1
JEE Main 2023 (Online) 13th April Evening Shift
+4
-1 Let $$S=\left\{z \in \mathbb{C}: \bar{z}=i\left(z^{2}+\operatorname{Re}(\bar{z})\right)\right\}$$. Then $$\sum_\limits{z \in \mathrm{S}}|z|^{2}$$ is equal to :

A
$$\frac{7}{2}$$
B
4
C
3
D
$$\frac{5}{2}$$
2
JEE Main 2023 (Online) 12th April Morning Shift
+4
-1 Let $$\mathrm{C}$$ be the circle in the complex plane with centre $$\mathrm{z}_{0}=\frac{1}{2}(1+3 i)$$ and radius $$r=1$$. Let $$\mathrm{z}_{1}=1+\mathrm{i}$$ and the complex number $$z_{2}$$ be outside the circle $$C$$ such that $$\left|z_{1}-z_{0}\right|\left|z_{2}-z_{0}\right|=1$$. If $$z_{0}, z_{1}$$ and $$z_{2}$$ are collinear, then the smaller value of $$\left|z_{2}\right|^{2}$$ is equal to :

A
$$\frac{3}{2}$$
B
$$\frac{5}{2}$$
C
$$\frac{13}{2}$$
D
$$\frac{7}{2}$$
3
JEE Main 2023 (Online) 11th April Evening Shift
+4
-1 For $$a \in \mathbb{C}$$, let $$\mathrm{A}=\{z \in \mathbb{C}: \operatorname{Re}(a+\bar{z}) > \operatorname{Im}(\bar{a}+z)\}$$ and $$\mathrm{B}=\{z \in \mathbb{C}: \operatorname{Re}(a+\bar{z})<\operatorname{Im}(\bar{a}+z)\}$$. Then among the two statements :

(S1): If $$\operatorname{Re}(a), \operatorname{Im}(a) > 0$$, then the set A contains all the real numbers

(S2) : If $$\operatorname{Re}(a), \operatorname{Im}(a) < 0$$, then the set B contains all the real numbers,

A
both are false
B
only (S1) is true
C
only (S2) is true
D
both are true
4
JEE Main 2023 (Online) 11th April Morning Shift
+4
-1 Let $$w_{1}$$ be the point obtained by the rotation of $$z_{1}=5+4 i$$ about the origin through a right angle in the anticlockwise direction, and $$w_{2}$$ be the point obtained by the rotation of $$z_{2}=3+5 i$$ about the origin through a right angle in the clockwise direction. Then the principal argument of $$w_{1}-w_{2}$$ is equal to :

A
$$-\pi+\tan ^{-1} \frac{8}{9}$$
B
$$-\pi+\tan ^{-1} \frac{33}{5}$$
C
$$\pi-\tan ^{-1} \frac{8}{9}$$
D
$$\pi-\tan ^{-1} \frac{33}{5}$$
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