ZIMSEC O Level Mathematics Paper 2 June 2017

Question 201

[2 marks]Similarity, Consumer Arithmetic and Sets

Two similar square-based pyramids have base areas of 9 cm

^2

and 25 cm

^2

. Find the ratio of their volumes, in the form

a : b

, where

a

and

b

are integers such that

a < b

Answer this when you sit the paper.

Question 202

[3 marks]Similarity, Consumer Arithmetic and Sets

Anesu changed 500 South African rands into United States dollars when the bank exchange rate was US$1 = R12,50. The bank charged 3% of the amount that had been changed as commission. Calculate the bank's commission, in US$.

Answer this when you sit the paper.

Question 203

[2 marks]Similarity, Consumer Arithmetic and Sets

Anesu changed 500 South African rands at the rate US$1 = R12,50, and the bank charged commission of US$1,20 on the exchange. Calculate the amount, in United States dollars, that Anesu received.

Answer this when you sit the paper.

Question 204

[2 marks]Similarity, Consumer Arithmetic and Sets

The Venn diagram shows the universal set

\xi

with subsets P and Q. There are 14 elements in P only, 10 in both P and Q,

w

in Q only, and 16 in neither set. Find

n(Q^1)

, where

Q^1

is the complement of set Q.

Answer this when you sit the paper.

Question 205

[3 marks]Similarity, Consumer Arithmetic and Sets

In the Venn diagram, the universal set

\xi

has subsets P and Q, with 14 elements in P only, 10 in both P and Q,

w

in Q only, and 16 in neither set. Find the value of

w

if the number of elements in

\xi

is twice the number of elements in Q.

Answer this when you sit the paper.

Question 301

[2 marks]Number Bases, Algebraic Fractions and Mensuration

It is given that

244_n + 32_n = 331_n

. Find the value of

n

Answer this when you sit the paper.

Question 302

[3 marks]Number Bases, Algebraic Fractions and Mensuration

Simplify

\dfrac{m^2 - m - 12}{m^3 - 9m}

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Question 303

[2 marks]Number Bases, Algebraic Fractions and Mensuration

The length of each side of an equilateral triangle is 8 cm. Calculate the area of the triangle, in cm

^2

, correct to 3 significant figures.

Answer this when you sit the paper.

Question 304

[2 marks]Number Bases, Algebraic Fractions and Mensuration

An equilateral triangle of side 8 cm has an area of 27,7128 cm

^2

. Express this area in square metres, correct to 3 significant figures.

Answer this when you sit the paper.

Question 401

[3 marks]Variation, Change of Subject and Circle Geometry

H varies directly as

\sqrt{Q}

and H

= 51

when Q

= 289

. Find the value of Q when H

= 81

Answer this when you sit the paper.

Question 402

[3 marks]Variation, Change of Subject and Circle Geometry

Make

m

the subject of the formula

T = 2\pi\sqrt{\dfrac{em}{g}}

A $m = \dfrac{gT}{2\pi e}$ , dividing by $2\pi$ and cancelling the root
B $m = \dfrac{gT^2}{4\pi^2 e}$ , dividing by $2\pi$ and then squaring
C $m = \dfrac{4\pi^2 g}{T^2 e}$ , squaring and then inverting the result
D $m = \dfrac{eT^2}{4\pi^2 g}$ , dividing by $2\pi$ and then squaring

Question 403

[1 marks]Variation, Change of Subject and Circle Geometry

In the diagram, R, T and X lie on the circumference of a circle centre O, the diameter XR is produced to P, and PT is a tangent to the circle at T. Given that R

\hat{\text{X}}

= 2y^\circ

, write down R

\hat{\text{T}}

P in terms of

y

Answer this when you sit the paper.

Question 404

[3 marks]Variation, Change of Subject and Circle Geometry

In the diagram, R, T and X lie on the circumference of a circle centre O, the diameter XR is produced to P, and PT is a tangent to the circle at T, with R

\hat{\text{P}}

= y^\circ

and R

\hat{\text{X}}

= 2y^\circ

. Find the value of

y

Answer this when you sit the paper.

