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

done Conjugate, Modulus and Argument of complex number Total Questions - 87

• question_answer1)
  The complex numbers \[\sin x+i\cos 2x\] and \[\cos x-i\sin 2x\] are conjugate to each other for [IIT 1988]
  A)
  \[x=n\pi \] done clear
  B)
  \[x=\left( n+\frac{1}{2} \right)\pi \] done clear
  C)
  \[x=0\] done clear
  D)
  No value of x done clear
  View Solution play_arrow

• question_answer2)
  If  \[z\]  is a complex number, then \[(\overline{{{z}^{-1}}})(\overline{z})=\]
  A)
  1 done clear
  B)
  -1 done clear
  C)
  0 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer3)
  If \[z\] is a complex number such that \[{{z}^{2}}={{(\bar{z})}^{2}},\] then
  A)
  \[z\]is purely real done clear
  B)
  \[z\]is purely imaginary done clear
  C)
  Either \[z\]is purely real or purely imaginary done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer4)
  If  z  is a complex number, then \[z.\,\overline{z}=0\] if and only if
  A)
  \[z=0\] done clear
  B)
  \[\operatorname{Re}(z)=0\] done clear
  C)
  \[\operatorname{Im}\,(z)=0\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer5)
  If \[(a+ib)(c+id)(e+if)(g+ih)\]\[=A+iB,\] then \[({{a}^{2}}+{{b}^{2}})({{c}^{2}}+{{d}^{2}})({{e}^{2}}+{{f}^{2}})({{g}^{2}}+{{h}^{2}})\] = [MNR 1989]
  A)
  \[{{A}^{2}}+{{B}^{2}}\] done clear
  B)
  \[{{A}^{2}}-{{B}^{2}}\] done clear
  C)
  \[{{A}^{2}}\] done clear
  D)
  \[{{B}^{2}}\] done clear
  View Solution play_arrow

• question_answer6)
  The number of solutions of the equation \[{{z}^{2}}+\bar{z}=0\] is
  A)
  1 done clear
  B)
  2 done clear
  C)
  3 done clear
  D)
  4 done clear
  View Solution play_arrow

• question_answer7)
  For the complex number \[z\], one from \[z+\bar{z}\] and \[z\,\bar{z}\]  is [RPET 1987]
  A)
  A real number done clear
  B)
  A imaginary number done clear
  C)
  Both are real numbers done clear
  D)
  Both are imaginary numbers done clear
  View Solution play_arrow

• question_answer8)
  The values of \[x\] and \[y\] for which the numbers \[3+i{{x}^{2}}y\] and \[{{x}^{2}}+y+4i\] are conjugate complex can be
  A)
  \[(-2,-1)\]or \[(2,-1)\] done clear
  B)
  \[(-1,\text{ }2)\]or \[(-2,\text{ }1)\] done clear
  C)
  \[(1,\,2)\]or \[(-1,-2)\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer9)
  The conjugate of the complex number  \[\frac{2+5i}{4-3i}\] is [MP PET 1994]
  A)
  \[\frac{7-26i}{25}\] done clear
  B)
  \[\frac{-7-26i}{25}\] done clear
  C)
  \[\frac{-7+26i}{25}\] done clear
  D)
  \[\frac{7+26i}{25}\] done clear
  View Solution play_arrow

• question_answer10)
  \[(z+a)(\bar{z}+a)\], where \[a\] is real, is equivalent to
  A)
  \[|z-a|\] done clear
  B)
  \[{{z}^{2}}+{{a}^{2}}\] done clear
  C)
  \[|z+a{{|}^{2}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer11)
  If \[\frac{z-i}{z+i}(z\ne -i)\] is a purely imaginary number, then \[z.\bar{z}\] is equal to
  A)
  0 done clear
  B)
  1 done clear
  C)
  2 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer12)
  If \[\frac{c+i}{c-i}=a+ib\], where \[a,b,c\]are real, then \[{{a}^{2}}+{{b}^{2}}=\]  [MP PET 1996]
  A)
  1 done clear
  B)
  \[-1\] done clear
  C)
  \[{{c}^{2}}\] done clear
  D)
  \[-{{c}^{2}}\] done clear
  View Solution play_arrow

• question_answer13)
  If the conjugate of \[(x+iy)(1-2i)\]be \[1+i\], then [MP PET 1996]
  A)
  \[x=\frac{1}{5}\] done clear
  B)
    \[y=\frac{3}{5}\] done clear
  C)
  \[x+iy=\frac{1-i}{1-2i}\] done clear
  D)
    \[x-iy=\frac{1-i}{1+2i}\] done clear
  View Solution play_arrow

