CBSE Questions for Class 11 Engineering Maths Complex Numbers And Quadratic Equations Quiz 9 - MCQExams.com

If $$z_{1}=8 +4i,\ z_{2}=6+4i$$ and $$arg \left(\dfrac {z-z_{1}}{z-z_{2}}\right)=\dfrac {\pi}{4}$$, then $$z$$ satisfy 
  • $$|z-7-4i|=1$$
  • $$|z-7-5i|=\sqrt {2}$$
  • $$|z-4i|=8$$
  • $$|z-7i|=\sqrt {18}$$
If $$a, b \notin R$$, then $$|e^{a + ib}| $$ is equal to
  • $$e^a$$
  • $$e^b$$
  • $$1$$
  • None of these
If $$z_{1}$$ and $$z_{2}$$ two non-zero complex number such that $$|z_{1}+z_{2}|=|z_{1}|+|z_{2}|$$, then $$arg z_{1}-arg z_{2}$$ is equal to
  • $$-p$$
  • $$p/2$$
  • $$-p/2$$
  • $$0$$
$$z$$ is a complex number. If $$a = | x | + | y |$$ and
$$b = \sqrt { 2 } | x + i y |$$ then which of the following is
true
  • $$a \leq b$$
  • $$a > b$$
  • none of these
  • $$a - b + 2$$
If arg $$\left( z \right) < 0,$$ then arg $$\left( { - z} \right)-arg(z)$$
  • $$\pi $$
  • $$ - \pi $$
  • $$ - \dfrac{\pi }{2}$$
  • $$\dfrac{\pi }{2}$$
Number of complex numbers $$z$$ such that $$|z|=1$$ and $$\left|\dfrac {z}{z}+\dfrac {\bar {z}}{z}\right|=1$$ is
  • $$4$$
  • $$1$$
  • $$8$$
  • $$more\ then\ 8$$
The modulus of $$\dfrac { \left( 3+2i \right) ^{ 2 } }{ \left( 4-3i \right)  } $$ is:
  • $$\frac { 13 }{ 5 } $$
  • $$\frac { 11 }{ 5 } $$
  • $$\frac { 9 }{ 5 } $$
  • $$\frac { 7 }{ 5 } $$
Let P$$\left( x \right) ={ x }^{ 3 }-6{ x }^{ 2 }+Bx+C$$ has 1+5i as a zero and B,C real number, then value of (B+C) is
  • -70
  • 70
  • 24
  • 138
Arg $$\left\{ {\sin \frac{{8\pi }}{5} + i\left( {1 + \cos \frac{{8\pi }}{5}} \right)} \right\}$$ is equal to
  • $$\frac{{3\pi }}{{10}}$$
  • $$\frac{{7\pi }}{{10}}$$
  • $$\frac{{4\pi }}{5}$$
  • $$\frac{{3\pi }}{5}$$
A value of $$\theta $$ for which$$\dfrac { 2+3isin\theta  }{ 1-2isin\theta  } $$ is purely imaginary, is:
  • $${ sin }^{ -1 }\left( \dfrac { 1 }{ \sqrt { 3 } } \right) $$
  • $$\dfrac { \pi }{ 3 } $$
  • $$cos^{-1}\sqrt-1$$
  • $$None of these$$
Purely imaginary then find the sum of statement i $$a,b$$ 
  • $$\dfrac {5\pi}{6}$$
  • $$\pi$$
  • $$\dfrac {3\pi}{4}$$
  • $$\dfrac {2\pi}{3}$$
If $$\alpha$$ and $$\beta$$ are the roots of $${ 4x }^{ 2 }-16x+c=0,$$ c>0 such that $$1<\alpha <2<\beta <3$$, then the no.of integer values of c is 
  • $$17$$
  • $$14$$
  • $$18$$
  • $$15$$
The principle amplitude of $$(\sin 40^{o}+i \cos 40^{o})^{5}$$ is
  • $$70^{o}$$
  • $$-1100^{o}$$
  • $$70^{110}$$
  • $$70^{-70}$$
Let 'z' be a complex number satisfying $$|z-2-i|\le 5,$$ Then |z-14-6i| lies in 
  • {8,18}
  • {2,8}
  • {0,2}
  • {3,7}
If  $$w = \dfrac { z } { z - \dfrac { 1 } { 3 } i }$$  and  $$| w | = 1$$  then  $$z$$  lies on
  • a circle
  • an ellipse
  • a parabola
  • a straight line
If the roots of the equation $$mx^{2}+(2m-1)x+m-2=0$$ are rational, then if $$m\in I$$ it will be 
  • odd integer
  • even integer
  • zero only
  • none of these
The discriminant of $${ 2x }^{ 2 }-x-p=0$$ is $$49$$, then $$p=$$ ______
  • $$6$$
  • $$24$$
  • $$48$$
  • None of these
If $$|z-3i|<\sqrt{5}$$, then $$|i(z+1)+1|<2\sqrt{5}$$.
