ZIMSEC O Level Mathematics Paper 2 June 2013

Question 101

[2 marks]Factorisation and Algebraic Fractions

Find the exact value of

20,71 - 8,2 \times 1,1

Answer this when you sit the paper.

Question 102

[2 marks]Factorisation and Algebraic Fractions

Factorise completely

12m - 2n^2 + 6mn - 4n

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Question 103

[2 marks]Factorisation and Algebraic Fractions

Factorise completely

2a^2 - 5a + 3

A $(2a - 1)(a + 3)$
B $(2a - 1)(a - 3)$
C $(2a + 3)(a - 1)$
D $(2a - 3)(a - 1)$

Question 104

[3 marks]Factorisation and Algebraic Fractions

Express

\dfrac{x+1}{x-2} - \dfrac{x+2}{1-x}

as a single fraction in its lowest terms.

A $\dfrac{3 - 2x^2}{(x - 1)(x - 2)}$
B $\dfrac{2x^2 - 5}{(x + 1)(x - 2)}$
C $\dfrac{2x^2 - 5}{(x - 1)(x - 2)}$
D $\dfrac{2x^2 + 3}{(x - 1)(x - 2)}$

Question 201

[3 marks]Matrices, Simple Interest and Mensuration

Simplify

\dfrac{1}{3}\begin{pmatrix} -5 & -10 \\ 1 & -1 \end{pmatrix}\begin{pmatrix} 1 & 2 \\ 1 & -1 \end{pmatrix}

, writing the entries of the resulting matrix row by row.

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Question 202

[2 marks]Matrices, Simple Interest and Mensuration

Find the time, in years, in which $20 000 will earn $1 600 simple interest at 8% per annum.

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Question 203

[3 marks]Matrices, Simple Interest and Mensuration

A compact disc has inside diameter 6,6 cm and outside diameter 13 cm. The useful part is the ring between the two circles. Taking

\pi

to be

\dfrac{22}{7}

, calculate the area of the useful part, in cm

^2

Answer this when you sit the paper.

Question 204

[3 marks]Matrices, Simple Interest and Mensuration

A compact disc has inside diameter 6,6 cm and outside diameter 13 cm, and the useful part is the ring between the two circles. What percentage of the area of the disc is useful?

A $25,8\%$ , the share taken up by the small central hole
B $74,2\%$ , the share left once the central hole is taken out
C $49,2\%$ , from comparing the two diameters rather than the areas
D $83,4\%$ , from subtracting the diameters before squaring them

Question 301

[3 marks]Equations, Vectors and Inequalities

Solve the equation

\dfrac{2}{x + 2} + \dfrac{1}{x} = 1

A $x = 2$ or $x = -1$
B $x = 1$ or $x = -2$
C $x = -2$ or $x = 0$
D $x = 2$ or $x = 1$

Question 302

[2 marks]Equations, Vectors and Inequalities

Given

\mathbf{a} = \begin{pmatrix} 3 \\ 3 \end{pmatrix}

and

\mathbf{b} = \begin{pmatrix} -1 \\ 2 \end{pmatrix}

, find

\mathbf{a} + \mathbf{b}

as a column vector, top entry first.

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Question 303

[2 marks]Equations, Vectors and Inequalities

Given

\mathbf{a} = \begin{pmatrix} 3 \\ 3 \end{pmatrix}

and

\mathbf{b} = \begin{pmatrix} -1 \\ 2 \end{pmatrix}

, find

\mathbf{a} - 3\mathbf{b}

as a column vector, top entry first.

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Question 304

[2 marks]Equations, Vectors and Inequalities

A vector has components

\begin{pmatrix} 6 \\ -3 \end{pmatrix}

. Find its magnitude, correct to 3 significant figures.

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Question 305

[3 marks]Equations, Vectors and Inequalities

Solve the inequality

5 - 3x \le 7 < -2x + 19

A $-\frac{2}{3} \le x < 6$
B $-\frac{2}{3} \ge x > 6$
C $\frac{2}{3} \le x < 6$
D $-\frac{2}{3} \le x \le 6$

Question 401

[2 marks]Circle Geometry

In the diagram, ABCD is a cyclic quadrilateral with

A\hat{D}C = 58°

. Find

A\hat{B}C

A $122°$ , because opposite angles of a cyclic quadrilateral add up to $180°$
B $116°$ , because the exterior angle of a cyclic quadrilateral is twice the interior one
C $58°$ , because angles on the same arc are equal
D $29°$ , because the angle at the centre is twice the angle at the circumference

Question 402

[2 marks]Circle Geometry

In the diagram, TDE is a tangent to the circle at D, so E, D and T lie on one straight line. Given that

A\hat{D}E = 75°

and

A\hat{D}C = 58°

, find

C\hat{D}T

in degrees.

