ZIMSEC O Level Mathematics Paper 2 June 2016

Question 101

[2 marks]Fractions, Decimals and Percentages

Simplify

7\frac{1}{2} + \frac{1}{2} \times 8 - 2

Answer this when you sit the paper.

Question 102

[3 marks]Fractions, Decimals and Percentages

Find the Lowest Common Multiple (L.C.M) of 60 and 84.

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Question 103

[2 marks]Fractions, Decimals and Percentages

Mary, Peter and John share a total amount of $500 in the ratio 3:2:5. Calculate Mary's share.

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Question 104

[2 marks]Fractions, Decimals and Percentages

Mary's share of an amount is $150. She uses part of it to buy a pair of shoes costing $30. Calculate the percentage of Mary's share that is left.

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Question 201

[1 marks]Variation

Factorise completely

4c^2 + 12cn

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Question 202

[2 marks]Variation

Factorise completely

k^2 + kh - kp - hp

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Question 203

[2 marks]Variation

Solve the equation

\frac{12 - y}{3} = 5 - 2y

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Question 204

[2 marks]Variation

The mass,

m

grams, of a solid is inversely proportional to its volume,

v

^3

. Given that

m = 3,5

g when

v = 4

^3

, find the formula connecting

m

and

v

Answer this when you sit the paper.

Question 205

[1 marks]Variation

The mass

m

grams and volume

v

^3

of a solid are connected by

m = \frac{14}{v}

. Find

m

when

v = 6

^3

Answer this when you sit the paper.

Question 206

[2 marks]Variation

Make

x

the subject of the formula

wx^2 = n

A $x = \sqrt{\frac{n}{w}}$ , with the negative root dropped as though $x$ had to be positive
B $x = \pm\sqrt{\frac{n}{w}}$ , keeping both roots because squaring hides the sign of $x$
C $x = \pm\sqrt{nw}$ , multiplying by $w$ instead of dividing when clearing it
D $x = \frac{n}{w}$ , dividing by $w$ but leaving the square on $x$ undone

Question 301

[2 marks]Quadratic Equations

In triangle LMN the angle at M is

90^\circ

, MN

= x

cm, ML

= (x + 3)

cm and LN

= 7

cm. Which equation does Pythagoras' theorem give for this triangle?

A $(x + 3)^2 + 7^2 = x^2$ , treating MN as the side facing the right angle
B $x^2 + (x + 3)^2 = 49$ , since LN is the side facing the right angle at M
C $x^2 + (x + 3)^2 = 7$ , using the length of LN itself in place of its square
D $x^2 + 7^2 = (x + 3)^2$ , treating ML as the side facing the right angle

Question 302

[3 marks]Quadratic Equations

Solve the equation

x^2 + 3x - 20 = 0

, giving the positive answer correct to 2 significant figures.

Answer this when you sit the paper.

Question 303

[2 marks]Quadratic Equations

In a right-angled triangle LMN, MN

= x

cm, ML

= (x + 3)

cm and LN

= 7

cm, where

x = 3,2

correct to 2 significant figures. Calculate the perimeter of triangle LMN.

Answer this when you sit the paper.

Question 304

[1 marks]Quadratic Equations

In a class of 40 pupils, 6 like Mathematics only, 25 like both Mathematics and Physics, and 5 like Physics only. Taking M as the set who like Mathematics and P as the set who like Physics, find

n(M \cup P)

Answer this when you sit the paper.

Question 305

[2 marks]Quadratic Equations

In a class of 40 pupils, 6 like Mathematics only, 25 like both Mathematics and Physics, and 5 like Physics only. Find the number of pupils who do not like either subject.

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Question 306

[1 marks]Quadratic Equations

In a class of 40 pupils, 6 like Mathematics only, 25 like both Mathematics and Physics, and 5 like Physics only. Taking M as the set who like Mathematics and P as the set who like Physics, find

n(M' \cap P)

Answer this when you sit the paper.

Question 401

[3 marks]Circle Geometry

J, K, M and L lie on the circumference of a circle. The chords JM and KL cross at H, with

J\hat{H}K = 97^\circ

and

J\hat{K}H = 48^\circ

. Find

H\hat{L}M

A $48^\circ$ , copying the angle at K because both stand on a chord of the circle
B $83^\circ$ , taking the supplement of the angle at H rather than working in the triangle
C $97^\circ$ , copying the angle at H because it is vertically opposite the required angle
D $35^\circ$ , since the angle at J found from triangle JKH stands on the same arc as it

Question 402

[1 marks]Circle Geometry

J, K, M and L lie on the circumference of a circle, and the chords JM and KL cross at H. Name, in the correct order, the triangle which is similar to triangle JKH.

