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Find the value of determinat `|{:(sqrt(13)+sqrt(3),2sqrt(5),sqrt(5)),(sqrt(15)+sqrt(26),5,sqrt(10)),(3+sqrt(65),sqrt(15),5):}|`.

A

`sqrt3(5-sqrt6)`

B

`-5sqrt3(5-sqrt6)`

C

`5sqrt3(5-sqrt6)`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
B
`=|{:(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))`
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