Answer this when you sit the paper.

Question 403

[2 marks]Circle Geometry

In the diagram, TC and TDE are tangents to the circle,

C\hat{D}T = 47°

and

A\hat{T}D = 31°

. Find

A\hat{T}C

in degrees.

Answer this when you sit the paper.

Question 404

[2 marks]Circle Geometry

The surface area

A

of a solid cylinder of height

h

and base radius

r

is given by

A = 2\pi r(r + h)

. Make

h

the subject of the formula.

A $h = \dfrac{2\pi r^2 - A}{2\pi r}$
B $h = \dfrac{A - 2\pi r^2}{2\pi r}$
C $h = \dfrac{A}{2\pi r} + r$
D $h = \dfrac{A - 2\pi r}{2\pi r}$

Question 405

[2 marks]Circle Geometry

The surface area of a solid cylinder is

A = 2\pi r(r + h)

. Find

h

, in cm, when

A = 77

^2

r = 2,5

cm and

\pi

is taken as

\dfrac{22}{7}

Answer this when you sit the paper.

Question 501

[2 marks]Approximations, Bearings and Variation

The dimensions of a rectangle, measured to the nearest centimetre, are 42 cm by 81 cm. Calculate the least possible perimeter of the rectangle, in cm.

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Question 502

[3 marks]Approximations, Bearings and Variation

A, B and C are points on level ground with AB = 4 km, BC = 4,5 km and AC = 7 km. Calculate

B\hat{A}C

, giving the answer to the nearest degree.

Answer this when you sit the paper.

Question 503

[2 marks]Approximations, Bearings and Variation

In the diagram, the bearing of B from A is 016° and

B\hat{A}C = 37°

. Find the bearing of C from A, giving the answer as a three figure bearing.

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Question 504

[3 marks]Approximations, Bearings and Variation

The volume

V

of a gas at constant temperature is inversely proportional to its pressure

P

. Given that

V = 45

litres when

P = 600

Newtons per square metre, find

V

, in litres, when

P = 1 050

Newtons per square metre, correct to 3 significant figures.

Answer this when you sit the paper.

Question 601

[3 marks]Constructions and Loci

In quadrilateral ABCD, BC = 6 cm, DC = 9,5 cm and

B\hat{C}D = 60°

. Calculate BD, in cm, correct to 3 significant figures.

Answer this when you sit the paper.

Question 602

[2 marks]Constructions and Loci

A is a fixed point. Which of these describes the locus of points 4,5 cm from A?

AA pair of parallel lines each 4,5 cm from the point A
BThe perpendicular bisector of a 4,5 cm line through A
CA circle with centre A and radius 4,5 cm
DA straight line 4,5 cm long drawn from the point A

Question 603

[2 marks]Constructions and Loci

AD and DC are two straight lines meeting at D. Which of these describes the locus of points equidistant from the line AD and the line DC?

AThe perpendicular bisector of the line joining A and C
BThe bisector of the angle ADC formed by the two lines
CThe circle that passes through the points A, D and C
DThe line through D parallel to the longer of the two lines

Question 604

[2 marks]Constructions and Loci

D and C are two fixed points. Which of these describes the locus of points equidistant from D and from C?

AThe perpendicular bisector of the line segment DC
BTwo lines parallel to DC, one on each side of it
CThe bisector of the angle between DC and the horizontal
DA circle drawn on DC as diameter, with centre the midpoint

Question 701

[2 marks]Consumer Arithmetic

A piece of fleece material costs R20 in South Africa, and that price is equivalent to $2,50. How many rand are equivalent to one dollar?

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Question 702

[3 marks]Consumer Arithmetic

Chido bought 20 pieces of fleece in South Africa at $2,50 each and incurred $30 in travelling costs. She sells all 20 pieces at $5 each. Calculate her net profit, in dollars.

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Question 703

[3 marks]Consumer Arithmetic

Chido buys 20 pieces of fleece in South Africa at $2,50 each and pays $30 in travelling costs. Each piece makes one morning gown, and she sells each gown at $12. Calculate her net profit from the sale of 20 gowns, in dollars.

