ESAT ESAT Further Mathematics 2

Question 1

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

Accepted Answer

y = Ce^(2x)

Answer

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?

Accepted Answer

√2 exp(i·3π/4)

Answer

|-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?

Accepted Answer

4

Answer

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?

Accepted Answer

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

Answer

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)?

Accepted Answer

5 - 5i

Answer

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

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