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Let the quadratic equation (c-5)x^2 -2cx...

Let the quadratic equation `(c-5)x^2 -2cx+c-4=0` has one root in `(0,2)` and other root in `(2,3)` then find the number of intergral values of c in solution set (a) `18` (b) `12` (c) `11` (d) `10`

A

11

B

10

C

12

D

18

Text Solution

Verified by Experts

The correct Answer is:
A
Let `f(x)=(c-5)x^(2)-2cx+(c-4)=0.`
Then, according to problem, the graph of `g=f(x)` will be either of the two ways, show below.

In both cases `f(0).f(2)lt0and f(2)f(3)lt0`
Now, consider `" "f(0)f92)lt0`
`implies(c-4)[4(c-5)-4c+(c-4)]lt0`
`implies(c-4)(c-24)lt0`
`implies c in(4,24)" "...(i)`

Similiarly, `f(2).f(3)lt0`
`implies[4(c-5)-4c+(c-4)]`
`[9(c-5)-6c+(c-4)]lt0`
`implies (c-24)(4c-49)lt0`

`impliesc in ((49)/(4),24)" "...(ii)`
From Eqs. (i) and (ii), we get
`c in ((49)/(4),24)`
`therefore` Integral vaues of c are 13,14 ....., 23, Thus, 11 integral values of c are possible.
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