1
JEE Main 2024 (Online) 29th January Evening Shift
+4
-1

Let $$\mathrm{r}$$ and $$\theta$$ respectively be the modulus and amplitude of the complex number $$z=2-i\left(2 \tan \frac{5 \pi}{8}\right)$$, then $$(\mathrm{r}, \theta)$$ is equal to

A
$$\left(2 \sec \frac{11 \pi}{8}, \frac{11 \pi}{8}\right)$$
B
$$\left(2 \sec \frac{3 \pi}{8}, \frac{3 \pi}{8}\right)$$
C
$$\left(2 \sec \frac{5 \pi}{8}, \frac{3 \pi}{8}\right)$$
D
$$\left(2 \sec \frac{3 \pi}{8}, \frac{5 \pi}{8}\right)$$
2
JEE Main 2024 (Online) 29th January Morning Shift
+4
-1

If $$z=\frac{1}{2}-2 i$$ is such that $$|z+1|=\alpha z+\beta(1+i), i=\sqrt{-1}$$ and $$\alpha, \beta \in \mathbb{R}$$, then $$\alpha+\beta$$ is equal to

A
2
B
$$-$$4
C
3
D
$$-$$1
3
JEE Main 2024 (Online) 27th January Morning Shift
+4
-1
If $S=\{z \in C:|z-i|=|z+i|=|z-1|\}$, then, $n(S)$ is :
A
1
B
2
C
3
D
0
4
JEE Main 2023 (Online) 15th April Morning Shift
+4
-1
If the set $\left\{\operatorname{Re}\left(\frac{z-\bar{z}+z \bar{z}}{2-3 z+5 \bar{z}}\right): z \in \mathbb{C}, \operatorname{Re}(z)=3\right\}$ is equal to

the interval $(\alpha, \beta]$, then $24(\beta-\alpha)$ is equal to :
A
36
B
27
C
42
D
30
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