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RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

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Table of contents
RS Aggarwal Exercise 1.2 Number System
RS Aggarwal Exercise 1.3 Number System
RS Aggarwal Exercise 1.4 Number System 
RS Aggarwal Exercise 1.5 Number System 
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RS Aggarwal Exercise 1.2 Number System

Q.1. Write actual division, find which of the following rational numbers are terminating decimals.
(i) 13/80
(ii) 7/24
(iii) 5/12
(iv) 31/375
(v) 16/125
Ans.
(i) 13/80
Denominator of 13/80 is 80.
And,
80 = 24×5
Therefore, 80 has no other factors than 2 and 5.
Thus, 13/80 is a terminating decimal.
(ii) 7/24
Denominator of 7/24 is 24.
And,
24 = 23×3
So, 24 has a prime factor 3, which is other than 2 and 5.
Thus, 7/24 is not a terminating decimal.
(iii) 5/12
Denominator of 5/12 is 12.
And,
12 = 22×3

So, 12 has a prime factor 3, which is other than 2 and 5.
Thus, 5/12 is not a terminating decimal.
(iv) 31/375
Denominator of 31/375 is 375.
375 = 53×3
So, the prime factors of 375 are 5 and 3.
Thus, 31/375 is not a terminating decimal.
(v) 16/125
Denominator of 16/125 is 125.
And,
125 = 53
Therefore, 125 has no other factors than 2 and 5.
Thus, 16/125 is a terminating decimal.

Question 2:
Write each of the following in decimal form and say what kind of decimal expansion each has.
(i) 5/8
(ii) 7/25
(iii) 3/11
(iv) 5/13
(v) 11/24
(vi) 261/400
(vii) 231/625
(viii) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Ans. (i) 5/8 = 0.625
By actual division, we have:
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
It is a terminating decimal expansion.
(ii) 7/25
7/25 = 0.28
By actual division, we have:
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
It is a terminating decimal expansion.
(iii) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
It is a non-terminating recurring decimal.
(iv) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
It is a non-terminating recurring decimal.
(v) 11/24
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
By actual division, we have:
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
It is nonterminating recurring decimal expansion.
(vi) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
It is a terminating decimal expansion.
(vii) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
It is a terminating decimal expansion.
(viii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
By actual division, we have:
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
It is non-terminating decimal expansion.

Q.3.

Express each of the following decimals in the form p/q, where p, q are integers and q ≠ 0.
(i)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

(ii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

(iii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

(iv)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

(v)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(vi)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(vii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(viii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(ix)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(x)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Ans.
(i) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9 
Let x = 0.222...                               .....(i)
Only one digit is repeated so, we multiply x by 10.
10x = 2.222...                                 .....(ii)
Subtracting (i) from (ii) we get
9x = 2
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(ii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Let x = 0.5353...                               .....(i)
Two digits are repeated so, we multiply x by 100.
100x = 53.5353...                                 .....(ii)
Subtracting (i) from (ii) we get
99x = 53
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9 
Let x = 2.9393...                               .....(i)
Two digits are repeated so, we multiply x by 100.
100x = 293.9393...                                 .....(ii)
Subtracting (i) from (ii) we get
99x = 291
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iv)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Let x = 18.4848...                               .....(i)
Two digits are repeated so, we multiply x by 100.
100x = 1848.4848...                                 .....(ii)
Subtracting (i) from (ii) we get
99x = 1830

RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(v)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Let x = 0.235235...                               .....(i)
Three digits are repeated so, we multiply x by 1000.
1000x = 235.235235...                                 .....(ii)
Subtracting (i) from (ii) we get
999x = 235
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(vi)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Let x = 0.003232...                               .....(i)
we multiply x by 100.
100x = 0.3232...                                 .....(ii)
Again multiplying by 100 as there are 2 repeating numbers after decimals we get
10000x = 32.3232...                           .....(iii)
Subtracting (ii) from (iii) we get
9900x = 32
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(vii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Let x = 1.32323...                               .....(i)
we multiply x by 10.
10x = 13.2323...                                 .....(ii)
Again multiplying by 100 as there are 2 repeating numbers after decimals we get
1000x = 1323.2323...                           .....(iii)
Subtracting (ii) from (iii) we get
990x=1310
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(viii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Let x = 0.3178178...                               .....(i)
we multiply x by 10.
10x = 3.178178...                                 .....(ii)
Again multiplying by 1000 as there are 3 repeating numbers after decimals we get
10000x = 3178.178178...                           .....(iii)
Subtracting (ii) from (iii) we get
9990x = 3175
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(ix)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Let x = 32.123535...                               .....(i)
we multiply x by 100.
100x = 3212.3535...                                 .....(ii)
Again multiplying by 100 as there are 2 repeating numbers after decimals we get
10000x = 321235.35...                           .....(iii)