Question 501

[3 marks]Approximation, Matrices and Quadratic Equations

The radius of a circle is 32 cm measured to the nearest centimetre. Taking

\pi

to be

\dfrac{22}{7}

, calculate the least possible value of the circumference of the circle, in centimetres.

Answer this when you sit the paper.

Question 502

[2 marks]Approximation, Matrices and Quadratic Equations

Evaluate

\begin{pmatrix} 3 & 1 \end{pmatrix}\begin{pmatrix} 4 \\ 3 \end{pmatrix}

Answer this when you sit the paper.

Question 503

[2 marks]Approximation, Matrices and Quadratic Equations

Matrix

\mathbf{G} = \begin{pmatrix} -2 & 5 \\ 0 & 6 \end{pmatrix}

. Find

\mathbf{G}^{-1}

, the inverse of matrix

\mathbf{G}

A $\begin{pmatrix} \frac{1}{2} & -\frac{5}{12} \\ 0 & -\frac{1}{6} \end{pmatrix}$ , using a determinant of $12$
B $\begin{pmatrix} -\frac{1}{6} & \frac{5}{12} \\ 0 & -\frac{1}{2} \end{pmatrix}$ , leaving the diagonal unswapped
C $\begin{pmatrix} -\frac{1}{2} & -\frac{5}{12} \\ 0 & \frac{1}{6} \end{pmatrix}$ , leaving the off-diagonal signs
D $\begin{pmatrix} -\frac{1}{2} & \frac{5}{12} \\ 0 & \frac{1}{6} \end{pmatrix}$ , using a determinant of $-12$

Question 504

[3 marks]Approximation, Matrices and Quadratic Equations

Solve the equation

3x^2 - 4x - 11 = 0

, giving the answers correct to 2 significant figures.

Answer this when you sit the paper.

Question 601

[3 marks]Constructions and Loci

In triangle ABC, AB

= 8

cm, AC

= 6,5

cm and B

\hat{\text{A}}

= 60^\circ

. Calculate the length of BC, in centimetres, correct to 1 decimal place.

Answer this when you sit the paper.

Question 602

[2 marks]Constructions and Loci

The perpendicular bisector of the side BC of a triangle is constructed with ruler and compasses. Describe the locus it represents.

AThe set of points that are the same distance from B as they are from C
BThe set of points that are a fixed distance from the mid-point of BC
CThe set of points that are the same distance from B as they are from A
DThe set of points that are the same distance from the lines AB and BC

Question 603

[2 marks]Constructions and Loci

Which construction gives the locus of the points inside triangle ABC that are equidistant from the lines AB and BC?

AThe perpendicular dropped from B onto the opposite side AC
BThe perpendicular bisector of the side AC, drawn with equal arcs
CThe line drawn from B parallel to the opposite side AC of the triangle
DThe bisector of angle ABC, drawn with compasses from the vertex B

Question 701

[2 marks]Inequalities and Linear Programming

A luxury bus has 100 units of seating area. An Ordinary seat takes up 1 unit of seating area and a First Class seat takes up 1,5 units. Let

x

be the number of Ordinary seats and

y

the number of First Class seats. Which inequality states this condition?

A $x + 1,5y \le 100$ , which doubles to give $2x + 3y \le 200$
B $1,5x + y \le 100$ , since an Ordinary seat is the larger of the two
C $x + 1,5y \ge 100$ , since the seating area must be filled completely
D $2x + 3y \le 100$ , taking the seating area in half units throughout

Question 702

[2 marks]Inequalities and Linear Programming

A luxury bus carries

x

Ordinary seats and

y

First Class seats. There must be at least 10 First Class seats, and at least twice as many Ordinary seats as First Class seats. Which pair of inequalities states both conditions?

A $y \ge 10$ and $2x \ge y$ , doubling the Ordinary seats in the second
B $x \ge 10$ and $y \ge 2x$ , taking $x$ as the First Class seat count
C $y \ge 10$ and $x \ge 2y$ , reading each condition as a minimum
D $y \le 10$ and $x \ge 2y$ , capping the number of First Class seats

Question 703

[2 marks]Inequalities and Linear Programming

For a luxury bus,

2x + 3y \le 200

y \ge 10

and

x \ge 2y

, where

x

and

y

are whole numbers of seats. Find the greatest number of First Class seats the bus can hold.