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Question 403

[3 marks]Circle Geometry

In triangle JKH, JH

= 25

cm, KH

= 20

cm and the angle between them,

J\hat{H}K

, is

97^\circ

. Calculate the length of JK, correct to 3 significant figures.

Answer this when you sit the paper.

Question 404

[2 marks]Circle Geometry

In triangle JKH, JH

= 25

cm, KH

= 20

cm and the angle between them,

J\hat{H}K

, is

97^\circ

. Calculate the area of triangle JKH, correct to 3 significant figures.

Answer this when you sit the paper.

Question 405

[3 marks]Circle Geometry

Triangle HLM is similar to triangle HJK, with HM

= 6

cm corresponding to HK

= 20

cm. Given that the area of triangle JKH is 248 cm

^2

, calculate the area of triangle HLM, correct to 3 significant figures.

Answer this when you sit the paper.

Question 501

[2 marks]Matrices

Matrix

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

. Find the value of

y

for which

A

has no inverse.

Answer this when you sit the paper.

Question 502

[2 marks]Matrices

Matrices

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

and

C = \begin{pmatrix} 5 & -4 \\ 2 & -9 \end{pmatrix}

. Evaluate

A + C

Answer this when you sit the paper.

Question 503

[2 marks]Matrices

Matrix

C = \begin{pmatrix} 5 & -4 \\ 2 & -9 \end{pmatrix}

. Evaluate

C^{-1}

A $\frac{1}{-37}\begin{pmatrix} -9 & 4 \\ -2 & 5 \end{pmatrix}$ , swapping the leading diagonal and negating the other
B $\frac{1}{-37}\begin{pmatrix} 5 & -4 \\ 2 & -9 \end{pmatrix}$ , dividing by the determinant but leaving the entries untouched
C $\frac{1}{-37}\begin{pmatrix} -9 & -4 \\ 2 & 5 \end{pmatrix}$ , swapping the diagonal but negating the wrong pair
D $\frac{1}{-53}\begin{pmatrix} -9 & 4 \\ -2 & 5 \end{pmatrix}$ , adding the two products instead of subtracting them

Question 504

[2 marks]Matrices

Matrices

B = \begin{pmatrix} -4 & 3 \end{pmatrix}

and

C = \begin{pmatrix} 5 & -4 \\ 2 & -9 \end{pmatrix}

. Evaluate

BC

Answer this when you sit the paper.

Question 505

[2 marks]Matrices

\log_x 27 = 1,5

, find

x

Answer this when you sit the paper.

Question 601

[2 marks]Constructions and Loci

In triangle ABC, AB

=

= 8

cm and

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

. Calculate the length of AC.

Answer this when you sit the paper.

Question 602

[1 marks]Constructions and Loci

Triangle ABC has AB

=

= 8

cm and

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

. N is marked on the bisector of

A\hat{B}C

so that BN

= 12

cm, and ABCN is completed. State the special name given to the quadrilateral ABCN.

AA trapezium, because the bisector runs parallel to one of the two equal sides
BA rhombus, because two of its sides are equal and its diagonals cross at right angles
CA kite, because BN is a line of symmetry, so AB $=$ CB and AN $=$ CN
DA parallelogram, because opposite sides are equal in pairs across the bisector

Question 603

[2 marks]Constructions and Loci

In triangle ABC the angle

A\hat{B}C

is bisected, and N lies on the bisector. Describe fully the locus represented by the bisector BN.

AThe locus of points equidistant from the two lines BA and BC, inside angle ABC
BThe locus of points equidistant from the two vertices A and C of the triangle
CThe locus of points at a fixed distance of 12 cm from the vertex B of the triangle
DThe locus of points equidistant from the vertex B and from the opposite side AC

Question 604

[3 marks]Constructions and Loci

In a quadrilateral ABCN, BC

= 8

cm, BN

= 12

cm and

C\hat{B}N = 30^\circ

. Calculate the length of CN, correct to 3 significant figures.

Answer this when you sit the paper.

Question 605

[2 marks]Constructions and Loci

In triangle BCN, BC

= 8

cm, BN

= 12

cm and CN

= 6,46

cm. Calculate the size of

B\hat{C}N

, giving the answer in degrees correct to 1 decimal place.

Answer this when you sit the paper.

