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Consider the system of equations a(1) ...

Consider the system of equations
`a_(1) x + b_(1) y + c_(1) z = 0`
`a_(2) x + b_(2) y + c_(2) z = 0`
`a_(3) x + b_(3) y + c_(3) z = 0`
If `|(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3))| =0`, then the system has

A

more than two solutions

B

one trivial and one non-trivial solutions

C

no solution

D

only trivial solution `(0, 0,0)`

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A
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Knowledge Check

  • Cosnsider the system of equation a_(1)x+b_(1)y+c_(1)z=0, a_(2)x+b_(2)y+c_(2)z=0, a_(3)x+b_(3)y+c_(3)z=0 if |{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|=0 , then the system has

    A
    More than two solutions
    B
    One trivial and one-non trivial solutions
    C
    No solution
    D
    Only trivial solution (0,0,0)

  • Consider the system of linear equations a_(1)x+b_(1)y+ c_(1)z+d_(1)=0 , a_(2)x+b_(2)y+ c_(2)z+d_(2)= 0 , a_(3)x+b_(3)y +c_(3)z+d_(3)=0 , Let us denote by Delta (a,b,c) the determinant |{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}| , if Delta (a,b,c) # 0, then the value of x in the unique solution of the above equations is

    A
    `(Delta(bcd))/(Delta(abc))`
    B
    `(-Delta(bcd))/(Delta(abc))`
    C
    `(Delta(acd))/(Delta(abc))`
    D
    `(Delta(abd))/(Delta(abc))`

  • If a_(1)b_(1)c_(1), a_(2)b_(2)c_(2) and a_(3)b_(3)c_(3) are 3 digit even natural numbers and Delta = |{:(c_(1),a_(1), b_(1)),(c_(2),a_(2), b_(2)),(c_(3),a_(3),b_(3)):}| , then Delta is:

    A
    divisible by 2 but not necessarily by 4
    B
    divisible by 4 but not necessarily by 8
    C
    divisible by 8
    D
    none of these

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