Question

Let z be a complex number such that the imaginary part of z is non zero and a = z² + z + 1 is real. Then a cannot take the value ______ 

Options

• -1

• `1/3`

• `1/2`

• `3/4`

Answer 1

Let z be a complex number such that the imaginary part of z is non zero and a = z² + z + 1 is real. Then a cannot take the value `underline(3/4)`.

Explanation:

Let z = x + iy ,y ≠ 0

a = z² + z + 1

⇒ (x² - y² + x + 1) + i(2xy + y) = a

⇒ x² - y² + x + 1 = a and 2xy + y = 0

⇒ x² - y² + x + 1 = a and y (2x + 1) = 0

⇒ x² - y² + x + 1 = a and x = `-1/2` ........[∵ y ≠ 0]

⇒ `y^2 = 3/4 - a`

Since, y ≠ 0

∴ a ≠ `3/4`