Complex Numbers and Quadratic Equations Test 34

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Complex Numbers and Quadratic Equations Test 34

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Weekly Quiz Competition

Question 1
1 / -0

The modulus of $\dfrac { \left( 3+2i \right) ^{ 2 } }{ \left( 4-3i \right) }$ is:

A
$\frac { 13 }{ 5 }$

B
$\frac { 11 }{ 5 }$

C
$\frac { 9 }{ 5 }$

D
$\frac { 7 }{ 5 }$

Solution
Question 2
1 / -0

Number of complex numbers $z$ such that $|z|=1$ and $\left|\dfrac {z}{z}+\dfrac {\bar {z}}{z}\right|=1$ is

A
$4$

B
$1$

C
$8$

D
$more\ then\ 8$

Solution

Since $z$ is equidistant from $1,-1,i$ , we can say that $z$ is center of circle with $1,-1,i$ lying on it's circumference and so there is one one circle possible passing through $3$ points. So, the answer is $1$ .
Question 3
1 / -0

If arg $\left( z \right) < 0,$ then arg $\left( { - z} \right)-arg(z)$

A
$\pi$

B
$- \pi$

C
$- \dfrac{\pi }{2}$

D
$\dfrac{\pi }{2}$

Solution
Question 4
1 / -0

Purely imaginary then find the sum of statement i $a,b$

A
$\dfrac {5\pi}{6}$

B
$\pi$

C
$\dfrac {3\pi}{4}$

D
$\dfrac {2\pi}{3}$
Question 5
1 / -0

If $\alpha$ and $\beta$ are the roots of ${ 4x }^{ 2 }-16x+c=0,$ c>0 such that $1<\alpha <2<\beta <3$ , then the no.of integer values of c is

A
$17$

B
$14$

C
$18$

D
$15$

Solution

Given, $\alpha$ and $\beta$ are roots of $4x^2-16x+x=0$
where $a=4$
$b=-16$
$x > 0$
now, $\alpha\beta =\dfrac{a}{a}$
$\Rightarrow \alpha\beta =\dfrac{c}{4}$
$\Rightarrow \beta =\dfrac{c}{4\alpha}$ ………(i)
we have $2 < \beta < 3$
$\Rightarrow 2 < \dfrac{c}{4\alpha} < 3$ [from $(1)$ ]
$\Rightarrow 8 < \dfrac{c}{\alpha} < 12$
$\Rightarrow 8\alpha < c < 12\alpha$ ………(ii)
and $1 < \alpha < 2$
$\Rightarrow \alpha > 1\Rightarrow 8\alpha > 8$ ……….(iii)
and $\alpha < 2$
$\Rightarrow 12\alpha < 24$ ………..(iv)
Substituting (iii) and (iv) in (ii) we get
$8 < 8\alpha < c < 12\alpha < 24$ $[\alpha =9, 10, 11, 12, 13, 14, 15, 16, 13, 18, 19, 20, 21, 22, 23]$
$\Rightarrow 8 < c < 24$
Therefore number of integer values of c is $15$ .
Question 6
1 / -0

$\sqrt { \left( \log _ { 3 } \tan x \right) }$ is real for:

A
$n \pi + \pi / 4 \leq x < n \pi + \pi / 2$

B
$n \pi < x < n \pi + \pi / 2$

C
$n \pi \pm \pi / 4 \leq x < n \pi \pm \pi / 2$

D
None of these.
Question 7
1 / -0

Arg $\left\{ {\sin \frac{{8\pi }}{5} + i\left( {1 + \cos \frac{{8\pi }}{5}} \right)} \right\}$ is equal to

A
$\frac{{3\pi }}{{10}}$

B
$\frac{{7\pi }}{{10}}$

C
$\frac{{4\pi }}{5}$

D
$\frac{{3\pi }}{5}$

Solution
Question 8
1 / -0

Let 'z' be a complex number satisfying $|z-2-i|\le 5,$ Then |z-14-6i| lies in

A
{8,18}

B
{2,8}

C
{0,2}

D
{3,7}

Solution

Here, $|z-2-i|\leq 5$ ……….. $(1)$
Now, $|z-14-6i|=|z-2-i-12-5i|=|(z-2-i)-(12+5i)|$
$\leq |z-2-i|+|12+5i|$
$\leq 5+|12+5i|$
$=5+\sqrt{(12)^2+(5)^2}$
$=5+\sqrt{144+25}$
$=5+\sqrt{169}$
$=5+13$
$=18$ .
$\therefore |z-14-6i|\leq 18$
and $|z-14-6i|=|z-2-i-12-5i|$
$=|-(12+5i)+(z-2-i)|$
$\geq ||-(12+5i)|-|z-2-i||$
$\geq |12+5i|-5$ $[\because |z-2-i|\leq 5$ $\Rightarrow -|z-2-i|\geq -5]$
$=13-5$
$=8$
$\therefore |z-14-6i|\geq 8$
$\therefore |z-14-6i|$ lies in $\{8, 18\}$ .
Question 9
1 / -0

