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

done Self Evaluation Test - Complex Numbers and Quadratic Equations Total Questions - 90

• question_answer1)
  If the roots of the quadratic equation\[{{x}^{2}}+px+q=0\] are \[\tan 30{}^\circ \] and \[\tan 15{}^\circ \] respectively, then the value of \[2+q-p\] is
  A)
  2   done clear
  B)
  3     done clear
  C)
  0                      done clear
  D)
  1 done clear
  View Solution play_arrow

• question_answer2)
  The greatest and the least absolute value of \[z+1,\] where \[|z+4|\le 3\] are respectively
  A)
  6 and 0             done clear
  B)
  10 and 6          done clear
  C)
  4 and 3             done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer3)
  If z and \[\omega \] are two non-zero complex numbers such that \[\left| z\omega  \right|=1\] and \[Arg(z)-Arg(\omega )=\frac{\pi }{2},\]then \[\bar{z}\omega \] is equal to
  A)
  \[-i\]                  done clear
  B)
  \[1\]    done clear
  C)
  \[-1\]     done clear
  D)
  \[i\] done clear
  View Solution play_arrow

• question_answer4)
  If \[\alpha ,\beta ,\gamma \] and a, b, c are complex numbers such that \[\frac{\alpha }{a}+\frac{\beta }{b}+\frac{\gamma }{c}=1+i\] and \[\frac{a}{\alpha }+\frac{b}{\beta }+\frac{c}{\gamma }=0,\] then the value of \[\frac{{{\alpha }^{2}}}{{{a}^{2}}}+\frac{{{\beta }^{2}}}{{{b}^{2}}}+\frac{{{\gamma }^{2}}}{{{c}^{2}}}\] is equal to
  A)
  \[0\]                  done clear
  B)
  \[-1\]    done clear
  C)
  \[2i\]    done clear
  D)
  \[-2i\] done clear
  View Solution play_arrow

• question_answer5)
  \[\sum\limits_{k=33}^{65}{\left( \sin \frac{2k\pi }{8}-i\cos \frac{2k\pi }{8} \right)}\]
  A)
  \[1+i\]               done clear
  B)
  \[1-i\] done clear
  C)
  \[1+\frac{i}{\sqrt{2}}\]  done clear
  D)
  \[\frac{1-i}{\sqrt{2}}\] done clear
  View Solution play_arrow

• question_answer6)
  If \[\alpha ,\beta \] are roots of \[a{{x}^{2}}+bx+b=0,\] then \[\sqrt{\frac{\alpha }{\beta }}+\sqrt{\frac{\beta }{\alpha }}+\sqrt{\frac{b}{a}}\] is (\[{{b}^{2}}\ge 4ab,\] a and b are of same sign)
  A)
  0          done clear
  B)
  1    done clear
  C)
  2                      done clear
  D)
  \[2\sqrt{\frac{b}{a}}\] done clear
  View Solution play_arrow

• question_answer7)
  If \[f({{x}_{1}})-f({{x}_{2}})=f\left( \frac{{{x}_{1}}-{{x}_{2}}}{1-{{x}_{1}}{{x}_{2}}} \right)\] for  then what is \[f(x)\] equal to?
  A)
  \[\ln \left( \frac{1-x}{1+x} \right)\] done clear
  B)
  \[\ln \left( \frac{2+x}{1-x} \right)\] done clear
  C)
  \[{{\tan }^{-1}}\left( \frac{1-x}{1+x} \right)\]     done clear
  D)
  \[{{\tan }^{-1}}\left( \frac{1+x}{1-x} \right)\] done clear
  View Solution play_arrow

• question_answer8)
  Let \[x+\frac{1}{x}=1\] and a, b and c are distinct positive integers such that \[\left( {{x}^{a}}+\frac{1}{{{x}^{a}}} \right)+\left( {{x}^{b}}+\frac{1}{{{x}^{b}}} \right)+\left( {{x}^{c}}+\frac{1}{{{x}^{c}}} \right)=0.\] Then the minimum value of \[(a+b+c)\] is
  A)
  7                      done clear
  B)
  8 done clear
  C)
  9                      done clear
  D)
  10 done clear
  View Solution play_arrow

• question_answer9)
  If z is a complex number such that \[z+|z|=8+12i,\] then the value of \[|{{z}^{2}}|\] is equal to
  A)
  228       done clear
  B)
  144  done clear
  C)
  121       done clear
  D)
  169 done clear
  View Solution play_arrow

• question_answer10)
  A value of b for which the equations \[{{x}^{2}}+bx-1=0\] \[{{x}^{2}}+x+b=0\] have one root in common is
  A)
  \[-\sqrt{2}\]        done clear
  B)
  \[-i\sqrt{3}\] done clear
  C)
  \[i\sqrt{5}\]        done clear
  D)
  \[\sqrt{2}\] done clear
  View Solution play_arrow

• question_answer11)
  If \[\omega =\frac{z}{z-\frac{1}{3}i}\] and \[|\omega |=1,\] then z lies on
  A)
  an ellipse       done clear
  B)
  a circle done clear
  C)
  a straight line    done clear
  D)
  a parabola done clear
  View Solution play_arrow

• question_answer12)
  \[{{z}_{1}}\] and \[{{z}_{2}}\] are the roots of \[3{{z}^{2}}+3z+b=0\]. If  \[O(0),\]\[A({{z}_{1}}),\] \[B({{z}_{2}})\] form an equilateral triangle, then the value of b is
  A)
  \[-1\]         done clear
  B)
  1 done clear
  C)
  0             done clear
  D)
  does not exist done clear
  View Solution play_arrow

