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Let \( \overline{z} \) denote the complex conjugate of a complex number \( z \) and let \( i=\sqrt{-1} \). In the set of complex numbers, the number of distinct roots of the equation \( \overline{z}-z^{2}=i(\overline{z}+z^{2}) \) is _________.

Let \( z \) be a complex number with non-zero imaginary part. If \( \dfrac{2+3z+4z^{2}}{\,2-3z+4z^{2}\,} \) is a real number, then the value of \( |z|^{2} \) is ____________.

For any complex number \( w = c + i d \), let \( \arg(w) \in (-\pi, \pi) \), where \( i = \sqrt{-1} \). Let \( \alpha \) and \( \beta \) be real numbers such that for all complex numbers \( z = x + i y \) satisfying \( \arg\!\left(\frac{z + \alpha}{z + \beta}\right) = \frac{\pi}{4} \), the ordered pair \( (x,y) \) lies on the circle \( x^2 + y^2 + 5x - 3y + 4 = 0 \). Then which of the following statements is (are) TRUE?

Let \( \overline{z} \) denote the complex conjugate of a complex number \( z \). If \( z \) is a non-zero complex number for which both real and imaginary parts of \( \left(\overline{z}^{\,2}+\dfrac{1}{z^{2}}\right) \) are integers, then which of the following is/are possible value(s) of \( |z| \)?

Let \( A = \left\{ \frac{1967 + 1686 i \sin\theta}{7 - 3 i \cos\theta} : \theta \in \mathbb{R} \right\} \). If \( A \) contains exactly one positive integer \( n \), then the value of \( n \) is _____.

Let \( z \) be a complex number satisfying \( |z|^3+2z^2+4\overline{z}-8=0 \), where \( \overline{z} \) denotes the complex conjugate of \( z \). Let the imaginary part of \( z \) be nonzero.
Match each entry in List-I to the correct entries in List-II.

The correct option is:

Let \( \theta_1,\theta_2,\ldots,\theta_{10} \) be positive valued angles (in radian) such that \( \theta_1+\theta_2+\ldots+\theta_{10}=2\pi \). Define the complex numbers \( z_1=e^{i\theta_1},\, z_k=z_{k-1}e^{i\theta_k} \) for \( k=2,3,\ldots,10 \), where \( i=\sqrt{-1} \). Consider the statements P and Q given below:
P: \( |z_2-z_1|+|z_3-z_2|+\ldots+|z_{10}-z_9|+|z_1-z_{10}| \le 2\pi \)
Q: \( |z_2^2-z_1^2|+|z_3^2-z_2^2|+\ldots+|z_{10}^2-z_9^2|+|z_1^2-z_{10}^2| \le 4\pi \)
Then,

For a non-zero complex number \( z \), let \( \arg(z) \) denote the principal argument of \( z \), with \( -\pi < \arg(z) \leq \pi \).
Let \( \omega \) be the cube root of unity for which \( 0<\arg(\omega)<\pi \). Let \( \alpha = \arg\left( \displaystyle\sum_{n=1}^{2025} (-\omega)^n \right) \). Then the value of \( \dfrac{3\alpha}{\pi} \) is __________________.

Let \( \mathbb{R} \) denote the set of all real numbers. Let \( z_{1}=1+2i \) and \( z_{2}=3i \) be two complex numbers, where \( i=\sqrt{-1} \). Let \( S=\{(x,y)\in \mathbb{R}\times \mathbb{R}: |x+iy-z_{1}|=2\,|x+iy-z_{2}|\} \).
Then which of the following statements is(are) TRUE?

Let \( A_{1}, A_{2}, A_{3}, \ldots, A_{8} \) be the vertices of a regular octagon that lie on a circle of radius \( 2 \). Let \( P \) be a point on the circle and let \( PA_{i} \) denote the distance between the points \( P \) and \( A_{i} \) for \( i=1,2,\ldots,8 \). If \( P \) varies over the circle, then the maximum value of the product \( PA_{1}\cdot PA_{2}\cdots PA_{8} \), is:

Let \( S = \{ a + b\sqrt{2} : a, b \in \mathbb{Z} \} \), \( T_1 = \{ (-1 + \sqrt{2})^n : n \in \mathbb{N} \} \), and \( T_2 = \{ (1 + \sqrt{2})^n : n \in \mathbb{N} \} \). Then which of the following statements is (are) TRUE?