ZIMSEC O Level Mathematics Paper 2 June 2012

Question 101

[2 marks]Algebraic Fractions

Express

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

as a single fraction in its simplest form.

Answer this when you sit the paper.

Question 102

[3 marks]Algebraic Fractions

Express

\frac{2a - 5}{a - 4} - \frac{1}{2}

as a single fraction in its simplest form.

A $\frac{2a - 6}{a - 6}$
B $\frac{3a - 14}{2a - 8}$
C $\frac{3a - 6}{2a - 8}$
D $\frac{a - 1}{2a - 8}$

Question 103

[2 marks]Algebraic Fractions

Find the numerical value of

d\left(n - d^2\right)

when

n = 4

and

d = \frac{1}{2}

Answer this when you sit the paper.

Question 104

[3 marks]Algebraic Fractions

By selling an article for 11,40 dollars, a shop owner made a loss of 5%. Calculate the cost price of the article, in dollars.

Answer this when you sit the paper.

Question 201

[2 marks]Algebraic Expressions

Factorise completely

2k^2 - 7k - 15

A $(2k + 3)(k - 5)$
B $(2k + 5)(k - 3)$
C $(2k - 3)(k + 5)$
D $(2k - 5)(k + 3)$

Question 202

[3 marks]Algebraic Expressions

Factorise completely

2am^2 - 2an^2 - bm^2 + bn^2

A $(2a - b)(n - m)(n + m)$
B $(2a - b)(m - n)(m - n)$
C $(2a + b)(m - n)(m + n)$
D $(2a - b)(m - n)(m + n)$

Question 203

[2 marks]Algebraic Expressions

Mary has

\$(3x - 4y)

and Diana has

\$(2y - x)

. Write down, in terms of

x

and

y

, the simplified expression for the amount of money that Mary has more than Diana.

Answer this when you sit the paper.

Question 204

[3 marks]Algebraic Expressions

Mary has

\$(3x - 4y)

and Diana has

\$(2y - x)

. Given that Mary has

\$12

and Diana has

\$8

, find the value of

x

and the value of

y

A $x = 12$ and $y = 10$
B $x = 28$ and $y = 18$
C $x = 18$ and $y = 28$
D $x = 20$ and $y = 12$

Question 301

[2 marks]Change of Subject of Formula

Given that

T = \frac{11v}{6} + 20

, make

v

the subject of the formula.

A $v = \frac{6T - 20}{11}$
B $v = \frac{T - 20}{66}$
C $v = \frac{11(T - 20)}{6}$
D $v = \frac{6(T - 20)}{11}$

Question 302

[3 marks]Change of Subject of Formula

Given that

T = \frac{11v}{6} + 20

, find

v

when

T = v

Answer this when you sit the paper.

Question 303

[3 marks]Change of Subject of Formula

In triangle AMC, AM = 7 cm and MC = 8 cm, and the area of triangle AMC is 25 cm

^2

. Calculate the acute angle

A\hat{M}C

, in degrees correct to one decimal place.

Answer this when you sit the paper.

Question 304

[2 marks]Change of Subject of Formula

Given that

\mathbf{Q} = \begin{pmatrix} 7 & 3 \\ 4 & 6 \end{pmatrix}

, find the inverse of

\mathbf{Q}

A $\frac{1}{30}\begin{pmatrix} 7 & -3 \\ -4 & 6 \end{pmatrix}$
B $\frac{1}{30}\begin{pmatrix} 6 & -3 \\ -4 & 7 \end{pmatrix}$
C $\frac{1}{30}\begin{pmatrix} 6 & -4 \\ -3 & 7 \end{pmatrix}$
D $\frac{1}{54}\begin{pmatrix} 6 & -3 \\ -4 & 7 \end{pmatrix}$

Question 305

[2 marks]Change of Subject of Formula

It is given that

\mathbf{P} = \begin{pmatrix} 4 & -3 \\ 5 & 2 \end{pmatrix}

and

\mathbf{Q} = \begin{pmatrix} 7 & 3 \\ 4 & 6 \end{pmatrix}

. Find the matrix

\mathbf{R}

such that

\mathbf{P} + \mathbf{R} = \mathbf{Q}

Answer this when you sit the paper.

Question 401

[2 marks]Constructions and Loci

A trapezium ABCD is constructed in which AB = 8,5 cm,

A\hat{B}C = 120^\circ

, BC = 6 cm, CD = 12 cm and AB is parallel to DC. Find the size of

C\hat{B}D

, in degrees.

