Mathematics Module Level II Volume 2, Study notes of Mathematics

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Ministry of Primary and

Secondary Education

Mathematics Module

Level II

Volume 2

Lifelong and

Continuing Education

Created by Olena Panasovskafrom the Noun Project

II

Acknowledgements

The Ministry of Primary and Secondary Education (MoPSE) wishes to acknowledge the Non Formal Education (PSNE) department of the Ministry for coordinating this programme and the Curriculum Development and Technical Services (CDTS) department for reviewing and editing the module. Special mention goes to Tapera. Vhulengoma – Materials Production Officer (CDTS) for compiling, editing and proof reading this module.

The writing of this Non-Formal Level 2 Maths Module was made possible by the contributions from the following Senior Teachers; Solomon Kamota, Catherine Mukuru, Thompson Mahachi, Macdonald Phiri and Elias Mukwekwe.

We also thank Dr Lovemore Ndlovu, the Consultant in the Open Distance Learning Project.

Above all, special thanks goes to UNICEF for providing funding for this Project.

III

How to use this module

As you start this journey of acquiring a qualification in Ordinary Level Mathematics through open distance learning, it is critical that you understand the need to manage your study time and balance it with your day-to-day activities. This module will provide you with the basic material to assist you towards your public examinations in Mathematics.

This module has been subdivided into two volumes, that is, Volume 1 Volume 2. You are advised to study Volume 1 first before going to Volume 2.

Wish you the best!

