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
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