Complex Numbers

Let the minimum value \(v_{0}\) of \(v=|z|^{2}+|z-3|^{2}+|z-6 i|^{2}, z \in \mathbb{C}\) is attained at \({ }{z}=z_{0}\). Then \(\left|2 z_{0}^{2}-\bar{z}_{0}^{3}+3\right|^{2}+v_{0}^{2}\) is equal to :

Let S be the set of all \((\alpha, \beta), \pi<\alpha, \beta<2 \pi\), for which the complex number \(\frac{1-i \sin \alpha}{1+2 i \sin \alpha}\) is purely imaginary and \(\frac{1+i \cos \beta}{1-2 i \cos \beta}\) is purely real. Let $$Z_{\a...

Let \(A = \left\{ {z \in C:\left| {{{z + 1} \over {z - 1}}} \right| < 1} \right\}\) and \(B = \left\{ {z \in C:\arg \left( {{{z - 1} \over {z + 1}}} \right) = {{2\pi } \over 3}} \right\}\). Then A \(\cap\) B is :

If \({z^2} + z + 1 = 0\), \(z \in C\), then \(\left| {\sum\limits_{n = 1}^{15} {{{\left( {{z^n} + {{( - 1)}^n}{1 \over {{z^n}}}} \right)}^2}} } \right|\) is equal to _________.

Let O be the origin and A be the point \({z_1} = 1 + 2i\). If B is the point \({z_2}\), \({\mathop{\rm Re}\nolimits} ({z_2}) < 0\), such that OAB is a right angled isosceles triangle with OB as hypotenuse, then which of the following is NOT...

If \(z=x+i y\) satisfies \(|z|-2=0\) and \(|z-i|-|z+5 i|=0\), then :

Let a circle C in complex plane pass through the points \({z_1} = 3 + 4i\), \({z_2} = 4 + 3i\) and \({z_3} = 5i\). If \(z( \ne {z_1})\) is a point on C such that the line through z and z1 is perpendicular to the line through z2 and z3, then...

Let z1 and z2 be two complex numbers such that \({\overline z _1} = i{\overline z _2}\) and \(\arg \left( {{{{z_1}} \over {{{\overline z }_2}}}} \right) = \pi\). Then :

For \(\mathrm{n} \in \mathbf{N}\), let \(\mathrm{S}_{\mathrm{n}}=\left\{z \in \mathbf{C}:|z-3+2 i|=\frac{\mathrm{n}}{4}\right\}\) and \(\mathrm{T}_{\mathrm{n}}=\left\{z \in \mathbf{C}:|z-2+3 i|=\frac{1}{\mathrm{n}}\right\}\). Then the numbe...

For \(z \in \mathbb{C}\) if the minimum value of \((|z-3 \sqrt{2}|+|z-p \sqrt{2} i|)\) is \(5 \sqrt{2}\), then a value Question: of \(p\) is _____________.

Let \(A = \{ z \in C:1 \le |z - (1 + i)| \le 2\}\)
and \(B = \{ z \in A:|z - (1 - i)| = 1\}\). Then, B :

Let S = {z \(\in\) C : |z \(-\) 3| \(\le\) 1 and z(4 + 3i) + \(\overline z\)(4 \(-\) 3i) \(\le\) 24}. If \(\alpha\) + i\(\beta\) is the point in S which is closest to 4i, then 25(\(\alpha\) + \(\beta\)) is equal to ___________.

A point z moves in the complex plane such that \(\arg \left( {{{z - 2} \over {z + 2}}} \right) = {\pi \over 4}\), then the minimum value of \({\left| {z - 9\sqrt 2 - 2i} \right|^2}\) is equal to _______________.

If z is a complex number such that \({{z - i} \over {z - 1}}\) is purely imaginary, then the minimum value of | z \(-\) (3 + 3i) | is :

Let C be the set of all complex numbers. Let\({S_1} = \{ z \in C||z - 3 - 2i{|^2} = 8\}\)\({S_2} = \{ z \in C|{\mathop{\rm Re}\nolimits} (z) \ge 5\}\) and \({S_3} = \{ z \in C||z - \overline z | \ge 8\}\).Then the number of elements in $...

If the real part of the complex number \(z = {{3 + 2i\cos \theta } \over {1 - 3i\cos \theta }},\theta \in \left( {0,{\pi \over 2}} \right)\) is zero, then the value of sin23\(\theta\) + cos2\(\theta\) is equal to _______________.

Let C be the set of all complex numbers. LetS1 = {z\(\in\)C : |z \(-\) 2| \(\le\) 1} and S2 = {z\(\in\)C : z(1 + i) + \(\overline z\)(1 \(-\) i) \(\ge\) 4}.Then, the maximum value of \({\left| {z - {5 \over 2}} \right|^2}\) for z\(\in\)S1 ...

If \(S = \left\{ {z \in C:{{z - i} \over {z + 2i}} \in R} \right\}\), then :

Let z1 and z2 be two complex numbers such that \(\arg ({z_1} - {z_2}) = {\pi \over 4}\) and z1, z2 satisfy the equation | z \(-\) 3 | = Re(z). Then the imaginary part of z1 + z2 is equal to ___________.

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

Let \(z = {{1 - i\sqrt 3 } \over 2}\), \(i = \sqrt { - 1}\). Then the value of $$21 + {\left( {z + {1 \over z}} \right)^3} + {\left( {{z^2} + {1 \over {{z^2}}}} \right)^3} + {\left( {{z^3} + {1 \over {{z^3}}}} \right)^3} + .... + {\left( {...

