Question

Statement 1: If $z_1,z_2$ are the roots of the quadratic equation $a{z}^2+bz+c=0$ such that $Im\left(z_1{z}_2\right)\ne 0$, then at least one of a, b, c is imaginary.
Statement 2: If quadratic equation having real coefficients has complex roots, then roots are always conjugate to each other.

• A. Both the statements are true, and Statement 2 is the correct explanation for Statement 1.
• B. Both the statements are true, but Statement 2 is not the correct explanation for Statement 1.
• C. Statement 1 is true and Statement 2 is false.
• D. Statement 1 is false and Statement 2 is true.

Answer 1

The correct option is A Both the statements are true, and Statement 2 is the correct explanation for Statement 1.

We have,
$a{z}^2+bz+c=0$
and $z_1,z_2$ [roots of (1)] are such that $Im\left(z_1{z}_2\right)\ne 0$. So, $z_1$ and $z_2$ are not conjugate of each other. That is complex roots of (1) are not conjugate of each other, which implies that coefficients a, b, c cannot all be real. Hence, at least one of $a,b,c$ is imaginary.