Total marks: 80 · Duration: 2 hours
Instructions to candidates
- Answer all questions.
- Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs.
- Write your answer to each question in the space provided. Show all necessary working clearly; no marks will be given for unsupported answers from a calculator.
- You should use a calculator where appropriate.
- Unless a question says otherwise, give non-exact numerical answers correct to 3 significant figures, and angles in degrees to 1 decimal place.
- Where a question asks for an exact answer, give it as a fraction, a surd, or in terms of π, e or ln.
- Angles in radians are written without a unit (for example, 1.2 means 1.2 radians). Make sure your calculator is in the correct mode.
- In this paper, a vector such as the one from A to N is written as the vector AN, and bold letters such as a and b are position vectors.
Information
- The total mark for this paper is 80.
- The number of marks for each question or part question is shown in brackets [ ].
- This mock follows the 2025–2027 Cambridge IGCSE Additional Mathematics (0606) format: Paper 2 is the calculator paper, 2 hours, 80 marks. Paper 1 is the same length but without a calculator.
Formulae you may need
- Quadratic equation: for ax² + bx + c = 0, x = (−b ± √(b² − 4ac)) / (2a)
- Binomial theorem: (a + b)ⁿ = aⁿ + ⁿC₁ aⁿ⁻¹ b + ⁿC₂ aⁿ⁻² b² + … + bⁿ, where ⁿCᵣ = n! / ((n − r)! r!)
- Arithmetic series: uₙ = a + (n − 1)d and Sₙ = ½n(2a + (n − 1)d)
- Geometric series: uₙ = arⁿ⁻¹, Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1, and S∞ = a/(1 − r) for |r| < 1
- Identities: sin²A + cos²A = 1, sec²A = 1 + tan²A, cosec²A = 1 + cot²A
- Triangle ABC: a/sin A = b/sin B = c/sin C, a² = b² + c² − 2bc cos A, area = ½ab sin C
- Circle sector, angle θ radians: arc length = rθ, area = ½r²θ
Paper
1 The polynomial p(x) = 2x³ + ax² + bx − 6, where a and b are constants, has (x − 2) as a factor. When p(x) is divided by (x + 1), the remainder is −12.
(a) Find the value of a and the value of b. [4]
(b) Show that p(x) = (x − 2)(2x² + x + 3), and hence show that the equation p(x) = 0 has exactly one real root. [3]
2 (a) Solve the equation |2x − 3| = |x + 1|. [3]
(b) Hence solve the inequality |2x − 3| > |x + 1|. [2]
3 (a) Solve the equation log₃(x + 4) + log₃(x − 2) = 3. [3]
(b) Solve the equation 2(4ˣ) − 7(2ˣ) + 3 = 0. [4]
4 The points A and B lie on a circle with centre O and radius 8 cm. The minor sector OAB has angle AOB = 1.2 radians.
(a) Find the length of the minor arc AB. [1]
(b) Find the area of the minor segment bounded by the chord AB and the arc AB. [3]
(c) Find the perimeter of this segment. [2]
5 The circle C has equation x² + y² − 6x + 4y − 12 = 0.
(a) Find the coordinates of the centre of C and the radius of C. [3]
(b) Show that the point P(6, 2) lies on C. [1]
(c) Find the equation of the tangent to C at P, giving your answer in the form ax + by = c, where a, b and c are integers. [3]
6 (a) The third term of an arithmetic progression is 11 and the sum of the first 10 terms is 185. Find the first term and the common difference. [4]
(b) A geometric progression has first term 48 and sum to infinity 64. Find the common ratio. [2]
(c) For the geometric progression in part (b), find the least number of terms for which the sum is greater than 63.9. [3]
7 (a) Solve 2 sin²x = 3 cos x for 0° ≤ x ≤ 360°. [4]
(b) Solve tan(2y − 0.3) = 1.5 for 0 ≤ y ≤ π radians. [3]
8 (a) Four-digit numbers are formed from the digits 1, 2, 3, 4, 5, 6 and 7. No digit may be used more than once in a number. Find how many of these numbers are even and greater than 5000. [3]
(b) A team of 5 is chosen from 6 boys and 4 girls. The team must include at least 2 girls. Find the number of different teams that can be chosen. [3]
9 Relative to an origin O, the points A and B have position vectors a and b, where a and b are not parallel. The point M is the midpoint of OA. The point N lies on OB such that ON : NB = 1 : 2. The lines AN and BM meet at the point X.
(a) Find, in terms of a and b, the vector AN and the vector BM. [2]
(b) Given that the vector AX = λ × (vector AN) and the vector BX = μ × (vector BM), find the value of λ and the value of μ, and hence find the vector OX in terms of a and b. [4]
10 A curve has equation y = (3x − 1)/(x² + 2).
(a) Show that dy/dx = (6 + 2x − 3x²)/(x² + 2)². [3]
(b) A point moves along the curve so that its x-coordinate is increasing at a constant rate of 0.5 units per second. Find the rate at which its y-coordinate is changing when x = 1. [2]
11 A curve has equation y = e²ˣ − 4x.
(a) Find dy/dx and d²y/dx². [2]
(b) Find the exact coordinates of the stationary point on the curve and determine its nature. [3]
(c) The curve lies entirely above the x-axis. Find the exact area of the region bounded by the curve, the x-axis and the lines x = 0 and x = 1. [3]
12 A particle P moves in a straight line. At time t seconds after leaving a fixed point O, its velocity, v m s⁻¹, is given by v = 2 + 4 cos 2t.
(a) Find the exact time at which P first comes to instantaneous rest. [2]
(b) Find the acceleration of P when t = 1. [2]
(c) Find the displacement of P from O when it first comes to instantaneous rest. [3]