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Consider the quadrationax^(2) - bx + c =0,a,b,c in N whichhas two distinct real roots belonging to theinterval (1,2). The least value of a is |
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Answer» 4 Let `f(x) = ax^(2) - bx + c `be thecorrespoding quadratic expression and `alpha,beta` be the roots of `f(x) = 0`. Then, `f(x) = a(x-alpha) (x -beta)` Now, `af(1) gt 0, af(2) gt 0,1lt(b)/(2a) LT2,b^(2) - 4ac gt 0` `rArr a(1-alpha) (1-beta) gt 0, a(2-alpha)(2-beta) gt 0, 2a lt b lt 4a, b^(2) - 4ax gt 0` `rArr a^(2) (1-alpha)(1-beta)(2-alpha) (2-beta) gt 0` `rArr a^(2) (1-alpha)(1-beta)(2-alpha)(2-beta) gt 0` As `f(x)` and `f(2)` both are integers and `f(1) gt 0`, and `f(2) gt 0`, so `f(1)f(2) gt 0` `rArr f(1) f(2) gt 1` `rArr 1ge a^(2)(alpha -1)(2-alpha)(beta-1)(2-beta)""(1)` Now, `((alpha-1)+(2-alpha))/(2)gt ((alpha -1)(2-alpha))^(1//2)` `rArr (alpha -1)(2-alpha) le (1)/(4)` Similarly, `(beta -1)(2-beta) le (1)/(4)` `rArr (alpha -1)(2-alpha) (beta- 1)(2-beta)lt (1)/(16)` As ` alpha ne beta`, so `alpha^(2) gt 1""("Using (1)")` `rArr a gt 5` `rArr b^(2) gt 20c and b gt 10 rArr b ge 11` Also, `b^(2) gt 100 rArr c gt 5 rArr c ge 6` |
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