Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced

# 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]

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## Mathematics for Airmen Group X

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

 1. What is a definite integral?
Ans. A definite integral is a mathematical concept used to calculate the accumulated value of a function over a given interval. It represents the area under the curve of the function within that interval.
 2. How is a definite integral calculated?
Ans. To calculate a definite integral, we need to find the antiderivative of the function and then evaluate it at the upper and lower limits of the interval. The difference between these two values gives us the value of the definite integral.
 3. What are the applications of definite integrals?
Ans. Definite integrals have various applications in different fields. Some common applications include calculating areas of irregular shapes, finding the average value of a function, determining displacement and velocity, finding the mass of an object with variable density, and analyzing population growth.
 4. How can definite integrals be used to calculate areas?
Ans. Definite integrals can be used to calculate areas by representing the area under a curve as the definite integral of the function that defines the curve. By integrating the function over a given interval, we can determine the area enclosed by the curve within that interval.
 5. Can definite integrals be used to solve real-world problems?
Ans. Yes, definite integrals are widely used to solve real-world problems in various fields such as physics, economics, engineering, and biology. They provide a mathematical tool to model and analyze real-world phenomena, allowing us to make predictions and understand complex systems.

## Mathematics for Airmen Group X

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