#### 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

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## Solutions for Chapter 1.6: Definite Integration

Below listed, you can find solutions for Chapter 1.6 of Maharashtra State Board SCERT Maharashtra Question Bank for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022.

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 1.6 Definite Integration Q.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

`int_0^2 e^x dx` = ______.

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)`

`int_"a"^"b" "f"(x) "d"x` = ______

`int_"b"^"a" "f"(x) "d"x`

`- int_"a"^"b" "f"(x) "d"x`

`- int_"b"^"a" "f"(x) "d"x`

`int_0^"a" "f"(x) "d"x`

`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 2022 Chapter 1.6 Definite Integration Q.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) dx` = ______

`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 2022 Chapter 1.6 Definite Integration Q.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 2022 Chapter 1.6 Definite Integration Q.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 the following integrals:**

`int_1^3 (root(3)(x + 5))/(root(3)(x + 5) + root(3)(9 - x))*dx`

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

Evaluate the following integrals : `int_0^1 log(1/x - 1)*dx`

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 1.6 Definite Integration Q.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 2022 Chapter 1.6 Definite Integration Q.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`

## Solutions for Chapter 1.6: Definite Integration

## SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 chapter 1.6 - Definite Integration

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Concepts covered in 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 chapter 1.6 Definite Integration are Fundamental Theorem of Integral Calculus, Properties of Definite Integrals.

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