• UNIT 16 - ALGEBRA 4 (Inequalities and Linear Programming) How to use this book? III
• 16.1 INTRODUCTION
• 16.2 SOLVING INEQUALITIES
• 16.2.1 Solving simple linear inequalities
• 16.2.2 Solving simultaneous inequalities
• 16.3 GRAPHICAL REPRESENTATION OF INEQUALITIES
• 16.3.1 Representing inequalities on a number line
• 16.3.2 Illustrating an inequality on the Cartesian plane
• 16.4 LINEAR PROGRAMMING
• 16.5 Summary
• 16.6 Further Reading
• 16.7 Assessment Test
• UNIT 17 - TRIGONOMETRY RATIOS
• 17.1 INTRODUCTION
• 17.2 PYTHAGORAS’ THEOREM
• 17.2.1 PYTHAGOREAN TRIPLES
• 17.3 TRIGONOMETRIC RATIOS
• 17.3.1 Tangent
• 17.3.2 Sine
• 17.3.3 Cosine
• 17.4 ANGLES OF ELEVATION AND DEPRESSION
• 17.5 Summary
• 17.6 Further Reading
• 17.7 Assessment Test
• UNIT 18 - VARIATION
• 18.1 INTRODUCTION
• 18.2 DIRECT VARIATION
• 18.3 INVERSE VARIATION/ INDIRECT VARIATION
• 18.4 JOINT VARIATION
• 18.5 PARTIAL VARIATION
• 18.6 Summary
• 18.7 Further Reading
• 18.8 Assessment Test
• UNIT 19 - TRIGONOMETRY V
• 19.1 INTRODUCTION
• 19.2 AREA OF A TRIANGLE USING THE SINE RATIO
• 19.3 THE SINE RULE
• 19.4 SINE RULE WITH BEARING
• 19.5 THE COSINE RULE
• 19.6 COSINE RULE WITH BEARING
• 19.7 Summary
• 19.8 Further Reading
• 19.9 Assessment Test
• UNIT 20 - TRAVEL GRAPHS
• 20.1 INTRODUCTION
• 20.2 SPEED-TIME GRAPHS
• 20.3 VELOCITY TIME GRAPH
• 20.4 DISTANCE TIME GRAPHS
• 20.5 Summary
• 20.6 Further Reading
• 20.7 Assessment Test
• UNIT 21 – PROBABILITY
• 21.1 INTRODUCTION
• 21.2 PROBABILITY DEFINITION
• 21.3 EXPERIMENTAL PROBABILITY
• 21.4 THEORETICAL PROBABILITY
• 21.5 MUTUALLY EXCLUSIVE EVENTS
• 21.6 INDEPENDENT EVENTS
• 21.7 COMBINED EVENTS
• 21.8 OUTCOME TABLES.
• 21.9 TREE DIAGRAM
• 21.10 Summary
• 21.11 Further Reading
• 21.12 Assessment Test
• UNIT 22 - VECTORS
• 22.1 INTRODUCTION
• 22.2 TYPES OF VECTORS
• 22.2.1 Vector notation
• 22.2.2 Translation vectors
• 22.2.3 Equal vectors
• 22.2.4 Negative Vectors VI
• 22.2.5 Parallel vectors
• 22.2.6 Position vectors
• 22.3 MAGNITUDE OF VECTORS
• 22.4 COMBINED VECTORS
• 22.4.1 Sum of vectors
• 22.5 VECTOR PROPERTIES OF PLANE SHAPES
• 22.6 VECTOR ALGEBRA (FINDING SCALARS)
• 22.7 Summary
• 22.8 Further Reading
• 22.9 Assessment Test
• UNIT 23 - TRANSFORMATION
• 23.1 INTRODUCTION
• 23.2 TRANSLATION
• 23.2.1 Vector notation of a translation vector
• 23.3 ROTATION
• 23.3.1 90 o rotation
• 23.3.2 180 o rotation
• 23.3.3 270 o rotation
• 23.4 REFLECTION
• 23.5 ENLARGEMENT
• 23.5.1 Positive enlargement scale factor
• 23.5.2 Negative enlargement scale factor
• 23.5.3 Fractional scale factor
• 23.6 Summary
• 23.7 Further Reading
• 23.8 Assessment Test
• UNIT 24 - FUNCTIONAL GRAPHS
• 24.1 INTRODUCTION
• 24.2 QUADRATIC FUNCTION/ GRAPHS
• 24.2.1 Examples of Quadratic Functions
• 24.2.2 Plotting quadratic graphs/functions
• 24.2.3 Area under the curve (under the quadratic graph)
• 24.2.4 Gradient of a Quadratic function
• 24.2.5 Roots of a quadratic function
• 24.2.6 Maximum and Minimum
• 24.2.7 Line of symmetry
• 24.3 INVERSE GRAPH/FUNCTION
• 24.3.1 Plotting an Inverse function
• 24.3.2 Area under the Inverse function/curve/graph
• 24.3.3 Gradient of an Inverse function VII
• 24.3.4 Intersection of a line and the Inverse function
• 24.4 Summary
• 24.5 Further Reading
• 24.6 Assessment Test
• UNIT 25 - GEOMETRY
• 25.1 INTRODUCTION
• 25.2 GEOMETRICAL CONSTRUCTION
• 25.2.1 Construction of Quadrilaterals
• 25.2.2 Construction of a perpendicular line from a point to a line
• 25.2.3 Construction of an inscribed circle of a triangle
• 25.2.4 Construction of a circumscribed circle of a triangle
• 25.3 LOCUS
• 25.3.1 The locus of points equidistant from a fixed point
• 25.3.2 The locus of points equidistant from two fixed points
• 25.3.3 The locus of points equidistant from a straight line.
• 25.3.4 The locus of points equidistant from two intersecting straight lines
• 25.4 Summary
• 25.5 Further Reading
• 25.6 Assessment Test
• UNIT 26 – STATISTICS
• 26.1 INTRODUCTION
• 26.2 STATISTICS – DATA COLLECTION
• 26.3 DATA CLASSIFICATION
• 26.4 HISTOGRAM AND FREQUENCY POLYGON
• 26.4.1 Histogram with equal class width
• 26.4.2 Frequency polygon
• 26.4.3 Histogram with unequal width
• 26.5 CUMULATIVE CURVE
• 26.6 MEASURES OF DISPERSION
• 26.7 Summary
• 26.8 Further Reading
• 26.9 Assessment Test
• UNIT 27 - TRANSFORMATION
• 27.1 INTRODUCTION
• 27.2 REFLECTION IN ANY LINE AND USE OF MATRICES
• 27.2.1 Reflection in any line (Geometrical solution)
• 27.2.2 Use of Matrices in Reflections
• 27.3 ROTATION BY DRAWING AND USE OF MATRICES
• 27.3.1 Rotation by drawing (Geometrical construction) VIII
• 27.3.2 Matrices in Rotation transformation
• 27.4 ENLARGEMENT USING MATRICES
• 27.5 STRETCH (ONE WAY AND TWO WAY)
• 27.6 SHEAR USING MATRICES AND GEOMETRICAL METHODS
• 27.7 COMBINATION OF TRANSFORMATIONS
• 27.8 Summary
• 27.9 Further Reading
• 27.10 Assessment Test
• GLOSSARY

ü represent inequalities on ;

• number line
• Cartesian plane

ü translate verbal constrains into linear inequalities

ü solve linear programming problems

ü find the maximum and the minimum of a linear programming problem

KEY TERMS

≤ – means less than or equal to

≥ – means greater than or equal to

Integers – these are positive and negative whole numbers.

