Additional Mathematics Formulas

Section One

Algebra, Sequences & Series

The foundation of CSEC Additional Mathematics. Master the algebraic toolkit — quadratics, polynomials, indices, logarithms, sequences — and the rest of the course follows naturally.

Quadratics

A quadratic takes the form ax² + bx + c = 0 where a ≠ 0. You will see it everywhere: algebra, parabolas, motion problems, and inside trigonometric equations.

Quadratic Formula

Use when the quadratic does not factorise cleanly, or when exact roots are required.

x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Discriminant

D > 0: two distinct real roots. D = 0: one real repeated root. D < 0: no real roots.

D = b^2 - 4ac

Sum & Product of Roots

For problems involving symmetric facts about the roots — sums, products, reciprocals.

\alpha + \beta = -\dfrac{b}{a}\,,\qquad \alpha\beta = \dfrac{c}{a}

Build a Quadratic from its Roots

The middle coefficient is the negative of the sum. A small sign — a frequent slip.

x^2 - (\alpha + \beta)\,x + \alpha\beta = 0

Completed Square Form

Vertex at (-h, k) — the minimum if a > 0, the maximum if a < 0. Axis of symmetry: x = -h.

f(x) = a(x + h)^2 + k\,,\quad h = \dfrac{b}{2a}\,,\quad k = c - \dfrac{b^2}{4a}

Polynomial Division, Factor & Remainder Theorems

Two theorems that let you handle cubics and quartics without grinding through long division every time.

Factor Theorem

To test whether a value is a root of a polynomial. Substitute and check for zero.

f(a) = 0 \;\;\Longrightarrow\;\; (x - a) \text{ is a factor of } f(x)

Remainder Theorem

For divisor (ax + b), evaluate at f\!\left(-\dfrac{b}{a}\right).

f(a) = r \;\;\Longrightarrow\;\; r \text{ is the remainder of } f(x) \div (x - a)

Long Division Layout

Dividend = Divisor × Quotient + Remainder. The cycle is Divide → Multiply → Subtract → Bring down, repeated until the remainder has lower degree than the divisor.

Equations Reducible to a Quadratic

Many high-degree or surd equations collapse to a clean quadratic after one good substitution.

The Substitution

Trigger: you see the shape (f(x))² + b f(x) + c = 0. Solve for u, then back-substitute for x.

\text{Let } u = f(x) \;\;\Longrightarrow\;\; au^2 + bu + c = 0

Example — Quartic

\begin{aligned}x^4 - 6x^2 + 8 &= 0 \quad\text{let } u = x^2 \\ u^2 - 6u + 8 &= 0 \\ (u-2)(u-4) &= 0 \\ x^2 = 2 \;\text{or}\; x^2 &= 4 \\ x = \pm\sqrt{2} \;\text{or}\; x &= \pm 2\end{aligned}

Example — Surd

\begin{aligned}x - 2\sqrt{x} + 1 &= 0 \quad\text{let } u = \sqrt{x} \\ u^2 - 2u + 1 &= 0 \\ (u-1)^2 &= 0 \\ u &= 1 \\ x &= 1\end{aligned}

Simultaneous Equations (Linear + Non-Linear)

Method

From the linear equation, express y in terms of x (or vice versa).
Substitute into the non-linear equation.
Solve the resulting quadratic.
Back-substitute each x-value into the linear equation to recover y.

Discriminant of the resulting quadratic

D > 0 → two intersection points · D = 0 → tangent (one contact) · D < 0 → no intersection.

Inequalities

Quadratic Inequality

Rearrange to zero, factorise and find roots α < β, sketch or use a sign diagram. For a > 0: f(x) < 0 ⇒ α < x < β; f(x) > 0 ⇒ x < α or x > β. For a < 0, the regions flip.

ax^2 + bx + c > 0 \quad\text{or}\quad ax^2 + bx + c < 0

Rational Inequality

Find critical values from numerator = 0 and denominator = 0, place on a number line, test signs in each region. The value making the denominator zero is always excluded.

\dfrac{ax + b}{cx + d} > 0 \;\;(\text{or } \geq,\, <,\, \leq)

Common pitfall

Never multiply both sides by (cx + d) — its sign is unknown. Always use a sign diagram instead.

Set-builder notation

{x : 2 < x < 3} reads “the set of all x such that x is between 2 and 3.” For two separate regions: {x : x < -1 or x > 4}.

