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Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE

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Q.1. Find the maximum area bounded by the curves y2 = 4ax, y = ax and y = x / a (1 ≤ a ≤ 2).

Ans. 84
Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE
The curves are y2 = 4ax and y = ax
At their point of intersection
a2x2 = 4ax
⇒ ax = 4, x = 0
x = 4  /  a
⇒ y = 4
i. e. Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE
Similarly for y2 = 4ax and y = x / a ,
Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE
⇒ x = 4a3
⇒ B(4a3, 4a2)
Area OAB = Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE
Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE
Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE
Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE
= 84

Q.2. Let f be a real valued function satisfying  Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE and Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE Find the area bounded by the curve y = f(x), the y–axis and the line y = 3. 

Ans. 3
Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE
Given, Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE
putting x = y =1, we get  f(1) = 0
Now, Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE
= Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE
⇒ f'(x)  = 3 x ⇒ f(x) = 3 lnx + c
Putting x =1 ⇒ c = 0 ⇒ f(x) = 3lnx = y  (say)
Required Area = Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE
= 3(e – 0) = 3e sq. units.

Q.3. Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE is equal to

Ans. 0
Given Limit = Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE
= Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE

Q.4. The value of Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE ,where [.] denotes the greatest integer function, is equal to;

Ans. 12
Let I = Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEEInteger Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE
= 1/3 (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8) = 12.

 

Q.5. If  Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE , then the value od P + Q + R is________.

Ans. 10
Put x + 7 = u4   ⇒ dx = 4u3 du
Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE
Rearranging like terms
Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE
⇒ P = 2, Q = 4, R = 4
⇒ P + Q + R = 10.

Q.6. If Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE  ,  then k is equal   s.

Ans. 1
Dividing the numerator and denominator by  x2, the   given  integral  becomes
Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE
Let Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE
Hence   k = 1.

Q.7. The value of Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE is ____________.

Ans. 9
Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE

Q.8.  If Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE, then Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE where  k  has the value equal to?

Ans. 9
Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE
Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE
Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE
Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE
⇒ k = 9

Q.9. Value of Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE is equal to?

Ans. 1
Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE
Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE
= 1

Q.10. Let  f be a function defined by Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE, where r = 3k, k∈ I then Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE is equal to? 

Ans. 5
Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE
Hence, f(x) is periodic with fundamental time period = 3
Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE 
= 50
Integer Answer Type Questions: Indefinite Integral - Notes | Study All Types of Questions for JEE - JEE
= 5

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