Complex Numbers

Let $Z_1, Z_2$ be the roots of the equation $Z^2+p Z+q=0$, where the coefficients $p$ and $q$ may be complex numbers and also let $A, B$ represent $Z_1, Z_2$ respectively in the complex plane. If $\angle A O B=\alpha \neq 0$ and $O A=O B$, ...

The total number of polynomials of the form $x^3+a x^2+b x+c$ which is divisible by $x^2+1$, where $a, b, c \in\{1,2,3, \ldots ., 10\}$ is

The expression $\sum_{k=1}^{32}(3 K+2)\left\{\sum_{r=1}^{10}\left(\sin \frac{2 r \pi}{11}-i \cos \frac{2 r \pi}{11}\right)\right\}^k$ represents

Let $\omega(\neq 1)$ be a cubic root of unity. Then the minimum value of the set $\left\{\mid a+b \omega+c \omega^2\right\}^2 ; a, b, c$ are distinct non-zero integers} equals

If $\left|Z_1\right|=\left|Z_2\right|=\left|Z_3\right|=1$ and $Z_1+Z_2+Z_3=0$, then the area of the triangle whose vertices are $Z_1, Z_2, Z_3$ is

If $z_1, z_2$ are complex numbers such that $\frac{2 z_1}{3 z_2}$ is a purely imaginary number, then the value of $\left|\frac{z_1-z_2}{z_1+z_2}\right|$ is

If \(\cos \theta+i \sin \theta, \theta \in \mathbb{R}\), is a root of the equation
\(a_0 x^n+a_1 x^{n-1}+\ldots .+a_{n-1} x+a_n=0, a_0, a_1, \ldots . a_n \in \mathbb{R}, a_0 \neq 0,\)
then the value of $$a_1 \sin \theta+a_2 \sin 2 \theta+\l...

If \(z_1\) and \(z_2\) be two roots of the equation \(z^2+a z+b=0, a^2<4 b\), then the origin, \(\mathrm{z}_1\) and \(\mathrm{z}_2\) form an equilateral triangle if

If z\(_1\) and z\(_2\) are two complex numbers satisfying the equation \(\left| {{{{z_1} + {z_2}} \over {{z_1} - {z_2}}}} \right| = 1\), then \({{{z_1}} \over {{z_2}}}\) may be

Reflection of the line \(\overline a z + a\overline z = 0\) in the real axis is given by :

If the vertices of a square are \({z_1},{z_2},{z_3}\) and \({z_4}\) taken in the anti-clockwise order, then \({z_3} =\)

Let z1 and z2 be two non-zero complex numbers. Then

If z = x \(-\) iy and \({z^{{1 \over 3}}} = p + iq(x,y,p,q \in R)\), then \({{\left( {{x \over p} + {y \over q}} \right)} \over {({p^2} + {q^2})}}\) is equal to

If \(|z - 25i| \le 15\), then Maximum arg(z) \(-\) Minimum arg(z) is equal to
(arg z is the principal value of argument of z)

If \(\left| {z + i} \right| - \left| {z - 1} \right| = \left| z \right| - 2 = 0\) for a complex number z, then z is equal to

Let C denote the set of all complex numbers. Define A = {(z, w) | z, w\(\in\)C and |z| = |w|}, B = {z, w} | z, w\(\in\)C and z2 = w2}. Then

If |z| = 1 and z \(\ne\) \(\pm\) 1, then all the points representing \({z \over {1 - {z^2}}}\) lie on

The number of complex numbers p such that \(\left| p \right| = 1\) and imaginary part of p4 is 0, is

The equation \(z\bar z + (2 - 3i)z + (2 + 3i)\bar z + 4 = 0\) represents a circle of radius

The general value of the real angle \(\theta\), which satisfies the equation, \((\cos \theta + i\sin \theta )(\cos 2\theta + i\sin 2\theta )...(\cos n\theta + i\sin n\theta ) = 1\) is given by, (assuming k is an integer)

Let z be a complex number such that the principal value of argument, arg z > 0. Then, arg z \(-\) arg(\(-\) z) is

The polar coordinate of a point P is \(\left( {2, - {\pi \over 4}} \right)\). The polar coordinate of the point Q which is such that line joining PQ is bisected perpendicularly by the initial line, is

For any non-zero complex number z, the minimum value of | z | + | z \(-\) 1 | is

If \(\theta \in R\) and \({{1 - i\cos \theta } \over {1 + 2i\cos \theta }}\) is real number, then \(\theta\) will be (when I : Set of integers)

