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Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

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Q. 1. Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE                         (1987 - 2 Marks)

Ans. Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. Given that,

Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Expanding along R1, we get

Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Multiply the above by π/2 when n is even. We get

Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 2. Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE               (1988 - 2 Marks)

Where [ ] denotes the greatest integer function, equals .......

Ans. 2 - √2

Solution. 

Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 3. Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE                (1989 - 2 Marks)

Ans. 4

Solution.

Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 4. Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE                (1993 -  2 Marks)

Ans. Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. 

Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 5. Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE                   (1994 -  2 Marks)

Ans. 1/2

Solution.

Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Adding (1) and (2), we get

Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q. 6. If for nonzero  Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE then  Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE             (1996 - 2 Marks)

Ans.  Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Solution.   Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE            …(1)

Integrating both sides within the limits 1 to 2, we get 

Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Replacing  Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Integrating both sides within the limits 1 to 2, we get 

Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Eliminate  Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE between (2) and (3) by multiplying (2) by a and (3) by b and subtracting

Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 7. Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE              (1996 - 1 Mark)

Ans. π2

Solution.

Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE     …(2)

Adding (1) and (2) we get

Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE  … (3)

[ Using above property again]

Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE    ....(4)

Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Adding (3) and (4) we get

Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q. 8. The value of Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE                   (1997 - 2 Marks)

Ans. 2

Solution.

Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 9. Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEthen one of the possible values of k is .......                 (1997 - 2 Marks)

Ans. 16

Solution.

Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


True / False

Q. 1. The value of the integral Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEequal to a.             (1997 - 2 Marks)

Ans. T

Solution.

Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE                … (1)
Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE             … (2)
Adding (1) and (2), we get

Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

∴ The given statement is true.

The document Fill in the Blanks: Definite Integrals and Applications of Integrals | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE is a part of the JEE Course Maths 35 Years JEE Main & Advanced Past year Papers.
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