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If a,b,c are rational numbers `(a gt b gt c gt 0)` and quadratic equation `(a+b-2x) x ^(2) + (b+c-2a) x+ (c+a-2b)=0` has a root in the interval `(-1,0)` then which of the following statement (s) is/are correct ?A. `c+alt2b`B. both roots are rationalC. the equation `ax^(2)+2bx+c=0` have both negative real rootsD. the equation `cx^(2)+2ax+b=0` have both positive roots |
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Answer» Correct Answer - D If `a, b, c` are the ……….. `f(-1)f(0)lt 0` `(2a-b-c)(c+a-2b)lt 0` `((a-b)+(a-c))(c+a-2b)lt 0` Since `f(1) = 0` one root is 1 other : `(c+a-2b)/(a+b-2c)` using `(c+a-2b)` both roots of `{:(ax^(2)+2bx+c=0),(cx^(2)+2ax+b=0):}}` are real and negative `alpham, beta = (-2b+- sqrt(4b^(2)-4ac))/(2a)` `= (-b+- sqrt(b^(2)-4ac))/(2a)` `{:(=(-b+-sqrt(b^(2)-4ac))/(a)),(= -b+- ("less than b")):}}-ve` |
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