if the system of equations `(a-t)x+by +cz=0` `bx+(c-t) y+az=0` `cx+ay+(b-t)z=0` has non-trivial solutions then product of all possible values of t isA. `|{:(a,,b,,c),(b,,c,,a),(c,,a,,b):}|`B. `a+b+c`C. `a^(2)+b^(2)+c^(2)`D. `1`

if the system of equations `(a-t)x+by +cz=0` `bx+(c-t) y+az=0` `cx+ay+

1.	if the system of equations `(a-t)x+by +cz=0` `bx+(c-t) y+az=0` `cx+ay+(b-t)z=0` has non-trivial solutions then product of all possible values of t isA. `\|{:(a,,b,,c),(b,,c,,a),(c,,a,,b):}\|`B. `a+b+c`C. `a^(2)+b^(2)+c^(2)`D. `1`
Answer» Correct Answer - A The given system of equations will have a non-trivial solutions if the determinant of coefficient is 0. `Delta= \|{:(a-t,,b,,c),(b,,c-t,,a),(c,,a,,b-t):}\|=0` `Delta=0` is a cubic equation in t, so it has 3 solutions say `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 is equal to -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):}\|`