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Mathematics
Algebra of complex number
The locus represented by $\left| z-1 \right|=\left| z+i \right|$ is
A) circle of radius 1 unit
B) An ellipse with foci at (1,0) and (0,1)
C) A straight line through the origin
D) A circle the line joining (1,0) and (0,1) as diameter
Mathematics
Algebra of complex number
What does ${i^4}$ equal?
Mathematics
Algebra of complex number
What is \[{\sin ^6}\theta \] in terms of non-exponential trigonometric function?
Mathematics
Algebra of complex number
let \[{z_1}\] and \[{z_2}\] be two complex numbers such that \[{z_1} + {z_2}\] and \[{z_1}{z_2}\] both are real, then
(1) \[{z_1} = - {z_2}\]
(2) \[{z_1} = {\text{bar }}{z_2}\]
(3) \[{z_1} = - {\text{bar }}{z_2}\]
(4) \[{z_1} = {z_2}\]
Mathematics
Algebra of complex number
How do you simplify $\dfrac{{ - 3 + 2i}}{{2 - 5i}}?$
Mathematics
Algebra of complex number
If a expression \[\dfrac{z-\alpha }{z+\alpha }\left( \alpha \in R \right)\] is a purely imaginary number and \[\left| z \right|=2\] then the value of \[\alpha \] is equal to
\[\begin{align}
  & A)1 \\
 & B)2 \\
 & C)\sqrt{2} \\
 & D)\dfrac{1}{2} \\
\end{align}\]

Mathematics
Algebra of complex number
If \[z = x - iy\] and \[{z^{\dfrac{1}{3}}} = p + iq\], then \[\dfrac{{\dfrac{x}{p} + \dfrac{y}{q}}}{{{p^2} + {q^2}}}\] is equal to
(A) \[1\]
(B) \[ - 1\]
(C) \[2\]
(D) \[ - 2\]
Mathematics
Algebra of complex number
The multiplicative inverse of \[\dfrac{{3 + 4i}}{{4 - 5i}}\] is
\[\left( 1 \right)\] \[\left( {\dfrac{{ - 8}}{{25}},\dfrac{{31}}{{25}}} \right)\]
\[\left( 2 \right)\] \[\left( {\dfrac{{ - 8}}{{25}},\dfrac{{ - 31}}{{25}}} \right)\]
\[\left( 3 \right)\] \[\left( {\dfrac{8}{{25}},\dfrac{{ - 31}}{{25}}} \right)\]
\[\left( 4 \right)\] \[\left( {\dfrac{{ - 8}}{{25}},\dfrac{{31}}{5}} \right)\]
Mathematics
Algebra of complex number
Let: ${z_1} = a + ib$, ${z_2} = c + id$. If the points represented by complex numbers \[{z_1},{z_2}\]and ${z_1} - {z_2}$are collinear, then
$A)ad + bc = 0$
$B)ad - bc = 0$
$C)ab + cd = 0$
$D)ab - cd = 0$
Mathematics
Algebra of complex number
The real value of $\theta $ for which the expression $\left( {1 + i\cos \theta } \right){\left( {1 - 2i\cos \theta } \right)^{ - 1}}$ is purely imaginary is
$1)$$n\pi $
$2)$ $n\pi \pm \dfrac{\pi }{6}$
$3)$$n\pi \pm \dfrac{{2\pi }}{3}$
$4)$$n\pi \pm \dfrac{\pi }{4}$

Mathematics
Algebra of complex number
Let \[z = \dfrac{{ - 1 + \sqrt {3i} }}{2}\], where \[i = \sqrt { - 1} \] and \[r,s \in \left\{ {1,2,3} \right\}\]. Let \[P = \left[ {\begin{array}{*{20}{c}}
  {{{\left( { - z} \right)}^r}}&{{z^{2s}}} \\
  {{z^{2s}}}&{{z^r}}
\end{array}} \right]\] and \[I\] be the identity matrix of order \[2\]. Then the total number of ordered pairs \[\left( {r,s} \right)\] for which \[{P^2} = - I\] is

Mathematics
Algebra of complex number
If \[{{\tanh }^{-1}}\left( x+iy \right)=\dfrac{1}{2}{{\tanh }^{-1}}\left( \dfrac{2x}{1+{{x}^{2}}+{{y}^{2}}} \right)+\dfrac{i}{2}{{\tan }^{-1}}\left( \dfrac{2y}{1-{{x}^{2}}-{{y}^{2}}} \right);x,y\in R\] then \[{{\tanh }^{-1}}\left( iy \right)\] is
1)\[{{\tanh }^{-1}}y\]
2) \[-i{{\tanh }^{-1}}y\]
3) \[i{{\tan }^{-1}}y\]
4) \[-{{\tan }^{-1}}y\]

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