CBSE Questions for Class 11 Engineering Maths Complex Numbers And Quadratic Equations Quiz 8 - MCQExams.com

The locus of $$z$$ such that $$\left| {\dfrac{{z + i}}{{z - 1}}} \right| = 2$$
  • straight line
  • circle with radius $$2$$
  • circle with radius $$\frac{{2\sqrt 2 }}{3}$$
  • none of these
$$\left(\dfrac{1 + i}{1 - i}\right)^4 + \left(\dfrac{1 - i}{1 + i}\right)^4 = $$ 
  • $$0$$
  • $$1$$
  • $$2$$
  • $$4$$
If $$|z_1+z_2|=|z_1|+|z_2|$$ where $$z_1$$ and $$z_2$$ are different non - zero complex number, then ?
  • $$Re\left(\dfrac{z_1}{z_2}\right)=0$$
  • $$Im\left(\dfrac{z_1}{z_2}\right)=0$$
  • $$z_1+z_2=0$$
  • None
Modulus of $$\dfrac{\cos \theta - i\sin \theta}{\sin \theta - i \cos \theta}$$ is
  • $$0$$
  • $$2\theta$$
  • $$\pi - 2\theta$$
  • None of these
If $$\cfrac{\pi}{3}$$ and $$\cfrac{\pi}{4}$$ are the arguments of $${z}_{1}$$ and $${\overline { z }  }_{ 2 }$$, then the value of arg $$({z}_{1}{z}_{2})$$ is
  • $$\cfrac{5\pi}{12}$$
  • $$\cfrac{\pi}{12}$$
  • $$\cfrac{7\pi}{12}$$
  • None of these
$$n\in N,\ { \left( \dfrac { 1+i }{ \sqrt { 2 }  }  \right)  }^{ 8n }+{ \left( \dfrac { 1-i }{ \sqrt { 2 }  }  \right)  }^{ 8n }=$$
  • $$0$$
  • $$1$$
  • $$2$$
  • $$-2$$
The roots of the equation $${x}^{\sqrt{x}}={(\sqrt{x})}^{x}$$ are
  • $$0$$ and $$1$$
  • $$0$$ and $$4$$
  • $$1$$ and $$4$$
  • $$0,1$$ and $$4$$
$$\left(\dfrac{1+\cos \dfrac{\pi}{8}+i\sin \dfrac{\pi}{8}}{1+\cos \dfrac{\pi}{8}-i\sin \dfrac{\pi}{8}}\right)^{8}=$$ ?
  • $$1+i$$
  • $$1-i$$
  • $$1$$
  • $$-1$$
The complex number $$z$$ satisfies $$z+|z|=2+8i$$. The value of $$|z|$$ is
  • $$10$$
  • $$13$$
  • $$17$$
  • $$23$$

For a complex number $$z$$, the minimum value of $$\left| z \right| + \left| {z - 1} \right|$$ is
  • 1
  • 2
  • 3
  • none of these

The value of $$\sum\limits_{n = 1}^{13} {\left( {{i^n} + {i^{n + 1}}} \right)} $$, where $$i = \sqrt { - 1} $$ equals:
  • $$i$$
  • $$i - 1$$
  • $$ - i$$
  • 0
The modulus of the complex quantity $$(2-3i)(-1+7i)$$.
  • $$5\sqrt{13}$$
  • $$5\sqrt{26}$$
  • $$13\sqrt{5}$$
  • $$26\sqrt{5}$$
If A and B be two complex numbers satisfying $$\dfrac{A}{B}+\dfrac{B}{A}=1$$. Then the two points represented by A and B and the origin form the vertices of
  • An equilateral triangle
  • An isosceles triangle which is not equilateral
  • An isosceles triangle which is not right angled
  • A right angled triangle
$$3+2\ i\ \sin \theta$$ will be real, if $$\theta=$$
  • $$2n \pi $$
  • $$n \pi +\pi/2$$
  • $$n\pi$$
  • $$none\ of\ these$$
If $$z_{1},\ z_{2}$$ are two complex numbers such that $$arg\left( { z }_{ 1 }+{ z }_{ 2 } \right) =0$$ and $$Im\left( { z }_{ 1 }{ z }_{ 2 } \right) =0$$, then
  • $$z_{1}=-z_{2}$$
  • $$z_{1}=z_{2}$$
  • $$z_{1}=\bar { { z }_{ 2 } } $$
  • $$none\ of\ these$$
The argument of the complex number $$\sin \dfrac {6\pi}{5}+i\left(1+\cos \dfrac {6\pi}{5}\right)$$ is
  • $$\dfrac {6\pi}{5}$$
  • $$\dfrac {5\pi}{6}$$
  • $$\dfrac {9\pi}{10}$$
  • $$\dfrac {2\pi}{5}$$
