Consider the quadratic equation (c−5)x2−2cx+(c−4)=0, c≠5  Let S be the set of all integral values of c for which one root of the equation lies in the interval (0, 2) and its other root lies in the interval (2, 3). Then the number of elements in S is

 Consider the quadratic equation (c5)x22cx+(c4)=0, c5  Let S be the set of all integral values of c for which one root of the equation lies in the interval (0, 2) and its other root lies in the interval (2, 3). Then the number of elements in S is

  1. A

    11

  2. B

    12

  3. C

    10

  4. D

    18

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    Solution:

     : Let f(x)=(c5)x22cx+c4

       f(0)=c4

          f(2)=(c5)44c+c4=c24f(3)=(c5)96c+c4=4c49

    Now, f(0)f(2)<0 and f(2)f(3)<0

    (c4)(c24)<0 and (c24)(4c49)<04<c<24 ...(i) and 49/4<c<24

    From  (i) and (ii), c∈(49/4,24)

     S={13,14,15,,23}

    So, number of elements in S is 11.

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