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

The value of $$(z+3) (\overline{z} +3)$$ is eqquivalent to
  • $$|z +3 | ^2$$
  • $$| z- 3|$$
  • $$z^2+3$$
  • None of these
If $$\dfrac {3+2i \sin x}{1-2i \sin x}$$ is purely imaginary then $$x=$$ ?
  • $$n \pi \pm \dfrac {\pi}{6}$$
  • $$n \pi \pm \dfrac {\pi}{3}$$
  • $$2n \pi \pm \dfrac {\pi}{3}$$
  • $$2n \pi \pm \dfrac {\pi}{6}$$
For $${ { Z }_{ 1 }=\sqrt [ 6 ]{ \dfrac { 1-i }{ 1+i\sqrt { 3 }  }  }  };\quad { { Z }_{ 2 }=\sqrt [ 6 ]{ \dfrac { 1-i }{ \sqrt { 3 } +i }  } ;\quad { { Z }_{ 3 }=\sqrt [ 6 ]{ \dfrac { 1-i }{ \sqrt { 3 } -i }  }  } }$$ which of the following holds good?
  • $$\sum { { \left| { Z }_{ 1 } \right| }^{ 2 } } =\dfrac { 3 }{ 2 } $$
  • $${ \left| { Z }_{ 1 } \right| }^{ 4 }+{ \left| { Z }_{ 2 } \right| }^{ 4 }={ \left| { Z }_{ 3 } \right| }^{ -8 }$$
  • $$\sum { { \left| { Z }_{ 1 } \right| }^{ 3 }+ } { \left| { Z }_{ 2 } \right| }^{ 3 }={ \left| { Z }_{ 3 } \right| }^{ -6 }$$
  • $${ \left| { Z }_{ 1 } \right| }^{ 4 }+{ \left| { Z }_{ 2 } \right| }^{ 4 }={ \left| { Z }_{ 3 } \right| }^{ 8 }$$
For a complex number $$z$$, the minimum value of $$\left | z \right |+\left | z-\cos\alpha-i\sin\alpha \right |$$ is
  • 0
  • 1
  • 2
  • None of these
$$|z-4| < |z-2|$$ represents the region given by?
  • $$Re(z) > 3$$
  • $$Re(z) < 0$$
  • $$Re(z) > 2$$
  • None of these
If $$P(x)=a{x}^{2}+bx+c$$ and $$Q(x)=-a{x}^{2}+dx+c$$ where $$ac\ne 0$$, then $$P(x).Q(x)=0$$ has
  • exactly one real root
  • atleast two real roots
  • exactly three real roots
  • all four are real roots
The modulus of $$\overline { 6+{ i }^{ 3 } } +\overline { 6+{ i } }+\overline { 6+{ i }^{ 2 } } $$ is
  • $$17$$
  • $$\sqrt{533}$$
  • $$\sqrt{456}$$
  • $$49$$
Let $$a,b,c,$$ be the sides of a triangle. No two of them are equal and $$\lambda\in R$$. If the roots of the equation $${x}^{2}+2(a+b+c)x+3\lambda (ab+bc+ca)=0$$ are real and distinct, then
  • $$\lambda< \cfrac{4}{3}$$
  • $$\lambda> \cfrac{5}{3}$$
  • $$\lambda \in \left( \cfrac { 1 }{ 3 } ,\cfrac { 5 }{ 3 } \right) $$
  • $$\lambda \in \left( \cfrac { 4 }{ 3 } ,\cfrac { 5 }{ 3 } \right) $$
If $$a + b + c = 0$$ then the roots of the equation $$4 a x ^ { 2 } + 3 b x+ 2 c = 0$$ where $$a , b , c \in R$$ are 
  • real and distinct
  • imaginary
  • real and equal
  • infinite
If $$a, b, c, d$$ be a form consecutive term of an increasing A.P., then the roots of the equation $$\left( {x - a} \right)\left( {x - c} \right) + 2\left( {x - b} \right)\left( {x - d} \right) = 0$$
  • Real & distinct
  • complex
  • equal roots
  • none of these
If  $$   | z _ { 1 } | < 1 \text { and } | \frac { z _ { 1 } - z _ { 2 } } { 1 - \overline { z } _ { 1 } z _ { 2 } } | < 1 , \text { then  } | z _ { 2 } | > 1$$
