Complex Numbers

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

If a and b are real numbers such that \({\left( {2 + \alpha } \right)^4} = a + b\alpha\) where \(\alpha = {{ - 1 + i\sqrt 3 } \over 2}\) then a + b is
equal to :

If \({\left( {{{1 + i} \over {1 - i}}} \right)^{{m \over 2}}} = {\left( {{{1 + i} \over {1 - i}}} \right)^{{n \over 3}}} = 1\), (m, n
\(\in\) N) then the
greatest common divisor of the least values of
m and n is _______ .

If z1
, z2
are complex numbers such that
Re(z1) = |z1 – 1|, Re(z2) = |z2 – 1| , and
arg(z1 - z2) = \({\pi \over 6}\), then Im(z1
+ z2
) is equal to :

The value of \({\left( {{{1 + \sin {{2\pi } \over 9} + i\cos {{2\pi } \over 9}} \over {1 + \sin {{2\pi } \over 9} - i\cos {{2\pi } \over 9}}}} \right)^3}\) is :

The imaginary part of
\({\left( {3 + 2\sqrt { - 54} } \right)^{{1 \over 2}}} - {\left( {3 - 2\sqrt { - 54} } \right)^{{1 \over 2}}}\) can be :

Let
A = \(\left\{ {\theta \in \left( { - {\pi \over 2},\pi } \right):{{3 + 2i\sin \theta } \over {1 - 2i\sin \theta }}is\,purely\,imaginary} \right\}\)
. Then the sum of the elements in A is :

Let \(\alpha\) and \(\beta\) be two roots of the equation x2 + 2x + 2 = 0 , then \(\alpha ^{15}\) + \(\beta ^{15}\) is equal to :

Let z0 be a root of the quadratic equation, x2 + x + 1 = 0, If z = 3 + 6iz\(_0^{81}\) \(-\) 3iz\(_0^{93}\), then arg z is equal to :

All the points in the set
\(S = \left\{ {{{\alpha + i} \over {\alpha - i}}:\alpha \in R} \right\}(i = \sqrt { - 1} )\) lie on a :

Let z \(\in\) C be such that |z| < 1.
If \(\omega = {{5 + 3z} \over {5(1 - z)}}\)z, then :

If \(\alpha\) and \(\beta\) be the roots of the equation
x2 – 2x + 2 = 0, then the least value of n for which \({\left( {{\alpha \over \beta }} \right)^n} = 1\) is :

If \(z = {{\sqrt 3 } \over 2} + {i \over 2}\left( {i = \sqrt { - 1} } \right)\),
then (1 + iz + z5 + iz8)9 is equal to :

If \({{z - \alpha } \over {z + \alpha }}\left( {\alpha \in R} \right)\) is a purely imaginary number and | z | = 2, then a value of \(\alpha\) is :

Let z1 and z2 be two complex numbers satisfying | z1 | = 9 and | z2 – 3 – 4i | = 4. Then the minimum value of
| z1 – z2 | is :

The equation |z – i| = |z – 1|, i = \(\sqrt { - 1}\), represents :

Let z \(\in\) C with Im(z) = 10 and it satisfies \({{2z - n} \over {2z + n}}\) = 2i - 1 for some natural number n. Then :

Let \({\left( { - 2 - {1 \over 3}i} \right)^3} = {{x + iy} \over {27}}\left( {i = \sqrt { - 1} } \right),\,\,\) where x and y are real numbers, then y \(-\) x equals :

Let z be a complex number such that |z| + z = 3 + i (where i = \(\sqrt { - 1}\)). Then |z| is equal to :

Let z1 and z2 be any two non-zero complex numbers such that   \(3\left| {{z_1}} \right| = 4\left| {{z_2}} \right|.\)  If  \(z = {{3{z_1}} \over {2{z_2}}} + {{2{z_2}} \over {3{z_1}}}\)  then :

Let \(z = {\left( {{{\sqrt 3 } \over 2} + {i \over 2}} \right)^5} + {\left( {{{\sqrt 3 } \over 2} - {i \over 2}} \right)^5}.\) If R(z) and 1(z) respectively denote the real and imaginary parts of z, then :

If a > 0 and z = \({{{{\left( {1 + i} \right)}^2}} \over {a - i}}\), has magnitude \(\sqrt {{2 \over 5}}\), then \(\overline z\) is equal to :

If z and w are two complex numbers such that |zw| = 1 and arg(z) – arg(w) = \({\pi \over 2}\)
, then :

