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

done De Moivre's theorem and Roots of unity Total Questions - 104

• question_answer1)
  \[\sqrt{i}=\]
  A)
  \[\frac{1\pm i}{\sqrt{2}}\] done clear
  B)
  \[\pm \frac{1-i}{\sqrt{2}}\] done clear
  C)
  \[\pm \frac{1+i}{\sqrt{2}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer2)
  If  \[{{x}_{r}}=\cos \left( \frac{\pi }{{{2}^{r}}} \right)+i\sin \left( \frac{\pi }{{{2}^{r}}} \right)\], then\[{{x}_{1}}.{{x}_{2}}......\infty \]is [RPET 1990, 2000; BIT Mesra 1996; Karnataka CET 2000]
  A)
  \[-3\] done clear
  B)
  \[-2\] done clear
  C)
  \[-1\] done clear
  D)
    0 done clear
  View Solution play_arrow

• question_answer3)
  \[\frac{{{(\cos \theta +i\sin \theta )}^{4}}}{{{(\sin \theta +i\cos \theta )}^{5}}}\] is equal to   [MNR 1985; UPSEAT 2000]
  A)
  \[\cos \theta -i\sin \theta \] done clear
  B)
  \[\cos 9\theta -i\sin 9\theta \] done clear
  C)
  \[\sin \theta -i\cos \theta \] done clear
  D)
  \[\sin 9\theta -i\cos 9\theta \] done clear
  View Solution play_arrow

• question_answer4)
  If  \[z={{\left( \frac{\sqrt{3}}{2}+\frac{i}{2} \right)}^{5}}+{{\left( \frac{\sqrt{3}}{2}-\frac{i}{2} \right)}^{5}}\], then [MP PET 1997]
  A)
  \[\operatorname{Re}(z)=0\] done clear
  B)
  \[\operatorname{Im}(z)=0\] done clear
  C)
  \[\operatorname{Re}(z)>0,\operatorname{Im}(z)>0\] done clear
  D)
  \[\operatorname{Re}(z)>0,\operatorname{Im}(z)<0\] done clear
  View Solution play_arrow

• question_answer5)
  The roots of \[{{(2-2i)}^{1/3}}\] are
  A)
   \[\sqrt{2}\left( \cos \frac{\pi }{12}-i\sin \frac{\pi }{12} \right),\sqrt{2}\left( -\sin \frac{\pi }{12}+i\cos \frac{\pi }{12} \right),-1-i\] done clear
  B)
   \[\sqrt{2}\left( \cos \frac{\pi }{12}+i\sin \frac{\pi }{12} \right),\sqrt{2}\left( -\sin \frac{\pi }{12}-i\cos \frac{\pi }{12} \right)\,,\,1+i\] done clear
  C)
  \[1+\sqrt{2}i,-1-i,-2-2i\] done clear
  D)
  None of the above done clear
  View Solution play_arrow

• question_answer6)
  The value of \[\frac{4(\cos {{75}^{o}}+i\sin {{75}^{o}})}{0.4(\cos {{30}^{o}}+i\sin {{30}^{o}})}\] is
  A)
  \[\frac{\sqrt{2}}{10}(1+i)\] done clear
  B)
  \[\frac{\sqrt{2}}{10}(1-i)\] done clear
  C)
  \[\frac{10}{\sqrt{2}}(1-i)\] done clear
  D)
  \[\frac{10}{\sqrt{2}}(1+i)\] done clear
  View Solution play_arrow

• question_answer7)
  The following in the form of \[A+iB\] \[{{(\cos 2\theta +i\sin 2\theta )}^{-5}}\] \[{{(\cos 3\theta -i\sin 3\theta )}^{6}}\]\[{{(\sin \theta -i\cos \theta )}^{3}}\] in the form of \[A+iB\] is [MNR 1991]
  A)
  \[(\cos 25\theta +i\sin 25\theta )\] done clear
  B)
  \[i(\cos 25\theta +i\sin 25\theta )\] done clear
  C)
  \[i\,(\cos 25\theta -i\sin 25\theta )\] done clear
  D)
  \[(\cos 25\theta -i\sin 25\theta )\] done clear
  View Solution play_arrow

• question_answer8)
  If \[a=\sqrt{2i}\] then which of the following is correct [Roorkee 1989]
  A)
  \[a=1+i\] done clear
  B)
  \[a=1-i\] done clear
  C)
  \[a=-(\sqrt{2})i\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer9)
  If \[(\cos \theta +i\sin \theta )(\cos 2\theta +i\sin 2\theta )........\] \[(\cos n\theta +i\sin n\theta )=1\], then the value of \[\theta \] is[Karnataka CET 1992; Kurukshetra CEE 2002]
  A)
  \[4m\pi \] done clear
  B)
  \[\frac{2m\pi }{n(n+1)}\] done clear
  C)
  \[\frac{4m\pi }{n(n+1)}\] done clear
  D)
  \[\frac{m\pi }{n(n+1)}\] done clear
  View Solution play_arrow

• question_answer10)
  \[{{\left( \frac{1+\cos \varphi +i\sin \varphi }{1+\cos \varphi -i\sin \varphi } \right)}^{n}}=\]
  A)
  \[\cos n\varphi -i\sin n\varphi \] done clear
  B)
  \[\cos n\varphi +i\sin n\varphi \] done clear
  C)
  \[\sin n\varphi +i\cos n\varphi \] done clear
  D)
  \[\sin n\varphi -i\cos n\varphi \] done clear
  View Solution play_arrow

• question_answer11)
  If  \[{{\left( \frac{1+\cos \theta +i\sin \theta }{i+\sin \theta +i\cos \theta } \right)}^{4}}=\cos n\theta +i\sin n\theta \], then \[n\] is equal to [EAMCET 1986]
  A)
  1 done clear
  B)
  2 done clear
  C)
  3 done clear
  D)
  4 done clear
  View Solution play_arrow

