Complex Numbers and Quadratic Equations Test -7

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Complex Numbers and Quadratic Equations Test -7

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Weekly Quiz Competition

Question 1
1 / -0

If $$z=2-3i$$ then $$z^2-4z+13=$$

A
$$0$$

B
$$1$$

C
$$2$$

D
$$3$$

Solution

$$z=2-3i$$
$$z^{2}=2^{2}-3^{2}-12i$$
$$=-5-12i$$
$$\therefore z^{2}-4z+13$$
$$=(-5-12i)-4(2-3i)+13$$
$$=-5-12i-8+12i +13$$
$$=-13+13$$
$$=0$$
Question 2
1 / -0

The complex number $$\displaystyle \frac{1+2i}{1-i}$$ lies in the quadrant :

A
I

B
II

C
III

D
IV

Solution

Let $$z =\dfrac{1+2i}{1-i}$$

$$\Rightarrow z =\dfrac{(1+2i)}{1-i}\times \dfrac{1+i}{1+i}$$

$$= \dfrac{1+2i+i+2i^2}{1-i^2}$$

$$=\dfrac{1+3i-2}{1+1}$$ ............ $$[\because i^2 = -1]$$

$$\Rightarrow z =\dfrac{-1+3i}{2}$$

$$ =-\dfrac{1}{2}+\dfrac{3}{2}i $$
$$=x+iy$$
Clearly $$x<0$$ and $$y>0$$
Hence $$z$$ lies in $$\text{II}$$ quadrant
Question 3
1 / -0

$$\sqrt{-3}\sqrt{-75}=$$

A
$$15$$

B
$$15i$$

C
$$-15$$

D
$$-15i$$

Solution

$$\sqrt{-3}\times\sqrt{-75}=\sqrt{3\times(-1)}\sqrt{75\times(-1)}$$
$$=\sqrt{3}i\times\sqrt{75}i$$
$$=\sqrt{225}i^{2}$$
$$= - 15$$
Question 4
1 / -0

The sum of two complex numbers $$a + ib$$ and $$c +id$$ is a real number if

A
$$a + c = 0$$

B
$$b + d = 0$$

C
$$a + b= 0$$

D
$$b + c = 0$$

Solution

It is given that

$$z_{1}=a+ib$$ and

$$z_{2}=c+id$$

Then
$$z_{1}+z_{2}=(a+c)+i(b+d)$$

Now
$$(z_{1}+z_{2})$$ is purely real.

Then the imaginary part has to be $$0$$.

Hence
$$b+d=0$$.
Question 5
1 / -0

The locus of complex number z such that z is purely real and real part is equal to - 2 is

A
Negative y-axis

B
Negative x-axis

C
The point (-2, 0)

D
The point (2, 0)

Solution

$$z=x+iy$$
$$(x,y)$$
$$z$$ is purely real and the real part equals $$-2$$
$$\therefore y=0$$ & $$x=-2$$
$$z=-2$$
Hence, this would be represented by the point (-2,0) on the Argand Plane.
Question 6
1 / -0

$$\dfrac{1}{i-1}+\dfrac{1}{i+1}$$ is

A
positive rational number

B
purely imaginary

C
positive Integer

D
negative integer

Solution

Let $$Z= \dfrac{1}{i-1}+\dfrac{1}{i+1}$$

$$=\dfrac{i+1+i-1}{(i-1)(i+1)}$$

$$=\dfrac{i+i}{(i^2-1^2)}$$

$$=\dfrac{2i}{-2}$$

$$ \therefore Z=-i$$
Question 7
1 / -0

If $$(x+iy)(2-3i)=4+i$$ then (x, y) =

A
$$\left ( 1,\dfrac{1}{13} \right )$$

B
$$\left ( -\dfrac{5}{13},\dfrac{14}{13} \right )$$

C
$$\left ( \dfrac{5}{13},\dfrac{14}{13} \right )$$

D
$$\left ( -\dfrac{5}{13},-\dfrac{14}{13} \right )$$

Solution

$$(x+iy)(2-3i)=4+i$$

$$2x-(3x)i+(2y)i-3yi^{2}=4+i$$

$$\underbrace{2x+3y}_{Real }+\underbrace{(2y-3x)}_{Imaginary }i=4+i$$

Comparing the real & imaginary parts,
$$2x+3y=4$$--------------------------(1)
$$2y-3x=1$$----------------------------(2)
Solving eq(1) & eq(2),
$$4x+6y=8$$
$$-9x+6y=3$$
$$13x=5\Rightarrow x=\dfrac{5}{13}$$
$$y=\dfrac{14}{13}$$
$$\therefore (x,y)=\left (\dfrac{5}{13},\dfrac{14}{13} \right )$$
Question 8
1 / -0

The argument of every complex number is

A
Double valued

B
Single valued

C
Many valued

D
Triple valued

Solution

$$z=x+iy$$
amplitude $$= tan^{-1}\frac{y}{x}$$
$$\Rightarrow $$ amplitude $$=\theta \pm 2k\pi $$ where $$\theta \epsilon \left [ -\pi ,\pi \right ]$$ $$\forall k\epsilon R$$
since $$k\epsilon R$$
$$\Rightarrow $$ Amplitude of any complex number is many valued.
Question 9
1 / -0

The sum of two complex numbers $$a + ib$$ and $$c+ id$$ is purely imaginary if

A
$$a + c = 0$$

B
$$a + d = 0$$

C
$$b + d = 0$$

D
$$b + c = 0$$

Solution

It is given that
$$z_{1}=a+ib$$ and
$$z_{2}=c+id$$
$$z_{1}+z_{2}=(a+c)+i(b+d)$$
$$z_{1}+z_{2}$$ is purely imaginary. (Given)
Then the real part has to be $$0$$.
Hence
$$a+c=0$$.
Question 10
1 / -0

For $$a > 0$$, arg $$(ia) =$$

A
$$\dfrac{\pi }{2}$$

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

C
$$\pi $$

D
$$-\pi $$

Solution

$$z= 0 + ia$$
$$\therefore arg(z) = arg (ia) = \tan^{-1} \cfrac a0$$
= $$\cfrac {\pi}{2}$$ ...(since a is greater than or equal to zero)

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Re-Attempt Weekly Quiz Competition

Complex Numbers and Quadratic Equations Test -7

Result

Complex Numbers and Quadratic Equations Test -7

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SHARING IS CARING

Question 1

$$0$$

$$1$$

$$2$$

$$3$$

Solution

Question 2

I

II

III

IV

Solution

Question 3

$$15$$

$$15i$$

$$-15$$

$$-15i$$

Solution

Question 4

$$a + c = 0$$

$$b + d = 0$$

$$a + b= 0$$

$$b + c = 0$$

Solution

Question 5

Negative y-axis

Negative x-axis

The point (-2, 0)

The point (2, 0)

Solution

Question 6

positive rational number

purely imaginary

positive Integer

negative integer

Solution

Question 7

$$\left ( 1,\dfrac{1}{13} \right )$$

$$\left ( -\dfrac{5}{13},\dfrac{14}{13} \right )$$

$$\left ( \dfrac{5}{13},\dfrac{14}{13} \right )$$

$$\left ( -\dfrac{5}{13},-\dfrac{14}{13} \right )$$

Solution

Question 8

Double valued

Single valued

Many valued

Triple valued

Solution

Question 9

$$a + c = 0$$

$$a + d = 0$$

$$b + d = 0$$

$$b + c = 0$$

Solution

Question 10

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

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

$$\pi $$

$$-\pi $$

Solution

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