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1

### JEE Main 2021 (Online) 27th August Morning Shift

If $$S = \left\{ {z \in C:{{z - i} \over {z + 2i}} \in R} \right\}$$, then :
A
S contains exactly two elements
B
S contains only one element
C
S is a circle in the complex plane
D
S is a straight line in the complex plane

## Explanation

Given $${{z - i} \over {z + 2i}} \in R$$

Then $$\arg \left( {{{z - i} \over {z + 2i}}} \right)$$ is 0 or $$\pi$$

$$\Rightarrow$$ S is straight line in complex
2

### JEE Main 2021 (Online) 26th August Evening Shift

If $${\left( {\sqrt 3 + i} \right)^{100}} = {2^{99}}(p + iq)$$, then p and q are roots of the equation :
A
$${x^2} - \left( {\sqrt 3 - 1} \right)x - \sqrt 3 = 0$$
B
$${x^2} + \left( {\sqrt 3 + 1} \right)x + \sqrt 3 = 0$$
C
$${x^2} + \left( {\sqrt 3 - 1} \right)x - \sqrt 3 = 0$$
D
$${x^2} - \left( {\sqrt 3 + 1} \right)x + \sqrt 3 = 0$$

## Explanation

$${\left( {2{e^{i\pi /6}}} \right)^{100}} = {2^{99}}(p + iq)$$

$${2^{100}}\left( {\cos {{50\pi } \over 3} + i\sin {{50\pi } \over 3}} \right) = {2^{99}}(p + iq)$$

$$p + iq = 2\left( {\cos {{2\pi } \over 3} + i\sin {{2\pi } \over 3}} \right)$$

p = $$-$$1, q = $$\sqrt 3$$

$${x^2} - (\sqrt 3 - 1)x - \sqrt 3 = 0$$
3

### JEE Main 2021 (Online) 26th August Morning Shift

The equation $$\arg \left( {{{z - 1} \over {z + 1}}} \right) = {\pi \over 4}$$ represents a circle with :
A
centre at (0, $$-$$1) and radius $$\sqrt 2$$
B
centre at (0, 1) and radius $$\sqrt 2$$
C
centre (0, 0) and radius $$\sqrt 2$$
D
centre at (0, 1) and radius 2

## Explanation

In $$\Delta$$OAC

$$\sin \left( {{\pi \over 4}} \right) = {1 \over {AC}}$$

$$\Rightarrow AC = \sqrt 2$$

Also, $$\tan {\pi \over 4} = {{OA} \over {OC}} = {1 \over {OC}}$$

$$\Rightarrow$$ OC = 1

$$\therefore$$ centre (0, 1); Radius = $$\sqrt 2$$
4

### JEE Main 2021 (Online) 27th July Morning Shift

Let C be the set of all complex numbers. Let

$${S_1} = \{ z \in C||z - 3 - 2i{|^2} = 8\}$$

$${S_2} = \{ z \in C|{\mathop{\rm Re}\nolimits} (z) \ge 5\}$$ and

$${S_3} = \{ z \in C||z - \overline z | \ge 8\}$$.

Then the number of elements in $${S_1} \cap {S_2} \cap {S_3}$$ is equal to :
A
1
B
0
C
2
D
Infinite

## Explanation

$${S_1}:|z - 3 - 2i{|^2} = 8$$

$$|z - 3 - 2i| = 2\sqrt 2$$

$${(x - 3)^2} + {(y - 2)^2} = {(2\sqrt 2 )^2}$$

$${S_2}:x \ge 5$$

$${S_3}:|z - \overline z | \ge 8$$

$$|2iy| \ge 8$$

$$2|y| \ge 8$$

$$\therefore$$ $$y \ge 4$$, $$y \le - 4$$

$$n\left( {{S_1} \cap {S_2} \cap {S_3}} \right) = 1$$

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