• question_answer12)
  The value of expression \[\left( \cos \frac{\pi }{2}+i\sin \frac{\pi }{2} \right)\] \[\,\left( \cos \frac{\pi }{{{2}^{2}}}+i\sin \frac{\pi }{{{2}^{2}}} \right)\]........to \[\infty \] is    [Kurukshetra CEE 1998]
  A)
  \[-1\] done clear
  B)
   \[1\] done clear
  C)
  0 done clear
  D)
  2 done clear
  View Solution play_arrow

• question_answer13)
  \[{{\left( \frac{\cos \theta +i\sin \theta }{\sin \theta +i\cos \theta } \right)}^{4}}\]equals [RPET 1996]
  A)
  \[\sin 8\theta -i\cos 8\theta \] done clear
  B)
  \[\cos 8\theta -i\sin 8\theta \] done clear
  C)
  \[\sin 8\theta +i\cos 8\theta \] done clear
  D)
  \[\cos 8\theta +i\sin 8\theta \] done clear
  View Solution play_arrow

• question_answer14)
  If \[\sin \alpha +\sin \beta +\sin \gamma =0=\]\[\cos \alpha +\cos \beta +\cos \gamma ,\] then the value of  \[{{\sin }^{2}}\alpha +{{\sin }^{2}}\beta +{{\sin }^{2}}\gamma \] is    [RPET 1999]
  A)
  2/3 done clear
  B)
  3/2 done clear
  C)
  1/2 done clear
  D)
  1 done clear
  View Solution play_arrow

• question_answer15)
  If \[\cos \alpha +\cos \beta +\cos \gamma =0=\]\[\sin \alpha +\sin \beta +\sin \gamma \] then \[\cos 2\alpha +\cos 2\beta +\cos 2\gamma \] equals [RPET 2000]
  A)
  \[2\cos (\alpha +\beta +\gamma )\] done clear
  B)
  \[\cos 2(\alpha +\beta +\gamma )\] done clear
  C)
  0 done clear
  D)
  1 done clear
  View Solution play_arrow

• question_answer16)
  \[{{(-\sqrt{3}+i)}^{53}}\] where \[{{i}^{2}}=-1\] is equal to [AMU 2000]
  A)
  \[{{2}^{53}}(\sqrt{3}+2i)\] done clear
  B)
    \[{{2}^{52}}(\sqrt{3}-i)\] done clear
  C)
  \[{{2}^{53}}\,\left( \frac{\sqrt{3}}{2}+\frac{1}{2}i \right)\] done clear
  D)
  \[{{2}^{53}}(\sqrt{3}-i)\] done clear
  View Solution play_arrow

• question_answer17)
  The value of \[{{\left[ \frac{1-\cos \frac{\pi }{10}+i\sin \frac{\pi }{10}}{1-\cos \frac{\pi }{10}-i\sin \frac{\pi }{10}} \right]}^{10}}=\] [Karnataka CET 2001]
  A)
  0 done clear
  B)
  - 1 done clear
  C)
  1 done clear
  D)
  2 done clear
  View Solution play_arrow

• question_answer18)
  We express \[\frac{{{(\cos 2\theta -i\sin 2\theta )}^{4}}{{(\cos 4\theta +i\sin 4\theta )}^{-5}}}{{{(\cos 3\theta +i\sin 3\theta )}^{-2}}{{(\cos 3\theta -i\sin 3\theta )}^{-9}}}\]  in the form of \[x+iy\], we get [Karnataka CET 2001]
  A)
  \[\cos 49\theta -i\,\sin 49\theta \] done clear
  B)
  \[\cos 23\theta -i\,\sin 23\theta \] done clear
  C)
  \[\cos 49\theta +i\,\sin 49\theta \] done clear
  D)
  \[\cos 21\theta +i\,\sin 21\theta \] done clear
  View Solution play_arrow

• question_answer19)
  \[{{(\sin \theta +i\,\cos \theta )}^{n}}\,\]is equal to [RPET 2001]
  A)
  \[\cos n\theta +i\,\sin n\theta \] done clear
  B)
  \[\sin n\theta +i\,\cos n\theta \] done clear
  C)
  \[\cos n\left( \frac{\pi }{2}-\theta  \right)+i\,\sin n\left( \frac{\pi }{2}-\theta  \right)\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer20)
  The value of \[\frac{(\cos \alpha +i\,\sin \alpha )\,(\cos \beta +i\,\sin \beta )}{(\cos \gamma +i\,\sin \gamma )\,(\cos \,\delta +i\,\sin \delta )}\] is [RPET 2001]
  A)
  \[\cos (\alpha +\beta -\gamma -\delta )-i\,\sin (\alpha +\beta -\gamma -\delta )\] done clear
  B)
  \[\cos (\alpha +\beta -\gamma -\delta )+i\,\sin (\alpha +\beta -\gamma -\delta )\] done clear
  C)
  \[\sin (\alpha +\beta -\gamma -\delta )-i\,\cos (\alpha +\beta -\gamma -\delta )\] done clear
  D)
  \[\sin (\alpha +\beta -\gamma -\delta )+i\,\cos (\alpha +\beta -\gamma -\delta )\] done clear
  View Solution play_arrow

• question_answer21)
  \[{{\left[ \frac{1+\cos (\pi /8)+i\,\sin (\pi /8)}{1+\cos (\pi /8)-i\,\sin (\pi /8)} \right]}^{8}}\] is equal to [RPET 2001]
  A)
  - 1 done clear
  B)
  0 done clear
  C)
  1 done clear
  D)
  2 done clear
  View Solution play_arrow

• question_answer22)
   If \[{{x}_{n}}=\cos \,\left( \frac{\pi }{{{4}^{n}}} \right)+i\,\sin \,\left( \frac{\pi }{{{4}^{n}}} \right)\,,\] then \[{{x}_{1}}.\,{{x}_{2}}.\,{{x}_{3}}....\infty =\] [EAMCET 2002]
  A)
  \[\frac{1+i\sqrt{3}}{2}\] done clear
  B)
  \[\frac{-1+i\sqrt{3}}{2}\] done clear
  C)
  \[\frac{1-i\sqrt{3}}{2}\] done clear
  D)
  \[\frac{-1-i\sqrt{3}}{2}\] done clear
  View Solution play_arrow

