ESAT Further Mathematics Flashcards

Which of the following is the general solution to the differential equation dy/dx = 2y?

Answer: y = Ce^(2x)

Separating variables gives dy/y = 2dx; integrating yields ln|y| = 2x + C, so y = Ae^(2x).

What is the polar form of the complex number -1 + i?

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).

How many distinct real roots does x⁴ - 5x² + 4 = 0 have?

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.

For f(x) = x³ - 3x, which statement correctly describes its turning points?

Answer: Local max at x = -1, local min at x = 1

f'(x) = 3x² - 3 = 0 gives x = ±1; f''(-1) = -6 0 (minimum).

What is the product of the complex numbers (2 + i) and (1 - 3i)?

Answer: 5 - 5i

(2+i)(1-3i) = 2 - 6i + i - 3i² = 2 - 5i + 3 = 5 - 5i, since i² = -1.

What is the volume of the solid formed when y = √x (0 ≤ x ≤ 4) is rotated 360° about the x-axis?

Answer: 8π

V = π∫₀⁴ (√x)² dx = π∫₀⁴ x dx = π[x²/2]₀⁴ = 8π.

Which technique is most appropriate for evaluating ∫x·e^x dx?

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.