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Prerequisite Knowledge Required from ‘O’ Level Add Math

1) Surds, Indices & Logarithms

Rules for Surds

1. a b ab× = √� × √� = √��

2. a a

bb=

3. aaaa ==× 2)(

4. x ya a= ⇒ x y=

5. ndcnba +=+ ⇒ a = c and b = d

Caution: √� + � ≠ √� + √�, √� − � ≠ √� - √�

Rationalising the Denominator (two types)

(a) E.g. 3 3 5 3 5

55 5 5= × =

(b) E.g. 2 2(5 2) 2(1 2) 2(1 2)

(5 2) 3(5 2) (5 2)(5 2)

+ + += = =

−− − +

Rules for Indices

Same base �� × � =��

�� ÷ � =���

Power (��)n = ��×

Same index � × �= (��)

� ÷ � =( �

�)n

Zero index �� = 1

Negative index �� = �

�

( �

�) - n = (

�

�)n

Fractional index ��� = (√�

�)�

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Rules for Logarithms

• For log� � to be defined, both a and x must be positive.

• Logarithm log�� � is written as lg x

• Logarithm log� � is written as ln x

Product of Logarithm log log loga a axy x y= +

Quotient of Logarithm log log loga a a

xx y

y= −

Power Law log logr

a ax r x=

Change of Base loglog

log

ca

c

bb

a=

Log of 1 to any base log 1 0a =

Log of a number to the same base as number log 1a a =

Exponential Graphs

The graphs of y = abx are shown below for different range of values of a & b.

NOTE: MUST DRAW THE ASYMPTOTES FOR THESE CASES!

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Logarithmic Graphs

The graphs of y = a ln (bx + c) are shown below for different range of values of a & b.

2) Quadratic Equations

Quadratic Equation: ax2 + bx + c = 0

Recall Quadratic Formula: x = ��±�� �!�"

#�

a) For roots α and β

• Sum of roots = α + β = a

b−

• Product of roots = αβ = a

c

• Equation is: α2 – (sum of roots)x + (product of roots) = 0

OR: 0))(( =−− βα xx

b) Useful:

αββαβα 2)( 222 −+=+

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Discriminant & type of roots

D = b2 – 4ac Coefficient of x

2 > 0.

y has a min. value

Coefficient of x2 < 0.

y has a max. value

Real & distinct

roots

b2 – 4ac> 0

Real & repeated

(equal) roots

b2 – 4ac= 0

No real roots

b2 – 4ac< 0

Question types

• Line does not intersect curve

• Curve lies entirely above x-axis (min. graph) e.g. y > 0 for all values of x, x2 – 5x

+3 > 0 always

• Curve lies entirely below x-axis (max. graph) e.g. y < 0 for all values of x, -5x2 – 5x

+3 < 0 always

Real roots

b2 – 4ac ≥ 0

For real roots, it can be distinct or equal roots

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Polynomials & Partial Fraction

Recall how long division is done.

Factor Theorem

A polynomial f(x) is divisible by (x – a) => remainder = 0 => (x – a) is a factor of f(x).

Partial Fractions

Proper rational algebraic expression, $(&)

'(&) is when degree of numerator f(x) < degree of denominator, g(x).

Note: To write as partial fractions, first check that the expression is proper.

dcx

B

bax

A

dcxbax

nmx

++

+=

+++

))((

22

2

)())(( dcx

C

dcx

B

bax

A

dcxbax

rqxpx

++

++

+=

++

++

2222

2

))(( cx

cBx

bax

A

cxbax

rqxpx

+

++

+=

++

++

3) Modulus Function

The absolute or modulus of a real number x is denoted by |x| and defined as:

The above is useful when solving modulus equation. Alternatively, you can square both sides to remove the

modulus sign. Refer to property 6 below.

Properties of modulus:

1. |a| ≥ 0

2. |- a| = |a|

3. |ab| = |a||b|

4. |�

�| =

|�|

|�|

5. |an| = |a|

n

6. |a | =|b| implies that a2 = b

2

Note: modulus is always non-negative.

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4) Coordinate Geometry

1. For 2 points A(x1, y1) and B(x2, y2),

• Length 2

122

12 )()( yyxxAB −+−=

• Mid-point of

++=

2,

2

2121 yyxxAB

• Gradient of 2 1

2 1

tany y

ABx x

θ−

= =−

2. Line L1 has gradient m1. Line L2 has gradient m2.

• 1L is parallel to L2 ⇒ 1 2.m km=

• 1L is perpendicular to L2 ⇒

.121 −=×mm

y

x

1 1( , )A x y

2 2( , )B x y

θ

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3. Areas of triangle:

Area of ∆ABC direction) clockwise-(anti 12

1

321

1321

yyyy

xxxx=

)(2

1122331133221 yxyxyxyxyxyx −−−++=

OR

1Area of ( )( )sin

2ABC AB AC θ∆ =

• Equation of straight line: y = mx + c, where m = gradient, c = y-intercept

• Equation of a horizontal line: y = a, where a is a constant.