  • True
  • False
If $$\left| z \right| =1$$ and $$\left| \omega -1 \right| =1$$ where $$z,\omega \in C$$ then the largest set of values of $${ \left| 2z-1 \right|  }^{ 2 }+{ \left| 2\omega -1 \right|  }^{ 2 }$$ equals 
  • $$\left[1,9\right]$$
  • $$\left[2,6\right]$$
  • $$\left[2,12\right]$$
  • $$\left[2,18\right]$$
If the quadratic equation $$4x^{2}-2(a+c-1)x+ac-b=0(a>b>c).$$
  • Both roots are greater than a
  • Both roots are less than c
  • Both roots lie between c/2 and a/2
  • Exactly one of the roots lie between c/2 and a/2
If the roots $$a^2x^2 +2bx+c^2 = 0$$ are imaginary then the roots of $$b(x^2+1)+2acx = 0$$ are 
  • complex number
  • real and unequal
  • real and equal
  • none
If $$a > b > c$$, $$a\neq 0$$ and the system of equations
$$ax+by+cz=0$$, $$bx+cy+az=0$$, $$cx+ay+bz=0$$ has non-trivial solutions, then the roots of the quadratic equation $$at^2+bt+c=0$$.
  • Are imaginary
  • Are real and equal
  • Are real and distinct
  • May be real of imaginary
If $$z$$ is a complex number such that $$|z-1|=1$$ then $$arg \left(\dfrac{1}{z}-\dfrac{1}{2}\right)$$ may be 
  • $$\dfrac{\pi}{6}$$
  • $$\dfrac{\pi}{2}$$
  • $$\dfrac{\pi}{4}$$
  • $$-\dfrac{\pi}{4}$$
If $$z=\dfrac{-2}{1+i\sqrt{3}}$$, then the value of arg(z) is?
  • $$\pi$$
  • $$\dfrac{\pi}{3}$$
  • $$\dfrac{2\pi}{3}$$
  • $$\dfrac{\pi}{4}$$
If $$\theta$$ real then the modulus of $$\dfrac{1}{1+\cos\theta+i\sin\theta}$$ is
  • $$\dfrac{1}{2}\sec\dfrac{\theta}{2}$$
  • $$\dfrac{1}{2}\cos\dfrac{\theta}{2}$$
  • $$\sec\dfrac{\theta}{2}$$
  • $$\cos\dfrac{\theta}{2}$$
The function of imaginary roots of the equation $$(x-1)(x-2)(3x+1)=32$$ is 
  • $$0$$
  • $$1$$
  • $$2$$
  • $$4$$
Let $$z,w$$ be complex numbers such that $$\vec {z}+i\vec {w}=$$ and $$zw=\pi$$ Then $$arg\ z$$ equals
  • $$\dfrac {\pi}{4}$$
  • $$\dfrac {5\pi}{4}$$
  • $$\dfrac {3\pi}{4}$$
  • $$\dfrac {\pi}{2}$$
If the roots of the equation $$bx^{2}+cx+a=0$$ be imaginary then for all real values of $$x$$ the expression $$3b^{2}x^{2}+6bcx+2c^{2}$$ is 
  • Less then $$4ab$$
  • Greater than $$-4ab$$
  • Less than $$-4ab$$
  • Greater then $$4ab$$
The equation $$x(x+2)(x^{2}-x)=-1$$, has 
  • All roots imaginary
  • All roots negative
  • Two roots real and two roots imaginary
  • All roots real
If $$\overline { \Delta  } =\begin{vmatrix} -1 & 2-3i & 5+4i \\ 2+3i & 8 & 1-i \\ 5-4i & 1+i & 3 \end{vmatrix}$$ then $$\Delta =$$
  • Purely real
  • Purely imaginary
  • Complex
  • $$0$$
Let $$z$$ be a complex number of maximum amplitude satisfying $$|z-3|=Re(z)$$, then $$|z-3|$$ is equal to
  • $$1$$
  • $$2$$
  • $$3/2$$
  • $$9$$
Let $$A$$ and $$B$$ represent $$z_{1}$$ and $$z_{2}$$ in the Argand plane and $$z_{1},z_{2}$$ be the roots of the equation $$z^{2}+pz+q=0$$ where $$p,q$$ are complex numbers. If $$O$$ is the origin $$OA=OB$$ and $$\angle AOB=\alpha$$ then $$p^{2}=$$
