ZIMSEC O Level Mathematics Paper 2 November 2009

Question 101

[2 marks]Factorisation, H.C.F and L.C.M

Simplify

8 - 24 \div 6 + 3 \times 4

Answer this when you sit the paper.

Question 102

[2 marks]Factorisation, H.C.F and L.C.M

Expand

(1 - 2x)(x + 3)

A $-2x^2 - 7x + 3$
B $-2x^2 - 5x + 3$
C $2x^2 + 5x - 3$
D $-2x^2 + 5x + 3$

Question 103

[3 marks]Factorisation, H.C.F and L.C.M

Find the L.C.M. of

15y^2

25xy^3

and

(x^3 - x^2)

A $75x^2y^3(x - 1)$
B $75xy^3(x - 1)$
C $5x^2y^3(x - 1)$
D $375x^3y^5(x - 1)$

Question 104

[3 marks]Factorisation, H.C.F and L.C.M

Evaluate

(\log_9 81) \times (2\log_4 8)

Answer this when you sit the paper.

Question 201

[2 marks]Sets

Solve the equation

7(h + 3) - 2(h - 4) = 4

Answer this when you sit the paper.

Question 202

[3 marks]Sets

Solve the equation

3^{(m + 4)} = 9^{(m - 1)}

Answer this when you sit the paper.

Question 203

[2 marks]Sets

Write down, in set notation, the set represented by the shaded region in the Venn diagram.

A $(A \cup B \cup C)'$
B $(B \cap C) \cup A'$
C $(B \cup C) \cap A'$
D $(B \cup C) \cap A$

Question 204

[2 marks]Sets

The universal set is

\xi = \{x : 2 \le x \le 20,\ x \text{ is an integer}\}

and

P = \{x : x \text{ is a prime number}\}

. List the elements of P.

A $\{1, 2, 3, 5, 7, 11, 13, 17, 19\}$
B $\{3, 5, 7, 11, 13, 17, 19\}$
C $\{2, 3, 5, 7, 11, 13, 17, 19, 20\}$
D $\{2, 3, 5, 7, 11, 13, 17, 19\}$

Question 205

[2 marks]Sets

With

\xi = \{x : 2 \le x \le 20,\ x \text{ is an integer}\}

P = \{x : x \text{ is a prime number}\}

and

Q = \{x : 4 \le x < 17\}

, find

n(Q' \cap P)

Answer this when you sit the paper.

Question 301

[2 marks]Matrices

Simplify

\dfrac{n - 3}{6} \div \dfrac{n^2 - 9}{4}

A $\dfrac{3(n + 3)}{2}$
B $\dfrac{2}{3(n + 3)}$
C $\dfrac{2}{3(n - 3)}$
D $\dfrac{(n - 3)^2(n + 3)}{24}$

Question 302

[2 marks]Matrices

Given that

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

and

\mathbf{B} = \begin{pmatrix} 4 & -1 \\ 5 & 6 \end{pmatrix}

, find

\mathbf{AB}

, giving the two entries in order.

Answer this when you sit the paper.

Question 303

[2 marks]Matrices

Find

\mathbf{B}^{-1}

for

\mathbf{B} = \begin{pmatrix} 4 & -1 \\ 5 & 6 \end{pmatrix}

A $\frac{1}{19}\begin{pmatrix} 6 & 1 \\ -5 & 4 \end{pmatrix}$
B $\frac{1}{29}\begin{pmatrix} 6 & -1 \\ 5 & 4 \end{pmatrix}$
C $\frac{1}{29}\begin{pmatrix} 6 & 1 \\ -5 & 4 \end{pmatrix}$
D $\frac{1}{29}\begin{pmatrix} 4 & 1 \\ -5 & 6 \end{pmatrix}$

Question 304

[2 marks]Matrices

The matrix

\begin{pmatrix} -2 & p \\ p + 3 & -4p \end{pmatrix}

is singular. Find the two possible values of

p

Answer this when you sit the paper.

Question 305

[2 marks]Matrices

A company invested money in a bank at 270% simple interest per annum. After 8 months the total value of the investment was 840 million dollars. Calculate the amount invested, in millions of dollars.

