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

The given equation is z² + z + 1 - a = 0. If the solution is not real then Δ = b² - 4ac of the quadratic ax² + bx + c = 0 is less than zero. → 1 - 4(1 - a) < 0 → 1 - 4 + 4a < 0 → 4a < 3 → a < 3/4. Hence, all values of a less than 3/4 will give non-real solutions. Options A, B and C are less than 3/4. Hence option D is correct.