Complex Numbers and Quadratic Equations

What is the value of $i^{101}$?

Find the imaginary part of the product $(1 + i)(1 + i)$.

What is the real part of the complex number $z = \\frac{1}{1+i}$?

Simplify the expression $(1 - i)^2$.

If $x + iy = (1+i)^2$, find the values of $x$ and $y$.

Find the modulus of $\frac{1}{2+i} - \frac{1}{2-i}$.

If the argument of $z$ is $\theta$, then the argument of its conjugate $\bar{z}$ is:

The complex number $z$ that satisfies $Arg(z) = \frac{\pi}{4}$ and $|z-1|=|z-5|$ is:

If $z = x + iy$ and $w = \frac{1-iz}{z-i}$, then $|w|=1$ implies that:

If $|z| = 3$, the area of the triangle whose vertices are $z$, $\omega z$, and $\omega^2 z$ (where $\omega$ is a cube root of unity) is:

Find the multiplicative inverse of $4 - 3i$.

The product of $i$ and $-i$ is:

Calculate the sum $i^5 + i^6 + i^7 + i^8$.

The modulus of the zero complex number $0 + 0i$ is:

If $z = 3 - 4i$, then calculate $|z|^2$.

Evaluate the value of $\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}$.

Find the smallest positive integer $n$ for which $\left(\frac{1 + i}{1 - i}\right)^n = 1$.

If $z$ is a complex number such that $\text{Re}\left(\frac{z-1}{z+1}\right) = 0$, what is the value of $|z|$?

If $1, \omega, \omega^2$ are the cube roots of unity, calculate $(1-\omega+\omega^2)(1-\omega^2+\omega^4)(1-\omega^4+\omega^8)$ up to $2n$ factors.

Determine the number of solutions of the equation $|z|^2 + \text{Re}(z^2) = 2$ where $z = x+iy$.

Find the modulus of $z = i^9$.

If $z = 4$, then find its conjugate $\bar{z}$.

The modulus of $z = \frac{1}{1 - i}$ is:

What is the conjugate of $z = (1 + i)(1 + 2i)$?

If $z = \frac{i+1}{i-1}$, then find the conjugate $\bar{z}$.

If $z = \frac{1+i}{2-i} + \frac{1-i}{2+i}$, then find $|z|$.

If $z = 1+i$, find the value of the complex number $z^2 \bar{z}$.

If $\text{Re}(\frac{z-1}{z+1}) = 0$, then the value of $|z|$ is:

If $|z| = \max\{|z-1|, |z+1|\}$, then the locus of $z$ is:

If $z = x+iy$ satisfies $|z-1| + |z+1| = 2$, then $z$ lies on:

If $z = r(\cos \theta + i \sin \theta)$, then $-z$ in polar form is:

What is the modulus of $z = \frac{1}{2} - i\frac{\sqrt{3}}{2}$?

If $z = 6$, what is its representation in the Argand plane?

The complex number $z = -1 - i\sqrt{3}$ has a principal argument of:

Find the polar form of $z = -\sqrt{3} - i$.

What is the modulus of the complex number $z = (1 + i)^{10}$?

The principal argument of $z = \frac{1 + i}{\sqrt{2}}$ is:

If $|z-2/z| = 1$, the maximum value of $|z|$ is:

The complex number $z = 1 + i\sqrt{3}$ is rotated about the origin by an angle of $60^\circ$ in the counter-clockwise direction. The new complex number is:

If $z_1, z_2$ are two complex numbers such that $|z_1 + z_2| = |z_1| + |z_2|$, then $\text{arg}(z_1) - \text{arg}(z_2)$ is:

Find the value of $k$ if the equation $x^2 + k = 0$ has roots $\pm 6i$.

What are the roots of $x^2 + 18 = 0$?

Find the roots of $x^2 + 2x + 17 = 0$.

Solve for $x$: $5x^2 + 1 = 0$.

If the roots of $ax^2 + bx + c = 0$ are complex, which of the following must be true for real $a, b, c$?

Find the complex roots of the equation $x^2 + 3 = 0$.

Solve $x^2 - x + 4 = 0$.

If $\alpha, \beta$ are the roots of $x^2 + px + q = 0$ and are non-real, what is the value of $(\alpha - \beta)^2$ in terms of $p$ and $q$?

If $z^2 + z + 1 = 0$, the value of $(z+1/z)^2 + (z^2+1/z^2)^2 + (z^3+1/z^3)^2$ is:

The minimum value of $|z|^2 + |z-1|^2$ where $z$ is a root of $x^2 + x + 1 = 0$ is:

Calculate the product $i \cdot i^2 \cdot i^3 \cdot i^4$.

Find the sum $i^{14} + i^{16}$.

Evaluate $i^8 - i^4$.

What is the value of $i^{11}$?

Calculate $i^n$ where $n$ is the sum of the first 4 positive integers.

For $n \in \mathbb{Z}$, find $i^{2n} + i^{2n+1} + i^{2n+2} + i^{2n+3}$.

Evaluate the ratio $\frac{i^4 + i^8 + i^{12} + i^{16}}{i^2 + i^6 + i^{10} + i^{14}}$.

Determine the value of the finite sum $\sum_{k=1}^{10} i^{k^k}$.

Find the smallest positive integer $n$ such that $\frac{(1+i)^n}{(1-i)^{n-2}} = 2i$.

Find the number of integers $n \in \{1, 2, \dots, 100\}$ such that $i^n + i^{-n} = -2$.

Find the result of $\sqrt{-400}$.

Evaluate $(\sqrt{-1})^2 + 1$.

Simplify $\sqrt{-75}$.

What is the square root of $-625$?

Solve the quadratic equation $x^2 + \frac{9}{16} = 0$.

Determine the value of the product $\sqrt{-1} \times \sqrt{-1} \times \sqrt{-1} \times \sqrt{-1}$.

Calculate the sum $\sqrt{-4} + \sqrt{-16}$.

What are the roots of $x^2 + 12 = 0$?

Simplify the expression $i^9 + i^{19}$.

Find the discriminant of $x^2 + x + 1 = 0$.

Simplify the expression $(1 + i)^2 - 2i$.

Expand and simplify $(1 + i)^2 + 2i$.

Find the product of the complex number $1 - 5i$ and its conjugate.

Find the product of the complex number $4 + 3i$ and its conjugate.

Evaluate the expression $(7 + 3i)^2 - (3 + 7i)^2$.

Find the value of $(1 + 2i)^3$.

If $z = x + iy$, what is the result of $z \cdot \bar{z}$?

The value of $(1+i)^{10} + (1-i)^{10}$ is:

The quadratic equation $x^2 + (1-2i)x - 2i = 0$ has roots:

If $|z| = 2$, then the points representing $z$ lie on a circle with center $(0,0)$ and radius: