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RD Sharma Solution CLass 12 Mathematics Chapter 15 Mean Value Theorems

Chapter 15 – Mean Value Theorems Exercise Ex. 15.1

Question 1
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 1
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 2
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 2
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 3
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 3
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 4
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 4
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 5
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 5
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 6
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 6
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 7

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 7

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 8

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 8

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 9

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 9

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 10

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 10

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 11

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 11

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 12

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 12

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 13

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 13

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 14

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 14

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 15

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 15

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 16

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 16

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 17

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 17

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 18

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 18

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 19

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 19

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 20

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 20

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 21

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 21

Here,

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 22

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 22

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 23

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 23

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 24

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 24

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 25

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 25

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 26

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 26

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 27

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 27

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 28

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 28

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 29

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 29

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 30

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 30

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 31

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 31

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 32

Verify Rolle’s theorem for function f(x) = sin x – sin 2x on [0, p]
on the indicated intervals.

Solution 32

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 33

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 33

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

x = 0 then y = 16

Therefore, the point on the curve is (0, 16)

Question 34

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 34

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

x = 0, then y = 0

Therefore, the point is (0, 0)

Question 35

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 35

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 36

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 36

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

x = 1/2, then y = – 27

Therefore, the point is (1/2, – 27)

Question 37
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 37
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 38

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 38

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 39

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 39

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Chapter 15 – Mean Value Theorems Exercise Ex. 15.2

Question 1
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 1
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 2

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 2

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 3
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 3
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 4
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 4
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 5
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 5
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 6
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 6
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 7
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 7
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 8

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 8

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 9
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 9
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 10

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 10

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 11
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 11
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 12
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 12
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 13

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 13

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 14
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 14
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 15
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 15
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 16
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 16
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 17

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 17

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 18

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 18

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 19

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 19

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 20
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 20
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 21
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 21
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 22
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 22
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 23

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 23

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 24
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 24
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 25

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 25

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 26

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 26

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Chapter 15 – Mean Value Theorems Exercise Ex. 15VSAQ

Question 1

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 1

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 2

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 2

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 3

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 3

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 4

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 4

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 5

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 5

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Chapter 15 – Mean Value Theorems Exercise MCQ

Question 1

If the polynomial equation a0xn + an-1xn – 1 + an – 2xn – 2 + ……+ a2x2 + a1x + a0 = 0 n being a positive integer, has two different real roots α and β, the equation n anxn-1 + (n – 1)an – 1xn – 2 + …. + a1 = 0 has

  1. exactly one root
  2. almost one root
  3. at least one root
  4. no root

Solution 1

Correct option: (c)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 2

If 4a + 2b + c = 0, then the equation 3ax2 + 2bx + c = 0 has at least one real root lying in the interval

  1. (0, 1)
  2. (1, 2)
  3. (0, 2)
  4. none of these

Solution 2

Correct option: (c)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 3

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

  1. 1
  2. Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
  3. 2
  4. none of these

Solution 3

Correct option: (b)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 4

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

  1. a < x1≤ b
  2. a ≤ x1< b
  3. a < x1< b
  4. a ≤ x1≤ b

Solution 4

Correct option: (c)

Using statement of Lagrange’s mean value theorem function is continuous on [a,b], differentiable on (a,b) then there exists c such that a < x1< b.

Question 5

Rolle’s theorem is applicable in case of ϕ(x) = asin x, a > 0 in

  1. any interval
  2. the interval [0, π]
  3. the interval (0, π/2)
  4. none of these

Solution 5

Correct option: (b)

ϕ(x) is continuous and differentiable function then using statement of Rolle’s theorem f(a)=f(b). Hence, here sin 0=0 also sin п=0. The answer is [0, Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems].

Question 6

The value of c in Rolle’s theorem when f(x) = 2x3 – 5x2 – 4x + 3, is x [1/3, 3]

  1. 2
  2. -1/3
  3. -2
  4. 2/3

Solution 6

Correct option: (a)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 7

When the tangent to the curve y = x log x is parallel to the chord joining the points (1,0) and (e, e), the value of x is

  1. e1/1 – e
  2. e(e – 1)(2e – 1)
  3. Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
  4. Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 7

Correct option: (a)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 8

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Solution 8

Correct answer: (c)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 9

The value of c in Lagrange’s mean value theorem for the function f(x) = x(x – 2) where x [1,2] is

  1. 1
  2. 1/2
  3. 2/3
  4. 3/2

Solution 9

Correct option: (d)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 10

The value of c in Rolle’s theorem for the function f(x) = x3 – 3x in the interval Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems is

  1. 1
  2. -1
  3. 3/2
  4. 1/3

Solution 10

Correct option: (a)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

Question 11

If f(x) = ex sin x in [0, π], then c in Rolle’s theorem is

  1. π/6
  2. π/4
  3. π/2
  4. 3π/4

Solution 11

Correct option: (d)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems

 

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