ZIMSEC O Level Mathematics Paper 2 November 2021

Question 101

[2 marks]Circle Geometry

A, B, C, D and E lie on a circle with centre O, and EC is a straight line through O. Given that

B\hat{C}E = 53^\circ

, calculate

B\hat{E}C

, in degrees.

Answer this when you sit the paper.

Question 102

[1 marks]Circle Geometry

A, B, C, D and E lie in that order on a circle, and

B\hat{C}E = 53^\circ

. Calculate

B\hat{A}E

, in degrees.

Answer this when you sit the paper.

Question 103

[2 marks]Circle Geometry

A, B, C, D and E lie on a circle with centre O, and EC is a straight line through O. Given that

E\hat{A}D = 48^\circ

, calculate

D\hat{O}C

, in degrees.

Answer this when you sit the paper.

Question 104

[2 marks]Circle Geometry

In the diagram, A, B, C, D and E lie on a circle with centre O. Which angle is equal to

A\hat{D}E

A $B\hat{A}E$ , since it stands on the chord BE from the other side of that chord
B $A\hat{C}D$ , since it stands on the same chord AD as the angle given
C $D\hat{E}C$ , since it stands on the chord DC from the opposite segment of the circle
D $A\hat{B}E$ , since it stands on the same chord AE and in the same segment

Question 201

[2 marks]Matrices

Given the formula

V = \frac{7}{3}\pi r^2 h

, calculate

V

when

r = 3

\pi = \frac{22}{7}

and

h = 2\frac{1}{2}

Answer this when you sit the paper.

Question 202

[2 marks]Matrices

Make

r

the subject of the formula

V = \frac{7}{3}\pi r^2 h

A $r = \dfrac{1}{3}\sqrt{\dfrac{V}{7\pi h}}$
B $r = \sqrt{\dfrac{3V}{7\pi h}}$
C $r = \dfrac{3V}{7\pi h}$
D $r = \sqrt{\dfrac{7\pi h}{3V}}$

Question 203

[2 marks]Matrices

Matrix

A = \begin{pmatrix} 5 & -2 \\ -6 & 3 \end{pmatrix}

and matrix

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

. Find

AB

, giving the two entries in order.

Answer this when you sit the paper.

Question 204

[3 marks]Matrices

Find

A^{-1}

, the inverse of

A = \begin{pmatrix} 5 & -2 \\ -6 & 3 \end{pmatrix}

A $\frac{1}{3}\begin{pmatrix} 5 & 2 \\ 6 & 3 \end{pmatrix}$
B $\frac{1}{27}\begin{pmatrix} 3 & 2 \\ 6 & 5 \end{pmatrix}$
C $\frac{1}{3}\begin{pmatrix} 3 & 2 \\ 6 & 5 \end{pmatrix}$
D $\frac{1}{3}\begin{pmatrix} 3 & -2 \\ -6 & 5 \end{pmatrix}$

Question 301

[3 marks]Constructions & Loci

Triangle ABC has AB = 8 cm, AC = 7 cm and

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

. Calculate the length of BC, in centimetres.

Answer this when you sit the paper.

Question 302

[2 marks]Constructions & Loci

What is the locus of the points that are 3 cm from the line AB?

AA pair of lines parallel to AB, one 3 cm on each side of it
BThe perpendicular bisector of AB, with a point marked 3 cm from A
CA single line parallel to AB and 3 cm from it on the side of C
DA circle of radius 3 cm drawn with its centre at the midpoint of AB

Question 303

[2 marks]Constructions & Loci

Using ruler and compasses only, how is an angle of

120^\circ

set up at A?

AConstruct a perpendicular at A, then bisect the right angle and add it
BConstruct $60^\circ$ at A, then bisect the angle left on the other side
CConstruct $60^\circ$ at A, then a further $60^\circ$ from the new arm
DConstruct $60^\circ$ at A, then bisect it and add the half to it

Question 304

[2 marks]Constructions & Loci

In triangle ABC, AB = 8 cm, AC = 7 cm and

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

. The perpendicular from A meets BC at D. Calculate AD, in centimetres, to 3 significant figures.

Answer this when you sit the paper.

Question 401

[2 marks]Sets

Calculate the simple interest obtained by investing $800 for 10 months at a rate of 36% per annum, in dollars.

Answer this when you sit the paper.

Question 402

[2 marks]Sets

f(x) = x^2 + 5x - 6

, find

f(2)

Answer this when you sit the paper.

Question 403

[3 marks]Sets

f(x) = x^2 + 5x - 6

, find the values of

x

for which

f(x) = 0

Answer this when you sit the paper.

