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If xneynez" and " |{:(x,x^(2),1+x^(3)),(...

If `xneynez" and " |{:(x,x^(2),1+x^(3)),(y,y^(2),1+y^(3)),(z,z^(2),1+z^(3)):}|=0,` then `xyz` =

A

` 1`

B

`2`

C

`-1`

D

`-2`

Text Solution

Verified by Experts

The correct Answer is:
C
Since each element of `C_(1)` is the sum of two determinants we get
`Delta =|{:(x^(3),,x^(2),,x),(y^(3),,y^(2),,y),(z^(3),,z^(2),,z):}|+ |{:(1,,x^(2),,x),(1,,y^(2),,y),(1,,z^(2),,z):}|`
`=xyz |{:(x^(2),,x,,1),(y^(2),,y,,1),(z^(2),,z,,1):}|+|{:(1,,x^(2),,x),(1,,y^(2),,y),(1,,z^(2),,z):}|`
`=-(xyz +1) |{:(1,,x,,x^(2)),(1,,y,,y^(2)),(1,,z,,z^(2)):}|`
`=-(xyz +1) (x-y)(y-z)(z-x)(x+y+z)`
Since `Delta =0 ,x y,z ` all are distinct we have xyz +1=0
`" or " xyz=-1`
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