Question 701

[1 marks]Vector Geometry

O, F and H are points with

\overrightarrow{OF} = 3p

and

\overrightarrow{OH} = 2q

. Express

\overrightarrow{FH}

in terms of

p

and

q

Answer this when you sit the paper.

Question 702

[2 marks]Vector Geometry

O, F, G and H are points with

\overrightarrow{OF} = 3p

\overrightarrow{OH} = 2q

and

\overrightarrow{FG} = h\overrightarrow{OH}

. Express

\overrightarrow{OG}

in terms of

p

q

and

h

Answer this when you sit the paper.

Question 703

[2 marks]Vector Geometry

O, F and H are points with

\overrightarrow{OF} = 3p

and

\overrightarrow{OH} = 2q

, and W lies on FH with

\overrightarrow{FW} = k\overrightarrow{FH}

. Express

\overrightarrow{OW}

in terms of

p

q

and

k

Answer this when you sit the paper.

Question 704

[2 marks]Vector Geometry

Points O and G satisfy

\overrightarrow{OG} = 3p + 2hq

, and W lies on OG with OW:OG

= 2:3

. Express

\overrightarrow{OW}

in terms of

p

q

and

h

Answer this when you sit the paper.

Question 705

[3 marks]Vector Geometry

The vector

\overrightarrow{OW}

can be written both as

3(1 - k)p + 2kq

and as

2p + \frac{4h}{3}q

, where

p

and

q

are not parallel. Find the numerical value of

k

Answer this when you sit the paper.

Question 706

[2 marks]Vector Geometry

W lies on FH with

\overrightarrow{FW} = \frac{1}{3}\overrightarrow{FH}

. Find the numerical value of

\frac{FW}{WH}

Answer this when you sit the paper.

Question 801

[1 marks]Statistics and Probability

In a grouped frequency distribution the class

400 \le S < 450

has frequency density 1,2. Find the frequency

x

of that class.

Answer this when you sit the paper.

Question 802

[2 marks]Statistics and Probability

In a grouped frequency distribution the class

200 \le S < 300

has frequency 380. Find the frequency density

y

of that class.

Answer this when you sit the paper.

Question 803

[1 marks]Statistics and Probability

The weekly sales of 1 000 traders fall into the classes

50 \le S < 100

(frequency density 1,6),

100 \le S < 200

(3,6),

200 \le S < 300

(3,8),

300 \le S < 400

(1,2) and

400 \le S < 450

(1,2). State the modal class.

A $100 \le S < 200$ , because it has a wide interval and a high frequency density
B $400 \le S < 450$ , because it is a narrow class and so packs its traders tightly
C $200 \le S < 300$ , because it has the highest frequency density of the five classes
D $50 \le S < 100$ , because the mode of a grouped distribution sits in the first class

Question 804

[2 marks]Statistics and Probability

A pie chart is drawn using frequency densities of 1,6, 3,6, 3,8, 1,2 and 1,2 for five classes of weekly sales. Calculate the angle at the centre, correct to the nearest degree, that would represent the class of density 1,6.

Answer this when you sit the paper.

Question 805

[3 marks]Statistics and Probability

The weekly sales of 1 000 traders are grouped as

50 \le S < 100

(80 traders),

100 \le S < 200

(360),

200 \le S < 300

(380),

300 \le S < 400

(120) and

400 \le S < 450

(60). Calculate an estimate of the mean sales, in dollars.

Answer this when you sit the paper.

Question 806

[3 marks]Statistics and Probability

Of 1 000 traders, 120 had weekly sales in the class

300 \le S < 400

and 60 in the class

400 \le S < 450

. Two traders are chosen at random. Find the probability that both had earnings greater than or equal to $300, correct to 3 significant figures.

Answer this when you sit the paper.

Question 901

[2 marks]Geometrical Transformation

Point D has coordinates

(1;3)

. Find the coordinates of

D_1

, the image of D under the transformation with matrix

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

Answer this when you sit the paper.

Question 902

[3 marks]Geometrical Transformation

Point F has coordinates

(2;5)

. Find the coordinates of

F_2

, the image of F under an enlargement of factor 2 and centre

(4;2)

Answer this when you sit the paper.

Question 903

[3 marks]Geometrical Transformation

Point E has coordinates

(1;5)

. Find the coordinates of

E_3

, the image of E after a shear of factor

-2

with the y-axis invariant.

Answer this when you sit the paper.

