Every board and O/A-Level mathematics paper draws from a surprisingly small, predictable set of formulas and concept groups. This guide organizes them the way they actually appear on exams — by topic, with the reasoning behind each formula — so you can revise in one focused pass instead of flipping between chapters the night before a test.
Algebra: The Foundation Layer
Quadratic formula: For ax² + bx + c = 0, x = (−b ± √(b² − 4ac)) / 2a. The discriminant (b² − 4ac) tells you the nature of the roots before you even solve: positive means two real roots, zero means one repeated root, negative means complex roots — examiners frequently ask you to identify this without fully solving the equation.
Factorization identities:
- (a + b)² = a² + 2ab + b²
- (a − b)² = a² − 2ab + b²
- a² − b² = (a + b)(a − b)
- a³ + b³ = (a + b)(a² − ab + b²)
- a³ − b³ = (a − b)(a² + ab + b²)
These identities are tested both directly and hidden inside longer simplification problems — recognizing the pattern quickly is the real skill being tested.
Simultaneous equations: Elimination and substitution both work, but substitution is faster when one equation already isolates a variable. Always check your solution by plugging both values back into the original equations — a two-second check that catches most arithmetic slips.
Geometry and Trigonometry
Pythagorean theorem: a² + b² = c², used constantly in coordinate geometry and word problems, not just triangle questions.
Key trigonometric ratios:
- sin θ = opposite / hypotenuse
- cos θ = adjacent / hypotenuse
- tan θ = opposite / adjacent = sin θ / cos θ
Trigonometric identities worth memorizing cold:
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = cosec²θ
Coordinate geometry essentials:
- Distance formula: √[(x₂−x₁)² + (y₂−y₁)²]
- Midpoint formula: ((x₁+x₂)/2, (y₁+y₂)/2)
- Slope: m = (y₂−y₁) / (x₂−x₁)
- Line equation (point-slope): y − y₁ = m(x − x₁)
Calculus (FSc / A-Level)
Differentiation rules:
- Power rule: d/dx(xⁿ) = nxⁿ⁻¹
- Product rule: d/dx(uv) = u’v + uv’
- Quotient rule: d/dx(u/v) = (u’v − uv’) / v²
- Chain rule: d/dx[f(g(x))] = f'(g(x)) · g'(x)
Common derivatives:
- d/dx(sin x) = cos x
- d/dx(cos x) = −sin x
- d/dx(eˣ) = eˣ
- d/dx(ln x) = 1/x
Integration basics:
- ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (for n ≠ −1)
- ∫1/x dx = ln|x| + C
- ∫eˣ dx = eˣ + C
- ∫sin x dx = −cos x + C
- ∫cos x dx = sin x + C
Applications examiners love testing: finding maxima/minima by setting the first derivative to zero, finding the area under a curve using definite integrals, and related-rates problems that combine differentiation with real-world scenarios (like a ladder sliding down a wall).
Statistics and Probability
Measures of central tendency:
- Mean = sum of values ÷ number of values
- Median = middle value when data is ordered
- Mode = most frequently occurring value
Standard deviation measures how spread out data is from the mean — a small standard deviation means data points cluster closely around the mean, while a large one means they’re spread widely. Understanding this conceptually matters more than memorizing the formula, since exams frequently ask you to interpret standard deviation rather than just calculate it.
Basic probability rules:
- P(A or B) = P(A) + P(B) − P(A and B)
- P(A and B) = P(A) × P(B), only when A and B are independent
- P(not A) = 1 − P(A)
Exam Technique Notes
- Show every step. Marking schemes award most marks for correct method, even when the final numerical answer is wrong due to a small slip.
- Label units and diagrams. Coordinate geometry graphs without labeled axes, or answers missing units, routinely lose marks that have nothing to do with actual math ability.
- Check your discriminant before solving. If a quadratic question asks about the “nature of roots,” you don’t need to fully solve it — the discriminant alone answers the question faster and more reliably.
- Simplify before substituting. In algebra and calculus problems, simplifying an expression first (factoring, canceling) before plugging in numbers reduces arithmetic errors significantly.
Frequently Asked Questions
Which formulas should I prioritize if I’m short on time? The quadratic formula, differentiation rules (power, product, chain), and the distance/slope formulas in coordinate geometry appear across the widest range of question types — prioritize these first.
Do I need to memorize every trigonometric identity? No — sin²θ + cos²θ = 1 and its two direct variants (1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ) cover the vast majority of identity-based questions; more obscure identities can usually be derived from these three.
How do I know if I should differentiate or integrate in a word problem? If the question describes a rate of change (speed, growth, how fast something is changing), differentiate. If it describes accumulation (total distance, total area, total quantity), integrate.
What’s the fastest way to check an algebra answer? Substitute your solution back into the original equation — if both sides match, your answer is correct; if not, you’ve caught an error before submitting.
How do I quickly tell if a system of equations has one solution, no solution, or infinite solutions? Compare the slopes: different slopes mean exactly one solution, same slope with different y-intercepts means no solution (parallel lines), and same slope with the same y-intercept means infinitely many solutions (the same line written differently).
What’s a fast way to double-check calculus differentiation answers? Estimate the derivative’s sign at a specific point by checking whether the original function is increasing or decreasing there — if your calculated derivative has the wrong sign at that point, you know there’s an error before checking the full working.