  • True
  • False
The value of the sum $$\sum _{ n=1 }^{ 13 }{ ({ i }^{ n }+{ i }^{ n+1 }) } $$ , where $$i=\sqrt { -1 } $$ , equals
  • $$i$$
  • $$i-1$$
  • $$-i$$
  • $$0$$
$$z_1$$ and $$z_2$$ are two non-zero complex numbers such that $$z_1=2+4i\\z_2=5-6i$$, then $$z_2-z_1$$ equals
  • $$3-10i$$
  • $$3+10i$$
  • $$7-2i$$
  • $$10-24i$$
IF $$z_1=1+i,z_2=1-i$$ find $$z_1z_2$$
  • $$z_1+z_2$$
  • $$z_1-z_2$$
  • $$z_1/z_2$$
  • None.
If  $$z = \cos \dfrac { \pi } { 4 } + i \sin \dfrac { \pi } { 6 } ,$$  then
  • $$| z | = 1 , \arg ( z ) = \dfrac { \pi } { 4 }$$
  • $$| z | = 1 , \arg ( z ) = \dfrac { \pi } { 6 }$$
  • $$| z | = \dfrac { \sqrt { 3 } } { 2 } , \arg ( z ) = \dfrac { 5 \pi } { 24 }$$
  • $$| z | = \dfrac { \sqrt { 3 } } { 2 } , \arg ( z ) = \tan ^ { - 1 } \dfrac { 1 } { \sqrt { 2 } }$$
The real value of $$'\theta '$$, for which the expression $$\frac { 1+i\cos { \theta  }  }{ 1-2i\cos { \theta  }  } $$ is a real number is
  • $$2n\pi +\frac { 3\pi }{ 2 } ,n\in I$$
  • $$2n\pi -\frac { 3\pi }{ 2 } ,n\in I$$
  • $$2n\pi \pm \frac { \pi }{ 2 } ,n\in I$$
  • $$2n\pi +\frac { \pi }{ 4 } ,n\in I$$
Given $$ z _ { 1 } + 3 z _ { 2 } - 4 z _ { 3 } = 0 $$ then $$ z _ { 1 } , z _ { 2 } , z _ { 3 } $$ are
  • collinear
  • can form sides of equilateral $$ \Delta $$
  • lie on circle
  • none of these
The greatest and least value of $$\left | z \right |$$ if $$z$$ satisfies $$\left | z - 5 + 5i \right |$$ $$\leq 5$$ are 
  • $$10$$ , $$5\sqrt{2}$$
  • $$5\sqrt{2}$$ , $$5$$
  • $$10$$ , $$0$$
  • $$5 + 5\sqrt{2}$$ , $$5\sqrt{2} - 5$$
If z be a complex number satisfying $$z^{4}+z^{3}+2z^{2}+z+1=0$$ then $$\left|z\right|=$$
  • $$\dfrac{1}{2}$$
  • $$\dfrac{3}{4}$$
  • $$1$$
  • none of these
The discriminant of the quadratic equation $$\left( { 2 }^{ \lambda  } \right) { x }^{ 2 }+\left( { a }^{ 2 } \right) x-{ 8 }^{ \lambda  }=0$$ where $$a,\lambda ,\in N$$ is surely 
  • A perfect square
  • A prime number
  • A composite number
  • An even number
Express the following complex numbers in the standard form $$ a+ib$$ :
$$ \left ( \dfrac{1}{1-4i}-\dfrac{2}{1+i} \right )\left ( \dfrac{3-4i}{5+i} \right )$$
  • $$ \dfrac{307}{442}+i \dfrac{599}{442}i$$
  • $$ \dfrac{307}{442}-i \dfrac{599}{442}i$$
  • $$ \dfrac{-307}{442}+i \dfrac{599}{442}i$$
  • None of the above
Let z be a complex number such that the principal value of argument, arg $$z < 0$$. Then $$arg(-z) - arg (z)$$ is
  • $$\dfrac{\pi}{2}$$
  • $$\pm \pi$$
  • $$\pi$$
  • $$-\pi$$
Express the following complex numbers in the standard form $$ a+ib$$ :
$$ \dfrac{\left ( 2+i \right )^{3}}{2+3i}$$
  • $$ \dfrac{37}{13}-\dfrac{16}{13}i$$
  • $$ \dfrac{-37}{13}+\dfrac{16}{13}i$$
  • $$ \dfrac{37}{13}+\dfrac{16}{13}i$$
  • None of the above
The imaginary roots of the equation $${ ({ x }^{ 2 }+2) }^{ 2 }+8{ x }^{ 2 }=6x({ x }^{ 2 }+2)$$ are ____________.
