Complex Numbers

For $n\in\mathbf N$, let $S_n=\left\{z\in\mathbf C:|z-3+2i|=\dfrac n4\right\}$ and $T_n=\left\{z\in\mathbf C:|z-2+3i|=\dfrac1n\right\}$. Then the number of elements in the set $\{n\in\mathbf N:S_n\cap T_n=\phi\}$ is :

Let $A=\begin{pmatrix}2&-1&-1\\1&0&-1\\1&-1&0\end{pmatrix}$ and $B=A-I$. If $\omega=\dfrac{\sqrt3i-1}2$, then the number of elements in the set $\left\{n\in\{1,2,\dots,100\}:A^n+(\omega B)^n=A+B\right\}$ is equal to ______.

For $z\in\mathbb C$ if the minimum value of $\left(\left|z-3\sqrt2\right|+\left|z-p\sqrt2i\right|\right)$ is $5\sqrt2$, then a value of $p$ 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 ?

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

Let the minimum value $v_0$ of $v=|z|^2+|z-3|^2+|z-6i|^2$, $z\in\mathbf C$ is attained at $z=z_0$. Then $\left|2z_0^2-\bar z_0^3+3\right|^2+v_0^2$ is equal to

Let $S=\left\{z\in\mathbf C:z^2+\bar z=0\right\}$. Then $\displaystyle\sum_{z\in S}\left(\operatorname{Re}(z)+\operatorname{Im}(z)\right)$ 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 $Z_{\alpha\beta}=\sin2\alpha+i\cos2\beta$, $(\alpha,\beta)\in S$. Then $\displaystyle\sum_{(\alpha,\beta)\in S}\left(iZ_{\alpha\beta}+\frac1{i\bar Z_{\alpha\beta}}\right)$ is equal to :

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 $z=2+3i$, then $z^5+(\bar z)^5$ is equal to :

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

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

Let $z_1=1+2i$, $z_2=2+i$, $\dfrac1{z_1}+\dfrac1{z_2}=\dfrac6w$, and $z=\dfrac{iw}{2-\bar w}$.

Consider the statements

(S$_1$) $|z|=\dfrac5{\sqrt{17}}$

(S$_2$) $\arg(z)+\arg(w)=\tan^{-1}\left(\dfrac53\right)$.

Then

Let $\theta_1, \theta_2, \ldots, \theta_{10}$ be positive valued angles (in radian) such that $\theta_1 + \theta_2 + \cdots + \theta_{10} = 2\pi$. 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:
$$P: |z_2 - z_1| + |z_3 - z_2| + \cdots + |z_{10} - z_9| + |z_1 - z_{10}| \le 2\pi$$
$$Q: |z_2^2 - z_1^2| + |z_3^2 - z_2^2| + \cdots + |z_{10}^2 - z_9^2| + |z_1^2 - z_{10}^2| \le 4\pi$$
Then,

For any complex number $w = c + id$, let $\arg(w) \in (-\pi, \pi]$, where $i = \sqrt{-1}$. Let $\alpha$ and $\beta$ be real numbers such that for all complex numbers $z = x + iy$ satisfying $\arg\left(\dfrac{z + \alpha}{z + \beta}\right) = \dfrac\pi4$, 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?

If the least and the largest real values of α, for which the equation $z+\alpha|z-1|+2i=0$ ($z\in\mathbb C$ and $i=\sqrt{-1}$) has a solution, are p and q respectively; then $4(p^2+q^2)$ is equal to _______.

Suppose $a, b$ denote the distinct real roots of the quadratic polynomial $x^2 + 20x - 2020$ and suppose $c, d$ denote the distinct complex roots of the quadratic polynomial $x^2 - 20x + 2020$. Then the value of

$$ac(a-c) + ad(a-d) + bc(b-c) + bd(b-d)$$

is

Let $S$ be the set of all complex numbers $z$ satisfying $|z^2 + z + 1| = 1$. Then which of the following statements is/are TRUE?

For a complex number $z$, let $\operatorname{Re}(z)$ denote the real part of $z$. Let $S$ be the set of all complex numbers $z$ satisfying $z^4 - |z|^4 = 4iz^2$, where $i = \sqrt{-1}$. Then the minimum possible value of $|z_1 - z_2|^2$, where $z_1, z_2 \in S$ with $\operatorname{Re}(z_1) > 0$ and $\operatorname{Re}(z_2) < 0$, is ______