1
JEE Main 2023 (Online) 25th January Morning Shift
MCQ (Single Correct Answer)
+4
-1

Let $$\mathrm{z_1=2+3i}$$ and $$\mathrm{z_2=3+4i}$$. The set $$\mathrm{S = \left\{ {z \in \mathbb{C}:{{\left| {z - {z_1}} \right|}^2} - {{\left| {z - {z_2}} \right|}^2} = {{\left| {{z_1} - {z_2}} \right|}^2}} \right\}}$$ represents a

A
hyperbola with the length of the transverse axis 7
B
hyperbola with eccentricity 2
C
straight line with the sum of its intercepts on the coordinate axes equals $$-18$$
D
straight line with the sum of its intercepts on the coordinate axes equals $$14$$
2
JEE Main 2023 (Online) 24th January Evening Shift
MCQ (Single Correct Answer)
+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
$${1 \over 2}\left( {\sqrt 3 + i} \right)$$
3
JEE Main 2023 (Online) 24th January Morning Shift
MCQ (Single Correct Answer)
+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
$${x^2} - 4x - 1 = 0$$
4
JEE Main 2022 (Online) 29th July Evening Shift
MCQ (Single Correct Answer)
+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
$$\sqrt{2}+1$$
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