  • $$2q\ \cos \left(\dfrac{\alpha}{2}\right)$$
  • $$4q\ \cos \left(\dfrac{\alpha}{2}\right)$$
  • $$4q\ \cos^{2} \left(\dfrac{\alpha}{2}\right)$$
  • $$4q^{2}\ \cos^{2} \left(\dfrac{\alpha}{2}\right)$$
If z be any complex number such that $$|3z-2|+|3z+2|=4$$, then locus of z is
  • An ellipse
  • A circle
  • A line-segment
  • None of these
The number of imaginar roots of the equation $$(x-1)(x-2)(3x-2)(3x+1)=32$$ is
  • $$Zero$$
  • $$1$$
  • $$2$$
  • $$4$$
If $$\mathrm{{z} _ { 1 }} = 10 + 6\mathrm{i} ,  \mathrm{{ z } _ { 2 }}= 4 + 6 \mathrm { i }$$ and $$\mathrm{ z}$$ is a complex number such that $$\operatorname { amp } \left( \dfrac { \mathrm { z } - \mathrm { z } _ { 1 } } { \mathrm { z } - \mathrm { z } _ { 2 } } \right) = \dfrac { \pi } { 4 }$$ , then the value of $$\left| \mathrm{z} - 7 - 9 \mathrm { i } \right|$$ is equal to
  • $$\sqrt { 2 }$$
  • $$2\sqrt { 2 }$$
  • $$3\sqrt { 2 }$$
  • $$2\sqrt { 3 }$$
$$z_0$$ is the roots of $$1+x+x^2 =0$$ and $$z=3+6iz^{81}_0 - 3z^{93}_0$$, then arg(z) is 
  • $$\dfrac{\pi}{4}$$
  • $$\dfrac{\pi}{6}$$
  • $$\dfrac{\pi}{9}$$
  • none of these
The modulus of the complex number $$z$$ such that $$\left| z + 3 - i\right | = 1$$ and $$\arg{z} = \pi$$ is equal to
  • $$1$$
  • $$2$$
  • $$4$$
  • $$3$$
The roots of the equation $$(3b+c-4a)x^2+(3c+a-4b)x+(3a+b-4c)= 0$$ are 
  • Irrational
  • Rational
  • Non-real
  • Imaginary
$$\frac { { z }_{ 2 }-{ 2z }_{ 2 } }{ { z }_{ 2 }-{ z }_{ 1 }{ z }_{ 2 } } $$ is unimodular then
  • $$|{ z }_{ 2 }|=2$$
  • $$|{ z }_{ 1 }|=1$$
  • Both A and B
  • None of these
In quadratic equation $$ax^2 + bx + c = 0$$, if discriminant $$D = b^2 - 4ac$$, then roots of quadritic equation are: 
  • real and distinct, if D > 0
  • real and equal (repeated rotos), if D = 0
  • non-real (imaginary), if D < 0
  • none of the above
If $$Z=\sin \frac {6\pi}5+i(1+\cos \frac {6\pi }5)$$ then
  • $$|Z|=-2\cos \frac {3\pi}5$$
  • $$Arg(Z)=\frac {\pi}5$$
  • $$Arg(Z)=\frac {9\pi }{10}$$
  • none of these
If $$z_1$$ and $$z_2$$ are two non zero complex numbers such that $$|z_1 + z_2| = |z_1| + |z_2|,$$ then arg $$z_1$$ - arg $$z_2$$ is equal
  • $$-\pi$$
  • $$\frac{-\pi}{2}$$
  • $$0$$
  • $$\frac{\pi}{2}$$
This equation $$(x-5)^{11}+(x-5^{2})^{11}+....+(x-5^{11})^{11}=0$$ has 
  • all the roots real
  • one real and 10 imaginary roots
  • real roots namely $$x=5,5^{2}....,5^{9},5^{10},5^{11}$$
  • none
Find the value of $$k$$ for which the roots are real and equal:
$$5{ x }^{ 2 }-4x+2+k(4{ x }^{ 2 }-2x-1)=0$$
  • $$1$$
  • $$-1$$
  • $$  \dfrac{6}{5}$$
  • $$ - \dfrac{6}{5}$$
