1
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
2
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)
3
JEE Main 2023 (Online) 25th January Morning Shift
+4
-1

Let $$\mathrm{z_1=2+3i}$$ and $$\mathrm{z_2=3+4i}$$. The set $$\mathrm{S = \left\{ {z \in \mathbb{C}:{{\left| {z - {z_1}} \right|}^2} - {{\left| {z - {z_2}} \right|}^2} = {{\left| {{z_1} - {z_2}} \right|}^2}} \right\}}$$ represents a

A
hyperbola with the length of the transverse axis 7
B
hyperbola with eccentricity 2
C
straight line with the sum of its intercepts on the coordinate axes equals $$-18$$
D
straight line with the sum of its intercepts on the coordinate axes equals $$14$$
4
JEE Main 2023 (Online) 24th January Evening Shift
+4
-1

The value of $${\left( {{{1 + \sin {{2\pi } \over 9} + i\cos {{2\pi } \over 9}} \over {1 + \sin {{2\pi } \over 9} - i\cos {{2\pi } \over 9}}}} \right)^3}$$ is

A
$$- {1 \over 2}\left( {1 - i\sqrt 3 } \right)$$
B
$$- {1 \over 2}\left( {\sqrt 3 - i} \right)$$
C
$${1 \over 2}\left( {1 - i\sqrt 3 } \right)$$
D
$${1 \over 2}\left( {\sqrt 3 + i} \right)$$
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