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If |{:(a,,a^(2),,1+a^(3)),(b,,b^(2),,1+b...

If `|{:(a,,a^(2),,1+a^(3)),(b,,b^(2),,1+b^(3)),(c,,c^(2),,1+c^(3)):}|=0` and the vectors
`overset(to)((A)) =(1, a, a^(2)) , overset(to)((B)) = (1, b, b^(2)) , overset(to)((C))=(1,c,c^(2))`
are non-coplanar then the value of `abc` = ….

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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