If z-{1} and z-{2} are two non zero complex numbers such that $$\left| z{  }_{ 1 }+z{  }{  }_{ 2 } \right| =\left| { z }_{ 1 } \right| +\left| { z }_{ 2 } \right| $$, then arg$$z_{1}$$-arg $$z_{2}$$ is equal to:
  • $$-\pi$$
  • $$-\frac\pi{2}$$
  • $$0$$
  • $$\frac\pi{2}$$
Argument and modulus of $$\left[ \frac { 1 + i } { 1 - i } \right] ^ { 2013 }$$ are respectively $$\ldots \ldots \ldots$$
  • $$\frac { - \pi } { 2 }$$ and 1
  • $$\frac { \pi } { 2 }$$ and $$\sqrt { 2 }$$
  • 0 and $$\sqrt { 2 }$$
  • $$\frac { \pi } { 2 }$$ and 1
If the roots of the equation $${ x }^{ 2 }-8x+({ a }^{ 2 }-6a)=0$$ are real, then
  • -2 < a < 8
  • 2 < a < 8
  • $$2\le a\le 8$$
  • $$-2\le a\le 8$$
If $$z$$ is a complex number of unit modulus and argument $$\theta$$, then $$arg\left(\dfrac{1+z}{1+\overline{z}}\right)$$ equals 
  • $$-\theta$$
  • $$\dfrac{\pi}{2}-\theta$$
  • $$\theta$$
  • $$\pi-\theta$$
If a and b are positive real numbers and each of the equations $${ x }^{ 2 }+3a{ x }^{ 2 }+b=0$$ and $${ x }^{ 2 }+bx+3a=0$$ has real roots, then the smallest value of (a+b) is 
  • 16/3
  • 6
  • 14/3
  • 4
Let $$z$$, $$w$$ be complex numbers such that $$ \overline { z } +i\overline { w } =0$$ and arg $$\left(ZW\right)= pi$$. Then, arg $$\left(z\right)$$ equals
  • $$\dfrac { \pi }{ 4 } $$
  • $$\dfrac { \pi }{ 2 } $$
  • $$\dfrac { \3pi }{ 4 } $$
  • $$\dfrac { \5pi }{ 4 } $$
Argument of the complex number  $$\left( \dfrac { - 1 - 3 i } { 2 + i } \right).$$
  • $$45 ^ { \circ }$$
  • $$135 ^ { \circ }$$
  • $$225 ^ { \circ }$$
  • $$240 ^ { \circ }$$
If $$z=-1$$, then principal value of arg $$\left({z}^{2/3}\right)$$ is
  • $$0,\dfrac{2\pi}{3},-\dfrac{2\pi}{3}$$
  • $$\dfrac{\pi}{3},2\pi$$
  • $$\dfrac{5\pi}{3}$$
  • $$-\pi,\pi$$
The amplitude of $${ e }^{ { e }^{ -i\theta  } }$$ is equal to
  • sin$$\theta $$
  • -sin$$\theta $$
  • $${ e }^{ \cos { \theta } }$$
  • $${ e }^{ \sin { \theta } }$$
If $$z=-1$$, then principal value of arg $$\left( {z}^{\dfrac{2}{3}} \right )$$ is 
  • $$0,\dfrac{2\pi}{3},-\dfrac{2\pi}{3}$$
  • $$\dfrac{\pi}{3},2\pi$$
  • $$\dfrac{5\pi}{3}$$
  • $$-\pi,\pi$$
$$ z _ { 0 } $$ is the roots of $$ 1 + x + x ^ { 2 } = 0 $$ and $$ z = 3 + 6 i z _ { 0 } ^ { 81 } - 3 z _ { 0 } ^ { 93 } . $$ Then $$ \arg ( z ) $$ is-
  • $$

    \frac { \pi } { 4 }

    $$
  • $$

    \frac { \pi } { 6 }

    $$
  • $$

    \frac { \pi } { 9 }

    $$
  • $$

    \frac { \pi } { 2 }

    $$
Find the modules and amplitude for each of the following complex numbers
  • 7-5i
  • $$\sqrt { 3 } +\sqrt { 2 } i$$
  • -8+15i
  • -3(1-i)
The value of 'a' fro which the equation $$a{x^2} - 2\sqrt 5 x + 4 = 0\;$$ has equal roots is 
  • $$\frac{5}{4}$$
  • $$\frac{4}{5}$$
  • $$\frac{-5}{4}$$
  • $$\frac{{ - 5}}{3}$$
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