More practice questions

5. Q5. Express [(1/3 + (7/3)i) + (4 + (1/3)i)] − (−4/3 + i) in the form a + ib (NCERT Exercise 4.1, Q7).

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   Real parts: 1/3 + 4 + 4/3 = 17/3. Imaginary parts: 7/3 + 1/3 − 1 = 5/3. So 17/3 + (5/3)i.

6. Q6. Evaluate [i¹⁸ + (1/i)²⁵]³ (NCERT Miscellaneous Exercise, Q1).

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   i¹⁸ = i^{16 + 2} = −1. 1/i = −i, and (−i)²⁵ = (−i)²⁴(−i) = −i, since (−i)⁴ = 1. So the bracket is −1 − i, and (−1 − i)³ = −(1 + i)³ = −(1 + 3i − 3 − i) = 2 − 2i.

7. Q7. Find i + i² + i³ + … + i¹⁰⁰.

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   Every block of four consecutive powers adds to i − 1 − i + 1 = 0, and 100 is a multiple of 4, so the sum is 0.

8. Q8. Find (1 + i)² and use it to find (1 + i)⁸.

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   (1 + i)² = 1 + 2i + i² = 2i. Then (1 + i)⁸ = (2i)⁴ = 16i⁴ = 16.

9. Q9. Find the real numbers x and y if (x − iy)(3 + 5i) is the conjugate of −6 − 24i (NCERT Miscellaneous Exercise, Q8).

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   The conjugate of a + ib is a − ib (NCERT Section 4.4), so the conjugate of −6 − 24i is −6 + 24i. (x − iy)(3 + 5i) = (3x + 5y) + i(5x − 3y). So 3x + 5y = −6 and 5x − 3y = 24. Multiply the first equation by 3 and the second by 5 and add: 34x = 102, so x = 3 and y = (−6 − 9)/5 = −3.

10. Q10. Express (3 + 2i)/(1 − i) in the form a + ib.

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    Multiply the top and the bottom by 1 + i: (3 + 3i + 2i + 2i²)/(1 + 1) = (1 + 5i)/2 = 1/2 + (5/2)i.