Consider the following two statement:
I. Any pair of consistent linear equations in two variables must have unique solutions.
II. There do not exist two consecutive integers, the sum of whose squares is 365.
Then

Options

A. Both I and II are true
B. both I and II are false
C. I is true and II false
D. I is false and II is true.

Explanation

To solve the question, we need to analyze both statements provided:

### Statement I:
"Any pair of consistent linear equations in two variables must have unique solutions."

1. **Understanding Consistent Linear Equations**:
- A pair of linear equations is said to be consistent if there is at least one solution. This means the lines represented by these equations intersect at least at one point.

2. **Types of Consistent Equations**:
- **Unique Solution**: If the two lines intersect at exactly one point, then there is a unique solution.
- **Infinitely Many Solutions**: If the two lines are coincident (i.e., they lie on top of each other), then there are infinitely many solutions.

3. **Conclusion for Statement I**:
- Therefore, the statement that "any pair of consistent linear equations in two variables must have unique solutions" is **false** because consistent equations can either have a unique solution or infinitely many solutions.

### Statement II:
"There do not exist two consecutive integers, the sum of whose squares is 365."

1. **Defining Consecutive Integers**:
- Let the first integer be \( x \) and the second consecutive integer be \( x + 1 \).

2. **Setting Up the Equation**:
- The sum of their squares can be expressed as:
\[ x^2 + (x + 1)^2 = 365 \]

3. **Expanding the Equation**:
- Expanding the equation gives:
\[ x^2 + (x^2 + 2x + 1) = 365 \]
- This simplifies to:
\[ 2x^2 + 2x + 1 = 365 \]

4. **Rearranging the Equation**:
- Rearranging gives:
\[ 2x^2 + 2x - 364 = 0 \]
- Dividing the entire equation by 2:
\[ x^2 + x - 182 = 0 \]

5. **Applying the Quadratic Formula**:
- Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- Here, \( a = 1, b = 1, c = -182 \).
- Calculate the discriminant:
\[ b^2 - 4ac = 1^2 - 4(1)(-182) = 1 + 728 = 729 \]
- Therefore, the roots are:
\[ x = \frac{-1 \pm \sqrt{729}}{2} = \frac{-1 \pm 27}{2} \]

6. **Finding the Values of x**:
- This gives two possible values:
- \( x = \frac{26}{2} = 13 \) (first integer)
- \( x + 1 = 14 \) (second integer)
- And the negative case:
- \( x = \frac{-28}{2} = -14 \) (first integer)
- \( x + 1 = -13 \) (second integer)

7. **Conclusion for Statement II**:
- Since we found two pairs of consecutive integers (13, 14) and (-14, -13) whose squares sum to 365, the statement "there do not exist two consecutive integers, the sum of whose squares is 365" is **false**.

### Final Conclusion:
Both statements are false.