If $$z=-1$$, then principal value of arg $$\left({z}^{2/3}\right)$$ is

$$0,\dfrac{2\pi}{3},-\dfrac{2\pi}{3}$$

MCQ Questions for Class 11 Engineering Maths Complex Numbers And Quadratic Equations Quiz 11

The value of

$(z+3) (\overline{z} +3)$ is eqquivalent to

0%
$|z +3 | ^2$
0%
$| z- 3|$
0%
$z^2+3$
0%
None of these

Explanation

$(2+3)(\bar{2}+3)$

We know that

$2.\bar{2}=|2|^{2}$

$(2+3).(\overline{2+3})$

$=12+31^{2}$

$\dfrac {3+2i \sin x}{1-2i \sin x}$ is purely imaginary then

$x=$ ?

0%
$n \pi \pm \dfrac {\pi}{6}$
0%
$n \pi \pm \dfrac {\pi}{3}$
0%
$2n \pi \pm \dfrac {\pi}{3}$
0%
$2n \pi \pm \dfrac {\pi}{6}$

Explanation

A complex number is said to be purely imaginary if $z+\overline { z } =0$

If $z=\dfrac { 3+2isinθ }{ 1−2isinθ }$

then $\overline { z } =\overline { \dfrac { 3+2isinθ }{ 1−2isinθ } } =\dfrac { 3-2isinθ }{ 1+2isinθ }$

$z+\overline { z } =\dfrac { 3+2isinθ }{ 1−2isinθ } +\dfrac { 3-2isinθ }{ 1+2isinθ }$

So, $\dfrac { 3+2isinθ }{ 1−2isinθ } +\dfrac { 3−2isinθ }{ 1+2isinθ } =0$

$\dfrac { (3+2isinθ)(1+2isinθ)+(3−2isinθ)(1−2isinθ) }{ (1−2isinθ)(1+2isinθ) } =0$

$3+6isinθ+2isinθ−4{ sin }^{ 2 }θ+3−6isinθ−2isinθ−4{ sin }^{ 2 }θ=0$

$6−{ 8sin }^{ 2 }θ=0$

${ sin }^{ 2 }θ=\dfrac { 3 }{ 4 }$

$sinθ=\dfrac { \sqrt { 3 } }{ 2 } =sin\dfrac { \pi }{ 3 }$

$θ=nπ+{ (−1) }^{ n }\left( \dfrac { \pi }{ 3 } \right)$

$sinθ=\dfrac { \sqrt { 3 } }{ 2 } =sin−\dfrac { \pi }{ 3 }$

$θ=nπ+{ (−1) }^{ n }\left( \dfrac { -\pi }{ 3 } \right) =nπ+{ (−1) }^{ n+1 }\left( \dfrac { \pi }{ 3 } \right)$

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?

0%
$\sum { { \left| { Z }_{ 1 } \right| }^{ 2 } } =\dfrac { 3 }{ 2 }$
0%
${ \left| { Z }_{ 1 } \right| }^{ 4 }+{ \left| { Z }_{ 2 } \right| }^{ 4 }={ \left| { Z }_{ 3 } \right| }^{ -8 }$
0%
$\sum { { \left| { Z }_{ 1 } \right| }^{ 3 }+ } { \left| { Z }_{ 2 } \right| }^{ 3 }={ \left| { Z }_{ 3 } \right| }^{ -6 }$
0%
${ \left| { Z }_{ 1 } \right| }^{ 4 }+{ \left| { Z }_{ 2 } \right| }^{ 4 }={ \left| { Z }_{ 3 } \right| }^{ 8 }$

For a complex number

$z$ , the minimum value of

$\left | z \right |+\left | z-\cos\alpha-i\sin\alpha \right |$ is

0%
0
0%
1
0%
2
0%
None of these

$|z-4| < |z-2|$ represents the region given by?

