`z=-sqrt(3)+i` find modulus and argument

Options

A. `1, 100^@`
B. `2, 150^@`
C. `3, 200^@`
D. `4, 250^@`

Explanation

To find the modulus and argument of the complex number \( z = -\sqrt{3} + i \), we will follow these steps:

### Step 1: Identify the real and imaginary parts
The complex number can be expressed as:
- Real part \( a = -\sqrt{3} \)
- Imaginary part \( b = 1 \)

### Step 2: Calculate the modulus
The modulus \( r \) of a complex number \( z = a + bi \) is given by the formula:
\[ r = \sqrt{a^2 + b^2} \]
Substituting the values of \( a \) and \( b \):
\[ r = \sqrt{(-\sqrt{3})^2 + (1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2 \]

### Step 3: Calculate the argument
The argument \( \theta \) of a complex number is given by:
\[ \tan \theta = \frac{b}{a} \]
Substituting the values of \( a \) and \( b \):
\[ \tan \theta = \frac{1}{-\sqrt{3}} = -\frac{1}{\sqrt{3}} \]
This value corresponds to an angle in the second quadrant since the real part is negative and the imaginary part is positive. The reference angle where \( \tan \theta = \frac{1}{\sqrt{3}} \) is \( 30^\circ \). Therefore, in the second quadrant, the angle is:
\[ \theta = 180^\circ - 30^\circ = 150^\circ \]

### Final Result
- Modulus \( r = 2 \)
- Argument \( \theta = 150^\circ \)