Surds

A surd is an irrational root left in exact form. CSEC expects clean manipulation and rationalised denominators in the final answer.

Multiplication

\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}

Division

\dfrac{\sqrt{a}}{\sqrt{b}} = \sqrt{\dfrac{a}{b}}

Simple Rationalisation

\dfrac{1}{\sqrt{a}} = \dfrac{\sqrt{a}}{a}

Rationalising a Binomial Denominator — by the Conjugate

The conjugate of a + √b is a - √b. Their product is a² - b — the surd disappears.

\begin{aligned} \dfrac{1}{a + \sqrt{b}} \;&=\; \dfrac{1}{a + \sqrt{b}} \times \dfrac{a - \sqrt{b}}{a - \sqrt{b}} \\[4pt] &=\; \dfrac{a - \sqrt{b}}{(a + \sqrt{b})(a - \sqrt{b})} \\[4pt] &=\; \dfrac{a - \sqrt{b}}{a^2 - b} \end{aligned}

Mirror Image

For a denominator of a - √b, multiply by (a + √b)/(a + √b). Same idea — flip the middle sign and the surd vanishes.

Indices — Laws of Exponents

Seven rules. Internalise them once; they carry you through every exponential and logarithmic equation in the syllabus.

Product

a^m \cdot a^n = a^{m+n}

Quotient

\dfrac{a^m}{a^n} = a^{m-n}

Power of a Power

(a^m)^n = a^{mn}

Identity

a^{\,1} = a

Zero Power

a^{\,0} = 1\,,\;\; a \ne 0

Negative Power

a^{-n} = \dfrac{1}{a^n}

Fractional Indices

Denominator is the root, numerator is the power.

a^{1/n} = \sqrt[n]{a}\,,\qquad a^{m/n} = \sqrt[n]{a^{\,m}} = \big(\sqrt[n]{a}\big)^{m}

Logarithms

A logarithm is an exponent in disguise. Master the bridge below, and every log law makes sense.

Log–Index Bridge

Convert between log form and exponential form. Example: log₂ 8 = 3 because 2³ = 8.

\log_a b = c \;\;\Longleftrightarrow\;\; a^c = b

Product

\log_a(PQ) = \log_a P + \log_a Q

Quotient

\log_a\!\left(\dfrac{P}{Q}\right) = \log_a P - \log_a Q

Power

\log_a(P^b) = b\log_a P

Log of the Base

\log_a a = 1

Log of 1

\log_a 1 = 0

Change of Base

\log_a b = \dfrac{\log_c b}{\log_c a}

Solving Exponential Equations

When both sides cannot be made the same base. Take logs of both sides.

a^x = b \;\;\Longrightarrow\;\; x = \dfrac{\log b}{\log a}

Linear Reduction (Log-Linear Form)

Some non-linear curves straighten out when you take logs. From a straight-line graph of log y vs something, you can read off the original equation.

Power Law

Plot log y on the vertical, log x on the horizontal. Gradient = n; intercept = log a.

y = ax^n \;\;\Longrightarrow\;\; \log y = n\log x + \log a

Exponential Law

Plot log y on the vertical, x on the horizontal. Gradient = log b; intercept = log a.

y = ab^x \;\;\Longrightarrow\;\; \log y = (\log b)\,x + \log a

Arithmetic Progression (A.P.)

An arithmetic progression adds the same fixed step every term. The step is the common difference d; the first term is a.

nth Term

T_n = a + (n - 1)\,d

Sum of First n Terms

The second form is faster when the last term T_n is already known.

\begin{aligned} S_n \;&=\; \dfrac{n}{2}\big[\,2a + (n-1)d\,\big] \\[4pt] &=\; \dfrac{n}{2}\,(a + T_n) \end{aligned}

Note

All arithmetic progressions diverge, except the trivial case d = 0. The terms never shrink — the sum keeps growing without bound.

Geometric Progression (G.P.)

A geometric progression multiplies by the same fixed ratio every term. The ratio is r; the first term is a.

nth Term

T_n = a\,r^{\,n - 1}

Sum of First n Terms

Choose whichever form keeps the numerator and denominator positive. Both give the same answer.

\begin{aligned} S_n \;&=\; \dfrac{a\,(r^n - 1)}{r - 1}\,,\;\; r > 1 \\[6pt] S_n \;&=\; \dfrac{a\,(1 - r^n)}{1 - r}\,,\;\; 0 < r < 1 \end{aligned}

Sum to Infinity

Valid only when |r| < 1. Outside that range the series diverges and S_∞ does not exist.

