TMUA Pure Mathematics 4 — Questions and Answers
Question 1: Find ∫(3x² + 2x − 1) dx.
- x³ + x² − x + C (Correct answer)
- 6x + 2 + C
- x³ + x² + C
- 3x³ + 2x² − x + C
Correct answer: x³ + x² − x + C
Integrating term-by-term: 3x²→x³, 2x→x², −1→−x, plus constant C.
Question 2: The line y = 2x + c is tangent to the circle x² + y² = 5. Find |c|.
- 5 (Correct answer)
- √5
- 2
- 10
Correct answer: 5
Distance from origin to the line 2x − y + c = 0 must equal √5: |c|/√5 = √5, so |c| = 5.
Question 3: If p(x) = x³ − 2x² − 5x + 6 and (x − 1) is a factor, fully factorise p(x).
- (x−1)(x−3)(x+2) (Correct answer)
- (x−1)(x+3)(x−2)
- (x−1)²(x−6)
- (x+1)(x−3)(x+2)
Correct answer: (x−1)(x−3)(x+2)
Dividing by (x−1) gives x² − x − 6 = (x−3)(x+2), so p(x) = (x−1)(x−3)(x+2).
Question 4: What is the range of f(x) = −x² + 4 for x ∈ ℝ?
- f(x) ≤ 4 (Correct answer)
- f(x) ≥ 4
- f(x) ≥ 0
- f(x) ≤ 0
Correct answer: f(x) ≤ 4
The parabola opens downward with vertex at (0, 4), so the maximum value is 4 and f(x) ≤ 4.
Question 5: Simplify (√5 + 2)(√5 − 2).
- 1 (Correct answer)
- 3
- 5 − 4
- 9
Correct answer: 1
Using the difference of two squares: (√5)² − 2² = 5 − 4 = 1.
Question 6: For the sequence defined by aₙ₊₁ = 3aₙ − 1 with a₁ = 2, find a₃.
- 14 (Correct answer)
- 5
- 15
- 10
Correct answer: 14
a₂ = 3(2) − 1 = 5; a₃ = 3(5) − 1 = 14.
Question 7: Find the value of k such that the equation kx² − 6x + 3 = 0 has exactly one solution.
- 3 (Correct answer)
- 6
- 9
- 1
Correct answer: 3
For one solution, discriminant = 36 − 12k = 0, so k = 3.