Let $$a, b, c\epsilon R_{0}$$ and $$1$$ be a root of the equation $$ax^{2} + bx + c = 0$$, then the equation $$4ax^{2} + 3bx + 2c = 0$$ has
  • Imaginary roots
  • Real and equal roots
  • Real and unequal roots
  • Rational roots
If z is a complex number such that $$|z|\ge 2$$, then the minimumm value of $$\left|z+\dfrac{1}{2}\right|$$:
  • is equal to $$\dfrac{5}{2}$$
  • lies in the interval $$(1,2)$$
  • is strictly greater then $$\dfrac{5}{2}$$
  • is strictly greater than $$\dfrac{3}{2}$$ but less than $$\dfrac{5}{2}$$
The complex no. $$\dfrac{1+2i}{1-i}$$ lies in which quadrant of the complex plane
  • first
  • second
  • third
  • fourth
If $$z \neq 0$$, then $$ \overset{100}{\underset{0}{\int}}arg(-|z|)dx =$$
  • $$0$$
  • Not defined
  • $$100$$
  • $$100\pi$$
If $$z$$ is purely real and $$Re(z)<0$$, then $$arg(x)$$ is
  • $$0$$
  • $${\pi}$$
  • $$-{\pi}$$
  • $$\dfrac{\pi}{2}$$
If $$Z$$ is a complex number such that $$|z| \ge 2$$,
then the minimum value of $$\left|z + \dfrac{1}{2}\right|$$
  • Is equal to $$\dfrac{5}{2}$$
  • Lies in the interval $$(1, 2)$$
  • Is strictly grater than $$\dfrac{5}{2}$$
  • Is strictly greater than $$\dfrac{3}{2}$$ but less than $$\dfrac{5}{2}$$
If $$|z|=1$$ and $$|\omega -1| =1$$ where $$z, \omega \in C$$, then the largest set of values of $$|2z - 1|^2 + | 2\omega -1|^2$$ equals  
  • $$[1, 9]$$
  • $$[2, 6]$$
  • $$[2, 12]$$
  • $$[2, 18]$$
If for complex number $$z_{1}and   z_{2}arg(z_{1})-arg(z_{2})=0then \mid z_{ 1}-z_{2}\mid $$ is equal to:
  • $$ \mid z_{1}+z_{2}\mid $$
  • $$ \mid z_{1}\mid +\mid z_{2}\mid$$
  • $$ \parallel z_{1}\mid -\mid z_{2}\parallel$$
  • 0
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 to
  • $$-\pi$$
  • $$\dfrac {\pi}{2}$$
  • $$-\dfrac {\pi}{2}$$
  • $$0$$
Argument and modules of $$[\dfrac{1+i}{1-i}]^{2\pi i}$$ are respectively................. 
  • $$\dfrac{-\pi}{2}$$ and $$1$$
  • $$\dfrac{\pi}{2}$$ and $$\sqrt{2}$$
  • $$0$$ and $$\sqrt{2}$$
  • $$\dfrac{\pi}{2}$$ and $$1$$

In Argand diagram, O, P, Q represents the origin, $$z$$ and $$z+iz$$
respectively. then $$\angle OPQ = $$
  • $$\dfrac{\pi }{4}$$
  • $$\dfrac{\pi }{6}$$
  • $$\dfrac{\pi }{2}$$
  • $$\dfrac{\pi }{3}$$
What is the modulus of following complex number:$$-2+2\sqrt { 3i } $$
  • 4
  • 5
  • 2
  • 3
If $$\theta$$ and $$\phi$$ are the roots of the equation $$8x^{2}+22x+5=0$$, then
  • both $$\sin^{-1}{\theta}$$ and $$\sin^{-1}{\phi}$$ are real
  • both $$\sec^{-1}{\theta}$$ and $$\sec^{-1}{\phi}$$ are real
  • both $$\tan^{-1}{\theta}$$ and $$\tan^{-1}{\phi}$$ are real
  • None of these
If root of the equation $${ \left( q-r \right) x }^{ 2 }+\left( r-p \right) x+\left( p-q \right) =0$$ are equal, then $$p,q,r$$ are in
  • $$AP$$
  • $$GP$$
  • $$HP$$
  • $$None\ of\ these$$
The integral values of $$'a'$$ for which the equation $$\cos ^{2}x-(a^{2}+a+5)|\cos x|+(a^{3}+3a^{2}+2a+6)=0$$ has real solution(s)
  • $$-3$$
  • $$-2$$
  • $$-1$$
  • $$0$$
If $$x_{r}=\cos(\dfrac{\pi}{3^{r}})+i\sin(\dfrac{\pi}{3^{r}})$$, then  $$x_{1}x_{2}x_{3}$$.... upto $$infinity=i$$.