Answer this when you sit the paper.

Question 401

[2 marks]Variation

In the diagram, square PQRS has its vertices at the midpoints of the sides of square ABCD, and

AB = 12

cm. Calculate the perimeter of PQRS, in centimetres, correct to 3 significant figures.

Answer this when you sit the paper.

Question 402

[2 marks]Variation

In the diagram, square PQRS has its vertices at the midpoints of the sides of square ABCD, and

AB = 12

cm. Calculate the area of triangle QRS, in square centimetres.

Answer this when you sit the paper.

Question 403

[1 marks]Variation

Sibongile's weekly wage W is partly constant and partly varies as the number of hours N of overtime she works per week. Express W in terms of N and the constants

h

and

k

A $W = h + \frac{k}{N}$
B $W = kN + hN$
C $W = hN + k$
D $W = h + kN$

Question 404

[1 marks]Variation

For the wage relation

W = h + kN

, it is given that

W = 80

when

N = 10

and

W = 60

when

N = 6

. Find the value of

h

Answer this when you sit the paper.

Question 405

[2 marks]Variation

For the wage relation

W = h + kN

, it is given that

W = 80

when

N = 10

and

W = 60

when

N = 6

. Find the value of

k

Answer this when you sit the paper.

Question 406

[3 marks]Variation

Sibongile's weekly wage in thousands of dollars is

W = 30 + 5N

, where N is her overtime hours, and her normal working time is 44 hours a week. Find the total number of hours she worked in a week in which she was paid 90 thousand dollars.

Answer this when you sit the paper.

Question 501

[2 marks]Change of Subject of Formula

The volume of material in a cylindrical tube is

V = \pi(R^2 - r^2)h

. Taking

\pi

to be

\frac{22}{7}

, find V, in cubic centimetres, when

R = 4

cm,

r = 3

cm and

h = 150

cm.

Answer this when you sit the paper.

Question 502

[3 marks]Change of Subject of Formula

Make R the subject of the formula

V = \pi(R^2 - r^2)h

A $R = \frac{V}{\pi h} + r^2$
B $R = \sqrt{\frac{V}{\pi h}} + r$
C $R = \sqrt{\frac{V}{\pi h} + r^2}$
D $R = \sqrt{\frac{V}{\pi h} - r^2}$

Question 503

[3 marks]Change of Subject of Formula

A solid cuboid measures 8 cm by 7 cm by

x

cm and has a total surface area of 442 cm

^2

. Calculate the value of

x

Answer this when you sit the paper.

Question 504

[2 marks]Change of Subject of Formula

A solid cuboid of density 0,7 g/cm

^3

measures 8 cm by 7 cm by 11 cm. Calculate the mass of the solid, in grams.

Answer this when you sit the paper.

Question 601

[2 marks]Inequalities and Linear Programming

Factorise completely

3p^2 + 7p - 6

A $(3p - 2)(p + 3)$
B $(3p + 2)(p - 3)$
C $(3p - 6)(p + 1)$
D $(p - 2)(3p + 3)$

Question 602

[2 marks]Inequalities and Linear Programming

On the graph, one boundary of the unshaded region R is the straight line through

(1; 2)

and

(3; 6)

. Write down the inequality this boundary gives for R.

A $x \le 2y$
B $y \le 2x + 1$
C $y \ge 2x$
D $y \le 2x$

Question 603

[2 marks]Inequalities and Linear Programming

On the graph, one boundary of the unshaded region R is the straight line through

(1; 2)

and

(3; 0)

. Write down the inequality this boundary gives for R.

A $x + y \ge 3$
B $x + y \le 3$
C $y - x \ge 3$
D $x + y \ge 5$

Question 604

[2 marks]Inequalities and Linear Programming

On the graph, one boundary of the unshaded region R is the straight line through

(3; 0)

and

(6; 5)

. Write down the inequality this boundary gives for R.

A $5x - 3y \ge 15$
B $3x - 5y \le 15$
C $5x + 3y \le 15$
D $5x - 3y \le 15$

Question 605

[2 marks]Inequalities and Linear Programming

Find the maximum value of

5y - x

for integer values of

x

and

y

in the unshaded region R shown.

