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question_answer1) The number of real values of \[a\] satisfying the equation \[{{a}^{2}}-2a\sin x+1=0\] is
A) Zero
B) One
C) Two
D) Infinite
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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) \[{{n}_{1}}={{n}_{2}}+1\]
B) \[{{n}_{1}}={{n}_{2}}-1\]
C) \[{{n}_{1}}={{n}_{2}}\]
D) \[{{n}_{1}}>0,{{n}_{2}}>0\]
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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) \[pqr={{r}^{2}}+{{p}^{2}}s\]
B) \[prs={{q}^{2}}+{{r}^{2}}p\]
C) \[qrs={{p}^{2}}+{{s}^{2}}q\]
D) \[pqs={{s}^{2}}+{{q}^{2}}r\]
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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
B) \[-160\]
C) 60
D) \[-60\]
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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) \[\frac{\pi }{3}+2n\pi ,n\in I\]
B) \[n\pi +\frac{\pi }{6},n\in I\]
C) \[n\pi -\frac{\pi }{3},n\in I\]
D) \[2n\pi -\frac{\pi }{3},n\in I\]
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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
B) - 3/2
C) 0
D) 1
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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) \[{{a}^{2}}-{{b}^{2}}\]
B) \[{{a}^{2}}+{{b}^{2}}\]
C) \[\sqrt{{{a}^{2}}+{{b}^{2}}}\]
D) \[\sqrt{{{a}^{2}}-{{b}^{2}}}\]
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question_answer8) If \[z\] is a complex number, then the minimum value of \[|z|+|z-1|\] is [Roorkee 1992]
A) 1
B) 0
C) 1/2
D) None of these
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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) \[({{a}^{2}}+{{b}^{2}})(|{{z}_{1}}|+|{{z}_{2}}|)\]
B) \[({{a}^{2}}+{{b}^{2}})(|{{z}_{1}}{{|}^{2}}+|{{z}_{2}}{{|}^{2}})\]
C) \[({{a}^{2}}+{{b}^{2}})(|{{z}_{1}}{{|}^{2}}-|{{z}_{2}}{{|}^{2}})\]
D) None of these
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question_answer10) The locus of \[z\]satisfying the inequality \[{{\log }_{1/3}}|z+1|\,>\] \[{{\log }_{1/3}}|z-1|\] is
A) \[R\,(z)<0\]
B) \[R\,(z)>0\]
C) \[I\,(z)<0\]
D) None of these
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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) \[|{{w}_{1}}|=1\]
B) \[|{{w}_{2}}|=1\]
C) \[R({{w}_{1}}\overline{{{w}_{2}}})=0,\]
D) All the above
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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 \[i\]
B) \[i\] or \[-i\]
C) 1 or - 1
D) \[i\]or -1
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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) \[\frac{1}{2}(\sqrt{{{a}^{2}}+1}+a)\]
B) \[\frac{1}{2}(\sqrt{{{a}^{2}}+2}+a)\]
C) \[\frac{1}{2}(\sqrt{{{a}^{2}}+4}+a)\]
D) None of these
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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
B) \[6\pm 8i\]
C) \[6+8i,\,6+17i\]
D) None of these
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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
B) Less than 1
C) Greater than 3
D) Equal to 3
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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) \[\sqrt{2}\]
B) \[2\sqrt{2}\]
C) \[3\sqrt{2}\]
D) \[2\sqrt{3}\]
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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) \[arg{{\left( \frac{{{z}_{2}}-{{z}_{1}}}{{{z}_{3}}-{{z}_{1}}} \right)}^{2}}\]
B) \[arg\left( \frac{{{z}_{2}}-{{z}_{1}}}{{{z}_{3}}-{{z}_{1}}} \right)\]
C) \[arg{{\left( \frac{{{z}_{3}}-{{z}_{1}}}{{{z}_{2}}-{{z}_{1}}} \right)}^{2}}\]
D) \[arg\left( \frac{{{z}_{3}}-{{z}_{1}}}{{{z}_{2}}-{{z}_{1}}} \right)\]
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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) \[w\]
B) \[-w\]
C) \[\overline{w}\]
D) \[-\overline{w}\]
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question_answer19) If \[|z-25i|\le 15\], then \[|\max .amp(z)-\min .amp(z)|=\]
A) \[{{\cos }^{-1}}\left( \frac{3}{5} \right)\]
B) \[\pi -2{{\cos }^{-1}}\left( \frac{3}{5} \right)\]
C) \[\frac{\pi }{2}+{{\cos }^{-1}}\left( \frac{3}{5} \right)\]