0%
$Re(z) > 3$
0%
$Re(z) < 0$
0%
$Re(z) > 2$
0%
None of these

$P(x)=a{x}^{2}+bx+c$ and

$Q(x)=-a{x}^{2}+dx+c$ where

$ac\ne 0$ , then

$P(x).Q(x)=0$ has

0%
exactly one real root
0%
atleast two real roots
0%
exactly three real roots
0%
all four are real roots

The modulus of

$\overline { 6+{ i }^{ 3 } } +\overline { 6+{ i } }+\overline { 6+{ i }^{ 2 } }$ is

0%
$17$
0%
$\sqrt{533}$
0%
$\sqrt{456}$
0%
$49$

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

0%
$\lambda< \cfrac{4}{3}$
0%
$\lambda> \cfrac{5}{3}$
0%
$\lambda \in \left( \cfrac { 1 }{ 3 } ,\cfrac { 5 }{ 3 } \right)$
0%
$\lambda \in \left( \cfrac { 4 }{ 3 } ,\cfrac { 5 }{ 3 } \right)$

Explanation

Given roots of equation

$x^2+2(a+b+c)x+3\lambda(ab+bc+ca)=0$ are real

So,

$D\ge0$

$\Rightarrow B^2-4AC\ge 0$

$\Rightarrow [2(a+b+c)]^2-4\times1\times3\lambda(ab+bc+ca)\ge 0$

$\Rightarrow 4(a+b+c)^2-12\lambda(ab+bc+ca)\ge 0$

$\Rightarrow 4(a+b+c)^2\ge12\lambda(ab+bc+ca)$

$\Rightarrow \lambda\le\dfrac{4(a+b+c)^2}{12(ab+bc+ca)}$

$\Rightarrow \lambda\le\dfrac{4(a^2+b^2+c^2)}{12(ab+bc+ca)}+\dfrac{8(ab+bc+ca)}{12(ab+bc+ca)}$

$\Rightarrow \lambda\le \dfrac{(a^2+b^2+c^2)}{3(ab+bc+ca)}+\dfrac{2}{3}.....(1)$

Now, we know that

$|a-b|<c\Rightarrow a^2+b^2-2ab<c^2$

$|b-c|<a\Rightarrow b^2+c^2-2bc<a^2$

$|c-a|<b\Rightarrow c^2+a^2-2ca<b^2$

On adding above, we get

$a^2+b^2-2ab+b^2+c^2-2bc+c^2+a^2-2ca<a^2+b^2+c^2$

$\Rightarrow a^2+b^2+c^2<2ab+2bc+2ca$

$\Rightarrow \dfrac{(a^2+b^2+c^2)}{(ab+bc+ca)}<2$

So, equation (1) becomes,

$\Rightarrow\lambda<\dfrac{2}{3}+\dfrac{2}{3}$

$\Rightarrow \lambda<\dfrac{4}{3}$

$a + b + c = 0$ then the roots of the equation

$4 a x ^ { 2 } + 3 b x+ 2 c = 0$ where

$a , b , c \in R$ are

0%
real and distinct
0%
imaginary
0%
real and equal
0%
infinite

$a, b, c, d$ be a form consecutive term of an increasing A.P., then the roots of the equation

$\left( {x - a} \right)\left( {x - c} \right) + 2\left( {x - b} \right)\left( {x - d} \right) = 0$

0%
Real & distinct
0%
complex
0%
equal roots
0%
none of these

$| z _ { 1 } | < 1 \text { and } | \frac { z _ { 1 } - z _ { 2 } } { 1 - \overline { z } _ { 1 } z _ { 2 } } | < 1 , \text { then } | z _ { 2 } | > 1$

0%
True
0%
False

Explanation

Given.

$|z _i|<1$

$\left|\frac{z_{1}-z_{2}}{1-z_{1} z_{2}}\right|<1 \\$

$\left|z_{1}-z_{2}\right|<\left|1-\bar{z}_{1} z_{2}\right| \\$

$\left|z_{1}-z_{2}\right|^{2}<\left|1-\bar{z}_{1} z_{2}\right|^{2}$

$\left(z_{1}-z_{2}\right)\left(\bar{z}_{1}-\bar{z}_{2}\right)<\left(1-\bar{z}_{1} z_{2}\right)\left(1-z_{1} \bar{z}_{2}\right)$