• question_answer14)
  The conjugate of \[\frac{{{(2+i)}^{2}}}{3+i},\] in the form of a +  ib, is  [Karnataka CET 2001; Pb. CET 2001]
  A)
  \[\frac{13}{2}+i\,\left( \frac{15}{2} \right)\] done clear
  B)
  \[\frac{13}{10}+i\left( \frac{-15}{2} \right)\] done clear
  C)
  \[\frac{13}{10}+i\,\left( \frac{-9}{10} \right)\] done clear
  D)
  \[\frac{13}{10}+i\,\left( \frac{9}{10} \right)\] done clear
  View Solution play_arrow

• question_answer15)
  If \[z=3+5i,\,\,\text{then }\,{{z}^{3}}+\bar{z}+198=\] [EAMCET 2002]
  A)
  \[-3-5i\] done clear
  B)
  \[-3+5i\] done clear
  C)
  \[3+5i\] done clear
  D)
  \[3-5i\] done clear
  View Solution play_arrow

• question_answer16)
  The conjugate of complex number \[\frac{2-3i}{4-i},\] is [MP PET 2003]
  A)
  \[\frac{3i}{4}\] done clear
  B)
  \[\frac{11+10i}{17}\] done clear
  C)
  \[\frac{11-10i}{17}\] done clear
  D)
  \[\frac{2+3i}{4i}\] done clear
  View Solution play_arrow

• question_answer17)
  Conjugate of 1 + i is [RPET 2003]
  A)
  i done clear
  B)
  1 done clear
  C)
  1 - i done clear
  D)
  1 + i done clear
  View Solution play_arrow

• question_answer18)
  The inequality \[|z-4|\,<\,|\,z-2|\]represents the region given by                               [IIT 1982; RPET 1995; AIEEE 2002]
  A)
  \[\operatorname{Re}(z)>0\] done clear
  B)
   \[\operatorname{Re}(z)<0\] done clear
  C)
  \[\operatorname{Re}(z)>2\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer19)
  If \[\frac{2{{z}_{1}}}{3{{z}_{2}}}\] is a purely imaginary number, then \[\left| \frac{{{z}_{1}}-{{z}_{2}}}{{{z}_{1}}+{{z}_{2}}} \right|\] =  [MP PET 1993]
  A)
  3/2 done clear
  B)
  1 done clear
  C)
  2/3 done clear
  D)
  4/9 done clear
  View Solution play_arrow

• question_answer20)
  If \[{{z}_{1}}\] and \[{{z}_{2}}\] are any two complex numbers then \[|{{z}_{1}}+{{z}_{2}}{{|}^{2}}\] \[+|{{z}_{1}}-{{z}_{2}}{{|}^{2}}\] is equal to    [MP PET 1993; RPET 1997]
  A)
  \[2|{{z}_{1}}{{|}^{2}}\,|{{z}_{2}}{{|}^{2}}\] done clear
  B)
  \[2|{{z}_{1}}{{|}^{2}}+\,2\,\,|{{z}_{2}}{{|}^{2}}\] done clear
  C)
  \[|{{z}_{1}}{{|}^{2}}+\,|{{z}_{2}}{{|}^{2}}\] done clear
  D)
  \[2|{{z}_{1}}|\,\,|{{z}_{2}}|\] done clear
  View Solution play_arrow

• question_answer21)
  If  z is a complex number such that \[\frac{z-1}{z+1}\] is purely imaginary, then [MP PET 1998, 2002]
  A)
  \[|z|\,=0\] done clear
  B)
  \[|z|\,=1\] done clear
  C)
  \[|z|\,>1\] done clear
  D)
  \[|z|\,<1\] done clear
  View Solution play_arrow

• question_answer22)
  If \[z\] is a complex number, then which of the following is not true [MP PET 1987]
  A)
  \[|{{z}^{2}}|\,=\,|z{{|}^{2}}\] done clear
  B)
  \[|{{z}^{2}}|\,=\,|\bar{z}{{|}^{2}}\] done clear
  C)
  \[z=\bar{z}\] done clear
  D)
  \[{{\bar{z}}^{2}}={{\bar{z}}^{2}}\] done clear
  View Solution play_arrow

• question_answer23)
  The maximum value of \[|z|\] where z satisfies the condition \[\left| z+\frac{2}{z} \right|=2\] is
  A)
  \[\sqrt{3}-1\] done clear
  B)
  \[\sqrt{3}+1\] done clear
  C)
  \[\sqrt{3}\] done clear
  D)
  \[\sqrt{2}+\sqrt{3}\] done clear
  View Solution play_arrow

• question_answer24)
  If  \[{{z}_{1}}\] and \[{{z}_{2}}\] are two complex numbers satisfying the equation  \[\left| \frac{{{z}_{1}}+{{z}_{2}}}{{{z}_{1}}-{{z}_{2}}} \right|\]=1, then \[\frac{{{z}_{1}}}{{{z}_{2}}}\] is a number which is
  A)
  Positive real done clear
  B)
  Negative real done clear
  C)
  Zero or purely imaginary done clear
  D)
   None of these done clear
  View Solution play_arrow

