A-Level Mathematics Core — Questions and Answers
Question 1: What is the derivative of sin(3x) with respect to x?
- 3cos(3x) (Correct answer)
- cos(3x)
- -3cos(3x)
- 3sin(3x)
Correct answer: 3cos(3x)
Using the chain rule, d/dx[sin(3x)] = cos(3x) × 3 = 3cos(3x). The chain rule requires multiplying by the derivative of the inner function.
Question 2: The equation x² + 6x + k = 0 has no real roots. Which inequality must k satisfy?
- k > 9 (Correct answer)
- k < 9
- k > 6
- k < -9
Correct answer: k > 9
For no real roots, the discriminant b² - 4ac < 0. Here, 36 - 4k < 0, so 4k > 36, giving k > 9.
Question 3: What is the integral of (2x + 1)⁴ with respect to x?
- (2x + 1)⁵ / 10 + C (Correct answer)
- (2x + 1)⁵ / 5 + C
- 4(2x + 1)³ + C
- (2x + 1)⁵ / 2 + C
Correct answer: (2x + 1)⁵ / 10 + C
Using reverse chain rule: ∫(2x+1)⁴ dx = (2x+1)⁵ / (5 × 2) + C = (2x+1)⁵ / 10 + C. We divide by both the new power and the coefficient of x.
Question 4: A geometric series has first term 8 and common ratio 0.5. What is the sum to infinity?
- 16 (Correct answer)
- 8
- 12
- 24
Correct answer: 16
The sum to infinity formula is S∞ = a / (1 - r) = 8 / (1 - 0.5) = 8 / 0.5 = 16. This converges because |r| < 1.
Question 5: What is the equation of the normal to the curve y = x³ at the point (2, 8)?
- y = -x/12 + 49/6 (Correct answer)
- y = 12x - 16
- y = -12x + 32
- y = x/12 + 47/6
Correct answer: y = -x/12 + 49/6
dy/dx = 3x². At x = 2, gradient = 12. The normal has gradient -1/12. Using y - 8 = -1/12(x - 2) gives y = -x/12 + 1/6 + 8 = -x/12 + 49/6.
Question 6: If log₂(x) + log₂(x - 2) = 3, what is the value of x?
- 4 (Correct answer)
- 2
- 6
- 8
Correct answer: 4
Combining logs: log₂[x(x-2)] = 3, so x(x-2) = 8. This gives x² - 2x - 8 = 0, factoring to (x-4)(x+2) = 0. Since x must be positive and x > 2, x = 4.
A-Level Mathematics Core
What is the derivative of sin(3x) with respect to x?