Answer this when you sit the paper.

Question 701

[1 marks]Polygons, Symmetry and Circles

In the hexagon PQRSTU, the lines PQ and UT are parallel and

U\hat{P}Q = y^\circ

. Write down an expression, in terms of

y

, for

P\hat{U}T

A $(180 + y)^\circ$
B $(90 - y)^\circ$
C $(180 - y)^\circ$
D $(360 - y)^\circ$

Question 702

[3 marks]Polygons, Symmetry and Circles

In the hexagon PQRSTU,

U\hat{P}Q = y^\circ

P\hat{Q}R = 130^\circ

Q\hat{R}S = 5y^\circ

R\hat{S}T = 155^\circ

S\hat{T}U = (180 - 2y)^\circ

and

P\hat{U}T = (180 - y)^\circ

. Using the sum of the interior angles, find the value of

y

Answer this when you sit the paper.

Question 703

[1 marks]Polygons, Symmetry and Circles

In the hexagon PQRSTU,

Q\hat{R}S = 5y^\circ

and

y = 25

. Write down the numerical value of

Q\hat{R}S

, in degrees.

Answer this when you sit the paper.

Question 704

[2 marks]Polygons, Symmetry and Circles

Expand and simplify

(2x - 5)(x + 3)

Answer this when you sit the paper.

Question 705

[3 marks]Polygons, Symmetry and Circles

Solve the equation

4x^2 - x - 6 = 0

and give the positive root correct to two decimal places.

Answer this when you sit the paper.

Question 706

[2 marks]Polygons, Symmetry and Circles

Solve the equation

4x^2 - x - 6 = 0

and give the negative root correct to two decimal places.

Answer this when you sit the paper.

Question 801

[2 marks]Constructions and Loci

A quadrilateral PQRS is to be drawn using a scale of 1 cm to represent 2 km, and

PS = 17

km. How long, in centimetres, is the line drawn for PS?

Answer this when you sit the paper.

Question 802

[2 marks]Constructions and Loci

In the quadrilateral PQRS, which construction gives the locus of points equidistant from the sides PS and SR?

AThe perpendicular bisector of the line joining P to R
BA circle centred on S and passing through both P and R
CThe bisector of angle PSR, drawn from the vertex S
DThe line drawn through S and parallel to the side PQ

Question 803

[1 marks]Constructions and Loci

On a drawing made to a scale of 1 cm to 2 km, which construction gives the locus of points 10 km from the point P?

AA circle of radius 5 cm with its centre at the point P
BA circle of radius 10 cm with its centre at the point P
CA line drawn 5 cm from P and parallel to the side PQ
DAn arc of radius 2 cm swung from the point P

Question 804

[2 marks]Constructions and Loci

In the quadrilateral PQRS,

PQ = 14

km,

QR = 12

km and

P\hat{Q}R = 90^\circ

. Calculate the length of PR, in kilometres, correct to 3 significant figures.

Answer this when you sit the paper.

Question 805

[2 marks]Constructions and Loci

On a drawing made to a scale of 1 cm to 2 km, which construction gives the locus of points 6 km from the side QR?

AA line drawn parallel to QR and 3 cm away from it
BA circle of radius 3 cm with its centre at the point Q
CThe perpendicular bisector of the side QR of the shape
DA line drawn parallel to QR and 6 cm away from it

Question 901

[2 marks]Geometrical Transformation

Triangle B, with vertices

(2; 0)

(4; 0)

and

(0; -4)

, is a reflection of triangle A, with vertices

(-2; 6)

(-2; 4)

and

(-6; 2)

. Write down the equation of the mirror line.

Answer this when you sit the paper.

Question 902

[1 marks]Geometrical Transformation

Triangle A is reflected onto triangle B in the line

y = x + 2

. Given that

(k; 8)

is one of the invariant points under this reflection, find the value of

k

Answer this when you sit the paper.

