O-Level Additional Mathematics — Questions and Answers
Question 1: What is the derivative of y = 3x⁴ - 2x² + 5?
- 12x³ - 4x (Correct answer)
- 12x³ - 4x + 5
- 3x³ - 2x
- 12x⁴ - 4x²
Correct answer: 12x³ - 4x
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
- x = 3, y = 4 and x = -4, y = -3 (Correct answer)
- x = 3, y = 4 only
- x = 5, y = 0 and x = 0, y = 5
- No real solutions exist
Correct answer: x = 3, y = 4 and x = -4, y = -3
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?
- 2
- 3 (Correct answer)
- 4
- 10
Correct answer: 3
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?
- (x + 3)/2
- (x - 3)/2 (Correct answer)
- 2x - 3
- 3x + 2
Correct answer: (x - 3)/2
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?
- x⁴ + 3x² + C (Correct answer)
- 12x² + 6 + C
- 4x⁴ + 6x² + C
- x⁴ + 3x² + 5
Correct answer: x⁴ + 3x² + C
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.
Question 6: What is the remainder when f(x) = x³ - 2x² + 3x - 4 is divided by (x - 2)?
- -2
- 0
- 2 (Correct answer)
- 6
Correct answer: 2
By the Remainder Theorem, the remainder when f(x) is divided by (x - a) is f(a). f(2) = 8 - 8 + 6 - 4 = 2.
O-Level Additional Mathematics
What is the derivative of y = 3x⁴ - 2x² + 5?