CBSE Questions for Class 11 Engineering Maths Complex Numbers And Quadratic Equations Quiz 6 - MCQExams.com

If quadratic equation $$\displaystyle 6x^{2}-7x+2=0$$ has one root $$\displaystyle \frac{1}{2}$$ then the second root will be--
  • $$\displaystyle -\frac{1}{2}$$
  • $$\displaystyle \frac{2}{3}$$
  • $$\displaystyle \frac{3}{2}$$
  • 1
The sum of the roots of $$ \displaystyle \frac{1}{x+a}+\frac{1}{x+b}=\frac{1}{c} $$   is zero. The product of the roots is
  • $$ \displaystyle -\frac{1}{2}(a^{2}+b^{2}) $$
  • 0
  • $$ \displaystyle \frac{1}{2}(a+b) $$
  • $$ \displaystyle 2(a^{2}+b^{2}) $$
Find the value of P for which the following equation has equal roots $$\displaystyle { px }^{ 2 }-8x+2p=0$$ 
  • $$\displaystyle \pm 2\sqrt { 2 } $$
  • $$\displaystyle \pm \sqrt { 2 } $$
  • $$\displaystyle \pm \sqrt { 3 } $$
  • $$\displaystyle \pm 2\sqrt { 3 } $$
The roots of the equation $${2x^{2 }+ 3x + 2} = 0$$ are
  • Real, rational, and equal
  • Real,rational and unequal
  • Real, irrational, and unequal
  • non real (imaginary)
Find the product. Write the answer in standard form.
$$i\left( 6-2i \right) \left( 7-5i \right) $$
  • $$52+16i$$
  • $$10{i}^{3}+44{i}^{2}+42i$$
  • $$-44-32i$$
  • $$44+32i$$
Which of the following statements has the truth value $$'F'$$?
  • A quadratic equation has always a real root
  • The number of ways of seating $$2$$ persons in two chairs out of $$n$$ persons in $$P(n,2)$$
  • The cube roots of unity are in GP
  • None of the above
The imaginary number $$i$$ is defined such that $$i^2=-1$$. What is the value of $$(1 - i \sqrt {5}) ( 1 + i\sqrt {5})$$?
  • $$\sqrt5$$
  • $$5$$
  • $$6$$
  • $$\sqrt6$$
The product of $$(3-2i)$$ and $$\left(\dfrac { 5 }{ 2 } -4i\right)$$, if $$i=\sqrt { -1 } $$ , is:
  • $$-\dfrac { 1 }{ 2 } -17i$$
  • $$14+\dfrac{9}{2}i$$
  • $$2-8i-14{i}^{2}$$
  • $$i\left(8+\dfrac{9}{2}\right)$$
The argument of $$\dfrac {1 + i\sqrt {3}}{\sqrt {3} + i}$$ is
  • $$\dfrac {\pi}{3}$$
  • $$\dfrac {\pi}{4}$$
  • $$\dfrac {2\pi}{3}$$
  • $$\dfrac {\pi}{6}$$
The principal amplitude of $$\displaystyle { \left( \sin { { 40 }^{ \circ  }+i\cos { { 40 }^{ \circ  } }  }  \right)  }^{ 5 }$$ is
  • $$\displaystyle { 70 }^{ \circ }$$
  • $$\displaystyle { -110 }^{ \circ }$$
  • $$\displaystyle { 110 }^{ \circ }$$
  • $$\displaystyle { -70 }^{ \circ }$$
Let $${ X }_{ n }=\left\{ z=x+iy:{ \left| z \right|  }^{ 2 } \le \dfrac { 1 }{ n }  \right\} $$ for all integers $$n\ge 1$$. Then, $$\displaystyle\bigcap _{ n=1 }^{ \infty  }{ { X }_{ n } } $$ is
  • A singleton set
  • Not a finite set
  • An empty set
  • A finite set with more than one element
The modulus of $$\dfrac { 1-i }{ 3+i } +\dfrac { 4i }{ 5 } $$ is
  • $$\sqrt { 5 } $$ unit
  • $$\dfrac { \sqrt { 11 } }{ 5 } $$ unit
  • $$\dfrac { \sqrt { 5 } }{ 5 } $$ unit
  • $$\dfrac { \sqrt { 12 } }{ 5 } $$ unit
Simplify $$(2+8i)(1-4i)-(3-2i)(6+4i)$$
 (Note$$:i=\sqrt{-1}$$)
  • $$8$$
  • $$26$$
  • $$34$$
  • $$50$$
$$\displaystyle \left|\dfrac{\sqrt{3}+i}{(1+i)(1+\sqrt{3}i)}\right|=$$
  • $$1$$
  • $$\sqrt{2}$$
  • $$\dfrac{1}{2}$$
  • $$\dfrac{1}{\sqrt{2}}$$
What is the approximate magnitude of $$8 + 4i$$?
