More practice questions

5. Q5. Plot z = 1 + 2i, iz and −z. How is each obtained from z?

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   z is (1, 2). iz = i + 2i² = −2 + i, the point (−2, 1): z turned a right angle anticlockwise about 0. −z = −1 − 2i, the point (−1, −2): z turned half a turn, the reflection of z in O.

6. Q6. Show that the points 3 + 4i, −4 + 3i and 5 all lie on one circle centred at 0, and give its radius.

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   |3 + 4i| = |−4 + 3i| = 5 and |5| = 5, so all three are 5 from 0: they lie on the circle of radius 5 centred at the origin.

7. Q7. For z = x + iy, where are the points with z = $\bar{z}$? With z = −$\bar{z}$?

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   z = $\bar{z}$ means y = −y, so y = 0: the real axis. z = −$\bar{z}$ means x = −x, so x = 0: the imaginary axis.

8. Q8. Find the number of non-zero integer solutions of |1 − i|^x = 2^x (NCERT Miscellaneous Exercise, Q12).

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   |1 − i| = √2, so the equation is (√2)^x = 2^x, that is 2^x/2 = 2^x, so x/2 = x and x = 0. There are no non-zero integer solutions: the number is 0.