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Q3 (3)   Prove that the following are irrationals :  6 + \sqrt 2

Let us assume  is rational. This means  can be written in the form  where p and q are co-prime integers. As p and q are integers  would be rational, this contradicts the fact that  is irrational. This contradiction arises because our initial assumption that  is rational was wrong. Therefore  is irrational.

 Q3 (2)   Prove that the following are irrationals :

(ii) 7 \sqrt 5

Let us assume  is rational. This means  can be wriiten in the form  where p and q are co-prime integers. As p and q are integers  would be rational, this contradicts the fact that  is irrational. This contradiction arises because our initial assumption that  is rational was wrong. Therefore  is irrational.

Q3  Prove that the following are irrationals : 

(i) 1/ \sqrt 2 

Let us assume  is rational. This means  can be written in the form  where p and q are co-prime integers. Since p and q are co-prime integers  will be rational, this contradicts the fact that   is irrational. This contradiction arises because our initial assumption that   is rational was wrong. Therefore  is irrational.

Q2  Prove that 3 + 2 \sqrt 5  is irrational.

Let us assume  is rational. This means  can be wriiten in the form  where p and q are co-prime integers. As p and q are integers  would be rational, this contradicts the fact that  is irrational. This contradiction arises because our initial assumption that  is rational was wrong. Therefore  is irrational.

Q1  Prove that \sqrt 5  is irrational.

Let us assume  is rational. It means  can be written in the form  where p and q are co-primes and  Squaring both sides we obtain From the above equation, we can see that p2 is divisible by 5, Therefore p will also be divisible by 5 as 5 is a prime number.  Therefore p can be written as 5r p = 5r  p2 = (5r)2 5q2 = 25r2 q2 = 5r2 From the above equation, we can see that q2 is divisible by 5,...
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