IF $z_1=1+i,z_2=1-i$ find $z_1z_2$

A
$z_1+z_2$

B
$z_1-z_2$

C
$z_1/z_2$

D
None.

Solution

$z_1=1+i$

$z_2=1-i$

$z_1z_2=1-(-1)=2$

$=1+1$

$=1-i+1+i$

$z_1z_2=z_1+z_2$
Question 10
1 / -0

The value of the sum $\sum _{ n=1 }^{ 13 }{ ({ i }^{ n }+{ i }^{ n+1 }) }$ , where $i=\sqrt { -1 }$ , equals

A
$i$

B
$i-1$

C
$-i$

D
$0$

Solution

As we know that $i+i^2+i^3+i^4=0$
$\displaystyle\sum ^{13}_{n=1} (i^n+i^{n+1})=\sum ^{13}_{n=1}(1+i)(i^n)=(1+i)\sum ^{13}_{n=1} i^{n}=(1+i)(0+0+i^{13})=(1+i)i=i+i^2=i-1$

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Complex Numbers and Quadratic Equations Test 34

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Question 1

135\frac { 13 }{ 5 } 513​

115\frac { 11 }{ 5 } 511​

95\frac { 9 }{ 5 } 59​

75\frac { 7 }{ 5 } 57​

Solution

Question 2

444

111

888

more then 8more\ then\ 8more then 8

Solution

Question 3

π\pi π

−π - \pi −π

−π2 - \dfrac{\pi }{2}−2π​

π2\dfrac{\pi }{2}2π​

Solution

Question 4

5π6\dfrac {5\pi}{6}65π​

π\piπ

3π4\dfrac {3\pi}{4}43π​

2π3\dfrac {2\pi}{3}32π​

Question 5

171717

141414

181818

151515

Solution

Question 6

nπ+π/4≤x<nπ+π/2n \pi + \pi / 4 \leq x < n \pi + \pi / 2nπ+π/4≤x<nπ+π/2

nπ<x<nπ+π/2n \pi < x < n \pi + \pi / 2nπ<x<nπ+π/2

nπ±π/4≤x<nπ±π/2n \pi \pm \pi / 4 \leq x < n \pi \pm \pi / 2nπ±π/4≤x<nπ±π/2

None of these.

Question 7

3π10\frac{{3\pi }}{{10}}103π​

7π10\frac{{7\pi }}{{10}}107π​

4π5\frac{{4\pi }}{5}54π​

3π5\frac{{3\pi }}{5}53π​

Solution

Question 8

{8,18}

{2,8}

{0,2}

{3,7}

Solution

Question 9

z1+z2z_1+z_2z1​+z2​

z1−z2z_1-z_2z1​−z2​

z1/z2z_1/z_2z1​/z2​

None.

Solution

Question 10

iii

i−1i-1i−1

−i-i−i

000

Solution

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$\frac { 13 }{ 5 }$

$\frac { 11 }{ 5 }$

$\frac { 9 }{ 5 }$

$\frac { 7 }{ 5 }$

$4$

$1$

$8$

$more\ then\ 8$

$\pi$

$- \pi$

$- \dfrac{\pi }{2}$

$\dfrac{\pi }{2}$

$\dfrac {5\pi}{6}$

$\pi$

$\dfrac {3\pi}{4}$

$\dfrac {2\pi}{3}$

$17$

$14$

$18$

$15$

$n \pi + \pi / 4 \leq x < n \pi + \pi / 2$

$n \pi < x < n \pi + \pi / 2$

$n \pi \pm \pi / 4 \leq x < n \pi \pm \pi / 2$

$\frac{{3\pi }}{{10}}$

$\frac{{7\pi }}{{10}}$

$\frac{{4\pi }}{5}$

$\frac{{3\pi }}{5}$

$z_1+z_2$

$z_1-z_2$

$z_1/z_2$

$i$

$i-1$

$-i$

$0$