Answer this when you sit the paper.

Question 704

[3 marks]Inequalities and Linear Programming

For a luxury bus,

2x + 3y \le 200

y \ge 10

and

x \ge 2y

, where

x

is the number of Ordinary seats and

y

the number of First Class seats, both whole numbers. The company charges $15 for each Ordinary seat and $25 for each First Class seat. Find the greatest possible amount of money, in dollars, the company would receive.

Answer this when you sit the paper.

Question 801

[2 marks]Logarithms, Trigonometry and Bearings

Evaluate

\log_7 7^{-2} - \log_5 \dfrac{1}{5}

Answer this when you sit the paper.

Question 802

[3 marks]Logarithms, Trigonometry and Bearings

In the diagram, EG

= 15

cm, E

\hat{\text{G}}

= 44^\circ

and E

\hat{\text{F}}

= 110^\circ

. Calculate EF, in centimetres, correct to 3 significant figures.

Answer this when you sit the paper.

Question 803

[3 marks]Logarithms, Trigonometry and Bearings

In the diagram, EH

= 11

cm, HG

= 8

cm and EG

= 15

cm. Calculate E

\hat{\text{H}}

G, giving the answer to the nearest degree.

Answer this when you sit the paper.

Question 804

[2 marks]Logarithms, Trigonometry and Bearings

In the diagram, EG

= 15

cm and E

\hat{\text{G}}

= 44^\circ

, with F on the line through G. Calculate the shortest distance from E to GF produced, in centimetres, correct to 3 significant figures.

Answer this when you sit the paper.

Question 805

[2 marks]Logarithms, Trigonometry and Bearings

In the diagram, E

\hat{\text{G}}

= 44^\circ

, E is due west of G, F lies south of the line EG, and E, F, G and H are on level ground. Find the bearing of F from G.

Answer this when you sit the paper.

Question 901

[3 marks]Geometrical Transformation

Triangle ABC has vertices at A

(1; 1)

, B

(3; 1)

and C

(2; 3)

. It is mapped onto triangle A

_1

_1

_1

by the matrix

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

. Write down the coordinates of B

_1

Answer this when you sit the paper.

Question 902

[3 marks]Geometrical Transformation

An enlargement of factor

-1\frac{1}{2}

, centre

(0; 0)

, maps triangle ABC onto triangle A

_2

_2

_2

. Given that C is the point

(2; 3)

, write down the coordinates of C

_2

Answer this when you sit the paper.

Question 903

[3 marks]Geometrical Transformation

Describe completely the transformation represented by the matrix

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

AA shear with the $x$ -axis invariant and shear factor 2
BA shear with the $y$ -axis invariant and shear factor 2
CA stretch parallel to the $y$ -axis with scale factor 2
DA rotation of $90^\circ$ about the origin, then a shear

Question 904

[2 marks]Geometrical Transformation

A translation

\begin{pmatrix} 3 \\ -4 \end{pmatrix}

maps the point B

(3; 1)

onto the point B

_3

. Write down the coordinates of B

_3

Answer this when you sit the paper.

Question 1001

[1 marks]Functional Graphs

For the function

y = \frac{1}{5}(3 - 2x - x^2)

, calculate the value of

y

when

x = 3

Answer this when you sit the paper.

Question 1002

[2 marks]Functional Graphs

Find the gradient of the curve

y = \frac{1}{5}(3 - 2x - x^2)

at the point where

x = 0

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Question 1003

[2 marks]Functional Graphs

The curve

y = \frac{1}{5}(3 - 2x - x^2)

cuts the

x

-axis at two points. What are their

x

-coordinates?

A $x = -3$ and $x = 1$ , the roots of $x^2 + 2x - 3 = 0$
B $x = -1$ and $x = 3$ , the roots of $x^2 - 2x - 3 = 0$
C $x = -3$ and $x = -1$ , the two negative roots of the quadratic
D $x = 3$ and $x = 5$ , found by putting $y = 1$ in the function

Question 1004

[3 marks]Functional Graphs

Solve

\frac{1}{5}(3 - 2x - x^2) = -0,5

, giving the positive root correct to 1 decimal place.

