RS Aggarwal Class 10 Solutions Chapter 1 Real Numbers Ex 1D
These Solutions are part of RS Aggarwal Solutions Class 10. Here we have given RS Aggarwal Solutions Class 10 Chapter 1 Real Numbers Ex 1D. You must go through NCERT Solutions for Class 10 Maths to get better score in CBSE Board exams along with RS Aggarwal Class 10 Solutions.
Question 1.
Solution:
(i) Rational numbers: Numbers in the form of \(\frac { p }{ q }\) where p and q are integers and q ≠ 0, are called rational numbers.
(ii) Irrational numbers : The numbers which are not rationals, are called irrational numbers. Irrational numbers can be expressed in decimal form as non terminating non-repeating decimal.
(iii) Real numbers : The numbers which are rational or irrational, are called real numbers.
Question 2.
Solution:
(i) \(\frac { 22 }{ 7 }\)
It is a rational number as it is in the form of \(\frac { p }{ q }\)
(ii) 3.1416
It is a rational number as it is a terminating decimal.
(iii) π
It is an irrational number as it is nonterminating non-repeating decimal.
(iv) \(3.\bar { 142857 }\)
It is a rational number as it is nonterminating repeating decimal.
(v) 5.636363… = 5.63
It is a rational number as it is nonterminating repeating decimal.
(vi) 2.040040004…
It is an irrational number as it is nonterminating non-repeating decimal.
(vii) 1.535335333…
It is an irrational number as it is non terminating non-repeating decimal.
(viii) 3.121221222…
It is an irrational number as it is nonterminating non-repeating decimal.
(ix) √21
It is an irrational number aS it is not in the form of \(\frac { p }{ q }\)
(x) \(\sqrt [ 3 ]{ 3 }\)
It is an irrational number as it is not in the form of \(\frac { p }{ q }\)
Question 3.
Solution:
(i) √6 is irrational.
Let √6 is not an irrational number, but it is a rational number in the simplest form of \(\frac { p }{ q }\)
√6 = \(\frac { p }{ q }\) (p and q have no common factors)
Squaring both sides,
6 = \(\frac { { p }^{ 2 } }{ { q }^{ 2 } }\)
p² = 6q²
p² is divisible by 6
=> p is divisible by 6
Let p = 6a for some integer a
6q² = 36a²
=> q² = 6a²
q² is also divisible by 6
=> q is divisible by 6
6 is common factors of p and q
But this contradicts the fact that p and q have no common factor
√6 is irrational
(ii) (2 – √3) is irrational
Let (2 – √3) is a rational and 2 is also rational, then
2 – (2 – √3 ) is rational (Difference two rationals is rational)
=> 2 – 2 + √3 is rational
=> √3 is rational
But it contradicts the fact
(2 – √3) is irrational
(iii) (3 + √2 ) is irrational
Let (3 + √2 ) is rational and 3 is also rational
(3 + √2 ) – 3 is rational (Difference of two rationals is rational)
=> 3 + √2 – 3 is rational
=> √2 is rational
But it contradicts the fact (3 + √2 ) is irrational
(iv) (2 + √5 ) is irrational
Let (2 + √5 ) is rational and 2 is also rational
(2 + √5) – 2 is rational (Difference of two rationals is rational)
=> 2 + √5 – 2 is rational
=> √5 is rational
But it contradicts the fact (2 + √5) is irrational
(v) (5 + 3√2 ) is irrational
Let (5 + 3√2 ) is rational and 5 is also rational
(5 + 3√2 ) – 5 is rational (Difference of two rationals is rational)
=>5 + 3√2 – 5 is rational
=> 3√2 is rational
Product of two rationals is rational
3 is rational and √2 is rational
√2 is rational
But it contradicts the fact
(5 + 3√2 ) is irrational
(vi) 3√7 is irrational
Let 3√7 is rational
3 is rational and √7 is rational (Product of two rationals is rational)
But √7 is rational, it contradicts the fact
3√7 is irrational
(vii) \(\frac { 3 }{ \surd 5 }\) is irrational
Let \(\frac { 3 }{ \surd 5 }\) is rational
\(\frac { 3\times \surd 5 }{ \surd 5\times \surd 5 } =\frac { 3\surd 5 }{ 5 }\) is rational
\(\frac { 3 }{ 5 }\) is rational and √5 is rational
But √5 is a rational, it contradicts the fact
\(\frac { 3 }{ \surd 5 }\) is irrational
(viii)(2 – 3√5) is irrational
Let 2 – 3√5 is rational, 2 is also rational
2 – (2 – 3√5) is rational (Difference of two rationals is rational)
2 – 2 + 3√5 is rational
=> 3√5 is rational
3 is rational and √5 is rational (Product of two rationals is rational)
√5 is rational
But it contradicts the fact
(2 – 3√5) is irrational
(ix) (√3 + √5) is irrational
Let √3 + √5 is rational
Squaring,
(√3 + √5)² is rational
=> 3 x 5 + 2√3 x √5 is rational
=> 8 + 2√15 is rational
=> 8 + 2√15 – 8 is rational (Difference of two rationals is rational)
=> 2√15 is rational
2 is rational and √15 is rational (Product of two rationals is rational)
√15 is rational
But it contradicts the fact
(√3 + √5) is irrational
Question 4.
