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

done Numerical Value Type Questions - Complex Numbers Quadratic Equations Total Questions - 15

• question_answer1)
  Let \[\alpha \] and \[\beta \] be two roots of the equation \[{{x}^{2}}+2x+2=0,\] then \[{{\alpha }^{15}}+{{\beta }^{15}}\] is equal to \[-{{(2)}^{k}},\] then k is
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• question_answer2)
  Let \[{{z}_{0}}\] be a root of the quadratic equation, \[{{x}^{2}}+x+1=0.\] If \[z=3+6i\,\,z_{0}^{81}-3i\,\,z_{0}^{93},\] if arg z is equal to \[\frac{\pi }{a},\] then a is
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• question_answer3)
  Let \[z={{\left( \frac{\sqrt{3}}{2}+\frac{i}{2} \right)}^{5}}+{{\left( \frac{\sqrt{3}}{2}-\frac{i}{2} \right)}^{5}}.\] and \[R(z)\] and \[I(z)\] respectively denote the real and imaginary parts of z, then the product of \[R(z)\] and \[I(z)\] is
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• question_answer4)
  Let z be a complex number such that \[\left| z \right|+z=3+i\] \[\left( where\,\,i=\sqrt{-1} \right).\] Then \[|z|\] is equal to
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• question_answer5)
  If \[\frac{z-\alpha }{z+\alpha }\,\,(\alpha \in R)\] is a purely imaginary number and \[\left| z \right|=2,\] then a value of \[\alpha \] is
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• question_answer6)
  Let \[{{z}_{1}}\] and \[{{z}_{2}}\] be two complex numbers satisfying \[|{{z}_{1}}|=9\] and \[|{{z}_{2}}-3-4i|=4.\] Then the minimum value of \[|{{z}_{1}}-{{z}_{2}}|\] is
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• question_answer7)
  If \[\alpha \] and \[\beta \] be the roots of the equation \[{{x}^{2}}-2x+2=0,\] then the least value of n for which \[{{\left( \frac{\alpha }{\beta } \right)}^{n}}=1\] is
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• question_answer8)
  All the points in the set \[S=\left\{ \frac{\alpha +i}{\alpha -i}:\alpha \in R \right\}\,\,\,(i=\sqrt{-1})\] lie on a circle, then it's radius is equal to
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• question_answer9)
  If m is chosen in the quadratic equation \[({{m}^{2}}+1){{x}^{2}}-3x+{{({{m}^{2}}+1)}^{2}}=0\] such that the sum of its roots is greatest and the absolute difference of the cubes of its roots is equal to \[a\sqrt{5},\] then a is
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• question_answer10)
  If \[z=x-i\,\,y\] and \[{{z}^{\frac{1}{3}}}=p+iq,\] then absolute value of \[{\left( \frac{x}{p}+\frac{y}{q} \right)}/{({{p}^{2}}+{{q}^{2}})}\;\] is equal to
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• question_answer11)
  If z and \[\omega \] are two non-zero complex numbers such that \[\left| z\omega  \right|=1\] and \[Arg(z)-Arg(\omega )=\frac{\pi }{2},\] and \[\bar{z}\omega \] is equal to \[{{i}^{k}},\] where k is smallest natural number then k is
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• question_answer12)
  If \[|z+4|\le 3,\] then the maximum value of \[|z+1|\]is
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• question_answer13)
  For the equation \[3{{x}^{2}}+px+3=0,\text{ }p>0,\] if one of the root is square of the other, then p is equal to
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• question_answer14)
  If one root of the equations \[a{{x}^{2}}+bx+c=0\] and \[b{{x}^{2}}+cx+a=0\] \[(a,b,c\in R)\] is common, then the value of \[{{\left( \frac{{{a}^{3}}+{{b}^{3}}+{{c}^{3}}}{abc} \right)}^{3}}\] is
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• question_answer15)
  If \[{{x}^{2}}-hx-21=0,\] \[{{x}^{2}}-3hx+35=0\,\,(h>0)\] has a common root, then the value of h is equal to
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