ZIMSEC O Level Mathematics Paper 2 June 2006

Question 101

[2 marks]Algebra

Express as a single fraction in its simplest form

\frac{3}{2} + \frac{4x - 1}{3}

Answer this when you sit the paper.

Question 102

[2 marks]Algebra

Given the formula

S = \frac{1}{2}n(r + l)

, calculate the value of

S

when

n = 13

r = 7

and

l = 11

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Question 103

[2 marks]Algebra

Make

r

the subject of the formula

S = \frac{1}{2}n(r + l)

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Question 104

[3 marks]Algebra

Calculate, in dollars, the principal that earns \$18 450 simple interest at 6% per annum in

1\frac{1}{2}

years.

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Question 105

[3 marks]Algebra

Solve the equation

5(y + 3) - 7 = 3(4 - y)

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Question 201

[2 marks]Ratios, Rates & Proportions

The chart shows the road distances, in kilometres, between five towns. Calculate

x

, the distance between Kadoma and Bulawayo.

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Question 202

[1 marks]Ratios, Rates & Proportions

The distance between Harare and Victoria Falls is 879 kilometres. Write this distance down correct to the nearest 10 kilometres.

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Question 203

[2 marks]Ratios, Rates & Proportions

A motorist travels the 440 kilometres from Bulawayo to Victoria Falls at an average speed of 80 km/h. Calculate the time taken, in hours.

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Question 204

[2 marks]Ratios, Rates & Proportions

A car uses 6,5 litres of fuel for every 100 kilometres travelled. Calculate the fuel, in litres, that it needs for a journey of 440 kilometres.

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Question 401

[2 marks]Constructions and Loci

A pentagon PQRST is constructed with

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

and

Q\hat{P}T = 135^\circ

. Calculate, in degrees, the sum of its three remaining interior angles.

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Question 402

[2 marks]Constructions and Loci

Two lines SR and ST meet at the point S. A point moves so that it stays the same distance from SR as it is from ST. On which line does that point lie?

AThe perpendicular bisector of the straight line joining the point R to the point T
BThe arc of a circle drawn with its centre at S and passing through both R and T
CThe bisector of the angle RST, drawn from S between the two lines
DThe line through S at right angles to RT, extended past S on both sides

Question 403

[2 marks]Constructions and Loci

In pentagon PQRST a line is drawn through R parallel to PQ, meeting ST at X. Why do triangle PRQ and triangle PXQ have equal areas?

AThey share the base PQ, and RX being parallel to PQ puts R and X the same perpendicular distance from PQ
BThey share the base PQ, and RX being parallel to PQ makes the slant sides RQ and XQ equal in length
CThey are congruent triangles, because a line drawn parallel to PQ maps triangle PRQ exactly onto triangle PXQ
DPQ is common to both and the angles at P are equal, so one triangle is an enlargement of the other about P

Question 404

[2 marks]Constructions and Loci

In the pentagon PQRST the vertex S satisfies RS = ST = 8 cm. Once R and T have been drawn, how is S located using ruler and compasses only?

ABy drawing a line through R parallel to PQ and stepping 8 cm along it from R towards ST
BBy the point where an arc of radius 8 cm centred on R meets an arc of radius 8 cm centred on T
CBy measuring 8 cm along the bisector of angle RPT from P, then marking the far end of it
DBy the point where the perpendicular bisector of RT meets a circle of radius 8 cm centred on P

Question 501

[2 marks]Sets

In a group of students, 8 study both Mathematics and Physics, and

x

of those 8 study Chemistry as well. Write down, in terms of

x

, the number who study Mathematics and Physics but not Chemistry.

Answer this when you sit the paper.

Question 502

[2 marks]Sets

18 students study Chemistry. Of these,

10 - x

study Chemistry and Mathematics only,

5 - x

study Chemistry and Physics only, and

x

study all three subjects. Find, in terms of

x

, the number who study Chemistry only.

Answer this when you sit the paper.

Question 503

[3 marks]Sets

The seven regions of the Venn diagram for 25 students hold

x - 3

8 - x

x - 1

10 - x

x

5 - x

and

x + 3

students. Form an equation in

x

and solve it.

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Question 504

[1 marks]Sets

10 students study Chemistry and Mathematics in total, and 3 of them study Physics as well. Find the number who study Mathematics and Chemistry only.

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Question 505

[1 marks]Sets

Two fair-sided coins are tossed at the same time. Write down the probability of getting two Heads or two Tails.

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Question 506

[2 marks]Sets

Solve the equation

2^{2n-1} = 64

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Question 701

[2 marks]Trigonometry, Bearing & Distances

The bearing of Q from P is

328^\circ

and the bearing of R from Q is

191^\circ

. Calculate

P\hat{Q}R

, in degrees.

