of 36

• date post

05-Apr-2018
• Category

## Documents

• view

221

0

Embed Size (px)

### Transcript of Form 4 Notes

• 7/31/2019 Form 4 Notes

1/36

MathematicForm 4

• 7/31/2019 Form 4 Notes

2/36

CHAPTER 1: STANDARD FORM AND SIGNIFICANT

FIGURES

Standard Form: a 10n

1 a < 10, n is an integer

215000 = 2.15 105

0.000324 = 3.24 10-4

Significant Figure: 1268954 = 1270000 to 3

significant figure

0.003674 = 0.00367 to 3 sig. fig.

Big number, n is

positive

Small number, n isnegative

• 7/31/2019 Form 4 Notes

3/36

CHAPTER 2: QUADRATIC EXPRESSION AND

EQUATIONS

Expansion: (2x 5)(x + 3) = 2x2 + 6x 5x 15

= 2x2 + x 15

Factorisation: 3x2 - x 2 = (3x + 2)(x 1)

2x

2

5x 3 = 0(2x + 1)(x 3) = 0

2x + 1 = 0, x =2

1

x 3 = 0, x = 3

• 7/31/2019 Form 4 Notes

4/36

CHAPTER 3: SET

The complement of set A

= A'

Symbol

- intersection

- union

- subset

- universal set

,{ }

- empty set

- is a member of

n(A) number of element in setA.

A complement of setA.

• 7/31/2019 Form 4 Notes

5/36

Operations on Set

(A B) C (P Q) R

• 7/31/2019 Form 4 Notes

6/36

Clone 2005

The Venn diagram shows the sets K, L and M with the

elements. It is given that the universal set = K L M andn(K) = n(L M).

K

L

M

a - 3

a - 2

8

2

2

2

Find the value of a

n(K) = 2 + 8 + 2 =

12

n(L M) = a 2 + 8

= a + 6

a + 6 = 12

a = 6

• 7/31/2019 Form 4 Notes

7/36

CHAPTER 4: MATHEMATICAL REASONING

(a) Statement

A mathematical sentence which is either true or false butnot both.

3 + 4 = 7 A true statement

32 = 6 A false statement

x + 5 = 8 Not a statement because it is not known

whether it is true or false.

• 7/31/2019 Form 4 Notes

8/36

(b) Implication

Ifa, then b

a antecedent

b consequent

Ifx is an even number, then x is divisible by two

Antecedent: x is an even number

Consequent: x is divisible by two.

• 7/31/2019 Form 4 Notes

9/36

p if and only ifq can be written in two implications:

1. Ifp, then q

2. Ifq, thenp

x + 2 = 5 if and only if x = 3

1. If x + 2 = 5, then x = 3

2. If x = 3, then x + 2 = 5

• 7/31/2019 Form 4 Notes

10/36

(c) Argument

Three types of argument:

Type IPremise 1: AllA are B

Premise 2 : CisA

Conclusion: Cis B

Type II

Premise 1: IfA, then B

Premise 2:A is true

Conclusion: B is true.

Type III

Premise 1: IfA, then BPremise 2: Not B is true.

Conclusion: NotA is true.

• 7/31/2019 Form 4 Notes

11/36

INDUCTION:

Make a general conclusion by induction for the numerical

sequence:

7 = 6(1)2 + 1

25 = 6(2)2 + 1

55 = 6(3)2 + 1

97 = 6(4)2 + 1

.

6(n)2 + 1, n = 1, 2, 3, .

• 7/31/2019 Form 4 Notes

12/36

CHAPTER 5: THE STRAIGHT LINE

m =12

12

xx

yy

(b) Equation of a straight lineGradient Form:

y = mx + c

c= y-intercept

• 7/31/2019 Form 4 Notes

13/36

Intercept Form:

-int ercept

-intercept

y

x =

Intercept Form: 1=+b

y

a

x

a

b

Find the equation of the straight line which passes through

the point A(1, 2) and has a gradient of 3.

Solution:

Equation of straight line: y = mx + c

Substitute A(1, 2) and m = 3, 2 = 3 + c

c = -1 Equation of straight line: y = 3x 1.Eqn

a =xintercept

b = yintercept

http://var/www/apps/conversion/current/tmp/scratch27822/C:/Documents%20and%20Settings/user/Local%20Settings/Temp/Rar\$DI00.968/Equation%20of%20straight%20line.gsphttp://var/www/apps/conversion/current/tmp/scratch27822/C:/Documents%20and%20Settings/user/Local%20Settings/Temp/Rar\$DI00.968/Equation%20of%20straight%20line.gsp
• 7/31/2019 Form 4 Notes

14/36

Parallel Lines

The gradient of two parallel lines are equal.

m1 = m2

OPQR is a parallelogram. Find

(a) the coordinates of Q

(b) the equation of QR5

Solution:

(a) Q(4, 7)

2

1

04

02=

c = 5, Eqn of QR: y = x + 521

(b) Gradient of OP = mOP = = mQR

• 7/31/2019 Form 4 Notes

15/36

CHAPTER 6: STATISTICS

(a) Mean

for ungrouped data.n

xx

= n = number of data

x = mid point

f

fx

= for grouped data.