  • 4.15
  • 8.94
  • 12.00
  • 18.64
  • 32.00
Find the value of $$k$$ for which the number lies between the roots of the equation $${ k }^{ 2 }+(1-2k)x+({ x }^{ 2 }-k-2)=0$$.
  • $$3< k< 4$$
  • $$3< k< 5$$
  • $$2< k< 6$$
  • $$2< k< 5$$
Given that $$4$$ is a root of the quadratic equation $${x}^{2}-5x+q=0$$. Find the value of $$q$$ and the other root.
  • $$4$$ and $$1$$ respectively
  • $$1$$ and $$4$$ respectively
  • $$4$$ and $$3$$ respectively
  • $$3$$ and $$1$$ respectively
The real part of $${ \left( 1-\cos { \theta  } +i\sin { \theta  }  \right)  }^{ -1 }$$ is
  • $$\cfrac{1}{2}$$
  • $$\cfrac { 1 }{ 1+\cos { \theta } } $$
  • $$\tan { \cfrac { \theta }{ 2 } } $$
  • $$\cot { \cfrac { \theta }{ 2 } } $$
If $$z = \dfrac {(\sqrt {3} + i)^{3} (3i + 4)^{2}}{(8 + 6i)^{2}}$$, then $$|z|$$ is equal to
  • $$0$$
  • $$1$$
  • $$2$$
  • $$3$$
If the roots of the equations $$ax^{2} + 2bx + c = 0$$ and $$bx^{2} - 2\sqrt {ac} x + b = 0$$ are simultaneously real, then
  • $$b^{2} = 4ac$$
  • $$b^{2} = ac$$
  • $$2b^{2} = 9ac$$
  • none
If the roots of the equation $${ x }^{ 2 }+2(3a+5)x+2(9{ a }^{ 2 }+25)=0$$ are real, then find $$a$$.
  • $$\cfrac{5}{3}$$
  • $$\cfrac{7}{3}$$
  • $$\cfrac{2}{3}$$
  • None of these
For what values of 'k', the equation $$x^{2} + 2(k - 4) x + 2k = 0$$ has equal roots?
  • $$8, 2$$
  • $$6, 4$$
  • $$12, 2$$
  • $$10, 4$$
What is the smallest integral value of $$k$$ such that $$2x (kx - 4) - x^{2} + 6 = 0$$ has no real roots?
  • $$-1$$
  • $$2$$
  • $$3$$
  • $$4$$
Perform the indicated operations:
$$(5+3i)(3-2i)$$
  • $$21-2i$$
  • $$19-3i$$
  • $$11-2i$$
  • $$21-i$$
The roots of the equation $$(b+c)x^2-(a+b+c)x+a=0$$ $$(a,b,c\ \epsilon \ Q, b+c \neq a)$$ are
  • irrational and different
  • rational and different
  • imaginary and different
  • real and equal
The number of real roots of $$\left (x+\dfrac{1}{x}\right)^2-4=0$$ is
  • $$0$$
  • $$2$$
  • $$4$$
  • none of these
If the argument of a complex number is $$\cfrac { \pi  }{ 2 } $$, then the number is:
  • Purely imaginary
  • Purely real
  • $$0$$
  • Neither real nor imaginary
Given : $$u = 1+i \sqrt{3}$$ and $$v = \sqrt{3} + i$$

Calculate $$\dfrac{u^3 }{ v^4}$$
  • $$(1/4) - i \sqrt{1/4}$$
  • $$(3/4) - i \sqrt{3}/4$$
  • $$(1/4) - i \sqrt{3}/4$$
  • none of these
If $$\overline { z } $$ lies in the third quadrant then $$z$$ lies in the
  • First quadrant
  • Second quadrant
  • Third quadrant
  • Fourth quadrant
If $$l,m,n $$ are real and $$l \neq m$$, the roots of the equation $$(l-m)x^2-5(l+m)x-2(l-m)=0$$ are-
  • complex
  • real and equal
  • real and unequal
  • none of these
If $$arg(z) < 0$$, then $$arg(-z)-arg(z)=$$
  • $$\pi$$
  • $$-\pi$$
  • $$\dfrac{\pi}{2}$$
  • $$-\dfrac{\pi}{2}$$
If Arg $$(z + i)\, -$$ Arg $$(z - i)$$ $$= \dfrac{\pi}{2}$$, then $$z$$ lies on a ..........
