revision of form 4 add maths

7/26/2019 Revision of Form 4 Add Maths

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Revision of Form 4 Add Maths

Chapter 1:

1. Relation

a) Domain ={ b) Codomain = {c) Object =d) Image =e) Range = {image} = {

2. Type of relation

(a) one to one (b) one to many

(c) many to one (d) many to many

3. Function

a) one to one * Inverse function one to one b) many to one

Remars:

1. x = object; f(x) = image

2. is mapped to it!s self f(x) =x

2. f "#$ % x

x+2 & x + 2 ≠ 0

3. ' x ( 3' % 1 x ( 3 % ) 1

4.

*. fg "#$ + gf "#$

,. f 2 "#$ % ff"#$

-. f

1

"#$ % y f "y$ % #

/. ff -1 "#$ % #

0#ample 1

Given tat f:x !" x # $ !% find(a) image of "%(b) object of &(c) x 'en x is maed to its sef (c) +,etc te gra f( x) for - . x . "/ 0ence% dertermine te range

of codomain/

4. Composition function

f g

a

b

c

d

e

z y x



1 y

f

f23

gf(x)

0#ample 1

Given function f ( x) = x # " and function g ( x) = ax 4 # b/ If tecomosite function gf is given by gf ( x) = 4 x 4 # 34 x 5 3"% finda) te vaues of a and b

b) gf (6)

0#ample 2

Given te functions g : x → x + 6 and fg : x → 4 x – "% find(a) f(x)

) te vaue of 1 'en gf(x) = 7/

8/ nverse Function

f(x) = y f --1(y) = x

91ame:

Given g # 3 : x x+5

1− x

% x ≠p/ ;ind

(a) p (b) g (x)

Chapter 2

3/ ;ind te roots < +ove te euationa) factori>ation ( )( )=-

b) ;ormua

1 = a

acbb

4

64 −±−

4/ ;orm euation < form ne' euation

0#ample 1

a) ;orm te uadratic euation 'ic as roots 2 6 and2

3/

b)Given tat te roots of uadratic euation ( )4

4 3 - x h x k − + =+

are 2" and $/ ;ind (i) te vaue of h (ii) te vaue of k /



c) α and ? are te roots of te euations 4 x 4 # x # 3 = -/

;ind a uadratic euation for root @ 4 and ? 4

4. 3 Types of roots

a) 4 rea and distinct<different roots/ 2 4ac b) 4 rea and eua roots<t'o same roots/ 2 4ac %

c) no rea root/ 2 4ac 5 0#ample

Ae uadratic euation ( 4 x # 7 )4 = ( p # 3-) x as t'o distinctroots/ ;ind te range of vaues of p.

+o' tat te straigt ine y = 4 # 1 does not meet te curve 414 # y4

5 , = - if //

Chapter 3 Function 6uadratic

3/ min < ma1 oints

0#ample

It is given tat te uadratic function f(1) = 2 4 x 4 534 x 24B(a) rite te euation of te a1is of symmetry%(b) +tate te ma1imum < minimum vaue

(c) s,etc te gra for domain - x 7/



Chapter 4

+imutaneous 9uations (sustitution 5 factorise to sove) 4 euation% 4 variabes2 2x + y = x2 + y = 52 find coordinate 4 intersection oint of 4 gra

Chapter *

78C09 A78 ;<ART=M

am # an % am(n am an % amn

"am$n % amn

"$ o>rithm a?"1$ lo>a a % 1 "2$ lo>a 1 %

"3$ lo>a#y % lo>a# ( lo>ay "4$ lo>a y

x

% lo>a# @ lo>ay

"*$ lo>a #n % n lo>a# ",$ lo>a %

a

b

c

c

*og

*og

"-$ lo>a %ab*og

3

0#p:

(a) 615" 26154 = " (b) 4124 E "154 = B3

a) ;ind te vaue of x if log " (4 x # 3) = 3 # log " ( x # 4)

b) +ove 7 --- (2

3) n 3 7-- for te argest ositive integer of n/

c) Given tat og U

P= 7 and og

P= "/ ;ind te vaue of og F P

U /

Chapter ,

(a) Distance bet'een (13% y3) and H(14% y4)

