# Chapter 2: Miscellaneous Exercise

Find the value of the following:(Class 12 ncert solution math chapter 2 Miscellaneous)

Question 1: Find the value of .

Solution:

Question 2: Find the value of .

Solution:

Question 3: Prove that .

Solution: Let

Then,

Therefore,

Thus,

Question 4: Prove that .

Solution: Let

Then,

Therefore,

Now, let

Then,

Therefore,

Thus, by using (1) and (2)

Question 5: Prove that .

Solution: Let

Then, Therefore,

—(i)

Now, let

Then,

Therefore,

—(ii)

Thus, by using (1) and (2)

Now, let

Then,

Therefore,

Hence proved.

Question 6: Prove that .

Solution: Let

Then,

Therefore,

Now, let

Then,

Therefore,

Now, let

Then,

Therefore,

Thus, by using (1) and (2)

Question 7: Prove that .

Solution: Let

Then,

Therefore,

Now, let

Then,

Therefore,

Thus, by using (1) and (2)

Question 8: Prove that

Solution:

Question 9: Prove that .

Solution: Let

Then,

Therefore,

Thus,

Question 10: Prove that .

Solution:

Thus,

Question 11: Prove that

Solution: Let

Thus,

Question 12: Prove that

Solution:

Now, let

Therefore,

Thus, by using (1) and (2)

Hence proved.

Question 13: Solve .

Solution: It is given that

Since,

Hence, Therefore,

Question 14: Solve

Solution: Since

Hence,

Question 15: Solve is equal to

(A)

(B)

(C)

(D)

Solution: Let

Therefore,

Now, let

Therefore,

Hence,

Thus,

Thus, the correct option is .

Question 16: Solve: , then is equal to

(A)

(B)

(C) 0

(D)

Solution: It is given that

Let

Hence,

From equation (1), we have

Put

Therefore,

When , it does not satisfy the equation.

Hence, is the only solution

Thus, the correct option is .

Question 17: Solve is equal to

(A)

(B)

(C)

(D)

Solution:Â  Â

Thus, the correct option is C.