the cross section of the container?

A $56,9$ cm $^2$ , dividing the volume by the depth, since volume equals base area times height
B $4\,608$ cm $^2$ , found by multiplying the volume of the water by the depth at which it stands
C $512$ cm $^2$ , since a column of water standing 9 cm deep has its volume as its base area
D $0,0176$ cm $^2$ , found by dividing the depth of the water by the volume that it occupies

Question 901

[2 marks]Mensuration & Quadratic Equations

PR is a chord of length 6 cm in a circle of centre O and radius 6 cm. Find the size, in degrees, of the angle POR at the centre.

Answer this when you sit the paper.

Question 902

[3 marks]Mensuration & Quadratic Equations

PR is a chord of length 6 cm in a circle of centre O and radius 6 cm. Taking

\pi

to be 3,142, calculate the area, in cm

^2

, of the minor segment cut off by PR, correct to 3 significant figures.

Answer this when you sit the paper.

Question 903

[2 marks]Mensuration & Quadratic Equations

In triangle PRS, PR = 6 cm, RS = 8 cm and

P\hat{R}S = 120^\circ

. Find

PS^2

A52, working $36 + 64 - 2(6)(8)\cos 120^\circ$ with $\cos 120^\circ$ taken as $+0,5$ instead
B100, using Pythagoras' theorem on the two given sides and leaving the angle at R out of it
C196, adding $2(6)(8)$ on to $36 + 64$ but dropping the cosine of the angle from that term
D148, since $36 + 64 - 2(6)(8)\cos 120^\circ = 100 + 48$ because $\cos 120^\circ = -0,5$

Question 904

[3 marks]Mensuration & Quadratic Equations

Solve the equation

3x^2 + 8x - 44 = 0

, giving both answers correct to 2 decimal places.

Answer this when you sit the paper.

Question 1001

[2 marks]Functional Graphs

A stone is thrown into the air and its height

h

metres after

t

seconds is given by

h = 60 + 30t - 5t^2

. Find the value of

h

when

t = 7

Answer this when you sit the paper.

Question 1002

[3 marks]Functional Graphs

A stone is thrown into the air and its height

h

metres after

t

seconds is given by

h = 60 + 30t - 5t^2

. Find the maximum height, in metres, reached by the stone.

Answer this when you sit the paper.

Question 1003

[3 marks]Functional Graphs

A stone is thrown into the air and its height

h

metres after

t

seconds is given by

h = 60 + 30t - 5t^2

. Find the velocity of the stone, in m/s, when

t = 2

Answer this when you sit the paper.

Question 1004

[3 marks]Functional Graphs

A stone is thrown into the air and its height

h

metres after

t

seconds is given by

h = 60 + 30t - 5t^2

. Find the two times, in seconds, at which the stone is 80 m above the ground, each correct to 2 significant figures.

Answer this when you sit the paper.

Question 1101

[2 marks]Measures & Mensuration

A wooden block of length 150 cm has a uniform cross section ABCD, a trapezium with AB parallel to DC, AB = 65 cm, AD = BC = 32,5 cm and perpendicular height 30 cm. The area of the trapezium is

1\,575

^2

. Calculate the length of CD, in cm.

Answer this when you sit the paper.

Question 1102

[2 marks]Measures & Mensuration

A wooden block of length 150 cm has a uniform cross section that is a trapezium of area

1\,575

^2

. Calculate the volume of the block, in cm

^3

Answer this when you sit the paper.

Question 1103

[3 marks]Measures & Mensuration

A wooden block has a volume of

236\,250

^3

and is made of wood of density 0,72 g/cm

^3

. Calculate the mass of the block, in kilograms.

Answer this when you sit the paper.

Question 1104

[3 marks]Measures & Mensuration

A wooden block of length 150 cm has a uniform cross section ABCD, a trapezium with AB = 65 cm, CD = 40 cm, AD = BC = 32,5 cm and area

1\,575

^2

. Calculate the total surface area of the block, in cm

^2

Answer this when you sit the paper.

