ESAT ESAT Further Mathematics 2 — Questions and Answers
Question 1: Which of the following is the general solution to the differential equation dy/dx = 2y?
- y = Ce^(2x) (Correct answer)
- y = Ce^(-2x)
- y = 2x + C
- y = x² + C
Correct answer: y = Ce^(2x)
Separating variables gives dy/y = 2dx; integrating yields ln|y| = 2x + C, so y = Ae^(2x).
Question 2: What is the polar form of the complex number -1 + i?
- √2 exp(i·3π/4) (Correct answer)
- √2 exp(iπ/4)
- 2 exp(i·3π/4)
- √2 exp(-iπ/4)
Correct answer: √2 exp(i·3π/4)
|-1+i| = √2 and arg(-1+i) = 3π/4 (second quadrant), so the polar form is √2 exp(i·3π/4).
Question 3: How many distinct real roots does x⁴ - 5x² + 4 = 0 have?
- 4 (Correct answer)
- 2
- 0
- 3
Correct answer: 4
Substituting u = x² gives (u-1)(u-4) = 0, so u = 1 or u = 4, yielding x = ±1 and x = ±2: four distinct real roots.
Question 4: For f(x) = x³ - 3x, which statement correctly describes its turning points?
- Local max at x = -1, local min at x = 1 (Correct answer)
- Local max at x = 1, local min at x = -1
- No turning points
- Local max at x = 0
Correct answer: Local max at x = -1, local min at x = 1
f'(x) = 3x² - 3 = 0 gives x = ±1; f''(-1) = -6 < 0 (maximum) and f''(1) = 6 > 0 (minimum).
Question 5: What is the product of the complex numbers (2 + i) and (1 - 3i)?
- 5 - 5i (Correct answer)
- 2 - 6i
- 5 + 5i
- -1 - 5i
Correct answer: 5 - 5i
(2+i)(1-3i) = 2 - 6i + i - 3i² = 2 - 5i + 3 = 5 - 5i, since i² = -1.
Question 6: What is the volume of the solid formed when y = √x (0 ≤ x ≤ 4) is rotated 360° about the x-axis?
- 8π (Correct answer)
- 4π
- 16π
- 32π
Correct answer: 8π
V = π∫₀⁴ (√x)² dx = π∫₀⁴ x dx = π[x²/2]₀⁴ = 8π.
Question 7: Which technique is most appropriate for evaluating ∫x·e^x dx?
- Integration by parts (Correct answer)
- Substitution
- Partial fractions
- Trigonometric substitution
Correct answer: Integration by parts
Integration by parts with u = x and dv = e^x dx gives ∫x·e^x dx = x·e^x - e^x + C.
ESAT ESAT Further Mathematics 2
Which of the following is the general solution to the differential equation dy/dx = 2y?