Match the following
`{:(,"Currents","r.m.s. values"),((1),x_(0) sin omega t,(i)" "x),((2),x_(0) sin omega t cos omega t,(ii)" "x_(0)/sqrt(2)),((3),x_(0) sin omega t+x_(0) cos omega t,(iii)" "x_(0)/((2sqrt(2)))):}`

Options

A. 1. (i), 2. (ii), 3. (iii)
B. 1. (ii), 2. (iii), 3. (i)
C. 1. (i), 2. (iii), 3. (ii)
D. None of these

Answer

To solve the problem of matching the given currents with their respective r.m.s. values, we will analyze each equation step by step.

### Step-by-Step Solution:

1. **Identify the first current equation:**
\[ x = x_0 \sin(\omega t) \]
- The peak value (amplitude) here is \( x_0 \).
- The formula for the r.m.s. value is given by:
\[ \text{r.m.s. value} = \frac{x_0}{\sqrt{2}} \]
- Therefore, the r.m.s. value for this equation is:
\[ \frac{x_0}{\sqrt{2}} \]
- This matches with option (ii).

2. **Identify the second current equation:**
\[ x = x_0 \sin(\omega t) \cos(\omega t) \]
- To find the peak value, we can use the identity:
\[ \sin(\omega t) \cos(\omega t) = \frac{1}{2} \sin(2\omega t) \]
- Thus, the peak value becomes:
\[ x_0 \cdot \frac{1}{2} = \frac{x_0}{2} \]
- The r.m.s. value is then:
\[ \text{r.m.s. value} = \frac{\frac{x_0}{2}}{\sqrt{2}} = \frac{x_0}{2\sqrt{2}} \]
- This matches with option (iii).

3. **Identify the third current equation:**
\[ x = x_0 \sin(\omega t) + x_0 \cos(\omega t) \]
- The r.m.s. value of the sum of two sinusoidal functions can be calculated using:
\[ \text{r.m.s. value} = \sqrt{(\text{r.m.s. of } \sin)^2 + (\text{r.m.s. of } \cos)^2} \]
- The r.m.s. value for both \( x_0 \sin(\omega t) \) and \( x_0 \cos(\omega t) \) is:
\[ \frac{x_0}{\sqrt{2}} \]
- Therefore, we have:
\[ \text{r.m.s. value} = \sqrt{\left(\frac{x_0}{\sqrt{2}}\right)^2 + \left(\frac{x_0}{\sqrt{2}}\right)^2} = \sqrt{\frac{x_0^2}{2} + \frac{x_0^2}{2}} = \sqrt{x_0^2} = x_0 \]
- This matches with option (i).

### Final Matching:
- **1 matches with (ii)**: \( x_0 \sin(\omega t) \) → \( \frac{x_0}{\sqrt{2}} \)
- **2 matches with (iii)**: \( x_0 \sin(\omega t) \cos(\omega t) \) → \( \frac{x_0}{2\sqrt{2}} \)
- **3 matches with (i)**: \( x_0 \sin(\omega t) + x_0 \cos(\omega t) \) → \( x_0 \)

### Summary of Matches:
- 1 → (ii)
- 2 → (iii)
- 3 → (i)