IGCSE Mathematics - Additional (0606) — May–June 2025, Paper 23

13 questions · every question answerable online with instant point-by-point AI marking against the official mark scheme.

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Question 1 · 2 marks · Quadratic functions - quadratic inequalities

Solve the inequality (x + 2)(4x − 5) ≤ 0.

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Question 2 · 3 marks · Differentiation - powers of x

Given that y = (4x^3 − 5) / x^2, show that dy/dx can be written as 2(2x^3 + 5) / x^3.

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Question 3 · 4 marks · Functions - transforming relationships to linear form

Variables x and y are such that when sqrt(y) is plotted against x^3 a straight line graph passing through the points (2, 5) and (10, 21) is obtained. Find y in terms of x.

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Question 4 · 8 marks · Functions - composite functions

The function f is defined by f(x) = 2x − 1 for x ∈ ℝ. Explain why the function f^2 can be formed.

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Question 5 · 4 marks · Trigonometry - periodicity of trigonometric functions

In this question, all angles are in radians. Write down the period of 5 tan(x/4) + 1.

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Question 6 · 6 marks · Permutations and combinations

Five of the digits 1 2 3 4 5 6 7 8 9 are used to make a 5-digit number. Find how many ways this can be done when the 5-digit number is greater than 50 000.

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Question 7 · 5 marks · Series - binomial expansions

Find the term independent of x in the expansion of (x^2 − 3/x^4)^15.

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Question 8 · 9 marks · Differentiation - product rule, exponential and trigonometric functions, small changes

It is given that y = e^(3x+2) tan x. Use calculus to find the approximate change in y as x increases from 0.1 to 0.1 + h, where h is small.

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Question 9 · 12 marks · Series - arithmetic and geometric progressions

The 1st term of an arithmetic progression is 9. The last term of this progression is 159. The sum of all the terms is 2604. The 12th term of this arithmetic progression is the 1st…

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Question 10 · 9 marks · Kinematics - vectors, position and velocity vectors

In this question, time is in seconds. At time t = 0, particle P starts from the point with position vector −30j. P travels with speed 58 m s^(-1) in the direction 20i + 21j. Find…

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Question 11 · 4 marks · Permutations and combinations - permutations

Given that ((n+1)P5) / 437 = (nP3), use an algebraic method to find the value of n.

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Question 12 · 10 marks · Trigonometry - solving trigonometric equations using identities

The function f is defined by f(x) = 15 cos^2(3x + 1.5) + 7 sin(3x + 1.5) − 13 for −0.3 ≤ x ≤ 0.5, where x is in radians. Solve the equation f(x) = 0.

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Question 13 · 4 marks · Indices and surds - exponential equations, hidden quadratics

Solve the equation 64^(x + 1/3) + 2^(3x) − 3 = 0.

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← May–June 2024 Paper 11 · May–June 2025 Paper 22 →

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