(b) Mode

Mode is the data with highest frequency.

• 7/31/2019 Form 4 Notes

16/36

(d) Class, Modal Class, Class Interval Size, Midpoint,

Cumulative frequency, Ogive

Example :

The table below shows the time taken by 80 studentsto type a document.

Time (min) Frequency

10-1415-1920-24

25-2930-3435-3940-4445-49

1712

21191262

Midpoint of modal class

= = 272

2925+

For the class 10 14 :Lower limit = 10 min

Upper limit = 14 min

Lower boundary = 9.5 minUpper boundary = 14.5 min

Class interval size = Upper

boundary lower boundary

= 14.5 9.5 = 5 minModal class = 25 29 min

• 7/31/2019 Form 4 Notes

17/36

Ogive

To draw an ogive, a table of upper boundary and cumulative

frequency has to be constructed.Time(min)

Frequency Upper boundary

Cumulativefrequency

5-9

10-1415-1920-2425-29

30-3435-3940-4445-49

0

17

1221

191262

9.5

14.519.524.529.5

34.539.544.549.5

0

182041

60727880

• 7/31/2019 Form 4 Notes

18/36

From the ogive :

Median = 29.5 minFirst quartile = 24. 5

min

Third quartile = 34 min

Interquartile range =34 24. 5 = 9.5 min.

• 7/31/2019 Form 4 Notes

19/36

Histogram, Frequency Polygon

(e) Histogram, Frequency Polygon

Example:The table shows the marks obtained by a group of

students in a test.

Marks Frequency

1 1011 2021 30

31 4041 50

28

16

204

• 7/31/2019 Form 4 Notes

20/36

CHAPTER 7: PROBABILITY

(b) Complementary Event

P(A ) = 1 P(A)

( )

( ) ( )

nAPA

nS=

(c) Probability of Combined Events

(i) For mutually exclusive events,AB = P(A orB) = P(AB) = P(A) + P(B)

(ii) For Independent Events.

P(A and B) = P(A B) = P(A) P(B)

Definition of Probability

(a) Probability that event A happen,

S = sample space

• 7/31/2019 Form 4 Notes

21/36

CHAPTER 8: CIRCLES III

Circle Theorems

O

A B

C

x

y O

O

Angle at the

centre = 2 angle

at the

circumferencex = 2y

AB

C

x

yO

O

D

Angles in the

same segment

are equal

x = y

OA B

C

9 0O

Angle in a semicircle

ACB = 90o

• 7/31/2019 Form 4 Notes

22/36

a

b

O

O

Sum of opposite

angles of a cyclicquadrilateral = 180o

a + b = 180o

a

b

O

O

The exterior angle of

a cyclic quadrilateralis equal to the

interior opposite

angle.

a = b

O

P Q

Angle between a

tangent and a radius =

90o

OPQ = 90o

• 7/31/2019 Form 4 Notes

23/36

x

y o

o

The angle between atangent and a chord is

equal to the angle in the

alternate segment.

x = y

P

T

S

O

IfPTand PS are tangents to acircle,

PT = PS

TPO = SPO TOP= SOP

• 7/31/2019 Form 4 Notes

24/36

CLONE 2004:

CDE is a tangent to the

circle. Find the value of x.

A. 16o B. 18o

C. 20o D. 22o

Solution:

G = 48oo

oo

GFD 66

2

48180=

=

x = 66o 48o = 18o Answer: B

• 7/31/2019 Form 4 Notes

25/36

CHAPTER 9: TRIGONOMETRY

sin =Opposite

hypotenuse

AB

AC

=

hypotenuse AC=

ABBC

=

Add Sugar To Coffee

• 7/31/2019 Form 4 Notes

26/36

CLONE 2005:

Find the value of cos .

A.12

5 B.

5

13

C.5

12 D.5

13

Solution:

cos = - cos EGF =5

13 Ans: B

• 7/31/2019 Form 4 Notes

27/36

TRIGONOMETRIC GRAPHS

y = cos x

y = sin x

• 7/31/2019 Form 4 Notes

28/36

y = tan x

• 7/31/2019 Form 4 Notes

29/36

CHAPTER 10: ANGLE OF ELEVATION AND

DEPRESSION

Angle of Elevation

The angle of elevation is the

angle betweeen the

horizontal line drawn from

the eye of an observer andthe line joining the eye of the

observer to an object which

is higher than the observer.

The angle of elevation ofB

fromA is BAC

• 7/31/2019 Form 4 Notes

30/36

Angle of Depression

The angle of depression is theangle between the horizontal line

from the eye of the observer an

the line joining the eye of the

observer to an object which islower than the observer.

The angle of depression ofB

fromA is BAC.

• 7/31/2019 Form 4 Notes

31/36

CHAPTER 11: LINES AND PLA