• question_answer23)
  \[\frac{{{(\cos \alpha +i\,\sin \alpha )}^{4}}}{{{(\sin \beta +i\,\cos \beta )}^{5}}}=\] [RPET 2002]
  A)
  \[\cos (4\alpha +5\beta )+i\,\sin (4\alpha +5\beta )\] done clear
  B)
  \[\cos (4\alpha +5\beta )-i\,\sin (4\alpha +5\beta )\] done clear
  C)
  \[\sin (4\alpha +5\beta )-i\cos (4\alpha +5\beta )\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer24)
  The value of i1/3 is [UPSEAT 2002]
  A)
  \[\frac{\sqrt{3}\,+i}{2}\] done clear
  B)
  \[\frac{\sqrt{3}\,-i}{2}\] done clear
  C)
  \[\frac{1+i\sqrt{3}}{2}\] done clear
  D)
  \[\frac{1-i\sqrt{3}}{2}\] done clear
  View Solution play_arrow

• question_answer25)
  Given \[z={{(1+i\sqrt{3})}^{100}},\] then \[\frac{\operatorname{Re}(z)}{\operatorname{Im}(z)}\] equals [AMU 2002]
  A)
  2100 done clear
  B)
  250 done clear
  C)
  \[\frac{1}{\sqrt{3}}\] done clear
  D)
  \[\sqrt{3}\] done clear
  View Solution play_arrow

• question_answer26)
  \[{{\left( \frac{1+\sin \theta +i\,\cos \theta }{1+\sin \theta -i\,\cos \theta } \right)}^{n}}\]= [Kerala (Engg.) 2002]
  A)
  \[\cos \left( \frac{n\pi }{2}-n\theta  \right)+i\,\sin \left( \frac{n\pi }{2}-n\theta  \right)\] done clear
  B)
  \[\cos \left( \frac{n\pi }{2}+n\theta  \right)+i\,\sin \left( \frac{n\pi }{2}+n\theta  \right)\] done clear
  C)
  \[\sin \left( \frac{n\pi }{2}-n\theta  \right)+i\,\cos \left( \frac{n\pi }{2}-n\theta  \right)\] done clear
  D)
  \[\cos \,n\left( \frac{\pi }{2}+2\theta  \right)+i\,\sin \,n\left( \frac{\pi }{2}+2\theta  \right)\] done clear
  View Solution play_arrow

• question_answer27)
  If n is a positive integer, then \[{{(1+i)}^{n}}+{{(1-i)}^{n}}\] is equal to [Orissa JEE 2003]
  A)
  \[{{(\sqrt{2})}^{n-2}}\cos \left( \frac{n\pi }{4} \right)\] done clear
  B)
  \[{{(\sqrt{2})}^{n-2}}\sin \left( \frac{n\pi }{4} \right)\] done clear
  C)
  \[{{(\sqrt{2})}^{n+2}}\cos \left( \frac{n\pi }{4} \right)\] done clear
  D)
  \[{{(\sqrt{2})}^{n+2}}\sin \left( \frac{n\pi }{4} \right)\] done clear
  View Solution play_arrow

• question_answer28)
  If \[\frac{1}{x}+x=2\cos \theta ,\] then \[{{x}^{n}}+\frac{1}{{{x}^{n}}}\] is equal to [UPSEAT 2001]
  A)
  \[2\cos n\theta \] done clear
  B)
  \[2\sin n\theta \] done clear
  C)
  \[\cos n\,\theta \] done clear
  D)
  \[\sin \,n\theta \] done clear
  View Solution play_arrow

• question_answer29)
  If \[i{{z}^{4}}+1=0\], then \[z\] can take the value [UPSEAT 2004]
  A)
  \[\frac{1+i}{\sqrt{2}}\] done clear
  B)
  \[\cos \frac{\pi }{8}+i\,\sin \frac{\pi }{8}\] done clear
  C)
  \[\frac{1}{4i}\] done clear
  D)
  i done clear
  View Solution play_arrow

• question_answer30)
  The two numbers such that each one is square of the other, are [MP PET 1987]
  A)
  \[\omega ,\,{{\omega }^{3}}\] done clear
  B)
  \[-i,\,\,i\] done clear
  C)
  \[-1,\,1\] done clear
  D)
  \[\omega ,\,\,{{\omega }^{2}}\] done clear
  View Solution play_arrow

• question_answer31)
  If \[\omega \] is a cube root of unity, then \[(1+\omega -{{\omega }^{2}})\] \[(1-\omega +{{\omega }^{2}})\] =                 [MNR 1990; MP PET 1993, 2002]
  A)
  1 done clear
  B)
  0 done clear
  C)
  2 done clear
  D)
  4 done clear
  View Solution play_arrow

• question_answer32)
  \[{{(27)}^{1/3}}=\]
  A)
  3 done clear
  B)
  \[3,\,\,3i,\,3{{i}^{2}}\] done clear
  C)
  \[3,\,3\omega ,\,3{{\omega }^{2}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer33)
  If \[n\] is a positive integer not a multiple of 3, then \[1+{{\omega }^{n}}+{{\omega }^{2n}}\] = [MP PET 2004]
  A)
  3 done clear
  B)
  1 done clear
  C)
  0 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer34)
  Square of either of the two imaginary cube roots of unity will be
  A)
  Real root of unity done clear
  B)
  Other imaginary cube root of unity done clear
  C)
  Sum of two imaginary roots of unity done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer35)
  If \[\omega \] is a cube root of unity, then \[{{(1+\omega )}^{3}}-{{(1+{{\omega }^{2}})}^{3}}=\]
  A)
  0 done clear
  B)
  \[\omega \] done clear
  C)
  \[{{\omega }^{2}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer36)
  If \[\alpha \]and \[\beta \] are imaginary cube roots of unity, then \[{{\alpha }^{4}}+{{\beta }^{4}}\] + \[\frac{1}{\alpha \beta }=\] [IIT 1977]
  A)
  3 done clear
  B)
  0 done clear
  C)
  1 done clear
  D)
  2 done clear
  View Solution play_arrow

