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Mathematics
Argument of a complex number
If (1+i)(1+2i)…(1+ni) = x+iy, then prove that
${{x}^{2}}+{{y}^{2}}=2\cdot 5\cdot 10\ldots \left( 1+{{n}^{2}} \right)$

Mathematics
Argument of a complex number
For any integer $n$, $\arg \left( {\dfrac{{{{\left( {\sqrt 3 + i} \right)}^{4n + 1}}}}{{{{\left( {1 - i\sqrt 3 } \right)}^{4n}}}}} \right)$ equals:
$\left( 1 \right)\dfrac{\pi }{3}$
$\left( 2 \right)\dfrac{\pi }{6}$
$\left( 3 \right)\dfrac{{2\pi }}{3}$
$\left( 4 \right)\dfrac{{5\pi }}{6}$

Mathematics
Argument of a complex number
If $\arg z<0$, then $\arg \left( -z \right)-\arg z=$
(A) $\pi$
(B) $-\pi$
(C) $\dfrac{\pi }{2}$
(D) $-\dfrac{\pi }{2}$
Mathematics
Argument of a complex number
If $\arg \left( {z - a} \right) = \dfrac{{pi}}{4}$, where a belongs to R, then the locus of z belongs to c is a
A) Hyperloop
B) Parabola
C) Ellipse
D) Straight line
Mathematics
Argument of a complex number
If $- \pi < \arg \left( z \right) < - \dfrac{\pi }{2}$ then $\arg \left( {\bar z} \right) - \arg \left( {\overline { - z} } \right) =$
(1) $\pi$
(2) $- \pi$
(3) $\dfrac{\pi }{2}$
(4) $\dfrac{{ - \pi }}{2}$
Mathematics
Argument of a complex number
If $\arg z = \theta$ , then $\arg \overline z =$
A) $\theta - \pi$
B) $\pi - \theta$
C) $\theta$
D) $- \theta$

Mathematics
Argument of a complex number
Let z and $w$ be complex numbers such that $\overline z + i\overline w = 0$ and $\arg zw = \pi$, then $\arg z =$

Mathematics
Argument of a complex number
The principal argument of $\dfrac{i-3}{i-1}$ is:
$\left( a \right)tan^{-1}\dfrac{1}{2}$
$\left( b \right) tan^{-1}\dfrac{3}{2}$
$\left( c \right) tan^{-1}\dfrac{5}{2}$
$\left( d \right) tan^{-1}\dfrac{7}{2}$
Mathematics
Argument of a complex number
The principal argument of $z=-3+3i$ is:
(a) $\dfrac{\pi }{4}$
(b) $-\dfrac{\pi }{4}$
(c) $\dfrac{3\pi }{4}$
(d) $-\dfrac{3\pi }{4}$
Mathematics
Argument of a complex number
The locus of the complex number $z$such that $\arg \left( {\dfrac{{z - 2}}{{z + 2}}} \right) = \dfrac{\pi }{3}$ is:
$A)\,A\,\,circle \\ B)\,A\,\,straight\,\,line \\ C)\,A\,\,parabola \\ D)\,An\,\,ellipse \\$
Mathematics
Argument of a complex number
Find the modulus, argument and the principal argument of the complex number
${{\left( \tan 1-i \right)}^{2}}$. 
A.$\text{Modulus}={{\sec }^{2}}1,\arg \left( z \right)=2n\pi +\left( 2-\pi \right),\text{pricipal }\arg \left( z \right)=\left( 2-\pi \right)$ 
B. $\text{Modulus}={{\operatorname{cosec}}^{2}}1,\arg \left( z \right)=2n\pi -\left( 2-\pi \right),\text{pricipal }\arg \left( z \right)=\left( -2-\pi \right)$
C. $\text{Modulus}={{\sec }^{2}}1,\arg \left( z \right)=2n\pi -\left( 2-\pi \right),\text{pricipal }\arg \left( z \right)=-\left( 2-\pi \right)$
D. $\text{Modulus}=\text{cose}{{\text{c}}^{2}}1,\arg \left( z \right)=2n\pi +\left( 2-\pi \right),\text{pricipal }\arg \left( z \right)=\left( 2-\pi \right)$

Mathematics
Argument of a complex number
Find the modulus and amplitude of the given complex number : $\sqrt 3 - i$
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