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For all \( z \in \mathbb{C} \) on the curve \( C_{1}: |z| = 4 \), let the locus of the point \( z + \frac{1}{z} \) be the curve \( C_{2} \). then:

Let \( S=\left\{ z\in\mathbb{C}-\{i,2i\}:\dfrac{z^{2}+8iz-15}{z^{2}-3iz-2}\in\mathbb{R}\right\}. \) If \( \alpha-\dfrac{13}{11}i\in S,\; \alpha\in\mathbb{R}-\{0\} \), then \( 242\alpha^{2} \) is equal to ______.

The complex number \( z=\frac{i-1}{\cos \frac{\pi}{3}+i\sin \frac{\pi}{3}} \) is equal to:

For two non-zero complex numbers \( z_1 \) and \( z_2 \), if \( \operatorname{Re}(z_1 z_2)=0 \) and \( \operatorname{Re}(z_1+z_2)=0 \), then which of the following are possible?
A. \( \operatorname{Im}(z_1) > 0 \) and \( \operatorname{Im}(z_2) > 0 \)
B. \( \operatorname{Im}(z_1) < 0 \) and \( \operatorname{Im}(z_2) > 0 \)
C. \( \operatorname{Im}(z_1) > 0 \) and \( \operatorname{Im}(z_2) < 0 \)
D. \( \operatorname{Im}(z_1) < 0 \) and \( \operatorname{Im}(z_2) < 0 \)
Choose the correct answer from the options given below:

Let \( A = \left\{ \ \theta \in (0, 2\pi) : \dfrac{1 + 2i \sin \theta}{1 - i \sin \theta} \ \text{is purely imaginary} \ \right\}. \) Then the sum of the elements in \( A \) is:

Let $\alpha = 8 - 14i$, $A = \left\{ z \in \mathbb{C} : \frac{\alpha z - \bar{\alpha}\bar{z}}{z^2 - (\bar{z})^2 - 112i} = 1 \right\}$ and $B = \{z \in \mathbb{C} : |z + 3i| = 4\}$. Then $\sum_{z \in A \cap B} (\text{Re } z - \text{Im } z)$ is equal to ________.

For \( a \in \mathbb{C} \), let \( A=\{ z \in \mathbb{C} : \operatorname{Re}(a+\bar z)>\operatorname{Im}(\bar{a}+z) \} \) and \( B=\{ z \in \mathbb{C} : \operatorname{Re}(a+\bar z)<\operatorname{Im}(\bar{a}+z) \} \). Then among the two statements:
(\( \mathrm{S1} \)) : If \( \operatorname{Re}(a), \operatorname{Im}(a) > 0 \), then the set \( A \) contains all the real numbers
(\( \mathrm{S2} \)) : If \( \operatorname{Re}(a), \operatorname{Im}(a) < 0 \), then the set \( B \) contains all the real numbers

The value of \( \left( \frac{1 + \sin\frac{2\pi}{9} + i\cos\frac{2\pi}{9}}{1 + \sin\frac{2\pi}{9} - i\cos\frac{2\pi}{9}} \right)^3 \) is:

Let \(S=\left\{z\in\mathbb{C}:\ \overline z=i\big(z^2+\operatorname{Re}(\overline z)\big)\right\}\). Then \(\displaystyle\sum_{z\in S}\lvert z\rvert^2\) is equal to :

Let \( p, q \in \) ℝ and \( (1-\sqrt{3}i)^{200} = 2^{199}(p+iq), \; i=\sqrt{-1} \). Then \( p+q+q^{2} \) and \( p-q+q^{2} \) are roots of the equation.

If the center and radius of the circle \( \left|\dfrac{z-2}{z-3}\right|=2 \) are respectively \( (\alpha,\beta) \) and \( \gamma \), then \( 3(\alpha+\beta+\gamma) \) is equal to:

If for \( z=\alpha+i\beta \), \( |z+2|=z+4(1+i) \), then \( \alpha+\beta \) and \( \alpha\beta \) are the roots of the equation.

Let \( z = 1 + i \) and \( z_{1} = \dfrac{1 + i\overline{z}}{\overline{z}(1 - z) + \dfrac{1}{\overline{z}}} \). Then \( \dfrac{12}{\pi} \arg(z_{1}) \) is equal to _______.

If the set \( \left\{\operatorname{Re}\!\left(\dfrac{z-\bar{z}+z\bar{z}}{2-3z+5\bar{z}}\right): z\in \mathbb{C}, \operatorname{Re}(z)=3 \right\} \) is equal to the interval \( (\alpha,\beta] \), then \( 24(\beta-\alpha) \) is equal to:

Let \( w_{1} \) be the point obtained by the rotation of \( z_{1} = 5 + 4i \) about the origin through a right angle in the anticlockwise direction, and \( w_{2} \) be the point obtained by the rotation of \( z_{2} = 3 + 5i \) about the origin through a right angle in the clockwise direction. Then the principal argument of \( w_{1} - w_{2} \) is equal to:

Let \( z_{1} = 2 + 3i \) and \( z_{2} = 3 + 4i \). The set \( S = \left\{ z \in \mathbb{C} : |z - z_{1}|^{2} - |z - z_{2}|^{2} = |z_{1} - z_{2}|^{2} \right\} \) represents a:

Let \( z \) be a complex number such that \( \left|\frac{z-2i}{z+i}\right| = 2, \, z \neq -i \). Then \( z \) lies on the circle of radius \( 2 \) and centre.

Let \( w= z\bar{z}+k_{1}z+k_{2}i{z}+\lambda(1+i),\ k_{1},k_{2}\in \mathbb{R} \). Let \( \operatorname{Re}(w)=0 \) be the circle \( C \) of radius \( 1 \) in the first quadrant touching the line \( y=1 \) and the y-axis. If the curve \( \operatorname{Im}(w)=0 \) intersects \( C \) at \( A \) and \( B \), then \( 30(AB)^{2} \) is equal to _____.

Let \( a, b \) be two real numbers such that \( ab < 0 \). If the complex number \( \frac{1 + ai}{\, b + i \,} \) is of unit modulus and \( a + ib \) lies on the circle \( |z - 1| = |2z| \), then a possible value of \( \frac{1 + [a]}{4b} \), where \( [t] \) is greatest integer function, is:

Let the complex number \( z=x+iy \) be such that \( \dfrac{2z-3i}{2z+i} \) is purely imaginary. If \( x+y^{2}=0 \), then \( y^{4}+y^{2}-y \) is equal to:

Let \( S = \left\{ z = x + iy : \dfrac{2z - 3i}{4z + 2i} \text{ is a real numbers} \right\} \). Then which of the following is NOT correct?

For \( \alpha, \beta, z \in \mathbb{C} \) and \( \lambda>1 \), if \( \sqrt{\lambda-1} \) is the radius of the circle \( |z-\alpha|^{2}+|z-\beta|^{2}=2\lambda \), then \( |\alpha-\beta| \) is equal to ________ .

The number of real roots of the equation \( x|x| - 5|x + 2| + 6 = 0 \), is:

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JEE Main 2023 24th Jan Morning Shift

Let \( p, q \in \) ℝ and \( (1-\sqrt{3}i)^{200} = 2^{199}(p+iq), \; i=\sqrt{-1} \). Then \( p+q+q^{2} \) and \( p-q+q^{2} \) are roots of the equation.