Answer this when you sit the paper.

Question 402

[2 marks]Constructions and Loci

A circle is drawn having the 12 cm line CD as its diameter. Which description of that circle as a locus is the complete one?

AThe locus of points at which CD subtends 90 degrees: centre the midpoint of CD, radius 6 cm
BThe locus of points equidistant from C and D: the perpendicular bisector of CD, drawn as a circle
CThe locus of points 12 cm from the midpoint of CD: a circle of centre that midpoint, radius 12 cm
DThe locus of points at which CD subtends 45 degrees: centre the midpoint of CD, radius 12 cm

Question 501

[2 marks]Circle Geometry

K is 5 km due east of M, D is 8 km due south of K and C is 10 km due east of D. Calculate the length of the straight line MC, in kilometres.

Answer this when you sit the paper.

Question 502

[3 marks]Circle Geometry

K is 5 km due east of M, D is 8 km due south of K and C is 10 km due east of D. Calculate the bearing of C from M, correct to the nearest degree.

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Question 503

[2 marks]Circle Geometry

P, Q, R and S are points on a circle. AT is a tangent to the circle at P, and R lies in the segment alternate to the angle

Q\hat{P}T

. Given that

P\hat{R}Q = 32^\circ

, find

Q\hat{P}T

A $16^\circ$
B $32^\circ$
C $58^\circ$
D $64^\circ$

Question 504

[2 marks]Circle Geometry

P, Q, R and S are points on a circle with centre O, and the line PS passes through O. Given that

P\hat{R}Q = 32^\circ

, find

Q\hat{P}S

, in degrees.

Answer this when you sit the paper.

Question 505

[2 marks]Circle Geometry

P, Q, R and S are points on a circle with centre O, the line PS passes through O, and QO is parallel to RS. Given that

Q\hat{P}S = 58^\circ

, find

P\hat{S}R

A $64^\circ$
B $116^\circ$
C $32^\circ$
D $58^\circ$

Question 601

[2 marks]Similarity and Congruency

Find the equation of the straight line which passes through

P(3;\ -4)

and

Q(-1;\ 2)

A $3x + 2y = -6$
B $2x + 3y = -6$
C $3x - 2y = -7$
D $3x + 2y = 1$

Question 602

[3 marks]Similarity and Congruency

Solve the equation

\frac{3}{2x - 5} - \frac{4}{x - 3} = 0

Answer this when you sit the paper.

Question 603

[2 marks]Similarity and Congruency

ABCD is a cyclic quadrilateral in which the diagonals AC and BD intersect at P. Name, in correct order, the triangle that is similar to triangle APD.

ATriangle DPC, since A corresponds to D, P to P and D to C
BTriangle CPB, since A corresponds to C, P to P and D to B
CTriangle BPC, since A corresponds to B, P to P and D to C
DTriangle BPA, since A corresponds to B, P to P and D to A

Question 604

[3 marks]Similarity and Congruency

ABCD is a cyclic quadrilateral in which the diagonals AC and BD intersect at P, with AP = 3 cm, PC = 4 cm and BP = 6 cm. Given that the area of triangle CPD is 8 cm

^2

, calculate the area of the quadrilateral ABCD, in cm

^2

Answer this when you sit the paper.

Question 701

[2 marks]Trigonometry, Bearing & Distances

The points H, G and D lie in a straight line on level ground. DE is a tree 6 m high standing at D, and the angle of elevation of E from G is

52^\circ

. HF is a tower standing at H, and the angle of depression of E from F is

24^\circ

. Find

G\hat{E}F

A $76^\circ$
B $104^\circ$
C $28^\circ$
D $52^\circ$

Question 702

[3 marks]Trigonometry, Bearing & Distances

The points H, G and D lie in a straight line on level ground. DE is a tree 6 m high standing at D, and the angle of elevation of E from G is

52^\circ

. Calculate the length of GE, in metres, correct to three significant figures.

Answer this when you sit the paper.

Question 703

[3 marks]Trigonometry, Bearing & Distances

In triangle FEG, EF = 9 m, GE = 7,61 m and

G\hat{E}F = 76^\circ

. Calculate the length of FG, in metres, correct to three significant figures.

Answer this when you sit the paper.

