For what value of `p`, is the system of equation `p^3x+(p+1)^3y=(p+2)^3 `and `px+(p+1)y=(p+2)` and `x+y=1` inconsistent

Options

A. `p = 0`
B. `p = 1`
C. `p = -1`
D. For all `pgt1`

Answer

C

Explanation

The given system of equations are :
`p^(3)x+(p+1)^(3)y=(p+2)^(3)" "...(1)`
`px+(p+1)y = (p+2)" "...(2)`
`x + y = 1" "...(3)`
This system is consistent, if values of x and y from first two equation satisfy the third equation.
which `rArr |{:(p^(3),(p+1)^(3),(p+2)^(3)),(p,(p+1),(p+2)),(1," " 1, " "1):}|=0`
Applying `C_(2) rarr C_(2) - C_(1)`
`rArr |{:(p^(3),(p+1)^(3)-p,(p+2)^(3)-p^(3)),(p," "1," "2),(1," " 0, " "0):}|=0`
`rArr 2(p+1)^(3)-2p^(3)-(p+2)^(3)+p^(3)=0`
`rArr 2(p^(3)+1+3p^(2)+3p)-2p^(3)-(p^(3)+8+12p+6p^(2))+p^(3)=0`
`rArr 2p^(3)+2+6p^(2)6p-2p^(3)-p^(3)-8-12p-6p^(2)+p^(3)=0`
`rArr -6-6p=0`
`rArr p = -1`