CIE A-Level Mathematics 9709 Paper 1 Pure Maths Practice Paper (Set B) | 75 marks, 1 h 50 min

AS Pure Mathematics practice

CIE A-Level · Mathematics 9709 · Paper 1 Pure Mathematics 1 (Set B)

Data and formulae

Questions

Total 75 marks · 1 h 50 min · show all working; exact values preferred

1. Given that f(x) = 2x² − 8x + 3.
(i) Express f(x) in the form a(x + b)² + c.
(ii) Hence state the coordinates of the vertex of y = f(x) and the minimum value of f(x). [5]

2. In the expansion of (1 + 3x)⁶, write down the first four terms in ascending powers of x, and state the coefficient of x². [5]

3. The points A(−1, 4) and B(5, 2) are given.
(i) Find the midpoint M of AB and the length AB in surd form.
(ii) Find the gradient of AB and hence the equation of the line through B perpendicular to AB, in the form y = mx + k. [6]

4. A sector has radius 6 cm and angle 1.2 rad.
(i) Find the arc length, the perimeter and the area of the sector (area correct to 3 decimal places).
(ii) Find the area of the segment cut off by the chord of the sector. [7]

5. Solve the equation 4 sin²x = 1 for 0° ≤ x ≤ 360°, giving exact answers in degrees. [7]

6. A curve has equation y = x³ − 6x² + 9x.
(i) Find dy/dx and the coordinates of the stationary points.
(ii) Use the second derivative to determine the nature of each stationary point, and state the corresponding y values. [7]

7. Evaluate ∫₁⁴ (3x² − 4x + 1) dx and state the geometrical meaning of the result. [8]

8. (i) An arithmetic progression has first term 7 and common difference 3. Find the 20th term and the sum of the first 20 terms.
(ii) A geometric progression is 12, 8, 16/3, … Find the common ratio and the sum to infinity. [8]

9. The functions f and g are defined by f(x) = 2x + 3 and g(x) = x² − 1.
(i) Find gf(x) in expanded form.
(ii) Find f⁻¹(x).
(iii) Solve the equation gf(x) = 0. [8]

10. Referred to an origin O, the position vectors of A and B are OA = 2i + j and OB = 6i − 5j.
(i) Find the vector AB.
(ii) Find |AB| in surd form.
(iii) Find the unit vector in the direction of AB.
(iv) M is the midpoint of AB; find the position vector OM. [9]

11. Given that f(x) = x³ − 4x² + x + 6.
(i) Show that (x + 1) is a factor of f(x).
(ii) Factorise f(x) completely and state all the roots of f(x) = 0. [5]

Answers and mark scheme

Q1（5 marks） Quadratics and completing the square

(i) f(x) = 2(x² − 4x) + 3 = 2[(x − 2)² − 4] + 3 = 2(x − 2)² − 5 (B1 factor 2, M1 complete the square, A1 constant).
(ii) Vertex (2, −5), minimum value −5 (A1, A1). Check: f(2) = 8 − 16 + 3 = −5.

Q2（5 marks） Binomial expansion

(1 + 3x)⁶ = 1 + 6(3x) + 15(3x)² + 20(3x)³ + … = 1 + 18x + 135x² + 540x³ + … (M1 ⁶Cᵣ, A1 18x, A1 135x², A1 540x³). Coefficient of x² = 135.

Q3（6 marks） Coordinate geometry

(i) M = ((−1 + 5)/2, (4 + 2)/2) = (2, 3) (M1 A1); AB = √(6² + (−2)²) = √40 = 2√10 (M1 A1).
(ii) m_AB = (2 − 4)/(5 − (−1)) = −1/3; perpendicular gradient = 3; y − 2 = 3(x − 5) → y = 3x − 13 (M1 A1).

Q4（7 marks） Circular measure

(i) Arc = rθ = 6 × 1.2 = 7.2 cm; perimeter = 2(6) + 7.2 = 19.2 cm; area = ½ r²θ = ½(36)(1.2) = 21.6 cm².
(ii) Triangle area = ½(36) sin 1.2 = 16.7767 cm²; segment = 21.6 − 16.7767 = 4.823 cm² (M1, M1, A1).

