JEE Complex Numbers PYQs With Video Solutions PDF

Let a complex number be $$w = 1 - \sqrt{3}i$$. Let another complex number $$z$$ be such that $$|zw| = 1$$ and $$\arg(z) - \arg(w) = \frac{\pi}{2}$$. Then the area of the triangle (in sq. units) with vertices origin, $$z$$ and $$w$$ is equal to

Let $$z$$ and $$w$$ be two complex numbers such that $$w = z\bar{z} - 2z + 2$$, $$\left|\frac{z+i}{z-3i}\right| = 1$$ and $$\text{Re}(w)$$ has minimum value. Then, the minimum value of $$n \in N$$ for which $$w^n$$ is real, is equal to ________.

If $$|z - 3 + 2i| \leq 4$$ then the difference between the greatest value and the least value of $$|z|$$ is:

Let $$a = \text{Im}\left(\frac{1+z^2}{2iz}\right)$$, where z is any non-zero complex number. The set $$A = \{a : |z| = 1$$ and $$z \neq \pm 1\}$$ is equal to:

Let $$S = \{z = x + iy : |z-1+i| \geq |z|, |z| < 2, |z+i| = |z-1|\}$$. Then the set of all values of x, for which $$w = 2x + iy \in S$$ for some $$y \in \mathbb{R}$$, is

If $$\alpha, \beta \in C$$ are the distinct roots of the equation $$x^2 - x + 1 = 0$$, then $$\alpha^{101} + \beta^{107}$$ is equal to:

The equation $$Im\left(\frac{iz - 2}{z - i}\right) + 1 = 0$$, $$z \in \mathbb{C}$$, $$z \neq i$$ represents a part of a circle having radius equal to:

Let $$z$$ be those complex numbers which satisfy $$|z + 5| \leq 4$$ and $$z(1 + i) + \bar{z}(1 - i) \geq -10$$, $$i = \sqrt{-1}$$. If the maximum value of $$|z + 1|^2$$ is $$\alpha + \beta\sqrt{2}$$, then the value of $$(\alpha + \beta)$$ is

If $$z \neq 0$$ be a complex number such that $$\left|z - \frac{1}{z}\right| = 2$$, then the maximum value of $$|z|$$ is

For all complex numbers z of the form $$1 + i\alpha$$, $$\alpha \in R$$, if $$z^2 = x + iy$$, then:

The value of $$\left(\frac{-1+i\sqrt{3}}{1-i}\right)^{30}$$ is:

Let the lines $$(2 - i)z = (2 + i)\bar{z}$$ and $$(2 + i)z + (i - 2)\bar{z} - 4i = 0$$, (here $$i^2 = -1$$) be normal to a circle $$C$$. If the line $$iz + \bar{z} + 1 + i = 0$$ is tangent to this circle $$C$$, then its radius is:

If $$z = 2 + 3i$$, then $$z^5 + \bar{z}^5$$ is equal to:

Let $$z = x + iy$$ be a non-zero complex number such that $$z^2 = i|z|^2$$, where $$i = \sqrt{-1}$$, then $$z$$ lies on the:

Let z be a complex number such that |z - 6| = 5 and |z + 2 - 6i| = 5. Then the value of $$z^{3}+3z^{2}-15z+141$$ is equal to

If the equation $$a|z|^2 +\overline{\bar{\alpha}z + \alpha\bar{z}} + d = 0$$ represents a circle where $$a, d$$ are real constants then which of the following condition is correct?

Let $$S = \{z \in \mathbb{C} : z^2 + 4z + 16 = 0\}$$. Then $$\displaystyle\sum_{z \in S} |z + \sqrt{3}\,i|^2$$ is equal to :

Let $$z_1$$ and $$z_2$$ be two complex numbers such that $$z_1 + z_2 = 5$$ and $$z_1^3 + z_2^3 = 20 + 15i$$. Then $$|z_1^4 + z_2^4|$$ equals

If $$z^2 + z + 1 = 0, z \in C$$, then $$\left|\sum_{n=1}^{15}\left(z^n + (-1)^n \frac{1}{z^n}\right)^2\right|$$ is equal to ______

The area (in sq. units) of the region $$S = \{z \in \mathbb{C} : |z - 1| \leq 2; (z + \bar{z}) + i(z - \bar{z}) \leq 2, \text{Im}(z) \geq 0\}$$ is

Let the circles $$C_1 : |z| = r$$ and $$C_2 : |z - 3 - 4i| = 5$$, $$z \in \mathbb{C}$$, be such that $$C_2$$ lies within $$C_1$$. If $$z_1$$ moves on $$C_1$$, $$z_2$$ moves on $$C_2$$ and $$\min|z_1 - z_2| = 2$$, then $$\max|z_1 - z_2|$$ is equal to :

If $$\alpha$$ denotes the number of solutions of $$|1 - i|^x = 2^x$$ and $$\beta = \frac{|z|}{\arg(z)}$$, where $$z = \frac{\pi}{4}(1+i)^4\left(\frac{1-\sqrt{\pi}\cdot i}{\sqrt{\pi}+i} + \frac{\sqrt{\pi}-i}{1+\sqrt{\pi}\cdot i}\right)$$, $$i = \sqrt{-1}$$, then the distance of the point $$(\alpha, \beta)$$ from the line $$4x - 3y = 7$$ is ______

If $$z$$ and $$\omega$$ are two complex numbers such that $$|z\omega| = 1$$ and $$\arg(z) - \arg(\omega) = \frac{3\pi}{2}$$, then $$\arg\left(\frac{1 - 2\bar{z}\omega}{1 + 3\bar{z}\omega}\right)$$ is:
(Here $$\arg(z)$$ denotes the principal argument of complex number $$z$$)

If the least and the largest real values of $$\alpha$$, for which the equation $$z + \alpha|z - 1| + 2i = 0$$ ($$z \in C$$ and $$i = \sqrt{-1}$$) has a solution, are $$p$$ and $$q$$ respectively; then $$4(p^2 + q^2)$$ is equal to ______.

Let $$i = \sqrt{-1}$$. If $$\frac{(-1 + i\sqrt{3})^{21}}{(1 - i)^{24}} + \frac{(1 + i\sqrt{3})^{21}}{(1 + i)^{24}} = k$$, and $$n = [|k|]$$ be the greatest integral part of $$|k|$$. Then $$\sum_{j=0}^{n+5} (j + 5)^2 - \sum_{j=0}^{n+5} (j + 5)$$ is equal to ______.

The equation $$\arg\left(\frac{z-1}{z+1}\right) = \frac{\pi}{4}$$ represents a circle with:

If $$(\sqrt{3} + i)^{100} = 2^{99}(p + iq)$$, then $$p$$ and $$q$$ are roots of the equation:

Let $$u = \frac{2z+i}{z-ki}$$, $$z = x + iy$$ and $$k \gt 0$$. If the curve represented by Re(u) + Im(u) = 1 intersects the y-axis at points P and Q where PQ = 5 then the value of k is

The equation $$|z - i| = |z - 1|$$, $$i = \sqrt{-1}$$, represents:

For all $$z \in C$$ on the curve $$C_1$$: $$|z| = 4$$, let the locus of the point $$z + \dfrac{1}{z}$$ be the curve $$C_2$$. Then