Question 1105

[2 marks]Measures & Mensuration

A block whose total surface area is

28\,650

^2

is to be varnished all over. One litre of varnish covers

2\,000

^2

, and varnish is sold only in 5-litre tins. Calculate the number of tins that must be bought.

Answer this when you sit the paper.

Question 1201

[3 marks]Geometrical Transformation

Quadrilateral E has vertices

(-8; -4)

(-4; -4)

(-6; -12)

and

(-10; -8)

, and is the image of quadrilateral A with vertices

(4; 2)

(2; 2)

(3; 6)

and

(5; 4)

. Which matrix represents the transformation that maps E onto A?

A $\begin{pmatrix} 0 & -\frac{1}{2} \\ -\frac{1}{2} & 0 \end{pmatrix}$ , a reflection in the line $y = -x$ together with a halving of every length
B $\begin{pmatrix} -2 & 0 \\ 0 & -2 \end{pmatrix}$ , an enlargement of scale factor $-2$ about the origin
C $\begin{pmatrix} \frac{1}{2} & 0 \\ 0 & \frac{1}{2} \end{pmatrix}$ , an enlargement of scale factor $\frac{1}{2}$ about the origin
D $\begin{pmatrix} -\frac{1}{2} & 0 \\ 0 & -\frac{1}{2} \end{pmatrix}$ , an enlargement of scale factor $-\frac{1}{2}$ about the origin

Question 1202

[3 marks]Geometrical Transformation

Quadrilateral A has vertices

(4; 2)

(2; 2)

(3; 6)

and

(5; 4)

. Its image T has vertices

(0; 6)

(0; 4)

(-4; 5)

and

(-2; 7)

, taken in the same order. Which single transformation maps A onto T?

AA rotation through $90^\circ$ clockwise about the point $(0;\,2)$ , which reverses the sense
BA rotation through $90^\circ$ anticlockwise (a positive quarter turn) about the point $(0;\,2)$
CA rotation through $90^\circ$ anticlockwise about the origin, with no further movement after it
DA rotation through $180^\circ$ about the point $(0;\,2)$ , a half turn that keeps every length

Question 1203

[3 marks]Geometrical Transformation

A one-way stretch is represented by the matrix

\begin{pmatrix} 1 & 0 \\ 0 & -1\frac{1}{2} \end{pmatrix}

. Find the image of the point

(3; 6)

under this stretch.

Answer this when you sit the paper.

Question 1204

[2 marks]Geometrical Transformation

A one-way stretch is represented by the matrix

\begin{pmatrix} 1 & 0 \\ 0 & -1\frac{1}{2} \end{pmatrix}

. Which line is left fixed by this stretch?

AThe line $y = x$ , because the matrix is diagonal and treats both axes in the very same way
BThe line $y = -1\frac{1}{2}x$ , because that gradient is the one the stretch factor names
CThe $x$ axis, $y = 0$ , because a point $(x; 0)$ is sent to $(x; 0)$ and so does not move
DThe $y$ axis, $x = 0$ , because the stretch acts parallel to that axis and so holds it fixed

Mathematics Paper 2 November 2008

Sit this paper online

The questions

Question 101

Question 102

Question 103

Question 104

Question 201

Question 202

Question 203

Question 204

Question 205

Question 301

Question 302

Question 303

Question 304

Question 401

Question 402

Question 403

Question 404

Question 405

Question 501

Question 502

Question 503

Question 504

Question 601

Question 602

Question 603

Question 701

Question 702

Question 703

Question 704

Question 801

Question 802

Question 803

Question 804

Question 901

Question 902

Question 903

Question 904

Question 1001

Question 1002

Question 1003

Question 1004

Question 1101

Question 1102

Question 1103

Question 1104

Question 1105

Question 1201

Question 1202

Question 1203

Question 1204

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