MSBSHSE Solutions For Class 9 Maths Part 1 Chapter 2 Real Numbers consists of accurate solutions, which help the students to quickly complete their homework and prepare well for the exams. These solutions provide students an advantage with practical questions. Each step in the solution is explained to match studentsâ€™ understanding. To score good marks in Class 9 Mathematics examination, it is advised they solve questions provided at the end of each chapter in the Maharashtra Board Textbooks for Class 9. In the Maharashtra State Board Solutions for Class 9 Chapter 2, students will learn and solve problems based on topics like properties of rational numbers, properties of irrational numbers, surds, comparison of quadratic surds, operations on quadratic surds and Rationalization of quadratic surds.

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Practice set 2.1 Page no: 21

**1. Classify the decimal form of the given rational numbers into terminating and non-terminating recurring type.**

**i. 13/5Â Â **

**iv. 17/125
v.Â 11/6**

**Solution:**

i.

âˆµÂ The division is exact

âˆ´Â it is a terminating decimal.

ii.

âˆµÂ The division never ends and the digits â€˜18â€™ is repeated endlessly

âˆ´Â it is a non-terminating recurring type decimal.

iii.

âˆµÂ The division is exact

âˆ´Â it is a terminating decimal.

iv.

âˆµÂ The division is exact

âˆ´Â it is a terminating decimal.

v.

âˆµÂ The division never ends and the digit â€˜3â€™ is repeated endlessly

âˆ´Â it is a non-terminating recurring type decimal.

**2. Write the following rational numbers in decimal form.**

**i.Â 127/200Â **

**ii.Â 25/99
iii.Â 23/7Â **

**iv. 4/5
v. 17/8Â **

**Solution:**

i.

ii.

iii.

iv.

v.

**3. Write the following rational numbers in form.
i.Â Â **

**ii.Â
iii.Â Â **

**iv.Â
v.Â **

**Solution:**

Practice set 2.2 Page no: 25

**1. Show that is 4âˆš2 an irrational number.**

**Solution:**

**2. Prove that 3 + âˆš5 is an irrational number.**

**Solution:**

**3. Represent the numbers âˆš5 and âˆš10 on a number line.**

**Solution:**

Given âˆš5

Given that âˆš10

**4. Write any three rational numbers between the two numbers given below.**

**(i) 0.3 and -0.5**

**Solution:**

**(ii) -2.3 and -2.33**

**Solution:**

**(iii) 5.2 and 5.3**

**Solution:**

**(iv) -4.5 and 4.6**

**Solution:**

Practice set 2.3 Page no: 30

**1. State the order of the surds given below.**

**Solution:**

**2. State which of the following are surds. Justify.
**

**Solution:**

iv.Â âˆš256 = âˆš162Â = 16

SurdsÂ are numbers left in root form (âˆš) to express its exact value. It has an infinite number of non-recurring decimals. Therefore, surds are irrational numbers.

It is not a surdÂ âˆµÂ it is a rational number.

**3. Classify the given pair of surds into like surds and unlike surds.**

**i. âˆš52, 5âˆš13
ii. âˆš68, 5âˆš3
iii. 4âˆš18, 7âˆš2
iv. 19âˆš12, 6âˆš3
v. 5âˆš22, 7âˆš33
vi. 5âˆš5, âˆš75**

**Solution:**

Two or more surds are said to be similar or like surds if they have the same surd-factor.

Two or more surds are said to be dissimilar or unlike when they are not similar.

Therefore,

i. GivenÂ âˆš52, 5âˆš13

given surd can be written as

âˆš52 = âˆš (2Ã—2Ã—13) = 2âˆš13

5âˆš13

âˆµÂ both surds haveÂ same surd-factor that is âˆš13.

âˆ´Â they are like surds.

ii.Â Given âˆš68, 5âˆš3

Given surd can be written as

âˆš68 = âˆš (2Ã—2Ã—17) = 2âˆš17

5âˆš3

âˆµÂ both surds haveÂ different surd-factors âˆš17 and âˆš3.

âˆ´Â they are unlike surds.

iii.Â Given 4âˆš18, 7âˆš2

Given surd can be written as

4âˆš18 = 4 âˆš (2Ã—3Ã—3) = 4Ã—3âˆš2 = 12âˆš2

7âˆš2

âˆµÂ both surds haveÂ same surd-factor i.e., âˆš2.