S_\infty = \dfrac{a}{1 - r}\,,\qquad -1 < r < 1

Convergence Rule

A geometric progression converges if and only if |r| < 1. Otherwise it diverges. Always check r before using S_∞.

Sigma Notation

Summation

k is the index, starting at the lower limit. n is the upper limit. a_k is the general term.

\sum_{k=1}^{n} a_k = a_1 + a_2 + a_3 + \dots + a_n

Constant Multiple

\sum_{i=1}^{n} k\,a_i \;=\; k\sum_{i=1}^{n} a_i

Compound Interest & Depreciation

Both are geometric progressions in disguise. Interest grows the principal; depreciation shrinks it. Same machinery, opposite direction.

Compound Interest

A = final amount, P = principal, r = rate (%) per period, T = number of periods.

A = P\!\left(1 + \dfrac{r}{100}\right)^{\!T}

Depreciation

A = depreciated value, P = original value, r = rate of decline (%), T = number of periods.

A = P\!\left(1 - \dfrac{r}{100}\right)^{\!T}

Section Two

Coordinate Geometry, Vectors & Trigonometry

Where geometry meets algebra. Lines, circles, vectors, and trigonometric ratios all describe shape and direction with equations — and CSEC tests them as a connected toolkit.

Straight Lines

Gradient

Rate of change of y with respect to x between two points.

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

Three Equivalent Forms of a Straight Line

Slope–Intercept

When you have the gradient m and y-intercept c directly.

y = mx + c

Point–Gradient

When you have one point on the line and the gradient.

y - y_1 = m(x - x_1)

General Form

The form most often required in the final answer.

ax + by + c = 0

Parallel & Perpendicular

Perpendicular gradients are negative reciprocals: m₂ = −1/m₁.

\text{Parallel: } m_1 = m_2 \qquad\qquad \text{Perpendicular: } m_1 \cdot m_2 = -1

Point of Intersection

Solve the two line equations simultaneously. The intersection point satisfies both equations at once.

Distance, Midpoint & Perpendicular Bisector

Length of a Line Segment

Pythagoras applied between two points — the line segment is the hypotenuse of a right triangle.

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

Midpoint

Average the x-coordinates and the y-coordinates separately.

M = \left(\dfrac{x_1 + x_2}{2},\;\; \dfrac{y_1 + y_2}{2}\right)

Perpendicular Bisector

Find the midpoint of the segment.
Find the gradient of the segment.
Take the perpendicular gradient $\;m_\perp = -\dfrac{1}{m}$ .
Apply point–gradient form using the midpoint and $m_\perp$ .

Circles

CSEC gives circles in two forms. Standard form shows the centre and radius directly; general form is the form you'll most often have to convert from.

Standard Form

Centre (a, b) — read straight from the equation. Radius r — the square root of the right-hand side.

(x - a)^2 + (y - b)^2 = r^2

General Form

Centre (-f, -g). Radius r = √(f² + g² − c).

x^2 + y^2 + 2fx + 2gy + c = 0

Conversion Shortcut

Compare coefficients directly: 2f is the coefficient of x, 2g is the coefficient of y, c is the constant. Then centre and radius come straight from the formulas above — no completing the square required.

Circle from Diameter Endpoints

Centre = midpoint of the two endpoints.
Radius = half the distance between them.
Substitute into the standard form $(x - a)^2 + (y - b)^2 = r^2$ .

Tangent & Normal to a Circle

A radius drawn to a point on the circle is perpendicular to the tangent at that point. That single fact powers every question on this topic.

Tangent at a Point

m_{\text{tangent}} = -\dfrac{1}{m_{\text{radius}}}

Find the centre of the circle.
Compute the gradient of the radius to the given point.
Tangent gradient is the negative reciprocal.
Apply point–gradient form.

Normal at a Point

The normal to a circle is the radius extended. It passes through the centre. Use point–gradient form with the point on the circle.

m_{\text{normal}} = m_{\text{radius}}

Line–Circle Intersection

Method

From the linear equation, express y in terms of x.
Substitute into the circle equation.
Solve the resulting quadratic.
Back-substitute to find the y-coordinates.

Reading the Discriminant

D > 0 → two intersection points · D = 0 → tangent (one point of contact) · D < 0 → the line misses the circle.

