Question

For the equation $a{x}^2+bx+c=0,$ $a,b$ and $c$ are real,

Statement 1: If the equation $a{x}^2+bx+c=0,0<a<b<c$, has non-real complex roots $z_1$ and $z_2$, then $|z_1|>1,|z_2|>1$.

Statement 2: Complex roots always occur in conjugate pairs.

• 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.

If roots of $a{x}^2+bx+c=0$, 0 < a < b < c, are nonreal, then they will be the conjugate of each other. Hence,
$z_2={\overline{z}}_1\Rightarrow |z_1|=|z_2|$
Now, $z_1{z}_2=\frac{c}{a}>1\Rightarrow |z_1{|}^2>1$
$\Rightarrow |z_1|>1$
$\Rightarrow |z_2|>1$