OCR A Level Mathematics B (MEI) – H640/01 | Pure
Mathematics & Mechanics — 250 Revision MCQs |
TEST BANK | LATEST UPDATE
These 250 questions span all key topics for OCR A Level Maths B (MEI) H640/01:
differentiation, integration, algebra, trigonometry, logarithms, vectors, sequences, and mechanics
(kinematics, Newton's laws, projectiles, statics, momentum, and energy). Use them for timed
practice — aim for 1–2 minutes per question. Good luck with your revision!
PURE MATHEMATICS
Q1. What is the gradient of the curve y = 3x⁴ − 5x² + 2 at x = 1?
A) 2 B) 4 (correct answer) C) 7 D) 10
Rationale: Differentiating gives dy/dx = 12x³ − 10x. At x = 1: 12(1) − 10(1) = 12 − 10 = 2.
Wait — rechecking: 12(1)³ − 10(1) = 12 − 10 = 2. Correction — the correct answer is 2. See
Q1 corrected below.
Q1 (corrected). What is the gradient of the curve y = 3x⁴ − 5x² + 2 at x = 1?
A) 10 B) 7 C) 2 (correct answer) D) 4
Rationale: Differentiating gives dy/dx = 12x³ − 10x. Substituting x = 1: 12(1)³ − 10(1) = 12 −
10 = 2.
Q2. The equation of a circle is x² + y² − 6x + 4y − 12 = 0. What is the radius?
A) 5 B) 7 C) √37 (correct answer) D) √29
Rationale: Complete the square: (x − 3)² + (y + 2)² = 9 + 4 + 12 = 25. Wait — 3² + 2² + 12 = 9
+ 4 + 12 = 25, so r = 5.
Q2 (corrected). The equation x² + y² − 6x + 4y − 12 = 0 has radius:
,A) √29 B) 5 (correct answer) C) √37 D) 25
Rationale: Rewriting by completing the square: (x − 3)² − 9 + (y + 2)² − 4 − 12 = 0, so (x −
3)² + (y + 2)² = 25. The radius is √25 = 5.
Q3. What is ∫(4x³ − 2x + 1) dx?
A) 12x² − 2 + C B) x⁴ − x² + x + C (correct answer) C) x⁴ − 2x² + x + C D) 4x⁴ − x² + x + C
Rationale: Integrate term by term: ∫4x³ dx = x⁴, ∫−2x dx = −x², ∫1 dx = x. Sum with constant
C gives x⁴ − x² + x + C.
Q4. Solve log₂(x) + log₂(x − 2) = 3.
A) x = 4 B) x = 4 (correct answer) C) x = 2 D) x = 8
Rationale: log₂(x(x − 2)) = 3 → x(x − 2) = 8 → x² − 2x − 8 = 0 → (x − 4)(x + 2) = 0. Since x >
2, x = 4.
Q5. Find the binomial expansion of (1 + 2x)⁵, the coefficient of x³.
A) 40 B) 80 (correct answer) C) 160 D) 32
Rationale: The term in x³ is C(5,3)(2x)³ = 10 × 8x³ = 80x³. So the coefficient is 80.
Q6. The function f(x) = x³ − 3x has a local maximum at:
A) x = 0 B) x = −√3 C) x = −1 (correct answer) D) x = 1
Rationale: f'(x) = 3x² − 3 = 0 → x² = 1 → x = ±1. f''(x) = 6x. At x = −1: f''(−1) = −6 < 0,
confirming a local maximum.
Q7. What is the discriminant of 3x² − 5x + 4?
A) 25 B) −23 C) −23 (correct answer) D) 73
,Rationale: Discriminant = b² − 4ac = 25 − 48 = −23. Since Δ < 0, the quadratic has no real
roots.
Q8. If sin θ = 3/5 and θ is in the first quadrant, what is cos θ?
A) 3/4 B) 4/5 (correct answer) C) 5/4 D) 1/5
Rationale: Using the Pythagorean identity: cos²θ = 1 − sin²θ = 1 − 9/25 = 16/25. In Q1, cos θ
= 4/5.
Q9. Find the sum to infinity of the geometric series 8 + 4 + 2 + …
A) 12 B) 14 C) 16 (correct answer) D) 20
Rationale: First term a = 8, common ratio r = 1/2. S∞ = a/(1 − r) = 8/(1 − 1/2) = 8/(1/2) = 16.
Q10. The curve y = e^(2x) passes through which point?
A) (0, 2) B) (0, 1) (correct answer) C) (1, e) D) (2, e)
Rationale: When x = 0: y = e^(2×0) = e⁰ = 1. The curve passes through (0, 1).
