Question

Assertion :$|z_1-a|<a,|z_2-b|<b,|z_3-c|<c$. where $a,b,c$ are positive real numbers, then $|z_1+z_2+z_3|$ is greater than $2|a+b+c|$. Reason: $|z_1\pm z_2|\le \left|z_1\right|+\left|z_2\right|$.

• 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 D Assertion is incorrect and Reason is correct

Given that, $|z_1-a|<a,|z_2-b|<b,|z_3-c|<c$
Where $a$, $b$, $c$ are positive real numbers.
As we know that, $|z_1\pm z_2|\le \left|z_1\right|+\left|z_2\right|$
Therefore, $\left|z_1\right|=|(z_1-a)+a|\le |z_1-a|+a<2a$
and $\left|z_2\right|=|(z_2-b)+b|\le |z_2-b|+b<2b$
and $\left|z_3\right|=|(z_3-c)+c|\le |z_3-c|+c<2c$
$\Rightarrow \left|z_1\right|+\left|z_2\right|+\left|z_3\right|<2(a+b+c)$ ....(1)
Since, $|z_1+z_2+z_3|\le \left|z_1\right|+\left|z_2\right|+\left|z_3\right|$
Therefore, $|z_1+z_2+z_3|<2(a+b+c)$ ....[from (1)]
Ans: D