CBSE Questions for Class 11 Engineering Maths Complex Numbers And Quadratic Equations Quiz 7 - MCQExams.com

If '$$\omega$$' is a complex cube root of unity,then $$\omega ^{ \begin{pmatrix} \frac { 1 }{ 3 }  & +\frac { 2 }{ 9 } +\frac { 4 }{ 27 } ...\infty  \end{pmatrix} }+\omega^{ \begin{pmatrix} \frac { 1 }{ 2 }  & +\frac { 3 }{ 8 } +\frac { 9 }{ 32 } ...\infty  \end{pmatrix} }=$$
  • $$1$$
  • $$-1$$
  • $$\omega$$
  • $$i$$
Find a complex number z satisfying the equation $$z+\sqrt{2}|z+1|+i=0$$
  • $$2-i$$
  • $$-2-i$$
  • $$\sqrt{2}-i$$
  • None of these
Consider the quadratic equation $$nx^{2} + 7\sqrt {n}x + n = 0$$, where $$n$$ is a positive integer. Which of the following statements are necessarily correct?
I. For any $$n$$, the roots are distinct.
II. There are infinitely many values of $$n$$ for which both roots are real.
III. The product of the roots is necessarily an integer.
  • III only
  • I and III only
  • II and III only
  • I, II and III
If $${z}_{1}=-3+5i;{z}_{2}=-5-3i$$ and $$z$$ is a complex number lying on the line segment joining $${z}_{1}$$ and $${z}_{2}$$, then $$arg(z)$$ can be:
  • $$-\cfrac { 3\pi }{ 4 } $$
  • $$-\cfrac { \pi }{ 4 } $$
  • $$\cfrac { \pi }{ 6 } $$
  • $$\cfrac { 5\pi }{ 6 } $$
If $$\dfrac {lz_{2}}{mz_{1}}$$ is purely imaginary number, then $$\left |\dfrac {\lambda z_{1} + \mu z_{2}}{\lambda z_{1} - \mu z_{2}}\right |$$ is equal to
  • $$\dfrac {l}{m}$$
  • $$\dfrac {\lambda}{\mu}$$
  • $$\dfrac {-\lambda}{\mu}$$
  • $$1$$
The complex number  $$\dfrac{1+2i}{1-i}$$ lies in which quadrant of the compiles plan
  • First
  • Second
  • Third
  • Fourth
If in applying the quardratic formula to a quadratic equation
$$f(x) = ax^2 + bx + c = 0$$, it happens that $$c = b^2/4a$$, then the graph of $$y = f(x)$$ will certainly:
  • have a maximum
  • have a minimum
  • tangent to the $$x$$-axis
  • be tangent to the $$y$$-axis
  • lie in one quadrant only
The real and imaginary part of the complex number $$1 + \sqrt {i}$$ where $$i = \sqrt {-1}$$ are
  • $$1 - \dfrac {1}{\sqrt {2}}$$ and $$-\dfrac {1}{\sqrt {2}}$$ respectively
  • $$1 - \dfrac {1}{\sqrt {2}}$$ and $$\dfrac {1}{\sqrt {2}}$$ respectively
  • $$1 + \dfrac {1}{\sqrt {2}}$$ and $$\dfrac {1}{\sqrt {2}}$$ respectively
  • $$1 + \dfrac {1}{\sqrt {2}}$$ and $$-\dfrac {1}{\sqrt {2}}$$ respectively
The given quadratic equations have real roots and the roots are equal and that is $$\dfrac{1}{\sqrt2}$$  :
$$2x^2 \, - \, 2\sqrt{2}x \, + \, 1 \, = \, 0$$
  • True
  • False
The given quadratic equation have real roots and roots are $$\dfrac{\sqrt5}{3}, \, -\sqrt5$$ :
 $$3x^2 \, + \, 2\sqrt{5x} \, - \, 5 \, = \, 0$$
  • True
  • False
The roots of the following quadratic equation are Real and equal. 
$$3x^2-4\sqrt3x+4=0$$
  • True
  • False
The given quadratic equations have real roots and the roots are $$-\sqrt2, \, \dfrac{-5}{\sqrt2}$$ :
$$\sqrt{2}x^2 \, + \, 7x \, + \, 5\sqrt{2} \, = \, 0$$
  • True
  • False
In the following, determine whether the given quadratic equations have real roots and if so, find the roots :
$$16x^2$$ = 24x + 1
  • $$\text{not real}$$
  • $$real,\dfrac{3 \, \pm \, \sqrt{10}}{4}$$
  • $$\dfrac{3 \, \pm \, \sqrt{10}}{2}$$
  • $$\dfrac{3 \, \pm \, \sqrt{13}}{4}$$
The given quadratic equations have real roots and the roots are equal and it is $$1$$:
$$x^2$$ - 2x + 1 = 0
  • True
  • False
The roots of the following quadratic equation Real and equal.
 $$3x^2$$ - 2$$\sqrt{6}x$$ + 2 = 0
  • True
  • False
The discrimination of the equation $$x^2 + 2x\sqrt3 + 3 = 0$$ is zero. Hence, its roots are:
  • Real and Equal
  • Rational and Equal
  • Rational and Unequal
  • Irrational and Unequal
  • Imaginary
The roots of the following quadratic equation are Not real
2$$(a^2 + b^2)x^2$$ + 2(a + b) x + 1 = 0
  • True
  • False
The complex number $$x+iy$$ whose modulus is unity, $$y\neq 0$$, can be represented as $$x+iy=\dfrac { a+i }{ a-i }$$,  where $$a$$ is real number.
  • True
  • False
If $$\mid{z_1}\mid=2$$, $$\mid{z_2}\mid=3$$, $$\mid{z_3}\mid=4$$ and $$\mid{z_1+z_2+z_3}\mid=2$$, then the value of $$\mid{4z_2z_3+9z_3z_1+16z_1z_2}\mid$$.
  • 24
  • 48
  • 96
  • 120
Evaluate:
$${ \left( \dfrac { cos\dfrac { \pi  }{ 8 } -isin\dfrac { \pi  }{ 8 }  }{ cos\dfrac { \pi  }{ 8 } +isin\dfrac { \pi  }{ 8 }  }  \right)  }^{ 4 }$$
  • $$1$$
  • $$-1$$
  • $$2$$
  • $$\dfrac { 1 }{ 2 } $$
Let $$z_1$$ and $$z_2$$ are two complex numbers such that $$(1-i)z_1=2z_2$$ and $$arg(z_1z_2)=\dfrac{\pi}{2}$$ then $$arg(z_2)$$ is equals to:
  • $$\dfrac{3 \pi}{8}$$
  • $$\dfrac{\pi}{8}$$
  • $$\dfrac{5 \pi}{8}$$
  • $$\dfrac{-7 \pi}{8}$$