âˆ´Â they are like surds.

iv.Â Given 19âˆš12, 6âˆš3

Given surd can be written as

19âˆš12 = 19âˆš (2Ã—2Ã—3) = 19Ã—2âˆš3 = 38âˆš3

6âˆš3

âˆµÂ both surds haveÂ same surd-factor i.e., âˆš3.

âˆ´Â they are like surds.

v.Â Given 5âˆš22, 7âˆš33

âˆµÂ both surds haveÂ different surd-factors âˆš22 and âˆš33.

âˆ´Â they are unlike surds.

vi.Â Given 5âˆš5, âˆš75

5âˆš5

Given surd can be written as

âˆš75 = âˆš (5Ã—5Ã—3) = 5âˆš3

âˆµÂ both surds haveÂ different surd-factors âˆš5 and âˆš3.

âˆ´Â they are unlike surds.

**4. Simplify the following surds.
i. âˆš27
ii. âˆš50
iii. âˆš250
iv. âˆš112
v. âˆš168**

**Solution:**

Practice set 2.4 Page no: 32

**1. Multiply
i. âˆš3(âˆš7 â€“ âˆš3)
ii. (âˆš5 â€“ âˆš7) âˆš2
iii. (3âˆš2 â€“ âˆš3) (4âˆš3 â€“ âˆš2)**

**Solution:**

i.Â Given âˆš3 (âˆš7 â€“ âˆš3)

=âˆš3 Ã— âˆš7 â€“ âˆš3 Ã— âˆš3

[âˆµ âˆša (âˆšb â€“ âˆšc) = âˆša Ã— âˆšb â€“ âˆša Ã— âˆšc]=âˆš21 â€“ 3

ii.Â Given (âˆš5 â€“ âˆš7) âˆš2

=âˆš5 Ã— âˆš2 â€“ âˆš7 Ã— âˆš2

[âˆµâˆša (âˆšb â€“ âˆšc) = âˆša Ã— âˆšb â€“ âˆša Ã— âˆšc]= âˆš10 â€“ âˆš14

iii.Â Given (3âˆš2 â€“ âˆš3) (4âˆš3 â€“ âˆš2)

=3âˆš2 (4âˆš3 â€“ âˆš2) â€“ âˆš3 (4âˆš3 â€“ âˆš2)

[âˆµ âˆša (âˆšb â€“ âˆšc) = âˆša Ã— âˆšb â€“ âˆša Ã— âˆšc]= 3âˆš2 Ã— 4âˆš3 â€“ 3âˆš2 Ã— âˆš2 â€“ âˆš3 Ã— 4âˆš3 + âˆš3 Ã— âˆš2

= 12âˆš6 â€“ 3 Ã— 2 â€“ 4 Ã— 3 + âˆš6

= 12âˆš6 â€“ 6 â€“ 12 + âˆš6

= 13âˆš6 â€“ 18

**2. Rationalize the denominator.
**

**iv.Â Â **

**Solution:**

Practice set 2.5 Page no: 33

**1. Find the value.
(i) |15 – 2|
(ii) |4 – 9|
(iii) |7| Ã— |-4|**

**Solution:**

i.Â Given |15 – 2|

Absolute value describes the distance of a number on the number line from 0 without considering which direction from zero the number lies. The absolute value of a number is never negative.

Therefore,

|15 – 2| = |13| = 13

ii. Given |4 – 9|

Absolute value describes the distance of a number on the number line from 0 without considering which direction from zero the number lies. The absolute value of a number is never negative.

Therefore,

|4 – 9| = |-5| = 5

iii.Â Given |7| Ã— |-4|

Absolute value describes the distance of a number on the number line from 0 without considering which direction from zero the number lies. The absolute value of a number is never negative.

Therefore,

|7| Ã— |-4| = 7 Ã— 4 = 28

**2. Solve.
i.Â |3x – 5| = 1
ii. |7 â€“ 2x| = 5
iii.Â
iv.Â **

**Solution:**

Problem set 2 Page no: 34

**1. Choose the correct alternative answer for the questions given below.
i. Which one of the following is an irrational number?
A.Â âˆš16/25
B.Â âˆš5
C. 3/9
D.Â âˆš196**

**Solution:**

B.Â âˆš5

**Explanation:**

An irrational number is a number that cannot be expressed as aÂ fractionÂ p/qÂ for anyÂ integersÂ p andÂ q and q â‰ 0.

