Upper Secondary
Additional Mathematics

Learn to solve quadratic equations using factorisation, completing the square, and the quadratic formula, and analyse graphs of parabolas.

Solve linear and quadratic equations and inequalities. Interpret solutions on number lines and understand how the discriminant affects the nature of quadratic roots.

Simplify and manipulate surds using standard rules. Learn how to rationalise denominators and work with conjugate surds to find exact values.

Factorise polynomials using the Remainder and Factor Theorems. Perform polynomial division and decompose rational expressions into partial fractions.

Expand powers of binomials using the Binomial Theorem. Find specific terms and apply combination formulas (nCr) to evaluate expressions efficiently.

Solve exponential and logarithmic equations using laws of indices and logarithms. Understand the relationship between logs and exponents, and sketch graphs with key features.

Understand how to calculate gradients, midpoints, and distances between points on a plane. Learn to derive equations of lines and circles, and apply geometric principles to prove collinearity, perpendicularity, and parallelism.

Formulas

Gradient Formula

$m=\dfrac{y_2-y_1}{x_2-x_1}$

Gradient of horizontal line is $0$Gradient of vertical line is undefined

Equation of straight line

$y-y_1=m(x-x_1)$

Parallel Lines Gradient

$m_1=m_2$

if line $l_1$ is parallel to line $l_2$

Perpendicular Lines Gradient

$m_1=-\dfrac{1}{m_2}$

if line $l_1$ is perpendicular to line $l_2$

Area of Rectilinear figure

$$\text{Area of } n\text{-sided polygon} = \frac{1}{2} \left| \begin{array}{ccccc} x_1 & x_2 & \cdots & x_n & x_1 \\ y_1 & y_2 & \cdots & y_n & y_1 \end{array} \right|$$

General formula for a figure with n sides: $\\(x_1y_2+x_2y_3+\ldots+x_ny_1)-(y_1x_2+y_2x_3+\ldots+y_nx_1)$

Length of AB

$AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

Midpoint of a line

$\left( \dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2} \right)$

Collinear points

$m_{AB}=m_{AC}=m_{BC}$

$A, B$ and $C$ are collinear if they lie on the same line

Angle to $\bm{x}$-axis

$m=-\dfrac{y_2-y_1}{x_2-x_1}=\tan \theta$

Equation of Circle

Standard form:$(x-a)^2+(y-b)^2=r^2$Centre of circle: $(a,b)$Radius of circle: $r$

Convert non-linear equations into linear form to simplify data analysis and graphing. Learn to plot relationships involving exponential, logarithmic, and power functions on straight line graphs.

Master the graphs of sine, cosine, and tangent functions. Identify amplitude, period, asymptotes, and interpret transformations of trigonometric graphs in real-life and mathematical contexts.

Formulas

$\bm{y=a\sin bx, a > 0}$

Range: $-a \leq a\sin bx \leq a$Period $=\dfrac{360^{\circ}}{b}$Amplitude $=|a|$Asymptotes: none

$\bm{y=a \cos bx, a > 0}$

Range: $-a \leq a\cos bx \leq a$Period $=\dfrac{360^{\circ}}{b}$Amplitude $=|a|$Asymptotes: none

$\bm{y=a\tan bx, a > 0}$

Range: any real numberPeriod $=\dfrac{180^{\circ}}{b}$Amplitude: undefinedAsymptotes: $x=\dfrac{90^{\circ}}{b}, x=\dfrac{270^{\circ}}{b}$

Trigonometric Ratios

TOA CAH SOH
$\sin \theta = \dfrac{\text{opposite}}{\text{hypotenuse}}$
$\cos \theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}$
$\tan \theta = \dfrac{\text{opposite}}{\text{adjacent}}$