Question 903

[3 marks]Geometrical Transformation

Describe fully the single transformation which maps triangle A, with vertices

(-2; 6)

(-2; 4)

and

(-6; 2)

, onto triangle C, with vertices

(6; 6)

(4; 6)

and

(2; 10)

AA reflection in the line y = x, then a translation up the page
BA rotation of 90 degrees clockwise about the point (2; 2)
CA rotation of 90 degrees anticlockwise about the point (2; 2)
DA rotation of 180 degrees about the centre point (2; 2)

Question 904

[1 marks]Geometrical Transformation

Triangle D, with vertices

(-5; 7)

(-5; 6)

and

(-7; 5)

, is the image of triangle A, with vertices

(-2; 6)

(-2; 4)

and

(-6; 2)

, under an enlargement with centre the origin followed by a translation. Write down the scale factor of the enlargement.

Answer this when you sit the paper.

Question 905

[2 marks]Geometrical Transformation

Triangle A, with vertices

(-2; 6)

(-2; 4)

and

(-6; 2)

, is enlarged with centre the origin and scale factor

\frac{1}{2}

, then translated onto triangle D, with vertices

(-5; 7)

(-5; 6)

and

(-7; 5)

. Write down the translation vector.

Answer this when you sit the paper.

Question 906

[3 marks]Geometrical Transformation

Describe fully the single transformation which maps triangle A, with vertices

(-2; 6)

(-2; 4)

and

(-6; 2)

, onto triangle E, with vertices

(3; 6)

(3; 4)

and

(9; 2)

AAn enlargement about the origin with scale factor -1,5, changing lengths in both directions
BA stretch with the y-axis invariant and stretch factor -1,5
CA shear with the y-axis invariant and shear factor -1,5
DA stretch with the x-axis invariant and stretch factor -1,5

Question 1001

[1 marks]Vector Geometry

In the diagram, OR is parallel to PQ and

\frac{PQ}{OR} = \frac{2}{3}

, with

\overrightarrow{OP} = \boldsymbol{p}

and

\overrightarrow{PQ} = \boldsymbol{q}

. Express

\overrightarrow{OR}

in terms of

\boldsymbol{p}

and/or

\boldsymbol{q}

A $\frac{3}{2}\boldsymbol{p}$
B $\boldsymbol{p} + \boldsymbol{q}$
C $\frac{3}{2}\boldsymbol{q}$
D $\frac{2}{3}\boldsymbol{q}$

Question 1002

[2 marks]Vector Geometry

In the diagram,

\overrightarrow{OP} = \boldsymbol{p}

\overrightarrow{PQ} = \boldsymbol{q}

and

\overrightarrow{OR} = \frac{3}{2}\boldsymbol{q}

. Express

\overrightarrow{RQ}

in terms of

\boldsymbol{p}

and/or

\boldsymbol{q}

A $\frac{1}{2}\boldsymbol{q} - \boldsymbol{p}$
B $\boldsymbol{p} + \frac{1}{2}\boldsymbol{q}$
C $\boldsymbol{p} - \frac{3}{2}\boldsymbol{q}$
D $\boldsymbol{p} - \frac{1}{2}\boldsymbol{q}$

Question 1003

[1 marks]Vector Geometry

In the diagram, OR is parallel to PQ with

\frac{PQ}{OR} = \frac{2}{3}

, and OP and RQ produced meet at S. Write down, in its lowest terms, the ratio

\frac{QS}{RS}

Answer this when you sit the paper.

Question 1004

[2 marks]Vector Geometry

In the diagram, triangles SPQ and SOR are similar with

PQ : OR = 2 : 3

. Write down, in its lowest terms, the ratio of the area of triangle PQS to the area of trapezium OPQR.

Answer this when you sit the paper.

Question 1005

[2 marks]Vector Geometry

A shop quotes curtaining material at

\$700\,000{-}00

per metre and labour at 10% of the total cost of the material. A customer bought 150 metres of material and had the curtains made at the shop. Calculate the amount, in dollars, that she paid altogether.

Answer this when you sit the paper.

Question 1006

[1 marks]Vector Geometry

Three weeks later the shop quotes curtaining material at

\$840\,000{-}00

per metre and labour at 15% of the total cost of the material. The customer bought another 150 metres and had the curtains made at the shop. Calculate the total amount, in dollars, she paid then.

