TMUA Pure Mathematics 5 — Questions and Answers
Question 1: What is the derivative of y = e^(2x) · sin(x)?
- e^(2x)(2sin(x) + cos(x)) (Correct answer)
- 2e^(2x)cos(x)
- e^(2x)(sin(x) + cos(x))
- e^(2x)(2sin(x) − cos(x))
Correct answer: e^(2x)(2sin(x) + cos(x))
By the product rule: dy/dx = 2e^(2x)sin(x) + e^(2x)cos(x) = e^(2x)(2sin(x) + cos(x)).
Question 2: Find the equation of the circle with centre (3, −2) and radius 5.
- (x−3)² + (y+2)² = 25 (Correct answer)
- (x+3)² + (y−2)² = 25
- (x−3)² + (y+2)² = 5
- (x−3)² + (y−2)² = 25
Correct answer: (x−3)² + (y+2)² = 25
Standard form (x − h)² + (y − k)² = r² with (h,k) = (3,−2) and r = 5 gives (x−3)² + (y+2)² = 25.
Question 3: Rationalise the denominator of 1/(3 − √2).
- (3 + √2)/7 (Correct answer)
- (3 − √2)/7
- (3 + √2)/11
- 1/(3 − √2)
Correct answer: (3 + √2)/7
Multiply numerator and denominator by (3 + √2): (3 + √2)/(9 − 2) = (3 + √2)/7.
Question 4: If log₃(x) = 4, what is x?
- 81 (Correct answer)
- 12
- 64
- 27
Correct answer: 81
log₃(x) = 4 means x = 3⁴ = 81.
Question 5: A function satisfies f(x) = f(−x) for all x. What type of symmetry does its graph have?
- Symmetry about the y-axis (Correct answer)
- Symmetry about the x-axis
- Rotational symmetry about the origin
- No symmetry
Correct answer: Symmetry about the y-axis
f(x) = f(−x) defines an even function, which is symmetric about the y-axis.
Question 6: Evaluate ∫₀² (3x² − 2) dx.
- 4 (Correct answer)
- 6
- 8
- 2
Correct answer: 4
[x³ − 2x]₀² = (8 − 4) − 0 = 4.
Question 7: How many distinct real roots does x⁴ − 5x² + 4 = 0 have?
- 4 (Correct answer)
- 2
- 0
- 3
Correct answer: 4
Let u = x²: u² − 5u + 4 = (u−1)(u−4) = 0, giving x² = 1 (x = ±1) and x² = 4 (x = ±2) — four distinct real roots.
TMUA Pure Mathematics 5
What is the derivative of y = e^(2x) · sin(x)?