Question 404

[2 marks]Sets

For

\xi = \{x : 1 \le x \le 12, \; x \in \mathbb{Z}\}

, with P the perfect squares in

\xi

and Q the even numbers in

\xi

, list all the elements of

P' \cap Q

Answer this when you sit the paper.

Question 405

[3 marks]Sets

For

\xi = \{x : 1 \le x \le 12, \; x \in \mathbb{Z}\}

, P is the set of perfect squares in

\xi

and Q is the set of even numbers in

\xi

. What is

P \cap Q

A $\{4\}$ , since 4 is the one number in $\xi$ that is a perfect square and even
B $\{1, 4, 9\}$ , since every perfect square in $\xi$ is also an even number
C $\{\;\}$ , since no number in $\xi$ is both a perfect square and an even number
D $\{2, 4, 6, 8, 10, 12\}$ , since every even number in $\xi$ is a perfect square too

Question 501

[2 marks]Factorisation

Evaluate

3\frac{1}{2} - 6\frac{1}{4} \times \frac{2}{5}

Answer this when you sit the paper.

Question 502

[2 marks]Factorisation

A lorry travels a distance of 123 km in

1\frac{1}{2}

hours. Find the average speed of the lorry, in km/h.

Answer this when you sit the paper.

Question 503

[3 marks]Factorisation

Solve the equation

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

Answer this when you sit the paper.

Question 504

[3 marks]Factorisation

Factorise completely

63 - 7p^2

Answer this when you sit the paper.

Question 601

[3 marks]Statistics & Probability

The marks of 20 learners in a test were: mark 10 scored by 0 learners, mark 11 by 6, mark 12 by 5, mark 13 by 4, mark 14 by 3 and mark 15 by 2. Calculate the mean mark.

Answer this when you sit the paper.

Question 602

[2 marks]Statistics & Probability

Of 20 learners, 6 scored 11 marks. The distribution is to be shown on a pie chart. Calculate the angle, in degrees, that represents the learners who scored 11 marks.

Answer this when you sit the paper.

Question 603

[2 marks]Statistics & Probability

In a group of 20 learners, the marks scored were 11 by 6 learners, 12 by 5, 13 by 4, 14 by 3 and 15 by 2. Two learners are chosen at random from the group. Calculate the probability that their marks are both less than 14.

Answer this when you sit the paper.

Question 604

[3 marks]Statistics & Probability

Solve

3x^2 - 5x - 7 = 0

, giving the answers to 2 decimal places.

A $x = -2,57$ or $x = 0,91$
B $x = 2,57$ or $x = -0,91$
C $x = 2,91$ or $x = -0,57$
D $x = 1,57$ or $x = -1,91$

Question 605

[2 marks]Statistics & Probability

Find the value of the discriminant

b^2 - 4ac

for the equation

3x^2 - 5x - 7 = 0

Answer this when you sit the paper.

Question 701

[2 marks]Functional Graphs

For the function

y = 1 - 2x - x^2

, a table of values gives

y = p

when

x = -1

and

y = q

when

x = 2

. Calculate the values of

p

and

q

Answer this when you sit the paper.

Question 702

[3 marks]Functional Graphs

What is the turning point of the curve

y = 1 - 2x - x^2

, and what kind of turning point is it?

A $(1; -2)$ , which is the maximum point of the curve
B $(-1; 2)$ , which is the minimum point of the curve
C $(-1; 2)$ , which is the maximum point of the curve
D $(-2; 1)$ , which is the maximum point of the curve

Question 703

[2 marks]Functional Graphs

Find the gradient of the curve

y = 1 - 2x - x^2

at the point where

x = 1

Answer this when you sit the paper.

Question 704

[3 marks]Functional Graphs

Estimate the roots of the equation

1 - 2x - x^2 = 0

, to 1 decimal place.

Answer this when you sit the paper.

Question 705

[2 marks]Functional Graphs

The curve

y = 1 - 2x - x^2

rises above the

x

axis between its two roots. Roughly what area does it enclose with that axis?

AAbout 3,8 square units
BAbout 1,9 square units
CAbout 5,7 square units
DAbout 2,4 square units

Question 801

[1 marks]Trigonometry, Bearing & Distances

In the diagram BED is a straight line and AB is parallel to DC. Which triangle is similar to triangle CDE?

ATriangle ACD
BTriangle ABE
CTriangle ABC
DTriangle BCD

Question 802

[2 marks]Trigonometry, Bearing & Distances

In the diagram BED is a straight line, AB is parallel to DC,

C\hat{B}D = 40^\circ

A\hat{C}B = 90^\circ

and

E\hat{C}D = 35^\circ

. Calculate

A\hat{B}E

, in degrees.