H =4

34

4

34 )()( y y x x −+−

(b) id oint H%

++

4%

4

4343 y y x x

J 'ic divides H in te ratio m : n! : n

" ( x # y ) $ ( x # y ) P1 1 2 2

J(1% y) =

+

+

+

+

!n

!yny

!n

!xnx 4343 %

(d) Kocus (using distance formua

JH =4

3

4

3 )()( y y x x −+−

"e$ Area of poly>on

(f) 0uation of strai>ht line



0#ample

Diagram 6 so's te triange %"$ 'ere % is te origin/ Joint &ies on te straigt ine "$/

(a) Cacuate te area% in unit4% of triange %"$/(b) ;ind te euation of te erendicuar bisector of ine segment

"$/

(c) Given tat te engt $& is4

5 of te distance of oint $

from te erendicuar bisector of te ine segment "$% find tecoordinates of oint & /

(d) oint P moves suc tat its distance from oint $ is a'ayst'ice its distance from oint & / ;ind te euation of te ocus of P /

0#ample

Given te straigt ines y + ax = "and 6 y + bx = 6 are erendicuar toeac oter/ 91ress a in terms of b/

Cater 8 9TAT9TC9

easurement of Centra Aendencyean

n

x x ∑=

;or ungroued data

∑∑=

f

fx x

;or ungroued data 'it freuency/

∑∑= f

fx x i

;or groued data% 1i = mid2oint

Median

Ae data in te centre 'en arranged in order (ascending or descending)/

Formula

= K 5&

f

' n

!

×−

43

K = Ko'er boundary of median cass/n = Aota freuency; = cumuative freuency before te median cass

f m = freuency of median cassC = cass interva si>eHy Ogive

n

n 2

* + , i a n

& - ! - l a . i / + ' 0 + - + n c y

U p p + 0 b o - n , a 0

(c) ode

Date 'it te igest freuencyHy 0istogram :

' 0 + - + n c y

& l a b o - n , a 0 y * o , +

easurement of Disersion

(a) Interuartie Range;ormua :

L3 =&

f

' n 3

4

×−

+3

363

3

(26%4)

H($%2B)



L" =&

f

' n 3

4

×−

+"

"6"

"

;>ive :

)

& - ! - l a . i / + f 0 + 1 - + n c y

U p p + 0 b o - n , a 0 441

5

5

6

( ( n

1

6 ( ( n

Interuartie range = L" # L3

(b) Mariance% +tandard DeviationMariance = (standard deviation)4

σ

=

44

xn

x−∑

;or ungroued data

σ

=

44

x f

fx−

∑∑

;or groued data

c) range and interuartie range0#ample

Aabe 4 so's te number of story boo,s read by a grou of students in a certain scoo/

Number of story boo,s read

- 3 4 "

Number of students 8 & " x

(a) +tate te argest ossibe vaue of x given tat te mode is 3/(b) +tate te argest ossibe vaue of x given tat te median is 3/(c) Cacuate te vaue of x given tat te mean is 3/

Aabe 4 so's te mar,s scored by 3-- students in te dditionaatematics arc onty test/Mar 3-2

3&4-24&

"-2"&

6-26&

7-2&

$-2$&

8-28&

B-2B&

nume

r

$ B 33 38 47 36 34 8

0ffects of uniform chan>es in data on

+k -k ×k ÷k

ean % edian% odeRange% Interuartie

+tandard deviationMarianceAe mean of a set of numbers% 8% 36% 37% a% 4a% 68 and 74% is 48/(a) ;ind te vaue of a and te standard deviation of te set of

numbers/(b) If eac of te numbers in te set is divided by 4% find te

variance of te ne' set of numbers/

Chapter / Circular Measurement

CRCBAR M0A9BR0

a) Radian → Degree

θ r = θ × π

-3B-

b) Degree → Radian

θ o = θ × 3B-

π

radc) Kengt of arc s =r θd) Ae engt of te cord

e) rea of sector K = 4

3

r 4θ = 4

3

rsf) rea of segment

K = 4

3

r 4(θ r # sin θo)

