PREFACE
MATHS GYM is not for students who dislike mental effort, a friendly and
apt warning to you. After all, the three indispensable keys to learning mathematics
are an understanding of concepts, a willingness to remember and a tireless effort to
practise. This book provides you close to 300 problems to practise, enhance your
understanding and help you memorise mathematical principles. Some of the more
challenging problems in this book are there to prove that you are capable of solving
them. This book is a mathematical gymnasium where “exercises” can be both beneficial
and fun.
If you work out the problems in the practices from Sessions 1 to 10 on your own,
you will likely do quite well in the coming examinations. You will find:
· Learning Objectives that detail what you should know about that topic.
· Useful Notes in the form of practical advice, reminders, worked examples and
detailed explanations to help you recap what you have learnt about the topic
and prepare you for the mental workout.
· Practice questions that train you specifically on a particular topic so that you
become familiar with the concepts taught.
· Challenge Yourself ! has questions designed to challenge you to work through
the varied interesting sums and to apply what you have learnt.
If you are a Secondary Two student and can work through the problems in
Challenge Yourself ! with little or no help, you must be a budding mathematician.
You are more than ready for Secondary Three Mathematics. If you continue to work
at it, your talent will be in great demand when you leave school. Congratulations!
Ee Teck Ee
CONTENTS
SESSION 1
Ratio, Rate and Proportion ------------------------------------------------· Map Scales
· Direct and Inverse Proportion
1
SESSION 2
Algebraic Manipulation ----------------------------------------------------· Changing the Subjects of Formulae
· Four Frequently Used Products
15
SESSION 3
Functions and Graphs ------------------------------------------------------· Graphs of Linear Equations in Two Unknowns
· Graphs of Quadratic Functions and Their Properties
27
SESSION 4
Solutions of Equations ------------------------------------------------------· Solutions of Simultaneous Linear Equations
· Solutions of Quadratic Equations
47
SESSION 5
Set Language and Notation -------------------------------------------------
61
SESSION 6
Congruence and Similarity -------------------------------------------------
74
SESSION 7
Pythagoras’ Theorem --------------------------------------------------------
92
SESSION 8
Mensuration ------------------------------------------------------------------- 107
· Solids with a Non-uniform Cross-section
SESSION 9
Data Analysis ------------------------------------------------------------------ 122
SESSION 10
Probability --------------------------------------------------------------------- 138
SESSION 11
Challenge Yourself! ----------------------------------------------------------- 152
ANSWERS
Detailed answers, all worked out. ------------------------------------ A1 - A51
Calculators should be used when necessary. If the degree of accuracy is not specified in
any question, and if the answer is not exact, give the answer to three significant figures.
For p, use either the calculator value or 3.142, unless the question requires the answer
in terms of p.
SESSION
1
Ratio, Rate and Proportion
Learning
Objectives
•
•
Map scales (distance and area)
Direct and inverse proportion
USEFUL NOTES
A
1
Map Scales
All maps are scale drawings of pieces of land. The linear scale is given either as a
ratio of two lengths or as a representative fraction (R.F.).
Example:
On a map, the scale is 5 cm to 1 km. Find the R.F.
5 cm
R.F. = ____
1 km
5 cm
= _________
100 000 cm
1
= _____
20 000 or 1 : 20 000
2
The area scale of a map is the square of its linear scale.
3
QUICK AND EASY
If the R.F. of a map is 1 : n, then
(i) 1 unit length on the map represents n unit length on the actual ground,
(ii) 1 unit length on the actual ground is represented by __1n unit length on the map,
(iii) 1 unit area on the map represents n2 unit area on the actual ground,
1
(iv) 1 unit area on the actual ground is represented by __
unit area on the map.
n2
B
1
Direct and Inverse Proportion
When two variables are related so that their ratio remains constant, each of them is
said to vary directly as (or directly proportional to) the other.
The constant is called the constant of variation.
In a direct variation, the variables change in the same manner. If one gets larger, the
other gets larger in the same proportion. If one gets smaller, the other gets smaller in
the same proportion.
1
Session 1
2
When two variables are related so that their product remains constant, each of them
is said to vary inversely as (or inversely proportional to) the other.
In an inverse variation, the variables change in the opposite manner. If one gets
larger, the other gets smaller in the reverse proportion. If one gets smaller, the other
gets larger in the reverse proportion.
In an inverse variation, each variable varies directly as the reciprocal of the other.
3
In a joint variation, one variable varies as the product of two or more variables.
2
Session 1
PRACTICE 1
PRACTICE
1.
(i)
Write a formula that shows the connection between density (D), mass (M )
and volume (V ).
(ii) State how M varies with D.
(iii) State how M varies with V.
(iv) State how D varies with V.
2.
(i)
Write a formula that expresses simple interest (I ) in terms of the principal (P),
rate per unit time (R) and time (T ).
(ii) If the principal is constant, how does the simple interest vary with the rate and
time?
(iii) If the rate and time are constant, how does the principal vary with the simple
interest?
3
Session 1
3.
Write down the value of y missing from the table below if
(i) y varies inversely as x,
3
5
x
(ii) y varies directly as x,
2
y
(iii) y varies directly as x2.
4.
x varies inversely as y and directly as z.
(i) If x = 20 when z = 5 and y = 25, find the value of x when y = 50 and z = 0.5.
(ii) How does x vary if y is multiplied by 12 and z is divided by 36?
(iii) Find a formula that connects x, y and z.
(iv) If y is increased by 25% and z is decreased by 25%, what is the percentage
change in x?
4
Session 1
5.
x varies directly as y but inversely as z. If y is increased by 30% and z is decreased
by 30%, how does x vary?
6.
The volume of a cylinder varies directly as the square of the base radius as well as
the height. How is the volume changed if the base radius is decreased by 10% but
the height is increased by 20%?
7.
If y varies directly as __1x and y = 4 when x = 25, express y in terms of x and find the
value of x when y = 20.
5
Session 1