From A Math to H2 Math: A Worked-Example Preparation Guide

For the 2026 syllabuses, A Math is organised into algebra, geometry and trigonometry, and calculus. H2 adds areas such as inverse and composite functions, sequences and series, three-dimensional vectors, complex numbers, and probability and statistics; it also extends calculus. Matrices and probability should not be listed as strands of the A Math syllabus itself.

Both courses assess reasoning, communication and application. A Math is not merely memorising formulae, and H2 is not merely a longer list of harder calculations. The examples below isolate two useful extensions rather than representing the full difficulty range of either course.

Sources: SEAB 4049, pages 3–8 and SEAB 9758, pages 3 and 6–15. For a fuller topic reference, use the H2 curriculum breakdown.

In the 2026 examination, A Math has two compulsory papers, each 2 hours 15 minutes, 90 marks and 50% of the total. H2 has two compulsory papers, each 3 hours, 100 marks and 50%. H2 Paper 1 is Pure Mathematics; Paper 2 has 40 marks of Pure Mathematics and 60 marks of Probability and Statistics.

H2 expects access to an approved graphing calculator. Practise interpreting its output and following instructions about analytical working; obtaining a number does not answer every kind of question. See SEAB 4049, page 5 and SEAB 9758, pages 4–5.

Build endurance progressively through mixed practice. Keep probability and statistics in your eventual revision plan rather than assuming strong calculus covers the whole H2 course.

A Math foundation. Find the minimum of $x^2-6x+13$ for real $x$.

$$x^2-6x+13=(x-3)^2+4.$$

The square is non-negative, so the minimum is ${4}$ at $x=3$.

H2 extension. Let $f(x)=(x-3)^2+4$ with domain $x\ge3$. Find its range and inverse function.

The range is $[4,\infty)$. On $[3,\infty)$, the function is one-to-one: as $x$ increases, $x-3$ is non-negative and its square increases. From $y=(x-3)^2+4$,

$$x-3=\sqrt{y-4},\qquad x=3+\sqrt{y-4}.$$

The positive square root is required by $x\ge3$. Interchanging the input and output labels gives

$$f^{-1}(x)=3+\sqrt{x-4},\qquad x\ge4.$$

Its range is $[3,\infty)$. As a check, $f(5)=8$ and $f^{-1}(8)=5$.

What develops: Completing the square still does the algebraic work, but you must now connect domain, range and a unique inverse. Without the restriction, $f(2)=f(4)=5$, so the function on all real numbers has no inverse function.

Wrong turn: Keeping both square-root branches does not give a single-valued inverse. If the original domain were $x\le3$, the inverse would instead use ${3-\sqrt{x-4}}$, still with inverse domain $x\ge4$. The domain decides the branch.

A Math foundation. Find the tangent to $y=x^2+1$ at $x=2$.

The point is $(2,5)$ and $\frac{dy}{dx}=2x$, so the gradient is ${4}$. Hence

$$y-5=4(x-2),\qquad y=4x-3.$$

H2 extension. A curve is defined by $x=t^2+1$ and $y=t^3-t$. Find its tangent at $t=2$.

The point is $(5,6)$. Differentiate each coordinate with respect to the same parameter:

$$\frac{dx}{dt}=2t,\qquad\frac{dy}{dt}=3t^2-1.$$

By the chain rule, where $\frac{dx}{dt}\ne0$,

$$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}=\frac{3t^2-1}{2t}.$$

At $t=2$, the gradient is ${11/4}$. Therefore

$$y-6=\frac{11}{4}(x-5).$$

What develops: You still need a point and a gradient. The new decision is which variable each derivative refers to. The value $dy/dt=11$ is not the tangent gradient with respect to $x$; divide by $dx/dt=4$.

Restriction: This quotient cannot be used at $t=0$ because $dx/dt=0$. That does not make the whole curve invalid; that point needs separate analysis. Here the requested point has $t=2$, so the quotient is valid.

Use the A Math differentiation guide if finding a derivative or forming a tangent equation is still difficult. Do not read difficulty with the new parameter idea as proof that all your calculus is weak.

Try these tasks without copying a solution. Mark each “independent”, “needed a hint” or “needs repair”. This is a preparation checklist, not an admissions test.

• Factorise, simplify algebraic fractions and state excluded values.

• Complete the square and connect the result to a graph's turning point.
• Apply index and logarithm laws, checking real-number domains.
• Use trigonometric identities and find all solutions in a stated interval; distinguish degrees from radians.
• Differentiate standard A Math functions and form a tangent equation.
• Integrate a familiar expression and differentiate the result to check it.
• Explain why a method fits the question and write a conclusion that answers it.

You do not need to know parametric differentiation before learning it in H2. The paired examples help you separate missing foundations from genuinely new content. The A Math formula guide is a reference for repairing the foundations.

1. Diagnose before pre-learning. Attempt the A Math halves of both examples. Use the A Math improvement diagnostic if the errors are hard to classify.
2. Repair algebra and graphs. Work on fractions, quadratics, logarithms and the meaning of a graph. Retest with changed numbers and no solution visible.
3. Reconnect trigonometry and calculus. Review identities, angle units, derivatives, integrals and their interpretations. Explain each choice in writing.
4. Preview one H2 extension at a time. Try the inverse-function example, then the parametric one. Compare what stayed the same with what needed a new idea.
5. Follow your school programme once JC starts. Use the H2 revision-order guide to identify dependencies when gaps appear. Learn how to use the MF27 reference booklet alongside actual questions.

This is a suggested preparation sequence, not a SEAB teaching order or a fixed holiday timetable. Move on when you can explain and reproduce the foundation independently; ask for help when the same gap returns.

SEAB's H2 syllabus assumes O-Level Mathematics knowledge and lists assumed A Math content on pages 14–15. That tells you what the course builds on; it does not set a universal college admission grade. See SEAB 9758, page 6 and the assumed-knowledge appendix.

Check your intended college's current subject-combination requirements directly before deciding what you are eligible to take. This guide does not prescribe a minimum grade or promise entry into H2 Math.

If you are still in secondary school and the foundation examples expose repeated gaps, review A Math tuition at Tim Gan Math for the secondary-level course. Bring your attempted questions to any discussion about support.

If you are moving into JC and need a structured route through the new material, explore H2 Math tuition at Tim Gan Math. Use your readiness checklist to explain which foundations are secure and which new ideas need guidance.