Argument of complex numbers Practice Questions - NDA Mathematics Question Bank

Question 1 of 3 NDA MCQ

What is the principal argument of 1/(1 + i) where i = √−1 ?

Detailed Explanation

Question 2 of 3 NDA MCQ

What is the argument of the complex number

Detailed Explanation

Option (a) is correct.

Explanation:

Calculate the argument of the complex number

\[ z = \frac{1 - i\sqrt{3}}{1 + i\sqrt{3}} \]

Multiply numerator and denominator by the conjugate of the denominator:

\[ z = \frac{(1 - i\sqrt{3})(1 - i\sqrt{3})}{(1 + i\sqrt{3})(1 - i\sqrt{3})} \]

Numerator simplifies to:

\[ 1 - 2i\sqrt{3} - (-3) = 4 - 2i\sqrt{3} \]

Denominator simplifies to:

\[ 1 + 3 = 4 \]

Thus,

\[ z = 1 - \frac{i\sqrt{3}}{2} \]

Now, Find the argument: For \( z = x + iy \), where \( x = 1 \) and \( y = -\sqrt{3} \), Using \( \tan^{-1} \left( \frac{y}{x} \right) \):

\[ \tan^{-1} \left( -\frac{\sqrt{3}}{1} \right) = -60^{\circ} \] \

Converting to standard position by adding \( 360^{\circ} \):

\[ -60^\circ + 360^\circ = 300^\circ = 240^\circ \] \

Thus, the argument is \( 240^\circ \).

📌 Hints / Properties Used:

📊 Visual Diagram Suggestion:

A diagram showing the complex number in the Argand plane, illustrating the angle \( 240^\circ \) with respect to the positive x-axis.

📖 Factual Verification & Reference:

Verified against standard textbooks on Complex Numbers.

Question 3 of 3 NDA MCQ

If z = (1 + i√3)/(1 − i√3), where i = √(−1), then what is the argument of z?

Detailed Explanation

Argument of complex numbers