Subtracting (ii) from (iii) we get
9900x=318023
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(x)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Let x = 0.40777...                               .....(i)
we multiply x by 100.
100x = 40.7777...                                 .....(ii)
Again multiplying by 10 as there is 1 repeating number after decimals we get
1000x = 407.777...                           .....(iii)
Subtracting (ii) from (iii) we get
900x = 367
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

Q.4. ExpressRS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9as a fraction in simplest form.
Ans.
Given:RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Let
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
First we take x and convert it into p/q
100x = 236.3636...        ...(iii)
Subtracting (i) from (iii) we get
99x=234
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Similarly, multiply y with 100 as there are 2 decimal places which are repeating themselves.
100y = 23.2323...                ...(iv)
Subtracting (ii) from (iv) we get
99y = 23
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Adding x and y we get

RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

Q.5. Express in the form of p/q RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Ans.

RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
x = 0.3838...                                  ...(i)
Multiply with 100 as there are 2 repeating digits after decimals
100x = 38.3838...                          ...(ii)
Subtracting (i) from (ii) we get
99x = 38
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Similarly, we take
y = 1.2727...                                  ...(iii)
Multiply y with 100 as there are 2 repeating digits after decimal.
100y = 127.2727...                       ...(iv)
Subtract (iii) from (iv) we get
99y = 126
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9


RS Aggarwal Exercise 1.3 Number System

Q.1. What are irrationl numbers? How do they differ from rational numbers? Give examples.
Ans. A number that can neither be expressed as a terminating decimal nor be expressed as a repeating decimal is called an irrational number. A rational number, on the other hand, is always a terminating decimal, and if not, it is a repeating decimal.
Examples of irrational numbers:
0.101001000...
0.232332333...

Q.2. Classify the following numbers as rational or irrational. give reasons to support your answer.
(i) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(ii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iv)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(v)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(vi) 4.1276
(vii) 22/7
(viii) 1.232332333..
(ix) 3.040040004....
(x) 2.356565656...
(xi) 6.834834...
Ans. RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
It is an irrational number.
(ii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9 = 19
So, it is rational.
(iii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
It is an irrational number.
(iv)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
So, it is rational.
(v) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
It is an irrational number
(vi) 4.1276
It is a terminating decimal. Hence, it is rational.

(vii) 22/7
22/7 is a rational number because it can be expressed in the p/q form.
(viii) 1.232332333...is an irrational number because it is a non−terminating, non−repeating decimal.
(ix) 3.040040004... is an irrational number because it is a non−terminating, non−repeating decimal.
(x) 2.356565656... is a rational number because it is repeating.
(xi) 6.834834... is a rational number because it is repeating.

Q.3. Let x be a rational number and y be an irrational number. Is x + y necessarily an irrational number? Give a example in support of your answer.
Ans.
x be a rational number and y be an irrational number then x + y necessarily will be an irrational number.
Example: 5 is a rational number but √2 is irrational.
So, 5 + √2 will be an irrational number.

Q.4. Let a be a rational number and b be an irrational number. Is ab necessarily an irrational number? Justify your answer with an example.
Ans. a be a rational number and b be an irrational number then ab necessarily will be an irrational number.
Example: 6 is a rational number but √5 is irrational. And 6√5 is also an irrational number.

Q.5. Is the product of two irrationals always irrational? Justify your answer.
Ans. Product of two irrational numbers is not always an irrational number.
Example: √5 is irrational number. And √5 × √5 = 5 is a rational number. But the product of another two irrational numbers √2 and √3 is √6 which is also an irrational numbers.


Q.6. Give an example of two irrational numbers whose
(i) difference is an irrational number.
(ii) difference is a rational number.
(iii) sum is an irrational number.
(iv) sum is a rational number.
(v) product is an irrational number.
(vi) product is a rational number.
(vii) quotient is an irrational number.
(viii) quotient is a rational number.
Ans.
(i) 2 irrational numbers with difference an irrational number will be 3−√5 and 3+√5.
(ii) 2 irrational numbers with difference is a rational number will be 5+√3 and 2+√3
(iii) 2 irrational numbers with sum an irrational number 7+√5 and √6 − 8

(iv) 2 irrational numbers with sum a rational number is 3−√2 and 3 + √2
(v) 2 irrational numbers with product an irrational number will be 6 + √3 and 7−√3
(vi) 2 irrational numbers with product a rational number will be (5 + √7) and (5 − √7)
(vii) 2 irrational numbers with quotient an irrational number will be √15 and √5
(viii) 2 irrational numbers with quotient a rational number will be √63 and √7.