Vectors

A vector carries both magnitude and direction. CSEC works exclusively in two dimensions and accepts either column form or $\mathbf{i}, \mathbf{j}$ form.

Two Equivalent Notations

i = (1, 0)ᵀ along the x-axis, j = (0, 1)ᵀ along the y-axis.

\mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix} = x\mathbf{i} + y\mathbf{j}

Magnitude (Modulus)

Pythagoras on the components. This is the vector's length.

|\mathbf{v}| = \sqrt{x^2 + y^2}

Unit Vector

A vector of length 1 in the same direction as v.

\hat{\mathbf{v}} = \dfrac{\mathbf{v}}{|\mathbf{v}|}

Equality of Vectors

Each component must match — gives you a pair of simultaneous equations.

\mathbf{a} = \mathbf{b} \;\;\Longleftrightarrow\;\; a_1 = b_1 \;\text{and}\; a_2 = b_2

Operations on Vectors

Operate on each component independently.

\mathbf{a} + \mathbf{b} = \begin{pmatrix} a_1 + b_1 \\ a_2 + b_2 \end{pmatrix}\;,\qquad \mathbf{a} - \mathbf{b} = \begin{pmatrix} a_1 - b_1 \\ a_2 - b_2 \end{pmatrix}\;,\qquad k\mathbf{a} = \begin{pmatrix} k a_1 \\ k a_2 \end{pmatrix}

Displacement Vector

Read as "End minus start." The displacement from A to B when a, b are the position vectors.

\overrightarrow{AB} = \mathbf{b} - \mathbf{a}

Dot Product & the Angle Between Vectors

Dot Product

Compute from components; or from magnitudes and the included angle. Setting them equal is how you solve for θ.

\begin{aligned} \mathbf{a} \cdot \mathbf{b} \;&=\; a_1 b_1 + a_2 b_2 \\[4pt] &=\; |\mathbf{a}|\,|\mathbf{b}|\,\cos\theta \end{aligned}

Angle Between Two Vectors

Always returns 0 ≤ θ ≤ 180°. If cos θ is negative, θ is obtuse.

\cos\theta = \dfrac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}|\,|\mathbf{b}|}

Parallel & Perpendicular Vectors

Perpendicular: cos 90° = 0. Parallel: one vector is a scalar multiple of the other.

\text{Perpendicular: } \mathbf{a} \cdot \mathbf{b} = 0 \qquad\qquad \text{Parallel: } \mathbf{a} = k\mathbf{b}

Radians, Arc Length & Sector Area

CSEC trigonometry works in radians by default. Always confirm the units before plugging in.

Degrees ↔ Radians

Multiply degrees by π/180 to get radians. Multiply radians by 180/π to get degrees.

\pi \text{ rad} = 180°\,,\quad 1\text{ rad} = \dfrac{180°}{\pi}\,,\quad 1° = \dfrac{\pi}{180}\text{ rad}

Arc Length

θ in radians. The arc is the fraction θ/(2π) of the full circumference.

s = r\theta \quad\text{or}\quad \ell = r\theta

Sector Area

θ in radians. The sector is the fraction θ/(2π) of the full area πr².

A = \tfrac{1}{2}\, r^2 \theta

Segment Area

The region between a chord and its arc. Sector area minus the triangle inside it.

A_{\text{segment}} = \tfrac{1}{2}\, r^2 (\theta - \sin\theta)

Exact Values & the CAST Diagram

Exact Values

Standard Angles

Angle	$0$ (0°)	$\dfrac{\pi}{6}$ (30°)	$\dfrac{\pi}{4}$ (45°)	$\dfrac{\pi}{3}$ (60°)	$\dfrac{\pi}{2}$ (90°)
$\sin\theta$	$0$	$\dfrac{1}{2}$	$\dfrac{\sqrt{2}}{2}$	$\dfrac{\sqrt{3}}{2}$	$1$
$\cos\theta$	$1$	$\dfrac{\sqrt{3}}{2}$	$\dfrac{\sqrt{2}}{2}$	$\dfrac{1}{2}$	$0$
$\tan\theta$	$0$	$\dfrac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$	undefined

CAST — Signs by Quadrant

Quadrant 2 · 90°–180°

Sine positive

$\sin\theta > 0$

$\cos\theta < 0$

$\tan\theta < 0$

Quadrant 1 · 0°–90°

All positive

$\sin\theta > 0$

$\cos\theta > 0$

$\tan\theta > 0$

Quadrant 3 · 180°–270°

Tan positive

$\sin\theta < 0$

$\cos\theta < 0$

$\tan\theta > 0$

Quadrant 4 · 270°–360°

Cosine positive

$\sin\theta < 0$

$\cos\theta > 0$

$\tan\theta < 0$

Related Angles

Once you find the principal (acute) angle, use CAST to locate the other solutions in the range:
Quadrant 2: 180° − θ · Quadrant 3: 180° + θ · Quadrant 4: 360° − θ.

