Method 1: Using Cartesian form of z

1. Let the complex number z be represented as z = x + iy, where x and y are real numbers.
2. The problem states that the imaginary part of z is non-zero, which means y ≠ 0.
3. We are given the expression $a = z^2 + z + 1$, where a is a real number.
4. Substitute z = x + iy into the expression for a: $a = (x + iy)^2 + (x + iy) + 1$ $a = (x^2 - y^2 + 2ixy) + (x + iy) + 1$
5. Group the real and imaginary parts: $a = (x^2 - y^2 + x + 1) + i(2xy + y)$
6. Since a is a real number, its imaginary part must be zero. $Im(a) = 2xy + y = 0$ $y(2x + 1) = 0$
7. We are given that y ≠ 0. Therefore, the other factor must be zero: $2x + 1 = 0$ $x = -{1 \over 2}$
8. Now, we can express a (which is the real part of the expression) in terms of y, by substituting x = -1/2: $a = Re(a) = x^2 - y^2 + x + 1$ $a = \left(-{1 \over 2}\right)^2 - y^2 + \left(-{1 \over 2}\right) + 1$ $a = {1 \over 4} - y^2 - {1 \over 2} + 1$ $a = {1 - 2 + 4 \over 4} - y^2$ $a = {3 \over 4} - y^2$
9. We know that y ≠ 0, which implies $y^2 > 0$.
10. Multiplying by -1 reverses the inequality sign: $-y^2 < 0$.
11. Adding 3/4 to both sides: ${3 \over 4} - y^2 < {3 \over 4}$ $a < {3 \over 4}$
12. This means a can be any real number strictly less than 3/4. Now we check the given options:
    • A: -1. Since -1 < 3/4, this value is possible.
    • B: 1/3. Since 1/3 = 4/12 and 3/4 = 9/12, we have 1/3 < 3/4. This value is possible.
    • C: 1/2. Since 1/2 = 2/4, we have 1/2 < 3/4. This value is possible.
    • D: 3/4. The value a must be strictly less than 3/4. Therefore, a cannot be equal to 3/4. If a = 3/4, then $y^2 = 0$, which means y=0, contradicting the given condition y ≠ 0.

Method 2: Using Quadratic Equation

1. The given relation is $a = z^2 + z + 1$.
2. We can rearrange this into a quadratic equation in z: $z^2 + z + (1 - a) = 0$
3. We are given that a is a real number. Therefore, the coefficients of this quadratic equation (1, 1, and 1-a) are all real.
4. The problem states that the imaginary part of z is non-zero, which means z is a non-real complex number.
5. A quadratic equation with real coefficients has non-real roots if and only if its discriminant is negative.
6. The discriminant D of this quadratic equation is: $D = b^2 - 4ac = (1)^2 - 4(1)(1 - a)$ $D = 1 - 4 + 4a$ $D = 4a - 3$
7. For z to be non-real, we must have D < 0: $4a - 3 < 0$ $4a < 3$ $a < {3 \over 4}$
8. This result shows that a must be strictly less than 3/4. Examining the options, the only value that does not satisfy this condition is a = 3/4.

Conclusion: Both methods lead to the same result that a cannot take the value 3/4.