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Question 704

[2 marks]Consumer Arithmetic

Chido buys 20 pieces of material in Zimbabwe at $5 a piece, with no travelling costs, and makes 20 gowns which she sells at $12 each. Calculate her profit, in dollars.

Answer this when you sit the paper.

Question 705

[2 marks]Consumer Arithmetic

Making 20 gowns from material bought in South Africa returns a profit of $160, while making them from material bought in Zimbabwe returns $140. Find the difference in the profits realised, and say which is the better buy.

A$40 more from buying the material in South Africa
B$60 more from buying the material in Zimbabwe
C$20 more from buying the material in South Africa
D$20 more from buying the material in Zimbabwe

Question 801

[2 marks]Functional Graphs

Write down the roots of the equation

(x + 1)^2(x - 2) = 0

Answer this when you sit the paper.

Question 802

[2 marks]Functional Graphs

The curve

y = (x + 1)^2(x - 2)

has two points at which its gradient is zero. Write down the coordinates of the lower of the two, its minimum point.

Answer this when you sit the paper.

Question 803

[1 marks]Functional Graphs

For which values of

x

is the function

y = (x + 1)^2(x - 2)

positive?

A $x < -1$ only, where the squared factor is largest
B $-1 < x < 2$ , the whole stretch between the two roots
C $x < -1$ or $x > 2$ , on both sides of the two roots
D $x > 2$ only, since the squared factor is not negative

Question 804

[2 marks]Functional Graphs

Using the graph of

y = (x + 1)^2(x - 2)

, find the gradient of the curve at the point where

x = 1,5

Answer this when you sit the paper.

Question 805

[3 marks]Functional Graphs

Solve the equation

(x + 1)^2(x - 2) = -2

, giving each root correct to 1 decimal place where it is not exact.

A $x = -1,7$ or $x = 1,7$ only
B $x = -2,0$ , $x = 0$ or $x = 2,0$
C $x = -1,0$ , $x = 0$ or $x = 2,0$
D $x = -1,7$ , $x = 0$ or $x = 1,7$

Question 806

[2 marks]Functional Graphs

Using the graph of

y = (x + 1)^2(x - 2)

, find the area, in square units, bounded by the curve, the

x

-axis, the

y

-axis and the line

x = 1

Answer this when you sit the paper.

Question 901

[2 marks]Geometrical Transformation

The point

(8; 4)

is mapped by the translation

\begin{pmatrix} -9 \\ 2 \end{pmatrix}

. Write down the coordinates of its image.

Answer this when you sit the paper.

Question 902

[2 marks]Geometrical Transformation

The point

(5; 2)

is reflected in the line

y = -x

. Write down the coordinates of its image.

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Question 903

[3 marks]Geometrical Transformation

Triangle A has vertices

(2; 2)

(5; 2)

and

(8; 4)

. Its image has vertices

(-2; 4)

(4; 4)

and

(10; 8)

. Describe completely the single transformation which maps A onto its image.

AA rotation of $180°$ about the point $(6; 0)$
BAn enlargement, centre $(6; 0)$ , scale factor 2
CAn enlargement, centre $(0; 0)$ , scale factor 2
DA stretch parallel to the $x$ -axis, factor 2

Question 904

[2 marks]Geometrical Transformation

The point

(8; 4)

is mapped by the transformation with matrix

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

. Write down the coordinates of its image.

Answer this when you sit the paper.

Question 905

[3 marks]Geometrical Transformation

The point

(8; 4)

is rotated through 90° clockwise about the point

(2; -2)

. Write down the coordinates of its image.

Answer this when you sit the paper.

Question 1001

[2 marks]Measures and Mensuration

In the diagram, the block is sawn along the plane WXYZ. That plane is a rectangle whose depth WZ is 10 cm and whose slant side WX rises 4 cm over a run of 10 cm. Calculate the area of the plane WXYZ, in cm

^2

, correct to 3 significant figures.

Answer this when you sit the paper.

Question 1002

[2 marks]Measures and Mensuration

In the diagram, triangle ZCY is right angled at C with ZC = 10 cm and CY = 4 cm. Calculate the area of triangle ZCY, in cm

^2

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Question 1003

[2 marks]Measures and Mensuration

In the diagram, triangle ZCY is right angled at C with ZC = 10 cm and CY = 4 cm. Calculate angle CZY, in degrees, correct to 1 decimal place.