  • $$1+i$$
  • $$2\pm i$$
  • $$-1\pm i$$
  • $$none of these$$
Express the following complex numbers in the standard form $$ a+ib$$ :
$$ \dfrac{3-4i}{\left ( 4-2i \right )\left ( 1+i \right )}$$
  • $$ \dfrac{1}{4}+\dfrac{3}{4}i$$
  • $$ \dfrac{1}{4}-\dfrac{3}{4}i$$
  • $$ \dfrac{-1}{4}-\dfrac{3}{4}i$$
  • None of the above
Find the modulus and argument of the following complex numbers and hence express each of them in the polar form:
$$1-i$$
  • $$\sqrt{2}(cos\,\pi /4+i\, sin\, \pi /4)$$
  • $$\sqrt{2}(cos\,\pi /3-i\, sin\, \pi /3)$$
  • $$\sqrt{2}(cos\,\pi /4-i\, sin\, \pi /4)$$
  • $$\sqrt{2}(cos\,\pi /3+i\, sin\, \pi /3)$$
If $$a$$ and $$b$$ are the nonzero distinct roots of $$x^2 + ax + b =0$$, then the minimum vlue of $$x^2 + ax+b$$ is
  • $$\dfrac{2}{3}$$
  • $$\dfrac{9}{4}$$
  • $$\dfrac{-9}{4}$$
  • $$\dfrac{-2}{3}$$
Express the following complex numbers in the standard from $$ a+ib$$ :
$$ \dfrac{5+\sqrt{2}i}{1-\sqrt{2}i}$$
  • $$ 1-2\sqrt{2}i$$
  • $$ 1+\sqrt{2}i$$
  • $$ 1+2\sqrt{2}i$$
  • $$ 1-\sqrt{2}i$$
Let z be a complex number such that $$\left|\dfrac{z-i}{z+2i}\right|=1$$ and $$|z|=\dfrac{5}{2}$$. Then the value of $$|z+3i|$$ is?
