prerequisite knowledge required from ‘o’ level add · pdf fileprerequisite...
Post on 17-Mar-2018
223 Views
Preview:
TRANSCRIPT
1
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 ��� = (√�
�)�
2
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!
3
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 −+=+
4
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
5
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.
6
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
7
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
8
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
9
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
10
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
11
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
12
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.
top related