1.

If `a gt 0` and discriminant of `ax^(2) + 2bx + c` is negative, then `Delta = |(a,b,ax +b),(b,c,bx +c),(ax +b,bx +c,0)|`, isA. positiveB. `(ac -b^(2)) (ax^(2) + 2bx +c)`C. negativeD. 0

Answer» Correct Answer - C
It is given that the discriminant of `ax^(2) + 2bx + c` is negative.
`:. 4b^(2) - 4ac lt 0 rArr b^(2) - ac lt 0 rArr ac -b^(2) gt 0`
Also, `a gt 0` and discriminant is negative.
`:. Ax^(2) + 2bx + c gt 0` for all `x in R`
Now,
`Delta = |(a,b,ax + b),(b,c,bx + c),(ax +b,bx+c,0)|`
`rArr Delta = |(a,b,0),(b,c,0),(ax +b,bx +c,- x (ax +b) - (bx +c))|`
`rArr Delta = |(a,b,0),(b,c,0),(ax +b,bx +c,-(ax^(2) + 2bx +c))|`
`rArr Delta = -(ax^(2) + 2bx + c) (ac -b^(2)) lt 0` [Using (i) and (ii)]


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