Form 2

01 Directed Numbers

  1. Short Notes
  2. PT3 Focus Practice (with Solution)

02 Squares, Square Roots, Cubes and Cube Roots

  1. Short Notes
  2. PT3 Focus Practice (with Solution)

03 Algebraic Expressions (II)

  1. Short Notes
  2. PT3 Focus Practice (with Solution)


04 Linear Equations (I)

  1. Short Notes
  2. PT3 Focus Practice (with Solution)

Form 3

01 Lines and Angles (II)

  1. Short Notes
  2. PT3 Focus Practice (with Solution)

02 Polygons (II)

  1. Short Notes
  2. PT3 Focus Practice (with Solution)

03 Circles (II)

  1. Short Notes
  2. PT3 Focus Practice (with Solution)

04 Statistics (II)

  1. Short Notes
  2. PT3 Focus Practice (with Solution)


05 Indices

  1. Short Notes
  2. PT3 Focus Practice (with Solution)

06 Algebraic Expression III

  1. Short Notes
  2. PT3 Focus Practice (with Solution)

07 Algebraic Formulae

  1. Short Notes
  2. PT3 Focus Practice (with Solution)

08 Solid Geometri III

  1. Short Notes
  2. PT3 Focus Practice (with Solution)

09 Scale Drawings

  1. Short Notes
  2. PT3 Focus Practice (with Solution)
    • Question 1 - 5

10 Transformations II

  1. Short Notes
  2. PT3 Focus Practice (with Solution)
    • Question 1 - 5

11 Linear Equations II

  1. Short Notes
  2. PT3 Focus Practice (with Solution)
    • Question 1 - 5

12 Linear Inequalities

  1. Short Notes
  2. PT3 Focus Practice (with Solution)
    • Question 1 - 5

13 Graphs of Functions

  1. Short Notes
  2. PT3 Focus Practice (with Solution)
    • Question 1 - 5

14 Ratio, Rates and Proportions II

  1. Short Notes
  2. PT3 Focus Practice (with Solution)
    • Question 1 - 5

15 Trigonometry

  1. Short Notes
  2. PT3 Focus Practice (with Solution)
    • Question 1 - 5



5.1 Indices

5.1.1 Indices
1.      A number expressed in the form an is known as an index notation.
2.      an is read as ‘a to the power of n’ where a is the base and n is the index.
Example:
3.      If a is a real number and n is a positive integer, then


4.      The value of a real number in index notation can be found by repeated multiplication.
Example:
           64 = 6 × 6 × 6 × 6
            = 1296

5.      A number can be expressed in index notation by dividing the number repeatedly by the base.
Example:
243 = 3 × 3 × 3 × 3 × 3
       = 35



4.1 Linear Equations I

4.1.1 Equality
1. An equation is a mathematical statement that joins two equal quantities together by an equality sign ‘=’.
Example: 1km = 1000 m

2. If two quantities are unequal, the symbol ‘≠’ (is not equal) is used.
Example: 9 ÷ 4 ≠ 3


4.1.2 Linear Equations in One Unknown
1. A linear algebraic term is a term with one unknown and the power of unknown is one.
Example: 8x, -7y, 0.5y, 3a, …..

2. A linear algebraic expression contains two or more linear algebraic terms which are joined by a plus or minus sign.
Example:
3x – 4y, 4x + 9, 6x – 2y + 5, ……

3. A linear equation is an equation involving numbers and linear algebraic terms.
Example:
5x – 4 = 11, 4x + 7 = 15, 3y – 2 = 7


4.1.3 Solutions of Linear Equations in One Unknown
1. Solving an equation is a process of finding the values of the unknown in the equation.
2. The number that satisfies the equation is called the solution or root of the equation.
Example 1:
x + 4 = 12
      x = 12 – 4 ← (When +4 is moved to the right of the equation, it becomes –4)
      x = 8

Example 2:
x – 7 = 11
      x = 11 + 7 ← (When –7 is moved to the right of the equation, it becomes +7)
      x = 18

Example 3:
8x=16 x= 16 8 when the multiplier 8 is moved to the right of the equation, it becomes the divisor 8. x=2

Example 4:
x 5 =3 x=3×5 the divisor 5 becomes the multiplier 5 when moved  to the right of the equation. x=15


3.1 Circles II

(A) Properties of Circles Involving Symmetry, Chords and Arcs




(B) Identify the Properties of Chords

(1) A radius that is perpendicular to a chord divides the chord into two equal parts.