  • True
  • False
If $$z=(3+7i)(p+iq)$$ where $$p,q\in I-\left\{ 0 \right\} $$, is purely imaginary then minimum value of $${ \left| z \right|  }^{ 2 }$$ is
  • $$0$$
  • $$58$$
  • $$\dfrac{3364}{3}$$
  • $$3364$$
The figure formed by four points $$1+0i;-1+0i,3+4i$$ and $$\cfrac{25}{-3-4i}$$ on the argand plane is
  • parallelogram but not a rectangle
  • a trapesium which is not equilateral
  • cyclic quadrilateral
  • none of these
The amplitude of $$\cfrac { 1+\sqrt { 3i }  }{ \sqrt { 3 } +1 } $$ is
  • $$\cfrac { \pi }{ 3 } $$
  • $$-\cfrac { \pi }{ 3 } $$
  • $$\cfrac { \pi }{ 6 } $$
  • $$-\cfrac { \pi }{ 6 } $$
If $$Z = \frac{{1 - \sqrt 3 i}}{{1 + \sqrt 3 i}}$$ then find $$arg(z).$$
  • $$- \dfrac{2\pi }{3} $$
  • $$ \dfrac{2\pi }{5} $$
  • $$ \dfrac{\pi }{3} $$
  • $$ \dfrac{2\pi }{3} $$
If $$\left| {z + 2 - i} \right| = 5$$ then the maximum value of $$\left| {3z + 9 - 7i} \right|$$ is 
  • 18
  • 19
  • 20
  • 8
The set of the possible values of $$a$$ for which the expression $$x^{2}-ax+1-2a^{2}$$ is always positive is $$(\alpha, \beta)$$, then
  • $$\alpha-2\beta=2$$
  • $$\alpha+2\beta=2$$
  • $$\beta+2\alpha=2$$
  • $$\beta-2\alpha=2$$
If $$\left|z\right|=1$$ and $$\varpi=\dfrac{z-1}{z+1}$$, where $$z\neq-1$$, then $$Re\left(\varpi\right)$$ is
  • $$0$$
  • $$-\dfrac{1}{|z+1|^{2}}$$
  • $$\dfrac{1}{\sqrt{2}}{\left|z+1\right|^{2}}$$
  • $$\dfrac{\sqrt{2}}{|z+1|^{2}}$$
If $$a,b,c$$ are distinct, positive  in $$H.P$$, then quadratic equation $${ ax }^{ 2 }+2bx+c=0$$ has 
  • Two non-real roots such that their sum is real
  • Two distinct real roots
  • Two non real conjugate roots
  • Two equal real roots
The root of the equation $${\left| x \right|^2} + \left| x \right| - 6 = 0$$ are -
  • only one real number
  • real and sum =$$1 \, or \, -1$$
  • real and sum =$$0$$
  • real and product =$$0$$
If $$z=(3+7i)(p+iq)$$, where $$p,q\in I-\left\{ 0 \right\} $$, is a purely imaginary, then minimum value of $${ \left| z \right|  }^{ 2 }$$ is
  • $$0$$
  • $$58$$
  • $$\cfrac{3364}{3}$$
  • $$3364$$
If $$z=\dfrac{\sqrt{3}}{2}+\dfrac{1}{2}i$$ then $$z\bar{z}$$ is
  • $$-1$$
  • $$0$$
  • $$1$$
  • $$2$$
Solve $$i^{57}+\dfrac{1}{i^{125}}$$
  • $$0$$
  • $$2i$$
  • $$-2i$$
  • $$2$$
$$(y^{2}+1)^{2}-y^{2}=0$$ has .
  • No real roots
  • One real root
  • Two real roots
  • Three real roots
  • None of these
If $$z_{1} and z_{2} are on straight line$$ $$\left| \frac { 1 } { 2 } \left( z _ { 1 } + z _ { 2 } \right) + \sqrt { z _ { 1 } z _ { 2 } } \right| + \left| \frac { 1 } { 2 } \left( z _ { 1 } + z _ { 2 } \right) - \sqrt { z _ { 1 } z _ { 2 } } \right| =$$
  • $$\left| z _ { 1 } + z _ { 2 } \right|$$
  • $$\left| z _ { 1 } - z _ { 2 } \right|$$
  • $$\left| z _ { 1 } \right| + \left| z _ { 2 } \right|$$
  • $$\left| z _ { 1 } \right| - \left| z _ { 2 } \right|$$
The value of $$2x^{4}+5x^{3}+7x^{2}-x+41$$, when $$x=-2-\sqrt{3i}$$ is:
  • -4
  • 4
  • -6
  • 6
The argument of the complex number $$\sin \dfrac{{6\pi }}{5} + i\left( {1 + \cos \dfrac{{6\pi }}{5}} \right)$$ is 
  • $$\dfrac{{6\pi }}{5}$$
  • $$\dfrac{{5\pi }}{5}$$
  • $$\dfrac{{9\pi }}{10}$$
  • $$\dfrac{{7\pi }}{10}$$
If $$Re(\dfrac{z+2i}{z+4})=0$$ then z lies on a circle with center:
  • (-2,-1)
  • (-2,1)
  • (2,-1)
  • (2,1)
If the six solutions of $$x^6 = -64$$ are written in the form $$a + bi$$, where $$a$$ and $$b$$ are real, then the product those solution with $$a < 0$$, is
  • $$4$$
  • $$8$$
  • $$16$$
  • $$64$$
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