1.

`[[a,b,ax+by],[b,c,bx+cy],[ax+by,bx+cy,0]]=(b^2-ac)(ax^2+2bxy+cy^2)`A. zeroB. positiveC. negativeD. `b^(2) + ac`

Answer» Correct Answer - C
We have,
`Delta = |(a,b,ax + by),(b,c,bx + cy),(ax + by,bx + cy,0)|`
`rArr Delta = |(a,b,ax + by),(b,c,bx + cy),(0,0,-(ax^(2) + 2bxy + cy^(2)))| " " [("Applying " R_(3) rarr),(R_(3) - xR_(1) - yR_(2))]`
`rArr Delta = (b^(2) - ac) (ax^(2) + 2bxy + cy^(2))`
Now, `b^(2) - ac lt 0 and a gt 0`
`rArr` Discriminant of `ax^(2) + 2bxy + cy^(2)` is negative and `a gt 0`
`rArr ax^(2) + 2bxy+ cy^(2) gt 0` for all x,`y in R`
`rArr Delta = (b^(2) -ac) (ax^(2) + 2bxy + cy^(2)) lt 0`


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