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if thesystemof equations (a-t)x+by +cz=0 bx+(c-t) y+az=0 cx+ay+(b-t)z=0 hasnon-trivial solutions thenproduct of allpossible values of t is |
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Answer» `|{:(a,,b,,c),(b,,c,,a),(c,,a,,b):}|` `DELTA= |{:(a-t,,b,,c),(b,,c-t,,a),(c,,a,,b-t):}|=0` `Delta=0` is acubicequation in t, so ithas 3solutionssay `t_(1), t_(2)" and" t_(3)` LET `Delta =p_(0)t_(3)+p_(1)t^(2) +p_(2)t+p_(3)` Clearly ,Po= coeff . of `t^(3)` which isequalto -1 , so `t_(1) t_(2)t_(3) =-(P_(3))/((-1))=P_(3)` = constant TERM in the expansion of `Delta i.e, Delta _((t=0))` hence `t_(1)t_(2)t_(3)= |{:(a,,b,,c),(b,,c,,a),(c,,a,,b):}|` |
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