Form 5 add math

01 Progression 1.

Formula (PDF Copy) 2.

Arithmetic Progression a) Characteristics of arithmetic progressions b) The steps to prove whether a given number sequence is an arithmetic progression c) The nth term of an arithmetic progression  Example 1  Example 2  Example 3  Example 4 d) The number of terms in an arithmetic progression e) The consecutive terms of an arithmetic progression f) The sum of the first n terms of an arithmetic progressions g) SPM practice 1 (Arithmetic Progression) - 15 Questions  Download PDF Copy - Without Answer Space  Download PDF Copy - With Answer Space  Step by Step Answer 3.

GeometricProgression a) Characteristics of geometric progressions b) The steps to prove whether a given number sequence is a geometric progression c) The nth term of a geometric progression d) The number of terms in a geometry progression  Example 1  Example 2  Example 3 e) The consecutive terms of a geometric progression f) The sum of the first n terms of a geometric progression g) The sum to infinity of a geometry progression  Sum to infinity example 1 : Recurring Decimal h) SPM practice 2 (Geometric Progression) - 15 Questions  Download PDF Copy - Without Answer Space  Download PDF Copy - With Answer Space  Step by Step Answer 4.

Mind Map 02 Linear Law 1.

Formula (PDF Copy) 2.

Revise of Important Concept (Straight Line)  Equation of a Straight Line  Gradient of a Straight Line  Mid Point of Two Points on the Straight Line  Distance of Two Points on the Straight Line 3.

Lines of Best Fit  Steps to draw a Line of Best Fit

4.

5.   6.     7.    8.       9.

Equations of Line of Best Fit Applications of Linear Law to Non-linear Relations Linear versus Non-Linear Function Tips to reduce Non-Linear Function to Linear Function SPM Practice 1 (Reduce Non-linear Function to Linear Function) Download PDF Copy-Without Answer Space Download PDF Copy-With Answer Space Step by Step Answer [Example (a)-(f)] Step by Step Answer [Example (g)-(l)] SPM Practice 2 (SPM Paper 1-10 questions) Download PDF Copy-Without Answer Space Download PDF Copy-With Answer Space Step by Step Answer SPM Practice 3 (SPM Paper 2 - 4 questions) Download PDF Copy-Without Answer Space Download PDF Copy-With Answer Space Step by Step answer (Question 1) Step by Step Answer (Question 2) Step by Step Answer (Question 3) Step by Step Answer (Question 4) Mind Map

03 Integration 1.

Indefinite Integral a) Basic Integration b) Integration by Substitution c) Finding Equation of a Curve from its Gradient Function 2.

Definite Integrals a) Part 1 b) Part 2 3.

Integration as the Summation of Areas 4.

Integration as the Summation of Volumes 5.

Further Practice 6.

Mind Map 04 Vectors 1.

Vector 2.

Addition and Subtraction of Vectors 3.

Vectors in Cartesian Plane 4.

Further Practice a) Short Questions  Example 1 & 2

b) Long Questions  Example 1 5.

Mind Map

05 Trigonometric Functions 1.

Positive and Negative Angles 2.

Six Trigonometric Functions of any Angle a) The definitions of sin, cos, tan, cosec, sec and cot b) Special Angles 3.

Graphs of the Sine, Cosine and Tangent Functions 4.

Basic Trigonometric Identities 5.

Formulae of sin (A±B), cos (A±B), tan (A±B), sin 2A, cos 2A, tan 2A 6.

Solving Trigonometric Equation a) Basic Equation in sin x/cos x/tan x/cosec x/sec x/cot x b) Factorization c) Form Quadratic Equation in sin x/cos x/tan x/cosec x/sec x/cot x d) Using sin 2x 7.

Proving Trigonometric Identities 8.

Solve by drawing triangle 9.

Further Practice a) Short Questions  Example 1 - 4 b) Long Questions 10. Mind Map 06 Permutations and Combinations 1.

Permutations Part 1 2.

Permutations Part 2 3.

Combinations 4.

Mind Map 07 Probability 1.

Probability 2.

Probability of Mutually Exculsive Events 3.

Probability of Independent Events 4.

Mind Map 08 Probability Distributions 1.

Binomial Distribution 2.

Normal Distribution 3.

Mind Map

09 Motion Along a Straight Line 1.

Displacement 2.

Velocity 3.

Acceleration 4.

Mind Map 10 Linear Programming 1.

Graph of Linear Inequality 2.

Linear Programming 3.

Mind Map

Characteristics Of Arithmetic Progressions An arithmetic progression is a progression in which the difference between any term and the immediate term before is a constant. The constant is called the common difference ( d ).

d=Tn−Tn−1 or d=Tn+1−Tn

Example Determine whether the following number sequences is an arithmetic progression (AP) or not. (a) -5, -3, -1, 1, ..... (b) 10, 7, 4, 1, -2, .... (c) 2, 8, 15, 23, .... (d) 3, 6, 12, 24, .......

Hint : For an arithmetic progression, you always plus or minus a fixed number

The steps to prove whether a given number sequence is an arithmetic progression The steps to prove whether a given number sequence is an arithmetic progression.

