Sets

We are given a set containing 5 odd numbers. For each element in the set, we have two choices: either include it in the subset or exclude it. Since there are 5 elements, the total number of subsets is: However, we need to exclude the empty set, which does not contain any odd numbers. Therefore, the number of subsets of that contain only odd numbers is: Thus, the correct answer is (A) 31.

Given function:

Step 1: Domain of
For a square root function to be defined, the expression inside the root must be non-negative:
Step 2: Domain of
For the logarithmic term to be defined and non-zero in the denominator:

Step 3: Combine all conditions
So, domain is:

Correct Answer:

We are given the following equations:

Next, we differentiate both sides of the first equation with respect to :

Using the product rule, we get:

Now, we differentiate both sides of the second equation with respect to :

Using the product rule, we get:

Now, to find , we divide :

Substituting the expressions for and , we get:

Next, we substitute the values of and at (1,1) into the equation to evaluate :

Dividing the second equation by the first equation, we get:

Simplifying the expression for using trigonometric identities:

Since , we substitute:

Therefore, at (1,1), is equal to 0, which corresponds to option (A) 0.

1. Understand the problem:

We have set A = {2, 3, ..., 18} and a relation R defined on A × A such that (a, b)R(c, d) if and only if ad = bc. We need to find the number of ordered pairs in the equivalence class of (3, 2).

2. Find the equivalence class of (3, 2):

All pairs (x, y) must satisfy 3y = 2x ⇒ x/y = 3/2. So we need all pairs in A × A where the ratio x/y = 3/2.

3. List all valid pairs:

Possible pairs (x, y) where both x and y ∈ A and x/y = 3/2:

(3, 2), (6, 4), (9, 6), (12, 8), (15, 10), (18, 12)

4. Count the pairs:

There are 6 such ordered pairs.

Correct Answer: (C) 6