If \({Z_r} = \sin {{2\pi r} \over {11}} - i\cos {{2\pi r} \over {11}}\), then \(\sum\limits_{r = 0}^{10} {{Z_r}}\) is equal to

If z1 and z2 be two non-zero complex numbers such that \({{{z_1}} \over {{z_2}}} + {{{z_2}} \over {{z_1}}} = 1\), then the origin and the points represented by z1 and z2

Let z1 and z2 be complex numbers such that z1 \(\ne\) z2 and |z1| = |z2|. If Re(z1) > 0 and Im(z2) < 0, then \({{{z_1} + {z_2}} \over {{z_1} - {z_2}}}\) is

If \({a_r} = {(\cos 2r\pi + i\sin 2r\pi )^{1/9}}\), then the value of \(\left| {\matrix{ {{a_1}} & {{a_2}} & {{a_3}} \cr {{a_4}} & {{a_5}} & {{a_6}} \cr {{a_7}} & {{a_8}} & {{a_9}} \cr } } \right|\) is equal to

The expression \({{{{(1 + i)}^n}} \over {{{(1 - i)}^{n - 2}}}}\) equals

Let z = x + iy, where x and y are real. The points (x, y) in the X-Y plane for which \({{{z + i} \over {z - i}}}\) is purely imaginary, lie on

The complex number z satisfying the equation | z \(-\) 1 | = | z + 1 | = 1 is

If \(\omega\) is an imaginary cube root of unity, then the value of (2 \(-\) \(\omega\)) (2 \(-\) \(\omega\)2) + 2(3 \(-\) \(\omega\))(3 \(-\) \(\omega\)2) + ... + (n \(-\) 1) (n \(-\) \(\omega\))(n \(-\) \(\omega\)2) is

If z = sin\(\theta\) \(-\) icos\(\theta\), then for any integer n,

The value of \(\sum\limits_{n = 1}^{13} {({i^n} + {i^{n + 1}})}\), \(i = \sqrt { - 1}\) is

If \(\omega\) \(\ne\) 1 is a cube root of unity, then the sum of the series \(S = 1 + 2\omega + 3{\omega ^2} + \,\,.....\,\, + 3n{\omega ^{3n - 1}}\) is

If \(x + {1 \over x} = 2\cos \theta\), then for any integer n, \({x^n} + {1 \over {{x^n}}} =\)

For the real parameter t, the locus of the complex number \(z = (1 - {t^2}) + i\sqrt {1 + {t^2}}\) in the complex plane is

If \(- \pi < \arg (z) < - {\pi \over 2}\), then \(\arg \overline z - \arg ( - \overline z )\) is

If \(z = {4 \over {1 - i}}\), then \(\overline z\) is (where \(\overline z\) is complex conjugate of z)

For any complex number z, the minimum value of \(|z| + |z - 1|\) is

The modulus of \({{1 - i} \over {3 + i}} + {{4i} \over 5}\) is

If \(i = \sqrt { - 1}\) and n is positive integer, then \({i^n} + {i^{n + 1}} + {i^{n + 2}} + {i^{n + 3}}\) is equal to

Prove that if the ratio \({{z - i} \over {z - 1}}\) is purely imaginary, the point z lies on the circle in the Argand plane whose centre is at the point \({1 \over 2}(1 + i)\) and radius is \({1 \over {\sqrt 2 }}\).

If 1, \(\omega\), \(\omega\)2 are cube roots of unity, then \(\left| {\matrix{ 1 & {{\omega ^n}} & {{\omega ^{2n}}} \cr {{\omega ^{2n}}} & 1 & {{\omega ^n}} \cr {{\omega ^n}} & {{\omega ^{2n}}} & 1 \cr } } \right|\) has val...

For two complex numbers z1, z2 the relation \(\left| {{z_1} + {z_2}} \right| = \left| {{z_1}} \right| + \left| {{z_2}} \right|\) holds if

A and B are two points on the Argand plane such that the segment AB is bisected at the point (0, 0). If the point A, which is in the third quadrant has principal amplitude \(\theta\), then the principal amplitude of the point B is

The principal amplitude of \({(\sin 40^\circ + i\cos 40^\circ )^5}\) is

Let \(\alpha\), \(\beta\) be the roots of \({x^2} - 2x\cos \phi + 1 = 0\), then the equation whose roots are \({\alpha ^n},{\beta ^n}\) is

The value of \({(1 - \omega + {\omega ^2})^5} + {(1 + \omega - {\omega ^2})^5}\), where \(\omega\) and \(\omega\)2 are the complex cube roots of unity is