• question_answer37)
  If w is a complex cube root of unity, then \[(1-\omega )(1-{{\omega }^{2}})\] \[(1-{{\omega }^{4}})(1-{{\omega }^{8}})=\]
  A)
  0 done clear
  B)
  1 done clear
  C)
  - 1 done clear
  D)
  9 done clear
  View Solution play_arrow

• question_answer38)
  If \[\omega \] is a cube root of unity, then the value of \[{{(1-\omega +{{\omega }^{2}})}^{5}}+{{(1+\omega -{{\omega }^{2}})}^{5}}=\] [IIT 1965; MP PET 1997; RPET 1997]
  A)
  16 done clear
  B)
  32 done clear
  C)
  48 done clear
  D)
  - 32 done clear
  View Solution play_arrow

• question_answer39)
  If  \[x=a,y=b\omega ,z=c{{\omega }^{2}}\], where \[\omega \] is a complex cube root of unity, then \[\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=\] [AMU 1983]
  A)
  3 done clear
  B)
  1 done clear
  C)
  0 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer40)
  If \[\omega \] is a complex cube root of unity, then \[(x-y)(x\omega -y)\] \[(x{{\omega }^{2}}-y)=\]
  A)
  \[{{x}^{2}}+{{y}^{2}}\] done clear
  B)
  \[{{x}^{2}}-{{y}^{2}}\] done clear
  C)
  \[{{x}^{3}}-{{y}^{3}}\] done clear
  D)
  \[{{x}^{3}}+{{y}^{3}}\] done clear
  View Solution play_arrow

• question_answer41)
  If  \[\omega \] is a complex cube root of unity, then \[(1+\omega )(1+{{\omega }^{2}})\] \[(1+{{\omega }^{4}})(1+{{\omega }^{8}})...\]to \[2n\] factors = [AMU 2000]
  A)
   0 done clear
  B)
  1 done clear
  C)
  \[-1\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer42)
  The product of all the roots of \[{{\left( \cos \frac{\pi }{3}+i\sin \frac{\pi }{3} \right)}^{3/4}}\]  is  [MNR 1984; EAMCET 1985]
  A)
  \[-1\] done clear
  B)
  1 done clear
  C)
  \[\frac{3}{2}\] done clear
  D)
  \[-\frac{1}{2}\] done clear
  View Solution play_arrow

• question_answer43)
  If  \[\omega \] is a cube root of unity, then a root of the equation \[\left| \begin{matrix}    x+1 & \omega  & {{\omega }^{2}}  \\    \omega  & x+{{\omega }^{2}} & 1  \\    {{\omega }^{2}} & 1 & x+\omega   \\ \end{matrix} \right|=0\] is   [MNR 1990; MP PET 1999]
  A)
  \[x=1\] done clear
  B)
  \[x=\omega \] done clear
  C)
  \[x={{\omega }^{2}}\] done clear
  D)
  \[x=0\] done clear
  View Solution play_arrow

• question_answer44)
  If \[x=a+b,y=a\alpha +b\beta \] and \[z=a\beta +b\alpha ,\] where \[\alpha \]and \[\beta \] are complex cube roots of unity, then \[xyz\] = [IIT 1978; Roorkee 1989; RPET 1997]
  A)
  \[{{a}^{2}}+{{b}^{2}}\] done clear
  B)
  \[{{a}^{3}}+{{b}^{3}}\] done clear
  C)
  \[{{a}^{3}}{{b}^{3}}\] done clear
  D)
  \[{{a}^{3}}-{{b}^{3}}\] done clear
  View Solution play_arrow

• question_answer45)
  If \[x=a+b,y=a\omega +b{{\omega }^{2}},z=a{{\omega }^{2}}+b\omega \],  then the value of \[{{x}^{3}}+{{y}^{3}}+{{z}^{3}}\] is equal to        [Roorkee 1977; IIT 1970]
  A)
  \[{{a}^{3}}+{{b}^{3}}\] done clear
  B)
  \[3({{a}^{3}}+{{b}^{3}})\] done clear
  C)
  \[3({{a}^{2}}+{{b}^{2}})\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer46)
  The value of \[\frac{a+b\omega +c{{\omega }^{2}}}{b+c\omega +a{{\omega }^{2}}}+\frac{a+b\omega +c{{\omega }^{2}}}{c+a\omega +b{{\omega }^{2}}}\] will be [BIT Ranchi 1989; Orissa JEE 2003]
  A)
  1 done clear
  B)
  - 1 done clear
  C)
  2 done clear
  D)
  - 2 done clear
  View Solution play_arrow

• question_answer47)
  The cube roots of unity when represented on the Argand plane form the vertices of an       [IIT 1988; Pb. CET 2004]
  A)
  Equilateral triangle done clear
  B)
  Isosceles triangle done clear
  C)
  Right angled triangle done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer48)
  \[{{\left( -\frac{1}{2}+\frac{\sqrt{3}}{2}i \right)}^{1000}}=\]
  A)
  \[\frac{1}{2}+\frac{\sqrt{3}}{2}i\] done clear
  B)
  \[\frac{1}{2}-\frac{\sqrt{3}}{2}i\] done clear
  C)
  \[-\frac{1}{2}+\frac{\sqrt{3}}{2}i\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer49)
  If  \[\alpha ,\beta ,\gamma \] are the cube roots of  \[p(p<0)\], then for any \[x,y\] and \[z,\,\,\frac{x\alpha +y\beta +z\gamma }{x\beta +y\gamma +z\alpha }=\] [IIT 1989]
  A)
  \[\frac{1}{2}(-1+i\sqrt{3})\] done clear
  B)
  \[\frac{1}{2}(1+i\sqrt{3})\] done clear
  C)
  \[\frac{1}{2}(1-i\sqrt{3})\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer50)
  If  \[z=\frac{\sqrt{3}+i}{2}\], then the value of  \[{{z}^{69}}\] is [RPET 2002]
  A)
  \[-i\] done clear
  B)
  \[i\] done clear
  C)
  1 done clear
  D)
  \[-1\] done clear
  View Solution play_arrow

