NCERT Solutions of Real Numbers Class 10
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Real Numbers

NCERT Solution - Exercise - 1.3

Question 1 - Prove that Class 10 math NCERT Solution 1.3_1 is irrational.

Answer: Let us assume the contrary, i.e. Class 10 math NCERT Solution 1.3_2 is irrational.

Thus, there can be two integers a and b (bā‰ 0) and a and b are coprime so that;

Class 10 math NCERT Solution 1.3_3

Squaring on both sides, we get;

Class 10 math NCERT Solution 1.3_4

This means that a2 is divisible by 5 and hence a is also divisible by 5.

This contradicts our earlier assumption that a and b are coprime, because we have found 5 as at least one common factor of a and b.

This also contradicts our earlier assumption that Class 10 math NCERT Solution 1.3_2 is irrational.

Class 10 math NCERT Solution 1.3_5

Question 2 - Prove that Class 10 math NCERT Solution 1.3_6 is irrational.

Answer: Let us assume to the contrary, i.e. Class 10 math NCERT Solution 1.3_7 is irrational.

Thus, there can be two integers a and b (bā‰ 0) and a and b are coprime so that;

Class 10 math NCERT Solution 1.3_8
Class 10 math NCERT Solution 1.3_9

Since a and b are rational, so  Class 10 math NCERT Solution 1.3_10 is rational and hence, Class 10 math NCERT Solution 1.3_11 is rational.

But this contradicts the fact that Class 10 math NCERT Solution 1.3_11 is irrational.

This happened because of our faulty assumption.

Class 10 math NCERT Solution 1.3_12

Question 3 - Prove that following are irrationals:

Class 10 math NCERT Solution 1.3_13

Answer: Let us assume to the contrary, i.e. Class 10 math NCERT Solution 1.3_14 is rational.

Thus, there can be two integers a and b (bā‰ 0) and a and b are coprime so that;

Class 10 math NCERT Solution 1.3_15

Squaring on both sides, we get;

Class 10 math NCERT Solution 1.3_16

This means that b2 is divisible by 2 and hence a is also divisible by 2.

This contradicts our earlier assumption that a and b are co-prime, because 2 is at least one common factor of a and b.

This also contradicts our earlier assumption that Class 10 math NCERT Solution 1.3_14 is rational.

Hence, Class 10 math NCERT Solution 1.3_14 is irrational proved.


Class 10 math NCERT Solution 1.3_17

Answer: Let us assume to the contrary, i.e. Class 10 math NCERT Solution 1.3_18 is rational.

There can be two integers a and b (bā‰ 0) and a and b are coprime, so that;

Class 10 math NCERT Solution 1.3_19

Squaring on both sides, we get;

Class 10 math NCERT Solution 1.3_20

This means that a2 is divisible by 245; which means that a is also divisible by 245.

This contradicts our earlier assumption that a and b are coprime, because 245 is at least one common factor of a and b.

This happened because of our faulty assumption and hence, Class 10 math NCERT Solution 1.3_18 is irrational proved.

Class 10 math NCERT Solution 1.3_21

Answer: Let us assume to the contrary, i.e. Class 10 math NCERT Solution 1.3_22 is rational.

Thus, there can be two integers a and b (bā‰ 0) and a and b are coprime so that;

Class 10 math NCERT Solution 1.3_23

Since a and b are rational, so Class 10 math NCERT Solution 1.3_24 is rational and hence, Class 10 math NCERT Solution 1.3_25 is rational.

But this contradicts the fact that Class 10 math NCERT Solution 1.3_25 is irrational.

This happened because of our faulty assumption.

Hence, Class 10 math NCERT Solution 1.3_26 is irrational proved.


    
  NCERT Solutions of Real Numbers Class 10
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