H2 Mathematics revision reference

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This selected lookup is for 2027 H2 Mathematics (9758). Choose a section to recall a formula, its notation and the conditions for using it. Use the lesson links for derivations and practice; examination instructions determine what is supplied.

Functions and graphs

Domain is the permitted input set; range is the set of resulting outputs. A function has an inverse function when it is one-to-one on its stated domain. The inverse swaps input and output; its domain is the original range. f⁻¹(x) is not 1/f(x).

In fg(x) = f(g(x)), apply g first. The composite exists on the stated domain only when all the outputs of g are valid inputs to f.

y = f(x) + a: move the graph up a.
y = f(x + a): move it left a.
y = af(x): multiply vertical coordinates by a.
y = f(ax), a ≠ 0: multiply horizontal coordinates by 1/a.

Use negative scale factors to account for reflections. For rational inequalities, mark denominator zeros as excluded and test signs on intervals; do not multiply by an expression whose sign is unknown without handling its cases.

Sequences and series

For first term a, common difference d, ratio r and positive integer n:

uₙ = a + (n-1)d, Sₙ = n/2{2a + (n-1)d},

uₙ = arⁿ⁻¹, Sₙ = (a(1-rⁿ))/(1-r) (r ≠ 1).

For r = 1, the geometric finite sum is na. If |r| < 1, the geometric sum to infinity is a/(1-r). The zero sequence a = 0 has sum zero even outside that ratio condition.

uₙ = Sₙ-Sₙ₋₁ for n ≥ 2, with u₁ = S₁. A recurrence gives the next term from earlier terms; retain its starting index and initial value. A proposed recurrence limit must satisfy the fixed-point equation and a convergence argument.

Vectors, lines and planes

For a = (a₁,a₂,a₃), its magnitude is |a| = square root of (a₁² + a₂² + a₃²). A nonzero vector’s unit vector is a/|a|.

a · b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos θ.

|a × b| = |a||b| sin θ.

Here θ is the angle between nonzero vectors. The scalar product is a number; the vector product is perpendicular to both vectors with direction set by the right-hand rule. Reversing its order reverses its sign. For unit n hat, a · n hat is a signed scalar component, while |a × n hat| is a perpendicular magnitude.

Line: r = a + td, with direction d ≠ 0.
Plane: (r-a) · n = 0, with normal n ≠ 0.
Point p to plane distance: (|(p-a) · n|)/|n|.
Point p to line distance: (|(p-a) × d|)/|d|.

For smaller angles between lines or between planes, take the absolute scalar product of their directions or normals before dividing by the magnitudes. For the angle α between a line and a plane:

sin α = (|d · n|)/(|d||n|).

Use all three coordinate equations when testing an intersection. Parallel directions do not establish that two lines coincide; nonparallel lines in space need not intersect. This edition excludes triple products and shortest-distance/common-perpendicular problems between skew lines.

Complex numbers

Write z = x + iy, with i² = -1. Then z bar = x-iy, |z| = square root of (x² + y²) and zz bar = |z|². To divide by nonzero w, multiply numerator and denominator by w bar.

An Argand diagram puts real parts on the horizontal axis and imaginary parts on the vertical axis. The principal argument is the angle in (-π,π] for a nonzero complex number. Use both coordinate signs to choose the quadrant; zero has no argument.

For a polynomial with real coefficients, nonreal roots occur in conjugate pairs. Conjugation reflects in the real axis; multiplication by i rotates 90° anticlockwise about the origin. This edition excludes polar and exponential forms of complex numbers.

Differentiation and series approximations

Use radians for trigonometric derivatives.

(xⁿ)' = nxⁿ⁻¹ where defined; (e^x)' = e^x; (ln x)' = 1/x for x > 0.
(sin x)' = cos x; (cos x)' = - sin x; (tan x)' = sec² x.
Product rule: (uv)' = u'v + uv'.
Quotient rule: (u/v)' = (u'v-uv')/v², v ≠ 0.
Chain rule: multiply the derivative of the outer function by that of the inner function.
Parametric gradient: dy/(dx) = (dy/dt)/(dx/dt) where dx/dt ≠ 0.

For an implicit equation, differentiate every term with respect to x, treating y as a function of x. At (a,f(a)), the tangent is y-f(a) = f'(a)(x-a). A normal’s gradient is -1/f'(a) when nonzero; a horizontal tangent has a vertical normal.

