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Question

If z and ω are complex numbers such that zω=1, argz-argω=3π2 then find arg1-2z¯ω1+3z¯ω


A

π4

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B

-π4

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C

3π4

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D

-3π4

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Solution

The correct option is D

-3π4


Explanation for the correct option.

Step 1. Assume complex numbers z and ω.

It is given that zω=1, so let z=r and ω=1r.

Also, it is given that argz-argω=3π2, so let argz=θ, then argω=θ-3π2.

Now, the complex number z and ω can be given as:

z=reiθ;ω=1reiθ-3π2

Now the conjugate of z=reiθ is given as: z¯=re-iθ.

Step 2. Find z¯ω.

Multiply the complex number z¯ and ω.

z¯ω=re-iθ×1reiθ-3π2=e-iθ+iθ-i3π2=ei-3π2=cos-3π2+isin-3π2eiθ=cosθ+isin(θ)=0+i(1)=i

Step 3. Find the complex number 1-2z¯ω1+3z¯ω.

In the complex number 1-2z¯ω1+3z¯ω substitute i for z¯ω and simplify.

1-2z¯ω1+3z¯ω=1-2i1+3i[z¯ω=i]=1-2i1-3i1+3i1-3i=1-3i-2i+6i21-9i2=1-5i+6-11-9-1[i2=-1]=1-5i-61+9=-5-5i10=-12-12i

Step 4. Find the argument.

The complex number 1-2z¯ω1+3z¯ωi is the same as -12-12i.

The complex number -12-12i is in third quadrant.

So, the argument is given as:

arg-12-12i=-π+tan-1-12-12=-π+tan-11=-π+π4=-3π4

So, arg1-2z¯ω1+3z¯ω=-3π4.

Hence, the correct option is D.


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