Question 7: Examine whether the following numbers are rational or irrational.
(i) 3 + √3
(ii) √7 − 2
(iii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iv) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(v) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(vi) √8 × √2
Ans. (i) Let us assume, to the contrary, that 3 + √3 is rational.

Then, 3+√3 = pq, where p and q are coprime and q ≠ 0.
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Since, p and q are are integers.
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9 is rational.

So, √3 is also rational.
But this contradicts the fact that √3 is irrational.
This contradiction has arisen because of our incorrect assumption that 3 + √3 is rational.
Hence, 3 + √3 is irrational.
(ii) Let us assume, to the contrary, that √7−2 is rational.
Then, √7 − 2 = p/q,  where p and q are coprime and q ≠ 0.
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Since, p and q are are integers.
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9is rational.
So, √7 is also rational.
But this contradicts the fact that √7 is irrational.
This contradiction has arisen because of our incorrect assumption that √7 − 2 is rational.
Hence, √7 − 2 is irrational.
(iii) As,RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
= 5, which is an integer
Hence, RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9 is rational.
(iv) As, RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
= 49, which is an integer
Hence, RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9is rational.

(v) As,RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9which is rational
Hence, RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9 is rational.
(vi) As, √8 × √2
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
= 4, which is an integer
Hence, √8 × √2 is rational.

Q.8. Insert a rational and an irrational number between 2 and 2.5.
Ans. 
As, few rational numbers between 2 and 2.5 are: 2.1, 2.2, 2.3, 2.4, ...
And,
Since, 2 = √4 and 2.5= RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
So, irrational number between 2 ans 2.5 are:RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Hence, a rational and an irrational number can be 2.1 and √5, respectively.
Disclaimer: There are infinite rational and irrational numbers between any two rational numbers.

Q.9. How many irrational numbers lie between √2 and √3? Find any three irrational numbers lying between √2 and √3.
Ans. There are infinite number of irrational numbers lying between √2 and √3.
As, √2 = 1.414 and √3 = 1.732
So, the three irrational numbers lying between √2 and √3 are:
1.420420042000..., 1.505005000... and 1.616116111...

Q.10. Find two rational and two irrational number between 0.5 and 0.55.
Ans. The two rational numbers between 0.5 and 0.55 are: 0.51 and 0.52
The two irrational numbers between 0.5 and 0.55 are: 0.505005000... and
0.5101100111000...
Disclaimer: There are infinite number of rational and irrational numbers between 0.5 and 0.55.

Q.11. Find three different irrational numbers between the rational numbers RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Ans. As, RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
So, the three different irrational numbers are: 0.72020020002..., 0.7515511555111... and 0.808008000...
Disclaimer: There are an infinite number of irrational numbers between two rational numbers.

Q.12. Find two rational numbers of the form p/q between the numbers 0.2121121112... and 0.2020020002... .
Ans. The rational numbers between the numbers 0.2121121112... and 0.2020020002... are:
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Disclaimer: There are an infinite number of rational numbers between two irrational numbers.

Q.13. Find two irrational numbers between 0.16 and 0.17.
Ans. The two irrational numbers between 0.16 and 0.17 are 0.161161116... and 0.1606006000...
Disclaimer: There are an infinite number of irrational numbers between two rational numbers.

Q.14. State in each case, whether the given statement is true of false.
(i) The sum of two rational numbers is rational.

(ii) The sum of two irrational numbers is irrational.
(iii) The product of two rational numbers is rational.
(iv) The product of two irrational number is irrational.
(v) The sum of a rational number and an irrational number is irrational.
(vi) The product of a nonzero rational number and an irrational number is a rational number.
(vii) Every real number is rational.
(viii) Every real number is either rational or irrational.
(ix) π is irrational and 22/7 is rational.
Ans.
(i) True
(ii) False
Example: (2 + √3) + (2 − √3) = 4
Here, 4 is a rational number.
(iii) True
(iv) False
Example: √ 3× √3=3
Here, 3 is a rational number.
(v) True
(vi) False
Example: (4)×√5 = 4√5
Here, 4√5 is an irrational number.
(vii) False
Real numbers can be divided into rational and irrational numbers.
(viii) True
(ix) True


