1
JEE Main 2022 (Online) 27th July Evening Shift
+4
-1

Let S be the set of all $$(\alpha, \beta), \pi<\alpha, \beta<2 \pi$$, for which the complex number $$\frac{1-i \sin \alpha}{1+2 i \sin \alpha}$$ is purely imaginary and $$\frac{1+i \cos \beta}{1-2 i \cos \beta}$$ is purely real. Let $$Z_{\alpha \beta}=\sin 2 \alpha+i \cos 2 \beta,(\alpha, \beta) \in S$$. Then $$\sum\limits_{(\alpha, \beta) \in S}\left(i Z_{\alpha \beta}+\frac{1}{i \bar{Z}_{\alpha \beta}}\right)$$ is equal to :

A
3
B
3 i
C
1
D
2 $$-$$ i
2
JEE Main 2022 (Online) 27th July Morning Shift
+4
-1

Let the minimum value $$v_{0}$$ of $$v=|z|^{2}+|z-3|^{2}+|z-6 i|^{2}, z \in \mathbb{C}$$ is attained at $${ }{z}=z_{0}$$. Then $$\left|2 z_{0}^{2}-\bar{z}_{0}^{3}+3\right|^{2}+v_{0}^{2}$$ is equal to :

A
1000
B
1024
C
1105
D
1196
3
JEE Main 2022 (Online) 26th July Evening Shift
+4
-1

If $$z=x+i y$$ satisfies $$|z|-2=0$$ and $$|z-i|-|z+5 i|=0$$, then :

A
$$x+2 y-4=0$$
B
$$x^{2}+y-4=0$$
C
$$x+2 y+4=0$$
D
$$x^{2}-y+3=0$$
4
JEE Main 2022 (Online) 26th July Morning Shift
+4
-1

Let O be the origin and A be the point $${z_1} = 1 + 2i$$. If B is the point $${z_2}$$, $${\mathop{\rm Re}\nolimits} ({z_2}) < 0$$, such that OAB is a right angled isosceles triangle with OB as hypotenuse, then which of the following is NOT true?

A
$$\arg {z_2} = \pi - {\tan ^{ - 1}}3$$
B
$$\arg ({z_1} - 2{z_2}) = - {\tan ^{ - 1}}{4 \over 3}$$
C
$$|{z_2}| = \sqrt {10}$$
D
$$|2{z_1} - {z_2}| = 5$$
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