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Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

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Q. 1. Let f: R → R be a continuous function which satisfies            (2009)

Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Then the value of f (ln 5) is                           (2009)

Ans. 0

Solution. 

Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Integrating both sides with respect to x, we get

Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 2. For any real number x, let [x] denote the largest integer less than or equal to x. Let f be a real valued function defined on the interval [–10, 10] by

Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Then the value of   Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE               (2010)

Ans. 4

Solution. 

Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

The graph of this function is as below

Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Clearly f(x) is periodic with period 2

Also cos πx is periodic with period 2

∴ f ( x) cosπx is periodic with period 2

Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 3. Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE               (JEE Adv. 2014)

Ans. 2

Solution. 

Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q. 4. Let f : R → R be a function defined by Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEwhere [x] is the greatest integer less than or equal to x, if  Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE , then the value of (4I – 1) is            (JEE Adv. 2015)

Ans. 0

Solution. 

Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Also 0 < x2 < 1 ⇒ f(x2) = [x2] = 0
1 < x2 < 2 ⇒ f(x2) = [x2] = 1
2 < x2 < 3 ⇒ f(x2) = 0 (using definition of f)
3 < x2 < 4 ⇒ f(x2) = 0 (using definition of f)

Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 5. Let  Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE for all x ∈ R and Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE a continuous function. F or Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE  is the area of the region bounded by  x = 0, y = 0, y = f(x) and x = a, then f(0) is          (JEE Adv. 2015)

Ans. 3

Solution. 

Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 6. Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEwhere tan–1x takes only principal values, then the value of  Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE is             (JEE Adv. 2015)

Ans. 9

Solution. 

Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q. 7. Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE be a continuous odd function, which vanishes exactly at one point and f (1) = 1/2. Suppose that   Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE then the value of  Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE               (JEE Adv. 2015)

Ans. 7

Solution. 

Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

f(t) being odd function

∴ Using L Hospital’s rule, we get

Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q. 8. The total number of distinct x ∈ [0, 1] for which Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE                 (JEE Adv. 2016)

Ans. 1

Solution. 

Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

∴ f is decreasing on [0, 1]

Also f(0) = 1

Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

∴ f(x) crosses x-axis exactly once in [0, 1]

∴ f(x) = 0 has exactly one root in [0, 1]

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