Complex Numbers and Quadratic Equations Test 35

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Complex Numbers and Quadratic Equations Test 35

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Weekly Quiz Competition

Question 1
1 / -0

$$z_1$$ and $$z_2$$ are two non-zero complex numbers such that $$z_1=2+4i\\z_2=5-6i$$, then $$z_2-z_1$$ equals

A
$$3-10i$$

B
$$3+10i$$

C
$$7-2i$$

D
$$10-24i$$

Solution

$$z_1=2+4i\\\\z_2=5-6i\\\\z_2-z_1=5-6i-2-4i=3-10i$$
Question 2
1 / -0

The imaginary part of $$t ; t \in R$$ is

A
$$0$$

B
$$1$$

C
$$2$$

D
$$-1$$

Solution
Question 3
1 / -0

If $$z = \cos \dfrac { \pi } { 4 } + i \sin \dfrac { \pi } { 6 } ,$$ then

A
$$| z | = 1 , \arg ( z ) = \dfrac { \pi } { 4 }$$

B
$$| z | = 1 , \arg ( z ) = \dfrac { \pi } { 6 }$$

C
$$| z | = \dfrac { \sqrt { 3 } } { 2 } , \arg ( z ) = \dfrac { 5 \pi } { 24 }$$

D
$$| z | = \dfrac { \sqrt { 3 } } { 2 } , \arg ( z ) = \tan ^ { - 1 } \dfrac { 1 } { \sqrt { 2 } }$$

Solution

$$z = \cos \dfrac{\pi}{4} + i \sin \dfrac{\pi}{6}$$

$$z = \dfrac{1}{\sqrt{2}} + i \dfrac{1}{2}$$

$$\theta = \tan^{-1} \dfrac{\dfrac{1}{2}}{\dfrac{1}{\sqrt{2}}} = \tan^{-1} \left(\dfrac{\sqrt{2}}{2} \right)$$

$$\theta = \tan^{-1} \left(\dfrac{1}{\sqrt{2}}\right)$$

$$\therefore arg z = \tan^{-1} \dfrac{1}{\sqrt{2}}$$
$$|z| = \sqrt{\left(\dfrac{1}{\sqrt{2}}\right)^2 + \left(\dfrac{1}{2}\right)^2} = \sqrt{\dfrac{1}{2} + \dfrac{1}{4}}$$

$$= \sqrt{\dfrac{2 + 1}{4}}$$

$$= \dfrac{\sqrt{3}}{2}$$

$$z = \cos \dfrac{\pi}{4} + i \sin \dfrac{pi}{4}$$

$$arg z = angle = \dfrac{\pi}{4}$$

and $$|z| = \cos^2 \dfrac{\pi}{4} + \sin^2 \dfrac{\pi}{4} = 1$$
Question 4
1 / -0

Let z be a complex number such that the principal value of argument, arg $$z < 0$$. Then $$arg(-z) - arg (z)$$ is

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

B
$$\pm \pi$$

C
$$\pi$$

D
$$-\pi$$

Solution

Consider
$$z=|z|e^{iθ}$$

Now,
$$−z=|z|e^{i(θ±π)}=|z|e^{i\bar{\theta}}$$

where$$ \bar{\theta}=arg(−z)$$

Then
$$arg(−z)−arg(z)=\bar{\theta}−θ=±π$$

This has been verified numerically for random values of $$z$$.
Question 5
1 / -0

Given $$ z _ { 1 } + 3 z _ { 2 } - 4 z _ { 3 } = 0 $$ then $$ z _ { 1 } , z _ { 2 } , z _ { 3 } $$ are

A
collinear

B
can form sides of equilateral $$ \Delta $$

C
lie on circle

D
none of these

Solution
Question 6
1 / -0

If z be a complex number satisfying $$z^{4}+z^{3}+2z^{2}+z+1=0$$ then $$\left|z\right|=$$

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

B
$$\dfrac{3}{4}$$

C
$$1$$

D
none of these

Solution

$$z^{4}+z^{3}+2z^{2}+z+1=0$$
$$\Rightarrow z^{4}+z^{2}+z^{3}+z+z^{2}+1=0$$ by rearranging
$$\Rightarrow z^{2}(z^{2}+1)+z(z^{2}+1)+1(z^{2}+1)=0$$
$$\Rightarrow z^{2}(z^{2}+1)+(z^{2}+1)(z+1)=0$$
$$\Rightarrow (z^{2}+1)(z^{2}+z+1)=0$$
$$\Rightarrow z^{2}+1=0, z^{2}+2+1=0$$
$$\Rightarrow z=\pm i, z=\frac{-1\pm i\sqrt{3}}{2}$$
$$\therefore z=i, -i, \frac{-1+i\sqrt{3}}{2},\frac{-1-i\sqrt{3}}{2}$$
$$\therefore |z|=+1$$
Question 7
1 / -0

Express the following complex numbers in the standard form $$ a+ib$$ :
$$ \left ( \dfrac{1}{1-4i}-\dfrac{2}{1+i} \right )\left ( \dfrac{3-4i}{5+i} \right )$$

A
$$ \dfrac{307}{442}+i \dfrac{599}{442}i$$

B
$$ \dfrac{307}{442}-i \dfrac{599}{442}i$$

C
$$ \dfrac{-307}{442}+i \dfrac{599}{442}i$$

D
None of the above

Solution
Question 8
1 / -0

Express the following complex numbers in the standard form $$ a+ib$$ :
$$ \dfrac{\left ( 2+i \right )^{3}}{2+3i}$$

A
$$ \dfrac{37}{13}-\dfrac{16}{13}i$$

B
$$ \dfrac{-37}{13}+\dfrac{16}{13}i$$

C
$$ \dfrac{37}{13}+\dfrac{16}{13}i$$

D
None of the above

Solution
Question 9
1 / -0

Find the modulus and argument of the following complex numbers and hence express each of them in the polar form:
$$1-i$$

A
$$\sqrt{2}(cos\,\pi /4+i\, sin\, \pi /4)$$

B
$$\sqrt{2}(cos\,\pi /3-i\, sin\, \pi /3)$$

C
$$\sqrt{2}(cos\,\pi /4-i\, sin\, \pi /4)$$

D
$$\sqrt{2}(cos\,\pi /3+i\, sin\, \pi /3)$$

Solution

Let $$z=1-i$$. Then, $$\left | z \right |=\sqrt{1^{2}+(-1)^{2}}=\sqrt{2}$$.

Let $$\alpha $$ be the acute angle given by $$tan\, \alpha =\dfrac{\left | Im(z) \right |}{\left | Re(z) \right |}$$.

Then,
$$tan\, \alpha =\dfrac{|-1|}{|1|}=1\Rightarrow \alpha =\dfrac{\pi }{4}$$

Clearly, z lies in the fourth quadrant. Therefore, $$arg(z)= -\alpha =-\dfrac{\pi }{4}$$.
Question 10
1 / -0

Express the following complex numbers in the standard form $$ a+ib$$ :
$$ \dfrac{3-4i}{\left ( 4-2i \right )\left ( 1+i \right )}$$

A
$$ \dfrac{1}{4}+\dfrac{3}{4}i$$

B
$$ \dfrac{1}{4}-\dfrac{3}{4}i$$

C
$$ \dfrac{-1}{4}-\dfrac{3}{4}i$$

D
None of the above

Solution