RS Aggarwal Exercise 1.4 Number System 

Q.1. Add:
(i) (2√3 − 5√2) and (√3+2√2)
(ii) (2√2 + 5√3 − 7√5)and(3√3 − √2 + √5)
(iii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Ans.
(i) 2√3 − 5√2 + √ 3 + 2√2
= (2√3 + √3) + (2√2 − 5√2)
= 3√3− 3√2
(ii) 2√2+ 5√3 − 7√5 + 3√3 − √2 + √5
= 2√2 - √2 + 5√3 + 3√3 + √5  − 7√5
= √2 + 8√3 −6√5
(iii) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

Q.2. Multiply:
(i) 3√5 by 2√5
(ii) 6√15 by 4√3
(iii) 2√6 by 3√3
(iv) 3√8 by 3√2
(v) √10 by √40
(vi) 3√28 by 2√7

Ans.
(i) 3√5 × 2√5 = 3 × 2 × √5 x √5 = 6 × 5 = 30
(ii) 6√15 × 4√3 = 6 × 4 × √5 × √3 × √3 = 24 × 3 × √5 = 72√5
(iii) 2√6 × 3√3 = 2 × 3 × √2 × √3 × √3 = 6 × 3 × √2 = 18√2
(iv) 3√8 × 3√2 = 3 × 3 × √2 × √2 × √2 × √2 = 9 × 4 = 36

(v) √10 × √40 = √2 × √5 × √2 × √2 × √2 × √5 = √2 × √2 × √2 × √2 × √5 × √5 = 2 × 2 × 5 = 20

(vi) 3√28 × 2√7 = RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9 × √7 = 6 × 7 × √4 = 42 × 2 = 84

Q.3. Divide:
(i) 16√6 by 4√2

(ii) 12√5 by 4√3
(iii) 18√21 by 6√7
Ans. 
(i) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9 
(ii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

Q.4. Simplify
(i) (3 − √11) (3 + √11)
(ii) (−3 + √5) (−3 − √5)
(iii) (3 − √3)2
(iv) (√5 − √3)2

(v) (5 + √7) (2 + √5)
(vi) (√5 – √2) (√2 - √3)
Ans. (i) (3 − √11) (3 + √11)
=3− (√11)2    [(a − b)( a + b) =a− b2]
=9 − 11
= −2
(ii) (−3 + √5) (−3 − √5)
=(−3)2−(√5)2   [(a + b)(a − b) = a− b2]
=9 − 5
=4
(iii) (3 − √3)2
=32+(√3)− 2 × 3 × √3   [(a−b)2=a2+b2−2ab]

=9 + 3 − 6√3
=12 − 6√3
(iv) (√5 − √3)2
=(√5)2+ (√3)− 2 × √5√3   [(a−b)= a+ b− 2ab]
=5 + 3 − 2√15 = 8 − 2√15

= √5 × √2 − √5 × √3 - √2 × √2 + √2 × √3
= √10 − √15 − 2 + √6
(v) (5 + √7) (2 + √5)
(5 + √7) (2 + √5)
= 5 × 2 + 5 × √5+ √7 × 2 +√7 × √5
=10 + 5√5 + 2√7 + 3√5
(vi) (√5 − √2) (√2 − √3)
(√5 - √2)(√2 − √3)
= √5 × √2 – √5 × √3 − √2 × √2 +  √2  × √3

= √10 − √15 − 2 + √6

Q.5. Simplify (3 + √3) (2 + √2)2.
Ans. (3 + √3) (2+√2)2
= (3 + 3) [22+(√2)+ 2 × 2√2]
=(3 + √3) [4 + 2 + 4√2]
= (3 + √3) [6 + 4√2]
=3 × 6 + 3 × 4√2 + √3 × 6 + √3 × 4√2
=18 + 12√2 + 6√3 + 4√6

Q.6. Examine whether the following numbers are rational or irrational:
(i) (5 − √5) (5 + √5)
(ii) (√3 + 2)2
(iii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iv) √8 + 4√32 − 6√2
Ans. (i) (5 − √5) (5 + √5)
=5− (√5)2   [(a − b)(a + b) = a− b2]
= 25 − 5
= 20, which is an integer
Hence, (5 − √5) (5 + √5) is rational.
(ii) (√3 + 2)2
= (√3)2+22+2×√3×2    [(a + b)2=a+ b+ 2ab]
= 3 + 4 + 4√3
= 7 + 4√3
Since, the sum and product of rational numbers and an irrational number is always an irrational.
⇒7 + 4√3 is irrational.
Hence, (√3 + 2)2 is irrational.
(iii) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
= −1, which is an integer
Hence, RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9is rational.

