Mock Test 1 — Complex Numbers

The value of the finite sum $\sum_{n=1}^{100} i^n$ is equal to:

If $z = \frac{4 + 3i}{1 + i}$, then the absolute modulus value $|z|$ is:

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

If $\omega$ is a non-real cube root of unity, the value of the expression $(1+\omega)(1+\omega^2)(1+\omega^4)(1+\omega^8)$ is:

The locus represented by the complex equation $|z - 4i| + |z + 4i| = 10$ is an:

If $z^2 = -i$, then the value solutions for $z$ can be written as:

The value of the product $(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}) \times (\cos\frac{\pi}{6} + i\sin\frac{\pi}{6})$ is equal to:

If $\text{Re}\left(\frac{z-1}{z+1}\right) = 0$, then the locus of the moving point $z$ is a:

The value of the complex exponent expression $e^{i\pi/2}$ matches which unit algebraic term?

If the complex numbers $z_1, z_2, z_3$ form the vertices of an equilateral triangle inscribed inside the unit circle $|z|=1$, their sum $z_1 + z_2 + z_3$ must equal:

Evaluate the value of the integer product: $(1-\omega)(1-\omega^2)(1-\omega^4)(1-\omega^8)$ where $\omega$ is a cube root of unity.

Find the number of distinct real solution roots that satisfy the equation $|x + i| = \sqrt{5}$ for $x \in \mathbb{R}$.

Determine the radius value $r$ of the circular locus path given by the expression $z\bar{z} - 2z - 2\bar{z} - 7 = 0$.

Assertion (A): The upper bound statement $|z_1 + z_2| = |z_1| + |z_2|$ is true if the vector rays point in the same direction.
Reason (R): The Triangle Inequality holds across all fields, and equality requires the arguments of the two numbers to be completely equal ($\arg z_1 = \arg z_2$).

Assertion (A): The expression $\sqrt{-4} \times \sqrt{-9}$ evaluates to $\sqrt{36} = 6$.
Reason (R): The product identity $\sqrt{a}\sqrt{b} = \sqrt{ab}$ is strictly valid only when at least one of the real radicand variables is non-negative.