Additional Mathematics

Formulae List

Form 5

(prepared by:BEHPC)

1

01 Progressions

Arithmetic Progression Geometric Progression

1. Common Difference

2. The nth term

3. Sum of the first n terms

or

where l = last term

1. Common Ratio

2. The nth term

3. Sum of the first n terms

4. Sum to infinity:

If is given as a function of n, then use:

First term,

Second term,

Sum of the terms from mth term to nth term:)

(For example –The sum from 3rd term to 7th term, )

2

02 Linear Law1. Drawing lines of best fit

2. Equations of lines of best fit

3. To reduce non-linear functions to linear form

A set of two variables are related non linearly can be converted to a linear equation. The line of best fit can be written in the form

whereX and Y are in terms of x and/or ym is the gradient,c is the Y-intercept

The graph of can be used to find the values of constants of the non-linear equation and others information relating the two variables.

Recall:(1) Equation of a straight line if two points

are given:

(2) Equation of a straight line if m and c are given:

Tips:

(1) The equation must have one constant (without x and y).

(2) Y cannot have constant, but can have x and y.

(3) X cannot have y., but can have x and constant.

3

Line of best fit has 2 characteristics: it passes through as many points as possible,the number of points which are not on the line of best fit are equally distributed on the both sides of the line.

Steps to draw a line of best fit: Construct a table consisting the given variables.Plot a graph of Y against X , using the scale specified AND draw a line of best fit.Calculate the gradient, m, and get the Y-intercept, c, from the graph.Re-write the original equation given and reduce it to linear form.Compare with values of m and c obtained, find the values of the unknowns required.

4

The table shows some of examples of non-linear equations can be reduced to the linear form:

Non-linear equation Linear equation Y X m c

x p q

q p

xy p q

q p

x p q

q p

p q

x

x

q

x +1

5

03 Integration

4. Finding Equation Of A Curve From Its Gradient Function:

Gradient functionThe equation of the curve

Differentiation

Integration

dxxfy )('

6

Integration is the inverse process of differentiation.If , then where c = constant.

Indefinite integrals:(Refer to the examples below after note 6)(a)

(b)

(c)

3. Definite integrals

The laws of definite integrals:(a)

(b) (c) (d) (e)

5. Integration As The Summation Of Areas:

Area bounded by the curve y= f(x),the lines x = a, x = b and the x-axis

b

a

ydxA

Area of the region between a curve y = f(x) and a straight line y = g(x)

b

a

b

a

dxxgdxxfA )()(

7

Area bounded by the curve x = f(y), the lines y = a, y = b and the y-axis.

6. Integration As The Summation Of Volumes

Refer to Note 3:

Example:

Example:

1.

The volume of the solid generated when the region enclosed by the curve y = f(x), the x-axis, the line x = a and the line x = b is revolved through 360° about the x-axis is given by

b

ax dxyV 2

Example:

Example:

Example:

8

Example:

The volume of the solid generated when the region enclosed by the curve x = f(y), the y-axis, the line y = a and the line y = b is revolved through 360 ° about the y-axis is given by

b

ay dyxV 2

Example: Example:

Example:

9

04 Vectors

10

Vector is a quantity that has both magnitude and direction. Scalar is a quantity that has magnitude only.A vector can be presented by a line segment with an arrow, known as a directed line segment.

Negative vector of has the same magnitude as but its direction is opposite to .

A zero vector ia a vector whose magnitude is zero. It is denoted by .Two vectors are equal if both the vectors have the same magnitude and direction.When a vector is multiplied by a scalar , the product is . Its magnitude is times the magnitude of the vector .The vector is parallel to the vector if and only if , where is a constant.If the vectors and are not parallel and , then and .Addition of vectors

Triangle Law Parallelogram Law

The subtraction of the vector from the vector is written as . This operation can be considered as the addition of the vector with the negative vector of , i.e. .

11

Column vector:

A unit vector is a vector whose magnitude is one unit.The magnitude of the vector can be calculated using the Pythagoras’ Theorem.

(i) To show is parallel to ,To show A, B and C are collinear,

To find the ratio of AB :BC

12

Use

05 Trigonometric Functions

Positive and Negative AnglesPositive angles are angles measure in an anticlockwise rotate from the positive x-axis about the origin, O.

Negative angles are angles measured in a clockwise rotation from the positive x-axis about the origin O.

Six Trigonometric Functions of Any Angle

Complementary Angle

Negative Angle

Positive/Negative sign at different quadrant

Just rememberthe positive

ratio!!!

13

S A

T C

1

14

Value of Special Angle 30 o and 60 o

Value of Special Angle 45 o

Value of Special Angle 0 o , 90 o , 180 o , 270 o , 360 o .

q 0o 90o 180o 270o 360o

sin q 0 1 0 -1 0

q 0o 90o 180o 270o 360o

cos q 1 0 -1 0 1

q 0o 90o 180o 270o 360o

tan q 0 ¥ 0 ¥ 0

Alternative Way:q 0o 30o 45o 60o 90o

0 1

1 0

0 1 ¥

Steps to solve simple trigonometric equation:(1) Determine the range of values of the required

angles.(2) Find a basic angle by using calculator.(3) Determine the quadrants the angle should be.(4) Determine the values of angles in those

quadrants.

15

Graphs of The Functions of Sine, Cosine and Tangent

Graph of. y = sin x

x 0o 90o 180o 270o 360o

sin 0 1 0 -1 0

Graph of. y = cos x

x 0o 90o 180o 270o 360o

cos x 1 0 -1 0 1

Graph of. y = tan x

x 0o 90o 180o 270o 360o

tan x 0 ¥ 0 ¥ 0

Basic Trigonometric Identities:

Compound AnglesFormulae:

Double Angle Formulae:

or

or

16

Half Angle Formulae:

or

or

06 Permutations and Combinations

Multiplication Principle / Rule Permutations

17

For right angled triangle, use:

If an operation can be carried out in r ways and another operation can be carried out in s ways, then the number of ways to carry out both the operations consecutively is r s, i.e. rs.

The rs multiplication principle can be expanded to three or more operations. If the numbers of ways for the occurrence of events A, B and C are r, s and p respectively, the number of ways for the occurrence of all the three events consecutively is r s p, i.e. rsp.

The number of permutations of n different objects is n!, where

n!, is read as n factorial.

Permutation of n Different Objects, Taken r at a Time

CombinationsThe number of permutations of n different objects, taken r at a time is given by :

A permutation of n different objects, taken r at a time, is an arrangement of a set of r objects chosen from n objects. The order of the objects in the chosen set is taken into consideration.

The number of permutations of n different objects, taken all at a time, is :

Note: (i)

(ii)

(iii)

The number of combinations of r objects chosen from n different objects is given by :

A combination of r bjects chosen from n different objects is a selection of a set of r objects chosen from n objects. The order of the objects in the chosen set is not taken into consideration.

Note: (i)

(ii)

(iii)

(iv)

18

07 Probability1. The probability for the occurrence of an event A in the sample space S is

2. (a) The range of values of a probability is .(b) If P(A) = 1, event A is sure to occur.(c) If P(A) = 0, event A will not occur.

3. The complement of an event A is denoted by and the probability of a complementary event is given by

4. The probability for the occurrence of events A or B or both is

5.

If the events A and B are mutually exclusive, then and . Thus,

6. The probability of the combination of two independent events, A and B, if the occurrence or non-occurrence of one event does not affect of the other is given by

7. The concept of the probability of two independent events can be expanded to three or more independent events. If A, B and C are three independent events, the probability for the occurrence of events A, B and C is

8. A tree diagram can be constructed to show all the possible outcomes of an experiment.

08 Probability Distributions

NORMAL DISTRIBUTIONS

Continuous random variable is a variable that can take any infinite value in a certain range.A normal distribution is a probability distribution of continuous random variables (only quantities that can be measured).The distribution is denoted by where = mean and = variance.

A normal distribution with =0 and = 1 is known as a standard normal distribution and is denoted by N (0,1).

A normal random variable, X, can be converted into standard normal random variable by using .

Z = standard score or z-scoreX = value of a normal random

variable m= mean of a normal

distribution s = standard deviation of a

normal distribution

PROBABILITY DISTRIBUTIONS

BINOMIAL DISTRIBUTIONS

1. A random variable that has finite and countable values is known as a discrete random variable.

2. For a Binomial Distribution, the probability of obtaining r numbers of successes out of n experiments is given by

whereP = probability X = discrete random variabler = number of success (0, 1, 2, 3, …,n)n = number of trialsp = probability of success in an experiment (0 < p <1)q = probability of failure in an experiment ( )

3. A binomial probability distribution can be plotted as a graph.

4. Determine the mean, variance and standard deviation of binomial distribution

If X is a binomial discrete random variable such that X~B (n, p), then

Mean of X,

Variance of X,

Standard deviation of X,

Normal Distribution1. A continuous random variable, X, is normally distributed if the graph of its probability function

has the following properties.