(iv) √8 + 4√32 −6√2
=2√2 + 4 × 4√2 − 6√2
=2√2 + 16√2 − 6√2
=12√2
Since, the product of a rational number and an irrational number is always an irrational.
Hence, √8 + 4√32 − 6√2 is rational.

Q.7. On her birthday Reema distributed chocolates in an orphanage. The total number of chocolates she distributed is given by (5+ √11) (5− √11).
(i) Find the number of chocolates distributed by her.
(ii) Write the moral values depicted here by Reema.
Ans.

(i) As, (5 + √11) (5 − √11)
=52−(√11)2 [(a+b)(a−b)=a2−b2]
=25 − 11
=14
Hence, the number of chocolates distributed by Reema is 14.
(ii) The moral values depicted here by Reema is helpfulness and caring.
Disclaimer: The moral values may vary from person to person.

Q.8. Simplify
(i) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(ii) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iii) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Ans.
(i) 3√45 − √125 + √200 − √50
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
= 3×3√5 – 5√5 + 10√2 − 5√2
= 9√5 − 5√5 + 5√2
=4√5 + 5√2
(ii) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iii) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
= 6√2 + 20√2 − 3√2
= 23√2


RS Aggarwal Exercise 1.5 Number System 

Q.1. Represent √5 on the number line.
Ans.
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

To represent √5 on the number line, follow the following steps of construction:
(i) Mark points 0 and 2 as O and P, respectively.

(ii) At point A, draw AB ⊥ OA such that AB = 1 units.

(iii) Join OB.

(iv) With O as centre and radius OB, draw an arc intersecting the number line at point P.

Thus, point represents √5 on the number line.

Justification:
In right ΔOAB,

Using Pythagoras theorem,
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

Q.2. Locate √3 on the number line

Ans.

RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

To represent √3 on the number line, follow the following steps of construction:

(i) Mark points 0 and 1 as O and A, respectively.

(ii) At point A, draw AB ⊥ OA such that AB = 1 units.
(iii) Join OB.
(iv) At point B, draw DB ⊥ OA such that DB = 1 units.
(v) Join OD.

(vi) With O as centre and radius OD, draw an arc intersecting the number line at point Q.

Thus, point Q represents √3 on the number line.
Justification:

In right Δ OAB,

Using Pythagoras theorem,

RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

Again, in right ΔODB,
Using Pythagoras theorem,
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

Q.3. Locate √10 on the number line.

Ans.

RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

To represent √10 on the number line, follow the following steps of construction:

(i) Mark points 0 and 3 as O and B, respectively.

(ii) At point A, draw AB ⊥ OA such that AB = 1 units.

(iii) Join OA.

(iv) With O as centre and radius OA, draw an arc intersecting the number line at point P.
Thus, point P represents √10 on the number line.

Justification:

In right Δ OAB,
Using Pythagoras theorem,
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

Q.4. Locate √8 on the number line.
Ans.
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
To represent √8 on the number line, follow the following steps of construction:
(i) Mark points 0 and 2 as O and B, respectively.
(ii) At point B, draw AB ⊥ OA such that AB = 2 units.

(iii) Join OA.
(iv) With O as centre and radius OA, draw an arc intersecting the number line at point P.
Thus, point P represents √8 on the number line.
Justification:

In right ΔOAB,
Using Pythagoras theorem,

RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

Q.5. Represent RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9 geometrically on the number line.

Ans.
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

To representRS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9on the number line, follow the following steps of construction:
(i) Mark two points A and B on a given line such that AB = 4.7 units.
(ii) From B, mark a point C on the same given line such that BC = 1 unit.

(iii) Find the mid point of AC and mark it as O.
(iv) With O as centre and radius OC, draw a semi-circle touching the given line at points A and C.
(v) At point B, draw a line perpendicular to AC intersecting the semi-circle at point D.
(vi) With B as centre and radius BD, draw an arc intersecting the given line at point E.
Thus, let us treat the given line as the number line, with B as 0, C as 1, and so on, then point E represents RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9.
Justification:
Here, in semi-circle, radii OA = OC = OD = RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
And, OB = AB − AO = 4.7 − 2.85 = 1.85 units
In a right angled triangle OBD,
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

Q.6. Represent RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9 on the number line.
Ans.
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

