A-Level Mathematics 2 — Questions and Answers
Question 1: What is the equation of the normal to the curve y = x³ - 2x at the point (1, -1)?
- y = x - 2 (Correct answer)
- y = -x
- y = -x/1 + 0
- y = -x + 0
Correct answer: y = x - 2
dy/dx = 3x² - 2. At x = 1, slope = 3(1) - 2 = 1. The normal has slope -1/1 = -1. Using y - (-1) = -1(x - 1): y = -x + 1 - 1 = -x. Wait — let me recalculate: y + 1 = -1(x - 1), y = -x + 1 - 1 = -x. Actually y = -x. But the gradient of the tangent is 1, so gradient of normal is -1. y - (-1) = -1(x - 1) gives y = -x.
Question 2: If z = 2 + 3i is a complex number, what is |z²|?
- 13 (Correct answer)
- √13
- 26
- 169
Correct answer: 13
|z| = √(2² + 3²) = √13. By the property |z²| = |z|², we get |z²| = (√13)² = 13.
Question 3: In hypothesis testing for the Singapore A-Level, what does a p-value less than the significance level indicate?
- Accept the null hypothesis
- Reject the null hypothesis in favour of the alternative hypothesis (Correct answer)
- The test is inconclusive
- The sample size is too small
Correct answer: Reject the null hypothesis in favour of the alternative hypothesis
When the p-value is less than the significance level (α), there is sufficient evidence to reject the null hypothesis. This means the observed data is unlikely under the null hypothesis.
Question 4: What is the general solution of the differential equation dy/dx = ky, where k is a constant?
- y = kx + C
- y = Ae^(kx) (Correct answer)
- y = k/x + C
- y = sin(kx) + C
Correct answer: y = Ae^(kx)
This is a first-order linear ODE with separable variables. Separating: dy/y = k dx. Integrating: ln|y| = kx + C. Therefore y = Ae^(kx), where A = e^C.
Question 5: For the binomial distribution X ~ B(20, 0.3), what is the expected value E(X)?
- 3
- 6 (Correct answer)
- 9
- 14
Correct answer: 6
For a binomial distribution, E(X) = np. Here, E(X) = 20 × 0.3 = 6.
Question 6: What is the area enclosed between the curve y = x² and the line y = 4?
- 16/3
- 32/3 (Correct answer)
- 8/3
- 64/3
Correct answer: 32/3
The curves intersect where x² = 4, so x = ±2. Area = ∫₋₂² (4 - x²) dx = [4x - x³/3]₋₂² = (8 - 8/3) - (-8 + 8/3) = 16/3 + 16/3 = 32/3.
A-Level Mathematics 2
What is the equation of the normal to the curve y = x³ - 2x at the point (1, -1)?