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

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

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 ha...

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

If 1, $$\omega$$, $$\omega$$2 are cube roots of unity, then $$\left| {\matrix{ 1 & {{\omega ^n}} & {{\omega ^{2n}}} \cr {{\omega ^{2n}}} & 1 &...

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

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

For any complex number z, the minimum value of $$|z| + |z - 1|$$ is

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

If $$ - \pi

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

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

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 z = x $$-$$ iy and $${z^{{1 \over 3}}} = p + iq(x,y,p,q \in R)$$, then $${{\left( {{x \over p} + {y \over q}} \right)} \over {({p^2} + {q^2})}}$$ i...

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

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...

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

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

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...

For any non-zero complex number z, the minimum value of | z | + | z $$-$$ 1 | 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 ...

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 repre...

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}} &...

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} -...

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

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 poi...

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

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

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)

The complex number z satisfying the equation | z $$-$$ 1 | = | z + 1 | = 1 is