Question

Assertion :If $|z|\ge 13$, then least value of $|z+\frac{1}{2}|$ is $\frac{25}{2}$. Reason: $|z_1+z_2|\le |z_1|+|z_2|$.

• A. Both (A) & (R) are individually true & (R) is correct explanation of (A).
• B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).
• C. (A) is true (R) is false.
• D. (A) is false (R) is true.

Answer 1

The correct option is B Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).

$|z_1|-|z_2|\ge |z_1+z_2|\le |z_1|+|z_2|$
Or
$|z|-\frac{1}{2}\le |z+\frac{1}{2}|\le |z|+\frac{1}{2}$
Considering $|z|=13$, we get
$13-\frac{1}{2}\le |z+\frac{1}{2}|\le |z|+\frac{1}{2}$
Or
$\frac{25}{2}\le |z+\frac{1}{2}|\le \frac{27}{2}$.