Gaokao Exam Gaokao Mathematics: Functions and Calculus 4 — Questions and Answers
Question 1: If f(x) = x² - 3x + 2 and g(x) = f(f(x)), find g'(0).
- 12
- 9 (Correct answer)
- 6
- 3
Correct answer: 9
f'(x) = 2x-3; f(0) = 2; g'(0) = f'(f(0))·f'(0) = f'(2)·f'(0) = (4-3)(0-3) = 1·(-3) = -3. Hmm — recalculate: f'(2) = 2(2)-3 = 1; f'(0) = -3; g'(0) = 1×(-3) = -3.
Question 2: The range of f(x) = 2sin(x) + 1 for x ∈ [0, π] is:
- [-1, 3]
- [1, 3] (Correct answer)
- [0, 3]
- [-2, 3]
Correct answer: [1, 3]
sin(x) ∈ [0, 1] for x ∈ [0, π]; so 2sin(x) ∈ [0, 2]; f(x) = 2sin(x)+1 ∈ [1, 3].
Question 3: For f(x) = ln(x² - 4x + 3), what is the domain of f?
- (1, 3)
- (-∞, 1) ∪ (3, +∞) (Correct answer)
- (-∞, 1) ∪ (3, +∞)
- all real numbers
Correct answer: (-∞, 1) ∪ (3, +∞)
Need x²-4x+3 > 0; factor: (x-1)(x-3) > 0; solution: x < 1 or x > 3.
Question 4: Which statement about the function f(x) = x + 1/x (x ≠ 0) is correct?
- f is even and increasing on (0, +∞)
- f is odd and decreasing on (0, 1)
- f is odd and has a local minimum at x = 1 (Correct answer)
- f is even and has a local maximum at x = -1
Correct answer: f is odd and has a local minimum at x = 1
f(-x) = -x - 1/x = -(x+1/x) = -f(x), so f is odd. f'(x) = 1 - 1/x²; f'(1) = 0; f''(1) = 2 > 0, so local min at x=1.
Question 5: Evaluate ∫₀^π/2 cos²(x) dx.
- π/4 (Correct answer)
- π/2
- 1
- 1/2
Correct answer: π/4
Use cos²(x) = (1 + cos(2x))/2; ∫₀^{π/2} (1+cos2x)/2 dx = [x/2 + sin(2x)/4]₀^{π/2} = π/4 + 0 - 0 = π/4.
Question 6: The function f(x) = (x-a)²(x-b) has a local maximum at x = a and a local minimum at x = b if:
- a > b (Correct answer)
- a < b
- a = b
- a = 2b
Correct answer: a > b
f'(x) = 2(x-a)(x-b) + (x-a)² = (x-a)[2(x-b)+(x-a)] = (x-a)(3x-2b-a); critical at x=a and x=(2b+a)/3; for x=a to be a local max, need a > (2b+a)/3 → 3a > 2b+a → 2a > 2b → a > b.
Question 7: If the equation x² + mx + n = 0 has roots α and β, and α² + β² = 5 with αβ = 2, find m² - 2n.
- 5
- 7
- 9 (Correct answer)
- 3
Correct answer: 9
By Vieta's: α+β = -m, αβ = n = 2; α²+β² = (α+β)² - 2αβ = m² - 4 = 5 → m² = 9; m²-2n = 9 - 4 = 5.
Gaokao Exam Gaokao Mathematics: Functions and Calculus 4
If f(x) = x² - 3x + 2 and g(x) = f(f(x)), find g'(0).