• question_answer13)
  If \[z=x+iy,\,\,{{z}^{1/3}}=a-ib,\] then \[\frac{x}{a}-\frac{y}{b}=k({{a}^{2}}-{{b}^{2}})\] where k is equal to
  A)
  1     done clear
  B)
  2     done clear
  C)
  3                      done clear
  D)
  4 done clear
  View Solution play_arrow

• question_answer14)
  Let \[{{A}_{0}}{{A}_{1}}{{A}_{2}}{{A}_{3}}{{A}_{4}}{{A}_{5}}\] be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments \[{{A}_{0}}{{A}_{1}},{{A}_{0}}{{A}_{2}}\] and \[{{A}_{0}}{{A}_{4}}\] is
  A)
  \[\frac{3}{4}\]  done clear
  B)
  \[3\sqrt{3}\] done clear
  C)
  3                      done clear
  D)
  \[\frac{3\sqrt{3}}{2}\] done clear
  View Solution play_arrow

• question_answer15)
  The values of k for which the equations \[{{x}^{2}}-kx-21=0\] and \[{{x}^{2}}-3kx+35=0\] will have a common roots are:
  A)
  \[k=\pm 4\]                    done clear
  B)
  \[k=\pm 1\] done clear
  C)
  \[k=\pm 3\]                   done clear
  D)
  \[k=0\] done clear
  View Solution play_arrow

• question_answer16)
  If \[\alpha ,\beta \] are real and \[{{\alpha }^{2}},{{\beta }^{2}}\] are the roots of the equation \[{{a}^{2}}{{x}^{2}}-x+1-{{a}^{2}}=0\left( \frac{1}{\sqrt{2}}<a<1 \right)\] and \[{{\beta }^{2}}\ne 1,\] then \[{{\beta }^{2}}=\]
  A)
  \[{{a}^{2}}\]                           done clear
  B)
  \[\frac{1-{{a}^{2}}}{{{a}^{2}}}\] done clear
  C)
  \[1-{{a}^{2}}\]           done clear
  D)
  \[1+{{a}^{2}}\] done clear
  View Solution play_arrow

• question_answer17)
  If one root of the equation \[(1-m){{x}^{2}}+1x+1=0\]is double the other and 1 is real, then what is the greatest value of m?
  A)
  \[-\frac{9}{8}\]   done clear
  B)
  \[\frac{9}{8}\]    done clear
  C)
  \[-\frac{8}{9}\]              done clear
  D)
  \[\frac{8}{9}\] done clear
  View Solution play_arrow

• question_answer18)
  What is \[\frac{{{(1+i)}^{4n+5}}}{{{(1-i)}^{4n+3}}}\] equal to, where n is a natural number and \[i=\sqrt{-1}\]?
  A)
  2                      done clear
  B)
  \[2i\]   done clear
  C)
  \[-2\]    done clear
  D)
  i done clear
  View Solution play_arrow

• question_answer19)
  If \[(a+ib)(c+id)(e+if)(g+ih)=A+iB,\] then \[({{a}^{2}}+{{b}^{2}})({{c}^{2}}+{{d}^{2}})({{e}^{2}}+{{f}^{2}})({{g}^{2}}+{{h}^{2}})=\]
  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_answer20)
  If centre of a regular hexagon is at origin and one of the vertex on arg and diagram is \[1+2i,\] then its perimeter is
  A)
  \[2\sqrt{5}\]       done clear
  B)
  \[6\sqrt{2}\] done clear
  C)
  \[4\sqrt{5}\]       done clear
  D)
  \[6\sqrt{5}\] done clear
  View Solution play_arrow

• question_answer21)
  If \[\left| {{z}_{1}} \right|=\left| {{z}_{2}} \right|=.......\left| {{z}_{n}} \right|=1,\] then the value of \[\left| {{z}_{1}}+{{z}_{2}}+....{{z}_{n}} \right|-\left| \frac{1}{{{z}_{1}}}+\frac{1}{{{z}_{2}}}+......+\frac{1}{{{z}_{n}}} \right|\] is,
  A)
  0                      done clear
  B)
  1 done clear
  C)
  \[-1\]                 done clear
  D)
  None done clear
  View Solution play_arrow

• question_answer22)
  If \[|z|=\max \,\{|z-1|,\,|z+1|\}\] then
  A)
  \[|z+\bar{z}|=\frac{1}{2}\] done clear
  B)
  \[z+\bar{z}=1\] done clear
  C)
  \[|z+\bar{z}|=1\]      done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer23)
  The value of a for which the sum of the squares of   the   roots   of   the   equation \[2{{x}^{2}}-2(a-2)x-(a+1)=0\] is least, is
  A)
  1                      done clear
  B)
  \[3/2\]   done clear
  C)
  2                      done clear
  D)
  None done clear
  View Solution play_arrow

• question_answer24)

  If \[A=\left| x\in IR:{{x}^{2}}+6x-7<0\} \right.\] and \[B=\{x\in IR:{{x}^{2}}+9x+14>0\},\] then which of the following is/ are correct?

