A-Level Mathematics Core 2 — Questions and Answers
Question 1: What is the value of ∫₀² (3x² - 2x) dx?
- 4 (Correct answer)
- 6
- 2
- 8
Correct answer: 4
Integrating: [x³ - x²]₀² = (8 - 4) - (0 - 0) = 4.
Question 2: The binomial expansion of (1 + 2x)⁵ has a term in x³. What is its coefficient?
- 80 (Correct answer)
- 40
- 160
- 10
Correct answer: 80
Using ⁵C₃ × (2x)³ = 10 × 8x³ = 80x³. The binomial coefficient ⁵C₃ = 10 and (2x)³ = 8x³.
Question 3: A circle has equation x² + y² - 6x + 4y - 12 = 0. What is its radius?
- 5 (Correct answer)
- √12
- 25
- 7
Correct answer: 5
Completing the square: (x-3)² + (y+2)² = 9 + 4 + 12 = 25. So the radius is √25 = 5.
Question 4: What is the exact value of sin(π/3)?
- √3/2 (Correct answer)
- 1/2
- √2/2
- 1/√3
Correct answer: √3/2
π/3 radians = 60°. From the standard triangle, sin(60°) = √3/2. This is a key exact trigonometric value required for A-Level.
Question 5: The vector a = 3i + 4j. What is the unit vector in the direction of a?
- (3i + 4j)/5 (Correct answer)
- (3i + 4j)/7
- (3i + 4j)/√7
- 3i/5 + 4j/3
Correct answer: (3i + 4j)/5
|a| = √(9 + 16) = √25 = 5. The unit vector is a/|a| = (3i + 4j)/5 = 0.6i + 0.8j.
Question 6: Solve 2sin(x) - 1 = 0 for 0 ≤ x ≤ 2π. How many solutions are there?
- 2 (Correct answer)
- 1
- 3
- 4
Correct answer: 2
sin(x) = 1/2 gives x = π/6 and x = 5π/6 in the range [0, 2π]. Sine is positive in the first and second quadrants, giving exactly 2 solutions.
A-Level Mathematics Core 2
What is the value of ∫₀² (3x² - 2x) dx?