• question_answer51)
  The roots of the equation \[{{x}^{4}}-1=0\],  are [MP PET 1986]
  A)
  \[1,\,1,i,-i\] done clear
  B)
  \[1,\,-1,i,-i\] done clear
  C)
  \[1,-1,\omega ,{{\omega }^{2}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer52)
  If  \[\omega \] is a complex cube root of unity, then for positive integral value of\[n\], the product of \[\omega .{{\omega }^{2}}.{{\omega }^{3}}........{{\omega }^{n}}\], will be [Roorkee 1991]
  A)
  \[\frac{1-i\sqrt{3}}{2}\] done clear
  B)
  \[-\frac{1-i\sqrt{3}}{2}\] done clear
  C)
  1 done clear
  D)
  (b) and (c) both done clear
  View Solution play_arrow

• question_answer53)
  One of the cube roots of unity is [MP PET 1994, 2003]
  A)
  \[\frac{-1+i\sqrt{3}}{2}\] done clear
  B)
  \[\frac{1+i\sqrt{3}}{2}\] done clear
  C)
  \[\frac{1-i\sqrt{3}}{2}\] done clear
  D)
  \[\frac{\sqrt{3}-i}{2}\] done clear
  View Solution play_arrow

• question_answer54)
  If  \[\omega (\ne 1)\]is a cube root of unity and \[{{(1+\omega )}^{7}}=A+B\omega \],  then \[A\] and \[B\] are respectively, the numbers [IIT 1995]
  A)
  0, 1 done clear
  B)
  1, 0 done clear
  C)
  1, 1 done clear
  D)
  \[-1,\ 1\] done clear
  View Solution play_arrow

• question_answer55)
  If \[\omega (\ne 1)\] is a cube root of unity, then \[\left| \begin{matrix}    1 & 1+i+{{\omega }^{2}} & {{\omega }^{2}}  \\    1-i & -1 & {{\omega }^{2}}-1  \\    -i & -i+\omega -1 & -1  \\ \end{matrix} \right|\] is equal to [IIT 1995]
  A)
  0 done clear
  B)
  1 done clear
  C)
  \[\omega \] done clear
  D)
  \[i\] done clear
  View Solution play_arrow

• question_answer56)
  The \[{{n}^{th}}\]roots of unity are in [Orissa JEE 2004]
  A)
  A.P. done clear
  B)
  G.P. done clear
  C)
  H.P. done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer57)
  If  \[1,\omega ,{{\omega }^{2}}\] are the three cube roots of unity, then \[{{(3+{{\omega }^{2}}+{{\omega }^{4}})}^{6}}=\] [MP PET 1995]
  A)
  64 done clear
  B)
  729 done clear
  C)
  2 done clear
  D)
  0 done clear
  View Solution play_arrow

• question_answer58)
  \[(1-\omega +{{\omega }^{2}})(1-{{\omega }^{2}}+{{\omega }^{4}})(1-{{\omega }^{4}}+{{\omega }^{8}})...........\]to \[2n\] factors is [EAMCET 1988]
  A)
  \[{{2}^{n}}\] done clear
  B)
  \[{{2}^{2n}}\] done clear
  C)
  0 done clear
  D)
  1 done clear
  View Solution play_arrow

• question_answer59)
  Let \[\Delta =\left| \,\begin{matrix}    1 & \omega  & 2{{\omega }^{2}}  \\    2 & 2{{\omega }^{2}} & 4{{\omega }^{3}}  \\    3 & 3{{\omega }^{3}} & 6{{\omega }^{4}}  \\ \end{matrix}\, \right|\] where \[\omega \] is the cube root of unity, then
  A)
  \[\Delta =0\] done clear
  B)
  \[\Delta =1\] done clear
  C)
  \[\Delta =2\] done clear
  D)
  \[\Delta =3\] done clear
  View Solution play_arrow

• question_answer60)
  If \[n\] is a positive integer greater than unity and \[z\] is a complex number satisfying the equation \[{{z}^{n}}={{(z+1)}^{n}}\],  then
  A)
  \[\operatorname{Re}(z)<0\] done clear
  B)
  \[\operatorname{Re}(z)>0\] done clear
  C)
  \[\operatorname{Re}(z)=0\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer61)
  If \[\omega \] is an nth root of unity, other than unity, then the value of \[1+\omega +{{\omega }^{2}}+...+{{\omega }^{n-1}}\] is     [Karnataka CET 1999]
  A)
  0 done clear
  B)
  1 done clear
  C)
  \[-1\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer62)
  If \[{{z}_{1}},{{z}_{2}},{{z}_{3}}......{{n}_{n}}\] are nth, roots of unity, then for \[k=1,\,2,.....,n\]
  A)
  \[|{{z}_{k}}|=k|{{z}_{k+1}}|\] done clear
  B)
  \[|{{z}_{k+1}}|=k|{{z}_{k}}|\] done clear
  C)
  \[|{{z}_{k+1}}|\,=\,|{{z}_{k}}|+|{{z}_{k+1}}|\] done clear
  D)
  \[|{{z}_{k}}|=|{{z}_{k+1}}|\] done clear
  View Solution play_arrow