Let $$z$$ be a complex number such that $$\left| z+\dfrac { 1 }{ z }  \right| =2$$. 

If $$\left| z \right| ={ r }_{ 1 }$$ and $$\left| \dfrac { 1 }{ z }  \right| =$$ $${r}_{2}$$ for $$\arg z=\dfrac { \pi  }{ 4 }$$ then 

$$\left| { r }_{ 1 }-{ r }_{ 2 } \right| =$$
  • $$\dfrac { 1 }{ \sqrt { 2 } }$$
  • $$1$$
  • $$\sqrt { 2 }$$
  • $$2$$
If $${z}_{1}=1+2i,\ {z}_{2}=2+3i,\ {z}_{3}=3+4i$$, then $${z}_{1},\ {z}_{2}$$ and $${z}_{3}$$ are collinear.
  • True
  • False
If $${z_1}$$ and $${z_2}$$ are two non-zero complex number such that $$\left| {{{{z_1}} \over {{z_2}}}} \right|$$ = 2 and $$\arg \left( {{z_1}{z_2}} \right) = {{3\pi } \over 2}$$ , then $${{\overline {{z_1}} } \over {{z_2}}}$$ is equal to 
  • 2i
  • -2
  • -2i
  • 2
When will the quadratic equation $$ax^2+bx+c=0$$ have Real Roots?
  • $$b^2- 4ac \ge 0$$
  • $$b^2-4ac \le 0$$
  • $$b^2-4ac < 0$$
  • None of the above.