Step 1: List down any three consecutive terms. [Example: T1,T2,T3 .]

Step 2: Calculate the values of T3−T2 and T2−T1 .

Step 3: If T3−T2=T2−T1=d, then the number sequence is an arithmetic progression.

[Try Question 8 and 9 in SPM Practice 1 (Arithmetic Progression)] Example : Prove whether the following number sequence is an arithmetic progression (a) 7, 10, 13,.... (b) -20, -15, -9 ,......

The nth term of an arithmetic progression

The nth term of an Arithmetic Progression Tn=a+(n−1)d a = first term d = common difference n = the number of term Tn = the nth term

The nth term of AP example 1 Example 1 If the 20th term of an arithmetic progression is 14 and the 40th term is -6, Find (a) the first term and the common difference, (b) the 10th term.

The nth term of AP Example 2 Example 2 The 3rd term and the 7th term of an arithmetic progression are 20 and 12 respectively. (a) Calculate the 20th term. (b) Find the term whose value is -34.

The nth term of AP Example 3 Example 3 The volume of water in a tank is 75 litres on the first day. Subsequently, 15 litres of water is added to the tank everyday.

Calculate the volume, in litres, of water in the tank at the end of the 12th day.

The nth term of AP Example 4 Example 4 The first three terms of an arithmetic progression are 72, 65 and 58.

The nth term of this progression is negative.

Find the least value of n.

The number of terms in an arithmetic progression Hint : You can find the number of term in an arithmetic progression if you know the last term Find the number of terms for each of the following arithmetic progressions. (a) 5, 9, 13, 17... , 121 (b) 1, 1.25, 1.5, 1.75,...,8

Three consecutive terms of an Arithmetic Progression If a, b, c are three consecutive terms of an arithmetic progression, then c–b=b-a Example: If x + 1, 2x + 3 and 6 are three consecutive terms of an arithmetic progression, find the value of x and its common difference.

The sum of the first n terms of an arithmetic progressions Sum of the First n terms of an Arithmetic Progressions

Sn=n2[2a+(n−1)d] Sn=n2(a+l)

a = first term d = common difference n = the number of term Sn = the sum of first n terms

Example Calculate the sum of each of the following arithmetic progressions. (a) -11, -8, -5, ... up to the first 15 terms. (b) 8, 1012, 13,... up to the first 13 terms. (c) 5, 7, 9,....., 75 [hint : The last term is given, you can find the number of term, n]

SPM Practice 1 (Arithmetic Progression): Question 1 Question 1 The third and eighth terms of an arithmetic progression are -5 and 15 respectively. Find (a) The first term and the common difference (b) The sum of the first 10 terms.

[Hint : Solving simultaneous equation to find a and d using Tn formula]

SPM Practice 1 (Arithmetic Progression): Question 2 Question 2 The first three terms of an arithmetic progression are 2k, 3k+3, 5k+1.Find (a) the value of k, (b the sum of the first 15 terms of the progression.

Remember if a, b, c are three consecutive term of arithmetic progression, then c - b = b - a

SPM Practice 1 (Arithmetic Progression): Question 3

Question 3 Given an arithmetic progression p+9, 2p+10, 7p-1,…., where p is a constant. Find (a) the value of p, (b) the sum of the next five terms.

Remember if a, b, c are three consecutive term of arithmetic progression, then c - b = b - a

Characteristics of a geometry progressions Characteristics of Geometric Progressions

Geometric progression is a progression in which the ratio of any term to the immediate term before is a constant. The constant is called common ratio (r).

Example Determine whether or not each of the following number sequences is a geometric progression (GP). (a) 1, 4, 16, 64 … (b) 10, -5, 2.5, -1.25 … (c) 2, 4, 12, 48,……. (d) -6, 1, 8, 15,.......

[Hint : For geometry progression, you times a fixed number every time to get the next term]

The steps to prove whether a given number sequence is a geometric progression The steps to prove whether a given number sequence is a geometric progression.

Step 1: List down any three consecutive terms. [Example: T1, T2, T3.]

Step 2: Calculate the values of T3T2 and T2T1 .

Step 3: If T3T2=T2T1=r, then the number sequence is a geometric progression.

Step 4: If T3T2≠T2T1, then the number sequence is not a geometric progression.

The nth term of a geometric progression r = common ratio n = the number of term Tn = the nth term

Example Find the given term for each of the following geometric progressions. (a) 8, 4, 2, ...... T8 (b)1627, 89, 43, ...; T6

The number of term in a geometry progression Hint : You can find the number of term in an arithmetic progression if you know the last term Find the number of terms for each of the following geometric progressions. (a) 2, 4, 8,……, 8192 (b) 14, 16, 19, ..., 16729 (c) −12, 1, −2, ..., 64

The nth term of geometric progression example 1 Example 1 The 6th term of a geometric progression is 32 and the 3rd term is 4. Find the 1st term and the common ratio. Solving the simultaneous equation of a and r. Using the formula

The nth term of geometric progression example 2 Example 2 In a geometric progression, the sum of the second term and the third term is 12 and the sum of the third term and the fourth term is 4, find the first term and the common ratio.

[Solving simultaneous equation to find a and r]