Consider the quadratic equation (c -- 5)x2 -- 2cx + (c -- 4) = 0, c [ ] 5.

Question

Consider the quadratic equation (c -- 5)x2 -- 2cx + (c -- 4) = 0, c $$ \ne $$ 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 -

Accepted Answer

Let f(x) = (c - 5)x2 - 2cx + c - 4

$$ \therefore $$  f(0)f(2) < 0      . . . . .(1)

\& f(2)f(3) < 0      . . . .    .(2)

from (1) and (2)

(c - 4)(c - 24) < 0

\& (c - 24)(4c - 49) < 0

$$ \Rightarrow $$  $${{49} \over 4}$$ < c < 24

$$ \therefore $$  s = {113, 14, 15, . . . . . 23}

Number of elements in set S = 11

Solutions · 1