Q11. Differentiate y = sin(3x) + cos(x).
A) 3cos(3x) + sin(x) B) 3cos(3x) − sin(x) (correct answer) C) cos(3x) − sin(x) D) −3sin(3x) −
sin(x)
Rationale: d/dx[sin(3x)] = 3cos(3x) by chain rule. d/dx[cos(x)] = −sin(x). Sum: 3cos(3x) −
sin(x).
Q12. What is the value of ∫₀¹ e^x dx?
A) 1 B) e C) e − 1 (correct answer) D) e + 1
Rationale: [e^x]₀¹ = e¹ − e⁰ = e − 1.
, Q13. Solve the inequality 2x² − 5x − 3 > 0.
A) −1/2 < x < 3 B) x < −1/2 or x > 3 (correct answer) C) x > 3 only D) −3 < x < 1/2
Rationale: Factorising: (2x + 1)(x − 3) > 0. Roots at x = −1/2 and x = 3. The parabola opens
upward, so the inequality holds for x < −1/2 or x > 3.
Q14. What is the equation of the tangent to y = x² at the point (3, 9)?
A) y = 6x + 9 B) y = 6x − 9 (correct answer) C) y = 3x − 9 D) y = 9x − 18
Rationale: dy/dx = 2x. At x = 3: gradient = 6. Tangent: y − 9 = 6(x − 3) → y = 6x − 9.
Q15. The vectors a = (2, −1, 3) and b = (1, 4, −2). Find a · b.
A) 10 B) −8 C) −8 (correct answer) D) 8
Rationale: a · b = (2)(1) + (−1)(4) + (3)(−2) = 2 − 4 − 6 = −8.
Q16. What is the exact value of tan(60°)?
A) 1/√2 B) 1/2 C) √3 (correct answer) D) √2
Rationale: From the 30-60-90 triangle: tan(60°) = opposite/adjacent = √3/1 = √3.
Q17. Find dy/dx when y = ln(5x).
A) 5/x B) ln(5) C) 1/x (correct answer) D) 5 ln(x)
Rationale: ln(5x) = ln 5 + ln x. Differentiating: d/dx[ln x] = 1/x. The constant ln 5 vanishes,
giving dy/dx = 1/x.
Q18. Which of the following is an arithmetic sequence?
Mathematics & Mechanics — 250 Revision MCQs |
TEST BANK | LATEST UPDATE
These 250 questions span all key topics for OCR A Level Maths B (MEI) H640/01:
differentiation, integration, algebra, trigonometry, logarithms, vectors, sequences, and mechanics
(kinematics, Newton's laws, projectiles, statics, momentum, and energy). Use them for timed
practice — aim for 1–2 minutes per question. Good luck with your revision!
PURE MATHEMATICS
Q1. What is the gradient of the curve y = 3x⁴ − 5x² + 2 at x = 1?
A) 2 B) 4 (correct answer) C) 7 D) 10
Rationale: Differentiating gives dy/dx = 12x³ − 10x. At x = 1: 12(1) − 10(1) = 12 − 10 = 2.
Wait — rechecking: 12(1)³ − 10(1) = 12 − 10 = 2. Correction — the correct answer is 2. See
Q1 corrected below.
Q1 (corrected). What is the gradient of the curve y = 3x⁴ − 5x² + 2 at x = 1?
A) 10 B) 7 C) 2 (correct answer) D) 4
Rationale: Differentiating gives dy/dx = 12x³ − 10x. Substituting x = 1: 12(1)³ − 10(1) = 12 −
10 = 2.
Q2. The equation of a circle is x² + y² − 6x + 4y − 12 = 0. What is the radius?
A) 5 B) 7 C) √37 (correct answer) D) √29
Rationale: Complete the square: (x − 3)² + (y + 2)² = 9 + 4 + 12 = 25. Wait — 3² + 2² + 12 = 9
+ 4 + 12 = 25, so r = 5.
Q2 (corrected). The equation x² + y² − 6x + 4y − 12 = 0 has radius:
,A) √29 B) 5 (correct answer) C) √37 D) 25
Rationale: Rewriting by completing the square: (x − 3)² − 9 + (y + 2)² − 4 − 12 = 0, so (x −
3)² + (y + 2)² = 25. The radius is √25 = 5.
Q3. What is ∫(4x³ − 2x + 1) dx?
A) 12x² − 2 + C B) x⁴ − x² + x + C (correct answer) C) x⁴ − 2x² + x + C D) 4x⁴ − x² + x + C
Rationale: Integrate term by term: ∫4x³ dx = x⁴, ∫−2x dx = −x², ∫1 dx = x. Sum with constant
C gives x⁴ − x² + x + C.
