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Question

The magnitude and amplitude of (1+i3)(2+2i)3-i are respectively


A

2,3π4

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B

22,3π4

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C

22,π4

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D

22,π2

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Solution

The correct option is C

22,π4


Explanation of the correct option.

Step 1: Simplify the fraction.

Given : (1+i3)(2+2i)3-i

Multiply the numerator and denominator with 3+i,

(1+i3)(2+2i)×3+i3-i×3+i2(1+i)(3+i+i3-3)3+12(1+i)(i4)44i-42-2+2i

Step 2: Compute the magnitude.

Since magnitude of x+iy is given by x+iy=x2+y2,

-2+2i=4+4-2+2i=22

Step 3: Compute the amplitude.

Since amplitude of x+iy is given by θ=tan-1yx,

θ=tan-12-2θ=tan-1-1θ=π4

Therefore, magnitude and amplitude of (1+i3)(2+2i)3-i is 22 and π4 respectively.

Hence option C is the correct option.


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