Quick Answer: Hard A-Level Maths questions are hard because they give little guidance: you choose the method, link two topics and justify each step. Below are 10 original questions in the style of the harder questions on Pearson Edexcel, AQA and Cambridge International A Level papers: six pure, two statistics and two mechanics. Each has a full worked solution and the mistake that costs students the most marks.
This post is for Year 12 and Year 13 students aiming for an A or A*, and for parents who want to see what the top end of the course looks like. If you are still building up from GCSE, start with our hard GCSE maths questions with answers. All ten questions were written for this post and are not taken from past papers. Use a calculator where needed, and take g = 9.8 m/s².
10 Hard A-Level Maths Questions With Answers
1. Pure: proof by exhaustion
Prove that n² + 2 is never divisible by 4 for any integer n.
Answer:
Every integer is either even or odd.
If n is even, n = 2k, so n² + 2 = 4k² + 2. This leaves remainder 2 when divided by 4.
If n is odd, n = 2k + 1, so n² + 2 = 4k² + 4k + 3 = 4(k² + k) + 3. This leaves remainder 3.
In neither case is the remainder 0, so n² + 2 is never divisible by 4. ∎
Common mistake: Checking n = 1, 2, 3, 4 and stopping. A proof by exhaustion must cover every case, so you split all integers into even and odd, not a list of examples.
Typical of: Edexcel and AQA, where proof is examined explicitly in pure papers.
2. Pure: integration by parts twice
Find the exact value of ∫₀^(π/2) x² sin x dx.
Answer:
Let u = x², dv/dx = sin x. Then ∫x² sin x dx = −x² cos x + ∫2x cos x dx.
Parts again with u = 2x, dv/dx = cos x: ∫2x cos x dx = 2x sin x − ∫2 sin x dx = 2x sin x + 2 cos x.
So ∫x² sin x dx = −x² cos x + 2x sin x + 2 cos x + c.
At x = π/2: 0 + π + 0 = π. At x = 0: 0 + 0 + 2 = 2.
Answer = π − 2 (about 1.14)
Common mistake: Sign errors. The integral of sin x is −cos x, and the minus sign in the parts formula then makes the second term positive. Writing each line in full, rather than doing signs in your head, avoids most slips.
Typical of: All three boards.
3. Pure: implicit differentiation
A curve has equation x² + xy + y² = 7.
(a) Show that (1, 2) lies on the curve and find the equation of the tangent there, in the form ax + by + c = 0. (b) Find the exact coordinates of the points where the tangent is horizontal.
Answer:
(a) 1 + 2 + 4 = 7 ✓. Differentiating: 2x + y + x(dy/dx) + 2y(dy/dx) = 0, so dy/dx = −(2x + y)/(x + 2y).
At (1, 2): dy/dx = −4/5. Tangent: y − 2 = −⅘(x − 1), which gives 4x + 5y − 14 = 0.
(b) Horizontal when 2x + y = 0, so y = −2x. Substitute: x² − 2x² + 4x² = 7, so 3x² = 7 and x = ±√21/3.
Points: (√21/3, −2√21/3) and (−√21/3, 2√21/3)
Common mistake: Differentiating xy as x(dy/dx) only. It needs the product rule: y + x(dy/dx). In (b), students also forget to substitute back into the curve equation.
Typical of: All three boards.
4. Pure: harmonic form (R cos(θ − α))
(a) Express 3 cos θ + 4 sin θ in the form R cos(θ − α), where R > 0 and 0° < α < 90°. (b) Solve 3 cos θ + 4 sin θ = 2 for 0° ≤ θ < 360°. (c) Find the maximum value of 1/(10 − 3 cos θ − 4 sin θ).
Answer:
(a) R cos α = 3 and R sin α = 4, so R = √(9 + 16) = 5 and tan α = 4/3, α = 53.13°. 5 cos(θ − 53.13°)
(b) cos(θ − 53.13°) = 0.4, so θ − 53.13° = 66.42° or −66.42° (equivalently 293.58°).
θ = 119.6° or θ = 346.7° (1 d.p.)
(c) The denominator is 10 − 5 cos(θ − α). Its smallest value is 10 − 5 = 5, so the maximum value is 1/5, when θ = 53.1°.
