Read the statements carefully and state 'T' for true and 'F' for false.
(i) The pair of linear equations `x + 2y = 5 and 7x + 3y = 13` has unique solution x = 2, y = 1.
(ii) `sqrt2x+ sqrt3y= 0, sqrt3x- sqrt8y= 0` has no solution.
(iii) The values of p and q for which the following system of equations `2x - y = 5, (p + q)x + (2p - q)y = 15` has infinite number of solutions, is p = 1 and q = 5.

Options

A. `{:(i,ii,iii),(T,F,T):}`
B. `{:(i,ii,iii),(T,T,F):}`
C. `{:(i,ii,iii),(F,T,T):}`
D. `{:(i,ii,iii),(F,F,T):}`

Answer

To solve the given statements, we will evaluate each one step by step.

### Statement (i):
**The pair of linear equations \(x + 2y = 5\) and \(7x + 3y = 13\) has a unique solution \(x = 2\), \(y = 1\).**

1. **Write the equations:**
\[ \text{Equation 1: } x + 2y = 5 \quad (1) \]
\[ \text{Equation 2: } 7x + 3y = 13 \quad (2) \]

2. **Substitute \(x = 2\) and \(y = 1\) into both equations to check if they hold true:**
- For Equation 1:
\[ 2 + 2(1) = 2 + 2 = 4 \quad \text{(not equal to 5, so false)} \]
- For Equation 2:
\[ 7(2) + 3(1) = 14 + 3 = 17 \quad \text{(not equal to 13, so false)} \]

3. **Conclusion for Statement (i):**
The statement is **False (F)** because the proposed solution does not satisfy either equation.

### Statement (ii):
**\(\sqrt{2}x + \sqrt{3}y = 0\) and \(\sqrt{3}x - \sqrt{8}y = 0\) has no solution.**

1. **Write the equations:**
\[ \text{Equation 1: } \sqrt{2}x + \sqrt{3}y = 0 \quad (3) \]
\[ \text{Equation 2: } \sqrt{3}x - \sqrt{8}y = 0 \quad (4) \]

2. **From Equation (4), express \(x\) in terms of \(y\):**
\[ \sqrt{3}x = \sqrt{8}y \implies x = \frac{\sqrt{8}}{\sqrt{3}}y = \frac{2\sqrt{2}}{\sqrt{3}}y \]

3. **Substitute \(x\) into Equation (3):**
\[ \sqrt{2}\left(\frac{2\sqrt{2}}{\sqrt{3}}y\right) + \sqrt{3}y = 0 \]
\[ \frac{2 \cdot 2}{\sqrt{3}}y + \sqrt{3}y = 0 \]
\[ \left(\frac{4}{\sqrt{3}} + \sqrt{3}\right)y = 0 \]
This implies \(y = 0\).

4. **Find \(x\) when \(y = 0\):**
\[ x = \frac{2\sqrt{2}}{\sqrt{3}}(0) = 0 \]

5. **Conclusion for Statement (ii):**
The statement is **False (F)** because the system has a unique solution \((x, y) = (0, 0)\).

### Statement (iii):
**The values of \(p\) and \(q\) for which the following system of equations \(2x - y = 5\) and \((p + q)x + (2p - q)y = 15\) has infinite number of solutions is \(p = 1\) and \(q = 5\).**

1. **Write the equations:**
\[ \text{Equation 1: } 2x - y = 5 \quad (5) \]
\[ \text{Equation 2: } (p + q)x + (2p - q)y = 15 \quad (6) \]

2. **For infinite solutions, the ratios must be equal:**
\[ \frac{2}{p + q} = \frac{-1}{2p - q} = \frac{5}{15} = \frac{1}{3} \]

3. **Set up the equations:**
- From \(\frac{2}{p + q} = \frac{1}{3}\):
\[ 2 \cdot 3 = p + q \implies p + q = 6 \quad (7) \]
- From \(\frac{-1}{2p - q} = \frac{1}{3}\):
\[ -1 \cdot 3 = 2p - q \implies 2p - q = -3 \quad (8) \]

4. **Solve the system of equations (7) and (8):**
- From (7): \(q = 6 - p\)
- Substitute into (8):
\[ 2p - (6 - p) = -3 \]
\[ 2p - 6 + p = -3 \implies 3p - 6 = -3 \implies 3p = 3 \implies p = 1 \]
- Substitute \(p = 1\) back into (7):
\[ 1 + q = 6 \implies q = 5 \]

5. **Conclusion for Statement (iii):**
The statement is **True (T)** because the values \(p = 1\) and \(q = 5\) yield infinite solutions.

### Final Summary:
- Statement (i): **F**
- Statement (ii): **F**
- Statement (iii): **T**