CIE A-Level Mathematics 9709 Paper 1 (Pure Mathematics 1) Practice Paper

CIE A-Level Mathematics 9709 Paper 1 (Pure Mathematics 1) Practice Paper — 10 questions, 75 marks, 1 h 50 min

2026-10-29

CIE A-Level Mathematics 9709 Paper 1 (Pure Mathematics 1) Practice Paper — 10 questions, 75 marks, 1 h 50 min

TUTORZONE original practice paper · Cambridge A-Level Mathematics 9709 Paper 1 (Pure Mathematics 1)

Written to the Cambridge International A-Level Mathematics 9709 Paper 1 (Pure Mathematics 1) format: 10 questions, 75 marks, 1 hour 50 minutes, worth 30% of the full A-Level.

Paper at a glance

Item	Detail
Paper	Paper 1 Pure Mathematics 1
Questions	10 questions (25 parts)
Marks	75 marks
Time	1 hour 50 minutes
Weighting	30% of the A-Level
Calculator	Not permitted

Questions

Answer all ten questions and show your working. Give non-exact numerical answers to 3 significant figures unless a different accuracy is stated.

1. (6 marks)Quadratics and completing the square

(a) Solve the equation 2x² − 7x + 3 = 0. (3 marks)

(b) Express 2x² − 12x + 7 in the form a(x + b)² + c, and state the minimum value of 2x² − 12x + 7. (3 marks)

2. (7 marks)Binomial expansion

(a) Find the first three terms, in ascending powers of x, of the expansion of (1 + 2x)⁸. (3 marks)

(b) Find the coefficient of x³ in the expansion. (2 marks)

3. (7 marks)Differentiation and stationary points

(a) A curve has equation y = x³ − 6x² + 9x + 2. Find dy/dx. (2 marks)

(b) Find the coordinates of the stationary points of the curve. (3 marks)

4. (8 marks)Equation of a circle

(a) The circle with equation x² + y² − 6x + 10y + 18 = 0 has centre C and radius r. Find the coordinates of C and the value of r. (4 marks)

(b) The point P has coordinates (10, −5). Find the shortest distance from P to the circle. (4 marks)

5. (8 marks)Trigonometric equations

(a) Show that the equation 3sin²θ − 2cosθ − 2 = 0 may be written as 3cos²θ + 2cosθ − 1 = 0. (3 marks)

(b) Hence solve 3cos²θ + 2cosθ − 1 = 0 for cosθ. (2 marks)

6. (8 marks)Definite integration and areas

(a) The curve y = x² + 2 crosses the lines x = 1 and x = 3. Find the area of the region bounded by the curve, the x-axis and these two lines. (4 marks)

(b) Find the area of the region enclosed between y = x² + 2 and y = 3x + 2 for 0 ≤ x ≤ 3. (4 marks)

7. (8 marks)Geometric progressions

(a) A geometric progression has first term 3 and common ratio 2/3. Find the sum to infinity. (3 marks)

(b) Find the sum of the first 5 terms, giving your answer as an exact fraction. (3 marks)

8. (8 marks)Factor theorem and factorisation

(a) The polynomial f(x) = 2x³ + ax² + bx − 6 has factors (x − 1) and (x + 2). Find the values of a and b. (5 marks)

(b) Hence factorise f(x) completely. (3 marks)

9. (8 marks)Chain rule and connected rates of change

(a) Given that y = (3x − 1)⁶, find dy/dx and hence the value of dy/dx at x = 1. (4 marks)

(b) The radius of a circle increases at 0.5 cm per second. Find the rate of increase of the area when the radius is 6 cm, giving your answer in exact form. (4 marks)

10. (7 marks)Vectors

(a) The points A and B have coordinates (2, −1) and (5, 3). Find the vector AB and its magnitude. (3 marks)

(b) Find a unit vector in the direction of AB. (2 marks)

Answers and mark scheme

Reference answers with the mark allocation: M marks for method, A marks for accuracy, B marks for independent accuracy.

Question 1 · 6 marks

(a) x = 1/2 or x = 3

M1 factorises correctly as (2x − 1)(x − 3), or uses the formula with a = 2, b = −7, c = 3; A1 both roots correct.

(b) 2(x − 3)² − 11; minimum value −11 (at x = 3)

M1 takes out the factor 2 and completes the square; A1 a = 2, b = −3, c = −11; A1 minimum value −11 stated (allow “minimum at x = 3”).