Answer this when you sit the paper.

Question 1007

[3 marks]Vector Geometry

A customer paid 115 500 000 dollars for a set of curtains and, three weeks later, 144 900 000 dollars for the same order. Calculate the percentage increase, correct to three significant figures.

Answer this when you sit the paper.

Question 1101

[1 marks]Functional Graphs

A boy on a swing has velocity

v = t^2 - 4t + 4

m/s at time

t

seconds. In a table of values,

p

is the velocity when

t = 0,5

. Find the value of

p

Answer this when you sit the paper.

Question 1102

[2 marks]Functional Graphs

The curve

v = t^2 - 4t + 4

is drawn for

0 \le t \le 4

. Write down the coordinates of its lowest point.

Answer this when you sit the paper.

Question 1103

[2 marks]Functional Graphs

A boy on a swing has velocity

v = t^2 - 4t + 4

m/s at time

t

seconds. Find his acceleration, in m/s

^2

, when

t = 3

Answer this when you sit the paper.

Question 1104

[2 marks]Functional Graphs

A boy on a swing has velocity

v = t^2 - 4t + 4

m/s at time

t

seconds, for

0 \le t \le 4

. His velocity is 1,5 m/s at two times. Find the later of the two, in seconds, correct to 1 decimal place.

Answer this when you sit the paper.

Question 1105

[2 marks]Functional Graphs

A boy on a swing has velocity

v = t^2 - 4t + 4

m/s and a falling object has velocity

v = 10t

m/s, both at time

t

seconds. At about what time did the boy and the object have the same speed?

AAbout 2,0 seconds after the start
BAbout 0,3 seconds after the start
CAbout 3,5 seconds after the start
DAbout 1,4 seconds after the start

Question 1106

[2 marks]Functional Graphs

A falling object has velocity

v = 10t

m/s at time

t

seconds. It collided with a boy on a swing 0,3 seconds after the start. How far had the object travelled by then, and why?

A3 m, which is the speed the object had reached at that time
B0,45 m, the area under the line $v = 10t$ up to then
C0,9 m, which is twice the area under the line $v = 10t$
D1,5 m, the distance the boy on the swing had covered

Question 1201

[3 marks]Statistics and Probability

Natsai's average mark for two tests is 63,5%. Find his mark in the third test, as a percentage, if his average for the three tests is 66%.

Answer this when you sit the paper.

Question 1202

[2 marks]Statistics and Probability

The frequency polygon shows the age distribution of the people living in a village, with ages from 0 to 80 years. Find the number of people living in this village.

Answer this when you sit the paper.

Question 1203

[2 marks]Statistics and Probability

The frequency polygon shows the age distribution of the people living in a village. State the modal age group.

AThe 40 to 50 age group
BThe 10 to 20 age group
CThe 60 to 70 age group
DThe 70 to 80 age group

Question 1204

[2 marks]Statistics and Probability

The frequency polygon shows the age distribution of the 33 people living in a village. Calculate the percentage of people who are older than 50 years, correct to 3 significant figures.

Answer this when you sit the paper.

Question 1205

[3 marks]Statistics and Probability

The frequency polygon shows the age distribution of the 33 people living in a village. Two people are chosen at random. Calculate the probability that they were each older than 10 years but less than or equal to 30 years, giving your answer as a fraction in its lowest terms.

Answer this when you sit the paper.

Mathematics Paper 2 November 2009

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 305

Question 401

Question 402

Question 403

Question 404

Question 405

Question 406

Question 501

Question 502

Question 503

Question 504

Question 601

Question 602

Question 603

Question 604

Question 605

Question 701

Question 702

Question 703

Question 704

Question 705

Question 706

Question 801

Question 802

Question 803

Question 804

Question 805

Question 901

Question 902

Question 903

Question 904

Question 905

Question 906

Question 1001

Question 1002

Question 1003

Question 1004

Question 1005

Question 1006

Question 1007

Question 1101

Question 1102

Question 1103

Question 1104

Question 1105

Question 1106

Question 1201

Question 1202

Question 1203

Question 1204

Question 1205

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