i) Find the two consecutive positive even integers, the sum of whose square is 340.
ii) Prove that there is a unique pair of consecutive odd positive integers such that sum of their squares is 290 find them.
iii) Find all the numbers which exceeds their square root by 12.
iv) Find the quadratic equation for which sum of the roots is 1 and sum of the squares of the roots is 13.

Answer

Let's solve the questions step by step.

### i) Find the two consecutive positive even integers, the sum of whose squares is 340.

1. **Let the first even integer be \( x \)**.
- The next consecutive even integer will be \( x + 2 \).

2. **Set up the equation based on the problem statement**:
\[ x^2 + (x + 2)^2 = 340 \]

3. **Expand the equation**:
\[ x^2 + (x^2 + 4x + 4) = 340 \]
\[ 2x^2 + 4x + 4 = 340 \]

4. **Rearrange the equation**:
\[ 2x^2 + 4x + 4 - 340 = 0 \]
\[ 2x^2 + 4x - 336 = 0 \]

5. **Divide the entire equation by 2**:
\[ x^2 + 2x - 168 = 0 \]

6. **Factor the quadratic equation**:
\[ (x + 14)(x - 12) = 0 \]

7. **Find the roots**:
\[ x + 14 = 0 \quad \Rightarrow \quad x = -14 \quad (\text{not positive}) \]
\[ x - 12 = 0 \quad \Rightarrow \quad x = 12 \quad (\text{positive}) \]

8. **Determine the two consecutive even integers**:
- The first integer is \( 12 \) and the second is \( 12 + 2 = 14 \).

**Final Answer**: The two consecutive positive even integers are **12 and 14**.