A-Level Mathematics Core

Question 1

What is the derivative of sin(3x) with respect to x?

Accepted Answer

3cos(3x)

Answer

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?

Accepted Answer

k > 9

Answer

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?

Accepted Answer

(2x + 1)⁵ / 10 + C

Answer

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?

Accepted Answer

16

Answer

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)?

Accepted Answer

y = -x/12 + 49/6

Answer

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.

A-Level Revision Practice Test