1
JEE Main 2022 (Online) 29th July Evening Shift
+4
-1

Let $$\mathrm{S}=\{z=x+i y:|z-1+i| \geq|z|,|z|<2,|z+i|=|z-1|\}$$. Then the set of all values of $$x$$, for which $$w=2 x+i y \in \mathrm{S}$$ for some $$y \in \mathbb{R}$$, is :

A
$$\left(-\sqrt{2}, \frac{1}{2 \sqrt{2}}\right]$$
B
$$\left(-\frac{1}{\sqrt{2}}, \frac{1}{4}\right]$$
C
$$\left(-\sqrt{2}, \frac{1}{2}\right]$$
D
$$\left(-\frac{1}{\sqrt{2}}, \frac{1}{2 \sqrt{2}}\right]$$
2
JEE Main 2022 (Online) 29th July Morning Shift
+4
-1

If $$z=2+3 i$$, then $$z^{5}+(\bar{z})^{5}$$ is equal to :

A
244
B
224
C
245
D
265
3
JEE Main 2022 (Online) 28th July Morning Shift
+4
-1

Let $$S_{1}=\left\{z_{1} \in \mathbf{C}:\left|z_{1}-3\right|=\frac{1}{2}\right\}$$ and $$S_{2}=\left\{z_{2} \in \mathbf{C}:\left|z_{2}-\right| z_{2}+1||=\left|z_{2}+\right| z_{2}-1||\right\}$$. Then, for $$z_{1} \in S_{1}$$ and $$z_{2} \in S_{2}$$, the least value of $$\left|z_{2}-z_{1}\right|$$ is :

A
0
B
$$\frac{1}{2}$$
C
$$\frac{3}{2}$$
D
$$\frac{5}{2}$$
4
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
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