Its curve has a bell shape and it is symmetrical at the line x = m. Its curve has a maximum value at x = m. The area enclosed by the normal curve and the x-axis is 1.

09 Motion Along a Straight Line

9.1 Displacement

1. (a) Positive displacement means that the particle is at the right-hand side of O. (b) Negative displacement means that the particle is at the left-hand side of O. (c) Zero displacement means that the particle is at O.

2. Total distance travelled in the first n seconds is the total distance travelled by a particle from time t = 0 to t = n.

3. Distance travelled during the nth second is the distance travelled by the particle from time t = (n – 1) to t = n.Thus, Distance travelled during the nth second =

9.2 Velocity

1. Instantaneous velocity, v, is the rate of change of displacement, s, with respect to time, t and it is

given by

2. (a) When a particle moves to the right, it has a positive velocity.(b) When a particle moves to the left, it has a negative velocity.(c) When a particle is instantaneously at rest, it has a zero velocity.

3. The displacement of particle is a maximum (to the right or to the left of the fixed point O) when its velocity is zero.

4. A particle reverses its direction when it comes to instantaneous rest, i.e. v = 0.

5. Displacement, s, is given by the integration of the instantaneous velocity, v, with respect to time, t, i.e.

s vdt

6. Distance travelled during the nth second =

7. The displacement of particle is a maximum (to the right or to the left of the fixed point O) when its velocity is zero.

8. A particle reverses its direction when it comes to instantaneous rest, i.e. v = 0.

9.3 Acceleration

1. Acceleration is the rate of change of velocity.

2. (a) Positive acceleration means that the velocity is increasing with respect to time. (b) Negative acceleration or deceleration means that the velocity is decreasing with respect to

time. (c) Zero acceleration means that the velocity is a constant (uniform velocity).

3. The velocity of a particle is a maximum when its acceleration is zero. 4. The following conclusion can be made.

vs

dsvdt

s v dt

a

dvadt

2

2d sadt

v a dt

Important tips:

Initial displacement / initial velocity / initial acceleration

(a) at the right-hand side of O

(b) 4 m to the right of O

(a) at the left-hand side of O

(b) 4 m to the left of O

when the particle moves to the right

when the particle moves to the left

when the particle is instantaneously at rest

when particle reverses its direction

when the particle returns to O

maximum displacement of the particle or

maximum velocity of the particle or

velocity is increasing / positive acceleration

velocity is decreasing / negative acceleration

uniform velocity(velocity is a constant) / zero acceleration

when particle M and particle N meet

10 Linear Programming1. Linear programming is a method of solving problems involving two variables that can be

represented by a mathematical model by using inequalities as constraints.2.

3. Just remember how to shade the regions which are greater than:

(a) (b)

(c) (b = y-intercept) (d) (a = x-intercept)

For or , use solid line ().

For > or < , use dashed line (----).

xO

y

xO

y

xO

y

y bx

O

y x a

a

b

(e) (m = positive value) (f) (m = negative value) (Eg: ) (Eg: )

4. A constraint is an inequality that represents a condition that must be satisfied in order for a problem to be solved.

Constraint Inequality1. y is more than x2. y is less than x3. y is not more than x4. y is not less than x5. y is at least k times of x6. y is at most k times of x

7. The total x and y is not more than k

8. The smallest value of y is k9. The greatest value of y is k10. x exceeds two times of y at least k

11. The ratio of y to x is k or more

5. Important keywords:

not less than at least smallest value minimum value

xO

y

xO

y

6. Steps to solve a linear programming problem through graphical method:Step 1: Determine the two variables, x and y.Step 2: From the given constraints, interpret the problem and form inequalities that satisfy all

the constraints.Step 3: Draw the straight lines for each inequality.Step 4: Determine the region which satisfies all the inequalities.Step 5: Form the optimal function .Step 6: By using a ruler and set square, slide the line towards the region to find the maximum

or minimum point based the function.Step 7: Determine the optimal value (maximum or minimum value).

TO MY BELOVED STUDENTS!!!

“GOOD LUCK & ALL THE BEST “IN YOUR SPM EXAMINATION

WISHING YOU THE VERY BEST IN EVERYTHING AND MAY ALL THE NICEST THINGS YOU WISH FOR ALWAYS COME TO YOU.

BEST WISHES,PN BEH

not more thanat mostgreatest valuemaximum value