TIME

You should be done with this unit in 10 hours

STUDY SKILLS

The key skill to mastery of mathematical concepts is practice. You need to solve as

many problems on Inequalities and Linear Programming as possible for you to

grasp all the concepts in this topic. Revisit the units on Algebra before undertaking

this unit if you had not understood concepts on how to solve equations

Tip; in this unit, we are dealing with integers for the values of y and x

16.2.2 Solving simultaneous inequalities

Inequalities are solved the same way we do equations. In some of the previous

units we covered, we learnt how to solve equations. Do you still remember how to

solved them? It is advisable that you revisit the previous units you learnt on

Algebra in order to easily grasp concepts in this unit.

There are certain rules to be followed when solving inequalities, which are as

follows;

1. Any number can be added or subtracted to both sides without affecting the

inequality sign

2. Both sides of the inequality can be multiplied or divided by any positive value

without affecting the inequality sign

3. If both sides of the inequality are multiplied or divided by a negative number

the inequality sign must be reversed.

Let us start with solving simple linear inequalities

16.2.1 Solving simple linear inequalities

A linear inequality is an inequality with the highest power of the variable equal to

1. For example,

2x – 6 < 10

3 + 5y ≥ 4

Let us consider an example in which we add both sides

Worked Example [1]

Question

Solve 𝑥𝑥 – 3 ≥ 4

Solution

x – 3 ≥ 4

x – 3 + 3 ≥ 4 + 3

x + 0 ≥ 7

x ≥ 7

x – 3 ≥ 4 ( taking − 3 to the other side

of the inequality sing it becomes + 3

x ≥ 4 + 3

x ≥ 7

Let us consider the following example in which we subtract both sides

Solution

Here we are going to simply divide both sides by 5

5x ≥ −

5x

x ≥ −

Do this yourself;

In the space below, solve 3x < x + 4

Collect like terms to one side of the inequality sign

Now having done this lets consider an example in which we divide with a negative

number

Tip; Watch out for sign change

Worked Example [4]

Question

Solve −6x < 12

Solution

You can use one of the two methods below

Divide both sides by a negative

−6x < 12

678

67 <^

9: 67

x > −2 (we have reversed the sign)

Avoid dividing by a negative,

interchange

−6x < 12

Move −6x to the other side and it will

be a +6x

and 12 becomes −

− 12 < 6x ( now we divide by 6)

6 9:

7 <^

78 7

− 2 < x

These two answers are the same

Do this yourself;

In the space below, solve 5 – 2x > 9

Collect like terms

You could have collected like terms in one of the following ways

5 − 9 > 2x −2x > 9 − 5

Finish off working out the solutions for both methods below

If all this have been done correctly your answer should be x < −2 or to write −2 >

x

Let us now consider an example we will multiply with a positive number.

Remember, no sign change when dealing with positive numbers

Worked Example [6]

Question

Solve 6=^8 > 5

Solution

You can use one of the two methods below

Multiplying by a negative

8

–4( 6=^8 ) > 5 (–4) reverse the inequality

sign

x < –

Or avoid multiplying with the negative

and change side change sign

8

If you change sides it be like

8

Multiply both sides by a 4

–5(4) > =^8 (4)

− 20 > x

Both the answer mean the same thing

Do this yourself;

Solve 6 – 8 > < 0

Do your working on the space provided using the two methods shown above

Taking 6 to the other side of the

inequality sign

Taking − 8 > to the other side of the

inequality sign

If all is done correctly your answer should be x > 18 or 18 < x these two statements

mean the same thing

Now let us consider examples which involves multiple operations

Worked Example [7]

Question

Solve 6 – 4x < 2x + 2

Solution

You have to collect like terms

6 – 4x < 2x + 2

6 − 2 < 2x + 4x

4 < 6x

Now we have to divide both sides by 6

=

7 <^

78 7 =

7 < x^ (reducing)

:

> < x

Do this yourself;

Solve 5(x − 4) > 2(2x + 11)

Remove brackets and collect like terms (use the space below)

If all is done correctly you should obtain x > 44