1.

Let, z = -1-i√3, then argument of z is1. π/32. -π/33. -2π/34. 3π/2

Answer» Correct Answer - Option 3 : -2π/3

Concept:

Consider a complex number z = a + ib,

Polar form is given by z = r(cosθ + isin θ), where r = \(\rm \sqrt {a^2 +b^2}\) and θ = tan-1 \(\left(\rm\frac{Im(z)}{Re(z)} \right)\)

 

Quadrant

Sign of x and y

Principle value of Argument

I

x > 0, y > 0

\(\rm \tan^{-1} \frac{y}{x}\)

II

x < 0, y > 0

π - \(\rm \tan^{-1} \left|\frac{y}{x}\right|\)

III

x < 0, y < 0

-π + \(\rm \tan^{-1} \left|\frac{y}{x}\right|\)

IV

x > 0, y < 0

\(-\rm \tan^{-1} \left|\frac{y}{x}\right|\)

 

Calculation: 

We have, z = -1 - i√3

Here,  Re(z) = -1 and Im(z) = -√3.

∴ tanθ = \(\frac{-\sqrt3}{-1}\)

⇒ θ = tan-1 (\(\frac{\sqrt3}{1}\)) = π/3

Now,  z = -1 - i√3, represented by the points, P(-1,-√3), which lies in the 3rd quadrant.

∴ arg(z) = θ = (-π + θ)

= (-π + π /3)

= -2π/3

Hence, option (3) is correct.


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