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question_answer1) For all complex numbers \[{{z}_{1}},{{z}_{2}}\] satisfying \[|{{z}_{1}}|\,\,=12\] and \[|{{z}_{2}}-3-4i|\,\,=5\], find the minimum value of \[|{{z}_{1}}-{{z}_{2}}|\].
question_answer2) If the complex number z satisfies the condition\[\left| \text{ }z\text{ } \right|\,\underline{>\,\,}3\], then find the least value of\[\left| \text{ }z+\left( 1/z \right)\text{ } \right|\].
question_answer3) If \[z={{\left( \frac{1+i\sqrt{3}}{1+i} \right)}^{25}}\] and \[\arg (z)=\frac{\pi }{k}\] then find k.
question_answer4) The locus of \[z=x+iy\] satisfying \[\left| \frac{z-i}{z+i} \right|=2\] is a circle then find radius of circle.
question_answer5) If one of the vertices of the square circumscribing the circle \[\left| z-1 \right|=\sqrt{2}\] is \[2+\sqrt{3i}\] and centroid of the triangle formed by other vertices is \[k-\frac{\sqrt{3}}{3}\,i\] then find k.
question_answer6) If \[\frac{2{{z}_{1}}}{3{{z}_{2}}}\] is a purely imaginary number, then \[\left| \frac{{{z}_{1}}-{{z}_{2}}}{{{z}_{1}}+{{z}_{2}}} \right|=\]
question_answer7) If\[{{\left( \sqrt{8}+i \right)}^{50}}={{3}^{49}}\left( a+ib \right)\], then find value of\[{{a}^{2}}+{{b}^{2}}\].
question_answer8) If \[\left| {{z}_{1}} \right|=\left| {{z}_{2}} \right|=\left| {{z}_{3}} \right|=1\] and \[{{z}_{1}}+{{z}_{2}}+{{z}_{3}}=0\], If the area of the triangle whose vertices are \[{{z}_{1}},{{z}_{2}},\,{{z}_{3}}\] is\[\sqrt{3}\,k\], then find k.
question_answer9) If \[\left| {{z}_{1}} \right|=1,\,\left| {{z}_{2}} \right|=2,\,\left| {{z}_{3}} \right|=3\] and \[\left| 9{{z}_{1}}{{z}_{2}}+4{{z}_{1}}{{z}_{3}}+{{z}_{2}}{{z}_{3}} \right|=12\], then the value of \[\left| {{z}_{1}}+{{z}_{2}}+{{z}_{3}} \right|\] is equal to :
question_answer10) If Re \[\frac{{{\left( 1+i \right)}^{2}}}{3-i}\]is k then find \[\left[ k \right]\] (where \[\left[ . \right]\] denotes greatest integer function).
question_answer11) If m & M denotes the minimum and maximum value of\[\left| 2z+1 \right|\] where\[\left| z-2i \right|\,\,\underline{<}\,\,1\], then find\[{{\left( m+M \right)}^{2}}\].
question_answer12) If \[\left| \frac{{{z}_{1}}-3{{z}_{2}}}{3-{{z}_{1}}{{\overline{z}}_{2}}} \right|=1\] and \[\left| {{z}_{2}} \right|\ne 1\], then find \[\left| {{z}_{1}} \right|\].
question_answer13) If\[{{x}^{2}}-x+1=0\], then find value of\[\sum\limits_{n=1}^{5}{{{\left( {{x}^{n}}+\frac{1}{{{x}^{n}}} \right)}^{2}}}\].
question_answer14) If \[z=x-iy\], and \[{{z}^{1/3}}=p+iq\], then find value of \[\frac{\left( \frac{x}{p}+\frac{y}{q} \right)}{\left( {{p}^{2}}+{{q}^{2}} \right)}\].
question_answer15) If\[z=\frac{\pi }{4}{{\left( 1+i \right)}^{4}}\left( \frac{1-\sqrt{\pi }i}{\sqrt{\pi }+i}+\frac{\sqrt{\pi }-i}{1+\sqrt{\pi }i} \right)\], then find\[\left( \frac{\left| z \right|}{Amp\left( z \right)} \right)\].
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