A Level Mathematics (9709) topical questions with answers
Every A Level Mathematics topic, with free past-paper-style questions and full mark schemes. Pick a chapter to practise its questions, or generate a complete 9709-format paper in seconds, from easy warm-ups to A*-level "brutal" papers.
Paper 3 · Pure Mathematics 3 (75 marks · 1h 50m); Paper 4 · Mechanics (50 marks · 1h 15m); Paper 5 · Probability & Statistics 1 (50 marks · 1h 15m); Paper 6 · Probability & Statistics 2 (50 marks · 1h 15m).
Sample A Level Mathematics questions
A mixed set from across the syllabus. Questions marked A* challenge come from the hardest bank.
Question 1Structured · 12 marksA* challenge
For n ≥ 0, let In = ∫0π/4 tann x dx.
(a)
Show that, for n ≥ 2, In + In − 2 = 1n − 1.
[4]
(b)
Prove that, for n ≥ 2, 12(n + 1) < In < 12(n − 1).
[4]
(c)
Hence show that 982315 < π < 334105.
[4]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
tann − 2 x(tan2 x + 1) = tann − 2 x sec2 x
In + In − 2 = ∫ tann − 2 x sec2 x dx M1 substitution u = tan x (or recognising the derivative) M1 1n − 1tann − 1 x between 0 and π4A1 = 1n − 1A1
4
(b)
In + 2 < In < In − 2
0 < tan x < 1 on 0 < x < π4, so In + 2 < InB1 In + In − 2 = 1n − 1 with In < In − 2 gives In < 12(n − 1)M1 In + 2 + In = 1n + 1 with In + 2 < In gives In > 12(n + 1)M1 both inequalities fully justified A1
4
(c)
I8 = π4 − 76105; 118 < I8 < 114
I0 = π4 and repeated use of (a) M1 I8 = π4 − 76105A1 n = 8 in (b): 118 < π4 − 76105 < 114M1 rearranged to the given result A1
4
Question 2Structured · 8 marksA* challenge
A chord AB of a circle with centre O and radius r divides the circle into two regions whose areas are in the ratio 1 : 5. Angle AOB = θ radians.
(a)
Show that θ = sin θ + π3.
[3]
(b)
Show by calculation that the equation has a root between 1.9 and 2.0.
[2]
(c)
Use the iterative formula θn+1 = ½(θn + sin θn + π3) with θ1 = 2 to find θ correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
½r2(θ − sin θ) = 16πr2
segment area = ½r2θ − ½r2 sin θ M1 equated to πr26M1 rearranged to the given form A1
3
(b)
sign change of θ − sin θ − π3 between 1.9 and 2.0
evaluate θ − sin θ − π3 at both ends: −0.093 and 0.044 M1 sign change and conclusion A1
u = 3xM1 u2 − 28u + 27 = 0 A1 u = 1, u = 27 A1 x = 0, x = 3 A1
4
Question 6Structured · 5 marks
The number of calls to a helpline in a 5-minute period has the distribution Po(2.7).
(a)
Find the probability of exactly 3 calls in a 5-minute period.
[2]
(b)
Find the probability of at least 2 calls in a 10-minute period.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
0.22
e−2.7 × 2.73 ÷ 3! M1 0.22 A1
2
(b)
0.971
λ = 5.4 B1 1 − P(0) − P(1) M1 0.971 A1
3
Frequently asked questions
Are these real 9709 past paper questions?
No. They are original questions written to match the A Level Mathematics 9709 syllabus and the style of real Cambridge papers, so you can practise beyond the past papers you have already done.
Do the Mathematics questions have answers?
Yes. Every question on these pages has its answer, and every generated paper comes with a full Cambridge-style mark scheme you can download as a PDF.
Can I make a full A Level Mathematics practice paper?
Yes. Choose the paper (Paper 3 Pure Mathematics 3, Paper 4 Mechanics, Paper 5 Probability & Statistics 1, Paper 6 Probability & Statistics 2), the topics, the marks and the difficulty, and the generator builds a fresh paper with the same number and style of questions as the real exam.
Want a mentor, not just a paper?
Riyaz Shaikh has mentored IGCSE, IB and A Level students for 15+ years, one-on-one, online and in Mumbai. The hardest questions on this site are the ones he uses to find a student's weak points.
Original Cambridge-style practice questions written for mentorriyaz.in. These are not Cambridge past paper questions, and this site is not affiliated with or endorsed by Cambridge University Press & Assessment. Updated 2026-11-04. Paper generator · mentorriyaz.in