Q4. Solve log₂(x) + log₂(x − 2) = 3.
A) x = 4 B) x = 4 (correct answer) C) x = 2 D) x = 8
Rationale: log₂(x(x − 2)) = 3 → x(x − 2) = 8 → x² − 2x − 8 = 0 → (x − 4)(x + 2) = 0. Since x >
2, x = 4.
Q5. Find the binomial expansion of (1 + 2x)⁵, the coefficient of x³.
A) 40 B) 80 (correct answer) C) 160 D) 32
Rationale: The term in x³ is C(5,3)(2x)³ = 10 × 8x³ = 80x³. So the coefficient is 80.
Q6. The function f(x) = x³ − 3x has a local maximum at:
A) x = 0 B) x = −√3 C) x = −1 (correct answer) D) x = 1
Rationale: f'(x) = 3x² − 3 = 0 → x² = 1 → x = ±1. f''(x) = 6x. At x = −1: f''(−1) = −6 < 0,
confirming a local maximum.
Q7. What is the discriminant of 3x² − 5x + 4?
A) 25 B) −23 C) −23 (correct answer) D) 73
,Rationale: Discriminant = b² − 4ac = 25 − 48 = −23. Since Δ < 0, the quadratic has no real
roots.
Q8. If sin θ = 3/5 and θ is in the first quadrant, what is cos θ?
A) 3/4 B) 4/5 (correct answer) C) 5/4 D) 1/5
Rationale: Using the Pythagorean identity: cos²θ = 1 − sin²θ = 1 − 9/25 = 16/25. In Q1, cos θ
= 4/5.
Q9. Find the sum to infinity of the geometric series 8 + 4 + 2 + …
A) 12 B) 14 C) 16 (correct answer) D) 20
Rationale: First term a = 8, common ratio r = 1/2. S∞ = a/(1 − r) = 8/(1 − 1/2) = 8/(1/2) = 16.
Q10. The curve y = e^(2x) passes through which point?
A) (0, 2) B) (0, 1) (correct answer) C) (1, e) D) (2, e)
Rationale: When x = 0: y = e^(2×0) = e⁰ = 1. The curve passes through (0, 1).
Q11. Differentiate y = sin(3x) + cos(x).
A) 3cos(3x) + sin(x) B) 3cos(3x) − sin(x) (correct answer) C) cos(3x) − sin(x) D) −3sin(3x) −
sin(x)
Rationale: d/dx[sin(3x)] = 3cos(3x) by chain rule. d/dx[cos(x)] = −sin(x). Sum: 3cos(3x) −
sin(x).
Q12. What is the value of ∫₀¹ e^x dx?
A) 1 B) e C) e − 1 (correct answer) D) e + 1
Rationale: [e^x]₀¹ = e¹ − e⁰ = e − 1.
, Q13. Solve the inequality 2x² − 5x − 3 > 0.
A) −1/2 < x < 3 B) x < −1/2 or x > 3 (correct answer) C) x > 3 only D) −3 < x < 1/2
Rationale: Factorising: (2x + 1)(x − 3) > 0. Roots at x = −1/2 and x = 3. The parabola opens
upward, so the inequality holds for x < −1/2 or x > 3.
Q14. What is the equation of the tangent to y = x² at the point (3, 9)?
A) y = 6x + 9 B) y = 6x − 9 (correct answer) C) y = 3x − 9 D) y = 9x − 18
Rationale: dy/dx = 2x. At x = 3: gradient = 6. Tangent: y − 9 = 6(x − 3) → y = 6x − 9.
Q15. The vectors a = (2, −1, 3) and b = (1, 4, −2). Find a · b.
A) 10 B) −8 C) −8 (correct answer) D) 8
Rationale: a · b = (2)(1) + (−1)(4) + (3)(−2) = 2 − 4 − 6 = −8.
Q16. What is the exact value of tan(60°)?
A) 1/√2 B) 1/2 C) √3 (correct answer) D) √2
Rationale: From the 30-60-90 triangle: tan(60°) = opposite/adjacent = √3/1 = √3.
Q17. Find dy/dx when y = ln(5x).
A) 5/x B) ln(5) C) 1/x (correct answer) D) 5 ln(x)
Rationale: ln(5x) = ln 5 + ln x. Differentiating: d/dx[ln x] = 1/x. The constant ln 5 vanishes,
giving dy/dx = 1/x.
Q18. Which of the following is an arithmetic sequence?
