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The area of the polygon, whose vertices are the non-real roots of the equation \( \overline{z} = i z^2 \) is:

Let \( S=\{z\in \mathbb{C}: z^{2}+\overline{z}=0\} \). Then \( \sum\limits_{z\in S}(\mathrm{Re}(z)+\mathrm{Im}(z)) \) is equal to _______.

Let \( A = \left\{ z \in \mathbb{C} : \left| \frac{z + 1}{z - 1} \right| < 1 \right\} \) and \( B = \left\{ z \in \mathbb{C} : \arg \!\left( \frac{z - 1}{z + 1} \right) = \frac{2\pi}{3} \right\} \). Then \( A \cap B \) is:

Let \( \arg(z) \) represent the principal argument of the complex number \( z \). Then, \( |z| = 3 \) and \( \arg(z-1) - \arg(z+1) = \pi/4 \) intersect.

For \( n \in N \), let \( S_{n}=\{ z \in C : |z-3+2i|=\dfrac{n}{4} \} \) & \( T_{n}=\{ z \in C : |z-2+3i|=\dfrac{1}{n} \} \). Then the number of elements in the set \( \{ n \in N : S_{n} \cap T_{n}=\phi \} \) is :

If \( z^{2}+z+1=0, \; z\in \mathbb{C} \), then \( \left| \displaystyle \sum_{n=1}^{15}\left(z^{n}+(-1)^{n}\frac{1}{z^{n}}\right)^{2}\right|^{} \) is equal to ______.

Let \( S=\left\{x \in [-6,3]-\{-2,2\}:\; \dfrac{|x+3|-1}{|x|-2}\ge 0\right\} \) and \( T=\left\{x\in \mathbb{Z}: x^2-7|x|+9\le 0\right\}. \) Then the number of elements in \( S\cap T \) is:

Let a circle \( C \) in complex plane pass through the point \( z_1 = 3 + 4i \), \( z_2 = 4 + 3i \) and \( z_3 = 5i \). If \( z (\neq z_1) \) is a point on \( C \) such that the line through \( z \) and \( z_1 \) is perpendicular to the line through \( z_2 \) and \( z_3 \), then \( \arg(z) \) is equal to:

If \( z=2+3i \), then \( z^{5}+(\overline{z})^{5} \) is equal to:

Let for some real numbers \( \alpha \) and \( \beta \) \( a = \alpha - i\beta \). If the system of equations \( 4i x + (1+i)y = 0 \) and \( 8\left( \cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3} \right)x + \bar{a}y = 0 \) has more than one solution, then \( \frac{\alpha}{\beta} \) is equal to

Let the minimum value \( v_0 \) of \( v = |z|^{2} + |z-3|^{2} + |z-6i|^{2}, \, z \in \mathbb{C} \) is attained at \( z=z_0 \). Then \( \left|2z_0{^2}-\overline{z_0{^3}}+3\right|^{2}+v_0{^2} \) is equal to:

Let \( S \) be the set of all \( (\alpha,\beta),\, \pi<\alpha,\beta<2\pi \), for which the complex number \( \dfrac{1-i\sin\alpha}{1+2i\sin\alpha} \) is purely imaginary and \( \dfrac{1+i\cos\beta}{1-2i\cos\beta} \) is purely real. Let \( Z_{\alpha\beta}=\sin 2\alpha + i\cos 2\beta,\, (\alpha,\beta)\in S \). Then \(\displaystyle \sum_{(\alpha,\beta)\in S}\left(iZ_{\alpha\beta}+\frac{1}{i{\bar Z_{\alpha\beta}}}\right) \) is equal to:

Let \( \alpha \) and \( \beta \) be the roots of the equation \( x^2 + (2i - 1) = 0 \) Then, the value of \( \lvert \alpha^8 + \beta^8 \rvert \) is equal to:

Let \( S=\{z\in \mathbb{C}: |z-3|\le 1 \text{ and } \bar{z}(4+3i)+z(4-3i)\le 24\} \). If \( \alpha+i\beta \) is the point in \( S \) which is closest to \( 4i \), then \( 25(\alpha+\beta) \) is equal to _______.

If \( \alpha,\beta,\gamma,\delta \) are the roots of the equation \( x^{4}+x^{3}+x^{2}+x+1=0 \), then \( \alpha^{2021}+\beta^{2021}+\gamma^{2021}+\delta^{2021} \) is equal to :

Let \( S = \{ z = x + iy : |z - 1 + i| \ge |z|, \; |z| < 2, \; |z + i| = |z - 1| \}\). Then the set of all values of \( x \), for which \( w = 2x + iy \in S \) for some \( y \in \mathbb{R} \), is

Let \( O \) be the origin and \( A \) be the point \( z_1 = 1 + 2i \). If \( B \) is the point \( z_2 \), \( \operatorname{Re}(z_2) < 0 \), such that \( OAB \) is a right angled isosceles triangle with \( OB \) as hypotenuse, then which of the following is NOT true?

Let \( \alpha \) be a root of the equation \( 1 + x^2 + x^4 = 0 \). Then the value of \( \alpha^{1011} + \alpha^{2022} - \alpha^{3033} \) is equal to:

If \( z=x+iy \) satisfies \( |z|-2=0 \) and \( |z-i|-|z+5i|=0 \), then

Let \( z_{1} \) and \( z_{2} \) be two complex numbers such that \( \bar{z}_{1} = i \bar{z}_{2} \) and \( \arg\!\left(\frac{z_{1}}{z_{2}}\right) = \pi \). Then

The number of points of intersection of \( \lvert z - (4 + 3i) \rvert = 2 \) and \( \lvert z \rvert + \lvert z - 4 \rvert = 6 \), \( z \in \mathbb{C} \), is

Let \( S_1 = \{\, z_1 \in \mathbb{C} : |z_1 - 3| = \dfrac{1}{2} \,\} \) and \( S_2 = \{\, z_2 \in \mathbb{C} : |z_2 - |z_2 + 1|| = |z_2 + |z_2 - 1|| \,\} \). Then, for \( z_1 \in S_1 \) and \( z_2 \in S_2 \), the least value of \( |z_2 - z_1| \) is:

Let \( z=a+ib ,\; b\neq 0 \) be complex numbers satisfying \( z^{2}=\overline{z}\cdot 2^{\,1-|z|} \). Then the least value of \( n\in\mathbb{N} \), such that \( z^{n}=(z+1)^{n} \), is equal to _______________.

Let \( A = \{ z \in C : 1 \le |z - (1 + i)| \le 2 \} \) and \( B = \{ z \in A : |z - (1 - i)| = 1 \} \). Then, \( B \) :

For \( z \in C \) if the minimum value of \( \left(|z-3\sqrt{2}|+|z-p\sqrt{2}i|\right) \) is \( 5\sqrt{2} \), then a value of \( p \) is _________

The number of elements in the set \( \{ z = a + ib \in \mathbb{C} : a, b \in \mathbb{Z} \text{ and } 1 < | z - 3 + 2i | < 4 \} \) is ________.

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

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JEE Main 2022 25th June Evening Shift

Let \( z_{1} \) and \( z_{2} \) be two complex numbers such that \( \bar{z}_{1} = i \bar{z}_{2} \) and \( \arg\!\left(\frac{z_{1}}{z_{2}}\right) = \pi \). Then