To represent RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9 on the number line, follow the following steps of construction:
(i) Mark two points A and B on a given line such that AB = 10.5 units.
(ii) From B, mark a point C on the same given line such that BC = 1 unit.
(iii) Find the mid point of AC and mark it as O.
(iv) With O as centre and radius OC, draw a semi-circle touching the given line at points A and C.
(v) At point B, draw a line perpendicular to AC intersecting the semi-circle at point D.
(vi)  With B as centre and radius BD, draw an arc intersecting the given line at point E.
Thus, let us treat the given line as the number line, with B as 0, C as 1, and so on, then point E represents RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Justification:
Here, in semi-circle, radii OA = OC = OD = RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
And, OB = AB − AO = 10.5 − 5.75 = 4.75 units
In a right angled triangle OBD,
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

Q.7. Represent RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9 geometrically on the number line.
Ans.
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
To represent RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9 on the number line, follow the following steps of construction:
(i) Mark two points A and B on a given line such that AB = 7.28 units.
(ii) From B, mark a point C on the same given line such that BC = 1 unit.
(iii) Find the mid point of AC and mark it as O.
(iv) With O as centre and radius OC, draw a semi-circle touching the given line at points A and C.
(v) At point B, draw a line perpendicular to AC intersecting the semi-circle at point D.
(vi)  With B as centre and radius BD, draw an arc intersecting the given line at point E.
Thus, let us treat the given line as the number line, with B as 0, C as 1, and so on, then point E representsRS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Justification:
Here, in semi-circle, radii OA = OC = OD =RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
And, OB = AB − AO = 7.28 − 4.14 = 3.14 units
In a right angled triangle OBD,

RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

Q.8. Represent (1 + RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9) on the number line.
Ans.
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
To represent (1 + RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9) on the number line, follow the following steps of construction:
(i) Mark two points A and B on a given line such that AB = 9.5 units.
(ii) From B, mark a point C on the same given line such that BC = 1 unit.
(iii) Find the mid point of AC and mark it as O.
(iv) With O as centre and radius OC, draw a semi-circle touching the given line at points A and C.
(v) At point B, draw a line perpendicular to AC intersecting the semi-circle at point D.
(vi) With B as centre and radius BD, draw an arc intersecting the given line at point E.
(vii) From E, mark a point F on the same given line such that EF = 1 unit.
Thus, let us treat the given line as the number line, with B as 0, C as 1, E as RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9 and so on, then point F represents (1+RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9).
Justification:
Here, in semi-circle, radii OA = OC = OD = RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
And, OB = AB − AO = 9.5 − 5.25 = 4.25 units
In a right angled triangle OBD,
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

Q.9. Visualize the representation of 3.765 on the number line using successive magnification.
Ans.
3 < 3.765 < 4
Divide the gap between 3 and 4 on the number line into 10 equal parts.
Now, 3.7 < 3.765 < 3.8
In order to locate the point 3.765 on the number line, divide the gap between 3.7 and 3.8
into 10 equal parts.
Further, 3.76 < 3.765 < 3.77
So, to locate the point 3.765 on the number line, again divide the gap between 3.76 and 3.77 into 10 equal parts.

Now, the number 3.765 can be located on the number line. This can be shown as follows:
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Here, the marked point represents the point 3.765 on the number line.

Q.10. Visualize the representation of RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9 on the number line up to 4 decimal places.
Ans.
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9 = 4.6767  (Upto 4 decimal places)

4 < 4.6767 < 5
Divide the gap between 4 and 5 on the number line into 10 equal parts.
Now, 4.6 < 4.6767 < 4.7
In order to locate the point 4.6767 on the number line, divide the gap between 4.6 and 4.7
into 10 equal parts.
Further, 4.67 < 4.6767 < 4.68
To locate the point 4.6767 on the number line, again divide the gap between 4.67 and 4.68 into 10 equal parts.
Again, 4.676 < 4.6767 < 4.677
To locate the point 4.6767 on the number line, again divide the gap between 4.676 and 4.677 into 10 equal parts.
Now, the number 4.6767 can be located on the number line. This can be shown as follows:
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Here, the marked point represents the point RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9 on the number line up to 4 decimal places.


