Complex Numbers

Match each entry in List-I to the correct entry in List-II and choose the correct option.

Let $\mathbb R$ denote the set of all real numbers and let $i=\sqrt{-1}$. Consider the matrices
$$S=\begin{bmatrix}0&-1\\1&0\end{bmatrix}\quad\text{and}\quad T=\begin{bmatrix}1&1\\0&1\end{bmatrix}.$$
Let $a,b,c,d$ be real numbers such that
$$ST=\begin{bmatrix}a&b\\c&d\end{bmatrix}.$$
Let
$$H=\{x+iy:\ x,y\in\mathbb R\ \text{and}\ y>0\}.$$
Then which of the following statements is (are) TRUE ?

Let $x$ and $y$ be real numbers such that $50\left(\dfrac{2x}{1+3i}-\dfrac{y}{1-2i}\right)=31+17i$, $i=\sqrt{-1}$. Then the value of $10(x-3y)$ is :

Let the circles $C_1:|z|=r$ and $C_2:|z-3-4i|=5$, $z\in\mathbb C$, be such that $C_2$ lies within $C_1$. If $z_1$ moves on $C_1$, $z_2$ moves on $C_2$ and $\min|z_1-z_2|=2$, then $\max|z_1-z_2|$ is equal to :

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 $a,b\in\mathbb C$. Let $\alpha,\beta$ be the roots of the equation $x^2+ax+b=0$. If $\beta-\alpha=\sqrt{11}$ and $\beta^2-\alpha^2=3i\sqrt{11}$, then $\left(\beta^3-\alpha^3\right)^2$ 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 :

Let the set of all values of $k\in\mathbb R$ such that the equation $z\left(\bar z+2+i\right)+k(2+3i)=0$, $z\in\mathbb C$, has at least one solution, be the interval $[\alpha,\beta]$. Then $9(\alpha+\beta)$ is equal to:

Let $S=\left\{z\in\mathbb C:z^2+\sqrt6\,iz-3=0\right\}$. Then $\displaystyle\sum_{z\in S}z^8$ is equal to :

The number of values of $z\in\mathbb C$, satisfying the equations $|z-(4+8i)|=\sqrt{10}$ and $|z-(3+5i)|+|z-(5+11i)|=4\sqrt5$, 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?

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

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

Let $z$ be a complex number satisfying $|z|^3 + 2z^2 + 4\bar z - 8 = 0$, 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 $A_1, A_2, A_3, \ldots, A_8$ 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 $z$ 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 _____.