List of top Mathematics Questions on Complex numbers asked in CUET (PG)

If \(1, \alpha_1, \alpha_2, \alpha_3, \ldots, \alpha_{n-1}\) are n roots of the equation, \( x^n = 1 \) then the value of \( (1-\alpha_1)(1-\alpha_2)(1-\alpha_3)\ldots(1-\alpha_{n-1}) \) is

The complex number \(z_1, z_2\) and origin, form an equilateral triangle only if:

Let z, \(z_1, z_2\) be complex numbers. Then which of the following statements are True?
(A) \(e^z\) is never zero
(B) \(|e^{ix}|=1\) if x is real
(C) \(e^z = 1\) if z is an integral multiple of \(2\pi i\)
(D) \(e^{z_1} = e^{z_2}\) if and only if \(z_1 - z_2 = \frac{2\pi i n}{\sqrt{3}}\), where n is an integer
(E) \(|e^z|>e^z\) for \(z \neq 0\)
Choose the correct answer from the options given below:

Match List-I with List-II:

\[ \begin{array}{|l|l|} \hline \textbf{List-I (Equation)} & \textbf{List-II (Roots)} \\ \hline (A)\; (z + 1)^{2n} + (z - 1)^{2n} = 0 & (I)\; \cos\!\left(\tfrac{\pi}{5}\right) \pm i\sin\!\left(\tfrac{\pi}{5}\right), -1, \cos\!\left(\tfrac{3\pi}{5}\right) \pm i\sin\!\left(\tfrac{3\pi}{5}\right) \\ \hline (B)\; z^5 + z^4 + z^3 + z^2 + z + 1 = 0 & (II)\; \text{purely imaginary number} \\ \hline (C)\; (z-1)^5 + z^5 = 0 & (III)\; -1,\; \pm\frac{1}{2} \pm i\frac{\sqrt{3}}{2} \\ \hline (D)\; z^5 + 1 = 0 & (IV)\; \tfrac{1}{2}\left[1+i\cot\!\left(\tfrac{p\pi}{10}\right)\right], \; p = 1,3,5,7,9 \\ \hline \end{array} \]
Choose the correct answer from the options given below:

If 'a' is an imaginary cube root of unity, then \( (1-a+a^2)^5 + (1+a-a^2)^5 \) is equal to: