Additional Mathematics Notes
Reproduced from http://teach.sg
1
Quadratic Equations & Inequalities
Sum & Product Of Roots
b
Sum of roots: α + β = −
a
c
Product of roots: αβ =
a
Rationalise Denominator
Laws Of Logarithms
For
1. loga xn = n loga x
2. loga xy = loga x + loga y
3. loga x
= loga x − loga y
y
logc b
4. loga b =
logc a
1
5. loga b =
logb a
√
k
√ , multiply numerator and denominator by b.
a b
k
√ , multiply by the conjugate, which is
For √
a
b
+
√
√ c d
a b − c d.
3
Polynomials & Partial Fractions
Polynomial Division
Logarithms To Exponential
P (x) = divisor × Q(x) + R(x)
x = loga y ⇔ y = ax
x = ln y ⇔ y = ex
Quadratic Equation From Roots
Remainder Theorem
x2 − (sum of roots) x + (product of roots) = 0
If P (x) is divided by ax − b, remainder is f
divided by x − c, remainder is f(c).
2, 1 or 0 real roots
2 real roots: b2 − 4ac > 0
1 real root (2 equal roots): b2 − 4ac = 0
0 real roots: b2 − 4ac < 0
Factor Theorem
Curve Always Positive / Negative
a3 + b3 = (a + b)(a2 − ab + b2 )
a3 − b3 = (a − b)(a2 + ab + b2 )
b2
− 4ac < 0 (because curve has 0 real roots)
6
Trigonometric Functions, Identities
& Equations
Special Angles
If x − c is a factor of P (x), f(c) = 0.
Cubic Polynomials
b2
Line intersect curve (at 2 points):
− 4ac > 0
Line tangent to curve: b2 − 4ac = 0
Line does not intersect curve: b2 − 4ac < 0
*Line meets curve: b2 − 4ac ≥ 0
Indices & Surds
Indices
1. am × an = am+n
2. am ÷ an = am−n
3. (am )n = amn
4. a0 = 1 where a 6= 0
5. a−n = a1n
1
√
6. a n = n a
m
√
7. a n = ( n a)m
n
8. (a × b) = an × bn
n
9. ( ab )n = abn
Surds
√
√
1. a × a = a
√
√
√
2. √a × qb = ab
a
a
3. √ =
b
b
√
√
√
4. m a + n a = (m + n) a
√
√
√
5. m a − n a = (m − n) a
c 2015 Eugene Guo Youjun
0◦
θ
Partial Fractions
Line & Curve
2
b
. If P (x) is
a
sin θ
√
0
2
=0
cos θ
√
4
2
=1
30◦
√
1
2
=
4
1. cosec θ =
Binomial Expansions
n
n
(a+b)n = an +
1
an−1 b+
2
an−2 b2 +...+
n
r
an−r br +...+bn
General
Term
Tr+1 =
n n−r r
a
b
r
n choose r
n
n(n − 1)...(n − r + 1)
n!
=
=
r!(n − r)!
r!
r
5
Power, Exponential, Logarithmic &
Modulus Functions
√
√
3
2
1
2
3
2
Reciprocal Functions
Binomial Expansions
60◦
√
A
B
f (x)
1.
=
+
(ax + b)(cx + d)
ax + b
cx + d
A
B
C
f (x)
=
+
+
2.
(ax + b)(cx + d)2
ax + b
cx + d
(cx + d)2
f (x)
A
Bx + C
3.
=
+ 2
(ax + b)(x2 + c)
ax + b
x +c
tan θ
45◦
0
√1
3
1
sin θ
1
2. sec θ =
cos θ
1
3. cot θ =
tan θ
Negative Functions
1. sin(−θ) = − sin θ
2. cos(−θ) = cos θ
3. tan(−θ) = − tan θ
Tangent & Cotangent
1. tan θ =
sin θ
cos θ
For |a| = b ⇒ a = b or a = −b.
