IGCSE Additional Mathematics (0606)
Syllabus Notes (2025-2027)

Syllabus Alignment Notice: These notes strictly follow the Cambridge Assessment International Education (CAIE) syllabus for 2025, 2026, and 2027. Note that the Indices and surds topic has been moved to assumed knowledge, a new topic on Coordinate geometry of the circle has been added, and Paper 1 is now completely non-calculator.

1. Functions

Understanding terms and mapping notation is critical for algebraic fluency.

  • Domain: The set of all possible input values (independent variable \(x\)).
  • Range (Image set): The set of output values generated by the function mapping.
  • Function conditions: A mapping is a function if and only if each input maps to exactly one output (one-to-one or many-to-one mappings).
  • Inverse Function (\(f^{-1}\)): Only exists if the function is one-to-one. Graphical relationship: reflection across the line \(y = x\).
  • Composite Functions (\(gf(x)\)): Exists if and only if the Range of \(f \subseteq\) Domain of \(g\). Notice that order matters: \(fg(x) \neq gf(x)\).
Inverse verification: \(f(f^{-1}(x)) = x\)

2. Quadratic Functions

Quadratic equations follow the structure \(ax^2 + bx + c = 0\).

Completing the Square

Transform expressions to find turning points easily: \(f(x) = a(x-h)^2 + k\), where \((h, k)\) is the vertex.

The Discriminant

Condition Discriminant Status Graphical Interpretation
Two real and distinct roots \(b^2 - 4ac > 0\) Intersects x-axis at two separate points
Two real and equal roots \(b^2 - 4ac = 0\) Tangent to the x-axis / turning point touches axis
No real roots \(b^2 - 4ac < 0\) Does not intersect the x-axis entirely

3. Factors of Polynomials

Applies to polynomials such as cubics \(f(x) = ax^3 + bx^2 + cx + d\).

Remainder Theorem: If a polynomial \(f(x)\) is divided by \((ax - b)\), the remainder is \(f\left(\frac{b}{a}\right)\).
Factor Theorem: If \(f\left(\frac{b}{a}\right) = 0\), then \((ax - b)\) is a perfect factor of \(f(x)\).

To solve cubics, use the Factor Theorem to find the first root, apply algebraic long division or synthetic comparison to reduce to a quadratic piece, and factorize completely.

4. Equations, Inequalities and Graphs

Deals with modulus equations and inequalities of linear, quadratic, and cubic varieties.

The modulus function |ax + b| always returns a non-negative value. To solve algebraic statements like |ax+b| = |cx+d|, square both sides or solve the split conditions: \(ax+b = \pm(cx+d)\).

When plotting modular curves, sketch the base function and reflect any portions below the x-axis upward onto the positive quadrant, forming characteristic sharp custom corners (cusps).

5. Simultaneous Equations

Solving two equations simultaneously with up to two unknowns using direct elimination or substitution methods.

Example system: \(y - x + 3 = 0\) and \(x^2 - 3xy + y^2 + 19 = 0\)

Isolate a linear variable (\(y = x - 3\)) and substitute it directly into the non-linear equation to acquire a manageable single-variable quadratic expression.

6. Logarithmic and Exponential Functions

Logarithmic and exponential operations are mathematical inverses: \(y = a^x \iff x = \log_a y\).

Core Laws of Logarithms

  • Product Rule: \(\log_a (xy) = \log_a x + \log_a y\)
  • Quotient Rule: \(\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y\)
  • Power Rule: \(\log_a (x^n) = n \log_a x\)
  • Change of Base Rule: \(\log_a b = \frac{\log_c b}{\log_c a}\)

7. Straight-line Graphs

Linear geometry formulas based on coordinates \((x_1, y_1)\) and \((x_2, y_2)\):

  • Gradient: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
  • Parallel condition: \(m_1 = m_2\)
  • Perpendicular condition: \(m_1 \times m_2 = -1\)
  • Midpoint: \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\)
  • Distance: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

Linearisation: Transform non-linear relationships into straight-line form \(Y = mX + C\). For instance, transforming \(y = Ax^n\) yields \(\ln y = n\ln x + \ln A\), where tracking \(\ln y\) against \(\ln x\) yields a linear gradient of \(n\).