Special Angles

$\theta$	$30^{\circ} / \frac{\pi}{6}$	$45^{\circ} / \frac{\pi}{4}$	$60^{\circ} / \frac{\pi}{3}$
$\sin \theta$	$\dfrac{\sqrt{1}}{2}$	$\dfrac{\sqrt{2}}{2}$	$\dfrac{\sqrt{3}}{2}$
$\cos \theta$	$\dfrac{\sqrt{3}}{2}$	$\dfrac{\sqrt{2}}{2}$	$\dfrac{\sqrt{1}}{2}$
$\tan \theta$	$\dfrac{\sqrt{3}}{3}$	$1$	$\sqrt{3}$

Negative Angles

Identities
$\sin (-\theta) = -\sin \theta$
$\cos (-\theta) = \cos \theta$
$\tan (-\theta) = -\tan \theta$

Complementary Angles

Identities
$\sin (90^{\circ}-\theta) = \cos \theta$
$\cos (90^{\circ}-\theta) = \sin \theta$
$\tan (90^{\circ}-\theta) = \dfrac{1}{\tan \theta}$

Reciprocals

Identities
$\cosec \theta = \dfrac{1}{\sin \theta}, \sin \theta \neq 0$
$\sec \theta = \dfrac{1}{\cos \theta}, \cos \theta \neq 0$
$\cot \theta = \dfrac{1}{\tan \theta}, \tan \theta \neq 0$

Solve trigonometric equations using identities such as Pythagorean, reciprocal, and addition formulae. Understand inverse trigonometric functions and apply R-formulae to simplify and solve expressions.

Strengthen your deductive reasoning by using congruence and similarity tests (SSS, SAS, AAS, AA) and properties of polygons and circles. Prove geometric relationships involving triangles, parallelograms, cyclic quadrilaterals, and more.

Formulas

Proofs using Properties of Triangles

Standard form:$(x-a)^2+(y-b)^2=r^2$Centre of circle: $(a,b)$Radius of circle: $r$

Proof of Equilateral Traingle

3 equal sides: $AB = BC = AC$3 equal angles: $\angle ABC = \angle ACB = \angle BAC$

Proof of Isoceles Traingle

2 equal sides: $AB = BC$2 equal angles: $\angle BAC = \angle ACB$

Perpendicular Bisector

Perpendicular bisector is a line that separates a line in two equal parts and is perpendicular to that line.$AE = BE$$AB \perp CD$

Angle Bisector

Angle bisector is a line that divides an angle into two equal angles.$DE$ is an angle bisector of $\angle BAC$$\angle BAE = \angle CAE$

SSS Congruency Test

$AB = DF$$AC = EF$$BC = DE$

SAS Congruency Test

$AB = DF$$AC = EF$$\angle BAC = \angle DFE$

AAS Congruency Test

$AB = DF$$\angle ABC = \angle FDE$$\angle BAC = \angle DFE$

AA Similarity Test

$\angle CAB = \angle FDE$$\angle ABC = \angle DEF$

SSS Similarity Test

$\dfrac{AB}{DE}=\dfrac{BC}{EF}=\dfrac{AC}{DF}=k$

SAS Similarity Test

$\dfrac{BC}{EF}=\dfrac{AC}{DF}=k$$\angle ACB = \angle DFE$

Midpoint Theorem

If $D$ is the midpoint of $AC$ and $E$ is the midpoint of $BC$,$AB // DE$$AB = 2DE$

Parallelogram

2 pairs of equal and parallel opposite sides2 pairs of equal and opposite anglesdiagonals bisect each other

Rectangle

2 pairs of equal and parallel opposite sides4 right anglesdiagonals are equal in length and bisect each other

Rhombus

4 equal sidesopposite sides are parallel2 pairs of equal opposite anglesdiagonals bisect each other at right angles

Square

4 equal sidesopposite sides are parallel4 right anglesdiagonals are equal in length and bisect each other at right angles

Kite

2 pairs of equal adjacent sidesopposite angles are equalone diagonal bisects the other diagonal at right anglesone diagonal bisects a pair of opposite angles

Perpendicular Bisector of a Chord

$OC \perp AB$$AC = CB$

Equal Chord

$AB = CD$$OX = OY$

Tangents from an external point are equal

Tangent:Tangent to a circle at any point is perpendicular to the radius drawn to that point.Tangent from an external point:The lengths of tangents drawn from an external point to a circle are equal $(AC = BC)$$\angle OCA = \angle OCB$

Angle at Centre $= 2 \times$ Angle at Cicumference

The angle subtended at the centre of the circle is twice the angle subtended by the same arc at any point of the circumference.$\angle AOB = 2\times \angle ACB$

Right angle in a semicircle

angle subtended by the diameter and a point on the circumference is always $90^{\circ}$.