$z_{1} \bar{z}_{1}-z_{2} \bar{z}_{1}-\bar{z}_{2} z_{1}+z_{2} \bar{z}_{2} \quad<1-z_{1} \bar{z}_{2}-\bar{z}_{1} z_{2}+\left(z_{1} \bar{z}_{1}\right)\left(z_{2} \bar{z}_{2}\right)$

$\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2} \quad\left\langle\left.\quad1+| z_{1}\right|^{2}\left|z_{2}\right|^{2}\right.$

$\left|z_{1}\right|^{2}+\left.|z_{2}\right|^{2}+\left|z_{1}\right|^{2}\left|z_{2}\right|^{2}-1<0$

$\left.\left|z_{2}\right|^{2}\left((1-\left|z_{1}\right|^{2}\right)-1\left(1-z_{1}\right)^{2}\right)< 0$

$\left(\left.| z_2\right|^{2}-1\right)\left(1-\left.|z_{1}\right|^{2}\right)<0$

$\left.\left.(|z_2|^{2}-1)\right(|z_2|^{2}-1\right)>0$

$|z_1|^{2}-1<0$

$\Rightarrow\left|\left.z_{2}\right|^{2}-1<0\right.$

$\left|z_{2}\right|^{2}<1$

$\left|z_{2}\right|<1$

option

$B$ is Correct

$\left| z \right| =1$ 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

0%
$\left[1,9\right]$
0%
$\left[2,6\right]$
0%
$\left[2,12\right]$
0%
$\left[2,18\right]$

If the quadratic equation

$4x^{2}-2(a+c-1)x+ac-b=0(a>b>c).$

0%
Both roots are greater than a
0%
Both roots are less than c
0%
Both roots lie between c/2 and a/2
0%
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

0%
complex number
0%
real and unequal
0%
real and equal
0%
none

$a > b > c$ ,

$a\neq 0$ 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

$at^2+bt+c=0$ .

0%
Are imaginary
0%
Are real and equal
0%
Are real and distinct
0%
May be real of imaginary

$z$ is a complex number such that

$|z-1|=1$ then

$arg \left(\dfrac{1}{z}-\dfrac{1}{2}\right)$ may be

0%
$\dfrac{\pi}{6}$
0%
$\dfrac{\pi}{2}$
0%
$\dfrac{\pi}{4}$
0%
$-\dfrac{\pi}{4}$

$z=\dfrac{-2}{1+i\sqrt{3}}$ , then the value of arg(z) is?

0%
$\pi$
0%
$\dfrac{\pi}{3}$
0%
$\dfrac{2\pi}{3}$
0%
$\dfrac{\pi}{4}$

$\theta$ real then the modulus of

$\dfrac{1}{1+\cos\theta+i\sin\theta}$ is

0%
$\dfrac{1}{2}\sec\dfrac{\theta}{2}$
0%
$\dfrac{1}{2}\cos\dfrac{\theta}{2}$
0%
$\sec\dfrac{\theta}{2}$
0%
$\cos\dfrac{\theta}{2}$

The function of imaginary roots of the equation

$(x-1)(x-2)(3x+1)=32$ is

0%
$0$
0%
$1$
0%
$2$
0%
$4$

Let

$z,w$ be complex numbers such that

$\vec {z}+i\vec {w}=$ and

$zw=\pi$ Then

$arg\ z$ equals

0%
$\dfrac {\pi}{4}$
0%
$\dfrac {5\pi}{4}$
0%
$\dfrac {3\pi}{4}$
0%
$\dfrac {\pi}{2}$

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

0%
Less then $4ab$
0%
Greater than $-4ab$
0%
Less than $-4ab$
0%
Greater then $4ab$

Explanation

Given that the roots of equation

$bx^2+cx+a=0$ is imaginary, so

$c^2-4ab<0$

$-c^2 > -4ab$ ……

$(1)$

Let

$3b^2x^2+6bcx+2c^2=y$

$3b^2x^2+6bcx+2c^2-y=0$

For real values of

$x$ we have

$(6ab)^2-4\cdot(3b^2)\cdot(2c^2-y)\geq 0$

$\Rightarrow 36b^2c^2-12b^2(2c^2-y)\geq 0$

$3c^2-2c^2+y\geq 0$

$c^2+y\geq 0$

$y\geq -c^2$ …..

