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If the vector veca = hati + ahatj + a^(2...

If the vector `veca = hati + ahatj + a^(2) hatk , vecb = hati + b hatj + b^(2) hatk , vecc = hati + chatj + c^(2) hatk` are three non-coplanar vectors and `|{:(a , a^(2) , 1 + a^(3)) , (b , b^(2) , 1 + b^(3)) , (c , c ^(2) , 1 + c^(3)):}|` = 0 , then the value of abc is equal to

A

2

B

`-1`

C

`1`

D

0

Text Solution

Verified by Experts

The correct Answer is:
B
Since, `|(a,a^(2),1+a^(3)),(b,b^(2),1+b^(3)),(c,c^(2),a+c^(3))|=|(a,a^(2),1),(b,b^(2),1),(c,c^(2),1)|+|(a,a^(2),a^(3)),(b,b^(2),b^(3)),(c,c^(2),c^(3))|=0`
`implies |(a,a^(2),1),(b,b^(2),1),(c,c^(2),1)|=abc|(a,a^(2),1),(b,b^(2),1),(c,c^(2),1)|=0`
`implies (1+abc)|(a,a^(2),1),(b,b^(2),1),(c,c^(2),1)|=0" "[because|(a,a^(2),1),(b,b^(2),1),(c,c^(2),1)| ne 0]`
`implies 1+abc=0`
`implies abc=-1`
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