Questions 3,4 and 5 only. . ...........................

Dear Pushkar
 
Please ask one question only in one thread.
5.
We write 3a² -  1 , As
= 3 (a² -  1)+
= 3 k  +  2,  Here  k = a² − 1.
And we know  that the square of an integer must either be of the form 3 k or 3k + 1.
Hence, 3a²  -  1 = 3k + 2 cannot be a perfect square.
we can also check for different value of a , As :
At a  =  1 , we get
3a² -  1  =  3 ( 1 )2 - 1  =  2 , That is not a perfect square .
Or
At a  = 2 , we get
3a² -  1  =  3 ( 2 )² - 1  =  11 , That is not a perfect square .
Or
At a  =  5 , we get
3a2 -  1  =  3 ( 5 )² - 1  =  74 , That is not a perfect square .
So,
we can say 3a² - 1 never be a perfect square.