September H2 Math Revision Checklist Before the A Levels
September H2 Math Revision Checklist Before the A Levels
By September, H2 Mathematics revision should no longer be about simply covering more notes or completing more worksheets. The more useful question is whether you can recognise, start and complete unfamiliar questions without being told which method to use.
This checklist is designed as a quick diagnostic before the final A-Level revision stretch. Each section contains one question from a major H2 Mathematics area. They are deliberately not basic definition questions. The aim is to test whether your understanding is strong enough to be applied when the question is presented in a less obvious form.
Before starting: Compare the areas below with the full
H2 Math 2026 syllabus
Try each question without referring to notes. If you cannot decide how to begin within a few minutes, mark that topic for revision even if the eventual solution looks familiar.
1. Functions: Can You Work Backwards From a Composite?
Functions become more difficult when students have to think about restrictions, inverses and composites together rather than applying one formula directly.
SEPTEMBER SELF-TEST
The function f is defined by
f(x) = (2x – 1)/(x + 2), where x ≠ -2.
The function g is defined by
g(x) = √(x + 3), where x ≥ -3.
(a) Find f-1(x).
(b) State the range of f.
(c) Determine the largest possible domain of gf.
What this tests: Whether you can connect inverse functions, excluded values and composite-function domain restrictions rather than treating them as separate skills.
2. Equations and Inequalities: Can You Control the Signs?
Inequality questions often expose weak algebra because every critical point and sign change matters.
SEPTEMBER SELF-TEST
Solve
(x2 – 5x + 6) / (x2 – 1) ≤ 0.
Give your answer as a set of intervals and clearly indicate any excluded values.
What this tests: Factorisation, critical values, sign analysis and the distinction between a zero of the numerator and an undefined point.
3. Sequences and Series: Can You Connect Terms and Sums?
SEPTEMBER SELF-TEST
A sequence has partial sum
Sn = 3n2 + 2n + 4(1/2)n – 4.
(a) Find an expression for un.
(b) Determine whether the sequence un is increasing for all positive integers n.
(c) Find the least n for which un > 100.
What this tests: The relationship un = Sn – Sn-1, algebraic simplification and whether you can reason about a sequence after obtaining its general term.
4. Complex Numbers: Can You Move Between Representations?
SEPTEMBER SELF-TEST
The complex number z satisfies
z3 = -8i.
(a) Express -8i in the form reiθ.
(b) Find all possible values of z in exact polar form.
(c) Represent the three roots on an Argand diagram and explain their geometrical relationship.
What this tests: Polar form, arguments, roots of complex numbers and the geometry behind equally spaced roots.
5. Vectors: Can You Build the Geometry From the Algebra?
SEPTEMBER SELF-TEST
The line L has equation
r = (1, 2, -1) + λ(2, -1, 2).
The plane P has equation
2x + y – 2z = 7.
(a) Find the point where L intersects P.
(b) Find the acute angle between L and P.
(c) A point A has coordinates (3, 0, 4). Find the shortest distance from A to P.
What this tests: Line-plane intersection, scalar products, angles and distance. A strong student should know which vector plays which geometrical role before calculating.
6. Differentiation: Can You Recognise the Structure First?
SEPTEMBER SELF-TEST
The curve C is defined implicitly by
x2y + y3 = 10.
(a) Find dy/dx in terms of x and y.
(b) Hence find the gradient of C at the point (1, 2).
(c) Find the equation of the normal to C at this point.
What this tests: Implicit differentiation, product rule and whether you can move from a derivative to a geometrical result.
7. Applications of Differentiation: Can You Optimise a Model?
SEPTEMBER SELF-TEST
An open cylindrical container has volume 500π cm3. Its base costs twice as much per square centimetre to manufacture as its curved surface.
If the cost per unit area of the curved surface is k cents:
(a) show that the total manufacturing cost can be written as a function of radius r only;
(b) find the value of r that minimises the cost;
(c) justify that your value gives a minimum.
What this tests: Modelling, substitution, optimisation and justification rather than simply differentiating an expression already provided.
8. Integration: Can You Choose the Technique Yourself?
SEPTEMBER SELF-TEST
Evaluate
∫ x ln(x + 1) dx.
Your final answer should not contain any unevaluated integral.
What this tests: Whether you can recognise that integration by parts is useful, then manage the remaining algebra rather than stopping after setting up the formula.