Common mistake: Missing the second solution in (b) by only taking the principal value of arccos. In (c), students take the maximum of the denominator instead of the minimum.
Typical of: All three boards. Edexcel and Cambridge often add a modelling or "maximum value" part.
5. Pure: partial fractions and binomial expansion
(a) Express (7x + 1)/((1 + x)(1 − 2x)) in partial fractions. (b) Hence find the expansion in ascending powers of x up to x³, and state the values of x for which it is valid.
Answer:
(a) 7x + 1 = A(1 − 2x) + B(1 + x). Put x = −1: −6 = 3A, so A = −2. Put x = ½: 4.5 = 1.5B, so B = 3.
−2/(1 + x) + 3/(1 − 2x)
(b) −2(1 + x)⁻¹ = −2 + 2x − 2x² + 2x³ and 3(1 − 2x)⁻¹ = 3 + 6x + 12x² + 24x³.
1 + 8x + 10x² + 26x³, valid for |x| < ½
Common mistake: Giving the validity as |x| < 1. Both expansions must be valid, so you take the stricter condition from (1 − 2x)⁻¹, which is |2x| < 1.
Typical of: All three boards.
6. Pure: geometric series
The second term of a geometric series is 12 and its sum to infinity is 64.
(a) Find the two possible values of the common ratio r and the first term in each case. (b) For the series with the larger r, find the smallest n for which the sum of the first n terms exceeds 63.
Answer:
(a) ar = 12 and a/(1 − r) = 64. Substituting a = 12/r gives 12 = 64r(1 − r), so 16r² − 16r + 3 = 0 and (4r − 1)(4r − 3) = 0.
r = ¼ with a = 48, or r = ¾ with a = 16. Both satisfy |r| < 1, so both are valid.
(b) Sₙ = 64(1 − 0.75ⁿ) > 63, so 0.75ⁿ < 1/64. Taking logs: n ln 0.75 < ln(1/64). ln 0.75 is negative, so the inequality flips: n > 14.46. n = 15
Check: S₁₄ ≈ 62.86 and S₁₅ ≈ 63.14.
Common mistake: Forgetting to reverse the inequality when dividing by ln 0.75, which gives n < 14.46. Students also reject one value of r without a reason.
Typical of: All three boards.
7. Statistics: binomial hypothesis test
A seed company claims that 20% of its seeds fail to germinate. A gardener thinks the true proportion is lower. In a random sample of 25 seeds, 1 fails. Test the gardener's claim at the 5% significance level, and state the critical region.
Answer:
Let X be the number of seeds that fail. Under H₀, X ~ B(25, 0.2).
H₀: p = 0.2, H₁: p < 0.2 (one-tailed)
P(X ≤ 1) = 0.8²⁵ + 25(0.2)(0.8²⁴) = 0.0274
0.0274 < 0.05, so the result is significant. Reject H₀: there is evidence at the 5% level that fewer than 20% of seeds fail.
Critical region: P(X ≤ 1) = 0.0274 and P(X ≤ 2) = 0.0982, so the critical region is X ≤ 1 (actual significance level 2.74%).
Common mistake: Calculating P(X = 1) instead of P(X ≤ 1). A test needs the probability of a result at least as extreme as the one seen. Students also write a conclusion with no context, which loses the final mark.
Typical of: Edexcel and AQA (Year 2 Statistics) and Cambridge Probability & Statistics.
8. Statistics: normal distribution with two unknowns
X ~ N(μ, σ²). P(X < 30) = 0.1 and P(X > 50) = 0.2. Find μ and σ.
Answer:
P(Z < z) = 0.1 gives z = −1.2816. P(Z < z) = 0.8 gives z = 0.8416.
(30 − μ)/σ = −1.2816 and (50 − μ)/σ = 0.8416
Subtracting: 20/σ = 2.1232, so σ = 9.42. Then μ = 30 + 1.2816 × 9.42 = 42.1 (3 s.f.)
Common mistake: Using z = 0.2 or z = −0.8416 for P(X > 50) = 0.2. The upper tail of 0.2 means P(X < 50) = 0.8, which gives a positive z because 50 is above the mean.
Typical of: All three boards.
9. Mechanics: projectile from a cliff
A stone is thrown from the top of a cliff 25 m above the sea, at 20 m/s and 30° above the horizontal. Find (a) the time until it hits the sea, (b) its horizontal distance from the foot of the cliff and (c) its speed on impact.
