Modulus and argument

Let $z$ be a complex number such that $|z+2|=|z-2|$ and $\arg\left(\dfrac{z+3}{z-i}\right)=\dfrac\pi4$. Then $|z|^2$ is equal to:

Let $S=\{z\in\mathbb C:z^2+4z+16=0\}$. Then $\displaystyle\sum_{z\in S}\left|z+\sqrt3i\right|^2$ is equal to:

Let $z_1,z_2\in\mathbb C$ be the distinct solutions of the equation $z^2+4z-(1+12i)=0$. Then $|z_1|^2+|z_2|^2$ is equal to :

For a non-zero complex number $z$, let $\arg(z)$ denote the principal argument of $z$, with $-\pi < \arg(z) \le \pi$. Let $\omega$ be the cube root of unity for which $0 < \arg(\omega) < \pi$. Let

$$\alpha = \arg\left(\sum_{n=1}^{2025}(-\omega)^n\right).$$

Then the value of $\dfrac{3\alpha}{\pi}$ is _____.

Let $f(x) = x^4 + ax^3 + bx^2 + c$ be a polynomial with real coefficients such that $f(1) = -9$. Suppose that $i\sqrt{3}$ is a root of the equation $4x^3 + 3ax^2 + 2bx = 0$, where $i = \sqrt{-1}$. If $\alpha_1, \alpha_2, \alpha_3$, and $\alpha_4$ are all the roots of the equation $f(x) = 0$, then $|\alpha_1|^2 + |\alpha_2|^2 + |\alpha_3|^2 + |\alpha_4|^2$ is equal to _____.

Let $z$ be a complex number satisfying $|z|^3 + 2z^2 + 4\bar z - 8 = 0$, where $\bar z$ 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 $z$ be a complex number with non-zero imaginary part. If $$\frac{2+3z+4z^2}{2-3z+4z^2}$$ is a real number, then the value of $|z|^2$ is _____.

Let $\bar z$ 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 $$\bar z - z^2 = i(\bar z + z^2)$$ is _____.

Let $\bar z$ 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
$$(\bar z)^2 + \frac{1}{z^2}$$
are integers, then which of the following is/are possible value(s) of $|z|$?

Let a circle C in complex plane pass through the points $z_1=3+4i$, $z_2=4+3i$ and $z_3=5i$. If $z\,(\ne z_1)$ 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 :

Let $z_1$ and $z_2$ be two complex numbers such that $\bar z_1=i\bar z_2$ and $\arg\left(\dfrac{z_1}{\bar z_2}\right)=\pi$. Then

Let $A=\left\{z\in\mathbb C:\left|\dfrac{z+1}{z-1}\right|<1\right\}$ and $B=\left\{z\in\mathbb C:\arg\left(\dfrac{z-1}{z+1}\right)=\dfrac{2\pi}3\right\}$.

Then $A\cap B$ is :

Sum of squares of modulus of all the complex numbers $z$ satisfying $\bar z=iz^2+z^2-z$ is equal to ______.

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

Let $\arg(z)$ represent the principal argument of the complex number $z$. Then, $|z|=3$ and $\arg(z-1)-\arg(z+1)=\dfrac\pi4$ intersect

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 ?

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

Let $S_1=\left\{z_1\in\mathbf C:|z_1-3|=\dfrac12\right\}$ and $S_2=\left\{z_2\in\mathbf C:\big|z_2-|z_2+1|\big|=\big|z_2+|z_2-1|\big|\right\}$. Then, for $z_1\in S_1$ and $z_2\in S_2$, the least value of $|z_2-z_1|$ is :

Let $z=a+ib$, $b\ne0$ be complex numbers satisfying $z^2=\bar z\cdot2^{1-|z|}$. Then the least value of $n\in\mathbf N$, such that $z^n=(z+1)^n$, is equal to ______.

If z ≠ 0 be a complex number such that $\left|z-\dfrac1z\right|=2$, then the maximum value of |z| is :