• Equation of a vertical line: x = a, where a is a constant.

• An x-intercept is a point where the graph cuts the x-axis. It is found by letting y = 0.

• An y-intercept is a point where the graph cuts the y-axis. It is found by letting x = 0.

Sine Rule

sin sin

BC AB

θ φ=

Cosine Rule

2 2 2 2( )( ) cosBC AB AC AB AC θ= + −

5) Linear Law

Re-write Non-Linear equations into Linear equations of form y = mx + c

Non-linear equations Linear Equations Y-axis X-axis Gradient,

m

Y-intercept,

C

1 y = axm

ln y = m ln x + ln a ln y ln x m ln a

2 y = abx ln y = x ln b + ln a ln y x ln b ln a

θ

φ

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6) Trigometric Function

Trigonometric Ratio of Special Angles

30° 45° 60°

sin √12=12

√22

√32

cos √32

√22

√12=12

tan 1

√3=√33

1 √31

Note:

• For ease of remembering, observe that value of sin increases from √�

# to

√#

# to

√,

#.

• Always write the value of the special angles in terms of the surd form instead of 0.866 for √,

# or 0.707 for

√#

#

Complementary Angles

sin (90° - θ) = cos θ cos (90° - θ) = sin θ

tan (90° - θ) = cot θ cot (90° - θ) = tan θ

sec(90° - θ)=�

-./(0�°�2)

=�

3452

= cosecθ

cosec(90° - θ)=�

/67(0�°�2)

=�

"832

= secθ

Negative Angles

A positive angle is an anti-clockwise rotation from the positive x-axis about the origin.

A negative angle is a clockwise rotation from the positive x-axis about the origin.

For any angle θ,

cos (-θ) = cos θ

sin (-θ) = - sin θ

tan (-θ) = - tan θ

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Small Angle Approximation

Note:

From the power series expansions of sin x, cos x and tan x respectively, when x is small and measured in

radians,

• sin x ≈ x

• 2

cos 12

xx ≈ −

• tan x ≈ x.

a. Quotient Relationships

• θθ

θcos

sintan =

• θθ

θsin

coscot =

• 1

seccos

θθ

=

• 1

cossin

ecθθ

=

b. Pythagoras’ Trigonometric Identities

• 1cossin 22 =+ θθ

• θθ 22 sec1tan =+

• θθ 22 coscot1 ec=+

c. R-Formula

For a > 0, b > 0 & acute angle α,

a cos θ ± b sin θ R cos (θ∓ α )

a sin θ ± b cos θ R sin (θ ± α )

where R = √�2 + �2 , tan α =��

The following are provided in the MF15:

- Addition Formula

- Double Angle Formula

- Factor Formula

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d. Graphs of sin, cos and tan functions

y = sin x y = cos x y = tan x

Amplitude = 1 Amplitude = 1 Amplitude undefined

Period = 360º (2 ) Period = 360º (2 ) Period = 180º ( )

7) Tangent and Normal

For a curve y = f(x),

(i) Gradient of tangent = )('or xfdx

dy

(ii) Gradient of tangent at (a, b) = ).('or afdx

dy

ax=

(iii) Equation of tangent at (a, b): ))((' axafby −=− .

(iv) Equation of normal at (a, b): ).()('

1ax

afby −−=−

8) Properties of Circle

Symmetrical Properties of Circles

A line through the centre of circle and

perpendicular to chord will bisect the chord.

If OMQ = 90º then MP = MQ or vice versa

Note: This property is true for all isosceles or

equilateral triangle.

Equal chords are equidistant from the centre.

If PQ = RS then OX = OY or vice versa

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A tangent to a circle is perpendicular to the

radius.

Tangents from an external point to a circle are

equal and subtend equal angles at the centre.

OT bisects POQ & PTQ

i.e. TP = TQ, POT = QOT &

PTO = QTO

Angle Properties of Circles

Angle at Centre = Twice Angle at

circumference

POR = 2 × PQR

Note: refer to three different diagrams

Angle in semi-circle

PQR = 90º

Note: PR is diameter of circle

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Angles in the same segment

PQR = PRS

QPR = QSR

Note: angles at the circumference are

subtended by the same arc

Opp. Angles in cyclic quad

SRQ + SPQ = 180º

RSP + RQP = 180º

Note: all 4 sides must touch the

circumference.