IGCSE Additional Mathematics 0606 Paper 2 Practice Paper | 10 Pure Mathematics Questions, 100 Marks

2026-11-01

IGCSE Additional Mathematics

IGCSE Additional Mathematics 0606 Paper 2 practice paper: circular measure and sector area, permutations and combinations, arithmetic and geometric sequences, indicial and logarithmic equations, matrices and inverses, stationary points and tangents, definite integrals and the area between curves, and trigonometric equations in R sin form; 10 questions for 100 marks in 2 hours with full answers and marking notes.

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Section A: Circular measure, permutations and combinations, arithmetic sequences (Questions 1-3, 30 marks)

Answer all questions. Show all working; answers without working score no marks.

#	Question	Answer and marking notes
1	(a) A sector has radius 9 cm and angle 2.4 radians. Find its arc length and area. (4 marks) (b) Find the perimeter of the sector. (2 marks) (c) A second sector has radius 12 cm and angle 0.9 radians. Find its area. (4 marks)	(a) Arc length = r times theta = 9 times 2.4 = 21.6 cm; area = half r squared theta = 0.5 times 81 times 2.4 = 97.2 square cm. (b) Perimeter = 2r plus arc = 18 plus 21.6 = 39.6 cm. (c) Area = 0.5 times 144 times 0.9 = 64.8 square cm.
2	(a) Three people are chosen from 4 boys and 3 girls. Find the number of possible committees. (3 marks) (b) Find how many of these committees contain at least one girl. (3 marks) (c) Five different books are arranged on a shelf with two named books adjacent. Find the number of arrangements. (4 marks)	(a) The number is C(7, 3) = 35. (b) Subtract the all-boy committees: 35 minus 4 = 31. (c) Treat the two books as one unit, arrange 4 units and then the internal order: 4 factorial times 2 = 48.
3	(a) An arithmetic sequence has first term 7 and common difference minus 3. Find the 20th term. (3 marks) (b) Find the sum of the first 20 terms. (4 marks) (c) Find which term is first equal to minus 20. (3 marks)	(a) T(20) = 7 plus 19 times minus 3 = -50. (b) S(20) = 10 times (7 plus minus 50) = -430. (c) Solving 7 minus 3(n minus 1) = minus 20 gives n = 10.

Section B: Geometric sequences, indicial and logarithmic equations (Questions 4-6, 30 marks)

Answer all questions. Show all working; answers without working score no marks.

#	Question	Answer and marking notes
4	(a) A geometric sequence has first term 3 and common ratio 1.5. Find the 6th term. (3 marks) (b) Find the sum of the first 6 terms. (4 marks) (c) A second geometric sequence has first term 5 and common ratio 0.8. Find its sum to infinity. (3 marks)	(a) T(6) = 3 times 1.5 to the fifth power = 22.781. (b) S(6) = 3 times (1.5 to the sixth power minus 1) divided by 0.5 = 62.344. (c) S(infinity) = 5 divided by 0.2 = 25.
5	(a) Solve 3^(2x) minus 10 times 3^x plus 9 = 0. (4 marks) (b) Solve 4^(x + 1) = 8^(2x minus 3). (3 marks) (c) Evaluate 27^(2/3) times 16^(3/4). (3 marks)	(a) Let y = 3^x. Then y squared minus 10y plus 9 = 0, so y = 1 or 9 and x = 0 or 2. (b) Write both sides as powers of 2: 2x plus 2 = 6x minus 9, so x = 2.750. (c) 27^(2/3) = 9 and 16^(3/4) = 8, giving 72.
6	(a) Solve log base 10 of (x + 3) plus log base 10 of x = 1. (4 marks) (b) Solve log base 2 of (3x minus 1) = 4. (3 marks) (c) Given log base 3 of 2 = 0.631, find log base 3 of 8 and log base 3 of 1.5. (3 marks)	(a) x squared plus 3x minus 10 = 0, so x = 2 or x = minus 5; x must be positive, so x = 2. (b) 3x minus 1 = 16, so x = 5.667. (c) log base 3 of 8 = 1.893 and log base 3 of 1.5 = 0.369.

Section C: Matrices and differentiation (Questions 7-8, 20 marks)

Answer all questions. Show all working; answers without working score no marks.

#	Question	Answer and marking notes
7	(a) Given M with rows (3, 1) and (2, 4), find det M and the inverse of M. (4 marks) (b) Use matrices to solve 3x plus y = 7 and 2x plus 4y = 14. (4 marks) (c) Given N with rows (1, 2) and (0, 1), find N cubed. (2 marks)	(a) det M = 10 and the inverse is one tenth of the matrix with rows (4, minus 1) and (minus 2, 3). (b) x = 1.4 and y = 2.8. (c) N squared has rows (1, 4) and (0, 1), so N cubed has upper-right entry 6.
8	(a) Given y = x cubed minus 6x squared plus 9x plus 2, find dy/dx and the coordinates of all stationary points. (4 marks) (b) Use the second derivative to classify each stationary point. (3 marks) (c) Find the gradient and the equation of the tangent at x = 0. (3 marks)	(a) dy/dx = 3x squared minus 12x plus 9 = 3(x minus 1)(x minus 3); the stationary points are (1, 6) and (3, 2). (b) The second derivative is 6x minus 12, which is minus 6 at x = 1 (maximum) and 6 at x = 3 (minimum). (c) At x = 0 the gradient is 9 and the tangent is y = 9x plus 2.

Section D: Integration and trigonometric equations (Questions 9-10, 20 marks)

Answer all questions. Show all working; answers without working score no marks.

#	Question	Answer and marking notes
9	(a) Evaluate the integral from 1 to 4 of (3x squared minus 2x plus 1) dx. (4 marks) (b) Find the area enclosed by y = x squared and y = 4. (4 marks) (c) Find the integral of (2x + 1) cubed dx. (2 marks)	(a) The antiderivative is x cubed minus x squared plus x; substitution gives 51. (b) The curves meet at x = minus 2 and x = 2, and the area is 32/3, about 10.667. (c) The integral is (2x + 1)^4 divided by 8 plus a constant.
10	(a) Solve 2 sin squared x plus sin x minus 1 = 0 for x between 0 and 360 degrees. (5 marks) (b) Write 3 sin theta plus 4 cos theta as R sin(theta + alpha), where R is positive and alpha lies between 0 and 90 degrees, and state the maximum value. (5 marks)	(a) With s = sin x, 2s squared plus s minus 1 = 0 gives s = 0.5 or s = minus 1, so x = 30, 150 or 270 degrees. (b) R = 5 and alpha is about 53.1 degrees, so the expression equals 5 sin(theta + 53.1 degrees) with maximum value 5.

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