Question 904

[3 marks]Geometrical Transformation

Describe fully the transformation defined by the matrix

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

AA rotation through $180^\circ$ about the origin, since the matrix carries a negative entry
BA rotation through $90^\circ$ clockwise about the origin, reading the columns the wrong way
CA reflection in the line $y = x$ , since the two coordinates appear to trade places
DA rotation through $90^\circ$ anticlockwise about the origin, since it sends $(1;0)$ to $(0;1)$

Question 1001

[2 marks]Inequalities and Linear Programming

On a school trip the number of pupils must be at least the number of teachers, and must not be greater than three times the number of teachers. Taking

x

as the number of teachers and

y

as the number of pupils, which pair of inequalities states these conditions?

A $y \ge x$ and $y \le 3x$ , reading pupils as at least the teachers and at most three times them
B $y \ge x$ and $y \ge 3x$ , treating three times the teachers as a lower limit on the pupils
C $y \le x$ and $y \ge 3x$ , with both inequality signs turned the wrong way round
D $x \ge y$ and $x \le 3y$ , with the roles of teachers and pupils interchanged

Question 1002

[1 marks]Inequalities and Linear Programming

A bus can take up to 60 passengers. If

x

teachers and

y

pupils travel on it, write down an inequality that satisfies this condition.

Answer this when you sit the paper.

Question 1003

[2 marks]Inequalities and Linear Programming

Transport costs $4,00 per teacher and $2,00 per pupil, and the money collected must cover the $120,00 hire of the bus. If

x

teachers and

y

pupils travel, write down an inequality that satisfies this condition.

Answer this when you sit the paper.

Question 1004

[3 marks]Inequalities and Linear Programming

The numbers of teachers

x

and pupils

y

on a trip satisfy

y \ge x

y \le 3x

x + y \le 60

and

4x + 2y \ge 120

. Find the least number of teachers.

Answer this when you sit the paper.

Question 1005

[3 marks]Inequalities and Linear Programming

The numbers of teachers

x

and pupils

y

on a trip satisfy

y \ge x

y \le 3x

x + y \le 60

and

4x + 2y \ge 120

. Find the greatest number of pupils that could go on the trip.

Answer this when you sit the paper.

Question 1101

[1 marks]Functional Graphs

For the function

y = x(4 - x)

, calculate the value of

y

when

x = -1

Answer this when you sit the paper.

Question 1102

[1 marks]Functional Graphs

For the function

y = x(4 - x)

, calculate the value of

y

when

x = 3

Answer this when you sit the paper.

Question 1103

[2 marks]Functional Graphs

The curve

y = x(4 - x)

has a maximum point. Write down the coordinates of that maximum point.

Answer this when you sit the paper.

Question 1104

[2 marks]Functional Graphs

Find the gradient of the curve

y = x(4 - x)

when

x = 3

Answer this when you sit the paper.

Question 1105

[3 marks]Functional Graphs

Solve the equation

x(4 - x) = x

Answer this when you sit the paper.

Question 1106

[3 marks]Functional Graphs

Estimate the area of the region bounded by the curve

y = x(4 - x)

and the line

y = x

, in square units.

Answer this when you sit the paper.

Question 1201

[3 marks]Measures and Mensuration

The diagram shows the transverse cross-section of a steel bar. The measurements are in centimetres. Calculate the area of the cross-section.

Answer this when you sit the paper.

Question 1202

[2 marks]Measures and Mensuration

A steel bar 2 m long has a uniform cross-section of area 550 cm

^2

. Calculate the volume of the bar, in cm

^3

Answer this when you sit the paper.

Question 1203

[2 marks]Measures and Mensuration

A steel bar has a volume of 110 000 cm

^3

and a density of 7 800 kg/m

^3

. Find the mass of the bar, in kilograms.

Answer this when you sit the paper.

Question 1204

[3 marks]Measures and Mensuration

The diagram shows the transverse cross-section of a steel bar which is 2 m long, with measurements in centimetres and a cross-sectional area of 550 cm

^2

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

^2

Answer this when you sit the paper.

Question 1205

[2 marks]Measures and Mensuration

A steel bar has a total surface area of 27 100 cm

^2

. It is to be coated with zinc at a cost of $10,00 per square metre. Calculate the cost of coating the bar.

Answer this when you sit the paper.

Mathematics Paper 2 June 2016

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 206

Question 301

Question 302

Question 303

Question 304

Question 305

Question 306

Question 401

Question 402

Question 403

Question 404

Question 405

Question 501

Question 502

Question 503

Question 504

Question 505

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 806

Question 901

Question 902

Question 903

Question 904

Question 1001

Question 1002

Question 1003

Question 1004

Question 1005

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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