Complex Numbers and Quadratic Equations Test -3

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Complex Numbers and Quadratic Equations Test -3

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Weekly Quiz Competition

Question 1
1 / -0

Amplitude of $$\displaystyle \frac{1 +\sqrt 3i}{ \sqrt3 + i} $$is

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

B
$$\displaystyle \frac {\pi}{2}$$

C
0

D
$$\displaystyle \frac {\pi}{6}$$

Solution

$$\displaystyle Z= \frac{1 +\sqrt 3i}{ \sqrt3 + i} $$
$$\displaystyle\Rightarrow \left( \frac { 1+\sqrt { 3 } i }{ \sqrt { 3 } +i } \right) \left( \frac { \sqrt { 3 } -i }{ \sqrt { 3 } -i } \right) =\frac { 2\sqrt { 3 } +2i }{ 4 } $$
$$\therefore$$ Amplitude of $$Z$$ is $$\tan ^{ -1 }\left({\displaystyle \frac { 1 }{ \sqrt3 } } \right)=\displaystyle\frac{\pi}{6}$$
Hence, option D.
Question 2
1 / -0

The value of $$-i^{48}$$ is

A
$$-i$$

B
$$i$$

C
$$-1$$

D
$$1$$

Solution

We need to find value of $$-i^{48}$$
The power rise to the $$i$$ is even, then the value of $$i$$ is $$-1$$.
The power rise to the $$i$$ is odd, then the value of $$i$$ is $$-i$$.
So, the value of $$-i^{48}=-1$$
Question 3
1 / -0

For $$i=\sqrt{-1}$$, what is the sum $$\left(7+3i\right) + \left(-8+9i\right)$$?

A
$$-1+12i$$

B
$$-1-6i$$

C
$$15+12i$$

D
$$15-6i$$

Solution

Given: $$(7+3i)+(-8+9i)$$
$$= (7+(-8))+(3i+9i)$$
$$=(7-8)+12i$$
$$=-1+12i$$
Option A is correct.
Question 4
1 / -0

In the complex plane, what is the distance of $$4-2i$$ from the origin?

A
$$2$$

B
$$3.46$$

C
$$4.47$$

D
$$6$$

E
$$12$$

Solution

Let $$z=4-2i$$
In a complex plane, distance of $$z$$ from any point origin is given by $$|z|=\sqrt{(x_1)^2+(y_1)^2}$$
$$\therefore$$ Distance of $$z=4-2i$$ from origin is given by $$|z|=|4-2i|=\sqrt{4^2+(-2)^2}=\sqrt{20}\approx 4.47$$
Hence, the answer is $$4.47$$.
Question 5
1 / -0

If the roots of an equation $$p{ x }^{ 2 }+qx+r=0$$ are equal, then

A
$${ q }^{ 2 }=pr$$

B
$${ q }^{ 2 }=4pr$$

C
$${ p }^{ 2 }=4qr$$

D
$$p=qr$$

Solution

For equal roots Discriminant has to be $$0$$
So $${q}^{2}-4 pr = 0$$
$$\Rightarrow q^2 = 4pr$$
Question 6
1 / -0

In the complex plane, the number 4 + j3 is located in the

A
first quadrant

B
second quadrant

C
third quadrant

D
fourth quadrant

Solution

Since both the real and imaginary parts are positive , 4 + j3 lies in first quadrant of argand/complex plane.
Question 7
1 / -0

For the quadratic equation $$ax^2 + bx + c = 0, a, b, c, \in Q$$, If $$D = 0$$ then ...................
Choose the correct option in respect to the statements below.
(P) The roots of the equation are equal.
(Q) The roots of the equation are not equal.
(R) The roots of the equation are rational numbers.
(S) The roots of the equation has no roots.

A
Statements $$P$$ and $$R$$ are correct

B
Statements $$Q$$ and $$R$$ are correct

C
Only statement $$S$$ is correct

D
Only statement $$P$$ is correct

Solution

For the quadratic equation $$ax^2 + bx + c = 0, a, b, c, \varepsilon Q$$

Roots are given by

$$\alpha=\dfrac{-b + \sqrt{b^2 - 4ac}}{2a}$$ and

$$\beta=\dfrac{-b - \sqrt{b^2 - 4ac}}{2a}$$

If $$D = 0 $$ then $$\alpha=\dfrac{-b}{2a}=\beta$$ i.e. the roots are equal and real.

Since $$a, b$$ and $$c \in Q$$, the roots will be equal and rational.

Hence, statements P and R are correct.
Question 8
1 / -0

What is the modulus of $$\cfrac { \sqrt { 2 } +i }{ \sqrt { 2 } -i } $$ where $$i=\sqrt { -1 } $$

A
$$3$$

B
$$\dfrac{1}{2}$$

C
$$1$$

D
None of the above

Solution

$$\left| \dfrac { \sqrt { 2 } +i }{ \sqrt { 2 } -i } \right| =\dfrac { \sqrt { ({ \sqrt { 2 } })^{2}+{ 1 }^{ 2 } } }{ \sqrt { (\sqrt { 2 })^{ 2 }+(-1)^{ 2 } } } =1$$
Hence, option C is correct.
Question 9
1 / -0

If $$\cos { \left( \log { { i }^{ 4i } } \right) } =a+ib$$, then

A
$$a=1,b=-1$$

B
$$a=-1,b=1$$

C
$$a=1,b=0$$

D
$$a=1,b=2$$

Solution

Given, $$a+ib=\cos { \left( \log { { i }^{ 4i } } \right) } $$
$$=\cos { \left[ 4i\log { i } \right] } =\cos { \left[ 4i\log { \left( { e }^{ i\cfrac { \pi }{ 2 } } \right) } \right] } =\cos { \left[ \left( 4i \right) \left( i\cfrac { \pi }{ 2 } \right) \right] } =\cos { \left( -2\pi \right) } =1$$
$$\therefore a=1;b=0$$
Question 10
1 / -0

If ............, then the quadratic equation does not have real solution.

A
$$D=0$$

B
$$D> 0$$

C
$$D< 0$$

D
$$D\ge 0$$

Solution

The general form of quadratic equation is given by $$ax^2+bx+c=0$$ ... $$(i)$$

Let $$\alpha, \beta$$ be roots of quadratic equation $$(i)$$

$$\alpha =\dfrac{-b-\sqrt{b^2-4ac}}{2a}$$ and $$\beta =\dfrac{-b+\sqrt{b^2-4ac}}{2a}$$

The expression $$D=b^2-4ac$$ is the discriminant

If $$D>0$$, then $$\alpha$$ and $$\beta$$ are real.

If $$D=0$$, then $$\alpha=\beta$$

If $$D<0$$, then $$\alpha$$ and $$\beta$$ are not real.

So, when $$D< 0$$, the quadratic equation does not have any real roots.
Hence, option C is correct.

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Complex Numbers and Quadratic Equations Test -3

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Question 1

$$\displaystyle \frac {\pi}{3}$$

$$\displaystyle \frac {\pi}{2}$$

0

$$\displaystyle \frac {\pi}{6}$$

Solution

Question 2

$$-i$$

$$i$$

$$-1$$

$$1$$

Solution

Question 3

$$-1+12i$$

$$-1-6i$$

$$15+12i$$

$$15-6i$$

Solution

Question 4

$$2$$

$$3.46$$

$$4.47$$

$$6$$

$$12$$

Solution

Question 5

$${ q }^{ 2 }=pr$$

$${ q }^{ 2 }=4pr$$

$${ p }^{ 2 }=4qr$$

$$p=qr$$

Solution

Question 6

first quadrant

second quadrant

third quadrant

fourth quadrant

Solution

Question 7

Statements $$P$$ and $$R$$ are correct

Statements $$Q$$ and $$R$$ are correct

Only statement $$S$$ is correct

Only statement $$P$$ is correct

Solution

Question 8

$$3$$

$$\dfrac{1}{2}$$

$$1$$

None of the above

Solution

Question 9

$$a=1,b=-1$$

$$a=-1,b=1$$

$$a=1,b=0$$

$$a=1,b=2$$

Solution

Question 10

$$D=0$$

$$D> 0$$

$$D< 0$$

$$D\ge 0$$

Solution

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