 Hi, and welcome to the session. Let's discuss the following question. It says, find the modulus and the argument of the complex number. So, let us first understand what is the modulus and what is the argument of a complex number. If we have a complex number, z is equal to x plus iota y, then the modulus of z is equal to under root of x square plus y square and it is denoted by r. And the polar form of z is given by r cos theta plus iota r sin theta. And the argument of z is given by theta. So, this becomes the key idea for the question. Let's now proceed on to the solution. z is equal to minus 1 minus iota root 3. Now, x is equal to minus 1 and y is equal to minus root 3. So, the modulus of z is equal to root of minus 1 square plus minus root 3 square by the formula for modulus and it is equal to root 4 which is equal to 2. So, this is the modulus of z. Now, we have to find the argument of z. Now, x is equal to minus 1 and y is equal to minus root 3. So, let us first plot these points on the argon plane. Now, since x is equal to minus 1 and y is equal to minus root 3, since both the points are negative, they lie in the third quadrant. So, the point xy lies in the third quadrant and its distance from origin is 2 because the modulus is 2. Now, z is equal to minus 1 minus iota root 3 and in the polar form it can be written as r cos theta plus iota r sin theta. But r is 2. So, this is equal to 2 cos theta plus iota 2 sin theta and we need to obtain the value of theta to find the argument. Now, comparing the two values of z, what do we get? Minus 1 is equal to 2 cos theta and minus root 3 is equal to 2 sin theta. Now, first equation implies cos theta is equal to minus 1 by 2 and sin theta is equal to minus root 3 by 2 and to obtain argument we have to find the theta. So, we need to obtain the value of theta. Now, we know that cos pi by 3 is equal to 1 by 2 and sin pi by 3 is equal to root 3 by 2. Now, since this point is in third quadrant, so we have cos minus pi plus pi by 3 is equal to minus 1 by 2 and sin minus pi plus pi by 3 is equal to minus root 3 by 2 and this implies cos minus 2 pi by 3 is equal to minus 1 by 2 and the second equation implies sin minus 2 pi by 3 is equal to minus root 3 by 2. That means the value of theta is minus 2 pi by 3 and that is what we have to find. Hence, the modulus of z is 2 and the argument of z is minus 2 pi by 3 and this completes the question. Bye for now. Take care. Hope you enjoyed the session.