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Question 1004

[3 marks]Measures and Mensuration

The wedge WBXYCZ is a prism 10 cm deep. Its triangular cross section is right angled with sides 10 cm and 4 cm, and its sawn face measures 10 cm by 10,77 cm. Calculate the total surface area of the wedge, in cm

^2

, correct to 3 significant figures.

Answer this when you sit the paper.

Question 1005

[3 marks]Measures and Mensuration

A rectangular block of wood 25 cm long, 10 cm wide and 8 cm high is sawn to remove a wedge whose triangular cross section is right angled with sides 10 cm and 4 cm and whose depth is 10 cm. Calculate the percentage of the volume of the wood that is removed.

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Question 1101

[2 marks]Statistics and Probability

In a cumulative frequency record for 100 boys, the entry for

x \le 35

hours is 46. A further 23 boys watched for

35 < x \le 40

hours and 17 more for

40 < x \le 45

hours. Find the entry

q

for

x \le 45

Answer this when you sit the paper.

Question 1102

[2 marks]Statistics and Probability

One hundred boys watched soccer on TV. Of them, 1 watched for

x \le 15

hours, 2 for

15 < x \le 20

, 7 for

20 < x \le 25

and 11 for

25 < x \le 30

. Find the number of boys who watched for 30 hours and below.

Answer this when you sit the paper.

Question 1103

[3 marks]Statistics and Probability

For a distribution of 100 boys, the cumulative frequency is 46 at 35 hours and 69 at 40 hours. Estimate the median number of hours, correct to the nearest hour.

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Question 1104

[2 marks]Statistics and Probability

For a distribution of 100 boys, 3 watched soccer for

x \le 20

hours and 86 watched for

x \le 45

hours. Find the number of boys who watched for more than 20 hours but not more than 45 hours.

Answer this when you sit the paper.

Question 1105

[3 marks]Statistics and Probability

Of 100 boys, 21 watched soccer for 30 hours or less. Two boys are chosen at random from the group. Find the probability that both of them watched for 30 hours or less.

A $\dfrac{21}{50}$ , twice the chance for one single boy chosen
B $\dfrac{7}{165}$ , about 0,042 once the first boy is taken out
C $\dfrac{441}{10000}$ , treating the two picks as independent
D $\dfrac{21}{100} + \dfrac{20}{99}$ , adding the two separate chances

Question 1201

[3 marks]Inequalities and Linear Programming

A grocer's offer needs at least 1 kg of apples and more than 1 kg of grapes, with the total limited to 5 kg. Taking

x

as the mass of apples and

y

as the mass of grapes, write down the three inequalities.

A $x \ge 1$ , $y > 1$ and $x + y \ge 5$
B $x \ge 1$ , $y \ge 1$ and $x + y < 5$
C $x \ge 1$ , $y > 1$ and $x + y \le 5$
D $x > 1$ , $y \ge 1$ and $x + y \le 5$

Question 1202

[3 marks]Inequalities and Linear Programming

A shop makes 40c per kg profit on apples and 55c per kg on grapes. A purchase must satisfy

x \ge 1

y > 1

and

x + y \le 5

, where

x

kg is the mass of apples and

y

kg the mass of grapes. Which combination gives the shop its greatest profit?

A2,5 kg of apples and 2,5 kg of grapes
B1 kg of apples and 1 kg of grapes
C4 kg of apples and 1 kg of grapes
D1 kg of apples and 4 kg of grapes

Question 1203

[3 marks]Inequalities and Linear Programming

A shop makes 40c per kg profit on apples and 55c per kg on grapes. Calculate the profit, in cents, on a sale of 1 kg of apples and 4 kg of grapes.

Answer this when you sit the paper.

Question 1204

[2 marks]Inequalities and Linear Programming

A shop makes 40c per kg profit on apples and 55c per kg on grapes. Calculate the profit, in cents, on a sale of 2 kg of apples and 3 kg of grapes.

Answer this when you sit the paper.

Mathematics Paper 2 June 2013

Sit this paper online

The questions

Question 101

Question 102

Question 103

Question 104

Question 201

Question 202

Question 203

Question 204

Question 301

Question 302

Question 303

Question 304

Question 305

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 604

Question 701

Question 702

Question 703

Question 704

Question 705

Question 801

Question 802

Question 803

Question 804

Question 805

Question 806

Question 901

Question 902

Question 903

Question 904

Question 905

Question 1001

Question 1002

Question 1003

Question 1004

Question 1005

Question 1101

Question 1102

Question 1103

Question 1104

Question 1105

Question 1201

Question 1202

Question 1203

Question 1204

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