-
question_answer1)
The number of real values of \[a\] satisfying the equation \[{{a}^{2}}-2a\sin x+1=0\] is
A)
Zero done
clear
B)
One done
clear
C)
Two done
clear
D)
Infinite done
clear
View Solution play_arrow
-
question_answer2)
For positive integers \[{{n}_{1}},{{n}_{2}}\]the value of the expression \[{{(1+i)}^{{{n}_{1}}}}+{{(1+{{i}^{3}})}^{{{n}_{1}}}}+{{(1+{{i}^{5}})}^{{{n}_{2}}}}+{{(1+{{i}^{7}})}^{{{n}_{2}}}}\]where \[i=\sqrt{-1}\] is a real number if and only if [IIT 1996]
A)
\[{{n}_{1}}={{n}_{2}}+1\] done
clear
B)
\[{{n}_{1}}={{n}_{2}}-1\] done
clear
C)
\[{{n}_{1}}={{n}_{2}}\] done
clear
D)
\[{{n}_{1}}>0,{{n}_{2}}>0\] done
clear
View Solution play_arrow
-
question_answer3)
Given that the equation \[{{z}^{2}}+(p+iq)z+r+i\,s=0,\] where \[p,q,r,s\] are real and non-zero has a real root, then
A)
\[pqr={{r}^{2}}+{{p}^{2}}s\] done
clear
B)
\[prs={{q}^{2}}+{{r}^{2}}p\] done
clear
C)
\[qrs={{p}^{2}}+{{s}^{2}}q\] done
clear
D)
\[pqs={{s}^{2}}+{{q}^{2}}r\] done
clear
View Solution play_arrow
-
question_answer4)
If \[x=-5+2\sqrt{-4},\] then the value of the expression \[{{x}^{4}}+9{{x}^{3}}+35{{x}^{2}}-x+4\] is [IIT 1972]
A)
160 done
clear
B)
\[-160\] done
clear
C)
60 done
clear
D)
\[-60\] done
clear
View Solution play_arrow
-
question_answer5)
If \[\sqrt{3}+i=(a+ib)(c+id)\], then \[{{\tan }^{-1}}\left( \frac{b}{a} \right)+\] \[{{\tan }^{-1}}\left( \frac{d}{c} \right)\] has the value
A)
\[\frac{\pi }{3}+2n\pi ,n\in I\] done
clear
B)
\[n\pi +\frac{\pi }{6},n\in I\] done
clear
C)
\[n\pi -\frac{\pi }{3},n\in I\] done
clear
D)
\[2n\pi -\frac{\pi }{3},n\in I\] done
clear
View Solution play_arrow
-
question_answer6)
If \[a=\cos \alpha +i\,\sin \alpha ,\,\,b=\cos \beta +i\,\sin \beta ,\]\[c=\cos \gamma +i\,\sin \gamma \,\,\text{and}\,\,\frac{b}{c}+\frac{c}{a}+\frac{a}{b}=1,\] then \[\cos (\beta -\gamma )+\cos (\gamma -\alpha )+\cos (\alpha -\beta )\] is equal to [RPET 2001]
A)
3/2 done
clear
B)
- 3/2 done
clear
C)
0 done
clear
D)
1 done
clear
View Solution play_arrow
-
question_answer7)
If \[(1+i)(1+2i)(1+3i).....(1+ni)=a+ib\], then 2.5.10....\[(1+{{n}^{2}})\] is equal to [Karnataka CET 2002; Kerala (Engg.) 2002]
A)
\[{{a}^{2}}-{{b}^{2}}\] done
clear
B)
\[{{a}^{2}}+{{b}^{2}}\] done
clear
C)
\[\sqrt{{{a}^{2}}+{{b}^{2}}}\] done
clear
D)
\[\sqrt{{{a}^{2}}-{{b}^{2}}}\] done
clear
View Solution play_arrow
-
question_answer8)
If \[z\] is a complex number, then the minimum value of \[|z|+|z-1|\] is [Roorkee 1992]
A)
1 done
clear
B)
0 done
clear
C)
1/2 done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer9)
For any two complex numbers \[{{z}_{1}}\]and\[{{z}_{2}}\] and any real numbers a and b; \[|(a{{z}_{1}}-b{{z}_{2}}){{|}^{2}}+|(b{{z}_{1}}+a{{z}_{2}}){{|}^{2}}=\] [IIT 1988]
A)
\[({{a}^{2}}+{{b}^{2}})(|{{z}_{1}}|+|{{z}_{2}}|)\] done
clear
B)
