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The magnitude and amplitude of \frac{(1+i \sqrt 3 )(2+2i)}{(\sqrt 3 - i)}    are respectively:

  • Option 1)

    2,\:\:\frac{3 \pi}{4}

  • Option 2)

    2\sqrt 2,\:\:\frac{3 \pi}{4}

  • Option 3)

    2\sqrt 2,\:\:\frac{ \pi}{4}

  • Option 4)

    2\sqrt 2,\:\:\frac{\pi}{2}

 

Answers (1)

best_answer

As learnt in concept

Property of Modulus of z(Complex Number) -

\left |\frac{z_{1}}{z_{2}} \right |=\frac{\left |z_{1} \right |}{\left |z_{2} \right |}

- wherein

|.| denotes modulus of complex number

 

 

Definition of Argument/Amplitude of z in Complex Numbers -

\theta =tan^{-1}|\frac{y}{x}|, z\neq 0

\boldsymbol{\theta,\pi-\theta,-\pi+\theta,-\theta} are Principal Argument if z lies in first, second, third or fourth quadrant respectively.

- wherein

 

 Magnitude \left | Z \right | =\frac{\left | (1+i\sqrt{3}) \right | \left | (2+2i) \right |}{\left | (\sqrt{3}-i) \right |}

=\frac{2\times2\sqrt{2}}{2}=2\sqrt{2}

amplitude is

tan^{-1}\sqrt{3}+tan^{-1} 1-tan ^{-1} \left ( \frac{-1}{\sqrt {3}} \right )

=\frac{\pi}{3}+\frac{\pi}{4} -(\frac{-\pi}{6})=\frac{3\pi}{4}


Option 1)

2,\:\:\frac{3 \pi}{4}

This is incorrect option

Option 2)

2\sqrt 2,\:\:\frac{3 \pi}{4}

This is correct option

Option 3)

2\sqrt 2,\:\:\frac{ \pi}{4}

This is incorrect option

Option 4)

2\sqrt 2,\:\:\frac{\pi}{2}

This is incorrect option s

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