9. Applications of Integration: Can You Translate a Diagram Into an Integral?
SEPTEMBER SELF-TEST
The region R is bounded by the curve
y = x(4 – x),
the x-axis and the lines x = 1 and x = 3.
The region is rotated through 2π radians about the line y = -2.
Find the exact volume of the solid formed.
What this tests: Whether you can identify the correct outer and inner radii when the axis of rotation is not the x-axis itself.
10. Differential Equations: Can You Use the Initial Condition Properly?
SEPTEMBER SELF-TEST
The population P of a colony satisfies
dP/dt = 0.2P(1 – P/5000),
where t is measured in years.
Given that P = 1000 when t = 0:
(a) solve the differential equation to obtain P in terms of t;
(b) find the time when the population first exceeds 4000.
What this tests: Separation of variables, partial fractions, constants of integration and interpreting the mathematical result in context.
11. Permutations and Probability: Can You Avoid Double Counting?
SEPTEMBER SELF-TEST
A committee of 5 is selected from 6 boys and 5 girls. Two particular boys, A and B, refuse to serve together. Three particular girls, C, D and E, must either all be selected or none of them selected.
Find the number of possible committees.
What this tests: Case construction and restrictions. If your solution becomes a long collection of unexplained combinations, check whether your cases are mutually exclusive and complete.
12. Random Variables: Can You Build a Distribution?
SEPTEMBER SELF-TEST
A game involves rolling a fair six-sided die twice. Let X be the number of rolls showing a value greater than 4, and let Y be the product of the two scores.
(a) Find P(X = 1 and Y > 15).
(b) Find E(X).
(c) Determine whether the events X = 1 and Y > 15 are independent.
What this tests: Sample-space reasoning, expectation and independence rather than the direct use of one standard probability formula.
13. Normal Distribution: Can You Work Backwards From a Probability?
SEPTEMBER SELF-TEST
The mass X grams of a product is normally distributed with mean μ and standard deviation 4.
It is known that 8% of products have mass less than 192 g.
(a) Find μ.
(b) Hence find the probability that the mass of a randomly selected product differs from μ by more than 6 g.
(c) In a box of 12 independently selected products, find the probability that at least 10 have masses within 6 g of μ.
What this tests: Inverse normal calculations followed by a change of model from a continuous normal variable to a binomial count.
14. Hypothesis Testing: Can You Interpret the Result?
SEPTEMBER SELF-TEST
A manufacturer claims that the mean lifetime of a component is 1200 hours. A technician believes that a new production process has increased the mean lifetime.
A random sample of 40 components has mean lifetime 1246 hours. The population standard deviation may be taken as 150 hours.
Test the technician’s claim at the 5% significance level. State your hypotheses clearly and write your conclusion in the context of the question.
What this tests: Choosing the direction of the test, setting hypotheses correctly, obtaining the relevant probability or critical value and interpreting statistical evidence without claiming more than the test establishes.
15. Correlation and Regression: Can You Separate Association From Prediction?
SEPTEMBER SELF-TEST
For a group of students, x represents the number of hours spent on revision and y represents their test score. A calculator gives
r = 0.82 and the regression line
y = 3.6x + 48.2.
The observed x-values lie between 2 and 12.
(a) Interpret the value of r.
(b) Estimate the score of a student who revises for 9 hours.
(c) Explain why using the same equation to predict the score of a student who revises for 25 hours may be unreliable.
(d) Explain why the information does not prove that increasing revision time causes higher scores.
What this tests: Interpretation, interpolation, extrapolation and the difference between correlation and causation.
How Should You Use This September Checklist?
Do not judge your revision only by whether you eventually obtained the correct answer. The more useful September test is how independently you reached it.
Green: You identified the approach quickly and completed most of the question without help.
Amber: You understood the solution once you saw a hint, but could not independently identify the first step.
Red: You could not recognise the relevant concept, or your algebra and working broke down before you could apply it.
Amber topics deserve almost as much attention as red ones. A common revision trap is to look at a worked solution, understand every line and assume that the topic is secure. In an A-Level paper, however, the question does not tell you which method to use.
September Is the Time to Find Weak Links
At this stage of the year, completing large numbers of practice papers can be useful, but only if the mistakes lead to targeted revision. If the same difficulty keeps appearing in functions, vectors, calculus or statistics, doing another full paper without fixing the underlying weakness may simply reproduce the same result.
Use your mistakes to identify the exact point where the solution broke down. Was the problem a missing concept, weak algebra, uncertainty over which method to choose, calculator usage, or an inability to connect two topics within the same question?