Angles in the same Segment

Angles in the same segment are angles at the semicircle subtended by the same arc.$\angle ACB = \angle ADB = \theta$

Angles in the opposite Segments

A cyclic quadrilateral is one where all 4 of its vertices lie on the circumference of the circle.$\angle ABC + \angle BCD + \angle CDA + \angle DAB = 360^{\circ}$Opposite angles of a cyclic quadrilateral add up to $180^{\circ}$$\angle ABC + \angle CDA = 180^{\circ}$$\angle DAB + \angle BCD = 180^{\circ}$

Learn to find derivatives using key rules and apply them to study gradients, curves, and function behavior.

Use derivatives to solve rate of change problems, find turning points, and analyse motion through graphs.

Understand integration as the reverse of differentiation and apply it to solve equations and evaluate areas.

Apply definite and indefinite integrals to calculate area under curves and between lines for real-world contexts.

Formulas

Indefinite integral of the function $f(x)$

$\int f(x)dx = F(x) + c$

where $c$ is a constant

Properties of Definite Integrals

Definite Integrals
$\int_a^b f(x) dx = F(b) - F(a)$
$\int_a^a f(x) dx = 0$
$\int_a^b f(x) dx = -\int_b^a f(x) dx$
$\int_a^b kf(x) dx = k\int_a^b f(x) dx$
$\int_a^c f(x) dx = \int_a^b f(x) dx + \int_b^c f(x) dx$
$\int_a^b [f(x) \pm g(x)] dx = \int_a^b f(x)dx \pm \int_a^b g(x)dx$

Area between $\bm{y=f(x)}$ and $\bm{x}$-axis

Area = $|\int_a^b f(x) dx|$

Area between $\bm{x=f(y)}$ and $\bm{y}$-axis

Area = $|\int_c^d f(y) dy|$

Area between $\bm{y=f(x)}$ and $\bm{y=g(x)}$

Curve $y=f(x)$ lies above the straight line $y=g(x)$,Area = $\int_a^b [f(x)-g(x)]dx$

Area between $\bm{y=f(x)}$ and $\bm{y=g(x)}$

Curve $y=f(x)$ lies below the straight line $y=g(x)$,Area = $\int_a^b [g(x)-f(x)]dx$

Use calculus to relate displacement, velocity, and acceleration, and interpret motion in physical problems.

Formulas

Displacement

$s = \int v dt$

If $s>0$, the particle is along the positive $s$-axis.If $s<0$, the particle is along the negative $s$-axis.If $s=0$, the particle is at a fixed point.

Velocity

$v = \dfrac{ds}{dt} \\ v = \int a dt$

If $v>0$, the particle is moving in the direction of the positive $s$-axis.If $v<0$, the particle is moving in the direction of the negative $s$-axis.If $v=0$, the particle is at instantaneous rest.

Acceleration

$a = \dfrac{dv}{dt}$

If $a>0$, the velocity of the particle is increasing with respect to time.If $a<0$, the velocity of the particle is decreasing with respect to time.If $a=0$, the particle moves at constant velocity.

Kinematics Summary

Initial $\Rightarrow t = 0$Instantaneously at rest $\Rightarrow v = 0$At the fixed point $\Rightarrow s = 0$Maximum or minimum displacement $\Rightarrow \frac{ds}{dt} = 0$Maximum or minimum velocity $\Rightarrow \frac{dv}{dt} = 0$