• question_answer63)
  If \[1,\omega ,{{\omega }^{2}}\] are three cube roots of unity, then \[{{(a+b\omega +c{{\omega }^{2}})}^{3}}\] + \[{{(a+b{{\omega }^{2}}+c\omega )}^{3}}\] is equal to, if \[a+b+c=0\] [West Bengal JEE 1992]
  A)
  \[27\,abc\] done clear
  B)
  0 done clear
  C)
  \[3\,abc\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer64)
  The common roots of the equations \[{{x}^{12}}-1=0\], \[{{x}^{4}}+{{x}^{2}}+1=0\] are [EAMCET 1989]
  A)
  \[\pm \omega \] done clear
  B)
  \[\pm {{\omega }^{2}}\] done clear
  C)
  \[\pm \omega ,\,\pm {{\omega }^{2}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer65)
  If \[{{z}_{1}},{{z}_{2}}{{z}_{3}},{{z}_{4}}\]are the roots of the equation \[{{z}^{4}}=1\], then the value of  \[\sum\limits_{i=1}^{4}{z_{i}^{3}}\]is  [Kurukshetra CEE 1996]
  A)
  0 done clear
  B)
  1 done clear
  C)
  \[i\] done clear
  D)
  \[1+i\] done clear
  View Solution play_arrow

• question_answer66)
  If \[\alpha \] is an imaginary cube root of unity, then for  \[n\in N\],  the value of \[{{\alpha }^{3n+1}}+{{\alpha }^{3n+3}}+{{\alpha }^{3n+5}}\] is [MP PET 1996; Pb. CET 2000]
  A)
  \[-1\] done clear
  B)
  0 done clear
  C)
  1 done clear
  D)
  3 done clear
  View Solution play_arrow

• question_answer67)
  \[{{\left( \frac{-1+i\sqrt{3}}{2} \right)}^{20}}+{{\left( \frac{-1-i\sqrt{3}}{2} \right)}^{20}}=\]
  A)
  \[20\sqrt{3}i\] done clear
  B)
  1 done clear
  C)
  \[\frac{1}{{{2}^{19}}}\] done clear
  D)
  \[-1\] done clear
  View Solution play_arrow

• question_answer68)
  If  \[\alpha \] and \[\beta \] are imaginary cube roots of unity, then the value of  \[{{\alpha }^{4}}+{{\beta }^{28}}+\frac{1}{\alpha \beta }\],is [MP PET 1998]
  A)
  1 done clear
  B)
  \[-1\] done clear
  C)
  0 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer69)
  If  \[\omega \] is the cube root of unity, then \[{{(3+5\omega +3{{\omega }^{2}})}^{2}}\] + \[{{(3+3\omega +5{{\omega }^{2}})}^{2}}\] = [MP PET 1999]
  A)
  4 done clear
  B)
  0 done clear
  C)
  - 4 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer70)
  If \[\omega \] is an imaginary cube root of unity, then the value of  \[\sin \,\left[ ({{\omega }^{10}}+{{\omega }^{23}})\,\pi -\frac{\pi }{4} \right]\] is        [IIT Screening 1994]
  A)
  \[-\sqrt{3}/2\] done clear
  B)
  \[-1/\sqrt{2}\] done clear
  C)
  \[1/\sqrt{2}\] done clear
  D)
  \[\sqrt{3}/2\] done clear
  View Solution play_arrow

• question_answer71)
  \[{{\left( \frac{\sqrt{3}+i}{2} \right)}^{6}}+{{\left( \frac{i-\sqrt{3}}{2} \right)}^{6}}\]is equal to [RPET 1997]
  A)
  \[-2\] done clear
  B)
  0 done clear
  C)
  2 done clear
  D)
  1 done clear
  View Solution play_arrow

• question_answer72)
  If \[\omega \] is an imaginary cube root of unity, \[{{(1+\omega -{{\omega }^{2}})}^{7}}\]equals [IIT 1998; MP PET 2000]
  A)
  \[128\omega \] done clear
  B)
  \[-128\omega \] done clear
  C)
  \[128{{\omega }^{2}}\] done clear
  D)
  \[-128{{\omega }^{2}}\] done clear
  View Solution play_arrow

• question_answer73)
  \[\frac{{{(-1+i\sqrt{3})}^{15}}}{{{(1-i)}^{20}}}+\frac{{{(-1-i\sqrt{3})}^{15}}}{{{(1+i)}^{20}}}\] is equal to [AMU 2000]
  A)
  - 64 done clear
  B)
  - 32 done clear
  C)
  - 16 done clear
  D)
  \[\frac{1}{16}\] done clear
  View Solution play_arrow

• question_answer74)
  If \[\pi /3\] is a complex root of the equation \[{{z}^{3}}=1\], then \[\omega +{{\omega }^{\left( \frac{1}{2}\,+\,\frac{3}{8}\,+\,\frac{9}{32}\,+\,\frac{27}{128}\,+... \right)}}\] is equal to [Roorkee 2000; AMU 2005]
  A)
  - 1 done clear
  B)
  0 done clear
  C)
  9 done clear
  D)
  i done clear
  View Solution play_arrow

• question_answer75)
  If cube root of 1 is \[\omega \], then the value of \[{{(3+\omega +3{{\omega }^{2}})}^{4}}\] is [MP PET 2001]
  A)
  0 done clear
  B)
  16 done clear
  C)
  \[16\,\omega \] done clear
  D)
  \[16\,{{\omega }^{2}}\] done clear
  View Solution play_arrow

• question_answer76)
  The value of \[(1-\omega +{{\omega }^{2}})\,{{(1-{{\omega }^{2}}+\omega )}^{6}}\], where \[\omega ,{{\omega }^{2}}\] are cube roots of unity [DCE 2001]
  A)
  128\[\omega \] done clear
  B)
  \[-128{{\omega }^{2}}\] done clear
  C)
  \[-128\omega \] done clear
  D)
  \[128{{\omega }^{2}}\] done clear
  View Solution play_arrow

• question_answer77)
  If \[1,\omega ,{{\omega }^{2}}\] are the cube roots of unity, then their product is [Karnataka CET 1999, 2001]
  A)
  0 done clear
  B)
  \[\omega \] done clear
  C)
  - 1 done clear
  D)
  1 done clear
  View Solution play_arrow

