Mathematics Flashcards

In the Singapore GCE A-Level H2 Mathematics syllabus, what is the derivative of ln(3x + 1)?

Answer: 3/(3x + 1)

Using the chain rule, d/dx[ln(3x + 1)] = 1/(3x + 1) × 3 = 3/(3x + 1). The chain rule requires multiplying by the derivative of the inner function.

What is the sum of the first 10 terms of the arithmetic series 3, 7, 11, 15, ...?

Answer: 210

Using S_n = n/2 × [2a + (n-1)d], where a = 3, d = 4, n = 10: S_10 = 10/2 × [2(3) + (9)(4)] = 5 × [6 + 36] = 5 × 42 = 210.

If P(A) = 0.4, P(B) = 0.5, and P(A ∩ B) = 0.2, are events A and B independent?

Answer: Yes, because P(A ∩ B) = P(A) × P(B)

Events are independent if P(A ∩ B) = P(A) × P(B). Here, P(A) × P(B) = 0.4 × 0.5 = 0.2 = P(A ∩ B). Since the condition is met, A and B are independent.

What is the integral of sin²(x) dx?

Answer: x/2 - sin(2x)/4 + C

Using the identity sin²(x) = (1 - cos(2x))/2, the integral becomes ∫(1/2 - cos(2x)/2) dx = x/2 - sin(2x)/4 + C.

A vector a = 2i + 3j - k. What is the magnitude of vector a?

Answer: √14

|a| = √(2² + 3² + (-1)²) = √(4 + 9 + 1) = √14. The magnitude is found by taking the square root of the sum of squares of all components.

In the Singapore A-Level syllabus, what does the Maclaurin series expansion represent?

Answer: A Taylor series centred at x = 0

A Maclaurin series is a special case of the Taylor series where the expansion is centred at x = 0. It expresses a function as f(x) = f(0) + f'(0)x + f''(0)x²/2! + ...