Maths PMR Form 3 Note

NOTES AND FORMULAE PMR MATHEMATICS

1.

Sphere

SOLID GEOMETRY

(a) (a)

Area Area and and per perim imet eter  er  Triangle 1

 A = 2

 = V  =

× base × height

 = V  =

Trapezium 1

 A = 2

=



× base area ×

 = Area of cross section V  =

× length

(a + b) ×   h 2.

Area = πr 2 Circumference = 2πr 

CIRCLE THEOREM

Angle at the centre = 2 * angle at the circumference  x = 2y

Sector  Area of sector =

θ   !"

× πr 2

#ength of arc = !"

1

rism

(sum of two

Circle

θ  

πr 

height

 parallel sides) × height 1 2



$ramid

1 bh 2

=

&

× 2πr 

C$linder  Cur%e surface area = 2πrh

Angles in the same segment are e+ual  x = y

Angle in a semicircle

∠ ACB = ,"o

Sphere Cur%e surface area = &πr 2 (b)

a + b = 1-"o

Soli Solidd and and 'olume ume Cube ' = x × x × x = x

Cuboid ' = l × b × h = lbh C$linder  V  =  = π r 2h

Cone  = V  =

Note prepared by Mr. Sim KY

1 

Sum of opposite angles of a c$clic +uadrilateral = 1-"o

π r 2h

The eterior angle of a c$clic +uadrilateral is e+ual to the interior opposite angle. b=a

Angle between a tangent and a radius = ,"o

∠OPQ = ,"o The angle between a tangent and a chord is e+ual to the angle in the alternate segment.  x = y

/f  PT  and PS  are tangents to a circle0

(a)  xm × x n = xm 6 n

 PT = PS  ∠TPO = ∠SPO ∠TOP  = ∠SOP 

(b)  xm ÷ xn = xm  n (c)

( xm)n = x m × n

1

(d)  x-n =

n

 x

. (a)

POLYGON

1

The sum of the interior angles of a n sided pol$gon = (n  2) × 1-"o

(e)

 x n

(b)

Sum of eterior angles of a pol$gon = !"o

(f)

 x n

(c)

ach eterior angle of a regular n sided pol$gon = !" "

(g)  x0 = 1

(d)

3egular pentagon

4.

n

n

=

 x

m

=

(

n

 x )

m

ALGEBRAIC FRACTION 1 1" − k 

press

2k 

−

2

!k 

as a fraction in its simplest

form. Solution 1

ach eterior angle = 42o ach interior angle = 1"-o (e)

2k 

=

1" − k 2

!k

2

1× k − (1" − k )

=

2

!k  &k − 1" !k

2

=

−

!k

2

5)

k =

−

k

5 

2

 

1 (n 6 2) = n  20 calculate the %alue 5

ach eterior angle = !"o

of n. Solution

3egular octagon

1 (n 6 2) = n  2 5 1 5 * (n 6 2) = 5(n  2) 5

ach interior angle = 12"o

2(k

LINEAR EQUATION

8i%en that

(f)

=

!k k − 1" + k

3egular heagon -.

−

n 6 2 = 5n  1" 2 6 1" = 5n  n

2n = 12

n=!

o

ach eterior angle = &5 ach interior angle = 15o &.

,.

(a)

Substitution 9ethod $ = 2  5 ::::::::(1) 2 6 $ = 4 ::::::::(2) Substitute (1) into (2) 2 6 2  5 = 4 & = 12 = Substitute  =  into (1)0 $ = ! 5= 1 (b) limination 9ethod Sol%e  6 2$ = 5 ::::::::::(1)    2$ = 4 ::::::::::(2) (1) 6 (2)0 & = 120 = Substitute into (1) , 6 2$ = 5 2$ = 5  , = 7& $ = 72

FACTORISATION

(a)  xy + xz = x(y + z) (b)  x2 – y2 = (x – y)(x + y) (c)  xy + xz + ay + az = x (y + z) + a (y + z) = (y + z)(x + a)

(d)  x2 6 & x 6  = ( x 6 )( x 6 1) 5.