\[({{a}^{2}}+{{b}^{2}})(|{{z}_{1}}{{|}^{2}}+|{{z}_{2}}{{|}^{2}})\] done
clear
C)
\[({{a}^{2}}+{{b}^{2}})(|{{z}_{1}}{{|}^{2}}-|{{z}_{2}}{{|}^{2}})\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer10)
The locus of \[z\]satisfying the inequality \[{{\log }_{1/3}}|z+1|\,>\] \[{{\log }_{1/3}}|z-1|\] is
A)
\[R\,(z)<0\] done
clear
B)
\[R\,(z)>0\] done
clear
C)
\[I\,(z)<0\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer11)
If \[{{z}_{1}}=a+ib\] and \[{{z}_{2}}=c+id\] are complex numbers such that \[|{{z}_{1}}|\,=\,|{{z}_{2}}|=1\] and \[R({{z}_{1}}\overline{{{z}_{2}}})=0,\] then the pair of complex numbers \[{{w}_{1}}=a+ic\] and \[{{w}_{2}}=b+id\] satisfies [IIT 1985]
A)
\[|{{w}_{1}}|=1\] done
clear
B)
\[|{{w}_{2}}|=1\] done
clear
C)
\[R({{w}_{1}}\overline{{{w}_{2}}})=0,\] done
clear
D)
All the above done
clear
View Solution play_arrow
-
question_answer12)
Let\[z\]and \[w\] be two complex numbers such that \[|z|\,\le 1,\] \[|w|\,\le 1\]and\[|z+iw|\,=\,|z-i\overline{w}|=2\]. Then \[z\] is equal to [IIT 1995]
A)
1 or \[i\] done
clear
B)
\[i\] or \[-i\] done
clear
C)
1 or - 1 done
clear
D)
\[i\]or -1 done
clear
View Solution play_arrow
-
question_answer13)
The maximum distance from the origin of coordinates to the point \[z\] satisfying the equation \[\left| z+\frac{1}{z} \right|=a\]is
A)
\[\frac{1}{2}(\sqrt{{{a}^{2}}+1}+a)\] done
clear
B)
\[\frac{1}{2}(\sqrt{{{a}^{2}}+2}+a)\] done
clear
C)
\[\frac{1}{2}(\sqrt{{{a}^{2}}+4}+a)\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer14)
Find the complex number z satisfying the equations \[\left| \frac{z-12}{z-8i} \right|=\frac{5}{3},\left| \frac{z-4}{z-8} \right|=1\] [Roorkee 1993]
A)
6 done
clear
B)
\[6\pm 8i\] done
clear
C)
\[6+8i,\,6+17i\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer15)
If \[{{z}_{1}},{{z}_{2}},{{z}_{3}}\] are complex numbers such that \[|{{z}_{1}}|\,=\,|{{z}_{2}}|\,=\] \[\,|{{z}_{3}}|\,=\] \[\left| \frac{1}{{{z}_{1}}}+\frac{1}{{{z}_{2}}}+\frac{1}{{{z}_{3}}} \right|=1\,,\] then\[\text{ }|{{z}_{1}}+{{z}_{2}}+{{z}_{3}}|\] is [MP PET 2004; IIT Screening 2000]
A)
Equal to 1 done
clear
B)
Less than 1 done
clear
C)
Greater than 3 done
clear
D)
Equal to 3 done
clear
View Solution play_arrow
-
question_answer16)
If \[{{z}_{1}}=10+6i,{{z}_{2}}=4+6i\] and \[z\] is a complex number such that \[amp\left( \frac{z-{{z}_{1}}}{z-{{z}_{2}}} \right)=\frac{\pi }{4},\] then the value of \[|z-7-9i|\] is equal to [IIT 1990]
A)
\[\sqrt{2}\] done
clear
B)
\[2\sqrt{2}\] done
clear
C)
\[3\sqrt{2}\] done
clear
D)
\[2\sqrt{3}\] done
clear
View Solution play_arrow
-
question_answer17)
If \[{{z}_{1}},{{z}_{2}},{{z}_{3}}\]be three non-zero complex number, such that \[{{z}_{2}}\ne {{z}_{1}},a=|{{z}_{1}}|,b=|{{z}_{2}}|\] and \[c=|{{z}_{3}}|\] suppose that \[\left| \begin{matrix} a & b & c \\ b & c & a \\ c & a & b \\ \end{matrix} \right|=0\], then \[arg\left( \frac{{{z}_{3}}}{{{z}_{2}}} \right)\] is equal to
