CBSE Questions for Class 11 Engineering Maths Complex Numbers And Quadratic Equations Quiz 2 - MCQExams.com

For the quadratic equation $$ax^2 + bx + c = 0, a, b, c, \in Q$$, If $$D = 0$$ then ...................
Choose the correct option in respect to the statements below.
(P) The roots of the equation are equal.
(Q) The roots of the equation are not equal.
(R) The roots of the equation are rational numbers.
(S) The roots of the equation has no roots.
  • Statements $$P$$ and $$R$$ are correct
  • Statements $$Q$$ and $$R$$ are correct
  • Only statement $$S$$ is correct
  • Only statement $$P$$ is correct
What is the modulus of $$\cfrac { \sqrt { 2 } +i }{ \sqrt { 2 } -i } $$ where $$i=\sqrt { -1 } $$
  • $$3$$
  • $$\dfrac{1}{2}$$
  • $$1$$
  • None of the above
In the complex plane, what is the distance of $$4-2i$$ from the origin?
  • $$2$$
  • $$3.46$$
  • $$4.47$$
  • $$6$$
  • $$12$$
If the roots of an equation  $$p{ x }^{ 2 }+qx+r=0$$  are equal, then
  • $${ q }^{ 2 }=pr$$
  • $${ q }^{ 2 }=4pr$$
  • $${ p }^{ 2 }=4qr$$
  • $$p=qr$$
In the complex plane, the number 4 + j3 is located in the
  • first quadrant
  • second quadrant
  • third quadrant
  • fourth quadrant
For a quadratic equation if $$D < 0$$ then which of the following is true?
  • Real roots do not exist
  • Roots are real and equal
  • Roots are rational and distinct
  • Roots are real and distinct
The roots of the equation $${ \left( z+\alpha \beta  \right)  }^{ 3 }={ \alpha  }^{ 3 }$$ represent the vertices of a triangle, one of whose sides is of length
  • $$\sqrt { 3 } \left| \alpha \beta \right|$$
  • $$\sqrt { 3 } \left| \alpha \right|$$
  • $$\sqrt { 3 } \left| \beta \right|$$
  • $$None\ of\ these$$
Put the following in the form of A + iB :
$$\dfrac{(3 \, - \, 2i)(2 \, + \, 3i)}{(1 \, + \, 2i)(2 \, - \, i)}$$
  • $$\dfrac{3}{4} \, + \, \dfrac{9}{4} \, i$$
  • $$\dfrac{63}{25} \, - \, \dfrac{16}{25} \, i$$
  • $$\dfrac{5}{4} \, + \, \dfrac{9}{4} \, i$$
  • $$\dfrac{1}{4} \, + \, \dfrac{7}{4} \, i$$
State true or false:
The following quadratic equations has real roots  
$$3a^2x^2 \, + \, 8abx \, + \, 4b^2 \, = \, 0, \, a ,b\, \neq \, 0$$
  • True
  • False
When will the quadratic equation $$ax^2+bx+c=0$$ NOT have Real Roots?
  • $$b^2 - 4ac \ge 0$$
  • $$b^2 - 4ac > 0$$
  • $$b^2 - 4ac < 0$$
  • None of these

If the value of '$$b^2-4ac$$' is greater than zero, the quadratic equation $$ax^2+bx+c=0$$ will have
  • Two Equal Real Roots.
  • Two Distinct Real Roots.
  • No Real Roots.
  • No Roots or Solutions.
If $$a, b, c$$ are real and $$b^2- 4ac $$ is perfect square then the roots of the equation $$ax^2+bx+c=0$$, will be:
  • Rational & distinct
  • Real & equal
  • Irrational & distanct
  • Imaginary & distinct

If the value of '$$b^2-4ac$$' is less than zero, the quadratic equation $$ax^2+bx+c=0$$ will have
  • Two Equal Real Roots.
  • Two Distinct Real Roots.
  • No Real Roots.
  • None of the above.
The number of solution of $$z^2 + \bar{z} = 0$$ is
  • $$5$$
  • $$4$$
  • $$2$$
  • $$3$$
$$arg\left( -\cfrac { 3 }{ 2 }  \right) $$ equals
  • $$\cfrac{\pi}{2}$$
  • $$-\cfrac{\pi}{2}$$
  • $$0$$
  • $$\pi $$
Find the which of the complex number has greatest modulus.
  • $$7-5i$$
  • $$\sqrt{3}+\sqrt{2}i$$
  • $$-8+15i$$
  • $$-3(1-i)$$
The roots of the equation $$x^{2}+2\sqrt {3}x+3=0$$ are 
  • real and unequal
  • rational and equal
  • irrational and equal
  • irrational and unequal
 For any complex number $$z$$ the minimum value of $$|z|+|z-2013i|$$ is...
  • $$2010$$
  • $$2011$$
  • $$2013$$
  • $$2012$$
Which of the following is true
  • $$(3 + \sqrt{-5})(3 - \sqrt{-5}) = 14$$
  • $$(-2 + \sqrt{-3})(-3 + 2\sqrt{-3}) = -7\sqrt{3}i$$
  • $$(2 + 3i)^2 = (-5 + 12i)$$
  • $$(\sqrt{5} - 7i)^2 = -44 - 14\sqrt{5}i$$
Argument and modulus of $$\left[\dfrac {1+i}{1-i}\right]^{2013}$$ are respectively ____
  • $$\dfrac {-\pi}{2}$$ and $$1$$
  • $$\dfrac {\pi}{2}$$ and $$\sqrt {2}$$
  • $$0$$ and $$\sqrt {2}$$
  • $$\dfrac {\pi}{2}$$ and $$1 $$
 If $$z_1=\sqrt { 3 } -i,z_2=1+i\sqrt { 3 } ,$$ then amp$$(z_1+z_2)=$$ 
  • $$\dfrac { \pi }{ 12 } $$
  • $$\dfrac { \pi }{ 15 } $$
  • $$\dfrac { \pi }{ 6 } $$
  • $$\dfrac { \pi }{ 4 } $$
If the roots of $$2x^2+3x+p=0$$ be equal, then the value of p is :
  • $$\dfrac{9}{8}$$
  • $$\dfrac{6}{5}$$
  • $$\dfrac{4}{3}$$
  • $$\dfrac{5}{4}$$
If $$2^{x^{2}} : 2^{2x}=8^{k}:1$$, then equation has only one solution if
  • $$ k > \dfrac{1}{3}$$
  • $$ k = \dfrac{1}{3}$$
  • $$ k < \dfrac{-1}{3}$$
  • $$ k = \dfrac{-1}{3}$$
If $$z_1=3+4i\\z_2=4-5i$$ Then find $$z_1+z_2$$
  • 7-i
  • 7+i
  • 7+9i
  • None of these
Find the least positive value of n, if $$(\dfrac{1+i}{1-i})^n=1$$
  • 1
  • 2
  • 3
  • 4
The complex numbers $$z_1=8+9i, z_2=4-6i$$ then $$z_1-z_2$$
  • $$4+15i$$
  • $$4-3i$$
  • $$12+3i$$
  • $$12-15i$$
If $$z$$ is a complex number such that $$|z|=1$$, then $$\left|\dfrac 1{\bar z}\right|$$ is 
  • $$0$$
  • $$-1$$
  • $$\sqrt{2}$$
  • $$1$$
If $$z_1=3+4i,z_2=2-i$$ find $$z_2-z_1$$
  • -1-5i
  • 2-5i
  • 1+5i
  • 1-5i
If $$\alpha \epsilon \left( -1,1 \right) $$ then roots of the quadratic equation $$\left( a-1 \right) { x }^{ 2 }+ax+\sqrt { 1-{ a }^{ 2 } } =0$$ are
  • real
  • imaginary
  • both equal
  • none of these
If $$z_1=4+i,z_2=4-i $$ find $$z_1z_2$$
  • $$17$$
  • $$16$$
  • $$17-i$$
  • $$16i$$
$$z_1=9+8i\ \ \  |z|=$$
  • $$\sqrt {145}$$
  • $$\sqrt {163}$$
  • $$\sqrt {117}$$
  • $$\sqrt {137}$$
If $$(x+iy)(2-3i)=4+i\left ( \dfrac{1}{2} \right )$$ then $$x + y =$$
  • $$\dfrac{3}{2}$$
  • $$\dfrac{1}{2}$$
  • $$0$$
  • $$\dfrac{2}{3}$$
If $$z_1$$ and $$z_2$$ are two complex numbers, then $$Re(z_1z_2)$$ is:
  • $$Re(z_1)Re(z_2)$$
  • $$Re(z_1).Re(z_2)-Im(z_1).Im(z_2)$$
  • $$Im(z_1).Re(z_2)$$
  • $$Re(z_1).Im(z_2)$$
$$\left| \dfrac { (3+i)(2-i) }{ 1+i }  \right|=$$
  • $$\sqrt{5}$$
  • $$5\sqrt{2}$$
  • $$\sqrt{10}$$
