More practice questions

5. Q5. Let z₁ = 2 − i and z₂ = −2 + i. Find (i) Re (z₁z₂/$\bar{z}_1$); (ii) Im (1/(z₁$\bar{z}_1$)) (NCERT Miscellaneous Exercise, Q7).

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   (i) z₁z₂ = (2 − i)(−2 + i) = −3 + 4i and $\bar{z}_1$ = 2 + i, so the quotient is (−3 + 4i)(2 − i)/5 = (−2 + 11i)/5; its real part is −2/5. (ii) z₁$\bar{z}_1$ = |z₁|² = 5, so 1/5 is real and its imaginary part is 0.

6. Q6. If x + iy = (a + ib)/(a − ib), with x, y, a and b real and a, b not both 0, prove that x² + y² = 1 (NCERT Example 8, page 85).

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   x² + y² = |x + iy|² = |a + ib|²/|a − ib|² = (a² + b²)/(a² + b²) = 1, because a number and its conjugate have the same modulus.

7. Q7. If α and β are different complex numbers with |β| = 1, find |(β − α)/(1 − $\bar{\alpha}$β)| (NCERT Miscellaneous Exercise, Q11).

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   Since |β| = 1, β $\bar{\beta}$ = 1, so 1 − $\bar{\alpha}$β = β $\bar{\beta}$ − $\bar{\alpha}$β = β($\bar{\beta}$ − $\bar{\alpha}$), and $\bar{\beta}$ − $\bar{\alpha}$ is the conjugate of β − α. Hence |1 − $\bar{\alpha}$β| = |β| |β − α| = |β − α|, which is not 0 because α ≠ β. The required value is 1.

8. Q8. For real numbers a, b, c, d, e, f, g, h, A and B, if (a + ib)(c + id)(e + if)(g + ih) = A + iB, show that the product of a² + b², c² + d², e² + f² and g² + h² is A² + B² (NCERT Miscellaneous Exercise, Q13).

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   Take the modulus of both sides and square: the modulus of a product is the product of the moduli, and |a + ib|² = a² + b² for each factor, while |A + iB|² = A² + B².

9. Q9. Find the least positive integer m with ((1 + i)/(1 − i))^m = 1 (NCERT Miscellaneous Exercise, Q14).

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   (1 + i)/(1 − i) = (1 + i)²/((1 − i)(1 + i)) = 2i/2 = i, and i^m = 1 first for m = 4.