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 ""_____"`

Options

A. 1
B. 2
C. 3
D. 4

Answer

B

Explanation

We have,
`|(x,x^(2),1 + x^(3)),(2x,4x^(2),1 + 8x^(3)),(3x,9x^(2),1 + 27 x^(3))| = 10`
`rArr x^(3) |(1,1,1 + x^(3)),(2,4,1 + 8x^(3)),(3,9,1 + 27 x^(3))| = 10`
`rArr x^(3) |(1,1,1),(2,4,1),(3,9,1)| + x^(3) |(1,1,x^(3)),(2,4,8x^(3)),(3,9,27x^(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`
Applying `R_(2) rarr R_(2) - R_(1), R_(3) rarr R_(3) - R_(1)` in first determinant and `C_(2) rarr C_(2) - C_(1), C_(3) rarr C_(3) - C_(1)` in the second determinant, we get
`rArr x^(3) |(1,1,1),(1,3,0),(2,8,0)| + x^(6) |(1,0,0),(2,2,6),(3,6,24)| = 10`
`rArr ( 8-6) x^(3) + (48 - 36) x^(6) = 10`
`rArr 2x^(3) + 12x^(6) = 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 rArr x = ((5)/(6))^(1//3) or, x = -1 [ :' x in R]`