H1 Mathematics revision reference

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This lookup is for 2027 H1 Mathematics (8865). It summarises selected relationships and their conditions, with links back to the teaching. It is not the official examination formula list.

Functions, logarithms and quadratics

A function assigns one output to each input. y = e^x is equivalent to x = ln y; a real logarithm has a positive argument.

For positive x,y:

ln(xy) = ln x + ln y, ln(x/y) = ln x- ln y, ln(x^a) = a ln x.

An exponential model y = Ae^kt grows for k > 0 and decays for k < 0 when A > 0. Interpret the variables and the unit of t before finding k.

For ax² + bx + c, with a ≠ 0, the discriminant D = b²-4ac gives two distinct real roots if D > 0, a repeated root if D = 0 and no real roots if D < 0. The expression is always positive if a > 0,D < 0, and always negative if a < 0,D < 0.

To solve a quadratic inequality, use roots and interval signs, not just the root values. For a graph-based solution, choose a window that shows the required intersections and check numerical answers by substitution.

H1 in this edition excludes domain-and-range problems, mapping notation and change-of-base logarithms. Those H2 or Additional Mathematics techniques are not added to this reference.

Differentiation

(d/(dx))xⁿ = nxⁿ⁻¹, (d/(dx))e^x = e^x, d/(dx) ln x = 1/x (x > 0).

Use the real domain where a power is differentiable. Apply constant multiples, sums and differences term by term. The chain rule multiplies the derivative of the outer function by that of the inner function; for example, (e^3x)' = 3e^3x.

At (a,f(a)), the tangent is:

y-f(a) = f'(a)(x-a).

The derivative is positive on an increasing interval and negative on a decreasing one. At a stationary point, f'(a) = 0; a change from positive to negative gives a local maximum, and negative to positive gives a local minimum. Alternatively, f''(a) < 0 or f''(a) > 0 discriminates when nonzero. A zero second derivative is inconclusive. Check endpoints as well as stationary points in an optimisation problem.

H1 excludes product and quotient rules, first-principles derivatives and implicit or parametric differentiation. The syllabus lists tangents to curves but not normals. Use the methods taught in the H1 lessons.

Integration and area

∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C (n ≠ -1),

∫ e^(ax + b) dx = 1/a e^(ax + b) + C (a ≠ 0),

∫(ax + b)ⁿ dx = ((ax + b)ⁿ⁺¹)/(a(n + 1)) + C (a ≠ 0, n ≠ -1).

The power n = -1 needs a logarithmic antiderivative: ∫1/x dx = ln |x| + C on an interval not containing zero. C is the constant of integration; find it from a supplied value and differentiate to check an antiderivative.

For an antiderivative F, ∫ₐ^b f(x) dx = F(b)-F(a). For an area between curves in the permitted region, integrate upper minus lower, splitting where that order changes. H1’s held syllabus excludes area below the x-axis and the trapezium rule.

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.

Paper 1

H1 Mathematics has one paper of 3 hours and 100 marks. Answer every question. Section A is Pure Mathematics and Section B is Probability and Statistics.

Section	Marks	Questions
A	40	about 5
B	60	6 to 8

At least one question applies mathematics to a real-world context and is worth at least 12 marks. It can appear in either section. 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.