List of top Mathematics Questions on Complex numbers asked in JEE Main

Let $z$ be the complex number satisfying $|z - 5| \le 3$ and having maximum positive principal argument. Then $34 \left| \frac{5z - 12}{5iz + 16} \right|^2$ is equal to :

If $x^2+x+1=0$, then the value of $\left(x + \frac{1}{x}\right)^2 + \left(x^2 + \frac{1}{x^2}\right)^2 + \left(x^3 + \frac{1}{x^3}\right)^2 + \dots + \left(x^{25} + \frac{1}{x^{25}}\right)^2$ is:

Let \[ S = \{ z \in \mathbb{C} : 4z^2 + \bar{z} = 0 \}. \] Then \[ \sum_{z \in S} |z|^2 \] is equal to

$z$ is a complex number satisfying \[ \left| \frac{z - 6i}{z - 2i} \right| = 1 \quad \text{and} \quad \left| \frac{z - 8 + 2i}{z + 2i} \right| = \frac{3}{5} \] then find $\sum |z|^2$.

Let $S=\left\{ z\in\mathbb{C}:\left|\frac{z-6i}{z-2i}\right|=1 \text{ and } \left|\frac{z-8+2i}{z+2i}\right|=\frac{3}{5} \right\}.$
Then $\sum_{z\in S}|z|^2$ is equal to

If $ z = \dfrac{\sqrt{3}}{2} + \dfrac{i}{2} $, $ i = \sqrt{-1} $, then \[ \left(z^{201} - i\right)^8 \] is equal to

If $(9+7\alpha-7\beta)^{20} + (9\alpha+7\beta-7)^{20} + (9\beta+7-7\alpha)^{20} + (14+7\alpha+7\beta)^{20}$ is $m^{10}$ then the value of m is : (where $\alpha = \frac{-1+i\sqrt{3}}{2}$ \& $\beta = \frac{-1-i\sqrt{3}}{2}$)

A building construction work can be completed by two masons A and B together in 22.5 days. Mason A alone can complete the construction work in 24 days less than mason B alone. Then mason A alone will complete the construction work in :

Let $ z $ be a complex number such that $ |z-6|=5 $ and $ |z+2-6i|=5 $. Then the value of $ z^3+3z^2-15z+14 $ is equal to:

If $ z = \dfrac{\sqrt{3}}{2} + \dfrac{i}{2} $, then the value of \[ \left(z^{201} - i\right)^8 \] is:

Let the curve $z(1+i) + z(1-i) = 4$, $z \in \mathbb{C}$, divide the region $|z-3| \le 1$ into two parts of areas $\alpha$ and $\beta$. Then $|\alpha - \beta|$ equals:

If $x^2 + x + 1 = 0$, find the value of
$\sum_{k=1}^{15} \left(x^k + \frac{1}{x^k}\right)^4$

If complex numbers $ z_1, z_2, \ldots , z_n $ satisfy the equation $ 4z^2 + \bar{z} = 0 $, then $ \sum_{i=1}^{n} |z_i|^2 $ is equal to:

If $ z_1, z_2, z_3 \in \mathbb{C} $ are the vertices of an equilateral triangle, whose centroid is $ z_0 $, then $ \sum_{k=1}^{3} (z_k - z_0)^2 $ is equal to

Let the product of $ \omega_1 = (8 + i) \sin \theta + (7 + 4i) \cos \theta $ and $ \omega_2 = (1 + 8i) \sin \theta + (4 + 7i) \cos \theta $ be $ \alpha + i\beta $, where $ i = \sqrt{-1} $. Let $ p $ and $ q $ be the maximum and the minimum values of $ \alpha + \beta $ respectively.

Let the curve $($$ z(1 + i) + \overline{z(1 - i)} = 4$, $\, z \in \mathbb{C}$ $)$, divide the region $|z - 3| \leq 1$ into two parts of areas $ \alpha $ and $ \beta $. Then $ |\alpha - \beta| $ equals:

If the locus of $ z \in \mathbb{C} $, such that $ \text{Re} \left( \frac{z - 1}{2z + i} \right) + \text{Re} \left( \frac{ \bar{z} - 1}{2 \bar{z} - i} \right) = 2, $ is a circle of radius $ r $ and center $ (a, b) $, then $ \frac{15ab}{r^2} \text{ is equal to:} $

Among the statements: (S1): The set $ \{ z \in \mathbb{C} - \{-i\} : |z| = 1 \text{ and } \frac{z - i}{z + i} \text{ is purely real} \} $ contains exactly two elements.
(S2): The set $ \{ z \in \mathbb{C} - \{-1\} : |z| = 1 \text{ and } \frac{z - 1}{z + 1} \text{ is purely imaginary} \} $ contains infinitely many elements. Then, which of the following is correct?

Let integers $ a, b \in [-3, 3] $ be such that $ a + b \neq 0 $. $\text{Then the number of all possible ordered pairs}$ $ (a, b) $, $\text{for which}$ \[ \left| \frac{z - a}{z + b} \right| = 1 \quad \text{and} \quad \left| \begin{matrix} z + 1 & \omega & \omega^2 \\ \omega^2 & 1 & z + \omega \\ \omega^2 & 1 & z + \omega \end{matrix} \right| = 1, \] $\text{is equal to:}$

If $ \alpha + i\beta $ and $ \gamma + i\delta $ are the roots of the equation $ x^2 - (3-2i)x - (2i-2) = 0 $, $ i = \sqrt{-1} $, then $ \alpha\gamma + \beta\delta $ is equal to:

Let the curve $z(1 + i) +$ $\overline{z(1 - i) = 4}$, z $\in$$ \mathbb{C}$, divide the region $|z - 3| \leq 1$ into two parts of areas $ \alpha $ and $ \beta $. Then $ |\alpha - \beta| $ equals:}

Let $ A = \left\{ x \in (0, \pi) \mid - \log\left(\frac{2}{\pi}\right)\sin x + \log\left(\frac{2}{\pi}\right)\cos x = 2 \right\} $ and

\[ B = \left\{ x \geq 0 : \sqrt{x}(\sqrt{x - 4}) - 3\sqrt{x - 2} + 6 = 0 \right\}. \]

Then $ n(A \cup B) $ is equal to:

If $ \alpha $ and $ \beta $ are the roots of the equation $ 2z^2 - 3z - 2i = 0 $, where $ i = \sqrt{-1} $, then \[ 16 \cdot {Re} \left( \frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{15} + \beta^{15}} \right) \cdot {Im} \left( \frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{15} + \beta^{15}} \right) \] is equal to:

Let $ z_1, z_2, z_3 $ be three complex numbers on the circle $ |z| = 1 $ with $ \arg(z_1) = -\frac{\pi}{4}, \arg(z_2) = 0 $ and $ \arg(z_3) = \frac{\pi}{4} $. If $ |z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1}|^2 = \alpha + \beta \sqrt{2} $, where $ \alpha, \beta \in \mathbb{Z} $, then the value of $ \alpha^2 + \beta^2 $ is :

The number of complex numbers $ z $, satisfying $ |z| = 1 $ and \[ \left| \frac{z}{\overline{z}} + \frac{\overline{z}}{z} \right| = 1, \] is: