MCQ Questions for Class 11 Commerce Applied Mathematics Number Theory Quiz 12

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

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Irrational
0%
Rational
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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
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None of these

If z is a complex number of unit modules and argument

$\theta$ , then the real part of

$\dfrac { z(1-\bar { z } ) }{ z(1+z) }$ is :

0%
${ -2sin }^{ 2 }\dfrac { \theta }{ 2 }$
0%
${ 2sin }^{ 2 }\dfrac { \theta }{ 2 }$
0%
$1+cos\dfrac { \theta }{ 2 }$
0%
$1-cos\dfrac { \theta }{ 2 }$

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 $x=5,5^{2}....,5^{9},5^{10},5^{11}$
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none

If z=

$\dfrac { 1 }{ { \left( 2+3i \right) }^{ 2 } } ,\quad then\left| z \right|$ =

0%
$\dfrac { 1 }{ 13 }$
0%
$\dfrac { 1 }{ 5 }$
0%
$\dfrac { 1 }{ 12 }$
0%
none of these

The complex number z satisfies the equation z + |z| = 2 + 8i. Then the value of |z| is

0%
15
0%
16
0%
17
0%
18

Explanation

$z+\left| z \right| =2+8i\quad \longrightarrow \left( 1 \right)$

let

$z=x+iy\quad \longrightarrow \left( 2 \right)$

then,

$\left| z \right| =\sqrt { { x }^{ 2 }+{ y }^{ 2 } } \quad \longrightarrow \left( 3 \right)$

$\therefore$ Substituting

$(2)$ &

$(3)$ in

$(1)$ we get

$x+iy+\sqrt { { x }^{ 2 }+{ y }^{ 2 } } =2+8i$

$\Rightarrow x+\sqrt { { x }^{ 2 }+{ y }^{ 2 } } +iy=2+8i$

Comparing

$LHS$ &

$RHS$ we get

$y=8\longrightarrow \left( 4 \right)$ and

$x+\sqrt { { x }^{ 2 }+{ y }^{ 2 } } =2\quad \longrightarrow \left( 5 \right)$

Substituting the value of

$y$ in the equation

$(5)$

We get

$x+\sqrt { { x }^{ 2 }+64 } =2$

$\sqrt { { x }^{ 2 }+64 } =2-x$

Squaring both sides,

${ x }^{ 2 }+64=4+{ x }^{ 2 }-4x$

$\Rightarrow 4x=-60$

$\Rightarrow x=-15$

$\therefore$

$z=-15+8i$

$\left| z \right| =\sqrt { 225+164 } =\sqrt { 289 } =17$ units.

The complex number

$z$ satisfies

$z+|z|=2+8i$ . The value of

$|z|$ is

0%
10
0%
13
0%
17
0%
23

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%
9
0%
4
0%
3

For two unimodular complex numbers

$z_1$ and

$z_2$ ,

$\left[ \begin{matrix} \bar { z_{ 1 } } & -z_{ 2 } \\ \bar { z_{ 1 } } & z_{ 1 } \end{matrix} \right] ^{ -1 } \left[ \begin{matrix} z_1 & z_2 \\- \bar { z_{ 2 } } & \bar{ z_1} \end{matrix} \right] ^{ -1 }$ is equal to

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$\begin{bmatrix} z_1 & z_2 \\ \bar {z_1} & \bar {z_2} \end{bmatrix}$
0%
$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$
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$\begin{bmatrix} \dfrac{1}{2} & 0 \\ 0 & \dfrac{1}{2} \end{bmatrix}$
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None of these

CBSE Questions for Class 11 Commerce Applied Mathematics Number Theory Quiz 12 - MCQExams.com

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Practice Class 11 Commerce Applied Mathematics Quiz Questions and Answers

CBSE Questions for Class 11 Commerce Applied Mathematics Number Theory Quiz 12 - MCQExams.com

0:0:1

Practice Class 11 Commerce Applied Mathematics Quiz Questions and Answers

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