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 < x^{2} < 1 ⇒ f(x^{2}) = [x^{2}] = 0
1 < x^{2} < 2 ⇒ f(x^{2}) = [x2] = 1
2 < x^{2} < 3 ⇒ f(x^{2}) = 0 (using definition of f)
3 < x^{2} < 4 ⇒ f(x^{2}) = 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^{–1}x 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 xaxis exactly once in [0, 1]
∴ f(x) = 0 has exactly one root in [0, 1]
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1. What is a definite integral? 
2. How is a definite integral calculated? 
3. What are the applications of definite integrals? 
4. How can definite integrals be used to calculate areas? 
5. Can definite integrals be used to solve realworld problems? 

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