Answer:
Initial components: horizontal 20 cos 30° = 17.32 m/s; vertical 20 sin 30° = 10 m/s upwards.
(a) Taking up as positive: −25 = 10t − 4.9t², so 4.9t² − 10t − 25 = 0.
t = (10 + √(100 + 490))/9.8 = 3.50 s (the negative root is rejected)
(b) 17.32 × 3.499 = 60.6 m
(c) By energy, or v² = u² + 2as vertically: v² = 20² + 2 × 9.8 × 25 = 890, so v = 29.8 m/s
Common mistake: Setting the displacement as +25 instead of −25 when up is positive. Pick one direction as positive and keep to it.
Typical of: Edexcel and AQA Mechanics, and Cambridge Mechanics.
10. Mechanics: connected particles with friction
Block A (3 kg) rests on a rough horizontal table (μ = 0.2). A light inextensible string over a smooth pulley at the edge joins A to block B (2 kg), which hangs 0.5 m above the floor. The system is released from rest.
Find (a) the acceleration and the tension, and (b) the total distance A moves before it stops, assuming it doesn't reach the pulley.
Answer:
Friction on A = 0.2 × 3 × 9.8 = 5.88 N.
(a) For B: 2g − T = 2a. For A: T − 5.88 = 3a. Adding: 19.6 − 5.88 = 5a, so a = 2.744 ≈ 2.74 m/s² and T = 2(9.8 − 2.744) = 14.1 N
(b) When B hits the floor: v² = 2 × 2.744 × 0.5 = 2.744, so v = 1.66 m/s. The string goes slack, and friction alone slows A at 0.2 × 9.8 = 1.96 m/s². Extra distance = 2.744 ÷ (2 × 1.96) = 0.70 m.
Total = 0.5 + 0.70 = 1.2 m
Common mistake: Assuming the tension equals B's weight (19.6 N). B is accelerating, so the tension must be less. In (b), students keep using 2.74 m/s² after B lands.
Typical of: All three boards.
How to Get Better at Hard A-Level Questions
- Learn proof as a skill. Practise writing complete arguments, not checking examples.
- Watch inequalities. Dividing by a negative (such as ln 0.75) flips the sign.
- Find every solution. For trig equations, sketch or use symmetry to check the interval.
- Write hypotheses and conclusions in context. These are easy marks that are often lost.
- Draw force diagrams. In mechanics, a clear diagram prevents most sign and direction errors.
How Improve ME Institute Helps
Want questions pitched at your exam? Our A-Level groups teach Mathematics and Further Mathematics in small groups of a maximum of six, matched to each student's board. With 40+ specialist tutors and 1,000+ students every year, we start every student with a free Diagnostic Test. For a different kind of challenge, read hard maths questions that actually make you better at maths.
Frequently Asked Questions
Q: How hard is A-Level Maths compared with GCSE?
A: It's a big step up. A-Level assumes fluent GCSE algebra and moves quickly into calculus, proof and modelling. Students who were secure at grade 8–9 usually adapt well, while gaps in algebra show up very quickly.
Q: What are the hardest A-Level Maths topics?
A: Students most often struggle with proof, integration techniques, parametric and implicit differentiation, hypothesis testing and mechanics problems involving friction or connected particles. They need understanding rather than memorised steps.
Q: Is the content the same on Edexcel, AQA and Cambridge?
A: Edexcel and AQA follow the same core content for A Level Mathematics in England, so the pure, statistics and mechanics topics are almost identical. Cambridge International A Level Mathematics (9709) has a different structure and route options, so check your syllabus.
Q: How many past papers should I do for an A*?
A: There is no fixed number. Most strong students work through several years of their board's papers under timed conditions, then redo the questions they got wrong a week later until they can do them without help.
Q: Do I need A-Level Further Maths for a maths-heavy degree?
A: Many competitive maths, engineering and physics courses prefer or require Further Maths. Check each university's entry requirements.
Ready for the Top of the Paper?
Pick the question you found hardest and try it again tomorrow without looking at the solution. If the same topics keep costing marks, a free Diagnostic Test at Improve ME Institute in Gold & Diamond Park, Al Quoz, will show where the marks are being lost.
