1. Find dy/dx when y = 4/(3x − 1)2.
- A−24/(3x − 1)3
- B−8/(3x − 1)3
- C24/(3x − 1)3
- D−24/(3x − 1)2
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Answer: A. y = 4(3x − 1)−2, so dy/dx = 4 × (−2)(3x − 1)−3 × 3 = −24/(3x − 1)3. Both the −2 and the chain-rule factor 3 are needed.
AS Level Mathematics chapter 1: Pure Mathematics 1 MCQs with answers
By Team Megalecture · Updated 4 Nov 2026
87 exam-style questions on Pure Mathematics 1, each with the answer and a short explanation. Ten of them are below to try. The chapter test draws ten fresh questions each time and is marked the moment you finish, with your score by topic.
87 questions · 8 topics · 10 per test
What chapter 1 covers
- Quadratics 11 questions · notes and questions
- Functions 11 questions · notes and questions
- Coordinate geometry 11 questions · notes and questions
- Circular measure 10 questions · notes and questions
- Trigonometry 11 questions · notes and questions
- Series 11 questions · notes and questions
- Differentiation 11 questions · notes and questions
- Integration 11 questions · notes and questions
Ten Pure Mathematics 1 questions with answers
Try each one, then open the answer. These ten stay the same; the test draws new ones every time.
2. Find the set of values of k for which the equation kx2 + 4x + k = 0 has no real roots.
- Ak > 2
- B−2 < k < 2
- Ck < −2 or k > 2
- Dk > 4
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Answer: C. No real roots needs b2 − 4ac < 0: 16 − 4k2 < 0, so k2 > 4, giving k < −2 or k > 2. Sketch y = k2 − 4 to get the outside region, not the inside.
3. Find the equation of the tangent to the circle (x − 1)2 + (y − 2)2 = 25 at the point (4, 6).
- A4x − 3y = −2
- B3x + 4y = 12
- C3x + 4y = 36
- D3x − 4y = −12
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Answer: C. The radius from (1, 2) to (4, 6) has gradient 4/3, so the tangent (perpendicular to the radius) has gradient −3/4. y − 6 = −3/4 (x − 4) gives 4y − 24 = −3x + 12, i.e. 3x + 4y = 36.
4. Simplify (1 − cos2θ)/(sin θ cos θ).
- Asin θ
- Bsin θ cos θ
- Ccos θ/sin θ
- Dtan θ
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Answer: D. Use sin2θ + cos2θ = 1: the numerator is sin2θ. Then sin2θ/(sin θ cos θ) = sin θ/cos θ = tan θ.
5. Find the set of values of x for which f(x) = x3 − 6x2 + 9x − 2 is an increasing function.
- A1 < x < 3
- Bx > 3
- Cx < 1 or x > 3
- Dx > 1
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Answer: C. f′(x) = 3x2 − 12x + 9 = 3(x − 1)(x − 3). Increasing where f′(x) > 0, which is outside the roots: x < 1 or x > 3.
6. Find ∫ 6/√x dx.
- A3√x + c
- B6√x + c
- C−3/x3/2 + c
- D12√x + c
Show answer
Answer: D. 6/√x = 6x−1/2. Integrating: 6x1/2/(1/2) + c = 12√x + c. Dividing by 1/2 doubles the coefficient.
7. The sum of the first n terms of an arithmetic progression is Sn = 2n2 + 3n. Find the common difference.
- A2
- B5
- C9
- D4
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Answer: D. u1 = S1 = 5 and u2 = S2 − S1 = 14 − 5 = 9, so d = 9 − 5 = 4. In general the n2 coefficient is d/2.
8. Convert 3π/8 radians to degrees.
- A135°
- B67.5°
- C60°
- D22.5°
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Answer: B. π rad = 180°, so 3π/8 = 3 × 180°/8 = 67.5°.
9. Find dy/dx when y = 3x4 − 2/x.
- A12x3 − 2/x2
- B12x3 + 2/x2
- C12x3 − 2
- D12x3 + 2/x
Show answer
Answer: B. Write −2/x as −2x−1. Its derivative is (−2)(−1)x−2 = +2/x2, so dy/dx = 12x3 + 2/x2. The two negatives make a plus.
10. A sector has radius 5 cm and angle 1.6 radians. Find the perimeter of the sector.
- A8 cm
- B13 cm
- C18 cm
- D26 cm
Show answer
Answer: C. Perimeter = two radii + arc = 2 × 5 + 5 × 1.6 = 10 + 8 = 18 cm. The common slip is forgetting one or both radii.
Read the notes for this chapter
Ways to practise this chapter
- Chapter test. Ten questions from Pure Mathematics 1, drawn fresh each time, about ten minutes.
- Practice mode. The right answer and the explanation after every question, with no clock. Exam mode keeps the marks until the end and runs a timer.
- Hints. One tap for a nudge, a second to remove two wrong options.
- My mistakes. Every question you get wrong is kept, and leaves the list once you get it right.
- Formula sheet. All AS Level Mathematics rules on one page.
Pure Mathematics 1: questions students ask
How many Pure Mathematics 1 MCQs are there?
87, each with the answer and a short explanation, across 8 topics: Quadratics, Functions, Coordinate geometry, Circular measure, Trigonometry, Series, Differentiation, Integration. Every chapter test draws ten of them, so you can take it many times.
Is the AS Level Mathematics chapter 1 test free?
Yes. The test, the answers and the explanations are free and need no sign-up. Sign in only if you want your scores kept across devices.
How is the chapter test marked?
The moment you press Finish: your score, the right answer and an explanation for every question, and the ones you got wrong saved in My mistakes to retry.