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Singapore GCE A-Level H2 Mathematics: Core Theorems & Vectors

Essential theorems, calculus techniques, vector geometry, complex numbers, and probability distributions for Singapore-Cambridge GCE A-Level H2 Mathematics (9758).

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Condition for Function to have an Inverse (

f^{- 1}

)

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Condition for Function to have an Inverse (

f^{- 1}

)

A function

f

has an inverse

f^{- 1}

if and only if

f

is a one-one (injective) function (i.e., horizontal line test: every horizontal line cuts the graph at most once).

Condition for Composite Function

g \circ f

to Exist

The composite function

g \circ f

exists if and only if the range of

f

is a subset of the domain of

g

R_{f} \subseteq D_{g}

Maclaurin Series Expansion Standard Formula

f (x) = f (0) + f^{'} (0) x + \frac{f ^{''} ( 0 )}{2 !} x^{2} + \frac{f ^{(3)} ( 0 )}{3 !} x^{3} + \dots + \frac{f ^{(n)} ( 0 )}{n !} x^{n} + \dots

Standard Series:

e^{x}

ln (1 + x)

sin x

cos x

e^{x} = 1 + x + \frac{x ^{2}}{2 !} + \dots

;

ln (1 + x) = x - \frac{x ^{2}}{2} + \frac{x ^{3}}{3} - \dots

(for

- 1 < x \leq 1

);

sin x = x - \frac{x ^{3}}{3 !} + \dots

;

cos x = 1 - \frac{x ^{2}}{2 !} + \dots

Vector Dot Product & Angle Formula

a \cdot b = ∣ a ∣∣ b ∣ cos θ = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}

\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|

. Orthogonal iff

a \cdot b = 0

Vector Cross Product Properties

∣ a \times b ∣ = ∣ a ∣∣ b ∣ sin θ

, pointing normal to both by right-hand rule. Area of triangle

= \frac{1}{2} ∣ a \times b ∣

; parallel iff

a \times b = 0

Vector Equation of a Straight Line

r = a + λ d

λ \in R

, where

a

is position vector of a point on line and

d

is direction vector.

Vector Equation of a Plane (Normal Form)

r \cdot n = d

, where

n

is a normal vector to the plane and

d = a \cdot n

for any point

a

lying on the plane.

Shortest Distance from Point

P

to Plane

r \cdot n = d

Distance

= \frac{∣ p \cdot n - d ∣}{∣ n ∣}

, where

p

is the position vector of point

P

Acute Angle Between Line and Plane

If line has direction

d

and plane has normal

n

, the angle

θ

between line and plane satisfies:

sin θ = \frac{∣ d \cdot n ∣}{∣ d ∣∣ n ∣}

Complex Number Polar and Exponential Forms

z = x + i y = r (cos θ + i sin θ) = r e^{i θ}

, where

r = ∣ z ∣ = x^{2} + y^{2}

and

θ = ar g (z) \in (- π, π]

De Moivre's Theorem

[r (cos θ + i sin θ)]^{n} = r^{n} (cos (n θ) + i sin (n θ)) = r^{n} e^{in θ}

for all

n \in Z

(and rational

n

for all roots).

Roots of Unity: Solutions to

z^{n} = 1

z_{k} = e^{i \frac{2 k π}{n}} = cos (\frac{2 k π}{n}) + i sin (\frac{2 k π}{n})

for

k = 0, 1, 2, \dots, n - 1

. Sum of all

n

-th roots equals

0

Conjugate Roots Theorem

For a polynomial

P (z) = 0

with real coefficients, if complex number

w = u + i v

(

v \neq = 0

) is a root, then its complex conjugate

w^{*} = u - i v

is also a root.

Integration by Parts Formula

\int u \frac{d v}{d x} d x = uv - \int v \frac{d u}{d x} d x

. Priority for choosing

u

: LIATE (Logarithmic, Inverse trig, Algebraic, Trig, Exponential).

Standard Derivative of Inverse Trigonometric Functions

\frac{d}{d x} [arcsin (\frac{x}{a})] = \frac{1}{a ^{2} - x ^{2}}

\frac{d}{d x} [arctan (\frac{x}{a})] = \frac{a}{a ^{2} + x ^{2}}

Standard Integrals Leading to Inverse Trig Functions

\int \frac{1}{a ^{2} - x ^{2}} d x = arcsin (\frac{x}{a}) + C

\int \frac{1}{a ^{2} + x ^{2}} d x = \frac{1}{a} arctan (\frac{x}{a}) + C

Separation of Variables for Differential Equations

Given

\frac{d y}{d x} = f (x) g (y)

, rewrite as

\frac{1}{g ( y )} d y = f (x) d x

and integrate both sides:

\int \frac{1}{g ( y )} d y = \int f (x) d x + C

Expectation and Variance of Linear Combinations

E (a X + bY) = a E (X) + b E (Y)

. If

X

and

Y

are independent:

Var (a X + bY) = a^{2} Var (X) + b^{2} Var (Y)

Binomial Distribution Properties:

X \sim B (n, p)

P (X = r) = (r n) p^{r} (1 - p)^{n - r}

E (X) = n p

Var (X) = n p (1 - p)

. Conditions: fixed

n

, binary outcomes, independent trials, constant

p

Poisson Distribution Properties:

X \sim Po (λ)

P (X = r) = \frac{e ^{- λ} λ ^{r}}{r !}

for

r = 0, 1, 2, \dots

E (X) = λ

Var (X) = λ

. Additive property:

X_{1} + X_{2} \sim Po (λ_{1} + λ_{2})

Normal Distribution Standardization:

X \sim N (μ, σ^{2})

Z = \frac{X - μ}{σ} \sim N (0, 1)

. Normal approximation to Binomial:

X \sim B (n, p) \approx N (n p, n p (1 - p))

n p > 5

and

n (1 - p) > 5

with continuity correction.

Central Limit Theorem (CLT)

For any population with mean

μ

and variance

σ^{2}

, sample mean

\overset{ˉ}{X} \approx N (μ, \frac{σ ^{2}}{n})

n \to \infty

(typically

n \geq 30

), regardless of population distribution.

Hypothesis Testing:

p

-value Rule

Reject null hypothesis

H_{0}

at significance level

α

p -value \leq α

. If

p -value > α

, do not reject

H_{0}

(insufficient evidence).

Correlation Coefficient

r

and Regression Lines

- 1 \leq r \leq 1

. Regression line of

y

x

(

y - \overset{y}{ˉ} = b (x - \overset{x}{ˉ})

) is used to predict

y

from

x

. It passes through the centroid

(\overset{x}{ˉ}, \overset{y}{ˉ})

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