1.

If the system of linear equations ax+by+cz=0 cx+ay+bz=0 bx+cy+az=0 where a,b,c in R are non-zero and distinct , has a non-zero solution, then :

Answer»

a+B+C=0
a,b,c are in A.P.
`1/a,1/b,1/c` are inA.P.
a,b,c are in G.P.

SOLUTION :`|(a,b,c),(c,a,b),(b,c,a)|=0`
`RARR (a^2-bc)-b(ac-b^2)+c(c^2-ab)=0 rArr a^3 -abc-abc+b^3+c^3-abc=0`
`rArr a^3+b^3+c^3-3abc=0 rArr (a+b+c)(a^2+b^2+c^2-ab-b-ca)=0`
`rArr` a+b+c=0


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