  1. \[(A\cap B)=(-2,1)\]

  2. \[(A\backslash B)=(-7,-2)\]

  Select the correct answer using the code given below:

  A)
  1 only               done clear
  B)
  2 Only done clear
  C)
  Both 1 and 2     done clear
  D)
  Neither 1 nor 2 done clear
  View Solution play_arrow

• question_answer25)
  If \[z=\frac{\pi }{4}{{(1+i)}^{4}}\left( \frac{1-\sqrt{\pi }i}{\sqrt{\pi }+i}+\frac{\sqrt{\pi }-i}{1+\sqrt{\pi }i} \right),\] then \[\left( \frac{|z|}{am{{p}^{(z)}}} \right)\] equals
  A)
  1          done clear
  B)
  \[\pi \]     done clear
  C)
  \[3\pi \]              done clear
  D)
  4 done clear
  View Solution play_arrow

• question_answer26)
  If \[{{m}_{1}},{{m}_{2}},{{m}_{3}}\] and \[{{m}_{4}}\] respectively denote the moduli of the complex numbers \[1+4i,\,\,3+i,\,\,1-i\] and \[2-3i,\] then the correct one, among the following is
  A)
  \[{{m}_{1}}<{{m}_{2}}<{{m}_{3}}<{{m}_{4}}\] done clear
  B)
  \[{{m}_{4}}<{{m}_{3}}<{{m}_{2}}<{{m}_{1}}\] done clear
  C)
  \[{{m}_{3}}<{{m}_{2}}<{{m}_{4}}<{{m}_{1}}\] done clear
  D)
  \[{{m}_{3}}<{{m}_{1}}<{{m}_{2}}<{{m}_{4}}\] done clear
  View Solution play_arrow

• question_answer27)
  If both the roots of \[k(6{{x}^{2}}+3)+rx+2{{x}^{2}}-1=0\]and \[6k(2{{x}^{2}}+1)+px+4{{x}^{2}}-2=0\] are common, then \[2r-p\]is equal to
  A)
  \[-1\]     done clear
  B)
  0    done clear
  C)
  1                      done clear
  D)
  2 done clear
  View Solution play_arrow

• question_answer28)
  For the complex numbers \[{{z}_{1}}\] and \[{{z}_{2}}\] if \[|1-{{\bar{z}}_{1}}{{z}_{2}}{{|}^{2}}-|{{z}_{1}}-{{z}_{2}}{{|}^{2}}=k(1-|{{z}_{1}}{{|}^{2}})(1-|{{z}_{2}}{{|}^{2}})\] then ?k? equals to
  A)
  1                      done clear
  B)
  \[-1\]   done clear
  C)
  2                      done clear
  D)
  \[-2\] done clear
  View Solution play_arrow

• question_answer29)
  The solution of \[2\sqrt{2}\,\,{{x}^{4}}=(\sqrt{3}-1)+i(\sqrt{3}+1)\] is
  A)
  \[\pm \,\,\left( \cos \frac{5\pi }{48}+i\sin \frac{5\pi }{48} \right)\] done clear
  B)
  \[\pm \,\,\left( \cos \frac{7\pi }{48}+i\sin \frac{7\pi }{48} \right)\] done clear
  C)
  \[\pm \,\,\left( \cos \frac{19\pi }{48}-i\sin \frac{19\pi }{48} \right)\] done clear
  D)
  None of these. done clear
  View Solution play_arrow

• question_answer30)
  If \[\alpha \] and \[\beta \] be the values of x in \[{{m}^{2}}({{x}^{2}}-x)+2mx+3=0\] and \[{{m}_{1}}\] and \[{{m}_{2}}\] be two values of m for which \[\alpha \] and \[\beta \] are connected by the relation \[\frac{\alpha }{\beta }+\frac{\beta }{\alpha }=\frac{4}{3}.\] Then the value of \[\frac{m_{1}^{2}}{{{m}_{2}}}+\frac{m_{2}^{2}}{{{m}_{1}}}\] is
  A)
  6 done clear
  B)
  68 done clear
  C)
  \[\frac{3}{68}\]              done clear
  D)
  \[-\frac{68}{3}\] done clear
  View Solution play_arrow

• question_answer31)
  If \[a,b,c\in \mathbf{R}\] and the equations  \[a\ne 0,\] has real roots \[\alpha \] and \[\beta \] satisfying \[\alpha <-1\] and \[\beta >1,\] then \[1+\frac{c}{a}+\left| \frac{b}{a} \right|\] is
  A)
  positive              done clear
  B)
  negative done clear
  C)
  zero                   done clear
  D)
  none done clear
  View Solution play_arrow

• question_answer32)
  If the point \[{{z}_{1}}=1+i\] where \[i=\sqrt{-1}\] is the reflection of a point \[{{z}_{2}}=x+iy\] in the line \[i\bar{z}-iz=5,\] then the point \[{{z}_{2}}\] is     
  A)
  \[1+4i\] done clear
  B)
  \[4+i\] done clear
  C)
  \[1-i\]     done clear
  D)
  \[-1-i\] done clear
  View Solution play_arrow

• question_answer33)
  If the roots of \[a{{x}^{2}}+bx+c=0\] are \[\sin \alpha \] and \[\cos \alpha \] for some \[\alpha ,\] then which one of the following is correct?
  A)
  \[{{a}^{2}}+{{b}^{2}}=2ac\] done clear
  B)
  \[{{b}^{2}}-{{c}^{2}}=2ab\] done clear
  C)
  \[{{b}^{2}}-{{a}^{2}}=2ac\] done clear
  D)
  \[{{b}^{2}}+{{c}^{2}}=2ab\] done clear
  View Solution play_arrow

