Maths

1. Number

Structure, Operations, and Accuracy

• Standard Form: Expressed as $a \times 10^n$ where $1 \leq a < 10$ and $n$ is an integer.
• Surds:
  • Multiplication rule: $\sqrt{a} \times \sqrt{b} = \sqrt{a b}$
  • Division rule: $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$
  • Rationalising denominators:
  • Single term: $\frac{a}{\sqrt{b}} = \frac{a \sqrt{b}}{b}$
  • Binomial denominator: Multiply numerator and denominator by the conjugate pair, e.g., $\frac{c}{a + \sqrt{b}} = \frac{c (a - \sqrt{b})}{a^2 - b}$
• Indices:
  • Multiplication: $a^m \times a^n = a^{m + n}$
  • Division: $a^m \div a^n = a^{m - n}$
  • Power of a power: $(a^m)^n = a^{m n}$
  • Negative index: $a^{-n} = \frac{1}{a^n}$
  • Fractional index: $a^{\frac{m}{n}} = (\sqrt[n]{a})^m$
• Recurring Decimals to Fractions:
  • Set $x$ equal to the recurring decimal.
  • Multiply by powers of $10$ to align repeating periods, then subtract equations to eliminate recurring digits and solve for $x$.
• Bounds and Error Intervals:
  • For a value rounded to the nearest unit $u$, the margin of error is $\pm \frac{u}{2}$.
  • $\text{Lower Bound} \leq x < \text{Upper Bound}$
  • Maximum value of a division: $\text{Maximum} = \frac{\text{Upper Bound of Numerator}}{\text{Lower Bound of Denominator}}$

2. Algebra

Expressions, Equations, and Inequalities

• Factorising Quadratics:
  • Expressions of the form $a x^2 + b x + c$.
  • Difference of two squares: $a^2 - b^2 = (a - b)(a + b)$.
• Completing the Square:
  • For $x^2 + b x + c$: $\left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2 + c$.
  • Turning point coordinates of $y = (x + p)^2 + q$ are $(-p, q)$.
• Quadratic Formula:
  • Used to solve $a x^2 + b x + c = 0$:
  • $x = \frac{-b \pm \sqrt{b^2 - 4 a c}}{2 a}$
• Simultaneous Equations:
  • Linear and non-linear systems: Substitute the linear equation into the quadratic equation to form a single quadratic equation, solve for one variable, then back-substitute.
• Inequalities:
  • Quadratic inequalities: Factorise to find critical values, sketch the parabola, and identify intervals where $y > 0$ or $y < 0$.

Functions and Sequences

• Functions:
  • Composite functions: $f(g(x))$ indicates applying function $g$ first, then function $f$.
  • Inverse functions: Solve $y = f(x)$ for $x$, then replace with $f^{-1}(x)$.
• Sequences:
  • Arithmetic: $n^{\text{th}} \text{ term} = d n + (a - d)$ where $d$ is common difference and $a$ is first term.
  • Quadratic: $n^{\text{th}} \text{ term} = a n^2 + b n + c$, where the second difference equals $2 a$.
  • Geometric: Terms have a common ratio $r$.
• Graph Transformations:
  • Translation vertically: $y = f(x) + a$
  • Translation horizontally: $y = f(x + a)$
  • Reflection in $x$-axis: $y = -f(x)$
  • Reflection in $y$-axis: $y = f(-x)$

3. Ratio, Proportion, and Rates of Change

Proportion and Growth

• Direct Proportion:
  • $y \propto x \implies y = k x$
• Inverse Proportion:
  • $y \propto \frac{1}{x} \implies y = \frac{k}{x}$
• Compound Interest and Exponential Decay:
  • Value after $n$ periods: $A = P \left(1 \pm \frac{r}{100}\right)^n$

Compound Measures and Kinematics

• Speed, Density, and Pressure:
  • $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$
  • $\text{Density} = \frac{\text{Mass}}{\text{Volume}}$
  • $\text{Pressure} = \frac{\text{Force}}{\text{Area}}$
• Gradients and Area Under Curves:
  • Gradient of distance-time graph represents speed.
  • Gradient of velocity-time graph represents acceleration.
  • Area under velocity-time graph represents total distance travelled (calculated using rectangles, triangles, or the trapezium rule).

4. Geometry and Measures

Trigonometry and Vectors

• Pythagoras' Theorem:
  • In right-angled triangles: $a^2 + b^2 = c^2$
  • In 3D contexts: $d^2 = x^2 + y^2 + z^2$
• Right-Angled Trigonometric Ratios:
  • $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
  • $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
  • $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$
• Non-Right-Angled Triangles:
  • Sine Rule (finding lengths): $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$
  • Sine Rule (finding angles): $\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}$
  • Cosine Rule (lengths): $a^2 = b^2 + c^2 - 2 b c \cos(A)$
  • Cosine Rule (angles): $\cos(A) = \frac{b^2 + c^2 - a^2}{2 b c}$
  • Area of a triangle: $\text{Area} = \frac{1}{2} a b \sin(C)$
• Vectors:
  • Vector arithmetic with column vectors: $\begin{pmatrix} x<em>1 \ y</em>1 \end{pmatrix} + \begin{pmatrix} x<em>2 \ y</em>2 \end{pmatrix} = \begin{pmatrix} x<em>1 + x</em>2 \ y<em>1 + y</em>2 \end{pmatrix}$
  • Parallel vectors: Vector $\mathbf{a}$ is parallel to $\mathbf{b}$ if $\mathbf{a} = k \mathbf{b}$ for scalar $k$.

Circle Theorems and Similarity

• Circle Theorems:
  • The angle subtended at the centre is twice the angle subtended at the circumference.
  • The angle in a semicircle is a right angle ($90^\circ$).
  • Angles in the same segment subtended by the same arc are equal.
  • Opposite angles of a cyclic quadrilateral sum to $180^\circ$.
  • The angle between a tangent and a chord equals the angle subtended in the alternate segment.
  • Tangents from an external point to a circle are equal in length and perpendicular to the radius at the point of contact.
• Similarity Ratios:
  • Length scale factor: $k$
  • Area scale factor: $k^2$
  • Volume scale factor: $k^3$

5. Probability and Statistics

Probability

• Mutually Exclusive and Independent Events:
  • Independent events: $P(A \cap B) = P(A) \times P(B)$
  • Mutually exclusive events: $P(A \cup B) = P(A) + P(B)$
• Conditional Probability:
  • $P(A|B) = \frac{P(A \cap B)}{P(B)}$
• Diagrams:
  • Tree diagrams: Multiply along branches to find combined probability, add down outcomes.
  • Venn diagrams: Used for set notation including intersection ($\cap$), union ($\cup$), and complement ($A'$).

Statistics

• Cumulative Frequency and Box Plots:
  • Cumulative frequency plotted against upper class boundaries.
  • Interquartile Range: $\text{IQR} = Q<em>3 - Q</em>1$
• Histograms:
  • Data plotted with unequal class intervals.
  • Area represents frequency: $\text{Frequency} = \text{Frequency Density} \times \text{Class Width}$
  • Vertical axis is frequency density: $\text{Frequency Density} = \frac{\text{Frequency}}{\text{Class Width}}$