Airforce X Y / Indian Navy SSR Exam  >  Airforce X Y / Indian Navy SSR Notes  >  Mathematics for Airmen Group X  >  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)

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

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

Ans. 0

Solution. 

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

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

Integrating both sides with respect to x, we get

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


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 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

Then the value of   Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR               (2010)

Ans. 4

Solution. 

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

The graph of this function is as below

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

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


Q. 3. Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR               (JEE Adv. 2014)

Ans. 2

Solution. 

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

Q. 4. Let f : R → R be a function defined by Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSRwhere [x] is the greatest integer less than or equal to x, if  Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR , 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 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
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 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

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


Q. 5. Let  Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR for all x ∈ R and Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR a continuous function. F or Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR  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 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR


Q. 6. Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSRwhere tan–1x takes only principal values, then the value of  Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR is             (JEE Adv. 2015)

Ans. 9

Solution. 

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

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

Q. 7. Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR 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 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR then the value of  Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR               (JEE Adv. 2015)

Ans. 7

Solution. 

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

f(t) being odd function

∴ Using L Hospital’s rule, we get

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

Q. 8. The total number of distinct x ∈ [0, 1] for which Integer Answer Type Questions: Definite Integrals and Applications of Integrals | JEE Advanced | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR                 (JEE Adv. 2016)

Ans. 1

Solution. 

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

∴ f is decreasing on [0, 1]

Also f(0) = 1

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

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

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

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

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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.
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