• question_answer25)
  The solution of the equation \[|z|-z=1+2i\] is [MP PET 1993]
  A)
  \[2-\frac{3}{2}i\] done clear
  B)
  \[\frac{3}{2}+2i\] done clear
  C)
  \[\frac{3}{2}-2i\] done clear
  D)
  \[-2+\frac{3}{2}i\] done clear
  View Solution play_arrow

• question_answer26)
  If \[{{z}_{1}}\text{ and }{{z}_{2}}\] be complex numbers such that \[{{z}_{1}}\ne {{z}_{2}}\] and \[|{{z}_{1}}|\,=\,|{{z}_{2}}|\]. If \[{{z}_{1}}\] has positive real part and \[{{z}_{2}}\] has negative imaginary part, then \[\frac{({{z}_{1}}+{{z}_{2}})}{({{z}_{1}}-{{z}_{2}})}\]may be [IIT 1986]
  A)
  Purely imaginary done clear
  B)
  Real and positive done clear
  C)
  Real and negative done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer27)
  The moduli of two complex numbers are less than unity, then the modulus of the sum of these complex numbers
  A)
  Less than unity done clear
  B)
  Greater than unity done clear
  C)
  Equal to unity done clear
  D)
  Any done clear
  View Solution play_arrow

• question_answer28)
  The product of two complex numbers each of unit modulus is also a complex number, of
  A)
  Unit modulus done clear
  B)
  Less than unit modulus done clear
  C)
  Greater than unit modulus done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer29)
  Let \[z\] be a complex number, then the equation \[{{z}^{4}}+z+2=0\] cannot have a root, such that
  A)
  \[|z|\,<1\] done clear
  B)
  \[|z|\,=1\] done clear
  C)
  \[|z|\,>1\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer30)
  If \[|{{z}_{1}}|=|{{z}_{2}}|=..........=|{{z}_{n}}|=1,\] then the value of \[|{{z}_{1}}+{{z}_{2}}+{{z}_{3}}+.............+{{z}_{n}}|\]=
  A)
  1 done clear
  B)
  \[|{{z}_{1}}|+|{{z}_{2}}|+.......+|{{z}_{n}}|\] done clear
  C)
  \[\left| \frac{1}{{{z}_{1}}}+\frac{1}{{{z}_{2}}}+.........+\frac{1}{{{z}_{n}}} \right|\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer31)
  For any complex number \[z,\bar{z}=\left( \frac{1}{z} \right)\]if and only if  [RPET 1985]
  A)
  \[z\] is a pure real number done clear
  B)
  \[|z|=1\] done clear
  C)
  \[z\]is a pure imaginary number done clear
  D)
  \[z=1\] done clear
  View Solution play_arrow

• question_answer32)
  If  \[{{z}_{1}}\] and \[{{z}_{2}}\] are two complex numbers, then  \[|{{z}_{1}}-{{z}_{2}}|\] is [MP PET 1994]
  A)
  \[\ge \,|{{z}_{1}}|-|{{z}_{2}}|\] done clear
  B)
  \[\le \,|{{z}_{1}}|-|{{z}_{2}}|\] done clear
  C)
  \[\ge \,|{{z}_{1}}|+|{{z}_{2}}|\] done clear
  D)
  \[\le \,|{{z}_{2}}|-|{{z}_{1}}|\] done clear
  View Solution play_arrow

• question_answer33)
  The values of  \[z\]for which \[|z+i|\,=\,|z-i|\] are [Bihar CEE 1994]
  A)
  Any real number done clear
  B)
  Any complex number done clear
  C)
  Any natural number done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer34)
  The value of  \[|z-5|\]if \[z=x+iy\],  is [RPET 1995]
  A)
  \[\sqrt{{{(x-5)}^{2}}+{{y}^{2}}}\] done clear
  B)
  \[{{x}^{2}}+\sqrt{{{(y-5)}^{2}}}\] done clear
  C)
  \[\sqrt{{{(x-y)}^{2}}+{{5}^{2}}}\] done clear
  D)
  \[\sqrt{{{x}^{2}}+{{(y-5)}^{2}}}\] done clear
  View Solution play_arrow

• question_answer35)
  \[\left| (1+i)\frac{(2+i)}{(3+i)} \right|=\] [MP PET 1995, 99]
  A)
  \[-\frac{1}{2}\] done clear
  B)
  \[\frac{1}{2}\] done clear
  C)
  1 done clear
  D)
  \[-1\] done clear
  View Solution play_arrow

