Find the modulus and argument of the following :
(i) `-sqrt (3)+i`
(ii) `-1-isqrt(3)`
(iii) `5+12i`
(iv) `3(cos300^(@)-isin30^(@))`

Explanation

To find the modulus and argument of the given complex numbers, we will follow these steps:

### (i) For the complex number \( z = -\sqrt{3} + i \)

1. **Modulus Calculation**:
\[ |z| = \sqrt{(-\sqrt{3})^2 + (1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2 \]

2. **Argument Calculation**:
\[ \text{Argument} = \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{1}{-\sqrt{3}}\right) \]
Since \( x < 0 \) and \( y > 0 \), the complex number is in the second quadrant. The reference angle is \( \frac{\pi}{6} \), so:
\[ \text{Argument} = \pi - \frac{\pi}{6} = \frac{5\pi}{6} \quad \text{or} \quad 150^\circ \]

### (ii) For the complex number \( z = -1 - i\sqrt{3} \)

1. **Modulus Calculation**:
\[ |z| = \sqrt{(-1)^2 + (-\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \]

2. **Argument Calculation**:
\[ \text{Argument} = \tan^{-1}\left(\frac{-\sqrt{3}}{-1}\right) = \tan^{-1}(\sqrt{3}) \]
Since \( x < 0 \) and \( y < 0 \), the complex number is in the third quadrant. The reference angle is \( \frac{\pi}{3} \), so:
\[ \text{Argument} = \pi + \frac{\pi}{3} = \frac{4\pi}{3} \quad \text{or} \quad 240^\circ \]

### (iii) For the complex number \( z = 5 + 12i \)

1. **Modulus Calculation**:
\[ |z| = \sqrt{(5)^2 + (12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]

2. **Argument Calculation**:
\[ \text{Argument} = \tan^{-1}\left(\frac{12}{5}\right) \]
This angle is in the first quadrant, so we can directly use:
\[ \text{Argument} = \tan^{-1}\left(\frac{12}{5}\right) \quad \text{(in radians or degrees)} \]

### (iv) For the complex number \( z = 3(\cos 300^\circ - i \sin 30^\circ) \)

1. **Convert to rectangular form**:
\[ z = 3\left(\cos 300^\circ - i \sin 30^\circ\right) = 3\left(\frac{1}{2} - i \cdot \frac{1}{2}\right) = \frac{3}{2} - \frac{3}{2}i \]

2. **Modulus Calculation**:
\[ |z| = \sqrt{\left(\frac{3}{2}\right)^2 + \left(-\frac{3}{2}\right)^2} = \sqrt{\frac{9}{4} + \frac{9}{4}} = \sqrt{\frac{18}{4}} = \frac{3\sqrt{2}}{2} \]

3. **Argument Calculation**:
\[ \text{Argument} = \tan^{-1}\left(\frac{-\frac{3}{2}}{\frac{3}{2}}\right) = \tan^{-1}(-1) \]
Since \( x > 0 \) and \( y < 0 \), the complex number is in the fourth quadrant:
\[ \text{Argument} = -\frac{\pi}{4} \quad \text{or} \quad 315^\circ \]

### Summary of Results:
1. \( |z| = 2 \), Argument = \( \frac{5\pi}{6} \) (or \( 150^\circ \))
2. \( |z| = 2 \), Argument = \( \frac{4\pi}{3} \) (or \( 240^\circ \))
3. \( |z| = 13 \), Argument = \( \tan^{-1}\left(\frac{12}{5}\right) \)
4. \( |z| = \frac{3\sqrt{2}}{2} \), Argument = \( -\frac{\pi}{4} \) (or \( 315^\circ \))