SOLUTIONS TO NCERT EXERCISE: CLASS XII: MATHEMATICSCOMPILED BY : M.SRINIVASAN, PGT(MATHS), ZIET, MUMBAI

MISC. EXERCISE

Q.1 Find derivative:

Using chain rule, we obtain

Q.2 Find derivative:

Q.3 Find the derivative:

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

Q.4 Find derivative of :

SOLUTIONS TO NCERT EXERCISE: CLASS XII: MATHEMATICSCOMPILED BY : M.SRINIVASAN, PGT(MATHS), ZIET, MUMBAI

Using chain rule, we obtain

Q. 5 Find the derivative of :

SOLUTIONS TO NCERT EXERCISE: CLASS XII: MATHEMATICSCOMPILED BY : M.SRINIVASAN, PGT(MATHS), ZIET, MUMBAI

Q.6 Find derivative of :

Therefore, equation (1) becomes

SOLUTIONS TO NCERT EXERCISE: CLASS XII: MATHEMATICSCOMPILED BY : M.SRINIVASAN, PGT(MATHS), ZIET, MUMBAI

Q.7 Find derivative of :

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

Q.8 Find derivative of : , for some constant a and b

By using chain rule, we obtainQ. 9 Find derivative:

Taking logarithm on both the sides, we obtain

SOLUTIONS TO NCERT EXERCISE: CLASS XII: MATHEMATICSCOMPILED BY : M.SRINIVASAN, PGT(MATHS), ZIET, MUMBAI

Differentiating both sides with respect to x, we obtain

Q.10 Find the derivative: , for some fixed and

Differentiating both sides with respect to x, we obtain

Differentiating both sides with respect to x, we obtain

SOLUTIONS TO NCERT EXERCISE: CLASS XII: MATHEMATICSCOMPILED BY : M.SRINIVASAN, PGT(MATHS), ZIET, MUMBAI

s = aa

Since a is constant, aa is also a constant.

∴From (1), (2), (3), (4), and (5), we obtain

Q.11 Find derivative: , for 

Differentiating both sides with respect to x, we obtain

Differentiating with respect to x, we obtain

SOLUTIONS TO NCERT EXERCISE: CLASS XII: MATHEMATICSCOMPILED BY : M.SRINIVASAN, PGT(MATHS), ZIET, MUMBAI

Also,

Differentiating both sides with respect to x, we obtain

Substituting the expressions of  in equation (1), we obtain

Q.12 Find , if

Q.13 Find , if

SOLUTIONS TO NCERT EXERCISE: CLASS XII: MATHEMATICSCOMPILED BY : M.SRINIVASAN, PGT(MATHS), ZIET, MUMBAI

Q.14 If , for, −1 < x <1, prove that

It is given that,

Differentiating both sides with respect to x, we obtain

SOLUTIONS TO NCERT EXERCISE: CLASS XII: MATHEMATICSCOMPILED BY : M.SRINIVASAN, PGT(MATHS), ZIET, MUMBAI

Q.15 If , for some   prove that

 is a constant independent of a and b.

It is given that,

Differentiating both sides with respect to x, we obtain

SOLUTIONS TO NCERT EXERCISE: CLASS XII: MATHEMATICSCOMPILED BY : M.SRINIVASAN, PGT(MATHS), ZIET, MUMBAI

Q.16 If with prove that

SOLUTIONS TO NCERT EXERCISE: CLASS XII: MATHEMATICSCOMPILED BY : M.SRINIVASAN, PGT(MATHS), ZIET, MUMBAI

Then, equation (1) reduces to

Hence, proved.

Q.17 If and , find

Q.18 If , show that  exists for all real x, and find it.

SOLUTIONS TO NCERT EXERCISE: CLASS XII: MATHEMATICSCOMPILED BY : M.SRINIVASAN, PGT(MATHS), ZIET, MUMBAI

It is known that, 

Therefore, when x ≥ 0, 

In this case,   and hence, 

When x < 0, 

In this case,   and hence, 

Thus, for ,  exists for all real x and is given by,

Q.19 Using mathematical induction prove that for all positive integers n.

For n = 1,

∴P(n) is true for n = 1

Let P(k) is true for some positive integer k.

That is, 

It has to be proved that P(k + 1) is also true.

SOLUTIONS TO NCERT EXERCISE: CLASS XII: MATHEMATICSCOMPILED BY : M.SRINIVASAN, PGT(MATHS), ZIET, MUMBAI

Thus, P(k + 1) is true whenever P (k) is true.

Therefore, by the principle of mathematical induction, the statement P(n) is true for every positive integer n.

Q.20 Using the fact that sin (A + B) = sin A cos B + cos A sin B and the differentiation, obtain the sum formula for cosines.

Differentiating both sides with respect to x, we obtain

Q.22 If, prove that

SOLUTIONS TO NCERT EXERCISE: CLASS XII: MATHEMATICSCOMPILED BY : M.SRINIVASAN, PGT(MATHS), ZIET, MUMBAI

Thus, 

Q.23 If , show that 

It is given that,

SOLUTIONS TO NCERT EXERCISE: CLASS XII: MATHEMATICSCOMPILED BY : M.SRINIVASAN, PGT(MATHS), ZIET, MUMBAI