A-Level H2 Mathematics 1 — Questions and Answers
Question 1: Find dy/dx if y = 3x⁴ - 5x² + 2x - 7.
- 12x³ - 10x + 2 (Correct answer)
- 12x³ - 5x + 2
- 4x³ - 10x + 2
- 12x³ - 10x + 7
Correct answer: 12x³ - 10x + 2
Differentiating term by term: d/dx(3x⁴) = 12x³, d/dx(-5x²) = -10x, d/dx(2x) = 2, d/dx(-7) = 0. Result: 12x³ - 10x + 2.
Question 2: Evaluate ∫(2x + 3) dx
- 2x² + 3x + C
- x² + 3x + C (Correct answer)
- x² + 3 + C
- 2x + 3x + C
Correct answer: x² + 3x + C
∫(2x + 3) dx = 2x²/2 + 3x + C = x² + 3x + C. Remember to include the constant of integration C.
Question 3: What is the sum to infinity of a geometric series with first term 4 and common ratio 1/3?
- 6 (Correct answer)
- 8
- 12
- 5
Correct answer: 6
S∞ = a/(1-r) = 4/(1 - 1/3) = 4/(2/3) = 4 × 3/2 = 6. Valid since |r| = 1/3 < 1.
Question 4: Solve: e^(2x) = 5
- x = ln5/2 (Correct answer)
- x = ln5
- x = 2ln5
- x = ln(5/2)
Correct answer: x = ln5/2
Taking natural log of both sides: 2x = ln5, so x = ln5/2.
Question 5: What is the equation of the normal to the curve y = x² at the point (2, 4)?
- y = 4x - 4
- y = -x/4 + 9/2 (Correct answer)
- y = 4x - 12
- y = x/4 + 7/2
Correct answer: y = -x/4 + 9/2
dy/dx = 2x. At x=2: gradient of tangent = 4. Gradient of normal = -1/4. Equation: y - 4 = -1/4(x - 2) → y = -x/4 + 1/2 + 4 = -x/4 + 9/2.
Question 6: Find the range of values of k for which x² - 6x + k = 0 has no real roots.
- k > 9 (Correct answer)
- k < 9
- k = 9
- k ≥ 9
Correct answer: k > 9
For no real roots: discriminant < 0. b² - 4ac < 0. (-6)² - 4(1)(k) < 0. 36 - 4k < 0. k > 9.
A-Level H2 Mathematics 1
Find dy/dx if y = 3x⁴ - 5x² + 2x - 7.