0#ample

Diagram 3- so's a suare "$&7 'it sides 7 cm in engt/ "P& is asector 'it its centre at $ and "$& is a semicirce/

"

$

&

Diagram 3

P

4 8

(a) Cacuate(i) te area of te segment "P& % 4 !a0kP(ii) te erimeter of te saded regions% 4 !a0kP(iii) te area of te saded regions% 4 !a0kP



(b) Given tat $48 is a sector 'it an ange 9 at its centre% $ andte engt of te arc "P is $ cm% find(i) te ange 9 in radians% 3 !a0k P(ii) te engt of te arc 48 if te area of "P48 is 34/$ cm4

" !a0kP

Chapter

8FF0R07TAT;73/ Differentiation by ;irst Jrincie

x y

xha,

,x,y

δ

δ

δ -→=

(a) ,x

,

(a) = - (b) ,x

,

(1n) = n1n23

(c) ,x

,

(a1n) = an1n23

(d) Differentiation of roduct

,x

,

(uv) = u ,x

,/

5 v ,x

,-

(e) Differentiation of Luotient

4

/

-/

/

-

,x

, ,x,/

,x,- −

=

(f) Differentiation of Comosite ;unction

,x

,

(a15b)n = an(a15b)n23 4/ 9uation tangent at J 9uation norma at J

"/ +tationary oint < turning oint→ ,x,y

= -Ma#imum point: Minimum point:

,x

,y

= - and4

4

,x

y,

- ,x

,y

= - and4

4

,x

y,

Q -

Rate of Chan>e y

,.

,x

,x

,y

,.

,y×=

9mall chan>es:

x,x

,y y δ δ /≈

yne' = yinitia 5δy91:

(a) ;ind te euation of te norma to te curve" 44 y x x= − at te

oint (3% 23)/

(b) It is given tat te euation of a curve is4 $ y x x= − /

;ind(i) te turning oint of te curve/ " !a0kP

(ii) te vaue of x if

4

4 B -

, y ,y y x

,x ,x+ + =

(c) Ae radius of a serica baoon is increasing at te rate of x cms23/ Given tat te rate of cange of te voume of te

baoon is 47π c!" 23 'en its radius is 7 cm/ ;ind te vaue

of x. "6

": 0 π =

P



(d) If y = "14 5 71 5 4% find te sma cange in y 'en 1 cangefrom " to "/-4

Chapter 1

9olution Trian>lea) Ao find un,no'n sides

a) se Cosine Rue to find te un,no'n sides or anges of atriange

c)d) mbiguous case (mbiguous means aving more tan one

vaue)

0#ample

Ae diagram so's a triange ;<3%

a) Dra' and abe anoter triange ;3< suc tat ST U = 8cm% SK =

&/7 cm and ∠ 3;< remains fi1ed at "7 o / b) Cacuate te obtuse ange of STK∠

c) ;ind TKT∠

Chapter 11 nde# 7umer

Diagram 37 so's te bar cart for te monty saes of five essentiaitems sod at a sundry so/ Aabe 37 so's teir rice in te year

4--- and 4--$% and te corresonding rice inde1 for te year 4--$ta,ing 4--- as te base year/

(a) ;ind te vaues of (i) x% (ii) y% (iii) z / " !a0kP(b) ;ind te comosite rice inde1 for coo,ing oi% rice% sat% sugar and

four in te year 4--$ based on te year 4---/ 4 !a0kP



(c) Ae tota monty sae for coo,ing oi% rice% sat% sugar and fourin te year 4--- is R 4 7--/ Cacuate te corresondingmonty sae for te same items in te year 4--$/ 4 !a0kP

(d) ;rom te year 4--$ to te year 4--8% te rice of te coo,ingoi% rice and sugar increased by 4V% 'ie te rice of bot sat

and four increased by 7 sen/ ;ind te comosite rice inde1 for ate five items in te year 4--8 ta,ing 4--$ as te base year/ "!a0kP

revision of form 4 add maths

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