A)
\[arg{{\left( \frac{{{z}_{2}}-{{z}_{1}}}{{{z}_{3}}-{{z}_{1}}} \right)}^{2}}\] done
clear
B)
\[arg\left( \frac{{{z}_{2}}-{{z}_{1}}}{{{z}_{3}}-{{z}_{1}}} \right)\] done
clear
C)
\[arg{{\left( \frac{{{z}_{3}}-{{z}_{1}}}{{{z}_{2}}-{{z}_{1}}} \right)}^{2}}\] done
clear
D)
\[arg\left( \frac{{{z}_{3}}-{{z}_{1}}}{{{z}_{2}}-{{z}_{1}}} \right)\] done
clear
View Solution play_arrow
-
question_answer18)
Let \[z\] and \[w\] be the two non-zero complex numbers such that \[|z|\,=\,|w|\] and \[arg\,z+arg\,w=\pi \]. Then \[z\] is equal to [IIT 1995; AIEEE 2002]
A)
\[w\] done
clear
B)
\[-w\] done
clear
C)
\[\overline{w}\] done
clear
D)
\[-\overline{w}\] done
clear
View Solution play_arrow
-
question_answer19)
If \[|z-25i|\le 15\], then \[|\max .amp(z)-\min .amp(z)|=\]
A)
\[{{\cos }^{-1}}\left( \frac{3}{5} \right)\] done
clear
B)
\[\pi -2{{\cos }^{-1}}\left( \frac{3}{5} \right)\] done
clear
C)
\[\frac{\pi }{2}+{{\cos }^{-1}}\left( \frac{3}{5} \right)\] done
clear
D)
\[{{\sin }^{-1}}\left( \frac{3}{5} \right)-{{\cos }^{-1}}\left( \frac{3}{5} \right)\] done
clear
View Solution play_arrow
-
question_answer20)
If \[{{z}_{1}},{{z}_{2}}\] and \[{{z}_{3}},{{z}_{4}}\] are two pairs of conjugate complex numbers, then \[arg\left( \frac{{{z}_{1}}}{{{z}_{4}}} \right)+arg\left( \frac{{{z}_{2}}}{{{z}_{3}}} \right)\] equals
A)
0 done
clear
B)
\[\frac{\pi }{2}\] done
clear
C)
\[\frac{3\pi }{2}\] done
clear
D)
\[\pi \] done
clear
View Solution play_arrow
-
question_answer21)
Let \[z,w\]be complex numbers such that \[\overline{z}+i\overline{w}=0\]and \[arg\,\,zw=\pi \]. Then arg z equals [AIEEE 2004]
A)
\[5\pi /4\] done
clear
B)
\[\pi /2\] done
clear
C)
\[3\pi /4\] done
clear
D)
\[\pi /4\] done
clear
View Solution play_arrow
-
question_answer22)
If \[{{(1+x)}^{n}}={{C}_{0}}+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}+.....+{{C}_{n}}{{x}^{n}},\] then the value of \[{{C}_{0}}-{{C}_{2}}+{{C}_{4}}-{{C}_{6}}+.....\]is
A)
\[{{2}^{n}}\] done
clear
B)
\[{{2}^{n}}\cos \frac{n\pi }{2}\] done
clear
C)
\[{{2}^{n}}\sin \frac{n\pi }{2}\] done
clear
D)
\[{{2}^{n/2}}\cos \frac{n\pi }{4}\] done
clear
View Solution play_arrow
-
question_answer23)
If \[x=\cos \theta +i\sin \theta \] and \[y=\cos \varphi +i\sin \varphi \], then \[{{x}^{m}}{{y}^{n}}+{{x}^{-m}}{{y}^{-n}}\] is equal to
A)
\[\cos (m\theta +n\varphi )\] done
clear
B)
\[\cos (m\theta +n\varphi )\] done
clear
C)
\[2\cos (m\theta +n\varphi )\] done
clear
D)
\[2\cos (m\theta -n\varphi )\] done
clear
View Solution play_arrow
-
question_answer24)
The value of \[\sum\limits_{r=1}^{8}{\left( \sin \frac{2r\pi }{9}+i\cos \frac{2r\pi }{9} \right)}\]is
A)
\[-1\] done
clear
B)
1 done
clear
C)
\[i\] done
clear
D)
\[-i\] done
