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The total number of distinct x in R for ...

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

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Given, `|{:(x, x^(2), 1+x^(3)),(2x, 4x^(2), 1+8x^(3)), (3x, 9x^(2), 1+27x^(3)):}|=10`
`rArr x*x^(2)|{:(1, 1, 1+x^(3)),(2, 4, 1+8x^(3)), (3, 9, 1+27x^(3)):}|=10`
Apply `R_(2) to R_(2) -2R_(1) " and "R_(3) to R_(3) -3R_(1)`, we get
`x^(3)|{:(1, 1, " " 1+x^(3)),(0, 2, -1+6x^(3)), (0, 6, -2+24x^(3)):}|=10`
`rArr x^(3)*|{:(2, 6x^(3)-1),(6,24x^(3)-2):}|=10`
`rArr x^(3) (48x^(3) -4 -36x^(3) + 6) = 10`
`rArr 12x^(6) + 2x^(3) = 10`
`rArr 6x^(6) + x^(3)-5 =0`
`rArr x^(3) = (5)/(6), -1`
`x = ((5)/(6))^(1//3), -1`
Hence, the number of real solutions is 2.
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