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RD Chapter 1 Real Numbers Ex 1.5 Solutions

Question - 1 : -
Show that the following numbers are irrational
(i) 1/√2
(ii) 7 √5
(iii) 6 + √2
(iv) 3 – √5

Answer - 1 : -


But itcontradics that because √5 is irrational
3 – √5 is irrational

Question - 2 : -
Prove that following numbers are irrationals :
(i) 2/√7
(ii) 3/2√5
(iii) 4 + √2
(iv) 5 √2

Answer - 2 : -


5 √2 is an irrational number

Question - 3 : - Show that 2 – √3 is an irrational number. [C.B.S.E. 2008]

Answer - 3 : -

Let 2 – √3 is not an irrational number
 
√3 is a rational number
But it contradicts because √3 is an irrational number
2 – √3 is an irrational number
Hence proved.

Question - 4 : - Show that 3 + √2 is an irrational number.

Answer - 4 : - Let 3 + √2 is a rational number

and √2 is irrational
But our suppositon is wrong
3 + √2 is an irrational number
 

Question - 5 : - Prove that 4 – 5√2 is an irrational number. [CBSE 2010]

Answer - 5 : -

Let 4 – 5 √2 is not are irrational number
and let 4 – 5 √2 is a rational number
and 4 – 5 √2 = a/b where a and b are positive prime integers
 
√2 is a rational number
But √2 is an irrational number
Our supposition is wrong
4 – 5 √2 is an irrational number

Question - 6 : - Show that 5 – 2 √3 is an irrational number.

Answer - 6 : -

Let 5 – 2 √3 is a rational number
Let 5 – 2 √3 = ab where a and b are positive integers
 
and √3 is a rational number
Our supposition is wrong
5 – 2 √3 is a rational number

Question - 7 : - Prove that 2 √3 – 1 is an irrational number. [CBSE 2010]

Answer - 7 : -

Let 2 √3 – 1 is not an irrational number
and let 2 √3 – 1 a ration number
and then 2 √3 – 1 = a/b where a, b positive prime integers
 
√3 is a rational number
But √3 is an irrational number
Our supposition is wrong
2 √3 – 1 is an irrational number

Question - 8 : - Prove that 2 – 3 √5 is an irrational number. [CBSE 2010]

Answer - 8 : -

Let 2 – 3 √5 is not an irrational number and let 2 – 3 √5 is a rational number
Let 2 – 3 √5 = a/b where a and b are positive prime integers
 
⟹2b−a3b=√5
√5 is a rational
But √5 is an irrational number
Our supposition is wrong
2 – 3 √5 is an irrational

Question - 9 : - Prove that √5 + √3 is irrational.

Answer - 9 : -

Let √5 + √3 is a rational number
and let √5 + √3 = a/b where a and b are co-primes
 
√3 is a rational number
But it contradics as √3 is irrational number
√5 + √3 is irrational

Question - 10 : - Prove that √2 + √3 is an irrational number.

Answer - 10 : -

Let us suppose that √2 + √3 is rational.
Let √2 + √3 = a, where a is rational.
Therefore, √2 = a – √3
Squaring on both sides, we get
 
which is a contradiction as the right hand side is a rational number while √3 is irrational.
Hence, √2 + √3 is irrational.

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