-
question_answer1)
If \[x+iy=\sqrt{\frac{a+ib}{c+id}},\]then \[{{({{x}^{2}}+{{y}^{2}})}^{2}}=\] [IIT 1979; RPET 1997; Karnataka CET 1999]
A)
\[\frac{{{a}^{2}}+{{b}^{2}}}{{{c}^{2}}+{{d}^{2}}}\] done
clear
B)
\[\frac{a+b}{c+d}\] done
clear
C)
\[\frac{{{c}^{2}}+{{d}^{2}}}{{{a}^{2}}+{{b}^{2}}}\] done
clear
D)
\[{{\left( \frac{{{a}^{2}}+{{b}^{2}}}{{{c}^{2}}+{{d}^{2}}} \right)}^{2}}\] done
clear
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question_answer2)
\[\sqrt{-8-6i}=\] [Roorkee 1979; RPET 1992]
A)
\[1\pm 3i\] done
clear
B)
\[\pm (1-3i)\] done
clear
C)
\[\pm (1+3i)\] done
clear
D)
\[\pm (3-i)\] done
clear
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question_answer3)
If \[{{(-7-24i)}^{1/2}}=x-iy,\] then \[{{x}^{2}}+{{y}^{2}}=\] [RPET 1989]
A)
15 done
clear
B)
25 done
clear
C)
- 25 done
clear
D)
None of these done
clear
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question_answer4)
If \[\sqrt{x+iy}=\pm (a+ib),\] then \[\sqrt{-x-iy}\] is equal to
A)
\[\pm (b+ia)\] done
clear
B)
\[\pm (a-ib)\] done
clear
C)
\[\pm (b-ia)\] done
clear
D)
None of these done
clear
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question_answer5)
The square root of 3 - 4i is [RPET 1999]
A)
\[\pm (2+i)\] done
clear
B)
\[\pm (2-i)\] done
clear
C)
\[\pm (1-2i)\] done
clear
D)
\[\pm (1+2i)\] done
clear
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question_answer6)
If \[\sqrt{a+ib}=x+iy\], then possible value of \[\sqrt{a-ib}\]is [Kerala (Engg.) 2002]
A)
\[{{x}^{2}}+{{y}^{2}}\] done
clear
B)
\[\sqrt{{{x}^{2}}+{{y}^{2}}}\] done
clear
C)
\[x+iy\] done
clear
D)
\[x-iy\] done
clear
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question_answer7)
The number of non-zero integral solutions of the equation \[|1-i{{|}^{x}}={{2}^{x}}\] is
A)
Infinite done
clear
B)
1 done
clear
C)
2 done
clear
D)
None of these done
clear
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question_answer8)
\[\frac{1+7i}{{{(2-i)}^{2}}}=\] [Roorkee 1981]
A)
\[\sqrt{2}\left( \cos \frac{3\pi }{4}+i\sin \frac{3\pi }{4} \right)\] done
clear
B)
\[\sqrt{2}\left( \cos \frac{\pi }{4}+i\sin \frac{\pi }{4} \right)\] done
clear
C)
\[\left( \cos \frac{3\pi }{4}+i\sin \frac{3\pi }{4} \right)\] done
clear
D)
None of these done
clear
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question_answer9)
If \[z=r{{e}^{i\theta }},\]then \[|{{e}^{iz}}|\]= [Kerala (Engg.) 2005]
A)
\[{{e}^{r\sin \theta }}\] done
clear
B)
\[{{e}^{-r\sin \theta }}\] done
clear
C)
\[{{e}^{-r\cos \theta }}\] done
clear
D)
\[{{e}^{r\cos \theta }}\] done
clear
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question_answer10)
\[\frac{1-i}{1+i}\]is equal to [RPET 1984]
A)
\[\cos \frac{\pi }{2}+i\sin \frac{\pi }{2}\] done
clear
B)
\[\cos \frac{\pi }{2}-i\sin \frac{\pi }{2}\] done
clear
C)
\[\sin \frac{\pi }{2}+i\cos \frac{\pi }{2}\] done
clear
D)
None of these done
clear
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question_answer11)
If \[-1+\sqrt{-3}=r{{e}^{i\theta }},\]then \[\theta \] is equal to [RPET 1989; MP PET 1999]
A)
\[\frac{\pi }{3}\] done
clear
B)
\[-\frac{\pi }{3}\] done
clear
C)
\[\frac{2\pi }{3}\] done
clear
D)
\[-\frac{2\pi }{3}\] done
clear
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question_answer12)
If \[y=\cos \theta +i\sin \theta \],then the value of \[y+\frac{1}{y}\] is [RPET 1995]
A)
\[2\cos \theta \] done
clear
B)
\[2\sin \theta \] done
clear
C)
\[2\text{cosec}\theta \] done
clear
D)
\[2\tan \theta \] done
clear
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question_answer13)
The value of \[{{(-i)}^{1/3}}\] is [Roorkee 1995]