• question_answer36)
  If \[{{z}_{1}},{{z}_{2}}\] are two complex numbers such that \[\left| \frac{{{z}_{1}}-{{z}_{2}}}{{{z}_{1}}+{{z}_{2}}} \right|=1\] and \[i{{z}_{1}}=k{{z}_{2}}\], where \[k\in R\], then the angle between \[{{z}_{1}}-{{z}_{2}}\] and \[{{z}_{1}}+{{z}_{2}}\] is
  A)
  \[{{\tan }^{-1}}\left( \frac{2k}{{{k}^{2}}+1} \right)\] done clear
  B)
  \[{{\tan }^{-1}}\left( \frac{2k}{1-{{k}^{2}}} \right)\] done clear
  C)
  - \[2{{\tan }^{-1}}k\] done clear
  D)
  \[2{{\tan }^{-1}}k\] done clear
  View Solution play_arrow

• question_answer37)
  Let z be a complex number (not lying on X-axis of maximum modulus such that \[\left| z+\frac{1}{z} \right|=1\]. Then
  A)
  \[\operatorname{Im}(z)=0\] done clear
  B)
  \[\operatorname{Re}(z)=0\] done clear
  C)
  \[amp(z)=\pi \] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer38)
  If z1, z2 are any two complex numbers, then \[|{{z}_{1}}+\sqrt{z_{1}^{2}-z_{2}^{2}}|\] \[+|{{z}_{1}}-\sqrt{z_{1}^{2}-z_{2}^{2}}|\] is equal to
  A)
  \[|{{z}_{1}}|\] done clear
  B)
  \[|{{z}_{2}}|\] done clear
  C)
  \[|{{z}_{1}}+{{z}_{2}}|\] done clear
  D)
  \[|{{z}_{1}}+{{z}_{2}}|+|{{z}_{1}}-{{z}_{2}}|\] done clear
  View Solution play_arrow

• question_answer39)
  \[\left| \frac{1}{2}({{z}_{1}}+{{z}_{2}})+\sqrt{{{z}_{1}}{{z}_{2}}} \right|+\left| \frac{1}{2}({{z}_{1}}+{{z}_{2}})-\sqrt{{{z}_{1}}{{z}_{2}}} \right|\] =
  A)
  \[|{{z}_{1}}+{{z}_{2}}|\] done clear
  B)
  \[|{{z}_{1}}-{{z}_{2}}|\] done clear
  C)
  \[|{{z}_{1}}+{{z}_{2}}|\] done clear
  D)
  \[|{{z}_{1}}|-|{{z}_{2}}|\] done clear
  View Solution play_arrow

• question_answer40)
  Modulus of  \[\left( \frac{3+2i}{3-2i} \right)\] is [RPET 1996]
  A)
  1 done clear
  B)
  1/2 done clear
  C)
  2 done clear
  D)
  \[\sqrt{2}\] done clear
  View Solution play_arrow

• question_answer41)
  If  \[|z|\,=1,(z\ne -1)\]and \[z=x+iy,\]then \[\left( \frac{z-1}{z+1} \right)\] is [RPET 1997]
  A)
  Purely real done clear
  B)
  Purely imaginary done clear
  C)
  Zero done clear
  D)
  Undefined done clear
  View Solution play_arrow

• question_answer42)
  The minimum value of  \[|2z-1|+|3z-2|\]is [RPET 1997]
  A)
  0 done clear
  B)
  \[1/2\] done clear
  C)
  \[1/3\] done clear
  D)
  2/3 done clear
  View Solution play_arrow

• question_answer43)
  If \[|z|\,=1\] and \[\omega =\frac{z-1}{z+1}\] (where \[z\ne -1)\], then \[\operatorname{Re}(\omega )\] is  [IIT Screening 2003]
  A)
  \[0\] done clear
  B)
  \[-\frac{1}{|z+1{{|}^{2}}}\] done clear
  C)
  \[\left| \frac{z}{z+1} \right|\,.\frac{1}{|z+1{{|}^{2}}}\] done clear
  D)
  \[\frac{\sqrt{2}}{|z+1{{|}^{2}}}\] done clear
  View Solution play_arrow

• question_answer44)
  A real value of x will satisfy the equation \[\left( \frac{3-4ix}{3+4ix} \right)=\] \[\alpha -i\beta \,(\alpha ,\beta \,\text{real),}\] if [Orissa JEE 2003]
  A)
  \[{{\alpha }^{2}}-{{\beta }^{2}}=-1\] done clear
  B)
  \[{{\alpha }^{2}}-{{\beta }^{2}}=1\] done clear
  C)
  \[{{\alpha }^{2}}+{{\beta }^{2}}=1\] done clear
  D)
  \[{{\alpha }^{2}}-{{\beta }^{2}}=2\] done clear
  View Solution play_arrow

• question_answer45)
  Let \[{{z}_{1}}\] be a complex number with \[|{{z}_{1}}|=1\] and \[{{z}_{2}}\]be any complex number, then \[\left| \frac{{{z}_{1}}-{{z}_{2}}}{1-{{z}_{1}}{{{\bar{z}}}_{2}}} \right|=\]    [Orissa JEE 2004]
  A)
  0 done clear
  B)
  1 done clear
  C)
  - 1 done clear
  D)
  2 done clear
  View Solution play_arrow

