1. Simplifying expressions Simplify algebraic expressions by combining like terms and using distributive property. Example: Simplify $3x + 5x - 2 = (3+5)x - 2 = 8x - 2$ 2. Rearranging formulae Isolate the desired variable by performing inverse operations on both sides. Example: Rearrange $A = 2l + 2w$ to find $l$: $A = 2l + 2w \Rightarrow 2l = A - 2w \Rightarrow l = \frac{A - 2w}{2}$ 3. Exponent laws and scientific notation Use laws like $a^m \times a^n = a^{m+n}$ and convert numbers to form $a \times 10^n$. Example: $2 \times 10^3 \times 3 \times 10^2 = 6 \times 10^{3+2} = 6 \times 10^5$ 4. Rational exponents Rewrite $a^{m/n} = \sqrt[n]{a^m}$. Example: $8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4$ 5. Solving linear equations Isolate variable using inverse operations. Example: Solve $2x + 3 = 7$: $2x = 4 \Rightarrow x = 2$ 6. Solving quadratic equations (a) Factorisation: Find factors that multiply to $c$ and add to $b$ in $ax^2 + bx + c=0$. Example: $x^2 + 5x + 6 = (x+2)(x+3) = 0 \Rightarrow x=-2,-3$ (b) Completing the square: Rewrite as $(x+p)^2 = q$. Example: $x^2 + 6x + 5=0 \Rightarrow (x+3)^2 = 4 \Rightarrow x=-3 \pm 2$ (c) Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Example: For $x^2 + 4x - 5=0$, $x=\frac{-4 \pm \sqrt{16+20}}{2} = \frac{-4 \pm 6}{2}$ 7. Linear simultaneous equations (a) Graphical: Plot lines and find intersection. (b) Substitution: Solve one for a variable and substitute. (c) Elimination: Add or subtract equations to eliminate a variable. Example: Solve $x+y=5$, $x-y=1$ by elimination: Adding gives $2x=6 \Rightarrow x=3$, then $y=2$ 8. Non-linear simultaneous equations Solve systems with one linear and one quadratic equation by substitution. Example: $y = 2x + 1$, $y = x^2$; set equal: $x^2 = 2x + 1$ solve quadratic. 9. Problem-solving using simultaneous equations Translate word problems into equations and solve simultaneously. 10. Linear inequalities Solve like equations but reverse inequality when multiplying/dividing by negative. Example: $2x - 3 > 5 \Rightarrow 2x > 8 \Rightarrow x > 4$ 11. Quadratic inequalities Solve quadratic equation, test intervals to find where inequality holds. 12. Regions of Cartesian plane Graph inequalities to shade regions satisfying conditions. 13. Feasible region Intersection of constraints in optimization problems. 14. Linear programming Maximize or minimize objective function over feasible region. 15. Distance formula Distance between $(x_1,y_1)$ and $(x_2,y_2)$ is $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. Example: Distance between (1,2) and (4,6) is $\sqrt{(4-1)^2 + (6-2)^2} = 5$ 16. Midpoint theorem Midpoint is $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$. Example: Midpoint of (2,3) and (4,7) is (3,5) 17. Gradient of parallel and perpendicular lines Parallel lines have equal gradients; perpendicular gradients multiply to -1. 18. Congruent triangles Triangles with equal sides and angles. 19. Proof using congruence Use criteria like SSS, SAS, ASA to prove triangle congruence. 20. Similar triangles Triangles with equal angles and proportional sides. 21. Trigonometric ratios Use $\sin = \frac{opposite}{hypotenuse}$, $\cos = \frac{adjacent}{hypotenuse}$, $\tan = \frac{opposite}{adjacent}$. Example: Find side opposite angle 30° in right triangle with hypotenuse 10: $10 \times \sin 30° = 5$ 22. True bearings Measure clockwise from north. 23. 3D trigonometry Use trigonometry in three dimensions for distances and angles. 24. Trigonometric functions Functions like $y=\sin x$, $y=\cos x$, $y=\tan x$. 25. Trigonometry with obtuse angles Use identities and unit circle. 26. Area of triangle $\frac{1}{2}ab\sin C$ 27. Sine rule $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ 28. Cosine rule $c^2 = a^2 + b^2 - 2ab \cos C$ 29. Pythagorean and Tan identities $\sin^2 x + \cos^2 x = 1$, $\tan x = \frac{\sin x}{\cos x}$ 30. Special angles and CAST diagram Use quadrant signs for trig functions. 31. Trigonometric equations Solve equations involving trig functions. 32. Function notation $f(x)$ represents function value at $x$. 33. Domain and range Domain: input values; Range: output values. 34. Graphing quadratic functions Use transformations of $y=x^2$. 35. Completing the square Rewrite quadratic in vertex form. 36. Axes intercepts Find where graph crosses axes. 37. Axis of symmetry and vertex Axis: $x = -\frac{b}{2a}$; vertex is point on parabola. 38. Finding quadratic function From points or vertex form. 39. Graphs of exponential functions $y = a^x$ with $a>0, a \neq 1$ 40. Exponential equations Solve using logarithms. 41. Reciprocal functions $y = \frac{1}{x}$ 42. Discrete vs continuous data Discrete: countable; continuous: measurable. 43. Measuring centre Mean, median, mode. 44. Cumulative data Data accumulated up to a point. 45. Measuring spread Range, interquartile range. 46. Box-and-whiskers Graphical summary of data. 47. Standard deviation Measure of spread around mean. 48. Normal distribution Bell-shaped curve. 49. Venn diagrams Visualize sets and their relations. 50. Intersection and union Intersection: common elements; union: all elements. 51. Theoretical probability Probability based on known outcomes. 52. Addition law $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ 53. Mutually exclusive events Events that cannot happen together. 54. Tree diagrams Visualize sequences of events. Each topic can be expanded with examples and detailed steps as needed for exam preparation.