Answer this when you sit the paper.

Question 803

[2 marks]Trigonometry, Bearing & Distances

In the diagram

A\hat{C}B = 90^\circ

C\hat{B}D = 40^\circ

and BC = 8 cm, with E on AC. Calculate EC, in centimetres.

Answer this when you sit the paper.

Question 804

[3 marks]Trigonometry, Bearing & Distances

In the diagram

B\hat{C}D = 125^\circ

C\hat{B}D = 40^\circ

and CD = 10 cm. Calculate BD, in centimetres.

Answer this when you sit the paper.

Question 805

[2 marks]Trigonometry, Bearing & Distances

In the diagram BC = 8 cm, CD = 10 cm and

B\hat{C}D = 125^\circ

. Calculate the area of triangle BCD, in square centimetres.

Answer this when you sit the paper.

Question 806

[2 marks]Trigonometry, Bearing & Distances

In the diagram CD = 10 cm and the angle between CD and CA is

35^\circ

. Calculate the shortest distance from D to AC, in centimetres.

Answer this when you sit the paper.

Question 901

[2 marks]Statistics & Probability

The heights of 50 plants were grouped as

10 < h \le 15

(frequency 10),

15 < h \le 20

(15),

20 < h \le 30

(m),

30 < h \le 35

(2) and

35 < h \le 50

(9). The class

20 < h \le 30

has frequency density 1,4. Find the value of

m

Answer this when you sit the paper.

Question 902

[2 marks]Statistics & Probability

For 50 plants grouped as

10 < h \le 15

(10 plants),

15 < h \le 20

(15),

20 < h \le 30

(14),

30 < h \le 35

(2) and

35 < h \le 50

(9), what is the modal class?

A $35 < h \le 50$ , because it covers the widest spread of heights in the table
B $10 < h \le 15$ , because it is the class in which the measurements begin
C $20 < h \le 30$ , because 14 plants fall in it, more than in any other class here
D $15 < h \le 20$ , because its bar is the tallest, its frequency density being 3

Question 903

[3 marks]Statistics & Probability

Fifty plants have heights grouped as

10 < h \le 15

(10 plants),

15 < h \le 20

(15),

20 < h \le 30

(14),

30 < h \le 35

(2) and

35 < h \le 50

(9). Calculate an estimate of the mean height, in centimetres.

Answer this when you sit the paper.

Question 904

[2 marks]Statistics & Probability

Of 50 plants, 2 have heights in

30 < h \le 35

and 9 in

35 < h \le 50

. Two plants are chosen at random. Calculate the probability that the heights of both are greater than 30 cm.

Answer this when you sit the paper.

Question 905

[3 marks]Statistics & Probability

Grouped heights have classes of different widths. Why is frequency density, rather than frequency, plotted up the vertical axis of a histogram?

ABecause dividing by the class width brings every bar down to the same height on the page
BBecause the area of each bar then shows the frequency, so wide and narrow classes compare fairly
CBecause frequency density gives the average height in each class, which is what a bar shows
DBecause the height of each bar then shows the frequency, whatever width the class happens to have

Question 1001

[2 marks]Geometrical Transformation

Triangle ABC has

A(-1; 3)

B(-2; 2)

and

C(-0,5; 1,5)

. It is mapped by a reflection onto triangle

A_1B_1C_1

with

A_1(-1; -1)

B_1(-2; 0)

and

C_1(-0,5; 0,5)

. Find the equation of the axis of reflection.

Answer this when you sit the paper.

Question 1002

[2 marks]Geometrical Transformation

In the diagram triangle ABC, with

A(-1; 3)

B(-2; 2)

and

C(-0,5; 1,5)

, is mapped onto triangle

A_2B_2C_2

, with

A_2(2; 4)

B_2(1; 3)

and

C_2(2,5; 2,5)

. Describe fully the single transformation.

AA translation by the vector $\begin{pmatrix} 1 \\ 3 \end{pmatrix}$
BA translation by the vector $\begin{pmatrix} -3 \\ -1 \end{pmatrix}$
CAn enlargement of scale factor 3 about the point $(1; 1)$
DA translation by the vector $\begin{pmatrix} 3 \\ 1 \end{pmatrix}$

Question 1003

[2 marks]Geometrical Transformation

Triangle ABC, with

A(-1; 3)

B(-2; 2)

and

C(-0,5; 1,5)

, is mapped by an enlargement onto triangle

A_3B_3C_3

, with

A_3(1; -3)

B_3(2; -2)

and

C_3(0,5; -1,5)

. Find the scale factor of the enlargement.

Answer this when you sit the paper.