You can also compare your revision coverage against the
complete H2 Math 2026 syllabus
so that weaker topics are not overlooked simply because they have appeared less often in your recent school papers.
What Should You Prioritise Before the A Levels?
Your September revision plan should become progressively more selective. Start with the topics where you cannot independently begin a question. Next, fix recurring algebraic and presentation mistakes. Finally, practise mixed and timed questions so that choosing the correct method becomes part of the challenge.
The objective is not to feel familiar with every chapter. It is to reach the point where an unfamiliar-looking question can still be broken into mathematical ideas that you recognise and know how to use.
Need more structured H2 Math revision?
MuscleMath helps JC students identify weak areas, strengthen mathematical understanding and practise the application of methods across unfamiliar H2 Mathematics questions.
September H2 Math Checklist Answer Key
Check the answers only after attempting each question independently. This answer key gives the final results rather than full worked solutions so that you can identify which topics require another attempt or more focused revision.
Important: A correct final answer does not always mean the topic is secure. If you needed a hint, checked a formula midway, or could not identify the method independently, mark the topic for further revision.
1. Functions
(a) f-1(x) = -(2x + 1)/(x – 2)
(b) Range of f: all real values except 2
(c) Largest possible domain of g∘f:
x < -2 or x ≥ -1
2. Equations and Inequalities
-1 < x < 1 or 2 ≤ x ≤ 3
3. Sequences and Series
(a) un = 6n – 1 – 4(1/2)n
(b) The sequence is increasing for all positive integers n.
(c) Least value: n = 17
4. Complex Numbers
(a) -8i = 8e-iπ/2
(b) The three roots may be written as:
2e-iπ/6,
2eiπ/2,
2ei7π/6
(c) The roots lie on the circle |z| = 2 and are equally spaced by an angle of 2π/3.
5. Vectors
(a) Point of intersection:
(-1, 3, -3)
(b) Acute angle between the line and plane:
sin-1(1/9) ≈ 6.38°
(c) Shortest distance from A to P:
3 units
6. Differentiation
(a)
dy/dx = -2xy/(x2 + 3y2)
(b) Gradient at (1, 2):
-4/13
(c) Equation of the normal:
y – 2 = (13/4)(x – 1)
7. Applications of Differentiation
Since πr2h = 500π,
h = 500/r2.
(a) Total cost:
C = 1000πk/r + 2πkr2
(b) Minimum occurs when
r3 = 250,
giving
r = 5∛2 cm.
(c) C” > 0 for r > 0 at the stationary point, confirming a minimum.
8. Integration
∫x ln(x + 1) dx
= (x2/2)ln(x + 1)
– x2/4
+ x/2
– (1/2)ln(x + 1)
+ C
9. Applications of Integration
Using washers about y = -2:
Outer radius = x(4 – x) + 2
Inner radius = 2
Volume = 282π/5 cubic units.
10. Differential Equations
(a)
P = 5000/(1 + 4e-0.2t)
(b)
P first exceeds 4000 when
t > 5ln(16).
Therefore this occurs after approximately
13.9 years.
11. Permutations and Probability
Considering separately the cases where C, D and E are all selected and where none is selected:
Number of possible committees = 63.
12. Random Variables
(a)
P(X = 1 and Y > 15) = 1/6
(b)
E(X) = 2/3
(c)
The events are not independent.
13. Normal Distribution
(a)
μ ≈ 197.62 g
(b)
P(|X – μ| > 6) ≈ 0.1336
(c)
Probability that at least 10 of 12 products are within 6 g of μ:
≈ 0.7907
14. Hypothesis Testing
H0: μ = 1200
H1: μ > 1200
Test statistic:
z ≈ 1.940
One-tailed p-value:
≈ 0.0262
Since p < 0.05, reject H0.
There is sufficient evidence at the 5% significance level to support the technician’s claim that the new process has increased the mean lifetime.
15. Correlation and Regression
(a)
r = 0.82 indicates a strong positive linear correlation between revision time and test score in this sample.
(b)
When x = 9:
y = 80.6.
(c)
x = 25 lies well outside the observed range of 2 to 12 hours, so this would be extrapolation and may be unreliable.
(d)
Correlation alone does not establish causation. Other variables may influence both revision time and test performance.
How many topics did you complete independently?
Do not use the score as a prediction of your A-Level grade. Instead, use every question you could not start or complete independently as a signal for what should move higher on your September revision list.
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