• question_answer34)
  If \[1,\omega ,{{\omega }^{2}}\] are the three cube roots of unity, then what is \[\frac{(a{{\omega }^{6}}+b{{\omega }^{4}}+c{{\omega }^{2}})}{(b+c{{\omega }^{10}}+a{{\omega }^{8}})}\] equal to?
  A)
  \[\frac{a}{b}\]                           done clear
  B)
  b      done clear
  C)
  \[\omega \]         done clear
  D)
  \[{{\omega }^{2}}\] done clear
  View Solution play_arrow

• question_answer35)
  If \[|z-2|=\min \{|z-1|,|z-5|\},\] where z is a complex number, then
  A)
  \[\operatorname{Re}(z)=\frac{3}{2}\] done clear
  B)
  \[\operatorname{Re}(z)=\frac{7}{2}\] done clear
  C)
  \[\operatorname{Re}(z)\in \left\{ \frac{3}{2},\frac{7}{2} \right\}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer36)
  Let \[{{x}_{1}}\] and \[{{y}_{1}}\] be real numbers. If \[{{z}_{1}}\] and \[{{z}_{2}}\] are complex numbers such that \[|{{z}_{1}}|=|{{z}_{2}}|=4,\] then \[|{{x}_{1}}{{z}_{1}}-{{y}_{1}}{{z}_{2}}{{|}^{2}}+|{{y}_{1}}{{z}_{1}}+{{x}_{1}}{{z}_{2}}{{|}^{2}}=\]
  A)
  \[32({{x}_{1}}^{2}+{{y}_{1}}^{2})\]                       done clear
  B)
  \[16({{x}_{1}}^{2}+{{y}_{1}}^{2})\] done clear
  C)
  \[4({{x}_{1}}^{2}+{{y}_{1}}^{2})\]             done clear
  D)
  \[32({{x}_{1}}^{2}+{{y}_{1}}^{2})|{{z}_{1}}+{{z}_{2}}{{|}^{2}}\] done clear
  View Solution play_arrow

• question_answer37)
  If \[\omega \] is imaginary cube root of unity, then \[\sin \left\{ ({{\omega }^{13}}+{{\omega }^{2}})\pi +\frac{\pi }{4} \right\}\] is equal to
  A)
  \[-\frac{\sqrt{3}}{2}\]     done clear
  B)
  \[-\frac{1}{\sqrt{2}}\] done clear
  C)
  \[\frac{1}{\sqrt{2}}\]                  done clear
  D)
  \[\frac{\sqrt{3}}{2}\] done clear
  View Solution play_arrow

• question_answer38)
  The principle value of the \[\arg \,(z)\] and \[|z|\] of the complex number \[z=1+\cos \left( \frac{11\pi }{9} \right)+i\sin \left( \frac{11\pi }{9} \right)\] are respectively.
  A)
  \[\frac{11\pi }{8},\,2\cos \left( \frac{\pi }{18} \right)\] done clear
  B)
  \[-\frac{7\pi }{18},-2\cos \left( \frac{11\pi }{18} \right)\] done clear
  C)
  \[\frac{2\pi }{9},2\cos \left( \frac{7\pi }{18} \right)\] done clear
  D)
  \[-\frac{\pi }{9},-2\cos \left( \frac{\pi }{18} \right)\] done clear
  View Solution play_arrow

• question_answer39)
  If \[\lambda \ne \mu \] and \[{{\lambda }^{2}}=5\lambda -3,\]\[{{\mu }^{2}}=5\mu -3,\] then the equation whose roots are \[\frac{\lambda }{\mu }\] and \[\frac{\mu }{\lambda }\] is
  A)
  \[{{x}^{2}}-5x+3=0\] done clear
  B)
  \[3{{x}^{2}}+19x+3=0\] done clear
  C)
  \[3{{x}^{2}}-19x+3=0\] done clear
  D)
  \[{{x}^{2}}+5x-3=0\] done clear
  View Solution play_arrow

• question_answer40)
  If \[z=\frac{-2\left( 1+2i \right)}{3+i}\] where \[i=\sqrt{-1},\] then argument \[\theta \,(-\pi <\theta \le \pi )\] of z is
  A)
  \[\frac{3\pi }{4}\]                       done clear
  B)
  \[\frac{\pi }{4}\] done clear
  C)
  \[\frac{5\pi }{6}\]                       done clear
  D)
  \[-\frac{3\pi }{4}\] done clear
  View Solution play_arrow

• question_answer41)
  Number of solutions of the equation, \[{{z}^{3}}+\frac{3{{\left| z \right|}^{2}}}{z}=0,\] where z is a complex number and \[|z|=\sqrt{3}\] is
  A)
  2                      done clear
  B)
  3     done clear
  C)
  6                      done clear
  D)
  4 done clear
  View Solution play_arrow

• question_answer42)
  If \[f(z)=\frac{7-z}{1-{{z}^{2}}},\] where \[z=1+2i,\] then \[|f(z)|\] is equal to:                                  
  A)
  \[\frac{|z|}{2}\]                         done clear
  B)
  \[|z|\] done clear
  C)
  \[2|z|\]              done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer43)
  Let \[a>0,\text{ }b>0\] and \[c>0\]. Then both the roots of the equation \[a{{x}^{2}}+bx+c=0\]                   
  A)
  are real and negative done clear
  B)
  have negative real parts done clear
  C)
  are rational numbers done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer44)
  The value of \[{{(1+2\omega +{{\omega }^{2}})}^{3n}}-{{(1+\omega +2{{\omega }^{2}})}^{3n}}\] is:
  A)
   0         done clear
  B)
  1      done clear
  C)
  \[\omega \]         done clear
  D)
  \[{{\omega }^{2}}\] done clear
  View Solution play_arrow