A stationary point has zero first derivative. Use a gradient sign change or a nonzero second derivative to classify it. Zero second derivative alone proves neither an extremum nor an inflexion. Check endpoints and the physical domain in optimisation and connected-rate problems. This edition excludes second derivatives of parametric functions and non-stationary inflexion points.

Maclaurin expansions, with omitted higher-order terms:

e^x = 1 + x + x²/(2!) + x³/(3!) + …,

sin x = x-x³/(3!) + …, cos x = 1-x²/(2!) + …,

ln(1 + x) = x-x²/2 + x³/3-… (|x| < 1),

(1 + x)ⁿ = 1 + nx + ((n(n-1))/2)x² + … (|x| < 1).

The binomial series terminates for a nonnegative integer n, when that finite identity has no |x| < 1 restriction. The exponential, sine and cosine series converge for all real x. Recheck the convergence condition after substitution. For small x in radians, sin x ≈ x, cos x ≈ 1-x²/2 and tan x ≈ x.

Read more: Differentiation and Series approximations.

Integration and differential equations

∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C for n ≠ -1.
∫1/x dx = ln |x| + C on an interval not containing zero.
∫ f'(x)[f(x)]ⁿ dx = [f(x)]ⁿ⁺¹/(n + 1) + C for n ≠ -1, on the real domain where defined.
For n = -1, the corresponding result is ln |f(x)| + C on an interval with f(x) ≠ 0.
∫ f'(x)e^(f(x)) dx = e^(f(x)) + C.
Integration by parts: ∫ u dv = uv-∫ v du.

For a > 0:

∫dx/(a² + x²) = 1/a tan⁻¹ (x/a) + C,

∫dx/(square root of (a²-x²)) = sin⁻¹ (x/a) + C (|x| < a).

Substitution requires changing the differential as well as the expression; change the limits too when evaluating entirely in the new variable.

A definite integral is signed. For geometric area, split at intersections and roots and integrate upper minus lower. A volume of revolution about the x-axis is π∫ₐ^b y² dx for a region from the axis to the curve; for a region between curves, subtract inner squared radius from outer squared radius. About the y-axis, integrate squared radii expressed in y. This edition excludes parametric area and volume-of-revolution problems.

For dy/(dx) = f(x)g(y), separate and integrate where g(y) ≠ 0. Check solutions with g(y) = 0 separately before dividing; these equilibrium solutions can otherwise be lost. A general solution contains an arbitrary constant; a particular solution satisfies the supplied condition. Check the original equation and interpret the permitted domain.

Read more: Integration and Differential equations.

Additional counting and discrete-variable results

For n objects with indistinguishable groups of sizes n₁,…,nₖ summing to n, distinct linear arrangements number n!/(n₁!… nₖ!). For n distinct objects around a circle, with rotations identified and reflections distinct, arrangements number (n-1)!.

For a discrete variable with values x and probabilities pₓ:

∑ pₓ = 1, E(X) = ∑ xpₓ,

Var(X) = E(X²)-[E(X)]².

Each probability is nonnegative. A transformation can merge outcomes; combine probabilities when different starting values give the same result.

Counting and probability

For integers 0 ≤ r ≤ n:

ⁿPᵣ = n!/((n-r)!), (n choose r) = ⁿCᵣ = n!/(r!(n-r)!).

Use permutations when order matters and combinations when it does not. These formulas select distinct objects without replacement; 0! = 1. Restrictions need their own counting argument.

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

P(A∪ B) = P(A) + P(B)-P(A∩ B),

P(A ∣ B) = (P(A∩ B))/(P(B)) (P(B) > 0).

Mutually exclusive means P(A∩ B) = 0. Independent means P(A∩ B) = P(A)P(B). Events of positive probability cannot be both. Conditioning changes the reference group; the denominator must describe that new group.

Binomial and normal models

For X∼B(n,p), there are a fixed number n of independent trials, two outcomes per trial and a constant success probability p.

P(X = r) = (n choose r) p^r(1-p)^(n-r),

E(X) = np, Var(X) = np(1-p).

Use integer r from 0 to n. A tail such as P(X ≥ k) equals 1-P(X ≤ k-1), not 1-P(X ≤ k).