• question_answer46)
  If  \[{{z}_{1}}\] and \[{{z}_{2}}\]  are two non-zero complex numbers such that \[|{{z}_{1}}+{{z}_{2}}|=|{{z}_{1}}|+|{{z}_{2}}|,\]then arg \[({{z}_{1}})-\]arg \[({{z}_{2}})\] is equal to                           [IIT  1979, 1987; EAMCET 1986; RPET 1997; MP PET 2001; AIEEE 2005]
  A)
  \[-\pi \] done clear
  B)
  \[-\frac{\pi }{2}\] done clear
  C)
  \[\frac{\pi }{2}\] done clear
  D)
  0 done clear
  View Solution play_arrow

• question_answer47)
  \[arg\,(5-\sqrt{3}i)=\]
  A)
  \[{{\tan }^{-1}}\frac{5}{\sqrt{3}}\] done clear
  B)
  \[{{\tan }^{-1}}\left( -\,\frac{5}{\sqrt{3}} \right)\] done clear
  C)
  \[{{\tan }^{-1}}\frac{\sqrt{3}}{5}\] done clear
  D)
  \[{{\tan }^{-1}}\left( -\frac{\sqrt{3}}{5} \right)\] done clear
  View Solution play_arrow

• question_answer48)
  Argument and modulus of \[\frac{1+i}{1-i}\]  are respectively [RPET 1984; MP PET 1987; Karnataka CET 2001]
  A)
  \[\frac{-\pi }{2}\]and 1 done clear
  B)
  \[\frac{\pi }{2}\]and \[\sqrt{2}\] done clear
  C)
  0 and \[\sqrt{2}\] done clear
  D)
  \[\frac{\pi }{2}\]and 1 done clear
  View Solution play_arrow

• question_answer49)
  If \[\bar{z}\] be the conjugate of the complex number \[z\], then  which of the following relations is false [MP PET 1987]
  A)
  \[|z|\,=\,|\bar{z}|\] done clear
  B)
  \[z.\,\bar{z}=|\bar{z}{{|}^{2}}\] done clear
  C)
  \[\overline{{{z}_{1}}+{{z}_{2}}}=\overline{{{z}_{1}}}+\overline{{{z}_{2}}}\] done clear
  D)
  \[arg\,z=arg\,\bar{z}\] done clear
  View Solution play_arrow

• question_answer50)
  If \[|z|\,=4\]  and \[arg\,\,z=\frac{5\pi }{6},\]then z = [MP PET 1987]
  A)
  \[2\sqrt{3}-2i\] done clear
  B)
  \[2\sqrt{3}+2i\] done clear
  C)
  \[-2\sqrt{3}+2i\] done clear
  D)
  \[-\sqrt{3}+i\] done clear
  View Solution play_arrow

• question_answer51)
  If\[z=\frac{1-i\sqrt{3}}{1+i\sqrt{3}},\]then \[arg(z)=\][Roorkee 1990; UPSEAT 2004]
  A)
  \[{{60}^{o}}\] done clear
  B)
  \[{{120}^{o}}\] done clear
  C)
  \[{{240}^{o}}\] done clear
  D)
  \[{{300}^{o}}\] done clear
  View Solution play_arrow

• question_answer52)
  If \[arg\,(z)=\theta \],  then \[arg\,(\overline{z})=\] [MP PET 1995]
  A)
  \[\theta \] done clear
  B)
  \[-\theta \] done clear
  C)
  \[\pi -\theta \] done clear
  D)
  \[\theta -\pi \] done clear
  View Solution play_arrow

• question_answer53)
  The amplitude of the complex number \[z=\sin \alpha +i(1-\cos \alpha )\] is
  A)
  \[2\sin \frac{\alpha }{2}\] done clear
  B)
  \[\frac{\alpha }{2}\] done clear
  C)
  \[\alpha \] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer54)
  The amplitude of  \[\frac{1+\sqrt{3}i}{\sqrt{3}+1}\] is [Karnataka CET 1992; Pb CET 2001]
  A)
  \[\frac{\pi }{3}\] done clear
  B)
  \[-\frac{\pi }{3}\] done clear
  C)
  \[\frac{\pi }{6}\] done clear
  D)
  \[-\frac{\pi }{6}\] done clear
  View Solution play_arrow

• question_answer55)
  The argument of the complex number \[-1+i\sqrt{3}\] is [MP PET 1994]
  A)
  \[-{{60}^{o}}\] done clear
  B)
  \[{{60}^{o}}\] done clear
  C)
  \[{{120}^{o}}\] done clear
  D)
  \[-{{120}^{o}}\] done clear
  View Solution play_arrow

