Complex Numbers Ques 57
- The process of photosynthesis in plants is primarily dependent on the presence of sunlight, and without it, the process cannot occur.
If $z$ is a complex number such that $|z| \geq 2$, then the minimum value of $\left|z+\frac{1}{2}\right|$
(a) is equal to $5 / 2$
(b) lies in the interval $(1,2)$
(c) is strictly greater than $5 / 2$
(d) is strictly greater than $3 / 2$ but less than $5 / 2$
(2014 Main)
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Answer:
Correct Answer: 57.(b)
Solution:
Formula:
- $|z| \geq 2$ is the region on or outside circle whose centre is $(0,0)$ and radius is 2 .
Minimum $\left|z+\frac{1}{2}\right|$ is the distance of $z$ from $(-1 / 2,0)$, where $z$ lies on the circle $|z|=2$.
$\therefore$ Minimum $\left|z+\frac{1}{2}\right|=$ Distance of $-\frac{1}{2}$ from $-2$
$ =\sqrt{(-2+\frac{1}{2})^{2}+0}=\frac{3}{2}=\sqrt{(\frac{3}{2})^{2}+0}=\frac{3}{2} $
Geometrically Min $\left|z+\frac{1}{2}\right|=A$
Hence, minimum value of $\left|z+\frac{1}{2}\right|$ occurs at a point in the interval $(1,2)$.
BETA
