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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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