Let z be a complex number such that the imaginary part of z is non-zero and a = z² + z + 1 is real.

Q: Why can't we just substitute $z = x + iy$ and equate the imaginary part to zero?

You absolutely can! If you substitute $z = x + iy$ into $z^2 + z + 1 = a$, you will get $(x^2 - y^2 + x + 1) + i(2xy + y) = 0$. Since $a$ is real, the imaginary part must be zero: $y(2x + 1) = 0$. Since $y \neq 0$, we must have $x = -1/2$. Substituting $x = -1/2$ back into the real part will give you the range of $a$. However, the discriminant method is much faster and less prone to algebraic errors!

Q: Wait, why is the discriminant strictly less than zero? Why not less than or equal to?

Great catch! The problem states that the imaginary part of $z$ is non-zero, which means $z$ cannot be a real number. If the discriminant were equal to zero, the roots would be real and equal. Since we need non-real roots, we must strictly enforce $D < 0$.

Q: Does $a$ being real guarantee the coefficients of the quadratic are real?

Yes, it does. Our equation is $z^2 + z + (1 - a) = 0$. The coefficients are $1$, $1$, and $(1 - a)$. Since $1$ is real and $a$ is given as real, $(1 - a)$ is also real. Therefore, all coefficients are real, allowing us to use the discriminant property for complex roots.

Question

Let $z$ be a complex number such that the imaginary part of $z$ is non-zero and $a = z^2 + z + 1$ is real. Then $a$ cannot take the value

Options:

(A) -1

(B) 1/3

(C) 1/2

(D) 3/4

Sigma Percentile · Accepted Answer

### Analyzing the Geometry of the Complex Plane

Imagine you are standing on the Argand plane. You have a complex number $z$, and you are told its imaginary part is non-zero.

This is a powerful geometric constraint. It means $z$ is not just a point on the horizontal real axis; it is a point that exists in the two-dimensional space of the complex plane, either above or below the axis.

### The Quadratic Bridge

We are given the expression $a = z^2 + z + 1$, which is purely real. This is a bridge between the world of complex numbers and the world of quadratic equations.

Let us rearrange this into the standard form:
$$z^2 + z + (1 - a) = 0$$

We have a quadratic in $z$ where the coefficients are $1$, $1$, and $(1 - a)$. Because $a$ is real, all these coefficients are real numbers.

### The Discriminant's Secret

In the realm of real-coefficient quadratic equations, the discriminant $D = B^2 - 4AC$ is the ultimate gatekeeper.

If $D > 0$, the roots are real and distinct. If $D = 0$, the roots are real and equal.

But if $D < 0$, the roots are non-real complex conjugates. Since our problem explicitly tells us that $z$ is non-real, we know with absolute certainty that the discriminant of our equation must be strictly less than zero.

### The Inequality Dance

Now, let us perform the calculation. The discriminant is $D = 1^2 - 4(1)(1 - a)$.

We set this to be less than zero:
$$1 - 4(1 - a) < 0$$

Expanding this, we get $1 - 4 + 4a < 0$, which simplifies to $-3 + 4a < 0$. Solving for $a$, we find:
$$a < \frac{3}{4}$$

This inequality defines the entire set of possible values for $a$. Any value of $a$ that is less than $\frac{3}{4}$ is perfectly valid.

However, the question asks us to identify a value that $a$ cannot take. Since $a$ must be strictly less than $\frac{3}{4}$, it can never be equal to $\frac{3}{4}$.

### Final Reflection

We have successfully navigated the relationship between complex numbers and quadratic theory. By recognizing that the non-real nature of $z$ forces the discriminant to be negative, we arrived at the answer with elegance.

**The value that $a$ cannot take is $\frac{3}{4}$.**

Remember, in JEE Advanced, it is rarely about brute force; it is about finding the right conceptual tool—in this case, the discriminant—to simplify the path forward.

Animated Solution for Mathematics - Complex Numbers: Let $z$ be a complex number such that the imaginary part of $z$ is non-zero and $a = z^{2} + z + 1$ is real. Then $a$ cannot take the value

Select Answer:

Visualized Solution

Visualizing $z$ in the Argand Plane

The Real Expression $a$

Identifying Coefficients

The Discriminant Tool

Setting up the Inequality

Expanding the Expression

Simplifying Further

The Range of $a$

Checking the Options

Final Conclusion

The Sigma Insight: Algebraic Operations on Complex Numbers

Analyzing the Geometry of the Complex Plane

The Quadratic Bridge

The Discriminant's Secret

The Inequality Dance

Final Reflection

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List-I

List-II

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