A)
\[\frac{1+\sqrt{3}i}{2}\] done
clear
B)
\[\frac{1-\sqrt{3}i}{2}\] done
clear
C)
\[\frac{-\sqrt{3}-i}{2}\] done
clear
D)
\[\frac{\sqrt{3}-i}{2}\] done
clear
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question_answer14)
If \[{{(1+i\sqrt{3})}^{9}}=a+ib,\] then \[b\] is equal to [RPET 1995]
A)
1 done
clear
B)
256 done
clear
C)
0 done
clear
D)
\[{{9}^{3}}\] done
clear
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question_answer15)
Real part of \[{{e}^{{{e}^{i\theta }}}}\]is [RPET 1995]
A)
\[{{e}^{\cos \theta }}[\cos (\sin \theta )]\] done
clear
B)
\[{{e}^{\cos \theta }}[\cos (\cos \theta )]\] done
clear
C)
\[{{e}^{\sin \theta }}[\sin (\cos \theta )]\] done
clear
D)
\[{{e}^{\sin \theta }}[\sin (\sin \theta )]\] done
clear
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question_answer16)
The amplitude of \[{{e}^{{{e}^{-i\theta }}}}\]is equal to [RPET 1997]
A)
\[\sin \theta \] done
clear
B)
\[-\sin \theta \] done
clear
C)
\[{{e}^{\cos \theta }}\] done
clear
D)
\[{{e}^{\sin \theta }}\] done
clear
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question_answer17)
If \[z=\frac{1+i\sqrt{3}}{\sqrt{3}+i},\] then\[{{(\bar{z})}^{100}}\] lies in [AMU 1999]
A)
I quadrant done
clear
B)
II quadrant done
clear
C)
III quadrant done
clear
D)
IV quadrant done
clear
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question_answer18)
If \[x+\frac{1}{x}=\sqrt{3},\] then x = [RPET 2002]
A)
\[\cos \frac{\pi }{3}+i\,\sin \frac{\pi }{3}\] done
clear
B)
\[\cos \frac{\pi }{2}+i\,\sin \frac{\pi }{2}\] done
clear
C)
\[\sin \frac{\pi }{6}+i\,\cos \frac{\pi }{6}\] done
clear
D)
\[\cos \frac{\pi }{6}+i\,\sin \frac{\pi }{6}\] done
clear
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question_answer19)
\[{{(-1+i\sqrt{3})}^{20}}\] is equal to [RPET 2003]
A)
\[{{2}^{20}}{{(-1+i\sqrt{3})}^{20}}\] done
clear
B)
\[{{2}^{20}}{{(1-i\sqrt{3})}^{20}}\] done
clear
C)
\[{{2}^{20}}{{(-1-i\sqrt{3})}^{20}}\] done
clear
D)
None of these done
clear
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question_answer20)
The imaginary part of \[{{\tan }^{-1}}\left( \frac{5i}{3} \right)\] is [RPET 1997]
A)
0 done
clear
B)
\[\infty \] done
clear
C)
\[\log 2\] done
clear
D)
\[\log 4\] done
clear
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question_answer21)
The real part of \[{{(1-i)}^{-i}}\]is [RPET 1999]
A)
\[{{e}^{-\pi /4}}\cos \left( \frac{1}{2}\log 2 \right)\] done
clear
B)
\[-{{e}^{-\pi /4}}\sin \left( \frac{1}{2}\log 2 \right)\] done
clear
C)
\[{{e}^{\pi /4}}\cos \left( \frac{1}{2}\log 2 \right)\] done
clear
D)
\[{{e}^{-\pi /4}}\sin \left( \frac{1}{2}\log 2 \right)\] done
clear
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question_answer22)
\[i\log \left( \frac{x-i}{x+i} \right)\] is equal to [RPET 2000]
A)
\[\pi +2{{\tan }^{-1}}x\] done
clear
B)
\[\pi -2{{\tan }^{-1}}x\] done
clear
C)
\[-\pi +2{{\tan }^{-1}}x\] done
clear
D)
\[-\pi -2{{\tan }^{-1}}x\] done
clear
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question_answer23)
If \[{{e}^{i\theta }}=\cos \theta +i\sin \theta \], then in \[\Delta ABC\] value of \[{{e}^{iA}}.{{e}^{iB}}.{{e}^{iC}}\] is [AMU 2005]
A)
-i done
clear
B)
1 done
clear
C)
-1 done
clear
D)
None of these done
clear
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question_answer24)
If \[z=\frac{7-i}{3-4i}\] then \[{{z}^{14}}=\] [Kerala (Engg.) 2005]
A)
\[{{2}^{7}}\] done
clear
B)
\[{{2}^{7}}i\] done
clear
C)
\[{{2}^{14}}i\] done
clear
D)
\[-{{2}^{7}}i\] done
clear
E)
\[-{{2}^{14}}\] done
clear
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