Question

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

• 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

Answer: (A)

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

Let us assume that there exists a $z$ which satisfies the given equation and $|z|<1$
Now, $z^4+z+2=0$
$\Rightarrow -2=z^4+z$
$\Rightarrow |-2|=|z^4+z|$
$\Rightarrow 2\le |z^4|+|z|$
$\Rightarrow 2<1+1$,because$|z|<1,$
This is not possible. Hence the equation does not have a root which satisfies $|z|<1$
Hence statement 1 is true and Statement 2 is the correct explanation for it.