  • $$5$$
The real part of $$\left ( \dfrac{1+i}{3-i} \right )^2=$$
  • $$1$$
  • $$16$$
  • $$16\omega ^2$$
  • $$\displaystyle \frac{-3}{25}$$
If $$z=-1+3i$$ then $$z^2+2z+10=$$
  • $$0$$
  • $$1$$
  • $$–1$$
  • $$2$$
In the argand diagram, the complex number z is in the fourth quadrant,  then $$\overline{z}$$, $$-z$$, $$\overline{-z}$$ are respectively are in quardrants
  • 1,3,2
  • 1,2,3
  • 3,2,1
  • 2,1,3
The value of $$1+(1+i)+(1+i)^2+(1+i)^3=$$
  • $$0$$
  • $$5i$$
  • $$4i$$
  • $$3i$$
If $$\left ( 5+3i \right )(x+iy)=3-4i$$ then $$34x = $$
  • $$1$$
  • $$2$$
  • $$3$$
  • $$4$$
If $$z_1$$, $$z_2$$ are the complex numbers such that $$|z_1+z_2|=|z_1|+|z_2|$$ then arg $$z_1 - $$ arg $$z_2$$ is
  • $$-\pi $$
  • $$\dfrac{-\pi }{2}$$
  • 0
  • $$\dfrac{\pi }{2}$$
The simplified value of $$\displaystyle \frac{1-i}{1+i}$$ is:
  • $$i$$
  • $$-i$$
  • $$1$$
  • $$-2i$$

The minimum value of $$|\mathrm{z}|+|\mathrm{z}-1|+|\mathrm{z}-2|$$ is
  • $$0$$
  • $$1$$
  • $$2$$
  • $$4$$
If $$\alpha $$ and $$\beta $$ are real then $$\left| \dfrac { \alpha +i\beta  }{ \beta +i\alpha  }  \right|=$$ 
  • Lies betwen 0 and 1
  • = 1
  • >1
  • 2
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 order among the following is :
  • $$m_1$$<$$m_2$$<$$m_3$$<$$m_4$$
  • $$m_4$$<$$m_3$$<$$m_2$$<$$m_1$$
  • $$m_3$$<$$m_2$$<$$m_4$$<$$m_1$$
  • $$m_3$$<$$m_1$$<$$m_2$$<$$m_4$$
The principal argument of $$z=-3+3i$$ is:
  • $$\dfrac{\pi }{4}$$
  • $$-\dfrac{\pi }{4}$$
  • $$\dfrac{3\pi }{4}$$
  • $$-\dfrac{3\pi }{4}$$
Assertion (A): The principal amplitude of complex number $$x + ix$$ is $$\cfrac{\pi }{4}$$.
Reason (R): The principal amplitude of a complex number $$x + iy$$ is $$\cfrac{\pi }{4}$$ if $$y = x$$.
  • Both A and R are true and R is the correct explanation of A
  • A is true R is false
  • A is false, R is true
  • Both A and R are false
The area of the triangle formed by the three complex numbers $$1 + i$$, $$i - 1$$ , $$2i$$ in the Argand diagram is:
  • $$\dfrac{1}{2}$$
  • $$1$$
  • $$\sqrt{2}$$
  • $$2$$
The modulus of $$(1 + i) (3 + 4i) =$$
  • $$\sqrt{50}$$
  • $$\sqrt{25}$$
  • $$10$$
  • $$10 \sqrt{2}$$
If $$z_1$$, $$z_2$$, $$z_3$$ are complex numbers such that $$\left| { z }_{ 1 } \right| =\left| { z }_{ 2 } \right| =\left| { z }_{ 3 } \right| =\left| \dfrac { 1 }{ { z }_{ 1 } } +\dfrac { 1 }{ { z }_{ 2 } } +\dfrac { 1 }{ { z }_{ 3 } }  \right| =1$$, then $$|z_1+z_2+z_3|$$ is:
  • Equal to 1
  • Less than 1
  • Greater than 3
  • Equal to 3
lf $$(x+iy)(2+cos\theta+isin\theta)=3$$ then $$x^{2}+y^{2}-4x+3$$ is
  • $$0$$
  • $$1$$
  • $$3$$
  • $$4$$
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