$(2)$

Comparing

$(1)$ and

$(2)$

we get

$y > (-4ab)$ .

The equation

$x(x+2)(x^{2}-x)=-1$ , has

0%
All roots imaginary
0%
All roots negative
0%
Two roots real and two roots imaginary
0%
All roots real

$\overline { \Delta } =\begin{vmatrix} -1 & 2-3i & 5+4i \\ 2+3i & 8 & 1-i \\ 5-4i & 1+i & 3 \end{vmatrix}$ then

$\Delta =$

0%
Purely real
0%
Purely imaginary
0%
Complex
0%
$0$

Let

$z$ be a complex number of maximum amplitude satisfying

$|z-3|=Re(z)$ , then

$|z-3|$ is equal to

0%
$1$
0%
$2$
0%
$3/2$
0%
$9$

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

$p^{2}=$

0%
$2q\ \cos \left(\dfrac{\alpha}{2}\right)$
0%
$4q\ \cos \left(\dfrac{\alpha}{2}\right)$
0%
$4q\ \cos^{2} \left(\dfrac{\alpha}{2}\right)$
0%
$4q^{2}\ \cos^{2} \left(\dfrac{\alpha}{2}\right)$

If z be any complex number such that

$|3z-2|+|3z+2|=4$ , then locus of z is

0%
An ellipse
0%
A circle
0%
A line-segment
0%
None of these

The number of imaginar roots of the equation

$(x-1)(x-2)(3x-2)(3x+1)=32$ is

0%
$Zero$
0%
$1$
0%
$2$
0%
$4$

$\mathrm{{z} _ { 1 }} = 10 + 6\mathrm{i} , \mathrm{{ z } _ { 2 }}= 4 + 6 \mathrm { i }$ 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

0%
$\sqrt { 2 }$
0%
$2\sqrt { 2 }$
0%
$3\sqrt { 2 }$
0%
$2\sqrt { 3 }$

$z_0$ is the roots of

$1+x+x^2 =0$ and

$z=3+6iz^{81}_0 - 3z^{93}_0$ , then arg(z) is

0%
$\dfrac{\pi}{4}$
0%
$\dfrac{\pi}{6}$
0%
$\dfrac{\pi}{9}$
0%
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

0%
$1$
0%
$2$
0%
$4$
0%
$3$

Explanation

$arg(z)=\pi$

Let

$z=x+iy$

$arg(x+iy)=\pi$

$\Rightarrow x < 0; y=0$

$|z+3-i|=1$

$|x+3-i|=1$

$(x+3)^2+1^2=1^2$

$\Rightarrow (x+3)^2=0$

$x=-3$

$|z|=|-3+0i|=3$ .

1214029_1281108_ans_f4566f4f191b44e88f3ebba64f1de415.jpg

The roots of the equation

$(3b+c-4a)x^2+(3c+a-4b)x+(3a+b-4c)= 0$ are

0%
Irrational
0%
Rational
0%
Non-real
0%
Imaginary

$\frac { { z }_{ 2 }-{ 2z }_{ 2 } }{ { z }_{ 2 }-{ z }_{ 1 }{ z }_{ 2 } }$ is unimodular then

0%
$|{ z }_{ 2 }|=2$
0%
$|{ z }_{ 1 }|=1$
0%
Both A and B
0%
None of these

In quadratic equation

$ax^2 + bx + c = 0$ , if discriminant

$D = b^2 - 4ac$ , then roots of quadritic equation are:

0%
real and distinct, if D > 0
0%
real and equal (repeated rotos), if D = 0
0%
non-real (imaginary), if D < 0
0%
none of the above

$Z=\sin \frac {6\pi}5+i(1+\cos \frac {6\pi }5)$ then

0%
$|Z|=-2\cos \frac {3\pi}5$
0%
$Arg(Z)=\frac {\pi}5$
0%
$Arg(Z)=\frac {9\pi }{10}$
0%
none of these

