JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank

done Square root, Representation and Logarithm of complex numbers Total Questions - 24

• 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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