1
JEE Main 2023 (Online) 24th January Evening Shift
+4
-1

The value of $${\left( {{{1 + \sin {{2\pi } \over 9} + i\cos {{2\pi } \over 9}} \over {1 + \sin {{2\pi } \over 9} - i\cos {{2\pi } \over 9}}}} \right)^3}$$ is

A
$$- {1 \over 2}\left( {1 - i\sqrt 3 } \right)$$
B
$$- {1 \over 2}\left( {\sqrt 3 - i} \right)$$
C
$${1 \over 2}\left( {1 - i\sqrt 3 } \right)$$
D
$${1 \over 2}\left( {\sqrt 3 + i} \right)$$
2
JEE Main 2023 (Online) 24th January Morning Shift
+4
-1

Let $$\mathrm{p,q\in\mathbb{R}}$$ and $${\left( {1 - \sqrt 3 i} \right)^{200}} = {2^{199}}(p + iq),i = \sqrt { - 1}$$ then $$\mathrm{p+q+q^2}$$ and $$\mathrm{p-q+q^2}$$ are roots of the equation.

A
$${x^2} + 4x - 1 = 0$$
B
$${x^2} - 4x + 1 = 0$$
C
$${x^2} + 4x + 1 = 0$$
D
$${x^2} - 4x - 1 = 0$$
3
JEE Main 2022 (Online) 29th July Evening Shift
+4
-1

If $$z \neq 0$$ be a complex number such that $$\left|z-\frac{1}{z}\right|=2$$, then the maximum value of $$|z|$$ is :

A
$$\sqrt{2}$$
B
1
C
$$\sqrt{2}-1$$
D
$$\sqrt{2}+1$$
4
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
$$\left(-\frac{1}{\sqrt{2}}, \frac{1}{2 \sqrt{2}}\right]$$
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