Answer this when you sit the paper.

Question 702

[3 marks]Trigonometry, Bearing & Distances

P and Q are two villages with PQ

= 3

km, and the bearing of Q from P is

328^\circ

. Calculate, in kilometres, the distance Q is north of P, correct to 3 significant figures.

Answer this when you sit the paper.

Question 703

[3 marks]Trigonometry, Bearing & Distances

In triangle PQR, PQ

= 3

km, QR

= 6

km and

P\hat{Q}R = 43^\circ

. Calculate PR, in kilometres, correct to 3 significant figures.

Answer this when you sit the paper.

Question 704

[2 marks]Trigonometry, Bearing & Distances

In triangle PQR, PQ

= 3

km, QR

= 6

km and PR

= 4,32

km. Use the cosine rule to calculate

Q\hat{P}R

, in degrees, correct to 1 decimal place.

Answer this when you sit the paper.

Question 705

[2 marks]Trigonometry, Bearing & Distances

The diagram shows villages P, Q and R. The bearing of Q from P is

328^\circ

and

Q\hat{P}R = 108,7^\circ

, with R lying to the south west of P. Find the bearing of R from P, correct to the nearest degree.

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Question 801

[1 marks]Travel Graphs

A particle moves in a straight line so that its velocity after

t

seconds is

v = 5 + 7t - 2t^2

m/s. Calculate

v

when

t = 4

Answer this when you sit the paper.

Question 802

[3 marks]Travel Graphs

A particle moves in a straight line so that its velocity after

t

seconds is

v = 5 + 7t - 2t^2

m/s. Find its maximum velocity, in m/s, correct to 3 significant figures.

Answer this when you sit the paper.

Question 803

[2 marks]Travel Graphs

A particle moves in a straight line so that its velocity after

t

seconds is

v = 5 + 7t - 2t^2

m/s. Find the value of

t

, in seconds, at which the particle is stationary, correct to 2 significant figures.

Answer this when you sit the paper.

Question 804

[2 marks]Travel Graphs

A particle moves in a straight line so that its velocity after

t

seconds is

v = 5 + 7t - 2t^2

m/s. Calculate its acceleration, in m/s squared, when

t = 5

Answer this when you sit the paper.

Question 805

[3 marks]Travel Graphs

A particle moves in a straight line so that its velocity after

t

seconds is

v = 5 + 7t - 2t^2

m/s. Estimate the distance, in metres, that it travels between

t = 2

and

t = 4

Answer this when you sit the paper.

Question 901

[2 marks]Quadratic Equations

In triangle ACD the angle at D is

90^\circ

A\hat{C}D = 60^\circ

and AC

= (x - 1)

cm. Find CD in terms of

x

Answer this when you sit the paper.

Question 902

[2 marks]Quadratic Equations

A triangle has base AB

= (2x + 3)

cm and perpendicular height CD

= \frac{x - 1}{2}

cm. Which expression gives its area, in cm

^2

A $\frac{1}{2}(2x^2 + x - 3)$ , from multiplying the base by the height without halving it
B $\frac{1}{4}(2x^2 + x - 3)$ , from halving the product of the base and the height
C $\frac{1}{4}(2x^2 + 5x + 3)$ , from adding the two brackets before multiplying them out
D $\frac{1}{2}(2x^2 - x - 3)$ , from taking the height as $(x - 1)$ instead of half of it

Question 903

[2 marks]Quadratic Equations

The area of a triangle is

\frac{1}{4}(2x^2 + x - 3)

^2

. Given that this area is 9 cm

^2

, write down the value of

2x^2 + x

Answer this when you sit the paper.

Question 904

[3 marks]Quadratic Equations

Solve the equation

2x^2 + x - 39 = 0

, giving the positive root correct to two decimal places.

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Question 905

[2 marks]Quadratic Equations

In a triangle AC

= (x - 1)

cm, where

x

is the positive root of

2x^2 + x - 39 = 0

and equals 4,17. Write down the length of AC, in centimetres, correct to two decimal places.

Answer this when you sit the paper.

Question 1001

[2 marks]Geometrical Transformation

Triangle ABC has vertices A

(0; 3)

, B

(2; 1)

and C

(4; 5)

. It is reflected in the line

y = 0

to give triangle

A_1B_1C_1

. Write down the coordinates of

C_1

Answer this when you sit the paper.

