 Hello friends, let's solve the following question. It says, find the modulus and the argument of the complex number. So, let us first write what is the modulus for a complex number z. If z is a complex number x plus iota y, then the modulus of z is equal to under root of x square plus y square and that is denoted by r. In the polar form of z is equal to r cos theta plus iota r sin theta and the argument of z is given by theta. So, this is the key idea behind this question. Let us now solve the problem. z is equal to minus root 3 plus iota. Let us now identify what is x and what is y. Now, x is minus root 3 and y is 1. So, the modulus of z is equal to under root of minus root 3 square plus 1 square and it is equal to root 4 which is equal to 2. So, this is the modulus of z. Now, we have to obtain the argument of z. That is we have to obtain theta. So, let us first plot the point minus root 3 and 1 on the argon plane. So, this is the argon plane and this is x axis and this one is y axis. Now, x is equal to minus root 3 and y is equal to 1. So, this point lies in the second quadrant and its distance from the origin is 2 because the modulus is 2. So, we need to obtain the value of theta. So, let us write z in the polar form and it is r cos theta plus iota r sin theta and r is 2. So, this is equal to 2 cos theta plus iota 2 sin theta and z is equal to minus root 3 plus iota. So, on comparing we get minus root 3 is equal to 2 cos theta and 1 is equal to 2 sin theta. Now, the first equation implies cos theta is equal to minus root 3 by 2 and sin theta is equal to 1 by 2. Now, we know that sin pi by 6 is equal to 1 by 2 and cos pi by 6 is equal to root 3 by 2. Right? So, cos pi by 2 is root 3 by 2 but we want to have the value of theta for which cos theta is equal to minus root 3 by 2. Right? Now, since this point is in second quadrant, so we have sin pi minus pi by 6 is equal to 1 by 2 and cos pi minus pi by 6 is equal to minus root 3 by 2. The first equation implies sin 5 pi by 6 is equal to 1 by 2 and the second equation implies cos 5 pi by 6 is equal to minus root 3 by 2. That means theta is minus 5 pi by 6. So, this implies theta is equal to 5 pi by 6 and this is the argument of z. Hence, the modulus is 2 and the argument of z is equal to 5 pi by 6 and this completes the question. Bye for now. Take care. Hope you enjoyed the session.