Find the value of determinant `|[sqrt((13))+sqrt(3),2sqrt(5),sqrt(5)],[sqrt((15))+sqrt((26)),5,sqrt((10))],[3+sqrt((65)),sqrt((15)),5]|`

Options

A. `sqrt3(5-sqrt6)`
B. `-5sqrt3(5-sqrt6)`
C. `5sqrt3(5-sqrt6)`
D. None of these

Answer

B

Explanation

`=|{:(sqrt(13),2sqrt(5),sqrt(5)),(sqrt(15),5,sqrt(10)),(3,sqrt(15),5):}| + |{:(sqrt(3) , 2sqrt(5) , sqrt(5)),(sqrt(26) , 5, sqrt(10)),(sqrt(65),sqrt(15),5):}|`
` = ( sqrt(5))^(2) |{:(sqrt(13),2,1),(sqrt(15),sqrt(5),sqrt(2)),(3,sqrt(3),5):}| + (sqrt(5))^(2) |{:(sqrt(3) ,2,1),(sqrt(26),sqrt(5),sqrt(2)),(sqrt(65),sqrt(3), sqrt(5)),(sqrt(65) , sqrt(3), sqrt(5)):}|`
`C_(1) to C_(1) - sqrt(3)C_(2) " " C_(1) to C_(1) - sqrt(13)C_(3)`
`=|{:(sqrt(13)-2sqrt(3),2,1),(0,sqrt(5),sqrt(2)),(0,sqrt(3),sqrt(5)):}| + 5 |{:(sqrt(3)-sqrt(13),2,1),(0,sqrt(5),sqrt(2)),(0,sqrt(3),sqrt(5)):}|`
`=5[(sqrt(13)-2sqrt(3))(5-sqrt(6))+(sqrt(3)-sqrt(13))(5-sqrt(6))]`
`=5[5-sqrt(6)][- sqrt(3)]`
` = - 5sqrt(3)(5-sqrt(6))`