Question 1002

[2 marks]Geometrical Transformation

Write down the matrix that represents a reflection in the line

y = 0

A $\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$
B $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$
C $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$
D $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$

Question 1003

[3 marks]Geometrical Transformation

Triangle ABC with vertices A

(0; 3)

, B

(2; 1)

and C

(4; 5)

is mapped onto triangle

A_2B_2C_2

with vertices

A_2(6; 3)

B_2(4; 1)

and

C_2(14; 5)

. Describe fully the single transformation.

AA shear with the $x$ -axis ( $y = 0$ ) invariant and shear factor 2
BA translation with vector $\begin{pmatrix} 6 \\ 0 \end{pmatrix}$ applied to each vertex
CA stretch parallel to the $x$ -axis with the $y$ -axis invariant and factor 3
DA shear with the $y$ -axis invariant and shear factor 2, taking B two units right

Question 1004

[3 marks]Geometrical Transformation

Triangle ABC has vertices A

(0; 3)

, B

(2; 1)

and C

(4; 5)

. It is enlarged with scale factor

-2

about the centre

(5; 0)

. Write down the coordinates of the image of A.

Answer this when you sit the paper.

Question 1005

[2 marks]Geometrical Transformation

A triangle is enlarged with scale factor

-2

. Find the ratio of the area of the original triangle to the area of its image, in the form

1 : n

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Question 1101

[2 marks]Statistics and Probability

The amounts contributed to charity by a mathematics class were grouped into five classes with frequencies 12, 11, 8, 10 and 4. Find the number of students in the class.

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Question 1102

[3 marks]Statistics and Probability

The amounts

x

contributed by 45 students were grouped as

15 < x \le 20

with 12 students,

20 < x \le 23

with 11,

23 < x \le 27

with 8,

27 < x \le 34

with 10 and

34 < x \le 40

with 4. Calculate the mean amount contributed, in dollars, correct to 3 significant figures.

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Question 1103

[3 marks]Statistics and Probability

A shape is made up of a rectangle 120 cm long and 70 cm wide with a semicircular top standing on the 120 cm side. Taking

\pi

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

^2

Answer this when you sit the paper.

Question 1104

[2 marks]Statistics and Probability

A scale drawing 120 cm long represents a billboard 3 metres long. The area of the drawing is 14 055,6 cm

^2

. Calculate the area of the actual billboard, in square metres, correct to 3 significant figures.

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Question 1105

[2 marks]Statistics and Probability

Painting costs \$4 000 per square metre. Calculate the cost, in dollars, of painting a billboard whose area is 8,784 75 square metres.

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Question 1201

[1 marks]Inequalities and Linear Programming

A factory makes

x

desks of type A and

y

desks of type B each year, and it produces at most 5 000 desks annually. Write down an inequality in

x

and

y

for this condition.

Answer this when you sit the paper.

Question 1202

[2 marks]Inequalities and Linear Programming

The number of type B desks,

y

, is not less than half the number of type A desks,

x

, and is also not more than the number of type A desks. Which pair of inequalities states these two conditions?

A $y > \frac{1}{2}x$ and $y < x$ , on the reading that neither boundary value is allowed
B $x \ge \frac{1}{2}y$ and $x \le y$ , with the roles of the two types of desk exchanged
C $y \ge \frac{1}{2}x$ and $y \le x$ , so $y$ lies between half of $x$ and $x$ itself
D $y \le \frac{1}{2}x$ and $y \ge x$ , reading not less than as an upper limit on $y$

Question 1203

[2 marks]Inequalities and Linear Programming

A type A desk costs \$4 000 to make and a type B desk costs \$9 200, and the factory has at least \$18 400 000 for its annual production. Dividing the resulting inequality through by 400 puts it in the form

10x + 23y \ge k

. Write down the value of

k

Answer this when you sit the paper.

Question 1204

[3 marks]Inequalities and Linear Programming

A region is bounded by the lines

x + y = 5\,000

2y = x

and

y = x

. Find the coordinates of the vertex where

x + y = 5\,000

meets

2y = x

, giving each value correct to the nearest whole number.

Answer this when you sit the paper.

Question 1205

[3 marks]Inequalities and Linear Programming

The profit is \$1 200 on each type A desk and \$1 000 on each type B desk. The greatest profit is reached at

x = 3\,333

type A desks and

y = 1\,667

type B desks. Calculate that greatest profit, in dollars.

Answer this when you sit the paper.

Mathematics Paper 2 June 2006

Sit this paper online

The questions

Question 101

Question 102

Question 103

Question 104

Question 105

Question 201

Question 202

Question 203

Question 204

Question 401

Question 402

Question 403

Question 404

Question 501

Question 502

Question 503

Question 504

Question 505

Question 506

Question 701

Question 702

Question 703

Question 704

Question 705

Question 801

Question 802

Question 803

Question 804

Question 805

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

Question 1205

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