Pythagorean & Quotient Identities

Pythagorean Identity

Rearrangements: sin²θ = 1 − cos²θ and cos²θ = 1 − sin²θ.

\sin^2\theta + \cos^2\theta \;\equiv\; 1

Quotient Identity

To eliminate tan θ from an equation, or to simplify expressions mixing sin, cos, tan.

\tan\theta \;\equiv\; \dfrac{\sin\theta}{\cos\theta}

Compound & Double Angle Formulae

Compound Angle — Sine

The signs match — straightforward.

\sin(A \pm B) \;=\; \sin A \cos B \;\pm\; \cos A \sin B

Compound Angle — Cosine

The signs are opposite. Most common slip in the entire topic.

\cos(A \pm B) \;=\; \cos A \cos B \;\mp\; \sin A \sin B

Compound Angle — Tangent

Top signs match, bottom signs flip.

\tan(A \pm B) \;=\; \dfrac{\tan A \pm \tan B}{1 \mp \tan A \tan B}

Double-Angle Formulae (set $B = A$ )

sin 2A

\sin 2A \;=\; 2\sin A \cos A

tan 2A

\tan 2A \;=\; \dfrac{2\tan A}{1 - \tan^2 A}

cos 2A — Three Equivalent Forms

When equation has both sin² and cos²

Use this form for identity proofs or when a mix of sin² and cos² is present.

\cos 2A = \cos^2 A - \sin^2 A

When other sin² terms appear

Substitute to collapse the equation into a quadratic in sin A.

\cos 2A = 1 - 2\sin^2 A

When other cos² terms appear

Substitute to collapse the equation into a quadratic in cos A.

\cos 2A = 2\cos^2 A - 1

Triangle Area & Trig Graphs

Area of a Triangle

When two sides a, b and the included angle C are known.

\text{Area} = \tfrac{1}{2}\, ab \sin C

Graphs of $\sin kx$ , $\cos kx$ , $\tan kx$

Key Properties

Function	Amplitude	Period	Key Feature
$\sin kx$	$1$	$\dfrac{2\pi}{k}$	Starts at 0, rises first
$\cos kx$	$1$	$\dfrac{2\pi}{k}$	Starts at 1 (the maximum)
$\tan kx$	undefined	$\dfrac{\pi}{k}$	Vertical asymptotes at odd multiples of $\dfrac{\pi}{2k}$

The role of k

k compresses the graph horizontally — doubling k halves the period and doubles the number of cycles in [0, 2π].

Section Three

Introductory Calculus

Calculus is the mathematics of change. Differentiation gives the gradient at a point; integration reverses that and recovers the original function — or the area beneath the curve.

Differentiation — The Basic Rules

Power Rule (Standard)

For any power of x. This is the rule you reach for every single day.

\dfrac{d}{dx}\big(x^n\big) = n\,x^{\,n-1}

Power Rule with Linear Inner

For brackets raised to a power. The factor of a comes from the chain rule on the inside.

\dfrac{d}{dx}\big(ax + b\big)^n = a\,n\,(ax + b)^{n - 1}

Constant

\dfrac{d}{dx}(c) = 0

Sum & Difference

\dfrac{d}{dx}\!\big[f \pm g\big] = f' \pm g'

Constant Multiple

\dfrac{d}{dx}\!\big[k f(x)\big] = k f'(x)

Trig Derivatives

Sine

For just sin x, the derivative is cos x.

\dfrac{d}{dx}\sin ax = a \cos ax

Cosine

For just cos x, the derivative is −sin x.

\dfrac{d}{dx}\cos ax = -a \sin ax

CSEC Scope

Differentiation of tan x is not part of the CSEC Add Math syllabus. Only polynomials, sine, and cosine are required.

Chain, Product & Quotient Rules

Chain Rule

For a function of a function. Let u be the inner expression, differentiate the outer in terms of u, then multiply by the derivative of u with respect to x.

