1
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
2
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
$$\pi-\tan ^{-1} \frac{33}{5}$$
3
JEE Main 2023 (Online) 10th April Evening Shift
+4
-1

Let $$S = \left\{ {z = x + iy:{{2z - 3i} \over {4z + 2i}}\,\mathrm{is\,a\,real\,number}} \right\}$$. Then which of the following is NOT correct?

A
$$y + {x^2} + {y^2} \ne - {1 \over 4}$$
B
$$(x,y) = \left( {0, - {1 \over 2}} \right)$$
C
$$x = 0$$
D
$$y \in \left( { - \infty , - {1 \over 2}} \right) \cup \left( { - {1 \over 2},\infty } \right)$$
4
JEE Main 2023 (Online) 10th April Morning Shift
+4
-1

Let the complex number $$z = x + iy$$ be such that $${{2z - 3i} \over {2z + i}}$$ is purely imaginary. If $${x} + {y^2} = 0$$, then $${y^4} + {y^2} - y$$ is equal to :

A
$${4 \over 3}$$
B
$${3 \over 2}$$
C
$${3 \over 4}$$
D
$${2 \over 3}$$
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