Since âˆš5 cannot be written asÂ p/qÂ it is an irrational number

Therefore âˆš5 is an irrational number.

**ii. Which of the following is an irrational number?
A. 0.17
B.Â
C.Â
D. 0.101001000….**

**Solution:**

D. 0.101001000….

**Explanation:**

An irrational number is a number that cannot be expressed as aÂ fractionÂ p/qÂ for anyÂ integersÂ p andÂ q and q â‰ 0.

0.101001000…. is an irrationalÂ number because it is a non-terminating and non-`repeating decimal.

Therefore, 0.101001000…. is an irrationalÂ number.

**iii. Decimal expansion of which of the following is non-terminating recurring?
A. 2/5
B.Â 3/16
C. 3/11
D.Â 137/25**

**Solution:**

C. 3/11

**Explanation:**

A non-terminating recurring decimal representation means that the number will have an infinite number of digits to the right of the decimal point and those digits will repeat themselves.

âˆµÂ it hasÂ an infinite number of digits to the right of the decimal point which are repeating themselvesÂ âˆ´Â it is a non-terminating recurring decimal.

**iv. Every point on the number line represent, which of the following numbers?
A. Natural numbers
B. Irrational numbers
C. Rational numbers
D. Real numbers.**

**Solution:**

D. Real numbers.

**Explanation:**

Every point of a number line is assumed to correspond to a real number, and every real number to a point.Â Therefore,Â every point on the number line represent a real number.

**v. The number 0.4 in p/q form is â€¦â€¦â€¦â€¦.
A. 4/9
B.Â 40/9
C. 3.6/9
D.Â 36/9**

**Solution:**

C. 3.6/9

**Explanation:**

**vi. What is âˆšn, if n is not a perfect square number?
A. Natural number
B. Rational number
C. Irrational number
D. Options A, B, C all are correct.**

**Solution:**

C. Irrational number

**Explanation:**

If n is not a perfect square number, then âˆšn cannot be expressed as ratio of a and b where a and b are integers and b â‰ 0

Therefore, âˆšn is an Irrational number

**vii. Which of the following is not a surd?
A. âˆš7
B.Â 3âˆš17
C. 3âˆš64
D. âˆš193**

**Solution:**

C. 3âˆš64

**Explanation:**

**viii. What is the order of the surdÂ ?
A. 3
B. 2
C. 6
D. 5**

**Solution:**

C. 6

**Explanation:**

**ix. Which one is the conjugate pair of 2âˆš5 + âˆš3?
A. -2âˆš5 + âˆš3
B. -2âˆš5 – âˆš3
C. 2âˆš3 + âˆš5
D. âˆš3 + 2âˆš5**

**Solution:**

A. -2âˆš5 + âˆš3

**Explanation:**

A math conjugate is formed by changing the sign between two terms in a binomial. For instance, the conjugate of x + y is x – y.

Now,

2âˆš5 + âˆš3 = âˆš3 + 2âˆš5

ItsÂ conjugate pair = âˆš3 – 2âˆš5 = -2âˆš5 + âˆš3

âˆ´Â TheÂ conjugate pair of 2âˆš5 + âˆš3 = -2âˆš5 + âˆš3

**x. The value ofÂ |12 â€“ (13 + 7) Ã— 4| is ………..
A. -68
B. 68
C. -32
D. 32**

**Solution:**

B. 68

**Explanation:**

|12 â€“ (13 + 7) Ã— 4| = |12 â€“ 20 Ã— 4|

(Solving it according to BODMAS)

â‡’ |12 â€“ (13 + 7) Ã— 4| = |12 â€“ 80|

â‡’ |12 â€“ (13 + 7) Ã— 4| = |-68|

â‡’ |12 â€“ (13 + 7) Ã— 4| = 68

Real numbers are simply the combination of rational and irrational numbers, in the number system. In general, all the arithmetic operations can be performed on these numbers and they can be represented in the number line. Students can depend on these Solutions to understand all the topics completely. Stay tuned to learn more about Real Numbers, MSBSHSE Exam pattern and other information.

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