Question 1004

[3 marks]Geometrical Transformation

In the diagram triangle ABC, with

A(-1; 3)

B(-2; 2)

and

C(-0,5; 1,5)

, is mapped onto triangle

A_4B_4C_4

, with

A_4(2; 3)

B_4(0; 2)

and

C_4(1; 1,5)

. Describe fully the single transformation.

AA one way stretch parallel to the $x$ axis with factor 1
BA rotation of $90^\circ$ about the origin, then a translation
CA shear with the $x$ axis invariant and shear factor 1
DA shear with the $y$ axis invariant and shear factor 1

Question 1005

[3 marks]Geometrical Transformation

The transformation with matrix

\begin{pmatrix} -2 & 0 \\ 0 & 3 \end{pmatrix}

maps

A(-1; 3)

onto

A_5

. Calculate the coordinates of

A_5

Answer this when you sit the paper.

Question 1101

[2 marks]Inequalities & Linear Programming

A publisher prints

x

pocket size copies and

y

medium size copies. At most 100 pocket size copies and at least 30 medium size copies are wanted. Which pair of inequalities states this?

A $x \le 100$ and $y \ge 30$
B $x \le 30$ and $y \ge 100$
C $x \ge 100$ and $y \le 30$
D $x < 100$ and $y > 30$

Question 1102

[2 marks]Inequalities & Linear Programming

A publisher prints

x

pocket size copies and

y

medium size copies. The medium size copies must number at most

\frac{3}{5}

of the pocket size copies. Which inequality states this?

A $y \ge \frac{3}{5}x$ , that is $5y \ge 3x$
B $y \le \frac{3}{5}x$ , that is $5y \le 3x$
C $x \le \frac{3}{5}y$ , that is $5x \le 3y$
D $y \le \frac{5}{3}x$ , that is $3y \le 5x$

Question 1103

[2 marks]Inequalities & Linear Programming

A pocket size copy costs $8,00 to produce and a medium size copy costs $10,00. Calculate the cost, in dollars, of producing 40 pocket size copies and 20 medium size copies.

Answer this when you sit the paper.

Question 1104

[3 marks]Inequalities & Linear Programming

A publisher prints

x

pocket size copies and

y

medium size copies, subject to

x \le 100

y \ge 30

5y \le 3x

and

4x + 5y \le 500

. Find the greatest total number of books, pocket size plus medium size, that can be printed.

Answer this when you sit the paper.

Question 1105

[3 marks]Inequalities & Linear Programming

A publisher works under

x \le 100

y \ge 30

5y \le 3x

and

4x + 5y \le 500

. Which of the four conditions plays no part in bounding the region of possible values?

A $x \le 100$ , since the cost condition holds $x$ to 87 at most
B $y \ge 30$ , since the ratio condition already forces $y$ above 30 anyway
C $5y \le 3x$ , since the cost condition already holds $y$ below that line
D $4x + 5y \le 500$ , since $x \le 100$ already limits what can be spent

Question 1201

[2 marks]Measures & Mensuration

A tumbler is a frustum of a cone whose base circle has diameter 4 cm. Taking

\pi

to be

\frac{22}{7}

, calculate the circumference of the base circle, in centimetres.

Answer this when you sit the paper.

Question 1202

[2 marks]Measures & Mensuration

A frustum of a cone has base radius 2 cm and top radius 3 cm, with the two circles 12 cm apart. Extending it downwards to the vertex E gives a small cone of height

h

cm below the base. Calculate

h

, in centimetres.

Answer this when you sit the paper.

Question 1203

[3 marks]Measures & Mensuration

A frustum has base radius 2 cm and top radius 3 cm, cut from a cone of height 36 cm by removing a cone of height 24 cm. Taking

\pi

to be

\frac{22}{7}

, calculate the volume of the frustum, in cubic centimetres, to 3 significant figures.

Answer this when you sit the paper.

Question 1204

[2 marks]Measures & Mensuration

A tumbler holds

238,857

^3

. Express this volume in litres, to 3 significant figures.

Answer this when you sit the paper.

Question 1205

[3 marks]Measures & Mensuration

The cone of height 36 cm and the small cone of height 24 cm cut from its point are similar. What is the ratio of the volume of the larger to the volume of the smaller?

A $3 : 2$ , the ratio of their radii themselves
B $9 : 4$ , the square of the ratio of their radii
C $36 : 24$ , the ratio of their vertical heights
D $27 : 8$ , the cube of the ratio of their radii

Mathematics Paper 2 November 2021

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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 401

Question 402

Question 403

Question 404

Question 405

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 801

Question 802

Question 803

Question 804

Question 805

Question 806

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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