1.

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

Answer»

4
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5

Solution :`az^(2) - bx + c =0`
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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