The least positive integer n for which \({\left( {{{1 + i\sqrt 3 } \over {1 - i\sqrt 3 }}} \right)^n} = 1,\) is :

The set of all \(\alpha\) \(\in\) R, for which w = \({{1 + \left( {1 - 8\alpha } \right)z} \over {1 - z}}\) is purely imaginary number, for all z \(\in\) C satisfying |z| = 1 and Re z \(\ne\) 1, is :

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

If \(\alpha ,\beta \in C\) are the distinct roots of the equation
x2 - x + 1 = 0, then \({\alpha ^{101}} + {\beta ^{107}}\) is equal to :

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

Let z\(\in\)C, the set of complex numbers. Then the equation, 2|z + 3i| \(-\) |z \(-\) i| = 0 represents :

Let \(\omega\) be a complex number such that 2\(\omega\) + 1 = z where z = \(\sqrt {-3}\). If
$$\left| {\matrix{
1 & 1 & 1 \cr
1 & { - {\omega ^2} - 1} & {{\omega ^2}} \cr
1 & {{\omega ^2}} & {{\omega ^7}} \cr

} } \right...

The point represented by 2 + i in the Argand plane moves 1 unit eastwards, then 2 units northwards and finally from there \(2\sqrt 2\) units in the south-westwardsdirection. Then its new position in the Argand plane is at the point represe...

A value of \(\theta \,\) for which \({{2 + 3i\sin \theta \,} \over {1 - 2i\,\,\sin \,\theta \,}}\) is purely imaginary, is :

A complex number z is said to be unimodular if \(\,\left| z \right| = 1\). Suppose \({z_1}\) and \({z_2}\) are complex numbers such that \({{{z_1} - 2{z_2}} \over {2 - {z_1}\overline {{z_2}} }}\) is unimodular and \({z_2}\) is not unimodula...

If z is a complex number such that \(\,\left| z \right| \ge 2\,\), then the minimum value of \(\,\,\left| {z + {1 \over 2}} \right|\) :

If z is a complex number of unit modulus and argument \(\theta\), then arg \(\left( {{{1 + z} \over {1 + \overline z }}} \right)\) equals :

If \(z \ne 1\) and \(\,{{{z^2}} \over {z - 1}}\,\) is real, then the point represented by the complex number z lies :

If \(\omega ( \ne 1)\) is a cube root of unity, and \({(1 + \omega )^7} = A + B\omega \,\). Then \((A,B)\) equals :

Let \(\alpha \,,\beta\) be real and z be a complex number. If \({z^2} + \alpha z + \beta = 0\) has two distinct roots on the line Re z = 1, then it is necessary that :

The number of complex numbers z such that \(\left| {z - 1} \right| = \left| {z + 1} \right| = \left| {z - i} \right|\) equals :

If \(\,\left| {z - {4 \over z}} \right| = 2,\) then the maximum value of \(\,\left| z \right|\) is equal to :

The conjugate of a complex number is \({1 \over {i - 1}}\) then that complex number is :

If \(\,\left| {z + 4} \right|\,\, \le \,\,3\,\), then the maximum value of \(\left| {z + 1} \right|\) is :

If \({z^2} + z + 1 = 0\), where z is complex number, then value of $${\left( {z + {1 \over z}} \right)^2} + {\left( {{z^2} + {1 \over {{z^2}}}} \right)^2} + {\left( {{z^3} + {1 \over {{z^3}}}} \right)^2} + .......... + {\left( {{z^6} + {1 \...

The value of \(\sum\limits_{k = 1}^{10} {\left( {\sin {{2k\pi } \over {11}} + i\,\,\cos {{2k\pi } \over {11}}} \right)}\) is :

If \(\,\omega = {z \over {z - {1 \over 3}i}}\,\) and \(\left| \omega \right| = 1\), then \(z\) lies on :

If the cube roots of unity are 1, \(\omega \,,\,{\omega ^2}\) then the roots of the equation \({(x - 1)^3}\) + 8 = 0, are :

If \({z_1}\) and \({z_2}\) are two non-zero complex numbers such that \(\,\left| {{z_1} + {z_2}} \right| = \left| {{z_1}} \right| + \left| {{z_2}} \right|\), then arg \({z_1}\) - arg \({z_2}\) is equal to :

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

If \(\,\left| {{z^2} - 1} \right| = {\left| z \right|^2} + 1\), then z lies on :

Let z and w be complex numbers such that \(\overline z + i\overline w = 0\) and arg zw = \(\pi\). Then arg z equals :