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If a , b , c are non-zero, then the syst...

If `a , b , c` are non-zero, then the system of equations `(alpha+a)x+alphay+alphaz=0,alphax+ , (alpha+b)y+alphaz=0,alphax+alphay+(alpha+c)z=0` has a non-trivial solution if `alpha^(-1)=` (A) ` -(a^(-1)+b^(-1)+c^(-1))` (B) `a+b+c` (C) `alpha+a+b+c=1` (D) none of these

A

`2alpha=a+b+c`

B

`alpha^(-1)=a+b+c`

C

`alpha+a+b+c=1`

D

`alpha^(-1)=-(a^(-1)+b^(-1)+c^(-1))`

Text Solution

Verified by Experts

The correct Answer is:
D
`(d)` The given system of equations will have a non-trivial solution if
`|{:(alpha+a,alpha,alpha),(alpha,alpha+b,alpha),(alpha,alpha,alpha+c):}|=0`
Operate `R_(2)toR_(2)-R_(1)` , `R_(3)-R_(3)-R_(1)`, then
`|{:(alpha+a,alpha,alpha),(-a,b,0),(-a,0,c):}|=0`
`impliesalpha(bc+ca+ab)+abc=0`
Since, `a,b,c ne 0`
`:.(1)/(alpha)=-((1)/(a)+(1)/(b)+(1)/(c ))`
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