Mathematics Flashcards

What is the equation of the normal to the curve y = x³ - 2x at the point (1, -1)?

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.

If z = 2 + 3i is a complex number, what is |z²|?

Answer: 13

|z| = √(2² + 3²) = √13. By the property |z²| = |z|², we get |z²| = (√13)² = 13.

In hypothesis testing for the Singapore A-Level, what does a p-value less than the significance level indicate?

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.

What is the general solution of the differential equation dy/dx = ky, where k is a constant?

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.

For the binomial distribution X ~ B(20, 0.3), what is the expected value E(X)?

Answer: 6

For a binomial distribution, E(X) = np. Here, E(X) = 20 × 0.3 = 6.

What is the area enclosed between the curve y = x² and the line y = 4?

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.