1
JEE Main 2023 (Online) 31st January Evening Shift
+4
-1 The complex number $z=\frac{i-1}{\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}}$ is equal to :
A
$\cos \frac{\pi}{12}-i \sin \frac{\pi}{12}$
B
$\sqrt{2}\left(\cos \frac{\pi}{12}+i \sin \frac{\pi}{12}\right)$
C
$\sqrt{2} i\left(\cos \frac{5 \pi}{12}-i \sin \frac{5 \pi}{12}\right)$
D
$\sqrt{2}\left(\cos \frac{5 \pi}{12}+i \sin \frac{5 \pi}{12}\right)$
2
JEE Main 2023 (Online) 31st January Morning Shift
+4
-1 For all $$z \in C$$ on the curve $$C_{1}:|z|=4$$, let the locus of the point $$z+\frac{1}{z}$$ be the curve $$\mathrm{C}_{2}$$. Then :

A
the curves $$C_{1}$$ and $$C_{2}$$ intersect at 4 points
B
the curve $$C_{2}$$ lies inside $$C_{1}$$
C
the curve $$C_{1}$$ lies inside $$C_{2}$$
D
the curves $$C_{1}$$ and $$C_{2}$$ intersect at 2 points
3
JEE Main 2023 (Online) 29th January Morning Shift
+4
-1

For two non-zero complex numbers $$z_{1}$$ and $$z_{2}$$, if $$\operatorname{Re}\left(z_{1} z_{2}\right)=0$$ and $$\operatorname{Re}\left(z_{1}+z_{2}\right)=0$$, then which of the following are possible?

A. $$\operatorname{Im}\left(z_{1}\right)>0$$ and $$\operatorname{Im}\left(z_{2}\right) > 0$$

B. $$\operatorname{Im}\left(z_{1}\right) < 0$$ and $$\operatorname{Im}\left(z_{2}\right) > 0$$

C. $$\operatorname{Im}\left(z_{1}\right) > 0$$ and $$\operatorname{Im}\left(z_{2}\right) < 0$$

D. $$\operatorname{Im}\left(z_{1}\right) < 0$$ and $$\operatorname{Im}\left(z_{2}\right) < 0$$

Choose the correct answer from the options given below:

A
A and C
B
A and B
C
B and D
D
B and C
4
JEE Main 2023 (Online) 25th January Evening Shift
+4
-1 Let $$z$$ be a complex number such that $$\left| {{{z - 2i} \over {z + i}}} \right| = 2,z \ne - i$$. Then $$z$$ lies on the circle of radius 2 and centre :

A
(0, $$-$$2)
B
(0, 0)
C
(0, 2)
D
(2, 0)
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