If z= 4 +i 6 , then - Vidyadip

MCQ

If $\text{z}=\cos\frac{\pi}{4}+\text{i}\sin\frac{\pi}{6},$ then

A
$|\text{z}|=1,\text{arg(z)}=\frac{\pi}{4}$
B
$|\text{z}|=1,\text{arg(z)}=\frac{\pi}{6}$
C
$|\text{z}|=\frac{\sqrt{3}}{2},\text{arg(z)}=\frac{5\pi}{24}$
✓
$|\text{z}|=\frac{\sqrt{3}}{2},\text{arg(z)}=\tan^{-1}\frac{1}{\sqrt{2}}$

✓

Answer

Correct option: D.

$|\text{z}|=\frac{\sqrt{3}}{2},\text{arg(z)}=\tan^{-1}\frac{1}{\sqrt{2}}$

$\text{z}=\cos\frac{\pi}{4}+\text{i}\sin\frac{\pi}{6}$
$\Rightarrow\text{z}=\frac{1}{\sqrt{2}}+\frac{1}{2}\text{i}$
$\Rightarrow|\text{z}|=\sqrt{\Big(\frac{1}{\sqrt{2}}\Big)^2+\frac{1}{4}}$
$\Rightarrow|\text{z}|=\sqrt{\frac{1}{2}+\frac{1}{4}}$
$\Rightarrow|\text{z}|=\sqrt{\frac{3}{4}}$
$\Rightarrow|\text{z}|=\frac{\sqrt{3}}{2}$
$\Rightarrow\tan\alpha=\Big|\frac{\text{Im(z)}}{\text{Re(z)}}\Big|$
$=\frac{1}{\sqrt{2}}$
$\Rightarrow\alpha=\tan^{-1}\Big(\frac{1}{\sqrt{2}}\Big)$
Since, the point z lies in the first quadrant.
Therefore, $|\text{z}|=\frac{\sqrt{3}}{2},\text{arg(z)}=\tan^{-1}\frac{1}{\sqrt{2}}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from Complex Numbers and Quadratic Equations→Complex Numbers and Quadratic Equations — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

If the equation of the locus of a point equidistant from the points $({a_1},{b_1})$ and $({a_2},{b_2})$ is $({a_1} - {a_2})x + ({b_1} - {b_2})y + c = 0$, then the value of $c$ is

$2 \sin A \sin B$ is equal to :

The coefficient of the term independent of $x$ in the expansion of $\Big(\frac{\sqrt{\text{x}}}{3}+\frac{3}{2\text{x}^2}\Big)10$ is:

The solution set of $|x-1|+|x-2|<3$ is...

The distance between two parallel lines $3x + 4y - 8 = 0$ and $3x + 4y - 3 = 0$, is given by

A debate club consists of $6$ girls and $4$ boys. A team of $4$ members is to be selected from this club including the selection of a captain (from among these $4$ memiers) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is

How many numbers of five digits can be formed from the numbers $2, 0, 4, 3, 8$ when repetition of digits is not allowed

Two aeroplanes $I$ and $II$ bomb a target in succession. The probabilities of $l$ and $II$ scoring a hit correctlyare $0.3$ and $0.2,$ respectively. The second plane will bomb only if the first misses the target. The probability that the target is hit by the second plane is

If $x=\{a, e, i, o, u\}$ and $y=\{a, o, u\}$, then the set $x-y$ is :

The length of the latus$-$rectum of the parabola $x^2 - 4x - 8y + 12 = 0$ is