Relations

Step 1: Given Function
The given function is , and we need to determine whether the function is increasing or decreasing in the specified intervals.

Step 2: First Derivative Calculation
To determine whether is increasing or decreasing in a given interval, we need to find the first derivative . Using the quotient rule for differentiation, we have:
Let's simplify the numerator: Expanding both terms: Subtracting the two expressions: Thus, the derivative becomes:

Step 3: Analyze the Sign of the Derivative
To determine whether the function is increasing or decreasing in an interval, we need to analyze the sign of . The denominator is always positive since the quadratic has no real roots and is always positive for all . Therefore, the sign of depends on the numerator , which can be factored as: The critical points are the values of where , which occur when or .

Step 4: Interval Analysis for Statement A
We are asked to check if is decreasing in the interval . To do this, we need to check the sign of in this interval. Since the critical points are and , we evaluate the sign of in the interval . Choosing a test point, say , we find: Simplifying: Since , the function is decreasing in the interval . Hence, statement (A) is true.

Step 5: Interval Analysis for Statement B
We are asked to check if is increasing in the interval . Again, we check the sign of in this interval. The critical points and are not in this interval, so we check the sign of for : Simplifying: Since , the function is increasing in the interval . Hence, statement (B) is true.

Final Answer:
Both statements (A) and (B) are correct:
- (A) is decreasing in the interval .
- (B) is increasing in the interval .

Step 1: Understand the set and the relation

- The set is , which has 6 elements.

- A relation on this set is a subset of the Cartesian product , i.e., a set of ordered pairs where .

- The total number of possible ordered pairs is .

- The relation must be:

\qquad - Reflexive: For every element , the pair must be in . So, must include , which are 6 pairs.

\qquad - Symmetric: If , then . This means that off-diagonal pairs (where ) come in pairs: if is in , so is .
- contains exactly 10 elements (i.e., 10 ordered pairs).

- We need to find the number of elements in , where is the set of all such relations . Step 2: Count the elements in

- Since is reflexive, it must contain the 6 diagonal pairs: .

- This accounts for 6 of the 10 elements in .

- The remaining elements in are off-diagonal pairs (i.e., pairs where ).

- Since has 10 elements total, the number of off-diagonal pairs is:
- Because is symmetric, off-diagonal pairs come in pairs: if is in , then must also be in . So, 4 off-diagonal pairs correspond to choosing pairs (where , and the pair represents both and ).

- Thus, the number of unordered pairs (where ) is:
So, each relation consists of:

- The 6 reflexive pairs (fixed due to reflexivity),

- 2 unordered pairs , each contributing the ordered pairs and , for a total of 4 ordered pairs.
This gives a total of ordered pairs, which satisfies the condition.
Step 3: Count the number of unordered pairs

- We need to choose 2 unordered pairs where , and .

- The number of ways to choose an unordered pair (where ) from a set of 6 elements is the number of ways to choose 2 elements from 6, which is given by the combination formula :
- So, there are 15 possible unordered pairs:
.
Step 4: Choose 2 unordered pairs for

- We need to select 2 unordered pairs from these 15 possible pairs to form .

- The pairs must be distinct (e.g., we can’t choose twice), because each pair corresponds to the distinct ordered pairs and , and is a set (no repeated elements).

- The number of ways to choose 2 distinct unordered pairs from 15 is:
Step 5: Define and find its size

- is the set of all relations on that are reflexive, symmetric, and have exactly 10 elements.

- Each is uniquely determined by choosing 2 unordered pairs , then including the corresponding ordered pairs and , along with the 6 reflexive pairs.

- From Step 4, the number of ways to choose 2 unordered pairs is 105.

- Thus, the number of such relations —which is the number of elements in —is 105.
Final Answer: The number of elements in is . Correct Answer Correct Answer: 105