• question_answer78)
  If \[z=\frac{\sqrt{3}+i}{-2}\], then \[{{z}^{69}}\] is equal to [RPET 2001]
  A)
  1 done clear
  B)
  - 1 done clear
  C)
  i done clear
  D)
  - i done clear
  View Solution play_arrow

• question_answer79)
  Let \[{{\omega }_{n}}=\cos \left( \frac{2\pi }{n} \right)+i\,\sin \left( \frac{2\pi }{n} \right)\,,\,{{i}^{2}}=-1\], then \[(x+y{{\omega }_{3}}+z{{\omega }_{3}}^{2})\] \[(x+y{{\omega }_{3}}^{2}+z{{\omega }_{3}})\] is equal to [AMU 2001]
  A)
  0 done clear
  B)
  \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}\] done clear
  C)
  \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-yz-zx-xy\]\[\] done clear
  D)
  \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+yz+zx+xy\] done clear
  View Solution play_arrow

• question_answer80)
  If \[z+{{z}^{-1}}=1,\,\text{then }\,{{z}^{100}}+{{z}^{-100}}\] is equal to [UPSEAT 2001]
  A)
  i done clear
  B)
  - i done clear
  C)
  1 done clear
  D)
  - 1 done clear
  View Solution play_arrow

• question_answer81)
   If \[\frac{1+\sqrt{3}\,i}{2}\] is a root of equation \[{{x}^{4}}-{{x}^{3}}+x-1=0\]  then its real roots are [EAMCET 2002]
  A)
  1, 1 done clear
  B)
  - 1, - 1 done clear
  C)
  1, - 1 done clear
  D)
  1, 2 done clear
  View Solution play_arrow

• question_answer82)
  If \[{{\left( \frac{1+i\sqrt{3}}{1-i\sqrt{3}} \right)}^{n}}\] is an integer, then n is  [UPSEAT 2002]
  A)
  1 done clear
  B)
  2 done clear
  C)
  3 done clear
  D)
  4 done clear
  View Solution play_arrow

• question_answer83)
  Find the value of \[{{(1+2\omega +{{\omega }^{2}})}^{3n}}-{{(1+\omega +2{{\omega }^{2}})}^{3n}}=\] [UPSEAT 2002]
  A)
  0 done clear
  B)
  1 done clear
  C)
  \[\omega \] done clear
  D)
  \[{{\omega }^{2}}\] done clear
  View Solution play_arrow

• question_answer84)
  If \[\omega \] is a non real cube root of unity, then \[(a+b)\] \[(a+b\omega )\] \[(a+b{{\omega }^{2}})\] is [Kerala (Engg.) 2002]
  A)
  \[{{a}^{3}}+{{b}^{3}}\] done clear
  B)
  \[{{a}^{3}}-{{b}^{3}}\] done clear
  C)
  \[{{a}^{2}}+{{b}^{2}}\] done clear
  D)
  \[{{a}^{2}}-{{b}^{2}}\] done clear
  View Solution play_arrow

• question_answer85)
  . Which of the following is a fourth root of \[\frac{1}{2}+\frac{i\sqrt{3}}{2}\] [Karnataka CET 2003]
  A)
  \[cis\left( \frac{\pi }{2} \right)\] done clear
  B)
  \[cis\left( \frac{\pi }{12} \right)\] done clear
  C)
  \[cis\left( \frac{\pi }{6} \right)\] done clear
  D)
  \[cis\left( \frac{\pi }{3} \right)\] done clear
  View Solution play_arrow

• question_answer86)
  The value of (8)1/3 is [RPET 2003]
  A)
  \[-1+i\sqrt{3}\] done clear
  B)
  \[-1-i\sqrt{3}\] done clear
  C)
  2 done clear
  D)
  All of these done clear
  View Solution play_arrow

• question_answer87)
  If \[\omega \] is a complex cube root of unity, then\[225+\]\[{{(3\omega +8{{\omega }^{2}})}^{2}}\]\[+{{(3{{\omega }^{2}}+8\omega )}^{2}}=\] [EAMCET 2003]
  A)
  72 done clear
  B)
  192 done clear
  C)
  200 done clear
  D)
  248 done clear
  View Solution play_arrow

• question_answer88)
  If \[1,\omega ,{{\omega }^{2}}\] are the cube roots of unity, then\[\Delta =\left| \,\begin{matrix}    1\,\,\,\, & {{\omega }^{n}} & {{\omega }^{2n}}  \\    {{\omega }^{n}}\,\, & \,\,\,{{\omega }^{2n}}\,\, & 1  \\    {{\omega }^{2n}}\, & 1\,\, & {{\omega }^{n}}  \\ \end{matrix} \right|\]= [AIEEE 2003]
  A)
  0 done clear
  B)
  1 done clear
  C)
  \[\omega \] done clear
  D)
  \[{{\omega }^{2}}\] done clear
  View Solution play_arrow

• question_answer89)
  If \[\omega =\frac{-1+\sqrt{3}i}{2}\]then \[{{(3+\omega +3{{\omega }^{2}})}^{4}}\]= [Karnataka CET 2004; Pb. CET 2000]
  A)
  16 done clear
  B)
  -16 done clear
  C)
  16 \[\omega \] done clear
  D)
  16\[{{\omega }^{2}}\] done clear
  View Solution play_arrow

• question_answer90)
  If \[1,\,\omega ,\,{{\omega }^{2}}\] are the roots of unity, then \[{{(1-2\omega +{{\omega }^{2}})}^{6}}\] is equal to [Pb. CET 2001]
  A)
  729 done clear
  B)
  246 done clear
  C)
  243 done clear
  D)
  81 done clear
  View Solution play_arrow

• question_answer91)
  If \[\omega \] is a complex cube root of unity, then the value of \[{{\omega }^{99}}+{{\omega }^{100}}+{{\omega }^{101}}\] is [Pb. CET 2004]
  A)
  1 done clear
  B)
  - 1 done clear
  C)
  3 done clear
  D)
  0 done clear
  View Solution play_arrow

