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Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals


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16 Questions MCQ Test Maths 35 Years JEE Main & Advanced Past year Papers | Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals

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Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals - Question 1

Let the definite integral be defined by the formula   For more accurate result for c ∈ (a, b), we can use  that for 

Q. 

Detailed Solution for Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals - Question 1

Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals - Question 2

Let the definite integral be defined by the formula   For more accurate result for c ∈ (a, b), we can use  that for 

Q.    then f(x) is of maximum degree

Detailed Solution for Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals - Question 2



[Using L’Hospital rule]

Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals - Question 3

Let the definite integral be defined by the formula   For more accurate result for c ∈ (a, b), we can use  that for 

Q.    and c is a point such that a < c < b, and (c, f(c)) is the point lying on the curve for which F(c) is maximum, then f '(c) is equal to

Detailed Solution for Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals - Question 3



∴ F (c) is max. at the point (c, f (c)) where F’ (c) = 0

(For 4-6). Given the implicit function y3 – 3y + x = 0 For x ∈ (–∞, –2) ∪ (2,∞) it is y = f (x) real valued differentiable function, and for x ∈ (–2, 2) it is y = g(x) real valued differentiable function.

Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals - Question 4

PASSAGE - 2 

Consider the functions defined implicitly by the equation y3 – 3y + x = 0 on various intervals in the real line. If x ∈(-∞, - 2) ∪ (2,∞) , the equation implicitly defines a unique real valued differentiable function y = f (x). If x ∈(-2, 2) , the equation implicitly defines a unique real valued differentiable function y = g(x) satisfying g(0) = 0.

Q. 

Detailed Solution for Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals - Question 4


Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals - Question 5

PASSAGE - 2 

Consider the functions defined implicitly by the equation y3 – 3y + x = 0 on various intervals in the real line. If x ∈(-∞, - 2) ∪ (2,∞) , the equation implicitly defines a unique real valued differentiable function y = f (x). If x ∈(-2, 2) , the equation implicitly defines a unique real valued differentiable function y = g(x) satisfying g(0) = 0.

Q. The area of the region bounded by the curve y = f (x), the x-axis, and the lines x = a and x = b, where -∞ < a <b <-2 , is

Detailed Solution for Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals - Question 5



Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals - Question 6

PASSAGE - 2 

Consider the functions defined implicitly by the equation y3 – 3y + x = 0 on various intervals in the real line. If x ∈(-∞, - 2) ∪ (2,∞) , the equation implicitly defines a unique real valued differentiable function y = f (x). If x ∈(-2, 2) , the equation implicitly defines a unique real valued differentiable function y = g(x) satisfying g(0) = 0.

Q. 

Detailed Solution for Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals - Question 6

For y = g(x), we have y3 – 3y + x = 0
⇒ [g (x) ]3 - 3[ g (x)] + x=0 ...(1)
Putting x = –x, we get
⇒ [g (-x)]3 - 3[ g (-x)] - x = 0 ...(2)
Adding equations (1) and (2) we get

For g(0) = 0, we should have g(x) + g(–x) = 0
[∵ From other factor we get g(0) = ±√3]

⇒ g(x) is an odd function

Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals - Question 7

PASSAGE - 3

Consider the function f : ( -∞,∞) →(-∞,∞) defined by

Q. Which of the following is true?

Detailed Solution for Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals - Question 7

⇒ (x2 + ax + 1) 2 f '(x) = 2a(x2- 1)
⇒ (x2 + ax +1)2 f "(x) + 2(x2 + ax+1)
(2 x + a) f '(x)=4ax ...(1)
Putting x = –1 in equation (1), we get
(2 - a2) f ''(-1) =-4a …(2)
Putting x = 1 in equation (1), we get
(2 + a)2 f "(1)=4a ...(3)
Adding equations (2) and (3), we get
(2 + a)2 f "(1) + (2 - a)2f "(-1)= 0

Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals - Question 8

PASSAGE - 3

Consider the function f : ( -∞,∞) →(-∞,∞) defined by

Q. Which of the following is true?

Detailed Solution for Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals - Question 8


f '(x) = 0 ⇒ x = –1, 1 are the critical points.