Answer this when you sit the paper.

Question 1005

[3 marks]Functional Graphs

Find the area bounded by the

x

-axis and the curve

y = \frac{1}{5}(3 - 2x - x^2)

, in square units, correct to 2 significant figures.

Answer this when you sit the paper.

Question 1101

[2 marks]Vector Geometry

In the diagram,

\overrightarrow{\text{OA}} = 10\mathbf{a}

and

\overrightarrow{\text{OB}} = 10\mathbf{b}

, and T lies on AB such that

\dfrac{\text{AT}}{\text{AB}} = \dfrac{3}{5}

. Express

\overrightarrow{\text{AT}}

in terms of

\mathbf{a}

and

\mathbf{b}

Answer this when you sit the paper.

Question 1102

[3 marks]Vector Geometry

In the diagram,

\overrightarrow{\text{OA}} = 10\mathbf{a}

and

\overrightarrow{\text{OB}} = 10\mathbf{b}

, M is the mid-point of OA, and T lies on AB such that

\dfrac{\text{AT}}{\text{AB}} = \dfrac{3}{5}

. Express

\overrightarrow{\text{MT}}

in terms of

\mathbf{a}

and

\mathbf{b}

Answer this when you sit the paper.

Question 1103

[2 marks]Vector Geometry

Vectors

\mathbf{a}

and

\mathbf{b}

are not parallel. It is given that

\overrightarrow{\text{OP}} = 10k\mathbf{b}

and also that

\overrightarrow{\text{OP}} = (5 - h)\mathbf{a} + 6h\mathbf{b}

. Find the value of

h

Answer this when you sit the paper.

Question 1104

[3 marks]Vector Geometry

Vectors

\mathbf{a}

and

\mathbf{b}

are not parallel. It is given that

\overrightarrow{\text{OP}} = 10k\mathbf{b}

and also that

\overrightarrow{\text{OP}} = (5 - h)\mathbf{a} + 6h\mathbf{b}

. Find the value of

k

Answer this when you sit the paper.

Question 1201

[2 marks]Statistics and Probability

The heights

h

, in centimetres, of 60 children fall in these classes:

110 < h \le 120

with frequency 12,

120 < h \le 125

with 18,

125 < h \le 130

with 8,

130 < h \le 145

with 12 and

145 < h \le 150

with 10. State the modal class.

A $130 < h \le 145$ , the class covering the widest range of heights
B $110 < h \le 120$ , the class in which the shortest children fall
C $120 < h \le 125$ , the class with the greatest frequency density
D $145 < h \le 150$ , the class with the largest upper class boundary

Question 1202

[2 marks]Statistics and Probability

In a grouped frequency distribution the class

125 < h \le 130

, where

h

is a height in centimetres, has a frequency of 8. Find its frequency density.

Answer this when you sit the paper.

Question 1203

[3 marks]Statistics and Probability

The heights

h

, in centimetres, of 60 children fall in these classes:

110 < h \le 120

, frequency 12;

120 < h \le 125

, frequency 18;

125 < h \le 130

, frequency 8;

130 < h \le 145

, frequency 12;

145 < h \le 150

, frequency 10. Calculate an estimate of the mean height, in centimetres, correct to 3 significant figures.

Answer this when you sit the paper.

Question 1204

[3 marks]Statistics and Probability

Of 60 children, 12 have a height of not more than 120 cm and 10 have a height greater than 145 cm. Two children are chosen at random, without replacement. Calculate the probability that one has a height of not more than 120 cm and the other has a height greater than 145 cm.

Answer this when you sit the paper.

Mathematics Paper 2 June 2017

Sit this paper online

The questions

Question 201

Question 202

Question 203

Question 204

Question 205

Question 301

Question 302

Question 303

Question 304

Question 401

Question 402

Question 403

Question 404

Question 501

Question 502

Question 503

Question 504

Question 601

Question 602

Question 603

Question 701

Question 702

Question 703

Question 704

Question 801

Question 802

Question 803

Question 804

Question 805

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 1201

Question 1202

Question 1203

Question 1204

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The answers, and why they are the answers

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