If the value of '$$b^2-4ac$$'is equal to zero, the quadratic equation $$ax^2+bx+c=0$$ will have
  • Two real roots which are equal
  • Two Distinct Real Roots.
  • No Real Roots.
  • No Roots or Solutions.
If z satisfies $$\left| {z - 1} \right| < \left| {z + 3} \right|$$ then $$w = 2z + 3 - i$$ , ( where $$w = 2z + 3 - i$$ ) satisfies:
  • $$\left| {w - 5 - i} \right| < \left| {w + 3i} \right|$$
  • $$\left| {w - 5} \right| < \left| {w + 3} \right|$$
  •  $$\left( {iw} \right) > 1$$
  • $$\left| {\arg \left( {w - 1} \right)} \right| < {\pi  \over 2}$$
If the equation $${ x }^{ 2 }+nx+n=0,n\epsilon I$$, has integral roots then $${ n }^{ 2 }-4n$$ can assume

  • no integral value
  • one integral value
  • two integral value
  • three integral value
Real part of  $$\dfrac{(1 + i)^2}{3 - i} =$$
  • $$-1/5$$
  • $$1/5$$
  • $$1/10$$
  • $$-1/10$$
If $$\dfrac{2z_1}{3z_2}$$ is a purely imaginary number,then $$\left|\dfrac{z_1-z_2}{z_1+z_2}\right|=$$
  • $$3/2$$
  • $$1$$
  • $$2/3$$
  • $$4/9$$
The value of $$\dfrac{1}{i} + \dfrac{1}{{{i^2}}} + \dfrac{1}{{{i^3}}} + ... + \dfrac{1}{{i^{102}}}$$ is equal to 
  • $$ - 1 - i$$
  • $$ - 1 + i$$
  • $$ 1 - i$$
  • $$1 + i$$
Find the real number $$x$$ if $$(x-2i)(1+i)$$ is purely imaginary.
  • $$2$$
  • $$-2$$
  • $$4$$
  • $$-4$$
$$i \, \log \left(\dfrac{x - i}{x + i}\right)$$ is equal to
  • $$2i\log (x-i)-i\log (x^2+1)$$
  • $$2i\log (x-i)+i\log (x^2+1)$$
  • $$2i\log (x+i)-3i\log (x^2+1)$$
  • $$2i\log (x-i)-i\log (x^2+i)$$
If $$i^2= -1$$, then $$1+ i^2+ i^4 +i^6+i^8 +.............to ( 2n +1)$$ terms is equal to
  • $$0$$
  • $$1$$
  • $$3i$$
  • $$4i$$
If roots of equation $$2{x}^{2}+bx+c=0;b,c\in R$$, are real & distinct then the roots of equation $$2{cx}^{2}+(b-4c)x+2c-b+1=0$$ are
  • imaginary
  • equal
  • real and distinct
  • cant say

