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]
327 docs|185 tests
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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 real-world problems? |
327 docs|185 tests
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