D) \[{{\sin }^{-1}}\left( \frac{3}{5} \right)-{{\cos }^{-1}}\left( \frac{3}{5} \right)\]
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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
B) \[\frac{\pi }{2}\]
C) \[\frac{3\pi }{2}\]
D) \[\pi \]
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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) \[5\pi /4\]
B) \[\pi /2\]
C) \[3\pi /4\]
D) \[\pi /4\]
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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) \[{{2}^{n}}\]
B) \[{{2}^{n}}\cos \frac{n\pi }{2}\]
C) \[{{2}^{n}}\sin \frac{n\pi }{2}\]
D) \[{{2}^{n/2}}\cos \frac{n\pi }{4}\]
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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) \[\cos (m\theta +n\varphi )\]
B) \[\cos (m\theta +n\varphi )\]
C) \[2\cos (m\theta +n\varphi )\]
D) \[2\cos (m\theta -n\varphi )\]
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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) \[-1\]
B) 1
C) \[i\]
D) \[-i\]
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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
B) Are similar
C) Are congruent
D) None of these
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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) \[-2,\,1-i\sqrt{3}\]
B) \[2,\,1+i\sqrt{3}\]
C) \[1+i\sqrt{3},-2\]
D) None of these
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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) \[{{z}_{1}}\,{{z}_{2}}\]
B) \[{{z}_{1}}\,\overline{{{z}_{2}}}\]
C) \[\overline{{{z}_{2}}}\,{{z}_{1}}\]
D) \[|{{z}_{1}}{{|}^{2}}=|{{z}_{2}}{{|}^{2}}\]
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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) \[a>2\]
B) \[a=2\]
C) \[a<2\]
D) None of these
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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
B) 2
C) 4
D) 8
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question_answer30) If \[{{z}^{2}}+z|z|+|z{{|}^{2}}=0\], then the locus of \[z\] is
A) A circle
B) A straight line
C) A pair of straight lines
D) None of these
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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
B) \[\cos (\alpha +\beta +\gamma )\]
C) \[3\cos (\alpha +\beta +\gamma )\]
D) \[3\sin (\alpha +\beta +\gamma )\]
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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) \[\cos \alpha +i\,\sin \alpha \]
B) \[\cos (\alpha /2)-i\sin (\alpha /2)\]
C) \[{{e}^{i\alpha /2}}\]
D) \[\sqrt[3]{{{e}^{i\alpha }}}\]
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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) \[-1,\,1+2\omega ,\,1+2{{\omega }^{2}}\]
B) \[-1,\,1-2\omega ,\,1-2{{\omega }^{2}}\]
C) \[-1,\,-1,\,-1\]
D) None of these
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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
B) 1
C) \[n\]
D) \[{{n}^{2}}\]
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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) \[\frac{1}{2}(n-1)n({{n}^{2}}+3n+4)\]
B) \[\frac{1}{4}(n-1)n({{n}^{2}}+3n+4)\]
C) \[\frac{1}{2}(n+1)n({{n}^{2}}+3n+4)\]
D) \[\frac{1}{4}(n+1)n({{n}^{2}}+3n+4)\]
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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) \[1-i\sqrt{3}\]
B) \[-1+i\sqrt{3}\]
C) \[i\sqrt{3}\]
D) \[-i\sqrt{3}\]
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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) \[{{x}^{2}}-x+2=0\]
B) \[{{x}^{2}}+x-2=0\]
C) \[{{x}^{2}}-x-2=0\]
D) \[{{x}^{2}}+x+2=0\]
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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
B) 4k + 2
C) 4k + 3
D) 4k
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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) \[{{\left[ \frac{n(n+1)}{2} \right]}^{2}}+n\]
B) \[{{\left[ \frac{n(n+1)}{2} \right]}^{2}}\]
C) \[{{\left[ \frac{n(n+1)}{2} \right]}^{2}}-n\]
D) None of these
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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
B) 5
C) 4
D) 3
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