Q5（7 marks） Trigonometric equations

sin²x = 1/4 → sin x = ±1/2 (M1). sin x = 1/2: x = 30°, 150°; sin x = −1/2: x = 210°, 330° (A1 A1).
Answer: x = 30°, 150°, 210°, 330° (final A1 only if all four are given).

Q6（7 marks） Differentiation

(i) dy/dx = 3x² − 12x + 9 = 3(x − 1)(x − 3) (M1 A1); stationary at x = 1, 3 (A1).
(ii) d²y/dx² = 6x − 12: at x = 1 it is −6 < 0 → maximum (1, 4); at x = 3 it is 6 > 0 → minimum (3, 0) (A1 A1 A1).

Q7（8 marks） Definite integration

∫(3x² − 4x + 1) dx = x³ − 2x² + x (M1 A1 A1). At x = 4: 64 − 32 + 4 = 36; at x = 1: 1 − 2 + 1 = 0 (M1). Integral = 36 − 0 = 36 (A1). Geometrically it is the area between the curve, the x-axis and the lines x = 1 and x = 4 (A1, the function is positive on the interval).

Q8（8 marks） Sequences and series

(i) u₂₀ = 7 + 19(3) = 64 (M1 A1); S₂₀ = 20/2 × (7 + 64) = 10 × 71 = 710 (M1 A1).
(ii) r = 8/12 = 2/3 (A1); S∞ = 12 ÷ (1 − 2/3) = 12 ÷ 1/3 = 36 (M1 A1).

Q9（8 marks） Functions

(i) gf(x) = (2x + 3)² − 1 = 4x² + 12x + 9 − 1 = 4x² + 12x + 8 (M1 A1 A1).
(ii) y = 2x + 3 → x = (y − 3)/2, so f⁻¹(x) = (x − 3)/2 (M1 A1).
(iii) 4x² + 12x + 8 = 0 → x² + 3x + 2 = 0 → (x + 1)(x + 2) = 0 → x = −1 or x = −2 (M1 A1).

Q10（9 marks） Vectors

(i) AB = OB − OA = (6 − 2)i + (−5 − 1)j = 4i − 6j (M1 A1).
(ii) |AB| = √(4² + (−6)²) = √52 = 2√13 (≈ 7.211) (M1 A1).
(iii) Unit vector = (4i − 6j)/2√13 = (2i − 3j)/√13 (M1 A1).
(iv) OM = (OA + OB)/2 = 4i − 2j (M1 A1).

Q11（5 marks） Polynomials and factor theorem

(i) f(−1) = (−1)³ − 4(−1)² + (−1) + 6 = −1 − 4 − 1 + 6 = 0, so (x + 1) is a factor (M1 A1).
(ii) By division or comparing coefficients: f(x) = (x + 1)(x² − 5x + 6) = (x + 1)(x − 2)(x − 3) (M1 A1); roots x = −1, 2, 3 (A1).

Marks by topic

Common mistakes

• Dropping the factor 2 when completing the square, or writing +5 instead of −5.
• Forgetting the 3ʳ in the binomial coefficients, or mixing ascending and descending powers.
• Using degrees in s = rθ with θ = 1.2 rad, or forgetting to subtract the triangle from the sector.
• Solving sin²x = 1/4 with only the positive root (30°, 150°) and missing 210°, 330°.
• Substituting the limits in the wrong order in a definite integral, giving a negative area.
• Writing AB = OA − OB, or forgetting to divide by |AB| for the unit vector.

FAQ

Is this the same as the official 9709 Paper 1?
The format (75 marks, 1 h 50 min, all questions compulsory, calculator allowed) follows the official 9709 syllabus. The questions are written by TutorZone; the answers use CIE M/A mark notation.

Why are exact values such as fractions and surds required?
CIE pure mathematics A marks are for exact answers. A decimal such as 7.21 usually earns the M mark only, so write the exact value and the decimal approximation.

Is this suitable for self-study?
Yes. Sit it closed-book in 110 minutes, then check every M1/A1 step against the answer file and redo each wrong question, noting whether the formula, substitution or unit was wrong.

TutorZone original practice paper; M/A notation follows CIE mark scheme conventions.

All numerical answers are computed and checked by code; report any error to tutorzoneee@gmail.com.