clear
View Solution play_arrow
-
question_answer25)
If \[a,b,c\] and\[u,v,w\] are complex numbers representing the vertices of two triangles such that \[c=(1-r)a+rb\] and \[w=(1-r)u+rv\], where r is a complex number, then the two triangles
A)
Have the same area done
clear
B)
Are similar done
clear
C)
Are congruent done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer26)
Suppose \[{{z}_{1}},\,{{z}_{2}},\,{{z}_{3}}\] are the vertices of an equilateral triangle inscribed in the circle \[|z|\,=2\]. If \[{{z}_{1}}=1+i\sqrt{3},\] then values of \[{{z}_{3}}\] and \[{{z}_{2}}\] are respectively [IIT 1994]
A)
\[-2,\,1-i\sqrt{3}\] done
clear
B)
\[2,\,1+i\sqrt{3}\] done
clear
C)
\[1+i\sqrt{3},-2\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer27)
If the complex number \[{{z}_{1}},{{z}_{2}}\] the origin form an equilateral triangle then \[z_{1}^{2}+z_{2}^{2}=\] [IIT 1983]
A)
\[{{z}_{1}}\,{{z}_{2}}\] done
clear
B)
\[{{z}_{1}}\,\overline{{{z}_{2}}}\] done
clear
C)
\[\overline{{{z}_{2}}}\,{{z}_{1}}\] done
clear
D)
\[|{{z}_{1}}{{|}^{2}}=|{{z}_{2}}{{|}^{2}}\] done
clear
View Solution play_arrow
-
question_answer28)
If at least one value of the complex number \[z=x+iy\] satisfy the condition \[|z+\sqrt{2}|={{a}^{2}}-3a+2\] and the inequality \[|z+i\sqrt{2}|<{{a}^{2}}\], then
A)
\[a>2\] done
clear
B)
\[a=2\] done
clear
C)
\[a<2\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer29)
If z, iz and \[z+iz\] are the vertices of a triangle whose area is 2 units, then the value of \[|z|\] is [RPET 2000]
A)
- 2 done
clear
B)
2 done
clear
C)
4 done
clear
D)
8 done
clear
View Solution play_arrow
-
question_answer30)
If \[{{z}^{2}}+z|z|+|z{{|}^{2}}=0\], then the locus of \[z\] is
A)
A circle done
clear
B)
A straight line done
clear
C)
A pair of straight lines done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer31)
If \[\cos \alpha +\cos \beta +\cos \gamma =\sin \alpha +\sin \beta +\sin \gamma =0\] then \[\cos 3\alpha +\cos 3\beta +\cos 3\gamma \] equals to [Karnataka CET 2000]
A)
0 done
clear
B)
\[\cos (\alpha +\beta +\gamma )\] done
clear
C)
\[3\cos (\alpha +\beta +\gamma )\] done
clear
D)
\[3\sin (\alpha +\beta +\gamma )\] done
clear
View Solution play_arrow
-
question_answer32)
If \[{{z}_{r}}=\cos \frac{r\alpha }{{{n}^{2}}}+i\sin \frac{r\alpha }{{{n}^{2}}},\] where r = 1, 2, 3,?.,n, then \[\underset{n\to \infty }{\mathop{\lim }}\,\,\,{{z}_{1}}{{z}_{2}}{{z}_{3}}...{{z}_{n}}\] is equal to [UPSEAT 2001]
A)
\[\cos \alpha +i\,\sin \alpha \] done
clear
B)
\[\cos (\alpha /2)-i\sin (\alpha /2)\] done
clear
C)
\[{{e}^{i\alpha /2}}\] done
clear
D)
\[\sqrt[3]{{{e}^{i\alpha }}}\] done
clear
View Solution play_arrow
-
question_answer33)
If the cube roots of unity be \[1,\omega ,{{\omega }^{2}},\] then the roots of the equation \[{{(x-1)}^{3}}+8=0\]are [IIT 1979; MNR 1986; DCE 2000; AIEEE 2005]