Question 2 · 7 marks

(a) 1 + 16x + 112x²

M1 uses the binomial coefficients ⁱC₁ and ⁱC₂; A1 16x; A1 +112x².

(b) 448

M1 ⁱC₃ × 2³ = 56 × 8; A1 448.

M1 substitutes x = 0.01 in the three terms; A1 1.1712.

Question 3 · 7 marks

(a) 3x² − 12x + 9

M1 differentiates term by term; A1 3x² − 12x + 9.

(b) (1, 6) and (3, 2)

M1 sets dy/dx = 0 and factorises 3(x − 1)(x − 3); A1 x = 1, 3; A1 both y-values correct.

M1 finds d²y/dx² = 6x − 12; A1 correct classification of both points.

Question 4 · 8 marks

(a) C(3, −5), r = 4

M1 completes the square in x and in y; A1 centre (3, −5); M1 r² = 9 + 25 − 18; A1 r = 4.

(b) 3 units

M1 distance CP = 7; M1 subtracts the radius; A1 3.

Question 5 · 8 marks

(a) 3(1 − cos²θ) − 2cosθ − 2 = 0, then collect terms

M1 uses sin²θ = 1 − cos²θ; A1 expands; A1 reaches the printed form with all signs correct.

(b) cosθ = 1/3 or cosθ = −1

M1 factorises (3cosθ − 1)(cosθ + 1); A1 both values.

B1 θ = 180°; M1 θ = 70.5°; A1 289.5° (all to 1 d.p., no extras in range).

Question 6 · 8 marks

(a) 38/3 ≈ 12.67 square units

M1 integrates to x³/3 + 2x; M1 substitutes 3 and 1; A1 15 − 7/3; A1 38/3.

(b) 9/2 = 4.5 square units

M1 forms the difference 3x − x²; M1 integrates; A1 27/2 − 9; A1 9/2.

Question 7 · 8 marks

(a) S∞ = 9

M1 uses a/(1 − r) with r = 2/3; A1 3 ÷ (1/3) = 9; A1 states the condition that r is between −1 and 1.

(b) 211/27 ≈ 7.815

M1 uses Sₙ = a(1 − rⁿ)/(1 − r); M1 substitutes n = 5 and r = 2/3; A1 211/27.

M1 solves or tests the inequality; A1 n = 12.

Question 8 · 8 marks

(a) a = 5, b = −1

M1 f(1) = 0 gives a + b = 4; M1 f(−2) = 0 gives 4a − 2b = 22; M1 solves the pair; A1 a = 5; A1 b = −1.

(b) f(x) = (x − 1)(x + 2)(2x + 3)

M1 divides out one factor; A1 reaches (x − 1)(x + 2)(2x + 3); A1 all three factors correct.

Question 9 · 8 marks

(a) dy/dx = 18(3x − 1)⁵; at x = 1 the value is 576

M1 uses the chain rule; A1 18(3x − 1)⁵; M1 substitutes x = 1; A1 576.

(b) 6π ≈ 18.85 cm² per second

M1 dA/dr = 2πr; M1 multiplies by dr/dt; A1 6π; A1 cm² per second with correct units.

Question 10 · 7 marks

(a) AB = (3, 4); magnitude 5

M1 subtracts coordinates; A1 (3, 4); A1 5.

(b) (3/5, 4/5)

M1 divides AB by 5; A1 (3/5, 4/5).

M1 sets up k/3 = 8/4; A1 k = 6.

Common mistakes

Taking the 2 out only after completing the square gives a wrong value of c, and the minimum value is then wrong.
Giving only one of the three roots in the range, or quoting a root outside 0° to 360°.
Integrating the wrong way round so the answer comes out negative — compare which curve is above first.
Forgetting the inner derivative 3 in the chain rule.
Quoting dA/dr without multiplying by dr/dt, or omitting the unit cm² per second.

Time allocation

Questions	Marks	Suggested time
Q1-Q2	13 marks	about 19 min
Q3-Q4	15 marks	about 22 min
Q5-Q6	16 marks	about 23 min
Q7-Q8	16 marks	about 23 min
Q9-Q10	15 marks	about 22 min

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Question paper PDF · Answers and mark scheme PDF

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