Question 704

[3 marks]Trigonometry, Bearing & Distances

The points H, G and D lie in a straight line on level ground, F is the top of a tower standing at H and E is the top of a tree standing at D. In triangle FEG, EF = 9 m, GE = 7,61 m and

G\hat{E}F = 76^\circ

, and the angle of depression of E from F is

24^\circ

. Find the angle of depression of G from F, in degrees correct to one decimal place.

Answer this when you sit the paper.

Question 801

[2 marks]Geometrical Transformation

Triangle A has vertices at

(3;\ 1)

(1;\ 2)

and

(2;\ 4)

. Triangle B is the image of triangle A under an anticlockwise rotation of

90^\circ

about

(-2;\ 2)

. Write down the coordinates of the image of the vertex

(3;\ 1)

Answer this when you sit the paper.

Question 802

[3 marks]Geometrical Transformation

A single transformation

P

maps triangle A, with vertices

(3;\ 1)

(1;\ 2)

and

(2;\ 4)

, onto triangle C, with vertices

(9;\ 1)

(3;\ 2)

and

(6;\ 4)

. Find the matrix which represents

P

A $\begin{pmatrix} 0 & 3 \\ 1 & 0 \end{pmatrix}$
B $\begin{pmatrix} 1 & 0 \\ 0 & 3 \end{pmatrix}$
C $\begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}$
D $\begin{pmatrix} 3 & 0 \\ 0 & 1 \end{pmatrix}$

Question 803

[3 marks]Geometrical Transformation

A single transformation

P

maps triangle A, with vertices

(3;\ 1)

(1;\ 2)

and

(2;\ 4)

, onto triangle C, with vertices

(9;\ 1)

(3;\ 2)

and

(6;\ 4)

. Describe fully the single transformation

P

AA stretch parallel to the $y$ -axis, scale factor 3, invariant line the $x$ -axis
BA shear parallel to the $x$ -axis with the $y$ -axis invariant and shear factor 3
CA stretch parallel to the $x$ -axis, scale factor 3, invariant line the $y$ -axis
DAn enlargement of scale factor 3 with the origin as centre, which trebles both coordinates

Question 804

[2 marks]Geometrical Transformation

Triangle A has vertices at

(3;\ 1)

(1;\ 2)

and

(2;\ 4)

. It is mapped onto triangle D by an enlargement of scale factor

-2

with the origin as the centre. Write down the coordinates of the image of the vertex

(2;\ 4)

Answer this when you sit the paper.

Question 901

[3 marks]Vector Geometry

The resistance,

R

newtons, to a train travelling at

v

km/h is given by

R = c + dv^2

, where

c

and

d

are constants. Given that

R = 4

when

v = 20

and that

R = 10

when

v = 40

, find the value of

c

and the value of

d

A $c = 0,005$ and $d = 2$
B $c = 2$ and $d = 0,005$
C $c = 3$ and $d = 0,005$
D $c = 2$ and $d = 0,05$

Question 902

[2 marks]Vector Geometry

The resistance,

R

newtons, to a train travelling at

v

km/h is given by

R = c + dv^2

, where

c = 2

and

d = 0,005

. Find

v

when

R = 3

Answer this when you sit the paper.

Question 903

[2 marks]Vector Geometry

OAB is a triangle in which K is the point on BA with

BK : KA = 1 : 2

. Given that

\overrightarrow{OA} = 3\mathbf{a}

and

\overrightarrow{OB} = 3\mathbf{b}

, express

\overrightarrow{BK}

in terms of

\mathbf{a}

and

\mathbf{b}

Answer this when you sit the paper.

Question 904

[2 marks]Vector Geometry

OAB is a triangle in which H lies on OA with

OH : HA = 1 : 2

and K lies on BA with

BK : KA = 1 : 2

. Given that

\overrightarrow{OA} = 3\mathbf{a}

and

\overrightarrow{OB} = 3\mathbf{b}

, express

\overrightarrow{HK}

in terms of

\mathbf{a}

and

\mathbf{b}

A $2\mathbf{b}$
B $2\mathbf{a}$
C $\mathbf{a} + \mathbf{b}$
D $2\mathbf{a} - 2\mathbf{b}$

Question 905

[2 marks]Vector Geometry

In triangle OAB,

\overrightarrow{HK} = 2\mathbf{b}

and

\overrightarrow{OB} = 3\mathbf{b}

. Write down the ratio

\frac{HK}{OB}

Answer this when you sit the paper.

Question 1001

[3 marks]Measures and Mensuration

Solve the equation

3q^2 - 5q - 5 = 0

, giving the positive root correct to two decimal places.

Answer this when you sit the paper.