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

The least positive integer n such that \({{{{(2i)}^n}} \over {{{(1 - i)}^{n - 2}}}},i = \sqrt { - 1}\) is a positive integer, is ___________.

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

Let $$S = \left\{ {n \in N\left| {{{\left( {\matrix{
0 & i \cr
1 & 0 \cr

} } \right)}^n}\left( {\matrix{
a & b \cr
c & d \cr

} } \right) = \left( {\matrix{
a & b \cr
c & d \cr

} } \right)\forall a,b,c,d \...

The equation of a circle is Re(z2) + 2(Im(z))2 + 2Re(z) = 0, where z = x + iy. A line which passes through the center of the given circle and the vertex of the parabola, x2 \(-\) 6x \(-\) y + 13 = 0, has y-intercept equal to ______________.

Let the lines (2 \(-\) i)z = (2 + i)\(\overline z\) and (2 \(+\) i)z + (i \(-\) 2)\(\overline z\) \(-\) 4i = 0, (here i2 = \(-\)1) be normal to a circle C. If the line iz + \(\overline z\) + 1 + i = 0 is tangent to this circle C, then it...

If \(\alpha\), \(\beta\) \(\in\) R are such that 1 \(-\) 2i (here i2 = \(-\)1) is a root of z2 + \(\alpha\)z + \(\beta\) = 0, then (\(\alpha\) \(-\) \(\beta\)) is equal to :

If the least and the largest real values of a, 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(p2 + q2) is equal to __________.

Let \(i = \sqrt { - 1}\). If \({{{{\left( { - 1 + i\sqrt 3 } \right)}^{21}}} \over {{{(1 - i)}^{24}}}} + {{{{\left( {1 + i\sqrt 3 } \right)}^{21}}} \over {{{(1 + i)}^{24}}}} = k\), and \(n = [|k|]\) be the greatest integral part of | k |. ...

Let n denote the number of solutions of the equation z2 + 3\(\overline z\) = 0, where z is a complex number. Then the value of \(\sum\limits_{k = 0}^\infty {{1 \over {{n^k}}}}\) is equal to :

If z and \(\omega\) are two complex numbers such that \(\left| {z\omega } \right| = 1\) and \(\arg (z) - \arg (\omega ) = {{3\pi } \over 2}\), then \(\arg \left( {{{1 - 2\overline z \omega } \over {1 + 3\overline z \omega }}} \right)\) is :...

If for the complex numbers z satisfying | z \(-\) 2 \(-\) 2i | \(\le\) 1, the maximum value of | 3iz + 6 | is attained at a + ib, then a + b is equal to ______________.

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

Let z1, z2 be the roots of the equation z2 + az + 12 = 0 and z1, z2 form an equilateral triangle with origin. Then, the value of |a| is :

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) = \({\pi \over 2}\). Then the area of the triangle with vertices origin, z and w is equal to :

The area of the triangle with vertices A(z), B(iz) and C(z + iz) is :

Let S1, S2 and S3 be three sets defined asS1 = {z\(\in\)C : |z \(-\) 1| \(\le\) \(\sqrt 2\)}S2 = {z\(\in\)C : Re((1 \(-\) i)z) \(\ge\) 1}S3 = {z\(\in\)C : Im(z) \(\le\) 1}Then the set S1 \(\cap\) S2 \(\cap\) S3 :

Let z and \(\omega\) be two complex numbers such that \(\omega = z\overline z - 2z + 2,\left| {{{z + i} \over {z - 3i}}} \right| = 1\) and Re(\(\omega\)) has minimum value. Then, the minimum value of n \(\in\) N for which \(\omega\)n is re...

Let a complex number z, |z| \(\ne\) 1, satisfy \({\log _{{1 \over {\sqrt 2 }}}}\left( {{{|z| + 11} \over {{{(|z| - 1)}^2}}}} \right) \le 2\). Then, the largest value of |z| is equal to ____________.

The least value of |z| where z is complex number which satisfies the inequality \(\exp \left( {{{(|z| + 3)(|z| - 1)} \over {||z| + 1|}}{{\log }_e}2} \right) \ge {\log _{\sqrt 2 }}|5\sqrt 7 + 9i|,i = \sqrt { - 1}\), is equal to :

Let z be complex number such that
\(\left| {{{z - i} \over {z + 2i}}} \right| = 1\) and |z| = \({5 \over 2}\). Then the value of |z + 3i| is :

If z be a complex number satisfying
|Re(z)| + |Im(z)| = 4, then |z| cannot be :

If the equation, x2 + bx + 45 = 0 (b \(\in\) R) has
conjugate complex roots and they satisfy
|z +1| = 2\(\sqrt {10}\) , then :

If \({\mathop{\rm Re}\nolimits} \left( {{{z - 1} \over {2z + i}}} \right) = 1\), where z = x + iy, then the point (x, y) lies on a :

If \({{3 + i\sin \theta } \over {4 - i\cos \theta }}\), \(\theta\) \(\in\) [0, 2\(\theta\)], is a real number, then an argument of sin\(\theta\) + icos\(\theta\) is :

The region represented by {z = x + iy \(\in\) C : |z| – Re(z) \(\le\) 1} is also given by the inequality :
{z = x + iy \(\in\) C : |z| – Re(z) \(\le\) 1}

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

If the four complex numbers \(z,\overline z ,\overline z - 2{\mathop{\rm Re}\nolimits} \left( {\overline z } \right)\) and \(z-2Re(z)\) represent the vertices of a square of
side 4 units in the Argand plane, then \(|z|\) is equal to :

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