  • Circle
  • Line
  • Coordinate axes
  • None of these
Determine the nature of the roots of the equation:
$$x^2-8x+12=0$$
  • Real and unequal
  • Real and equal
  • Imaginary
  • None of these
The complex number $$z$$ satisfying the equation $$|z - i| = |z + 1| = 1$$ is
  • $$0$$
  • $$1 + i$$
  • $$-1 + i$$
  • $$1 - i$$
If $$\left( \dfrac{1 + i}{1 - i} \right)^m = 1$$, then the least positive integral value of m is
  • 1
  • 4
  • 2
  • 3
The roots of the equation $${ x }^{ 2 }-8x+16=0$$
  • Are imaginary
  • Are distinct and real
  • Are equal and real
  • Cannot be ascertained
If $$p, q$$ are odd integers, then the roots of the equation $$2px^{2} + (2p + q) x + q = 0$$ are
  • Rational
  • Irrational
  • Non-real
  • Equal
The quadratic equation $${ x }^{ 2 }+bx+4=0$$ will have real roots if
  • $$b\le -4$$ only
  • $$b\ge 4$$ only
  • $$-4 < b < 4$$
  • $$b\le -4,b\ge 4$$
The principal argument of the complex number $$z=\cfrac { 1+\sin { \cfrac { \pi  }{ 3 }  } +i\cos { \cfrac { \pi  }{ 3 }  }  }{ 1+\sin { \cfrac { \pi  }{ 3 }  } -i\cos { \cfrac { \pi  }{ 3 }  }  } $$ is?
  • $$\cfrac { \pi }{ 3 } $$
  • $$\cfrac { \pi }{ 6 } $$
  • $$\cfrac { 2\pi }{ 3 } $$
  • $$\cfrac { \pi }{ 2 } $$
  • $$\cfrac { \pi }{ 4 } $$
If $$iz^{3} + z^{2} - z + i = 0$$, then $$|z|$$ is equal to
  • $$0$$
  • $$1$$
  • $$2$$
  • None of these
Let$$ z$$ = $$\cos\theta + i \sin\theta$$. Then the value of $$\sum\limits_{m=1}^15Im( z^{2m-1})$$ at $$\theta = 2^0$$ is 
  • $$\dfrac{1}{sin2^0}$$
  • $$\dfrac{1}{3sin2^0}$$
  • $$\dfrac{1}{2sin2^0}$$
  • $$\dfrac{1}{4sin2^0}$$
If $$z$$ is a complex number such that $$z + |z| = 8 + 12i$$, then the value of $$|z^{2}|$$ is
  • $$228$$
  • $$144$$
  • $$121$$
  • $$169$$
  • $$189$$
If $$z=1+i$$, then the argument of $${ z }^{ 2 }{ e }^{ z-i }$$ is
  • $$\dfrac { \pi }{ 2 } $$
  • $$\dfrac { \pi }{ 6 } $$
  • $$\dfrac { \pi }{ 4 } $$
  • $$\dfrac { \pi }{ 3 } $$
  • $$0$$
If $$z + \sqrt {2}|z + 1| + i = 0$$ and $$z = x + iy$$, then
  • $$x = -2$$
  • $$x = 2$$
  • $$y = -2$$
  • $$y = 1$$
The principal value of $$arg(z)$$ lies in the interval:
  • $$\left[ 0,\cfrac { \pi }{ 2 } \right] $$
  • $$(-\pi ,\pi ]$$
  • $$\left[ 0,\pi \right] $$
  • $$(-\pi ,0]$$
Let $$P(e^{i\theta_1})$$,  $$Q(e^{i\theta_2})$$  and  $$R(e^{i\theta_3})$$ be the vertices of a triangle PQR in the Argand Plane. The orthocenter of the triangle PQR is 
  • $$2e^(\theta_1+\theta_2+\theta_3)$$
  • $$\frac{2}{3}e^({\theta_1+\theta_2+\theta_3})$$
  • $$e^{\theta_1}$$+$$e^{\theta_2}$$+$$e^{\theta_3}$$
  • None of these
If $$(1 - i\sqrt {3})^{2} (z) (4i) = (1 + i\sqrt {3})$$, then $$Amp\ z$$ is
  • $$\pi$$
  • $$\dfrac {\pi}{2}$$
  • $$\dfrac {\pi}{3}$$
  • $$0$$
Which of the given alternatives represent a point in Argand plane, equidistant from roots of the equation $$(z+1)^4= 16z^4$$?
  • $$(0,0)$$
  • $$\left(-\dfrac{1}{3},0\right)$$
  • $$\left(\dfrac{1}{3},0\right)$$
  • $$\left(0,\dfrac{2}{\sqrt5}\right)$$
Let $$z,\omega$$ be complex numbers such that $$\vec{z}+i\vec{\omega}=0$$ and $$Arg(z\omega)=\pi$$ then $$Arg(z)=$$.
  • $$\displaystyle\frac{\pi}{4}$$
  • $$\displaystyle\frac{5\pi}{4}$$
  • $$\displaystyle\frac{3\pi}{4}$$
  • $$\displaystyle\frac{\pi}{2}$$
Study the statements carefully.
Statement I: Both the roots of the equation $$x^2-x+1=0$$ are real.
Statement II: The roots of the equation $$ax^2+bx+c=0$$ are real if and only if $$b^2-4ac \geq 0$$.
Which of the following options hold?
  • Both Statement I and Statement II are true
  • Both Statement I and Statement II are false
  • Statement I is true and Statement II is false
  • Statement I is false and Statement II is true
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