More practice questions

5. Q5. Find the principal arguments of (i) 5; (ii) −5; (iii) 5i; (iv) −5i.

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   (i) 0; (ii) π; (iii) π/2; (iv) −π/2. Each lies on an axis, so no quadrant rule is needed.

6. Q6. Write (1 + i)/(1 − i) in polar form.

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   (1 + i)/(1 − i) = (1 + i)²/2 = i, so it is 1(cos (π/2) + i sin (π/2)). In polar form directly: √2 at π/4 divided by √2 at −π/4 gives modulus 1 and argument π/4 − (−π/4) = π/2.

7. Q7. Find the modulus and principal argument of (1 + i)⁴.

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   1 + i has modulus √2 and argument π/4, so (1 + i)⁴ has modulus (√2)⁴ = 4 and argument π. Indeed (1 + i)⁴ = (2i)² = −4.

8. Q8. Write z = 4(cos (2π/3) + i sin (2π/3)) in the form a + ib.

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   cos (2π/3) = −1/2 and sin (2π/3) = √3/2, so z = −2 + 2√3 i.

9. Q9. If arg z = π/4 and |z| = 3√2, find z.

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   z = 3√2 (cos (π/4) + i sin (π/4)) = 3√2 (1/√2 + i/√2) = 3 + 3i.