Solution:
Let \(\frac { 1 }{ \surd 3 }\) is rational
= \(\frac { 1 }{ \surd 3 } \times \frac { \surd 3 }{ \surd 3 } =\frac { \surd 3 }{ 3 } = \frac { 1 }{ 3 } \surd 3\) is rational
\(\frac { 1 }{ 3 }\) is rational and √3 is rationals (Product of two rationals is rational)
√3 is rational But it contradicts the fact
\(\frac { 1 }{ \surd 3 }\) is irrational
Question 5.
Solution:
(i) We can take two numbers 3 + √2 and 3 – √2 which are irrationals
Sum = 3 + √2 + 3 – √2 = 6 Which is rational
3 + √2 and 3 – √2 are required numbers
(ii) We take two. numbers
5 + √3 and 5 – √3 which are irrationals
Now product = (5 + √3) (5 – √3)
= (5)² – (√3 )² = 25 – 3 = 22 which is rational
5 + √3 and 5 – √3 are the required numbers
Question 6.
Solution:
(i) True.
(ii) True.
(iii) False, as sum of two irrational can be rational number also such as
(3 + √2) + (3 – √2) = 3 + √2 + 3 – √2 = 6 which is rational.
(iv) False, as product of two irrational numbers can be rational also such as
(3 + √2)(3 – √2 ) = (3)2 – (√2 )2 = 9 – 2 = 7
which is rational
(v) True.
(vi) True.
Question 7.
Solution:
Let (2√3 – 1) is a rational number and 1 is a rational number also.
Then sum = 2√3 – 1 + 1 = 2√3
In 2√3, 2 is rational and √3 is rational (Product of two rational numbers is rational)
But √3 is rational number which contradicts the fact
(2√3 – 1) is an irrational.
Question 8.
Solution:
Let 4 – 5√2 is a rational number and 4 is also a rational number
Difference of two rational number is a rational numbers
4 – (4 – 5√2 ) is rational
=> 4 – 4 + 5√2 is rational
=> 5√2 is rational
Product of two rational number is rational
5 is rational and √2 is rational
But it contradicts the fact that √2 is rational √2 is irrational
Hence, 4 – 5√2 is irrational
Question 9.
Solution:
Let (5 – 2√3) is a rational number and 5 is also a rational number
Difference of two rational number is rational
=> 5 – (5 – 2√3) is rational
=> 5 – 5 + 2√3 or 2√3 is rational
Product of two rational number is rational
2 is rational and √3 is rational
But it contradicts the fact
(5 – 2√3) is an irrational number.
Question 10.
Solution:
Let 5√2 is a rational
Product of two rationals is a rational
5 is rational and √2 is rational
But it contradicts the fact
5√2 is an irrational.
Question 11.
Solution:
\(\frac { 2 }{ \surd 7 } =\frac { 2\surd 7 }{ \surd 7\times \surd 7 } =\frac { 2\surd 7 }{ 7 } =\frac { 2 }{ 7 } \surd 7\)
Let \(\frac { 2 }{ 7 } \surd 7\) is a rational number, then
\(\frac { 2 }{ 7 }\) is rational and √7 is rational
But it contradicts the fact \(\frac { 2 }{ \surd 7 }\) is an irrational number.
Hope given RS Aggarwal Solutions Class 10 Chapter 1 Real Numbers Ex 1D are helpful to complete your math homework.
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