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Mathematics
Properties of conjugate of a complex number
If $z=4-3\sqrt{2}i$ then the value of $z\overline{z}$
Mathematics
Properties of conjugate of a complex number
If $$\dfrac{\left(w-\overline wz\right)}{1-z}\;\mathrm{is}\;\mathrm{purely}\;\mathrm{real}\;\mathrm{where}\;w=\alpha+i\beta,\;\beta\neq0\;and\;z\neq1$$, then set of values of z is-
$$A.\;z:\left|z\right|=1\\B.\;z:z=\overline z\\C.\;z:z\neq1\\D.\;z:\left|z\right|=1,\;z\neq1$$
Mathematics
Properties of conjugate of a complex number
Solve the following and find the value of
$\log \left( a+ib \right)=\left( \text{where}a>0,b>0 \right)$
a) $\log \sqrt{{{a}^{2}}+{{b}^{2}}}$
b) $\dfrac{1}{2}\log \left( {{a}^{2}}+{{b}^{2}} \right)+i{{\tan }^{-1}}\left( \dfrac{b}{a} \right)$
c) $\dfrac{1}{2}\log \left( {{a}^{2}}+{{b}^{2}} \right)-i{{\tan }^{-1}}\left( \dfrac{b}{a} \right)$
d) $\dfrac{1}{2}\log \left( {{a}^{2}}+{{b}^{2}} \right)+i{{\tan }^{-1}}\left( \dfrac{a}{b} \right)$

Mathematics
Properties of conjugate of a complex number
Find the square root of the complex number – 21 – 20i.
(a) $\pm \left( 2+5i \right)$
(b) $\pm \left( 2-5i \right)$
(c) $\pm \left( 4+3i \right)$
(d) $\pm \left( 4-3i \right)$
Mathematics
Properties of conjugate of a complex number
Evaluate the given expression in cartesian form $\left( a+ib \right)$.
$\dfrac{i}{1+i}$
Mathematics
Properties of conjugate of a complex number
If w is the complex cube root of unity, then is \[{{\left( 2+5w+2{{w}^{2}} \right)}^{6}}\] = \[{{\left( 2+2w+5{{w}^{2}} \right)}^{6}}\] = 729?
(a) True
(b) False
Mathematics
Properties of conjugate of a complex number
Prove the following; If $x - iy = \sqrt {\dfrac{{a - ib}}{{c - id}}} $ prove that ${\left( {{x^2} + {y^2}} \right)^2} = \dfrac{{{a^2} + {b^2}}}{{{c^2} + {d^2}}}$.
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