8. Coordinate Geometry of the Circle

New Syllabus Topic: Standard Equation of a circle with center \((a, b)\) and radius \(r\): \[ (x - a)^2 + (y - b)^2 = r^2 \] General Form: \(x^2 + y^2 + 2gx + 2fy + c = 0\), with center \((-g, -f)\) and radius \(r = \sqrt{g^2 + f^2 - c}\).

Intersection calculations involve substituting linear equations into the circular format. If the resulting quadratic discriminant \(b^2 - 4ac = 0\), the straight line acts as a localized tangent.

9. Circular Measure

Calculations explicitly require working with angles configured in radians (\(\pi\text{ rad} = 180^\circ\)).

Arc Length: \(s = r\theta\)
Sector Area: \(A = \frac{1}{2}r^2\theta\)

Where \(\theta\) must be expressed strictly in radians.

10. Trigonometry

Extends to six standard functions: \(\sin x, \cos x, \tan x\), and their reciprocal counterparts:

\(\sec A = \frac{1}{\cos A} \qquad \csc A = \frac{1}{\sin A} \qquad \cot A = \frac{1}{\tan A}\)

Fundamental Identities

  • \(\sin^2 A + \cos^2 A = 1\)
  • \(\sec^2 A = 1 + \tan^2 A\)
  • \(\csc^2 A = 1 + \cot^2 A\)

11. Permutations and Combinations

Deals with arrangements where order matters vs. group selections where order is irrelevant.

  • Permutations (\(^nP_r\)): Ordering matters. Used for lists, seating positions, codes. \[ ^nP_r = \frac{n!}{(n-r)!} \]
  • Combinations (\(^nC_r\)): Choosing groups where position order doesn't matter. \[ ^nC_r = \frac{n!}{r!(n-r)!} \]

12. Series

Binomial Expansion

For any positive integer \(n\):

\[ (a + b)^n = a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \dots + b^n \]

Progressions

Series Type \(n\)-th Term (\(u_n\)) Sum of \(n\) terms (\(S_n\))
Arithmetic (AP) \(a + (n-1)d\) \(\frac{n}{2}(2a + (n-1)d)\)
Geometric (GP) \(ar^{n-1}\) \(\frac{a(1-r^n)}{1-r} \quad (r \neq 1)\)

Convergence of GP: A geometric series converges to a fixed sum to infinity if and only if \(|r| < 1\). The limiting equation is:

\[ S_\infty = \frac{a}{1-r} \]

13. Vectors in Two Dimensions

Vectors denote spatial displacements with independent magnitude and direction components.

Notation variants: \(\mathbf{r} = \begin{pmatrix} a \\ b \end{pmatrix} = a\mathbf{i} + b\mathbf{j}\). Magnitude is computed via standard Euclidean distance layout: \(|\mathbf{r}| = \sqrt{a^2 + b^2}\).

A unit vector sharing identical orientation direction components is evaluated using: \(\hat{\mathbf{r}} = \frac{\mathbf{r}}{|\mathbf{r}|}\).

14. Calculus

Note: No formulas are provided in the exam booklet for Calculus! You must memorize all tracking rules.

Differentiation Rules

Function \(f(x)\) Derivative \(f'(x)\)
\(x^n\) \(n x^{n-1}\)
\(\sin x\) \(\cos x\) (angles must be in radians)
\(\cos x\) \(-\sin x\)
\(\tan x\) \(\sec^2 x\)
\(e^{ax+b}\) \(a e^{ax+b}\)
\(\ln x\) \(\frac{1}{x}\)

Key Applications

  • Stationary Points: Set \(\frac{dy}{dx} = 0\). Evaluate stability with the second derivative test: \(\frac{d^2y}{dx^2} > 0\) implies a minimum, while \(\frac{d^2y}{dx^2} < 0\) indicates a localized maximum.
  • Integration Area: The bounded area enclosed below custom target contours tracking across Cartesian axes coordinates is determined using definite integration boundaries: \[ \text{Area} = \int_{a}^{b} y \, dx \]
  • Kinematics: Track equations relating shifting variables over time boundaries: \[ s (\text{displacement}) \xrightarrow{\frac{d}{dt}} v (\text{velocity}) \xrightarrow{\frac{d}{dt}} a (\text{acceleration}) \] Reverse directional track calculations rely on matching definite integration layouts.