A)
\[-1,\,1+2\omega ,\,1+2{{\omega }^{2}}\] done
clear
B)
\[-1,\,1-2\omega ,\,1-2{{\omega }^{2}}\] done
clear
C)
\[-1,\,-1,\,-1\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer34)
If \[1,\omega ,{{\omega }^{2}},{{\omega }^{3}}.......,{{\omega }^{n-1}}\] are the \[n,{{n}^{th}}\] roots of unity, then \[(1-\omega )(1-{{\omega }^{2}}).....(1-{{\omega }^{n-1}})\] equals [MNR 1992; IIT 1984; DCE 2001; MP PET 2004]
A)
0 done
clear
B)
1 done
clear
C)
\[n\] done
clear
D)
\[{{n}^{2}}\] done
clear
View Solution play_arrow
-
question_answer35)
The value of the expression \[1.(2-\omega )(2-{{\omega }^{2}})+2.(3-\omega )(3-{{\omega }^{2}})+.......\]\[....+(n-1).(n-\omega )(n-{{\omega }^{2}}),\]where \[\omega \] is an imaginary cube root of unity, is [IIT 1996]
A)
\[\frac{1}{2}(n-1)n({{n}^{2}}+3n+4)\] done
clear
B)
\[\frac{1}{4}(n-1)n({{n}^{2}}+3n+4)\] done
clear
C)
\[\frac{1}{2}(n+1)n({{n}^{2}}+3n+4)\] done
clear
D)
\[\frac{1}{4}(n+1)n({{n}^{2}}+3n+4)\] done
clear
View Solution play_arrow
-
question_answer36)
If \[i=\sqrt{-1},\] then \[4+5{{\left( -\frac{1}{2}+\frac{i\sqrt{3}}{2} \right)}^{334}}\] \[+3{{\left( -\frac{1}{2}+\frac{i\sqrt{3}}{2} \right)}^{365}}\]is equal to [IIT 1999]
A)
\[1-i\sqrt{3}\] done
clear
B)
\[-1+i\sqrt{3}\] done
clear
C)
\[i\sqrt{3}\] done
clear
D)
\[-i\sqrt{3}\] done
clear
View Solution play_arrow
-
question_answer37)
If \[a=\cos (2\pi /7)+i\,\sin (2\pi /7),\] then the quadratic equation whose roots are \[\alpha =a+{{a}^{2}}+{{a}^{4}}\] and \[\beta ={{a}^{3}}+{{a}^{5}}+{{a}^{6}}\] is [RPET 2000]
A)
\[{{x}^{2}}-x+2=0\] done
clear
B)
\[{{x}^{2}}+x-2=0\] done
clear
C)
\[{{x}^{2}}-x-2=0\] done
clear
D)
\[{{x}^{2}}+x+2=0\] done
clear
View Solution play_arrow
-
question_answer38)
Let \[{{z}_{1}}\] and \[{{z}_{2}}\] be nth roots of unity which are ends of a line segment that subtend a right angle at the origin. Then n must be of the form [IIT Screening 2001; Karnataka 2002]
A)
4k + 1 done
clear
B)
4k + 2 done
clear
C)
4k + 3 done
clear
D)
4k done
clear
View Solution play_arrow
-
question_answer39)
Let \[\omega \] is an imaginary cube roots of unity then the value of\[2(\omega +1)({{\omega }^{2}}+1)+3(2\omega +1)(2{{\omega }^{2}}+1)+.....\]\[+(n+1)(n\omega +1)(n{{\omega }^{2}}+1)\] is [Orissa JEE 2002]
A)
\[{{\left[ \frac{n(n+1)}{2} \right]}^{2}}+n\] done
clear
B)
\[{{\left[ \frac{n(n+1)}{2} \right]}^{2}}\] done
clear
C)
\[{{\left[ \frac{n(n+1)}{2} \right]}^{2}}-n\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer40)
\[\omega \] is an imaginary cube root of unity. If \[{{(1+{{\omega }^{2}})}^{m}}=\] \[{{(1+{{\omega }^{4}})}^{m}},\] then least positive integral value of m is [IIT Screening 2004]
A)
6 done
clear
B)
5 done
clear
C)
4 done
clear
D)
3 done
clear
View Solution play_arrow