Question 1002

[2 marks]Measures and Mensuration

Solve the equation

3q^2 - 5q - 5 = 0

, giving the negative root correct to two decimal places.

Answer this when you sit the paper.

Question 1003

[3 marks]Measures and Mensuration

A composite solid is made up of a cylinder of height 2,2 m with a cone standing on it, both having the same radius

r

metres. The slant height of the cone is 4,81 m and the total height of the solid is 5,5 m. Calculate

r

, in metres.

Answer this when you sit the paper.

Question 1004

[2 marks]Measures and Mensuration

A composite solid is made up of a cylinder of height 2,2 m with a cone standing on it, both of radius 3,5 m. The slant height of the cone is 4,81 m. Taking

\pi = \frac{22}{7}

, find the surface area of the solid excluding its base.

A $53$ m $^2$
B $101$ m $^2$
C $140$ m $^2$
D $48$ m $^2$

Question 1005

[2 marks]Measures and Mensuration

A composite solid is made up of a cylinder of height 2,2 m with a cone of height 3,3 m standing on it, both of radius 3,5 m. Taking

\pi = \frac{22}{7}

, calculate the volume of the solid, in m

^3

, correct to three significant figures.

Answer this when you sit the paper.

Question 1101

[2 marks]Functional Graphs

A table of values for

y = x^3 - 5x + 3

is drawn up for

x = -3, -2, -1, 0, 1, 2, 3

. The entry at

x = -2

m

and the entry at

x = 3

n

. Find

m

and

n

A $m = 5$ and $n = 9$
B $m = 21$ and $n = 15$
C $m = 5$ and $n = 15$
D $m = -5$ and $n = 15$

Question 1102

[3 marks]Functional Graphs

The curve

y = x^3 - 5x + 3

and the line

y = x + 3

are drawn on the same axes for

-3 \le x \le 3

. Write down the roots of the equation

x^3 - 5x + 3 = x + 3

A $x = 0$ , $x = 2,45$ and $x = -2,45$
B $x = 3$ , $x = 2,45$ and $x = -2,45$
C $x = 0$ , $x = 3$ and $x = -3$
D $x = 0$ , $x = 6$ and $x = -6$

Question 1103

[2 marks]Functional Graphs

Estimate the gradient of the curve

y = x^3 - 5x + 3

at the point where

x = -2

Answer this when you sit the paper.

Question 1201

[2 marks]Statistics and Probability

The heights of 100 pupils, in centimetres, have frequencies 2, 14, 24, 30, 16, 10, 3 and 1 in the eight classes from

110 < h \le 115

up to

145 < h \le 150

, each 5 cm wide. In the cumulative frequency table,

p

is the entry for

h \le 125

q

is the entry for

h \le 135

and

r

is the entry for

h \le 140

. Find

p

q

and

r

A $p = 40$ , $q = 70$ and $r = 86$
B $p = 38$ , $q = 84$ and $r = 94$
C $p = 40$ , $q = 86$ and $r = 96$
D $p = 24$ , $q = 16$ and $r = 10$

Question 1202

[2 marks]Statistics and Probability

For the heights of 100 pupils, the cumulative frequency is 40 at 125 cm and 70 at 130 cm. Estimate the median height, in centimetres, to the nearest centimetre.

Answer this when you sit the paper.

Question 1203

[3 marks]Statistics and Probability

The heights

h

cm of 100 pupils have frequencies 2, 14, 24, 30, 16, 10, 3 and 1 in the eight classes

110 < h \le 115

115 < h \le 120

120 < h \le 125

125 < h \le 130

130 < h \le 135

135 < h \le 140

140 < h \le 145

and

145 < h \le 150

. Calculate the approximate mean height, in centimetres, correct to the nearest centimetre.

Answer this when you sit the paper.

Question 1204

[3 marks]Statistics and Probability

Of 100 pupils, 30 have heights

h

cm with

125 < h \le 130

and 16 have

130 < h \le 135

. Two pupils are chosen at random from the group. Find the probability that both of them have a height of more than 125 cm but not more than 135 cm.

Answer this when you sit the paper.

Mathematics Paper 2 June 2012

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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 501

Question 502

Question 503

Question 504

Question 505

Question 601

Question 602

Question 603

Question 604

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 905

Question 1001

Question 1002

Question 1003

Question 1004

Question 1005

Question 1101

Question 1102

Question 1103

Question 1201

Question 1202

Question 1203

Question 1204

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