RD SHARMA Solutions for Class 12-science Maths Chapter 15 - Mean Value Theorems

Page / Exercise

Chapter 15 - Mean Value Theorems Exercise Ex. 15.1

Question 1(i)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 1(i)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 1(ii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 1(ii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 1(iii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 1(iii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 1(iv)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 1(iv)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 1(v)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 1(v)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 1(vi)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 1(vi)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 2(i)

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

Solution 2(i)

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

Question 2(ii)

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

Solution 2(ii)

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

Question 2(iii)

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

Solution 2(iii)

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

Question 2(iv)

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

Solution 2(iv)

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

Question 2(v)

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

Solution 2(v)

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

Question 2(vi)

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

Solution 2(vi)

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

Question 2(vii)

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

Solution 2(vii)

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

Question 2(viii)

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

Solution 2(viii)

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

Question 3(i)

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

Solution 3(i)

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

Question 3(ii)

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

Solution 3(ii)

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

Question 3(iii)

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

Solution 3(iii)

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

Question 3(iv)

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

Solution 3(iv)

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

Question 3(v)

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

Solution 3(v)

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

Question 3(vi)

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

Solution 3(vi)

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

Question 3(vii)

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

Solution 3(vii)

Here,

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

Question 3(viii)

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

Solution 3(viii)

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

Question 3(ix)

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

Solution 3(ix)

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

Question 3(x)

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

Solution 3(x)

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

Question 3(xi)

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

Solution 3(xi)

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

Question 3(xii)

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

Solution 3(xii)

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

Question 3(xiii)

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

Solution 3(xiii)

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

Question 3(xiv)

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

Solution 3(xiv)

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 3(xv)

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

Solution 3(xv)

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

Question 3(xvi)

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

Solution 3(xvi)

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

Question 3(xvii)

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

Solution 3(xvii)

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

Question 3(xviii)

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

Solution 3(xviii)

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

x = 0 then y = 16

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

Question 8(i)

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

Solution 8(i)

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

x = 0, then y = 0

Therefore, the point is (0, 0)

Question 8(ii)

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

Solution 8(ii)

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

Question 8(iii)

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

Solution 8(iii)

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 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

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 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

Chapter 15 - Mean Value Theorems Exercise Ex. 15.2

Question 1(i)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 1(i)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 1(ii)

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

Solution 1(ii)

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

Question 1(iii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 1(iii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 1(iv)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 1(iv)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 1(v)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 1(v)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 1(vi)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 1(vi)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 1(vii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 1(vii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 1(viii)

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

Solution 1(viii)

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

Question 1(ix)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 1(ix)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 1(x)

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

Solution 1(x)

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

Question 1(xi)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 1(xi)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 1(xii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 1(xii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 1(xiii)

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

Solution 1(xiii)

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

Question 1(xiv)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 1(xiv)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 1(xv)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 1(xv)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Question 1(xvi)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Mean Value Theorems
Solution 1(xvi)
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

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

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 

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