∴ x = – 1 is a point of local maximum
and x = 1 is a point of local minimum.

Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals - Question 9

PASSAGE - 3

Consider the function f : ( -∞,∞) →(-∞,∞) defined by

Q.  Which of the following is true?

Detailed Solution for Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals - Question 9

Now g '(x) >0 for e2x - 1> 0 ⇒ x > 0
and g '(x) <0 for e2x - 1 < 0 ⇒ x<0
∴ g'(x) is negative on (–∞ , 0) and posit ive on (0, ∞)

Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals - Question 10

PASSAGE - 4

f (x) = 1 + 2x + 3x2 + 4x3.
Let s be the sum of all distinct real roots of f (x) and let t = |s|.

Q. The real numbers lies in the interval

Detailed Solution for Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals - Question 10

f ( x) = 4x3 + 3x2 + 2x+1
∵ f (x) is a cubic polynomial
∴ It has at least one real root.
Also f '(x) =12x2 + 6x+ 2 = 2(6x2 +3x+1) 

∴ f (x) is strictly increasing function
⇒ There is only one real root of f (x) = 0

∴ Real root lies between   and hence 

Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals - Question 11

PASSAGE - 4

f (x) = 1 + 2x + 3x2 + 4x3.
Let s be the sum of all distinct real roots of f (x) and let t = |s|.

Q. The area bounded by the curve y = f (x) and the lines x = 0, y = 0 and x = t, lies in the interval

Detailed Solution for Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals - Question 11

y = f (x), x = 0,y= 0 and x = t bounds the area as shown in the figure

∴ Required area is given by

 

Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals - Question 12

PASSAGE - 4

f (x) = 1 + 2x + 3x2 + 4x3.
Let s be the sum of all distinct real roots of f (x) and let t = |s|.

Q. The function f'(x) is

Detailed Solution for Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals - Question 12

f '(x) = 2(6x2 + 3x+ 1)
f ''(x) = 6 4x+ 1) ⇒ Critical point x = – 14


Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals - Question 13

PASSAGE - 5

Given that for each dt exists. Let this limit be g(a). In addition, it is given that the function g(a) is differentiable on (0, 1).

Q. The value of  

Detailed Solution for Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals - Question 13




Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals - Question 14

PASSAGE - 5

Given that for each dt exists. Let this limit be g(a). In addition, it is given that the function g(a) is differentiable on (0, 1).

Q. 

Detailed Solution for Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals - Question 14



*Multiple options can be correct
Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals - Question 15

PASSAGE - 6

be a thrice differentiable function. Suppose that F(1) = 0, F(3) = –4 and F(x) < 0 for 

Q. The correct statement(s) is(are)

Detailed Solution for Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals - Question 15

f(x) = xF(x) ⇒ f ' (x) = F(x) + xF '(x)

f(2) = 2F(2) < 0,

(Q∵F '(x) < 0 ⇒ F is decreasing on 

F(3) = –4)
f '(x) = F(x) + x F '(x)

For the same reason given above and F '(x) < 0 given.


∴ f '(x) ≠ 0, x∈(1, 3)

*Multiple options can be correct
Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals - Question 16

PASSAGE - 6

be a thrice differentiable function. Suppose that F(1) = 0, F(3) = –4 and F(x) < 0 for 

Q. 

Detailed Solution for Test: Comprehension Based Questions: Definite Integrals and Applications of Integrals - Question 16



    ...(i)


⇒ 9(f ' (3) – F(3)) – (f ' (1) – F(1)) = 4
⇒ 9f ' (3) – 9 × (–4) – f ' (1) + 0 = 4
⇒ 9f ' (3) – f ' (1) + 32 = 0

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