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Test: MCQs (One or More Correct Option): Definite Integrals and Applications of Integrals | JEE Advanced


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13 Questions MCQ Test Maths 35 Years JEE Main & Advanced Past year Papers | Test: MCQs (One or More Correct Option): Definite Integrals and Applications of Integrals | JEE Advanced

Test: MCQs (One or More Correct Option): Definite Integrals and Applications of Integrals | JEE Advanced for JEE 2022 is part of Maths 35 Years JEE Main & Advanced Past year Papers preparation. The Test: MCQs (One or More Correct Option): Definite Integrals and Applications of Integrals | JEE Advanced questions and answers have been prepared according to the JEE exam syllabus.The Test: MCQs (One or More Correct Option): Definite Integrals and Applications of Integrals | JEE Advanced MCQs are made for JEE 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: MCQs (One or More Correct Option): Definite Integrals and Applications of Integrals | JEE Advanced below.
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Test: MCQs (One or More Correct Option): Definite Integrals and Applications of Integrals | JEE Advanced - Question 1

Detailed Solution for Test: MCQs (One or More Correct Option): Definite Integrals and Applications of Integrals | JEE Advanced - Question 1


Differentiating both sides w.r.t. x,

*Multiple options can be correct
Test: MCQs (One or More Correct Option): Definite Integrals and Applications of Integrals | JEE Advanced - Question 2

Let f(x) = x – [x], for every real number x, where [x] is the integral part of x. Then 

Detailed Solution for Test: MCQs (One or More Correct Option): Definite Integrals and Applications of Integrals | JEE Advanced - Question 2

[∵ x is an odd function]

Thus, putting value in equation (1) we get 

*Multiple options can be correct
Test: MCQs (One or More Correct Option): Definite Integrals and Applications of Integrals | JEE Advanced - Question 3

For which of the following values of m, is the area of the region bounded by the curve y = x –  x2 and the line y = mx equals 9/2?

Detailed Solution for Test: MCQs (One or More Correct Option): Definite Integrals and Applications of Integrals | JEE Advanced - Question 3


The two curves meet at

or (1 - m)3 = 27 ,
∴ m = -2 

But if m >1 then 1– m is – ive, then 

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Test: MCQs (One or More Correct Option): Definite Integrals and Applications of Integrals | JEE Advanced - Question 4

Let f (x) be a non-constant twice differentiable function definied on ( -∞,∞) such th at f (x) = f (1 – x) and 

Detailed Solution for Test: MCQs (One or More Correct Option): Definite Integrals and Applications of Integrals | JEE Advanced - Question 4

∴f (x) is a non constant twice differentiable function such that f (x) = f (1– x)  ⇒ f '(x) = – f ' (1 – x)    ...(1)

but given that 
Hence, f '(x) satisfies all conditions of Rolle's theorem for   So there exists at least one point   and at least one point 

Such that
f "(C1) = 0 and f "(C2) = 0 

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Test: MCQs (One or More Correct Option): Definite Integrals and Applications of Integrals | JEE Advanced - Question 5

Area of the region bounded by the curve y = ex and lines x = 0 and y = e is

Detailed Solution for Test: MCQs (One or More Correct Option): Definite Integrals and Applications of Integrals | JEE Advanced - Question 5

The area bounded by the curve y = ex and lines x = 0 and y = e is as shown in the graph.



Also required area


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Test: MCQs (One or More Correct Option): Definite Integrals and Applications of Integrals | JEE Advanced - Question 6

Detailed Solution for Test: MCQs (One or More Correct Option): Definite Integrals and Applications of Integrals | JEE Advanced - Question 6




Adding equations (1) and (2), we get
    [as integrand is an even function]





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Test: MCQs (One or More Correct Option): Definite Integrals and Applications of Integrals | JEE Advanced - Question 7

The value(s) of  

Detailed Solution for Test: MCQs (One or More Correct Option): Definite Integrals and Applications of Integrals | JEE Advanced - Question 7

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Test: MCQs (One or More Correct Option): Definite Integrals and Applications of Integrals | JEE Advanced - Question 8

Let f be a real-valued function defined on the interval (0, ∞) by  Then which of the following statement(s) is (are) true?

Detailed Solution for Test: MCQs (One or More Correct Option): Definite Integrals and Applications of Integrals | JEE Advanced - Question 8

We have



and f '(x) has finite  continuous

Which does not exist at the points where

∴ f '(x) is not differentiable.
∴ (a) is false but (b) is true



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Test: MCQs (One or More Correct Option): Definite Integrals and Applications of Integrals | JEE Advanced - Question 9

Let S be the area of the region enclosed by , y = 0, x = 0 and x = 1; then

Detailed Solution for Test: MCQs (One or More Correct Option): Definite Integrals and Applications of Integrals | JEE Advanced - Question 9

First of all let us draw a rough sketch of y = e–x.
At x = 0, y = 1 and at x = 1, y = 1/e


∴ is decreasing on (0, 1)
Hence its graph is as shown in figure given below

Now, S = area exclosed by curve = ABRO

and area of rectangle ORBM = 1/e 

Now S < area of rectangle APSO + area of rectangle CSRN

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Test: MCQs (One or More Correct Option): Definite Integrals and Applications of Integrals | JEE Advanced - Question 10

The option(s) with the values of a and L that satisfy the following equation is(are) 

Detailed Solution for Test: MCQs (One or More Correct Option): Definite Integrals and Applications of Integrals | JEE Advanced - Question 10





where ‘a’ can take any even
value.

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Test: MCQs (One or More Correct Option): Definite Integrals and Applications of Integrals | JEE Advanced - Question 11

Let f(x) = 7tan8x + 7tan6x – 3tan4x – 3tan2x for all  Then the correct expression(s) is(are)

Detailed Solution for Test: MCQs (One or More Correct Option): Definite Integrals and Applications of Integrals | JEE Advanced - Question 11

f(x) = 7 tan8x + 7tan6x – 3tan4x – 3tan2x                
= (7tan4x – 3) (tan4x + tan2x)                
= (7tan6x – 3tan2x) sec2x



= 1/12

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Test: MCQs (One or More Correct Option): Definite Integrals and Applications of Integrals | JEE Advanced - Question 12

then the possible values of m and M are

Detailed Solution for Test: MCQs (One or More Correct Option): Definite Integrals and Applications of Integrals | JEE Advanced - Question 12





∴ Only (d) is the correct option.

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Test: MCQs (One or More Correct Option): Definite Integrals and Applications of Integrals | JEE Advanced - Question 13

all x > 0. Then

Detailed Solution for Test: MCQs (One or More Correct Option): Definite Integrals and Applications of Integrals | JEE Advanced - Question 13







∴ f is an incr easing function.

Hence (b) and (c) are the correct options.

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