-
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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
View Solution play_arrow
-
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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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