NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.3 - Continuity and Differentiability
NCERT Solutions for Class 12 Maths Chapter 5 Miscellaneous Exercise: This page Consist of detail step by step NCERT Solutions For Class 12 Maths Chapter 5-Continuity and Differentiability miscellaneous exercise.
Krati Saraswat16 Jan, 2024
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NCERT Solutions for Class 12 Maths Chapter 5 Miscellaneous Exercise (Continuity and Differentiability)
NCERT Solutions for Class 12 Maths Chapter 5 Miscellaneous Exercise of Continuity and Differentiability is prepared by academic team of pw. We have prepared NCERT Solutions for all exercise of chapter-5. Given below is step by step solutions of all questions given in NCERT Solutions for Class 12 Maths Chapter 5 Miscellaneous Exercise of Continuity and Differentiability.NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.1
NCERT Solutions for Class 12 Maths Chapter 5 Miscellaneous Exercise Overview
These crucial subjects are covered in NCERT Solutions for Class 12 Maths Chapter 5. In order to completely comprehend the concepts presented in the chapter and make the best use of the offered answers, students are recommended to go over each topic in great detail. The Physics Wallah teachers have been working hard to help students comprehend the concepts discussed in this chapter, and the results are these solutions. Students should easily achieve excellent exam outcomes after reviewing and practicing these solutions.NCERT Solutions for Class 12 Maths Chapter 5 Miscellaneous Exercise PDF
Our team of teachers at Physics Wallah has developed thorough solutions for NCERT Solutions for Class 12 Maths Chapter 5 in order to help students understand and apply chapter concepts. The purpose of these questions is to help students comprehend explanations. You may obtain the NCERT Solutions for Class 12 Maths, Chapter 5 PDF by clicking on this link:NCERT Solutions Class 12 Maths Chapter 5 PDF Download Link
NCERT Solutions for Class 12 Maths Chapter 5 Miscellaneous Exercise
Solve The Following Questions of NCERT Solutions for Class 12 Maths Chapter 5 Miscellaneous Exercise of Continuity and Differentiability:
Question 1.
Solution :
Let
Using chain rule, we obtain
NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.2
Question 2.
Solution :
Let
NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.3
Question 3 .
Solution :
Let,
Taking logarithm on both the sides, we obtain
log y = 3 cos 2x log(5x)
Differentiating both sides with respect to x, we obtain
NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.4
Question 4.
Solution :
Let,
Using chain rule, we obtain
NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.5
Question 5.
Solution :
Let y =
NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.6
Question 6 .
Solution :
Let ,y =
.......(1)
NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.7
Question 7.
Solution :
Let, y =
Taking logarithm on both the sides, we obtain
log y = log x. log (log x)
Differentiating both sides with respect to x, we obtain
NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.8
Question 8 . Differentiate w.r.t. x the function cos (a cos x + b sin x), for some constant a and b. Solution : Let, y = cos (a cos x + b sin x) By using chain rule, we obtain
Question
9.
Solution :
Let, y =
Taking logarithm on both the sides, we obtain
Question
10.
, for some fixed a> 0 and x > 0
Solution :
Let y =
Since a is constant, aa is also a constant.
∴ ds/dx = 0 .....(5)
From (1), (2), (3), (4), and (5), we obtain
Question
11.
, for x > 3
Solution :
Question
12. Find dy/dx , if
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Solution :
Question 1
3. Find
Solution :
Question
14. If
Solution :
It is given that,
Differentiating both sides with respect to x, we obtain
Hence, proved.
Question
15.
Solution :
It is given that,
Differentiating both sides with respect to x, we obtain
= - c
which is constant and independent of a and b
Hence, proved.
Question
16. If cos y = x cos (a + y), with cos a ≠ ± 1, prove that prove that
Solution :
It is given cos y = x cos (a + y)
Hence, proved.
Question
17. If x = a (cos t + t sin t) and y = a (sin t – t cos t), find
Solution :
It is given that, x = a(cost + tsin t) and y = a (sin t - t cost)
Question
18. If f (x) = |x|
3
show that f''(x) exists for all real x, and find it.
Solution :
It is known that,
Therefore, when x ≥ 0, f(x) = |x|
3
= x
3
In this case, f'(x) = 3x
2
and hence, f''(x) = 6x
When x < 0, f(x) = |x|
3
= (-x
3
) = -x
3
In this case, f'(x) = -3x
2
and hence, f''(x) = -6x
Thus, for f(x) = |x|
3
, f''(x) exists for all real x and is given by,
Question
19. Using mathematical induction prove that
for all positive integers n.
Solution :
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.
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.
Hence, proved.
Question
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.
Solution :
sin (A + B) = sin A cos B + cos A sin B
Differentiating both sides with respect to x, we obtain
Question
21. Does there exist a function which is continuos everywhere but not differentiable at exactly two points? Justify your answer ?
Solution :
y={|x| −∞< x ≤ 1
2−x 1≤ x ≤ ∞
It can be seen from the above graph that, the given function is continuos everywhere but not differentiable at exactly two points which are 0 and 1.
Question
22. If
, prove that
Solution :
Question
23. If
, show that
Solution :
It is given that,
NCERT Solutions for Class 12 Maths Chapter 5 FAQs
What is the relationship between continuity and differentiability?
The relationship between continuous functions and differentiability is- all differentiable functions are continuous but not all continuous functions are differentiable.
What is important in continuity and differentiability class 12?
In continuity and differentiability class 12, we will learn important concepts of differentiability, continuity, and relationship between them.
What are the properties of continuity and differentiability?
Continuity of a function is the characteristic of a function by virtue of which, the graphical form of that function is a continuous wave. A differentiable function is a function whose derivative exists at each point in its domain.
What is the use of continuity in real life?
The water flow in the rivers is continuous. The flow of time in human life is continuous i.e. you are getting older continuously. And so on. Similarly, in mathematics, we have the notion of the continuity of a function.
Is continuity sufficient for differentiability?
In particular, any differentiable function must be continuous at every point in its domain. The converse does not hold: a continuous function need not be differentiable.
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