\dfrac{dy}{dx} \;=\; \dfrac{dy}{du} \times \dfrac{du}{dx}

Chain Rule — Linear Inner Shortcut

Example: d/dx (5x + 1)⁴ = 4(5x + 1)³ · 5 = 20(5x + 1)³.

\dfrac{d}{dx}\big[f(ax + b)\big] \;=\; a \cdot f'(ax + b)

Product Rule

For a product of two functions. The derivative of a product is not the product of derivatives.

\dfrac{d}{dx}(uv) \;=\; u\dfrac{dv}{dx} + v\dfrac{du}{dx}

Quotient Rule

For a quotient of two functions. Mind the order in the numerator — the minus sign matters.

\dfrac{d}{dx}\!\left(\dfrac{u}{v}\right) \;=\; \dfrac{v\,\dfrac{du}{dx} - u\,\dfrac{dv}{dx}}{v^2}

The Second Derivative

Second Derivative

The rate of change of the gradient itself — tells you how the curve is bending.

\dfrac{d^2 y}{dx^2} \;=\; f''(x) \;=\; \dfrac{d}{dx}\!\left(\dfrac{dy}{dx}\right)

Stationary Points & Their Nature

A stationary point is where the gradient is momentarily zero. From there it is either a maximum, a minimum, or a point of inflexion (excluded from CSEC).

Stationary Point

Differentiate, set f′(x) = 0, solve for x, then back-substitute to find the y-coordinate.

\dfrac{dy}{dx} = 0

Method 1 — Second Derivative Test

Second Derivative Test

If f″(x) = 0, fall back to Method 2.

f''(x) > 0 \Rightarrow \text{min},\quad f''(x) < 0 \Rightarrow \text{max}

A concave-up curve has f″(x) > 0 at its minimum. A concave-down curve has f″(x) < 0 at its maximum.

Method 2 — First Derivative Sign-Change Test

Procedure

Pick an x slightly less than the stationary point and slightly more. Check the sign of f′(x) in each.

Sign goes − to + → minimum · Sign goes + to − → maximum · No change → point of inflexion (not in CSEC).

Tangent & Normal to a Curve

Tangent at x = a

Use point–gradient form y − y₁ = m(x − x₁) with the point (a, f(a)).

m_{\text{tangent}} = f'(a)

Normal at x = a

Perpendicular to the tangent — negative reciprocal gradient. Same point on the curve.

m_{\text{normal}} = -\dfrac{1}{f'(a)}

Differentiation from First Principles

Definition of the Derivative

The limit of the gradient of a secant line as the two points squeeze together. The coordinate-geometry gradient taken at infinitesimally close points.

\dfrac{dy}{dx} \;=\; \lim_{h \to 0}\, \dfrac{f(x + h) - f(x)}{h}

Related Rates of Change

Chain Rule for Related Rates

For a circle of radius r, the area A = πr² changes with time via dA/dt = (dA/dr) · (dr/dt) = 2πr · (dr/dt).

\dfrac{dy}{dx} \;=\; \dfrac{dy}{du} \times \dfrac{du}{dx}

Integration — The Basic Rules

Integration reverses differentiation. It also accumulates change — giving the area under a curve.

Power Rule

Read: "Raise the power by one; divide by the new power." Always add the constant of integration c for an indefinite integral.

\int x^n\, dx \;=\; \dfrac{x^{\,n+1}}{n + 1} + c\,,\qquad n \ne -1

(ax + b)ⁿ Form

Power rule with an extra divide-by-the-inside-coefficient.

\int (ax + b)^n\, dx \;=\; \dfrac{(ax + b)^{\,n+1}}{a\,(n + 1)} + c\,,\qquad n \ne -1

Constant

\int k\, dx = kx + c

Sum & Difference

\int(f \pm g)\, dx = \int\! f\, dx \pm \int\! g\, dx

Constant Multiple

\int k\,f(x)\, dx = k\!\int f(x)\, dx

Trig Integrals

Integrating sin ax

\int \sin ax\, dx = -\dfrac{1}{a}\cos ax + c

Integrating cos ax

\int \cos ax\, dx = \dfrac{1}{a}\sin ax + c

Definite Integrals

Definite Integral

F(x) is any antiderivative. The constant of integration c cancels when you subtract.