• question_answer56)
   \[arg\left( \frac{3+i}{2-i}+\frac{3-i}{2+i} \right)\] is equal to
  A)
  \[\frac{\pi }{2}\] done clear
  B)
  \[-\frac{\pi }{2}\] done clear
  C)
  0 done clear
  D)
  \[\frac{\pi }{4}\] done clear
  View Solution play_arrow

• question_answer57)
  If \[{{z}_{1}}.{{z}_{2}}........{{z}_{n}}=z,\] then \[arg\,{{z}_{1}}+arg\,{{z}_{2}}+....\]+\[arg\,{{z}_{n}}\] and \[arg\]\[z\] differ by a
  A)
  Multiple of \[\pi \] done clear
  B)
  Multiple of\[\frac{\pi }{2}\] done clear
  C)
  Greater than \[\pi \] done clear
  D)
  Less than \[\pi \] done clear
  View Solution play_arrow

• question_answer58)
  Let \[z\]be a purely imaginary number such that \[\operatorname{Im}\,(z)>0\]. Then \[arg(z)\] is equal to
  A)
  \[\pi \] done clear
  B)
  \[\frac{\pi }{2}\] done clear
  C)
  0 done clear
  D)
  \[-\frac{\pi }{2}\] done clear
  View Solution play_arrow

• question_answer59)
  Let \[z\] be a purely imaginary number such that \[\operatorname{Im}(z)<0\]. Then  \[arg\,(z)\] is equal to
  A)
  \[\pi \] done clear
  B)
  \[\frac{\pi }{2}\] done clear
  C)
  0 done clear
  D)
  \[-\frac{\pi }{2}\] done clear
  View Solution play_arrow

• question_answer60)
  If \[z\] is a purely real number such that \[\operatorname{Re}(z)<0\],  then \[arg(z)\] is equal to
  A)
  \[\pi \] done clear
  B)
  \[\frac{\pi }{2}\] done clear
  C)
  0 done clear
  D)
  \[-\frac{\pi }{2}\] done clear
  View Solution play_arrow

• question_answer61)
  Let \[z\] be a complex number. Then the angle between vectors \[z\] and \[-iz\] is
  A)
  \[\pi \] done clear
  B)
  0 done clear
  C)
  \[-\frac{\pi }{2}\] done clear
  D)
  None of these done clear
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• question_answer62)
  For any two complex numbers \[{{z}_{1}},{{z}_{2}}\]we have \[|{{z}_{1}}+{{z}_{2}}{{|}^{2}}=\] \[|{{z}_{1}}{{|}^{2}}+|{{z}_{2}}{{|}^{2}}\] then
  A)
  \[\operatorname{Re}\left( \frac{{{z}_{1}}}{{{z}_{2}}} \right)=0\] done clear
  B)
  \[\operatorname{Im}\left( \frac{{{z}_{1}}}{{{z}_{2}}} \right)=0\] done clear
  C)
  \[\operatorname{Re}({{z}_{1}}{{z}_{2}})=0\] done clear
  D)
  \[\operatorname{Im}({{z}_{1}}{{z}_{2}})=0\] done clear
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• question_answer63)
  If for complex numbers \[{{z}_{1}}\] and \[{{z}_{2}}\], \[arg({{z}_{1}}/{{z}_{2}})=0,\] then \[|{{z}_{1}}-{{z}_{2}}|\] is equal to
  A)
  \[|{{z}_{1}}|+|{{z}_{2}}|\] done clear
  B)
  \[|{{z}_{1}}|-|{{z}_{2}}|\] done clear
  C)
  \[||{{z}_{1}}|-|{{z}_{2}}||\] done clear
  D)
  0 done clear
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• question_answer64)
  If  \[|{{z}_{1}}+{{z}_{2}}|=|{{z}_{1}}-{{z}_{2}}|\], then the difference in the amplitudes of \[{{z}_{1}}\] and \[{{z}_{2}}\] is [EAMCET 1985]
  A)
  \[\frac{\pi }{4}\] done clear
  B)
  \[\frac{\pi }{3}\] done clear
  C)
  \[\frac{\pi }{2}\] done clear
  D)
  0 done clear
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• question_answer65)
  If \[|{{z}_{1}}|\,=\,|{{z}_{2}}|\]and \[arg\,\,\left( \frac{{{z}_{1}}}{{{z}_{2}}} \right)=\pi \], then \[{{z}_{1}}+{{z}_{2}}\]is equal to
  A)
  0 done clear
  B)
  Purely imaginary done clear
  C)
  Purely real done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer66)
  If \[0<amp\text{ (z)}<\pi \text{,}\]then amp\[(z)\]- amp\[(-z)=\]
  A)
  0 done clear
  B)
  \[2\,amp\text{ }(z)\] done clear
  C)
  \[\pi \] done clear
  D)
  \[-\pi \] done clear
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• question_answer67)
  If \[z=1-\cos \alpha +i\sin \alpha \], then amp\[z\]=
  A)
    \[\frac{\alpha }{2}\] done clear
  B)
   \[-\frac{\alpha }{2}\] done clear
  C)
    \[\frac{\pi }{2}+\frac{\alpha }{2}\] done clear
  D)
   \[\frac{\pi }{2}-\frac{\alpha }{2}\] done clear
  View Solution play_arrow