RS Aggarwal Exercise 1.6 Number System

Q.1. Write the rationalising factor of the denominator in RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Ans.
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Here, the denominator i.e. 1 is a rational number. Thus, the rationalising factor of the denominator in RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Q.2. Rationalise the denominator of each of the following.
(i)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(ii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iv)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(v)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(vi)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(vii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(viii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(ix)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Ans.
(i)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9On multiplying the numerator and denominator of the given number by √7, we get:RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9(ii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
On multiplying the numerator and denominator of the given number by √3, we get:
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9(iii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
On multiplying the numerator and denominator of the given number by 2 - √3, we get:
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9(iv)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
On multiplying the numerator and denominator of the given number by √5 +2, we get:

RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9(v)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

On multiplying the numerator and denominator of the given number by 5-3√2, we get:
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9(vi)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Multiplying the numerator and denominator by √7+√6, we get
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9(vii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Multiplying the numerator and denominator by √11+√7, we get
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9(viii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Multiplying the numerator and denominator by 2+√2, we get
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9 RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9(ix)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Multiplying the numerator and denominator by 3-2√2, we get
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

Q.3. It being given that √2 = 1.414, √3 = 1.732, √5 = 2.236 and √10 = 3.162, find the value of three places of decimals, of each of the following.
(i)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(ii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

Ans.
(i)
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(ii)
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

(iii)
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

Q.4. Find rational numbers a and b such that

(i)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(ii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iv)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

Ans.

(i)
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(ii)
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9(iii)
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iv)
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

Q.5. It being given  that √3 = 1.732, √5 = 2.236, √6 = 2.449  and √10 = 3.162, find to three places of decimal, the value of each of the following.

(i)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(ii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iv)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(v)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(vi)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Ans.
(i)
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
= 0.213
(ii)
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
=3 × (2.236 − 1.732)
= 1.512
(iii)

RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9(iv)
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(v)
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
= 16.660
(vi)
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
= 4.441

Q.6. Simplify by rationalising the denominator.
(i)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(ii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Ans.
(i)
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9(ii)
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Q.7. Simplify
(i)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(ii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iv)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Ans.
(i)
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9(ii)
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9= 0(iii)
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
= 16 − √3
(iv)
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
= 0

Q.8. Prove that
(i)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(ii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Ans.
(i)
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
= 2/2
= 1
(ii)
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Q.9. Find the values of a and b if
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Ans.
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Comparing with the given expression, we get
a = 0 and b = 1
Thus, the values of a and b are 0 and 1, respectively.

Q.10. Simplify
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Ans.
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

Q.11. If x = 3 + 2√2, check whether RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9is rational or irrational.Ans.
x = 3 + 2√2    .....(1)
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Adding (1) and (2), we getRS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9which is a rational numberThus, RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9is rational.Q.12.
If x = 2 − √3, find value of RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Ans.
x = 2 − √3    .....(1)
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Subtracting (2) from (1), we getRS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Thus, the value of RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Q.13. If x = 9 − 4√5, find the value of RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Ans.
x = 9 − 4√5    .....(1)
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Adding (1) and (2), we getRS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Squaring on both sides, we getRS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Thus, the value of x2 + RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9 is 322.Q.14. If x = RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9 find the value of RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Ans.
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Adding (1) and (2), we get
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Thus, the value of x +RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9is 5.Q.15. If a = 3 − 2√2, find the value of a- RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Ans.
a = 3−2√2
⇒ a2 = (3−2√2)2
⇒ a2 = 9 + 8 − 12√2
⇒ a= 17 − 12√2    .....(1)
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9 RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Subtracting (2) from (1), we getRS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Thus, the value of a2 - RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Q.16. If x = √13 + 2√3, find the value of x − RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Ans.
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Subtracting (2) from (1), we get
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Thus, the value of x RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Q.17.

If x = 2 + √3, find the value of RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Ans.
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9 
Adding (1) and (2), we get
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Cubing both sides, we getRS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Thus, the value of RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Q.18. If RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9andRS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9show that RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Ans.
Disclaimer: The question is incorrect.
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9The question is incorrect. Kindly check the question.The question should have been to show that x − y = RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Q.19. If a = RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9and b = RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9show that 3a+ 4ab − 3b= RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Ans.
According to question,
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Now,3a+ 4ab − 3b2
= 3(a− b2) + 4ab
= 3 (a + b)(a − b) + 4ab
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Hence, 3a+ 4ab − 3b= RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Q.20.
If a = RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9and b =RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9find the value of a2 + b2 – 5ab.
Ans.
According to question,
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Now,RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Hence, the value of a2 + b2 – 5ab is 93.

Q.21.
If p = RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9and q = RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9find the value of p2 + q2.

Ans.
According to question,
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Now,p2 + q2 = (p+q)− 2pq
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Hence, the value of p2 + q2 is 47.