EXPANSION OF ALGERBRAIC EXPRESSIONS

(a) 22  ! 6    = 2 2  5 7  (b) (c) !.

( 6 )2 = 2 6 2 *  *  6 2 = 2 6 ! 6 , (  $)( 6 $) = 2 6 $  $  $2 = 2  $2

LAW OF INDICES

Note prepared by Mr. Sim KY

SIMULTANEOUS LINEAR EQUATIONS

1".

ALGEBRAIC FORMULAE

8i%en that ;  (m 6 2) = m0 epress m in terms of ;. Solution ;  (m 6 2) = m ;  m  2 = m ;  2 = m 6 m = &m m=

k  −

&

2

11.

LINEAR INEQUALITIES

1.

2.

.

Sol%e the linear ine+ualit$   2 < 1". Solution   2 < 1"  < 1" 6 2  < 12 <& #ist all integer %alues of  which satisf$ the linear ine+ualit$ 1   6 2 > & Solution 162>& Subtract 20 172622>&2 71   > 2 ∴  = 710 "0 1 Sol%e the simultaneous linear ine+ualities

A pie chart showing the fa%ourite drin;s of a group of students. 1.

TRIGONOMETRY

TBA SB CA

1 &p    p and p 6 2 ≥  p 2

Solution &p    p  p  1  p 6 2 ≥

12.

C 

&p  p  

1  p 2

p  

sin θ  =  

* 20 2p 6 & ≥ p

side

 A

G

2

9ean =

+ + &+ !+ 5

=

cos E tan E

1 

1&.

1

2

2



=

adIacent side opposite side h$potenuse adIacent side h$potenuse

=

=

!"F 2 2

2

 y =   x 2

G 2

1

H



GRAPHS OF FUNCTIONS

(i) #inear function.

 y = x

&.-

9ode =  9edian = & &0 50 !0 -0 ,0 1"0 there is no middle number0 the median is the mean of the two middle numbers. 9edian =



opposite side

&5F

1

sum of data

has fre+uenc$. 9ode is the data with the highest fre+uenc$ 9edian is the middle data which is arranged in ascending?descending order. 1. 0 0 &0 !0 -

.

adIacent "F side

sin E

number of data sum of(fre+uenc$ × data) 9ean = 0 when the data sum of fre+uenc$

2.

 cos θ  =

θ   B E

STATISTICS

2.

h$potenuse

opposite

2p  p ≥  7& p ≥ 7& ∴ The solution is 7&  p  1. 9ean =

  tan θ  =

(ii) Duadratic function.

!+=4 2

 y = x

A pictograp uses s$mbols to represent a set of data. ach s$mbol is used to represent certain fre+uenc$ of the data. @anuar$ ebruar$ 9arch 3epresents 5" boo;s

 y =  x2

2

(iii) Cubic function.  y =  x3

 y = x3

A !ar cart uses horizontal or %ertical bars to represent a set of data. The length or the height of each bar represents the fre+uenc$ of each data.

(i%)

3eciprocal  y  y =

O

a

 y =

 x

 y  x

O

&.

A pi" cart uses the sectors of a circle to represent the fre+uenc$?+uantiti$ of data.

Note prepared by Mr. Sim KY

15.

GEOMETRICAL CONSTRUCTIONS

:

a  x

 x

1!.

SCALE DRAWINGS

Scale of a drawing = 14.

The length of drawing The length of the actual obIect

LINES AND ANGLES

 x   !

 y  x = y

  = ! a b

 x + y = "#0$

1-.

COORDINATES

1. Jistance

=

(  2 − 1 ) 2 − ( $ 2 − $1 ) 2

 x1 + x2 , y1 + y2     2     2

2. 9idpoint0 ( x, y) =  1,.

TRANSFORMATIONS

1. Translation

 x   y÷   

2. 3eflection . 3otation (i) centre of rotation (ii) angle of rotation (iii) direction of rotation : eample ,"" cloc;wise ? ,"" anticloc;wise &. nlargement

(i) centre of enlargement (ii) scale factor k  =

length of  a side of  image length of  the corresponding side of  obIect k 2 =

area of image area of obIect

Note prepared by Mr. Sim KY