$z_1$ 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

0%
$-\pi$
0%
$\frac{-\pi}{2}$
0%
$0$
0%
$\frac{\pi}{2}$

This equation

$(x-5)^{11}+(x-5^{2})^{11}+....+(x-5^{11})^{11}=0$ has

0%
all the roots real
0%
one real and 10 imaginary roots
0%
real roots namely $x=5,5^{2}....,5^{9},5^{10},5^{11}$
0%
none

Find the value of

$k$ for which the roots are real and equal:

$5{ x }^{ 2 }-4x+2+k(4{ x }^{ 2 }-2x-1)=0$

0%
$1$
0%
$-1$
0%
$\dfrac{6}{5}$
0%
$- \dfrac{6}{5}$

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:

0%
$-\pi$
0%
$-\frac\pi{2}$
0%
$0$
0%
$\frac\pi{2}$

Argument and modulus of

$\left[ \frac { 1 + i } { 1 - i } \right] ^ { 2013 }$ are respectively

$\ldots \ldots \ldots$

0%
$\frac { - \pi } { 2 }$ and 1
0%
$\frac { \pi } { 2 }$ and $\sqrt { 2 }$
0%
0 and $\sqrt { 2 }$
0%
$\frac { \pi } { 2 }$ and 1

If the roots of the equation

${ x }^{ 2 }-8x+({ a }^{ 2 }-6a)=0$ are real, then

0%
-2 < a < 8
0%
2 < a < 8
0%
$2\le a\le 8$
0%
$-2\le a\le 8$

$z$ is a complex number of unit modulus and argument

$\theta$ , then

$arg\left(\dfrac{1+z}{1+\overline{z}}\right)$ equals

0%
$-\theta$
0%
$\dfrac{\pi}{2}-\theta$
0%
$\theta$
0%
$\pi-\theta$

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

0%
16/3
0%
6
0%
14/3
0%
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

0%
$\dfrac { \pi }{ 4 }$
0%
$\dfrac { \pi }{ 2 }$
0%
$\dfrac { \3pi }{ 4 }$
0%
$\dfrac { \5pi }{ 4 }$

Argument of the complex number

$\left( \dfrac { - 1 - 3 i } { 2 + i } \right).$

0%
$45 ^ { \circ }$
0%
$135 ^ { \circ }$
0%
$225 ^ { \circ }$
0%
$240 ^ { \circ }$

$z=-1$ , then principal value of arg

$\left({z}^{2/3}\right)$ is

0%
$0,\dfrac{2\pi}{3},-\dfrac{2\pi}{3}$
0%
$\dfrac{\pi}{3},2\pi$
0%
$\dfrac{5\pi}{3}$
0%
$-\pi,\pi$

The amplitude of

${ e }^{ { e }^{ -i\theta } }$ is equal to

0%
sin $\theta$
0%
-sin $\theta$
0%
${ e }^{ \cos { \theta } }$
0%
${ e }^{ \sin { \theta } }$

$z=-1$ , then principal value of arg

$\left( {z}^{\dfrac{2}{3}} \right )$ is

0%
$0,\dfrac{2\pi}{3},-\dfrac{2\pi}{3}$
0%
$\dfrac{\pi}{3},2\pi$
0%
$\dfrac{5\pi}{3}$
0%
$-\pi,\pi$

$z _ { 0 }$ 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-

0%
$\frac { \pi } { 4 }$
0%
$\frac { \pi } { 6 }$
0%
$\frac { \pi } { 9 }$
0%
$\frac { \pi } { 2 }$

Find the modules and amplitude for each of the following complex numbers

0%
7-5i
0%
$\sqrt { 3 } +\sqrt { 2 } i$
0%
-8+15i
0%
-3(1-i)

The value of 'a' fro which the equation

$a{x^2} - 2\sqrt 5 x + 4 = 0\;$ has equal roots is

0%
$\frac{5}{4}$
0%
$\frac{4}{5}$
0%
$\frac{-5}{4}$
0%
$\frac{{ - 5}}{3}$

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

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Practice Class 11 Engineering Maths Quiz Questions and Answers

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

0:0:1

Practice Class 11 Engineering Maths Quiz Questions and Answers

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