A-Level Mathematics 2

Question 1

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

Accepted Answer

y = x - 2

Answer

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|?

Accepted Answer

13

Answer

|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?

Accepted Answer

Reject the null hypothesis in favour of the alternative hypothesis

Answer

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?

Accepted Answer

y = Ae^(kx)

Answer

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)?

Accepted Answer

6

Answer

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

GCE A-Level Practice Test

GCE A-Level Practice Test

A-Level Mathematics 2