Mock Test 2 — Complex Numbers

If $z = \left(\frac{\sqrt{3}+i}{2}\right)^6 + \left(\frac{\sqrt{3}-i}{2}\right)^6$, then the value of $z$ simplifies to:

The minimum value of the expression $|z - 1| + |z - 5i|$ achieved across the Argand plane is exactly:

If $\alpha, \beta, \gamma$ are the roots of the cubic equation $x^3 - 3x^2 + 3x - 2 = 0$, then they match which geometric configuration?

The principal value of the complex logarithm expression $\ln(1-i)$ has an imaginary part equal to:

The locus represented by the equation $\arg\left(\frac{z-1}{z+1}\right) = \frac{\pi}{2}$ is a:

If $|z_1| = 1, |z_2| = 2, |z_3| = 3$ and $|9z_1z_2 + 4z_1z_3 + z_2z_3| = 12$, then the value of the modulus sum $|z_1 + z_2 + z_3|$ is:

If $\omega$ is a non-real cube root of unity, the value of the polynomial sum $\sum_{k=1}^{30} (1+\omega^k+\omega^{2k})$ is:

The complex number equation $x^2 + |x|^2 = 0$ has how many distinct solutions over the complex field?

The value of the product expression $\prod_{k=1}^{4} \left(2 - e^{i2\pi k/5}\right)$ can be evaluated using $n^{\text{th}}$ root identities. The value is:

If $|z-2| = \text{Re}(z)$, then the path traced by the moving complex point $z$ is a:

Evaluate the maximum value of the modulus $|z|$ if it satisfies the structural distance inequality $|z - 3/z| = 2$.

Find the number of real roots of the complex-coefficient polynomial equation $z^2 + 2iz + 3 = 0$.

Find the number of solutions to the equation $z^3 + \bar{z} = 0$.

Assertion (A): The complex value of $i^i$ is a purely real number.
Reason (R): Converting $i$ to Euler's form gives $e^{i\pi/2}$. Raising this expression to the power $i$ multiplies the exponents, which eliminates the imaginary unit because $i^2 = -1$.

Assertion (A): The roots of the equation $z^n = w$ always form a regular polygon centered at the origin on the Argand plane.
Reason (R): Multiplying a complex root by the $n^{\text{th}}$ roots of unity preserves its modulus and adds constant angle increments of $\frac{2\pi}{n}$ to its argument, which rotates the points uniformly around a circle.