1
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
$${2 \over 3}$$
2
JEE Main 2023 (Online) 8th April Evening Shift
+4
-1

Let $$A=\left\{\theta \in(0,2 \pi): \frac{1+2 i \sin \theta}{1-i \sin \theta}\right.$$ is purely imaginary $$\}$$. Then the sum of the elements in $$\mathrm{A}$$ is :

A
$$3 \pi$$
B
$$\pi$$
C
$$2 \pi$$
D
$$4 \pi$$
3
JEE Main 2023 (Online) 8th April Morning Shift
+4
-1

If for $$z=\alpha+i \beta,|z+2|=z+4(1+i)$$, then $$\alpha+\beta$$ and $$\alpha \beta$$ are the roots of the equation :

A
$$x^{2}+2 x-3=0$$
B
$$x^{2}+3 x-4=0$$
C
$$x^{2}+x-12=0$$
D
$$x^{2}+7 x+12=0$$
4
JEE Main 2023 (Online) 6th April Evening Shift
+4
-1

Let $$a \neq b$$ be two non-zero real numbers. Then the number of elements in the set $$X=\left\{z \in \mathbb{C}: \operatorname{Re}\left(a z^{2}+b z\right)=a\right.$$ and $$\left.\operatorname{Re}\left(b z^{2}+a z\right)=b\right\}$$ is equal to :

A
0
B
2
C
1
D
Infinite
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