• question_answer68)
  If \[{{z}_{1}},{{z}_{2}}\in C\], then \[amp\,\left( \frac{{{\text{z}}_{\text{1}}}}{{{{\text{\bar{z}}}}_{\text{2}}}} \right)=\]
  A)
  \[amp\,({{z}_{1}}{{\overline{z}}_{2}})\] done clear
  B)
  \[amp\,({{\overline{z}}_{1}}{{z}_{2}})\] done clear
  C)
  \[amp\,\left( \frac{{{z}_{2}}}{{{{\bar{z}}}_{1}}} \right)\] done clear
  D)
  \[amp\,\left( \frac{{{z}_{1}}}{{{z}_{2}}} \right)\] done clear
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• question_answer69)
  The argument of the complex number \[\frac{13-5i}{4-9i}\]is [MP PET 1997]
  A)
  \[\frac{\pi }{3}\] done clear
  B)
    \[\frac{\pi }{4}\] done clear
  C)
  \[\frac{\pi }{5}\] done clear
  D)
  \[\frac{\pi }{6}\] done clear
  View Solution play_arrow

• question_answer70)
  If  \[|{{z}_{1}}|\,=\,|{{z}_{2}}|\] and \[amp\,{{z}_{1}}+amp\,\,{{z}_{2}}=0\], then [MP PET 1999]
  A)
  \[{{z}_{1}}={{z}_{2}}\] done clear
  B)
  \[{{\bar{z}}_{1}}={{z}_{2}}\] done clear
  C)
  \[{{z}_{1}}+{{z}_{2}}=0\] done clear
  D)
  \[{{\bar{z}}_{1}}={{\bar{z}}_{2}}\] done clear
  View Solution play_arrow

• question_answer71)
  \[|{{z}_{1}}+{{z}_{2}}|\,=\,|{{z}_{1}}|+|{{z}_{2}}|\] is possible if [MP PET 1999; Pb. CET 2002]
  A)
  \[{{z}_{2}}={{\overline{z}}_{1}}\] done clear
  B)
  \[{{z}_{2}}=\frac{1}{{{z}_{1}}}\] done clear
  C)
  \[arg\,({{z}_{1}})=\]arg \[({{z}_{2}})\] done clear
  D)
  \[|{{z}_{1}}|\,=\,|{{z}_{2}}|\] done clear
  View Solution play_arrow

• question_answer72)
  Amplitude of  \[\left( \frac{1-i}{1+i} \right)\] is     [RPET 1996]
  A)
  -p/2 done clear
  B)
  p/2 done clear
  C)
    p/4 done clear
  D)
  p/6 done clear
  View Solution play_arrow

• question_answer73)
  Which of the following are correct for any two complex numbers \[{{z}_{1}}\] and \[{{z}_{2}}\] [Roorkee 1998]
  A)
  \[|{{z}_{1}}{{z}_{2}}|\,=\,|{{z}_{1}}||{{z}_{2}}|\] done clear
  B)
    \[arg\,\,({{z}_{1}}{{z}_{2}})=(arg\,{{z}_{1}})(arg\,{{z}_{2}})\] done clear
  C)
  \[|{{z}_{1}}+{{z}_{2}}|\,=\,|{{z}_{1}}|+|{{z}_{2}}|\] done clear
  D)
  \[|{{z}_{1}}-{{z}_{2}}|\,\ge \,|{{z}_{1}}|-|{{z}_{2}}|\] done clear
  View Solution play_arrow

• question_answer74)
   The amplitude of \[\frac{1+\sqrt{3}\,i}{\sqrt{3}+i}\] is [DCE 1999; Karnataka CET 2005]
  A)
  \[\frac{\pi }{6}\] done clear
  B)
  \[-\frac{\pi }{6}\] done clear
  C)
  \[\frac{\pi }{3}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer75)
  The amplitude of 0 is [RPET 2000]
  A)
  0 done clear
  B)
  \[\pi /2\] done clear
  C)
  \[\pi \] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer76)
  If \[arg\,z<0\] then \[arg\,(-z)-arg\,(z)\] is equal to [IIT Screening 2000]
  A)
  \[\pi \] done clear
  B)
  \[-\pi \] done clear
  C)
  \[-\frac{\pi }{2}\] done clear
  D)
  \[\frac{\pi }{2}\] done clear
  View Solution play_arrow

