Mathematics Core Flashcards

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

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.

The equation x² + 6x + k = 0 has no real roots. Which inequality must k satisfy?

Answer: k > 9

For no real roots, the discriminant b² - 4ac 36, giving k > 9.

What is the integral of (2x + 1)⁴ with respect to x?

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.

A geometric series has first term 8 and common ratio 0.5. What is the sum to infinity?

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.

What is the equation of the normal to the curve y = x³ at the point (2, 8)?

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.

If log₂(x) + log₂(x - 2) = 3, what is the value of x?

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.