A particle starts from a point $$z_0= I + i$$, where $$i
=\sqrt{-1}$$ It moves horizontally away from origin by $$2$$ units and then
vertically away from origin by $$3$$ units to reach a point$$ z_1$$. From $$z_1$$
particle moves $$\sqrt{5}$$ units in the direction of $$2\hat i + \hat j$$ and
then it moves through an angle of $$\cos e{c^{ - 1}}\sqrt 2 $$ in anticlockwise
direction of a circle with centre at origin to reach a point $$z_2$$ . The arg $$z_2$$ is given by
  • $${\sec ^{ - 1}}2$$
  • $${\cot ^{ - 1}}0$$
  • $${\sin ^{ - 1}}\left( {\dfrac{{\sqrt 3 - 1}}{{2\sqrt 2 }}} \right)$$
  • $${\cos ^{ - 1}}\left( {\dfrac{{ - 1}}{2}} \right)$$
The probability of choosing randomly a number c from the set $$\{1, 2, 3, ..........9\} $$ such that the quadratic equation $$x^2+ 4x +c=0$$ has real roots is:
  • $$\dfrac{1}{9}$$
  • $$\dfrac{2}{9}$$
  • $$\dfrac{3}{9}$$
  • $$\dfrac{4}{9}$$
If $$a,b,c$$ are integers and $${ b }^{ 2 }=4\left( ac+{ 5d }^{ 2 } \right)$$, $$\in$$, then roots of the quadratic equation $${ ax }^{ 2 }+bx+c=0$$ are 
  • Irrational
  • Rational and different
  • Complex conjugate
  • Rational and equal
If $$a+ ib= \sum_{k=1}^{101} i^k $$, then $$(a, b)$$ equals 
  • $$(0, 1)$$
  • $$(1, 0)$$
  • $$(0,- 1)$$
  • $$(1, 1)$$
If $$z = -3- i,$$ find $$|z|$$.
  • $$\sqrt{10}$$
  • $$\sqrt{9}$$
  • $$\sqrt{8}$$
  • $$\sqrt{7}$$
The modulus of the complex number $$z=\dfrac{1-i}{3-4i}$$ is
  • $$\dfrac{5}{\sqrt{2}}$$
  • $$\dfrac{\sqrt{2}}{5}$$
  • $$\sqrt{\dfrac{2}{5}}$$
  • none of these

(A) Which of the following quadratic polynomials has zeros
-9 and 9 
  • $$x-81=0$$
  • $${x^2} - 9=0$$
  • $${x^2} - 64=0$$
  • $${x^2} - 81=0$$
Let z be a complex number such that $$z\in c\ R$$ and $$\dfrac{1+z+z^2}{1-z+z^2}\in R$$, then  $$|z|=3$$.
  • True
  • False
If $$z=\dfrac{1+i}{\sqrt{2}}$$, then the value of $$z^{1929}$$ is
  • $$1+i$$
  • $$-1$$
  • $$\dfrac{1+i}{2}$$
  • $$\dfrac{1+i}{\sqrt{2}}$$
The equation $$4\sin^2x+4\sin x+a^2-3=0$$ possesses a solution if 'a' belongs to the interval.
  • $$(-1, 3)$$
  • $$(-3, 1)$$
  • $$[-2, 2]$$
  • $$R-(-2, 2)$$
If $$|Z|=2,|z_{2}|=3,|z_{3}=4|$$ and $$|z_{1}+z_{2}+z_{3}|=5$$ then $$|4z_{2}z_{3}+9z_{3}z_{1}+16z_{1}z_{2}|=$$
  • $$20$$
  • $$24$$
  • $$48$$
  • $$120$$
If $$\dfrac{x - 3}{3 + i} + \dfrac{y - 3}{3 - i} = i $$ where $$x , y \in R$$ then
  • $$x = 2$$ & $$y = -8$$
  • $$x = -2$$ & $$y = 8$$
  • $$x = -2$$ & $$y = -6$$
  • $$x = 2$$ & $$y = 8$$
The complex number $$\dfrac{1 + 2i}{1 - i}$$ lies in which quadrant of the complex plane.
  • First
  • Second
  • Third
  • Fourth
Given $$\left| z \right| =4$$ and $$Argz=\dfrac{5z}{6}$$, then $$z$$ is
  • $$2\sqrt{3}+2i$$
  • $$2\sqrt{3}-2i$$
  • $$-2\sqrt{3}+2i$$
  • $$-\sqrt{3}+i$$
If $$a < b < c < d$$, then the roots of the equation $$(x-a)(x-c)+2(x-b)(x-d)=0$$ are?
  • Real and distinct
  • Real and equal
  • Imaginary
  • None of these
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