• question_answer77)
  The amplitude of \[\frac{1+\sqrt{3}\,i}{\sqrt{3}-i}\] is [RPET 2001]
  A)
  0 done clear
  B)
  \[\pi /6\] done clear
  C)
  \[\pi /3\] done clear
  D)
  \[\pi /2\] done clear
  View Solution play_arrow

• question_answer78)
  If \[z=\frac{-2}{1+\sqrt{3}\,i}\] then the value of \[arg\,(z)\] is [ Orissa JEE 2002]
  A)
  \[\pi \] done clear
  B)
  \[\pi /3\] done clear
  C)
  \[2\pi /3\] done clear
  D)
  \[\pi /4\] done clear
  View Solution play_arrow

• question_answer79)
  If \[z=\cos \frac{\pi }{6}+i\sin \frac{\pi }{6}\] then [AMU 2002]
  A)
  \[|z|\,=1,\,\,\,\,arg\,z=\frac{\pi }{4}\]  done clear
  B)
  \[|z|\,=1,arg\,z=\frac{\pi }{6}\] done clear
  C)
  \[|z|\,=\frac{\sqrt{3}}{2},\,arg\,z=\frac{5\pi }{24}\]     done clear
  D)
  \[|z|\,=\frac{\sqrt{3}}{2},\,\,arg\,z={{\tan }^{-1}}\frac{1}{\sqrt{2}}\] done clear
  View Solution play_arrow

• question_answer80)
  The amplitude of \[\sin \frac{\pi }{5}+i\,\left( 1-\cos \frac{\pi }{5} \right)\]  [Karnataka CET 2003]
  A)
  \[\pi /5\] done clear
  B)
  \[2\pi /5\] done clear
  C)
  \[\pi /10\] done clear
  D)
  \[\pi /15\] done clear
  View Solution play_arrow

• question_answer81)
  Argument of \[-1-i\sqrt{3}\] is [RPET 2003]
  A)
  \[\frac{2\pi }{3}\] done clear
  B)
  \[\frac{\pi }{3}\] done clear
  C)
  \[-\frac{\pi }{3}\] done clear
  D)
  \[-\frac{2\pi }{3}\] done clear
  View Solution play_arrow

• question_answer82)
  If z and \[\omega \]are two non-zero complex numbers such that \[|z\omega |\,=1\] and \[arg(z)-arg(\omega )=\frac{\pi }{2},\] then \[\bar{z}\omega \] is equal to [AIEEE 2003]
  A)
  1 done clear
  B)
  - 1 done clear
  C)
  i done clear
  D)
  - i done clear
  View Solution play_arrow

• question_answer83)
  The sum of amplitude of z and another complex number is \[\pi \]. The other complex number can be written[Orissa JEE 2004]
  A)
  \[\bar{z}\] done clear
  B)
  \[-\overline{z}\] done clear
  C)
  z done clear
  D)
  \[-z\] done clear
  View Solution play_arrow

• question_answer84)
  The modulus and amplitude of \[\frac{1+2i}{1-{{(1-i)}^{2}}}\] are  [Karnataka CET 2005]
  A)
  \[\sqrt{2}\text{ and }\frac{\pi }{6}\] done clear
  B)
  1 and 0 done clear
  C)
  1 and \[\frac{\pi }{3}\] done clear
  D)
  1 and \[\frac{\pi }{4}\] done clear
  View Solution play_arrow

• question_answer85)
  If \[{{z}_{1}}=1+2i\] and \[{{z}_{2}}=3+5i\], and then \[\operatorname{Re}\,\left( \frac{{{{\bar{z}}}_{2}}{{z}_{1}}}{{{z}_{2}}} \right)\] is equal to [J & K 2005]
  A)
  \[\frac{-31}{17}\] done clear
  B)
  \[\frac{17}{22}\] done clear
  C)
  \[\frac{-17}{31}\] done clear
  D)
  \[\frac{22}{17}\] done clear
  View Solution play_arrow

• question_answer86)
  If \[(3+i)z=(3-i)\bar{z},\] then complex number z is [AMU 2005]
  A)
  \[x\,(3-i),\,x\in R\] done clear
  B)
  \[\frac{x}{3+i},\,x\in R\] done clear
  C)
  \[x(3+i),\,x\in R\] done clear
  D)
  \[x(-3+i),\,x\in R\] done clear
  View Solution play_arrow

• question_answer87)
  If \[{{(\sqrt{8}+i)}^{50}}={{3}^{49}}(a+ib)\] then \[{{a}^{2}}+{{b}^{2}}\] is [Kerala (Engg.) 2005]
  A)
  3 done clear
  B)
  8 done clear
  C)
  9 done clear
  D)
  \[\sqrt{8}\] done clear
  E)
  4 done clear
  View Solution play_arrow