• question_answer45)
  The set of all real numbers x for which \[{{x}^{2}}-[x+2]+x>0,\] is
  A)
  \[\left( -\infty ,-2 \right)\cup \left( 2,\infty  \right)\] done clear
  B)
  \[\left( -\infty ,-\sqrt{2} \right)\cup \left( \sqrt{2},\infty  \right)\] done clear
  C)
  \[\left( -\infty ,-1 \right)\cup \left( 1,\infty  \right)\] done clear
  D)
  \[\left( \sqrt{2},\infty  \right)\] done clear
  View Solution play_arrow

• question_answer46)
  If n is a positive integer grater than unity and z is a complex satisfying the equation \[{{z}^{n}}={{(z+1)}^{n}},\]then
  A)
  \[\operatorname{Re}(z)<2\]       done clear
  B)
  \[\operatorname{Re}(z)>0\] done clear
  C)
  \[\operatorname{Re}(z)=0\]         done clear
  D)
  z lies on \[x=-\frac{1}{2}\] done clear
  View Solution play_arrow

• question_answer47)
  Let Z and W be two complex numbers such that \[\left| Z \right|\le 1,\] \[\left| W \right|\le 1\] and \[\left| Z+i\,W \right|=\left| Z-i\overline{W} \right|=2.\] Then Z equals          
  A)
  1 or i      done clear
  B)
  i or \[-i\] done clear
  C)
  1 or \[-i\]            done clear
  D)
  i or \[-1\] done clear
  View Solution play_arrow

• question_answer48)
  For what value of \[\lambda \] the sum of the squares of the roots of \[{{x}^{2}}+(2+\lambda )x-\frac{1}{2}(1+\lambda )=0\] is minimum?
  A)
  \[3/2\]   done clear
  B)
  1    done clear
  C)
  \[1/2\]   done clear
  D)
  \[11/4\] done clear
  View Solution play_arrow

• question_answer49)
  If \[\operatorname{Re}\left( \frac{z-1}{z+1} \right)=0,\] where \[2=x+iy\] is a complex number, then which one of the following is correct?
  A)
  \[z=1+i\]         done clear
  B)
  \[\left| z \right|=2\] done clear
  C)
  \[z=1-i\] done clear
  D)
  \[\left| z \right|=1\] done clear
  View Solution play_arrow

• question_answer50)
  If \[\frac{1}{2-\sqrt{-2}}\] is one of the roots of \[a{{x}^{2}}+bx+c=0,\]where a, b, c are real, then what are the values of a, b, c respectively?
  A)
  \[6,-4,1\]            done clear
  B)
  \[4,6,-1\] done clear
  C)
  \[3,-2,1\]            done clear
  D)
  \[6,4,1\] done clear
  View Solution play_arrow

• question_answer51)
  If \[2x=3+5i,\] then what is the value of \[2{{x}^{3}}+2{{x}^{2}}-7x+72?\]
  A)
  4                      done clear
  B)
  \[-4\]   done clear
  C)
  8                      done clear
  D)
  \[-8\] done clear
  View Solution play_arrow

• question_answer52)
  If \[{{z}^{2}}+z+1=0,\] where z is complex number, then the value of \[{{\left( z+\frac{1}{z} \right)}^{2}}+{{\left( {{z}^{2}}+\frac{1}{{{z}^{2}}} \right)}^{2}}+{{\left( {{z}^{3}}+\frac{1}{{{z}^{3}}} \right)}^{2}}\] \[+....+{{\left( {{z}^{6}}+\frac{1}{{{z}^{6}}} \right)}^{2}}\]is
  A)
  18         done clear
  B)
  54    done clear
  C)
  6                      done clear
  D)
  12 done clear
  View Solution play_arrow

• question_answer53)
  \[A+iB\] form of \[\frac{(\cos x+i\sin x)(\cos y+i\sin y)}{(\cot u+i)(1+i\tan \,\,v)}\]is equal to:
  A)
  \[\sin u\,\,\cos v\,\,[\cos (x+y-u-v)+\]\[i\sin (x+y-u-v)]\] done clear
  B)
  \[\sin \,u\,\cos \,v[\cos \,(x+y+u+v)+\]\[i\sin (x+y+u+v)]\] done clear
  C)
  \[\sin \,\,u\,\,\,\cos \,\,v\,\,[\cos \,(x+y+u+v)-\]\[i\sin (x+y-u+v)]\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer54)
  If x be real and \[b<c,\]then \[\frac{{{x}^{2}}-bc}{2x-b-c}\] lies in
  A)
  \[(b,c)\]  done clear
  B)
  \[[b,c]\] done clear
  C)
  \[(-\infty ,b]\cup [c,\infty )\] done clear
  D)
  \[(-\infty ,b)\cup (c,\infty )\] done clear
  View Solution play_arrow

• question_answer55)
  The minimum value of \[\left| z \right|+|z-i|\] is
  A)
  0                      done clear
  B)
  1 done clear
  C)
  2                      done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer56)
  The real roots of the equation \[{{x}^{2}}+5|x|+4=0\] are
  A)
  \[\{-1,-4\}\]         done clear
  B)
  \[\{1,4\}\] done clear
  C)
  \[\{-4,4\}\]          done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer57)
  The locus of a point in the Argand plane that moves satisfying the equation \[|z-1+i|-|z-2-i|=3:\]
  A)
  is a circle with radius 3 and centre at \[z=\frac{3}{2}\] done clear
  B)
  is an ellipse with its foci at \[1-i\] and \[2+i\] and major axis \[=3\] done clear
  C)
  is a hyperbola with its foci at \[1-i\] and \[2+i\]and its transverse axis \[=3\] done clear
  D)
  None of the above done clear
  View Solution play_arrow

