if `Delta= |{:(3,,4,,5,,x),(4,,5,,6,,y),(5,,6,,7,,z),(x,,y,,z,,0):}|=0` then

Options

A. `x,y,z " are in A.P."`
B. `x,y,z" are in G.P "`
C. `x, y, z " are in H.P "`
D. none of these

Answer

A

Explanation

Applying `R_(1) to R_(1)+R_(3)-2R_(2)`we get
`Delta = |{:(0,,0,,0,,x+z-2y),(4,,5,,6,,y),(5,,6,,7,,z),(x,,y,,z,,0):}|`
`=-(x+z+2y) |{:(4,,5,,6),(5,,6,,7),(x,,y,,z):}|" ""[Expanding along "R_(1)"]"`
`=-(x+z-2y) |{:(0,,-1,,6),(0,,-1,,7),(x-2x+z,,y-z,,z):}|`
`"[Applying "C_(1) to C_(1)+C_(3) -2C_(2) " and " C_(2) to C_(2)-C_(3)"]"`
`=-(x+z-2y)^(2) |{:(-1,,6),(-1,,7):}|`
`=(x-2y+z)^(2)`
Hence `Delta =0 rArr x,y,z " are in A.P "`