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UPSC NDA I 2019 Mathematics Solved Paper

Question Numbers: 29-30

Direction: Consider the following for the next 02 (two) items:

A complex number is given by z.

z = \frac{{1 + 2i}}{{1 - {{\left( {1 - i} \right)}^2}}}

Question : 30 of 120

Marks: +1, -0

What is the principal argument of z?

Solution:

Concept:

The principal argument

\operatorname{Arg}(z)

of a complex number

z = a+ib

is the unique angle

\theta

(-\pi, \pi]

such that

\tan\theta = \frac{b}{a}

, considering the quadrant of

(a,b)

in the Argand plane.

Explanation:

Simplify the given expression:

z = \frac{1+2i}{1 - (1-i)^2}

Compute

(1-i)^2 = 1 - 2i + i^2 = 1 - 2i -1 = -2i

Then

1 - (1-i)^2 = 1 - (-2i) = 1+2i

Hence

z = \frac{1+2i}{1+2i} = 1

Write

z = 1 + i\cdot 0

, so

a = 1

b = 0

The argument is

\theta = \tan^{-1}\left(\frac{0}{1}\right) = \tan^{-1}(0) = 0

Since the point

(1,0)

lies on the positive real axis, the principal argument is

\operatorname{Arg}(z) = 0

Answer:

0

(Option A).

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