Question

Assertion :Let z be a complex number, then the equation $z^4+z+2=0$ cannot have a root, such that $\left|z\right|<1$. Reason: $|z_1+z_2|\le \left|z_1\right|+\left|z_2\right|$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect and Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$2\le {\left|z\right|}^5$
$z^4+z+2=0$ ...(1)
$z^4+z+2=0$ ...{ $\because \left|z_1{+z}_2\right|\le \left|z_1\right|+\left|z_2\right|$ }
$\Rightarrow 2\le {\left|z\right|}^5$
$\Rightarrow 2\le {\left|z\right|}^5<1$ ...{ $\because \left|z\right|<1$ }
$\Rightarrow 2<1$
Since $2<1$ is not possible
Therefore $z^4+z+2=0$ cannot have a root., such that $\left|z\right|<1$.
Ans; A