• question_answer58)
  The equation whose roots are the \[{{n}^{th}}\] power of the roots of the equation \[{{x}^{2}}-2x\,cos\theta +1=0\] is given by
  A)
  \[{{x}^{2}}+2x\,\cos \,\,n\theta +1=0\] done clear
  B)
  \[{{x}^{2}}-2x\,\cos \,\,n\theta +1=0\] done clear
  C)
  \[{{x}^{2}}-2x\,sin\,\,n\theta +1=0\] done clear
  D)
  \[{{x}^{2}}+2x\,sin\,\,n\theta +1=0\] done clear
  View Solution play_arrow

• question_answer59)
  If both the roots of the equation \[{{x}^{2}}-2kx+{{k}^{2}}-4=0\] lie between \[-3\] and 5, then which one of the following is correct?
  A)
  \[-2<k<2\]     done clear
  B)
  \[-5<k<3\] done clear
  C)
  \[-3<k<5\]     done clear
  D)
  \[-1<k<3\] done clear
  View Solution play_arrow

• question_answer60)
  What is the value of \[{{\left( -\sqrt{-1} \right)}^{4n+3}}+{{\left( {{i}^{41}}+{{i}^{-257}} \right)}^{9}},\] where \[n\in N\]?
  A)
  0          done clear
  B)
  1     done clear
  C)
  i                       done clear
  D)
  \[-i\] done clear
  View Solution play_arrow

• question_answer61)
  The points \[{{z}_{1}},{{z}_{2}},{{z}_{3}},{{z}_{4}}\] in a complex plane are vertices of a parallelogram taken in order, then
  A)
  \[{{z}_{1}}+{{z}_{4}}={{z}_{2}}+{{z}_{3}}\] done clear
  B)
  \[{{z}_{1}}+{{z}_{3}}={{z}_{2}}+{{z}_{4}}\] done clear
  C)
  \[{{z}_{1}}+{{z}_{2}}={{z}_{3}}+{{z}_{4}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer62)
  The greatest and the least value of \[|{{z}_{1}}+{{z}_{2}}|\] if \[{{z}_{1}}=24+7i\] and \[|{{z}_{2}}|=6\] respectively are
  A)
  \[25,\,\,19\]           done clear
  B)
  \[19,\,\,25\] done clear
  C)
  \[-19,\,\,-25\]         done clear
  D)
  \[-25,\,\,-19\] done clear
  View Solution play_arrow

• question_answer63)
  The solution set of \[\frac{{{x}^{2}}-3x+4}{x+1}>1,\]\[x\in R,\] is
  A)
  \[(3,+\infty )\]                done clear
  B)
  \[(-1,\,\,1)\cup (3,\,+\infty )\] done clear
  C)
  \[[-1,\,\,1]\cup (3,+\infty )\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer64)
  If the roots of the equations \[p{{x}^{2}}+2qx+r=0\]and \[q{{x}^{2}}-2\sqrt{pr}x+q=0\] be real, then
  A)
  \[p=q\]              done clear
  B)
  \[{{q}^{2}}=pr\] done clear
  C)
  \[{{p}^{2}}=qr\]          done clear
  D)
  \[{{r}^{2}}=pr\] done clear
  View Solution play_arrow

• question_answer65)
  Let \[\alpha ,\beta \] be the roots of \[{{x}^{2}}+x+1=0.\] Then the equation whose roots are \[{{\alpha }^{229}}\] and \[{{\alpha }^{1004}}\] is
  A)
  \[{{x}^{2}}-x-1=0\] done clear
  B)
  \[{{x}^{2}}-x+1=0\] done clear
  C)
  \[{{x}^{2}}+x-1=0\]   done clear
  D)
  \[{{x}^{2}}+x+1=0\] done clear
  View Solution play_arrow

• question_answer66)
  If z, \[\omega z\] ane \[\bar{\omega }z\] are the vertices of a triangle, then the area of the triangle will be (where \[\omega \] is cube root of unity):               
  A)
  \[\frac{3|z{{|}^{2}}}{2}\]       done clear
  B)
  \[\frac{3\sqrt{3}|z{{|}^{2}}}{2}\] done clear
  C)
  \[\frac{\sqrt{3}|z{{|}^{2}}}{2}\]          done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer67)
  If \[\omega \] is a complex cube root of unity and \[x={{\omega }^{2}}-\omega -2,\] then what is the value of \[{{x}^{2}}+4x+7\]?
  A)
  \[-2\]     done clear
  B)
  \[-1\]   done clear
  C)
  \[0\]      done clear
  D)
  \[1\] done clear
  View Solution play_arrow

• question_answer68)
  Consider \[f(x)={{x}^{2}}-3x+a+\frac{1}{a},\]\[a\in R-\{0\},\]such that \[f(3)>0\] and \[f(2)\le 0.\] If \[\alpha \] and \[\beta \] are the roots of equation \[f(x)=0\] then the value of \[{{\alpha }^{2}}+{{\beta }^{2}}\] is equal to
  A)
  greater than 11 done clear
  B)
  less than 5 done clear
  C)
  5 done clear
  D)
  depends upon a and a cannot be determined done clear
  View Solution play_arrow

