1
JEE Main 2022 (Online) 28th July Evening Shift
Numerical
+4
-1

Let $$\mathrm{z}=a+i b, b \neq 0$$ be complex numbers satisfying $$z^{2}=\bar{z} \cdot 2^{1-z}$$. Then the least value of $$n \in N$$, such that $$z^{n}=(z+1)^{n}$$, is equal to __________.

2
JEE Main 2022 (Online) 27th July Morning Shift
Numerical
+4
-1

Let $$S=\left\{z \in \mathbb{C}: z^{2}+\bar{z}=0\right\}$$. Then $$\sum\limits_{z \in S}(\operatorname{Re}(z)+\operatorname{Im}(z))$$ is equal to ______________.

3
JEE Main 2022 (Online) 29th June Morning Shift
Numerical
+4
-1

Let $$S = \{ z \in C:|z - 2| \le 1,\,z(1 + i) + \overline z (1 - i) \le 2\}$$. Let $$|z - 4i|$$ attains minimum and maximum values, respectively, at z1 $$\in$$ S and z2 $$\in$$ S. If $$5(|{z_1}{|^2} + |{z_2}{|^2}) = \alpha + \beta \sqrt 5$$, where $$\alpha$$ and $$\beta$$ are integers, then the value of $$\alpha$$ + $$\beta$$ is equal to ___________.

4
JEE Main 2022 (Online) 28th June Evening Shift
Numerical
+4
-1

Sum of squares of modulus of all the complex numbers z satisfying $$\overline z = i{z^2} + {z^2} - z$$ is equal to ___________.

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