Question

A is the region of the complex plane {z: z/4 and 4/$\bar{z}$ have real and imaginary part in (0, 1)}, then [p] (where p is the area of the region A and [.] denotes the greatest integer function) is

• A. 25
• B. 0
• C. 5
• D. 6

Answer 1

Answer: (B)

0

Answer 2

Sol. Re $\left( \frac{z}{4} \right)$Ī (0, 1), Im $\left( \frac{z}{4} \right)$Ī [0, 1)

means that if z = a + ib than a, b Ī (0, 4)

Now $\frac{4}{a - ib}$= $\frac{4a}{a^{2} + b^{2}}$+ $\frac{4bi}{a^{2} + b^{2}}$

Ž 0 < a, b < $\frac{a^{2} + b^{2}}{4}$

Ž (a –2)² + b² > 4 and a² + (b –2)² > 4

So we want area inside the square and outside the two circles

Ž area = 16 – 4p + (2p –4) = 12 –2p