Question

Let z be the complex number with modulus 2 and argument \[\dfrac{{2\pi }}{3}\]. Then z equals
A.\[ - 1 + i\sqrt 3 \]
B.\[\dfrac{{ - 1 + i\sqrt 3 }}{2}\]
C.\[ - 1 - i\sqrt 3 \]
D.\[\dfrac{{ - 1 - i\sqrt 3 }}{2}\]

Answer 1

Hint: This is a very basic question related to complex numbers. We just have to find which complex number satisfies the given condition of argument and modulus. For that we will find the modulus and argument by trial and error method. We will select option A and C first. Using the formula for the modulus and argument.

Complete step-by-step answer:
Given that z is the complex number.
We know that complex number is of the form \[z = x + iy\]
Now the modulus is given by, \[\left| z \right| = \sqrt {{x^2} + {y^2}} \]
Now we are given that modulus is 2 already. That means the value of this root should be 4.
So we can observe that, option A and C have 1 and \[\sqrt 3 \] as the value of x and y only the signs are somewhat different. So let’s combine them as,
\[\left| z \right| = \sqrt {{{\left( { - 1} \right)}^2} + {{\left( { \pm \sqrt 3 } \right)}^2}} \]
The square of positive as well as negative numbers is always positive only.
\[\left| z \right| = \sqrt {1 + 3} = \sqrt 4 = 2\]
Thus now we can say that, we need to check for the argument.
Argument of a complex number z is given by,
\[\arg \left( z \right) = {\tan ^{ - 1}}\left( {\dfrac{y}{x}} \right)\]
But it is already given as to be \[\dfrac{{2\pi }}{3}\]
So we can say that,
\[\dfrac{{2\pi }}{3} = {\tan ^{ - 1}}\left( {\dfrac{y}{x}} \right)\]
Taking tan function on both sides,
\[\tan \left( {\dfrac{{2\pi }}{3}} \right) = \tan \left( {{{\tan }^{ - 1}}\left( {\dfrac{y}{x}} \right)} \right)\]
\[\tan \left( {\dfrac{{2\pi }}{3}} \right) = \dfrac{y}{x}\]
We know that \[\tan \left( {\dfrac{{2\pi }}{3}} \right) = \sqrt 3 \]
\[\sqrt 3 = \dfrac{y}{x}\]
Now we need the value of y and x such that either both are positive or both are negative so that the ratio is positive.
And we find this in option C that is \[ - 1 - i\sqrt 3 \]
So we can eliminate option A simply.
So, the correct answer is “Option A”.

Note: Now here note that, we won’t touch option B and D because when we find the modulus only they do not satisfy the requirement. So we should not waste our time.
Rather when we find the argument do note that in general either the angle is clockwise or anticlockwise. We also know that the iy part of the complex number is imaginary and x is the real part. The modulus gives the length and the argument gives the angular distance.
Argument of a complex number z can also be written as \[\arg \left( z \right)\] and modulus as \[\left| z \right|\].