# RD Sharma Solution CLass 12 Mathematics Chapter 15 Mean Value Theorems

## Chapter 15 – Mean Value Theorems Exercise Ex. 15.1

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

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Here,

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

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

Solution 32

Question 33

Solution 33

x = 0 then y = 16

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

Question 34

Solution 34

x = 0, then y = 0

Therefore, the point is (0, 0)

Question 35

Solution 35

Question 36

Solution 36

x = 1/2, then y = – 27

Therefore, the point is (1/2, – 27)

Question 37
Solution 37
Question 38

Solution 38

Question 39

Solution 39

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

Question 14
Solution 14
Question 15
Solution 15
Question 16
Solution 16
Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20
Solution 20
Question 21
Solution 21
Question 22
Solution 22
Question 23

Solution 23

Question 24
Solution 24
Question 25

Solution 25

Question 26

Solution 26

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

## 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)

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)

Question 3

1. 1
2. 2
3. none of these

Solution 3

Correct option: (b)

Question 4

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, ].

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)

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)
Solution 7

Correct option: (a)

Question 8

Solution 8

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)

Question 10

The value of c in Rolle’s theorem for the function f(x) = x3 – 3x in the interval  is

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

Solution 10

Correct option: (a)

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)

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