Chapter 10 : Differentiability - Rd Sharma Solutions for Class 12-science Maths CBSE

Your CBSE Class 12 syllabus for Maths consists of topics such as linear programming, vector quantities, determinants, etc. which lay the foundation for further education in science, engineering, management, etc. Studying differential equations will be useful for exploring subjects such as Physics, Biology, Chemistry, etc. where your knowledge can be applied for scientific investigations.

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Chapter 10 - Differentiability Exercise Ex. 10.1

Question 1

Solution 1

Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5

Solution 5

Question 6

Find whether the following functions is differentiable at

 x = 1 and x = 2 or not:

 

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Chapter 10 - Differentiability Exercise Ex. 10.2

Question 1
Solution 1

Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7

Examine the differentiability of the function f defined by

 

 

Solution 7

Question 8

Solution 8

Question 9

Solution 9


Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Chapter 10 - Differentiability Exercise MCQ

Question 1

Let f(x) = |x| and g(x) = |x3|, then

  1. f(x) and g(x) both are continuous at x = 0
  2. f(x) and g(x) both are differentiable at x = 0
  3. f(x) is differentiable but g(x) is not differentiable at x = 0
  4. f(x) and g(x) both are not differentiable at x = 0
Solution 1

Correct option: (a)

Absolute value function is continuous on R.

Question 2

The function f(x) = sin-1 (cos x) is

  1. discontinuous at x = 0
  2. continuous at x = 0
  3. differentiable at x = 0
  4. none of these
Solution 2

Correct option: (b)

  

Question 3

The set of points where the function f(x) = x|x| is differentiable is

  1. (-∞, ∞)
  2. (-∞, 0) (0, ∞)
  3. (0, ∞)
  4. [0, ∞]
Solution 3

Correct option: (a)

  

Question 4

  1. continuous at x = -2
  2. not continuous at x = -2
  3. differentiable at x = -2
  4. continuous but not derivable at x = -2
Solution 4

Correct option: (b)

  

Question 5

Let f(x) = (x + |x|) |x|. Then, for all x

  1. f is continuous
  2. f is differentiable for some x
  3. f ' is continuous
  4. f" is continuous
Solution 5

Correct option: (a), (c)

  

 

Question 6

The function f (x) = e-|x| is

  1. continuous everywhere but not differentiable at x =0
  2. continuous and differentiable everywhere
  3. not continuous at x =0
  4. none of these
Solution 6

Correct option: (a)

  

Question 7

The function f (x) =|cos x| is

  1. everywhere continuous and differentiable
  2. everywhere continuous but not differentiable at (2n+1)π /2, n Z
  3. neither continuous nor differentiable at (2n +1) π/2,n Z
  4. none of these
Solution 7

Correct option : (b)

  

Question 8

  1. continuous on [-1,1] and differentiable on (-1,1)
  2. continuous on [-1,1] and differentiable (-1,0) ϕ(0,1)
  3. continuous and differentiable on [-1,1]
  4. none of these
Solution 8

Correct option: (b)

  

Question 9

If f (x) = a |sin x| +b e|x|+c|x|3 and if f (x) is differentiable at x =0,then

  1. a = b = c = 0
  2. a = 0, b = 0; c R
  3. b = c = 0; a R
  4. c = 0 , a = 0 , b R
Solution 9

Correct option: (b)

  

Question 10

  1. has no limit
  2. is discontinuous
  3. is continuous but not differentiable
  4. is differentiable
Solution 10

Correct option: (b)

  

Question 11

If f (x)= |loge x|, then

  1. f' (1+) =1
  2. f' (1-) =1
  3. f' (1) =1
  4. f' (1) = -1
Solution 11

Correct option: (a), (b)

  

Question 12

If f (x) = loge |x|, then

  1. f(x) is continuous and differentiable for all x in its domain
  2. f (x) is continuous for all for all x in its domain but not differentiable at x = ± 1
  3. f (x) is neither continuous nor differentiable at x =  ± 1
  4. none of these
Solution 12

Correct option: (b)

  

Question 13

Solution 13

Correct option: (b)

  

Question 14

The function f(x) =x -[x], where [.] denotes the greatest integer function is

  1. continuous everywhere
  2. continuous at integer point only
  3. continuous at non-integer points only
  4. differentiable everywhere
Solution 14

Correct option: (c)

  

Question 15

  1. a = 2
  2. a = 1
  3. a =0
  4. a =1/2
Solution 15

Correct option: (d)

  

Question 16

Let f (x) =|sin x|. Then,

  1. f (x) is everywhere differentiable.
  2. f (x) everywhere continuous but not differentiable at x = n π, n Z

  1. None of these
Solution 16

Correct option: (b)

  

Question 17

Let f (x) =|cos x|. Then,

  1. f (x) is everywhere differentiable
  2. f (x) everywhere continuous but not differentiable at x = n π, n Z

  1. None of these
Solution 17

Correct option: (c)

  

Question 18

The function f (x) =1+|cos x| is

  1. Continuous no where
  2. Continuous everywhere
  3. Not differentiable at x =0
  4. Not differentiable at x =n π, n Z
Solution 18

Correct option: (b)

  

Question 19

The function f (x) = |cos x| is

  1. Differentiable at x = (2n+1) π/2, n Z
  2. Continuous but not differentiable at x =(2n+1) π/2, n Z
  3. Neither differentiable nor continuous at x = n π, n Z
  4. None of these
Solution 19

Correct option: (b)

  

Question 20

  1. Continuous as well differentiable for all x R
  2. Continuous for all x but not continuous at some x.
  3. Differentiable for all but not continuous at same x'.
  4. None of these
Solution 20

Correct option: (a)

  

Question 21

Let f (x) = ɑ + b |x|+c|x|4, where ɑ, b and c are real constants. Then f (x) is differentiable at x =0 , if

  1. a =0
  2. b =0
  3. c =0
  4. none of these
Solution 21

Correct option: (b)

  

Question 22

If f (x) = |3-x|+(3+x) , where (x) denotes the least integer greater than or equal to x, then f (x) is

  1. continuous and differentiable at x =3
  2. continuous but not differentiable at x = 3
  3. differentiable but not continuous at x =3
  4. neither differentiable nor continuous at x =3
Solution 22

Correct option: (d)

  

Question 23

  1. continuous as well as differentiable at x =0
  2. continuous but not differentiable at x = 0
  3. differentiable but not continuous at x =0
  4. none of these
Solution 23

Correct option: (d)

  

Question 24

  1. continuous and differentiable
  2. differentiable but not continuous
  3. continuous but not differentiable
  4. neither continuous nor differentiable
Solution 24

Correct option: (a)

  

Question 25

The set of point where the function f (x) = |x-3| cos x is differentiable

  1. R
  2. R - {3}
  3. (0,∞ )
  4. None of these
Solution 25

Correct option: (b)

  

Question 26

  1. Continuous at x =1
  2. Differentiable at x = -1
  3. Everywhere continuous
  4. Everywhere differentiable
Solution 26

Correct option: (b)

  

Chapter 10 - Differentiability Exercise Ex. 10VSAQ

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6
Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

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