• question_answer92)
  The real part of  \[{{\sin }^{-1}}({{e}^{i\theta }})\]  is [RPET 1997]
  A)
  \[{{\cos }^{-1}}(\sqrt{\sin \theta })\] done clear
  B)
  \[{{\sinh }^{-1}}(\sqrt{\sin \theta })\] done clear
  C)
  \[{{\sin }^{-1}}(\sqrt{\sin \theta })\] done clear
  D)
  \[{{\sin }^{-1}}(\sqrt{\cos \theta })\] done clear
  View Solution play_arrow

• question_answer93)
  \[\sinh ix\] is [EAMCET 2002]
  A)
  \[i\sin (ix)\] done clear
  B)
  \[i\sin x\] done clear
  C)
  \[-i\sin x\] done clear
  D)
  \[\sin (ix)\] done clear
  View Solution play_arrow

• question_answer94)
  If \[\cos (u+iv)=\alpha +i\beta ,\] then \[{{\alpha }^{2}}+{{\beta }^{2}}+1\] equals [RPET 1999]
  A)
  \[{{\cos }^{2}}u+{{\sinh }^{2}}v\] done clear
  B)
  \[{{\sin }^{2}}u+{{\cosh }^{2}}v\] done clear
  C)
  \[{{\cos }^{2}}u+{{\cosh }^{2}}v\] done clear
  D)
  \[{{\sin }^{2}}u+{{\sinh }^{2}}v\] done clear
  View Solution play_arrow

• question_answer95)
  The value of \[\sec h(i\pi )\] is [RPET 1999]
  A)
  - 1 done clear
  B)
  i done clear
  C)
  0 done clear
  D)
  1 done clear
  View Solution play_arrow

• question_answer96)
  \[\cosh (\alpha +i\beta )-\cosh (\alpha -i\beta )\] is equal to [RPET 2000]
  A)
  \[2\,\,\sinh \,\alpha \,\,\sinh \,\beta \] done clear
  B)
  \[2\,\,\cosh \,\alpha \,\,\cosh \,\beta \] done clear
  C)
  \[2i\,\,\sinh \,\alpha \,\,\sin \,\beta \] done clear
  D)
  \[2\,\,\cosh \,\alpha \,\,\cos \,\beta \] done clear
  View Solution play_arrow

• question_answer97)
  The imaginary part of \[\cosh (\alpha +i\beta )\]is [RPET 2000]
  A)
  \[\cosh \,\alpha \,\,\cos \,\beta \] done clear
  B)
  \[\sinh \,\alpha \,\,\sin \,\beta \] done clear
  C)
  \[\cos \alpha \cosh \beta \] done clear
  D)
  \[\cos \alpha \cos \beta \] done clear
  View Solution play_arrow

• question_answer98)
  Which one is correct from the following [RPET 2001]
  A)
  \[\sin (ix)=i\,\sinh \,x\] done clear
  B)
  \[\cos (ix)=i\,\cosh \,x\] done clear
  C)
  \[\sin (ix)=-i\,\sinh \,x\] done clear
  D)
  \[\tan (ix)=-i\,\tanh \,x\] done clear
  View Solution play_arrow

• question_answer99)
  \[\cos (x+iy)\]is equal to [RPET 2001]
  A)
  \[\sin \,x\,\,\cosh \,y+i\,\cos \,x\,\,\sinh \,y\] done clear
  B)
  \[\cos \,x\,\,\cosh \,y+i\,\sin \,x\,\,\sinh \,y\] done clear
  C)
  \[\cos \,x\,\,\cosh \,y-i\,\sin \,x\,\,\sinh \,y\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer100)
  If \[\tan (u+iv)=i\], then the value of v is [RPET 2001]
  A)
  0 done clear
  B)
  \[\infty \] done clear
  C)
  1 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer101)
  If \[{{\tan }^{-1}}(\alpha +i\beta )=x+iy,\] then x = [RPET 2002]
  A)
  \[\frac{1}{2}{{\tan }^{-1}}\left( \frac{2\alpha }{1-{{\alpha }^{2}}-{{\beta }^{2}}} \right)\] done clear
  B)
  \[\frac{1}{2}{{\tan }^{-1}}\left( \frac{2\alpha }{1+{{\alpha }^{2}}+{{\beta }^{2}}} \right)\] done clear
  C)
  \[{{\tan }^{-1}}\left( \frac{2\alpha }{1-{{\alpha }^{2}}-{{\beta }^{2}}} \right)\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer102)
  If \[\omega \] is a cube root of unity but not equal to 1 then minimum value of \[|a+b\omega +c{{\omega }^{2}}|\] (where a, b, c are integers but not all equal) is [IIT Screening 2005]
  A)
  0 done clear
  B)
  \[\frac{\sqrt{3}}{2}\] done clear
  C)
  1 done clear
  D)
  2 done clear
  View Solution play_arrow

• question_answer103)
  If 1, \[\omega ,\,{{\omega }^{2}}\] are the cube roots of unity then \[{{\omega }^{2}}{{(1+\omega )}^{3}}-(1+{{\omega }^{2}})\omega =\] [Orissa JEE 2005]
  A)
  1 done clear
  B)
  -1 done clear
  C)
  i done clear
  D)
  0 done clear
  View Solution play_arrow

• question_answer104)
  Let \[x=\alpha +\beta ,\,y=\alpha \omega +\beta {{\omega }^{2}},\,z=\alpha {{\omega }^{2}}+\beta \omega ,\,\omega \] is an imaginary cube root of unity. Product of xyz is  [Orissa JEE 2005]
  A)
  \[{{\alpha }^{2}}+{{\beta }^{2}}\] done clear
  B)
  \[{{\alpha }^{2}}-{{\beta }^{2}}\] done clear
  C)
  \[{{\alpha }^{3}}+{{\beta }^{3}}\] done clear
  D)
  \[{{\alpha }^{3}}-{{\beta }^{3}}\] done clear
  View Solution play_arrow