2. cot θ =
cos θ
sin θ
Logarithm Definition
Trigonometric Identities
For loga y to be defined,
1. y > 0
2. a > 0, a 6= 1
1. sin2 θ + cos2 θ = 1
2. sec2 θ = 1 + tan2 θ
3. cosec2 θ = 1 + cot2 θ
Modulus Functions
Page 1
For Teach.sg
2
2
√
2
2
1
√
1
2
=
√
3
90◦
√
4
2
1
2
=1
√
0
2
=0
−
Addition Formulae
8
1. sin(A ± B) = sin A cos B ± cos A sin B
2. cos(A ± B) = cos A cos B ∓ sin A sin B
tan A ± tan B
3. tan(A ± B) =
1 ∓ tan A tan B
Differentiation Rules
Double Angle Formulae
1. sin 2A = 2 sin A cos A
2. cos 2A = cos2 A − sin2 A = 2 cos2 A − 1 = 1 − 2 sin2 A
2 tan A
3. tan 2A =
1 − tan2 A
R-Formulae
90◦ ,
For y = f(u) and u = g(x),
dy
du
dy
=
×
dx
du
dx
For a > 0, b >
<α<
1. a sin θ ± b cos θ = R sin(θ ± α)
2. a cos θ ± b sin θ = R cos(θ ∓ α)
√
b
where R = a2 + b2 , tan α = .
a
Further Differentiation Rules (Chain Rule)
Coordinate Geometry
Midpoint
x1 + x2 y1 + y2
,
2
2
m1 = m2
Perpendicular Lines
1
m2
m1 × m2 = −1
m1 = −
A=
=
1
2
d
(ax + b)n = an(ax + b)n−1
dx
d
sin(ax + b) = a cos(ax + b)
dx
d
cos(ax + b) = −a sin(ax + b)
dx
d
tan(ax + b) = a sec2 (ax + b)
dx
d ax+b
e
= aeax+b
dx
a
d
ln (ax + b) = ax+b
dx
Second Derivative Test
1. If
d2 y
< 0, it is a maximum point.
dx2
2. If
d2 y
> 0, it is a minimum point.
dx2
9
Integration
Integration Rules
1.
R
k dx = kx + c
xn+1
+ c, n 6= −1
2. xn dx =
n+1
R
3. sin x dx = − cos x + c
R
4. cos x dx = sin x + c
R
5. sec2 x dx = tan x + c
R
6. ex dx = ex + c
R 1
7. x dx = ln x + c
R
R
Note: kf(x) dx = k × f(x) dx
R
Further Integration Rules
x3
y3
x4
y4
x1
y1
R
(ax + b)n dx =
dv
du
d
(uv) = u
+v
dx
dx
dx
2.
R
sin(ax + b) dx = −
Quotient Rule
3.
R
cos(ax + b) dx =
v du − u dv
d u
= dx 2 dx
dx v
v
4.
R
sec2 (ax + b) dx =
5.
R
eax+b dx =
6.
R
1
ax+b
From y = f(x),
dy
is the gradient of the curve.
dx
1
|(x1 y2 + x2 y3 + x3 y4 + x4 y1 ) − (x2 y1 + x3 y2 + x4 y3 + x1 y4 )|
2
Note: coordinates should be in anti-clockwise direction
Circle
(x − a)2 + (y − b)2 = r2
(a, b): centre of circle
r: radius
Circle (Second Formula)
x2 + y 2 + 2gx + 2f y + c = 0
(−g, −f ): centre of circle
p
f 2 + g 2 − c: radius
c 2015 Eugene Guo Youjun
(ax + b)n+1
+ c, n 6= −1
a(n + 1)
1.
Gradient Of Curve
Area Of Quadrilateral
x2
y2
1.
2.
3.
4.
5.
6.
Product Rule
Parallel Lines
x
1
y1
d
c=0
1. dx
d n
2. dx
x = nxn−1
d
3. dx sin x = cos x
d
4. dx
cos x = − sin x
d
5. dx tan x = sec2 x
d x
6. dx
e = ex
d
ln x = x1
7. dx
d
d
Note: dx
kf(x) = k × dx
f(x)
Chain Rule
0, 0◦
7
Differentiation
Gradient Of Tangent & Normal
dy
,
dx
substitute x = k to get m (gradient of tangent).
1
−
is the gradient of the normal.
m
From
Increasing & Decreasing Functions
dy
> 0.
dx
dy
2. For decreasing functions,
< 0.
dx
1. For increasing functions,
Rates Of Change
dy
dy
If
is the rate of change of y with respect to x, and
and
dx
dt
dx
are the rates of change of y and x,
dt
dy
dy
dx
∴
=
×
dt
dt
dt
Page 2
dx =
cos(ax + b)
+c
a
sin(ax + b)
+c
a
tan(ax + b)
+c
a
eax+b
+c
a
ln(ax + b)
+c
a
Definite Integral
For
Z b
R
f(x) dx = F(x) + c,
f(x) dx = F(b) − F(a).
a
Kinematics
1. v =
ds
dt
dv
dt
Z
3. s =
v dt
2. a =
Z
4. v =
a dt
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