Let be a complex number such that the imaginary part of is non-zero and is real. Then, cannot take the value
- A.
- B.
- C.
- D.
Show answer and solution
Correct option: D
Solution
Let z = x + iy with y ≠ 0. Since a = z² + z + 1 is real, its imaginary part must be zero.
Im(z² + z + 1) = Im(x² - y² + 2xyi + x + iy + 1) = 2xy + y = y(2x + 1) = 0.
Since y ≠ 0, we need 2x + 1 = 0, so x = -1/2.
Now compute the real part: a = x² - y² + x + 1 = 1/4 - y² - 1/2 + 1 = 3/4 - y².
Since y ≠ 0, we have y² > 0, so a = 3/4 - y² < 3/4.
Therefore a can take any value less than 3/4, but cannot equal 3/4.
Im(z² + z + 1) = Im(x² - y² + 2xyi + x + iy + 1) = 2xy + y = y(2x + 1) = 0.
Since y ≠ 0, we need 2x + 1 = 0, so x = -1/2.
Now compute the real part: a = x² - y² + x + 1 = 1/4 - y² - 1/2 + 1 = 3/4 - y².
Since y ≠ 0, we have y² > 0, so a = 3/4 - y² < 3/4.
Therefore a can take any value less than 3/4, but cannot equal 3/4.
[AIEEE 2012]