# SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 chapter 2 - Definite Integration [Latest edition]

#### Chapters ## Chapter 2: Definite Integration

Q.1Q.2Q.3Q.4Q.5Q.6
Q.1

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 Chapter 2 Definite Integration Q.1

#### MCQ [1 Mark]

Q.1 | Q 1

Choose the correct alternative:

int_2^3 x^4  "d"x =

• 1/2

• 5/2

• 5/211

• 211/5

Q.1 | Q 2

Choose the correct alternative:

int_0^"a" 3x^5  "d"x = 8, then a =

• 2

• 0

• 8/3

• a

Q.1 | Q 3

Choose the correct alternative:

int_4^9 ("d"x)/sqrt(x) =

• 9

• 4

• 2

• 0

Q.1 | Q 4

Choose the correct alternative:

int_0^2 "e"^x  "d"x =

• e2 – 1

• 1 – e2

• e – 1

• 1 – e

Q.1 | Q 5

Choose the correct alternative:

int_(-2)^3 1/(x + 5)  "d"x =

• log (3/8)

• log (8/3)

• log (8/5)

• 0

Q.1 | Q 6

Choose the correct alternative:

int_2^3 x/(x^2 - 1)  "d"x =

• log (8/3)

• - log (8/3)

• 1/2 log(8/3)

• -1/2 log(8/3)

Q.1 | Q 7

Choose the correct alternative:

int_"a"^"b" "f"(x)  "d"x =

• int_"b"^"a" "f"(x)  "d"x

• - int_"b"^"a" "f"(x)  "d"x

• - int_"b"^"a" "f"(x)  "d"x

• int_0^"a" "f"(x)  "d"x

Q.1 | Q 8

Choose the correct alternative:

int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x))  "d"x =

• 7/2

• 5/2

• 7

• 2

Q.1 | Q 9

Choose the correct alternative:

int_(-9)^9 x^3/(4 - x^2)  "d"x =

• 0

• 3

• 9

• – 9

Q.2

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 Chapter 2 Definite Integration Q.2

#### Fill in the blanks [1 Mark]

Q.2 | Q 1

int_1^2 x^2  "d"x = ______

Q.2 | Q 2

int_0^"a" 4x^3  "d"x = 81, then a = ______

Q.2 | Q 3

int_0^1 "e"^(2x) "d"x = ______

Q.2 | Q 4

int_1^2 1/(2x + 3)  "d"x = ______

Q.2 | Q 5

int_2^4 x/(x^2 + 1)  "d"x = ______

Q.2 | Q 6

int_(-7)^7 x^3/(x^2 + 7)  "d"x = ______

Q.3

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 Chapter 2 Definite Integration Q.3

#### [1 Mark]

Q.3 | Q 1

State whether the following statement is True or False:

int_0^"a" 3x^2  "d"x = 27, then a = 2.5

• True

• False

Q.3 | Q 2

State whether the following statement is True or False:

int_0^1 1/(2x + 5)  "d"x = log(7/5)

• True

• False

Q.3 | Q 3

State whether the following statement is True or False:

int_2^3 x/(x^2 + 1)  "d"x = 1/2 log 2

• True

• False

Q.3 | Q 4

State whether the following statement is True or False:

int_"a"^"b" "f"(x)  "d"x = int_"a"^"b" "f"("a" + "b" - x)  "d"x

• True

• False

Q.3 | Q 5

State whether the following statement is True or False:

int_0^(2"a") "f"(x)  "d"x = int_0^"a" "f"(x)  "d"x + int_0^"a" "f"("a" - x)  "d"x

• True

• False

Q.3 | Q 6

State whether the following statement is True or False:

int_(-5)^5 x/(x^2 + 7)  "d"x = 10

• True

• False

Q.4

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 Chapter 2 Definite Integration Q.4

#### Solve the following [3 Marks]

Q.4 | Q 1

Evaluate int_0^1 (x^2 + 3x + 2)/sqrt(x)  "d"x

Q.4 | Q 2

Evaluate int_0^1 1/(sqrt(1 + x) + sqrt(x))  "d"x

Q.4 | Q 3

If int_0^"a" (2x + 1)  "d"x = 2, find a

Q.4 | Q 4

If int_1^"a" (3x^2 + 2x + 1)  "d"x = 11, find the real value of a

Q.4 | Q 5

Evaluate int_1^"e" 1/(x(1 + log x)^2)  "d"x

Q.4 | Q 6

Evaluate int_1^2 1/(x^2 + 6x + 5)  "d"x

Q.4 | Q 7

Evaluate int_1^2 (3x)/((9x^2 - 1))  "d"x

Q.4 | Q 8

Evaluate int_1^3 (root(3)(x + 5))/(root(3)(x + 5) + root(3)(9 - x))  "d"x

Q.4 | Q 9

Evaluate int_1^2 (sqrt(x))/(sqrt(3 - x) + sqrt(x))  "d"x

Q.4 | Q 10

Evaluate int_0^1 log(1 / x - 1)  "d"x

Q.5

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 Chapter 2 Definite Integration Q.5

#### [4 Marks]

Q.5 | Q 1

Evaluate int_2^3 x/((x + 2)(x + 3))  "d"x

Q.5 | Q 2

Evaluate int_1^2 "e"^(2x) (1/x - 1/(2x^2))  "d"x

Q.5 | Q 3

Evaluate int_1^3 log x  "d"x

Q.5 | Q 4

Evaluate int_1^3 x^2*log x  "d"x

Q.5 | Q 5

Evaluate int_0^1 "e"^(x^2)*"x"^3  "d"x

Q.5 | Q 6

Evaluate int_0^"a" x^2 ("a" - x)^(3/2)  "d"x

Q.5 | Q 7

Evaluate int_0^1 x(1 - x)^5  "d"x

Q.6

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 Chapter 2 Definite Integration Q.6

#### Activity [4 Marks]

Q.6 | Q 1

By completing the following activity, Evaluate int_1^2 (x + 3)/(x(x + 2))  "d"x

Solution: Let I = int_1^2 (x + 3)/(x(x + 2))  "d"x

Let (x + 3)/(x(x + 2)) = "A"/x + "B"/((x + 2))

∴ x + 3 = A(x + 2) + B.x

∴ A = square, B = square

∴ I = int_1^2[("( )")/x + ("( )")/((x + 2))] "d"x

∴ I = [square log x + square log(x + 2)]_1^2

∴ I = square

Q.6 | Q 2

By completing the following activity, Evaluate int_2^5 (sqrt(x))/(sqrt(x) + sqrt(7 - x))  "d"x.

Solution: Let I = int_2^5 (sqrt(x))/(sqrt(x) + sqrt(7 - x))  "d"x     ......(i)

Using the property, int_"a"^"b" "f"(x) "d"x = int_"a"^"b" "f"("a" + "b" - x)  "d"x, we get

I = int_2^5 ("(  )")/(sqrt(7 - x) + "(  )")  "d"x   ......(ii)

Adding equations (i) and (ii), we get

2I = int_2^5 (sqrt(x))/(sqrt(x) - sqrt(7 - x))  "d"x + (   )  "d"x

2I = int_2^5 (("(    )" + "(     )")/("(    )" + "(     )"))  "d"x

2I = square

∴ I =  square

## Chapter 2: Definite Integration

Q.1Q.2Q.3Q.4Q.5Q.6 ## SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 chapter 2 - Definite Integration

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Concepts covered in 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 chapter 2 Definite Integration are Fundamental Theorem of Integral Calculus, Properties of Definite Integrals.

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