#### Online Mock Tests

#### Chapters

Chapter 1.2: Matrices

Chapter 1.3: Differentiation

Chapter 1.4: Applications of Derivatives

Chapter 1.5: Integration

Chapter 1.6: Definite Integration

Chapter 1.7: Application of Definite Integration

Chapter 1.8: Differential Equation and Applications

Chapter 2.1: Commission, Brokerage and Discount

Chapter 2.2: Insurance and Annuity

Chapter 2.3: Linear Regression

Chapter 2.4: Time Series

Chapter 2.5: Index Numbers

Chapter 2.6: Linear Programming

Chapter 2.7: Assignment Problem and Sequencing

Chapter 2.8: Probability Distributions

## Chapter 2: Definite Integration

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

#### MCQ [1 Mark]

**Choose the correct alternative:**

`int_2^3 x^4 "d"x` =

`1/2`

`5/2`

`5/211`

**`211/5`**

**Choose the correct alternative:**

`int_0^"a" 3x^5 "d"x` = 8, then a =

2

0

`8/3`

a

**Choose the correct alternative:**

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

9

4

2

0

**Choose the correct alternative:**

`int_0^2 "e"^x "d"x` =

e

^{2}– 11 – e

^{2}e – 1

1 – e

**Choose the correct alternative:**

`int_(-2)^3 1/(x + 5) "d"x` =

`log (3/8)`

`log (8/3)`

`log (8/5)`

0

**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)`

**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`

**Choose the correct alternative:**

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

`7/2`

`5/2`

7

2

**Choose the correct alternative:**

`int_(-9)^9 x^3/(4 - x^2) "d"x` =

0

3

9

– 9

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

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

`int_1^2 x^2 "d"x` = ______

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

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

`int_1^2 1/(2x + 3) "d"x` = ______

`int_2^4 x/(x^2 + 1) "d"x` = ______

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

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

#### [1 Mark]

**State whether the following statement is True or False:**

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

True

False

**State whether the following statement is True or False:**

`int_0^1 1/(2x + 5) "d"x = log(7/5)`

True

False

**State whether the following statement is True or False: **

`int_2^3 x/(x^2 + 1) "d"x = 1/2 log 2`

True

False

**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

**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

**State whether the following statement is True or False:**

`int_(-5)^5 x/(x^2 + 7) "d"x` = 10

True

False

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

#### Solve the following [3 Marks]

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

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

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

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

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

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

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

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

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

Evaluate `int_0^1 log(1 / x - 1) "d"x`

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

#### [4 Marks]

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

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

Evaluate `int_1^3 log x "d"x`

Evaluate `int_1^3 x^2*log x "d"x`

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

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

Evaluate `int_0^1 x(1 - x)^5 "d"x`

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

#### Activity [4 Marks]

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`

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

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

SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 chapter 2 (Definite Integration) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Maharashtra State Board 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 solutions in a manner that help students grasp basic concepts better and faster.

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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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