If `a ,b ,c` are in G.P. with common ratio `r_1a n dalpha,beta,gamma` are in G.P. with common ratio `r_2` and equations `a x+alphay+z=0,b x+betay+z=0,c x+gammay+z=0` have only zero solution, then which of the following is not true? `a+b+c` b. `a b c` c.`1` d. none of these

Options

A. `r_(1) ne 1`
B. `r_(2) ne 1`
C. `r_(1) ne r_(2)`
D. none of these

Answer

D

Explanation

As a,b,c are in G.P. with common ration `r_(1) ` and `alpha , beta , gamma` are in G.P having common ratio `r_(2) , a ne 0 , alpha ne 0,b =ar_(1) c=ar_(1)^(2)`
`Beta =ar_(2) ,gamma =ar_(2)^(2)`
Also the system of equations has only zero (trivial) solution
`Delta |{:(a,,alpha,,1),(b,,beta,,1),(c,,gamma,,1):}|ne 0`
`rArr aalpha |{:(1,,1,,1),(r_(1),,r_(2),,1),(r_(1)^(2),,r_(2)^ (2),,1):}| ne 0`
`rArr aalpha (r_(1)-1) (r_(2) -1) (r_(1) -r_(2)) ne 0`
`rArr r_(1) ne 1 , r_(2) ne 1 " and " r_(1) ne r_(2)`