Q.22. Rationalise the denominator of each of the following.
(i)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9 
(ii) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Ans.
(i)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Hence, the rationalised form is RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9(ii) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Hence, the rationalised form is RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9(iii) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Hence, the rationalised form is RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Q.23. Given, √2 = 1.414 and √6 = 2.449, find the value of RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9correct to 3 places of decimal.
Ans.
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Hence, the value of RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9correct to 3 places of decimal is −1.465.Q.24. If x = RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9find the value of x3 – 2x2 – 7x + 5.
Ans.
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Now,
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Also,
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Now,
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Hence, the value of x– 2x2 – 7x + 5 is 3.

Q.25. Evaluate RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9 it being given that √5 = 2.236 and √10 = 3.162.Hint
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Ans.
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Hence,RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9


RS Aggarwal Exercise 1.7 Number System 

Q.1. Simplify
(i) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(ii) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iii) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iv) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Ans.
(i) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9=21
= 2
(ii) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iv) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
= (6)3
= 216

Q.2. Simplify:
(i)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(ii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Ans.
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

Q.3. Simplify:
(i) 31/4 × 51/4
(ii) 25/8 × 35/8
(iii) 61/2 × 71/2
Ans.
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

Q.4. Simplify:
(i) (34)1/4
(ii) (31/3)4
(iii) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Ans.
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

Q.5. Evaluate
(i)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(ii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iv)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(v)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

(vi)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Ans.
(i) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(ii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iv)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(v)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(vi)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

Q.6. If a = 2, b = 3, find the values of
(i) (ab + ba)–1

(ii) (aa + bb)–1
Ans.

(i) (ab + ba)–1

(a+ ba−1 = (2+ 32−1
= (8 + 9)−1

= (17)−1
= 1/17
(ii) (aa + bb)–1

(a+ bb)−1 = (2+ 33)−1
= (4 + 27)−1
= (31)−1
=1/31

Q.7. Simplify

(i)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(ii) (14641)0.25
(iii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iv)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Ans.
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(ii) (14641)0.25
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iii) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iv)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

Q.8. Evaluate
(i) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(ii) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iii) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iv) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Ans.
(i)
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
=4(6)2 + (4)3 + 2(3)
=144 + 64 + 6
=214
(ii) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9(iii) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iv) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

Q.9. Evaluate
(i)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(ii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iii) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iv) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Ans.
(i) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(ii) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iii) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
= 2
(iv) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

Q.10. Prove that
(i) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(ii) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iii) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Ans.
(i) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
LHS = RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
= √2
= RHS
∴ RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9(ii) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9= RHS∴RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9(iii) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

Q.11. Simplify RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9and express the result in the exponential form of x.
Ans.
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9 

RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

Hence, the result in the exponential form is RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Q.12. Simplify the product RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Ans.
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

Q.13. Simplify
(i) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(ii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Ans.
(i) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

(ii)
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iii)
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

Q.14. Find the value of x in each of the following.
(i)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(ii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iv)5x−3 × 32x−8 = 225 
(v)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Ans.
(i) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Hence, the value of x is 6.
(ii) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Hence, the value of x is 22.
(iii) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Hence, the value of x is 5.

(iv) 5x−3×32x−8 = 225
⇒5x−3 × 32x−8 = (15)2
⇒5x−3 × 32x − 8 = 52 × 32
⇒x − 3 = 2 and 2x − 8 = 2
⇒x = 2 + 3 and 2x = 2 + 8
⇒x = 5 and 2x = 10
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9⇒ x = 5 and x = 5⇒ x = 5
Hence, the value of x is 5.

(v)
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Hence, the value of x is RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Q.15.
Prove that
(i)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(ii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(iv)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Ans.
(i) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Hence, RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
(ii)RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

=x0
=1
= RHS
Hence,RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9(iii)
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
=xab−ac−ba+bc.xac−bc
=x−ac+bc.xac−bc
=x−ac+bc+ac−bc
=x0
= 1
= RHS
Hence, RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9(iv) RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
= 1
= RHS
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9


Q.16. If x is a positive real number and exponents are rational numbers, simplify
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
Ans.
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

Q.17. If RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9= 1/27, prove that m – n = 1.
Ans.
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9Hence, m - n = 1
Q.18. Write the following in ascending order of magnitude.
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

Ans.
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9
On Comparing (1), (2) and (3), we get
RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9

The document RS Aggarwal Solutions: Number System- 2 Notes | Study Mathematics (Maths) Class 9 - Class 9 is a part of the Class 9 Course Mathematics (Maths) Class 9.
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