O-Level Additional Mathematics

Question 1

What is the derivative of y = 3x⁴ - 2x² + 5?

Accepted Answer

12x³ - 4x

Answer

Using the power rule dy/dx = nxⁿ⁻¹: d/dx(3x⁴) = 12x³, d/dx(-2x²) = -4x, d/dx(5) = 0. So dy/dx = 12x³ - 4x.

Question 2

Solve the simultaneous equations: y = x + 1 and x² + y² = 25

Accepted Answer

x = 3, y = 4 and x = -4, y = -3

Answer

Substitute y = x + 1 into x² + y² = 25: x² + (x+1)² = 25, 2x² + 2x + 1 = 25, 2x² + 2x - 24 = 0, x² + x - 12 = 0, (x+4)(x-3) = 0. x = 3 → y = 4; x = -4 → y = -3.

Question 3

What is lg (logarithm base 10) of 1000?

Accepted Answer

3

Answer

lg 1000 = lg 10³ = 3. Since 10³ = 1000, the logarithm base 10 of 1000 is 3.

Question 4

If f(x) = 2x + 3, what is f⁻¹(x), the inverse function?

Accepted Answer

(x - 3)/2

Answer

To find the inverse: let y = 2x + 3. Swap x and y: x = 2y + 3. Solve for y: y = (x - 3)/2. So f⁻¹(x) = (x - 3)/2.

Question 5

What is the integral ∫(4x³ + 6x) dx?

Accepted Answer

x⁴ + 3x² + C

Answer

Using the power rule for integration: ∫4x³ dx = x⁴, ∫6x dx = 3x². So ∫(4x³ + 6x) dx = x⁴ + 3x² + C, where C is the constant of integration.

GCE O-Level Practice Test

GCE O-Level Practice Test

O-Level Additional Mathematics