For X∼N(μ,σ²) with σ > 0, standardise by:

Z = (X-μ)/σ∼N(0,1).

The second parameter is variance; a calculator’s normal-distribution input commonly asks for standard deviation σ. For a continuous normal variable, P(X = a) = 0. Subtract cumulative probabilities for an interval and use a complement for an upper tail. Neither H1 nor H2 in this edition includes normal approximation to a binomial distribution.

For constants a,b, E(aX + b) = aE(X) + b and Var(aX + b) = a²Var(X). For independent X,Y:

Var(aX + bY) = a²Var(X) + b²Var(Y).

The variance terms add even when b is negative. Two uses of the same random variable are not independent copies: X + X = 2X has variance 4Var(X).

Sampling and mean tests

Population is the group being studied; sample is the observed subset. A simple random sample of fixed size gives every subset of that size an equal chance of selection. An unbiased estimator has expectation equal to the population quantity it estimates.

For n independent observations with population mean μ and variance σ²:

E(X bar) = μ, Var(X bar) = σ²/n.

The sample mean is exactly normal for a normal population. For a sufficiently large sample, the Central Limit Theorem supplies an approximate normal model under its conditions. A large sample does not make the individual observations normally distributed.

For observed values x₁,…,xₙ, n > 1:

x bar = (∑ xᵢ)/n, s² = (∑ xᵢ²-nx bar²)/(n-1).

For a mean test with known population standard deviation σ > 0:

z = (x bar -μ₀)/(σ/square root of n).

Here μ₀ is the mean specified in the null hypothesis H₀. This statistic is exactly standard normal under a normal-population null, or approximately so for an appropriate large-sample model. In a large-sample test with unknown variance, estimate it by s²; this is an approximation, rather than a known-variance exact test.

Choose the alternative H₁ and significance level α before using the data. A p-value is the probability, under H₀, of a result at least as extreme in the direction or directions specified by H₁. It is not the probability that H₀ is true. Reject when the unrounded p-value is at most α; otherwise report insufficient evidence against H₀. State the conclusion in the problem’s context.

Correlation and regression

Correlation describes linear association. Its coefficient r lies between −1 and 1. A value near zero need not mean no relationship: a curved relationship can have little linear correlation.

For predicting y from x, use the regression line of y on x. Reversing that line does not generally give the regression of x on y. Interpolation predicts within the observed range; extrapolation predicts outside it and needs additional justification. Correlation alone does not establish causation.

H2 also uses square, reciprocal or logarithmic transformations to investigate a linear relationship. State the transformed axes, then convert estimates back to the original variables and respect their domains.

Papers

H2 Mathematics has two papers, each of 3 hours and 100 marks. Answer every question. Paper 1 is all Pure Mathematics. Paper 2 has Section A, Pure Mathematics, and Section B, Probability and Statistics.

Part	Marks	Questions
Paper 1	100	10 to 12
Paper 2, A	40	4 to 5
Paper 2, B	60	6 to 8

Each paper has at least one question worth at least 12 marks that applies mathematics to a real-world context: in Paper 1, and in Section B of Paper 2. Questions may combine topics. The list of formulae and results, MF27, is provided in the examination.

Graphing calculator and written working

The examination expects an approved graphing calculator without a computer algebra system. The calculator does the arithmetic; your written working shows the mathematics.

Show steps when the question asks. If a question says unsupported calculator answers are not allowed, write the mathematical steps. A list of calculator commands is not working.
Record what the calculator solved. Write the equation, system, function, integral or distribution you entered, then the result. For example, write X∼B(20,0.3) and P(X ≤ 4) = 0.238, not a command name.
Sketch a graph you used. When a solution comes from a graph, sketch it with labelled axes, the relevant curves and the points you read.
Keep precision until the end. Carry unrounded values through the working and round only the final answer to the accuracy the question asks for.
Working earns marks. A wrong answer with no working scores nothing. Correct working, including a clear record of calculator use, can earn method marks.

A context question also needs a conclusion in words. After a hypothesis test, a model or a prediction, state what the result means for the situation and any limitation of the model.

Mini Mathematics checks mark the result you enter. They do not mark written working, graph sketches or interpretation, so compare those with the worked solutions yourself.