\int_a^b f(x)\, dx \;=\; \Big[F(x)\Big]_a^b \;=\; F(b) - F(a)

Area & Volume by Integration

Area Under a Curve

CSEC restricts this to regions in the first quadrant, bounded by the curve, the x-axis, and the lines x = a and x = b.

A = \int_a^b f(x)\, dx

Volume of Revolution about the x-axis

Rotate the region under y = f(x) a full turn about the x-axis. Each thin strip becomes a disc of area πy². CSEC restricts this to polynomials up to degree 2.

V = \pi \int_a^b \big[f(x)\big]^2 dx

Equation of a Curve from its Gradient

Reverse the Differentiation

When given the gradient function and a point on the curve. Integrate, then substitute the point to find c.

y \;=\; \int \dfrac{dy}{dx}\, dx \;+\; c

Kinematics

Kinematics is calculus applied to motion. Differentiate to move forward through the chain; integrate to move backward.

Displacement

s(t)

position relative to a fixed origin

↑ integrate (add constant)differentiate ↓

Velocity

v = \dfrac{ds}{dt}

rate of change of displacement

↑ integrate (add constant)differentiate ↓

Acceleration

a = \dfrac{dv}{dt} = \dfrac{d^2 s}{dt^2}

rate of change of velocity

Forward — Differentiate

v = \dfrac{dx}{dt} = \dot{x}\,,\qquad a = \dfrac{d^2 x}{dt^2} = \dfrac{dv}{dt} = \ddot{x}

Reverse — Integrate

s = \int v\, dt\,,\qquad v = \int a\, dt

Displacement vs Total Distance

\displaystyle s = \int_a^b v(t)\, dt

can be negative when the particle moves backwards. For total distance use

|v(t)|

\displaystyle \text{Distance} = \int_a^b |v(t)|\, dt

. Find when v(t) = 0, split the integral there, and add the absolute values.

SUVAT — Equations of Motion (Optional)

SUVAT applies only when acceleration is constant. CSEC Add Math does not require these — every kinematics question can be solved using the calculus approach above. SUVAT can occasionally provide a faster route, so it is worth knowing.

Final Velocity

v = u + at

Displacement

s = ut + \tfrac{1}{2} at^2

Without Time

v^2 = u^2 + 2as

Section Four

Probability & Statistics

Statistics is half computation, half careful reading of data. Probability is half logic, half clean diagram work. Lock the formulas in, then practise reading questions slowly — that is where the marks slip.

Types of Data

Type	Meaning	Example
Qualitative	Non-numerical categories	Colours, gender, blood type
Quantitative — Discrete	Countable numbers	Number of students, goals scored
Quantitative — Continuous	Any value in a range	Height, weight, time, temperature

Mean, Median & Mode

Mean — Ungrouped

Sum of all the values, divided by how many there are.

\bar{x} \;=\; \dfrac{\displaystyle\sum_{i=1}^{n} x_i}{n}

Mean — Grouped

Use class midpoints for xᵢ. Each midpoint is weighted by its frequency.

\bar{x} \;=\; \dfrac{\displaystyle\sum_{i=1}^{n} f_i\, x_i}{\displaystyle\sum_{i=1}^{n} f_i}

Median & Mode

Mode: the value (or class) that occurs most frequently.

\text{Median position:} \;\; \tfrac{1}{2}(n + 1)^{\text{th}} \text{ term (raw)} \quad\text{or}\quad \tfrac{1}{2}n^{\text{th}} \text{ term (grouped / large data)}

Quartiles & Interquartile Range

Quartiles — Raw Data

For small, ordered lists of individual values.

\begin{aligned} Q_1 &= \tfrac{1}{4}(n + 1)^{\text{th}} \text{ term} \\ Q_2 &= \tfrac{1}{2}(n + 1)^{\text{th}} \text{ term} \\ Q_3 &= \tfrac{3}{4}(n + 1)^{\text{th}} \text{ term} \end{aligned}

Quartiles — Grouped / Large Data

For grouped frequency tables and large datasets — typically read from a cumulative frequency curve.

\begin{aligned} Q_1 &= \tfrac{1}{4}\,n^{\text{th}} \text{ term} \\ Q_2 &= \tfrac{1}{2}\,n^{\text{th}} \text{ term} \\ Q_3 &= \tfrac{3}{4}\,n^{\text{th}} \text{ term} \end{aligned}

Interquartile Range

The spread of the middle 50% of the data. Robust to extreme values.

