Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR PDF Download

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

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

Ans. 0

Solution. 

Integrating both sides with respect to x, we get

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

Then the value of                  (2010)

Ans. 4

Solution. 

The graph of this function is as below

Clearly f(x) is periodic with period 2

Also cos πx is periodic with period 2

∴ f ( x) cosπx is periodic with period 2

Q. 3.                (JEE Adv. 2014)

Ans. 2

Solution. 

Q. 4. Let f : R → R be a function defined by where [x] is the greatest integer less than or equal to x, if   , then the value of (4I – 1) is            (JEE Adv. 2015)

Ans. 0

Solution. 

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)

Q. 5. Let   for all x ∈ R and  a continuous function. F or   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. 

Q. 6. where tan^–1x takes only principal values, then the value of   is             (JEE Adv. 2015)

Ans. 9

Solution. 

Q. 7.  be a continuous odd function, which vanishes exactly at one point and f (1) = 1/2. Suppose that  
 then the value of                 (JEE Adv. 2015)

Ans. 7

Solution. 

f(t) being odd function

∴ Using L Hospital’s rule, we get

Q. 8. The total number of distinct x ∈ [0, 1] for which                  (JEE Adv. 2016)

Ans. 1

Solution. 

∴ f is decreasing on [0, 1]

Also f(0) = 1

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

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