Correct Option: A

Given equation is,
( 5 + 2√3 ) / (7 + 4√3 )
Now Multiply and divide by ( 7 - 4√3 ) in above given equation.
By Rationalization, we will get,
[ ( 5 + 2√3 ) / ( 7 + 4√3 ) ] x [ ( 7 - 4√3 ) / ( 7 - 4√3 ) ]
[ ( 5 + 2√3 ) x ( 7 - 4√3 ) ] / [ ( 7 + 4√3 ) x ( 7 - 4√3 ) ]
Apply the formula and multiplication rule of algebra,
( P + Q ) x (R - S ) = P x R - P x S + Q x R - Q x S .................(1)
( a + b ) x ( a - b) = a² - b².......................................................(2)
= ( 5 x 7 - 5 x 4√3 + 7 x 2√3 - 2√3 x 4√3 )/ ( 7² - (4√3)² )
= 35 - 20√3 + 14√3 - 8 x 3 / ( 49 - 16 x 3 )
= ( 35 - 24 - 6√3 )/(49 - 48 )
= 11 - 6√3
∵ ( 5 + 2√3 ) /( 7 + 4√3 ) = a + b√3
∴ a + b√3 = 11 - 6√3
on equating the above equation and we get,
or a = 11 and b = - 6