\text{IQR} \;=\; Q_3 - Q_1

Semi-IQR

Half the interquartile range.

\text{Semi-IQR} \;=\; \dfrac{Q_3 - Q_1}{2}

Variance & Standard Deviation

Variance — Ungrouped

The second form is the calculator-friendly shortcut.

\begin{aligned} S^2 \;&=\; \dfrac{\displaystyle\sum_{i=1}^{n} (x_i - \bar{x})^2}{n} \\[6pt] &=\; \dfrac{\displaystyle\sum_{i=1}^{n} x_i^{\,2}}{n} \;-\; (\bar{x})^2 \end{aligned}

Variance — Grouped

Use class midpoints for xᵢ, weighted by frequency.

S^2 \;=\; \dfrac{\displaystyle\sum_{i=1}^{n} f_i\, x_i^{\,2}}{\displaystyle\sum_{i=1}^{n} f_i} \;-\; (\bar{x})^2

Standard Deviation

Same units as the original data — easier to interpret than variance. Always find variance first.

S \;=\; \sqrt{S^2}

Box-and-Whisker Plots & Stem-and-Leaf

A box-and-whisker plot displays the five-number summary: Minimum, Q₁, Median (Q₂), Q₃, Maximum. The box spans Q₁ to Q₃; the line inside marks the median; the whiskers extend to the extremes.

Skewness from the box plot: the position of the median line inside the box tells you which way the data is pulled.

Stem-and-Leaf

Split each value into a stem (leading digits) and a leaf (final digit). Preserves the original values, so median, quartiles, and mode can all be read directly. Always include a key such as “

1\,|\,2

means

12

.”

Probability Foundations

An experiment produces an outcome. The sample space S is the set of all possible outcomes. An event A is a subset of the sample space.

Classical Probability

All outcomes must be equally likely — coins, fair dice, balls drawn at random.

P(A) \;=\; \dfrac{\text{number of outcomes in } A}{\text{total number of outcomes in } S}

Range

0 \le P(A) \le 1

Sample Space

\sum_{\text{outcomes}} P = 1

Complement

P(A') = 1 - P(A)

Addition Rule & Mutually Exclusive Events

Addition Rule

Adding P(A) and P(B) counts the overlap P(A ∩ B) twice. Subtracting it once removes the duplicate.

P(A \cup B) \;=\; P(A) + P(B) - P(A \cap B)

Mutually Exclusive Events

The events cannot both happen — rolling a 2 and rolling a 5 on the same die. No overlap, no subtraction.

P(A \cap B) = 0 \;\;\Longrightarrow\;\; P(A \cup B) = P(A) + P(B)

Conditional Probability

Read: "The probability of A, given that B has occurred." Knowing B restricts the sample space to B, and we recompute within that restricted space.

P(A | B) \;=\; \dfrac{P(A \cap B)}{P(B)}

Independent Events

Independence Condition

One event does not affect the other — like tossing two separate coins. Either condition can be used to prove independence.

P(A \cap B) = P(A) \times P(B) \qquad\text{or equivalently}\qquad P(A | B) = P(A)

Common Confusion

Mutually exclusive is not the same as independent. If two events are mutually exclusive (and both have non-zero probability), they cannot be independent — knowing one happened tells you the other did not.

Tree Diagrams, Venn Diagrams & Possibility Spaces

Tree Diagrams

Multiply along branches for “and” — joint probability of A then B.

Add across branches for “or” — total probability of an outcome reached by more than one path.

CSEC limits trees to two initial branches.

Venn Diagrams (Two Sets)

$A \cap B$ : inside both circles

$A \cap B'$ : inside A only

$A' \cap B$ : inside B only

$(A \cup B)'$ : outside both

Always fill in $P(A \cap B)$ first, then work outward.

Possibility Spaces

A grid of all outcome pairs from two experiments — e.g. rolling two dice gives a 6 × 6 grid.

$P(\text{event}) = \dfrac{\text{favourable squares}}{\text{total squares}}$

Quick Reference

Law	Formula
Complement	$P(A') = 1 - P(A)$
Addition (general)	$P(A \cup B) = P(A) + P(B) - P(A \cap B)$
Mutually exclusive	$P(A \cup B) = P(A) + P(B)$
Conditional	$P(A \| B) = \dfrac{P(A \cap B)}{P(B)}$
Independent	$P(A \cap B) = P(A) \times P(B)$