The total number of distinct `x in R` for which `|[x, x^2, 1+x^3] , [2x,4x^2,1+8x^3] , [3x, 9x^2,1+27x^3]|=10` is (A) 0 (B) 1 (C) 2 (D) 3

Answer

2

Explanation

` |{:(x,,x^(2),,1+x^(3)),(2x,,4x^(2),,1+8x^(3)),(3x,,9x^(2),,1+27x^(3)):}|=10`
`rArr x^(3) |{:(1,,1,,1+x^(3)),(2,,4,,1+8x^(3)),(3,,9,,1+27x^(3)):}|=10`
`rArr x^(3) |{:(1,,1,,1),(2,,4,,1),(3,,9,,1):}|+x^(6) |{:(1,,1,,1),(2,,4,,8),(3,,9,,27):}|=10`
`rArr x^(3) |{:(1,,0,,0),(2,,2,,-1),(3,,6,,-2):}|+x^(6) |{:(1,,0,,0),(2,,2,,6),(3,,6,,24):}|=10`
`rArr 6x^(6) +x^(3)-5=0`
`rArr (6x^(3) -5) (x^(3) +1) =0`
`rArr x^(3) =5//6 " or " x^(3) =-1`
Two real distinct values of x