• question_answer69)
  If z in any complex number satisfying  then which of the following is correct?
  A)
  \[\arg \,(z-1)=2argz\] done clear
  B)
  \[2\arg \,(z)=\frac{2}{3}arg({{z}^{2}}-z)\] done clear
  C)
  \[\arg (z-1)=\arg (z+1)\] done clear
  D)
  \[\arg z=2\arg (z+1)\] done clear
  View Solution play_arrow

• question_answer70)
  If  then
  A)
  \[y=6\]  done clear
  B)
  \[y=5\] done clear
  C)
  \[y=\sqrt{6}\]          done clear
  D)
  \[y=\sqrt{5}\] done clear
  View Solution play_arrow

• question_answer71)
  If \[0<a<b<c\] and the roots \[\alpha ,\beta \] of the equation \[a{{x}^{2}}+bx+c=0\] are imaginary then incorrect statement is
  A)
  \[|\alpha =|\beta |\]       done clear
  B)
  \[|\alpha |\,>1\] done clear
  C)
  \[|\beta |\,<1\]          done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer72)
  Let \[\lambda \in \mathbf{R}\] If the origin and the non-real roots of \[2{{z}^{2}}+2z+\lambda =0\] form the three vertices of an equilateral triangle in the arg and plane. Then \[\lambda \] is
  A)
  1                      done clear
  B)
  \[\frac{2}{3}\]    done clear
  C)
  2                      done clear
  D)
  \[-1\] done clear
  View Solution play_arrow

• question_answer73)
  If \[\alpha ,\beta \] be the roots of the equation \[{{x}^{2}}-px+q=0\] and \[{{\alpha }_{1}},\,\,{{\beta }_{1}}\] the roots of the equation \[{{x}^{2}}-qx+p=0,\] then the equation whose roots are   \[\frac{1}{{{\alpha }_{1}}\beta }+\frac{1}{\alpha {{\beta }_{1}}}\] and \[\frac{1}{\alpha {{\alpha }_{1}}}+\frac{1}{\beta {{\beta }_{1}}}\] is
  A)
  \[pq{{x}^{2}}-pqx+{{p}^{2}}+{{q}^{2}}+4pq=0\] done clear
  B)
  \[{{p}^{2}}{{q}^{2}}{{x}^{2}}-{{p}^{2}}{{q}^{2}}x+{{p}^{3}}+{{q}^{3}}-4pq=0\] done clear
  C)
  \[{{p}^{3}}{{q}^{3}}{{x}^{2}}-{{p}^{3}}{{q}^{3}}x+{{p}^{4}}+{{q}^{4}}-4{{p}^{2}}{{q}^{2}}=0\] done clear
  D)
  \[(p+q){{x}^{2}}-(p+q)x+{{p}^{2}}+{{q}^{2}}+pq=0\] done clear
  View Solution play_arrow

• question_answer74)
  If m and n are the roots of the equation \[(x+p)(x+q)-k=0,\] then the roots of the equation \[(x-m)(x-n)+k=0\] are
  A)
  p and q   done clear
  B)
  \[\frac{1}{p}\] and \[\frac{1}{q}\] done clear
  C)
  \[-p\]and \[-q\]     done clear
  D)
  \[p+q\] and \[p-q\] done clear
  View Solution play_arrow

• question_answer75)
  What is the argument of \[(1-\sin \theta )+i\cos \theta \]?
  A)
  \[\frac{\pi }{2}-\frac{\theta }{2}\]             done clear
  B)
  \[\frac{\pi }{2}+\frac{\theta }{2}\] done clear
  C)
  \[\frac{\pi }{4}-\frac{\theta }{2}\]            done clear
  D)
  \[\frac{\pi }{4}+\frac{\theta }{2}\] done clear
  View Solution play_arrow

• question_answer76)
  What is the real part of \[{{(\sin x+i\cos x)}^{3}}\]where\[i=\sqrt{-1}\] ?
  A)
  \[-\cos \,3x\]         done clear
  B)
  \[-\sin \,3x\] done clear
  C)
  \[\sin \,3x\]           done clear
  D)
  \[\cos \,3x\] done clear
  View Solution play_arrow

• question_answer77)
  If \[{{z}_{1}},{{z}_{2}}\] are the roots of the quadratic equation \[a{{z}^{2}}+bz+c=0\] such that \[\operatorname{Im}({{z}_{1}},{{z}_{2}})\ne 0\] then
  A)
  a, b, c are all real done clear
  B)
  at least one of a, b, c is real done clear
  C)
  at least one of a, b, c is imaginary done clear
  D)
  all of a, b, c are imaginary done clear
  View Solution play_arrow

• question_answer78)
  Suppose the quadratic equations \[{{x}^{2}}+px+q=0\] and \[{{x}^{2}}+rx+s=0\] are such that p, q, r, s are real and \[pr=2(q+s).\] Then
  A)
  Both the equations always have real roots done clear
  B)
  At least one equation always has real roots done clear
  C)
  Both the equation always have non real roots done clear
  D)
  At least one equation always has real and equal roots done clear
  View Solution play_arrow

• question_answer79)
  If \[\alpha \] and \[\beta \] \[(\alpha <\beta )\] are the roots of the equation \[{{x}^{2}}+bx+c=0,\] where, \[c<0<b,\] then
  A)
  \[0<\alpha <\beta \]      done clear
  B)
  \[\alpha <0<\beta <\,|\alpha |\] done clear
  C)
  \[\alpha <\beta <0\]      done clear
  D)
  \[\alpha <0<\,|\alpha |\,<\beta \] done clear
  View Solution play_arrow