(C) Properties of Angles in Circles

(1)


(2)


(3)



(4)


(5)





1.1 Angles and Lines II

Identifying Parallel, Transversals, Corresponding Angles, Alternate Angles and Interior Angles
(A) Parallel lines
v  Parallel lines are lines with the same direction. They remain the same distance apart and never meet.

(B) Transversal lines
v  A transversal is a straight line that intersects two or more straight lines.




(C) Alternate angles



(D) Corresponding angles



(E) Interior angles




PT3 Smart TIP

v  Alternate angles are easily identified by tracing out the pattern “Z” as shown.



v  Corresponding angles are easily identified by the pattern “F” as shown.





v  Interior angles are easily identified by the pattern “C” as shown.




Example 1:
In Diagram below, PQ is parallel to RS. Determine the value of y.
Solution:
Construct a line parallel to PQ and passing through W.
a = 40o and b = 50o ← (Alternate angles)
y = a + b
  = 40o + 50o    
   = 90o


Example 2:
In Diagram below, PSQ and STU are straight lines. Find the value of x.

Solution:
WSQ= 180 o 150 o            = 30 o Supplementary angle XTU=WSQ+x Corresponding angle        75 o = 30 o +x        x= 75 o 30 o       x= 45 o          x=45


2.1 Squares, Square Roots, Cube and Cube Roots

(A) Squares
The square of a number is the answer you get when you multiply a number by itself.

Example 1:
(a)    132 = 13 × 13 = 169
(b)   (–10)2 = (–10) × (–10) = 100
(c)    (0.4) 2 = 0.4 × 0.4 = 0.16
(d)   (–0.06)2 = (–0.06) × (–0.06) = 0.0036
(e)  ( 3 1 2 ) 2 = ( 7 2 ) 2 = 7 2 × 7 2 = 49 4 ( f )  ( 1 2 7 ) 2 = ( 9 7 ) 2 =( 9 7 )×( 9 7 )= 81 49



(B) Perfect Squares
1. Perfect squares are the squares of whole numbers.

2. Perfect squares are formed by multiplying a whole number by itself.
Example:
4 = 2 × 2          9 = 3 × 3          16 = 4 × 4

3. The first twelve perfect squares are:
= 12, 22, 32, 42, 52, 62, 72, 82, 92, 102, 112, 122
= 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144


(C) Square Roots
1. The square root of a positive number is a number multiplied by itself whose product is equal to the given number.

Example 2:
(a)  169 = 13×13 =13 (b)  25 64 = 5×5 8×8 = 5 8 (c)  72 98 = 72 36 98 49 = 6×6 7×7 = 6 7 (d)  3 1 16 = 49 16 = 7 4 =1 3 4 (e)  1.44 = 1 44 11 100 25 = 36 25 = 6 5 =1 1 5



1.1 Directed Numbers
1. Multiplication and division of like signs gives (+)

  ( + )×( + )=+             ( + )÷( + )=+     ( )×( )=+             ( )÷( )=+

2. Multiplication and division of unlike signs gives ()

  ( + )×( )=             ( + )÷( )=     ( )×( + )=             ( )÷( + )=

3. Multiplication of 3 integers.

  ( + )×( + )×( + )=( + )       ( + )×( + )×( )=( )   ( + )×( )×( )=( + )     

4. Division of 3 integers.

  ( + )÷( + )÷( + )=( + )       ( + )÷( + )÷( )=( )   ( + )÷( )÷( )=( + )     

5. BODMAS(Brackets Of Division, Multiplication, Addition and Subtraction)


           ·         Operations in the brackets should be carried out first.
           ·         Followed by × or ÷ from left to right.
           ·         Followed by + or from left to right.



Example 1:
(a)    –52 ÷ 13 – 15 × 4
(b)   63 ÷ (16 – 7) × (–2)
(c)    –30 + 9 × 7 – 16

Solution:
(a)
–52 ÷ 13 – 15 × 4
= (–52 ÷ 13) – (15 × 4) ← (calculate from left to right; ÷ and × are done first)
= –4 – 60
= –64

(b)
63 ÷ (16 – 7) × (–2)
= 63 ÷ 9 × (–2) ← (bracket is done first, then work from left to right)
= 7 × (–2)
= –14

(c)
–30 + 9 × 7 – 16
= –30 + (9 × 7) – 16 ← ( multiply first)
= –30 + 63 – 16
= 17