  • $$\dfrac{7}{2}$$
  • $$\dfrac{15}{4}$$
  • $$2\sqrt{3}$$
  • $$\sqrt{10}$$
The real part of $$(i - \sqrt{3})^{13}$$ is
  • $$2^{-10}\sqrt3$$
  • $$-2^{12}\sqrt3$$
  • $$2^{-12}\sqrt3$$
  • $$-2^{-12}\sqrt3$$
  • $$-2^{10}\sqrt3$$
$$(1-\sqrt{-1})(1+\sqrt{-1})(5-\sqrt{-7})(5+\sqrt{-7})=?$$
  • $$(25+7i)$$
  • $$(32+5i)$$
  • $$(29-3i)$$
  • $$none\ of\ these$$
$$arg (-1+i\sqrt{3})=?$$
  • $$\dfrac{\pi}{3}$$
  • $$\dfrac{2\pi}{3}$$
  • $$\pi$$
  • $$none\ of\ these$$
$$(2-3i)(-3+4i)=?$$
  • $$(6+17i)$$
  • $$(6-17i)$$
  • $$(-6+17i)$$
  • $$none\ of\ these$$
$$arg (1+i)=?$$
  • $$\pi$$
  • $$\dfrac{\pi}{2}$$
  • $$\dfrac{\pi}{3}$$
  • $$\dfrac{\pi}{4}$$
$$arg \left(\dfrac{2+6\sqrt{3}i}{5+\sqrt{3}i}\right)=?$$
  • $$\dfrac{\pi}{3}$$
  • $$\dfrac{\pi}{4}$$
  • $$\dfrac{2\pi}{3}$$
  • $$\pi$$
Let $$z, w$$ be complex numbers such that $$\bar z + i\bar w =0$$ and arg $$zw = \pi$$. then $$arg \ z$$ equals
  • $$\dfrac{\pi}{4}$$
  • $$\dfrac{\pi}{2}$$
  • $$\dfrac{3\pi}{4}$$
  • $$\dfrac{5\pi}{4}$$
For any complex numbers $$z_{1}$$ and $$z_{2}$$ compare List I with with List II and choose the correct answer, using codes given below:
List IList II
$$arg (z_{1},z_{2})$$$$\dfrac{\pi}{2}$$
$$arg \left(\dfrac{z_{1}}{z_{2}}\right)$$$$arg (z_{1}-arg (z_{2})$$
$$arg (z)+arg (\bar{z})$$$$arg (z_{1})+arg (z_{2})$$
$$arg (i)$$$$2\pi$$
  • $$(i)-(q), (ii)-(r), (iii)-(s), (iv)-(p)$$
  • $$(i)-(r), (ii)-(q), (iii)-(p), (iv)-(s)$$
  • $$(i)-(r), (ii)-(q), (iii)-(s), (iv)-(p)$$
  • $$none\ of\ these$$
The principal argument of the complex number 
$$[(1 + i)^5 (1 + \sqrt{3}i)^2] / [-2i(-\sqrt{3} +i)]$$ is
  • $$\frac{19\pi}{12}$$
  • $$-\frac{17\pi}{12}$$
  • $$-\frac{5\pi}{12}$$
  • $$\frac{5\pi}{12}$$
The argument and the principle value of the complex number $$\dfrac {2+i}{4i+(1+i)^2}$$ are
  • $$\tan^{-1}(-2)$$
  • $$-\tan^{-1} 2$$
  • $$\tan^{-1}\left(\dfrac {1}{2}\right)$$
  • $$-\tan^{-1}\left(\dfrac {1}{2}\right)$$
Compare List I with List II and choose the correct answer using codes given below:
List I (Complex number)List II (Its modulus)
$$(4-3i)$$$$10$$
$$(8+6i)$$$$\dfrac{1}{5}$$
$$\dfrac{1}{(3+4i)}$$$$1$$
$$\dfrac{(3-4i)}{(3+4i)}$$$$5$$
  • $$(i)-(p), (ii)-(s), (iii)-(r), (iv)-(q)$$
  • $$(i)-(s), (ii)-(p), (iii)-(q), (iv)-(r)$$
  • $$(i)-(s), (ii)-(p), (iii)-(r), (iv)-(q)$$
  • $$(i)-(r), (ii)-(p), (iii)-(s), (iv)-(q)$$
If $$ b_{1} b_{2}=2\left(c_{1}+c_{2}\right), $$ then at least one of the equations $$ x^{2}+b_{1} x $$ $$ +c_{1}=0 $$ and $$ x^{2}+b_{2} x+c_{2}=0 $$ has
  • imaginary roots
  • real roots
  • purely imaginary roots
  • none of these
Which of the following equations has no real roots.
  • $$x^{2} - 4x + 3\sqrt{2} = 0$$
  • $$x^{2} + 4x - 3\sqrt{2} = 0$$
  • $$x^{2} - 4x - 3\sqrt{2} = 0$$
  • $$3x^{2} + 4\sqrt{3}x + 4 = 0$$
The modulus and amplitude of the complex number $$\left[e^{{3}-i\dfrac{\pi}{4}}\right]^{3}$$ are respectively.
  • $$e^{6},\dfrac{-3\pi}{4}$$
  • $$e^{9},\dfrac{\pi}{2}$$
  • $$e^{9},\dfrac{-3\pi}{4}$$
  • $$e^{9},\dfrac{-\pi}{2}$$
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