• question_answer80)
  If \[z=1+i\tan \alpha \,\left( -\pi <\alpha <-\frac{\pi }{2} \right),\] then polar form of the complex number z is:
  A)
  \[\frac{1}{\cos \alpha }(\cos \alpha +i\sin \alpha )\] done clear
  B)
  \[\frac{1}{-\cos \alpha }[\cos \,(\pi +\alpha )+i\sin (\pi +\alpha )\] done clear
  C)
  \[\frac{1}{\cos \alpha }[\cos \,(2\pi +\alpha )+i\sin (2\pi +\alpha )]\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer81)
  If the roots of \[a{{x}^{2}}+bx+c=0\] are the reciprocals of those of \[\ell {{x}^{2}}+mx+n=0\] then \[a:b:c=\]
  A)
  \[n:m:\ell \]         done clear
  B)
  \[\ell :m:n\] done clear
  C)
  \[m:n:\ell \]         done clear
  D)
  \[n:\ell :m\] done clear
  View Solution play_arrow

• question_answer82)
  The value of \[Arg\left[ i\,\,\ln \left( \frac{a-ib}{a+ib} \right) \right],\] where a and b are real numbers, is
  A)
  \[0\] or \[\pi \]  done clear
  B)
  \[\frac{\pi }{2}\] done clear
  C)
  not defined      done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer83)
  The roots of the equation \[ab{{c}^{2}}{{x}^{2}}+3{{a}^{2}}cx+{{b}^{2}}cx-6{{a}^{2}}-ab+2{{b}^{2}}=0\] are
  A)
  non real done clear
  B)
  rational if a, b, c are rational done clear
  C)
  irrational if a, b, c are rational done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer84)
  What is \[{{\left[ \frac{\sin \frac{\pi }{6}+i\left( 1-\cos \frac{\pi }{6} \right)}{\sin \frac{\pi }{6}-i\left( 1-\cos \frac{\pi }{6} \right)} \right]}^{3}}\] where \[i=\sqrt{-1},\] equal to?
  A)
  1                      done clear
  B)
  \[-1\]   done clear
  C)
  \[i\]      done clear
  D)
  \[-i\] done clear
  View Solution play_arrow

• question_answer85)
  The value of the sum \[\sum\limits_{n=1}^{13}{\left( {{i}^{n}}+{{i}^{n+1}} \right)};\] where \[i=\sqrt{-1}\] is:
  A)
  \[i\]                   done clear
  B)
  \[-i\]    done clear
  C)
  \[0\]     done clear
  D)
  \[i-1\] done clear
  View Solution play_arrow

• question_answer86)
  If the roots of the equation \[{{x}^{2}}-ax+b=0\] are real and differ by a quantity which is less than \[c(c>0),\] then b lies between
  A)
  \[\frac{{{a}^{2}}-{{c}^{2}}}{4}\] and \[\frac{{{a}^{2}}}{4}\] done clear
  B)
  \[\frac{{{a}^{2}}+{{c}^{2}}}{4}\] and \[\frac{{{a}^{2}}}{4}\] done clear
  C)
  \[\frac{{{a}^{2}}-{{c}^{2}}}{2}\] and \[\frac{{{a}^{2}}}{4}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer87)
  Let \[z={{\log }_{2}}(1+i),\] then \[(z+\bar{z})+i(z-\bar{z})=\]
  A)
  \[\frac{\ln \,4+\pi }{\ln \,\,4}\]        done clear
  B)
  \[\frac{\pi -\ln \,4}{\ln \,\,2}\] done clear
  C)
  \[\frac{\ln \,4-\pi }{\ln \,\,4}\] done clear
  D)
  \[\frac{\pi +\ln \,\,4}{\ln \,\,2}\] done clear
  View Solution play_arrow

• question_answer88)
  For the equation \[\left| {{x}^{2}} \right|+\left| x \right|-6=0,\] the roots are
  A)
  One and only one real number done clear
  B)
  Real with sum one           done clear
  C)
  Real with sum zero done clear
  D)
  Real with product zero done clear
  View Solution play_arrow

• question_answer89)
  If \[{{z}_{1}},{{z}_{2}}\] and \[{{z}_{3}}\] are complex numbers such that \[\left| {{z}_{1}} \right|=\left| {{z}_{2}} \right|=\left| {{z}_{3}} \right|=\left| \frac{1}{{{z}_{1}}}+\frac{1}{{{z}_{2}}}+\frac{1}{{{z}_{3}}} \right|=1,\]  then \[\left| {{z}_{1}}+{{z}_{2}}+{{z}_{3}} \right|\]is
  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_answer90)
  If the roots of the equation \[a{{x}^{2}}-bx+c=0\] are \[\alpha ,\beta \] then the roots of the equation \[{{b}^{2}}c{{x}^{2}}-a{{b}^{2}}x+{{a}^{3}}=0\] are
  A)
  \[\frac{1}{{{\alpha }^{3}}+\alpha \beta },\frac{1}{{{\beta }^{3}}+\alpha \beta }\] done clear
  B)
  \[\frac{1}{{{\alpha }^{2}}+\alpha \beta },\frac{1}{{{\beta }^{2}}+\alpha \beta }\] done clear
  C)
  \[\frac{1}{{{\alpha }^{4}}+\alpha \beta },\frac{1}{{{\beta }^{4}}+\alpha \beta }\] done clear
  D)
  None of these done clear
  View Solution play_arrow