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Given that, for all real x, the expressi...

Given that, for all real x, the expression `(x^2+2x+4)/(x^2-2x+4)` lies between `1/3` and 3. The values between which the expression `(9.3^(2x)+6.3^x+4)/(9.3^(2x)-6.3^x+4)` lies are

A

`1/3` and `3`

B

`3/2` and 2

C

`-1` and 1

D

0 and 2

Text Solution

Verified by Experts

The correct Answer is:
B
We have `1/3lt(x^(2)-2x+4)/(x^(2)+2x+4)lt3,AA xepsilonR`
`1/2lt(x^(2)+2x+4)/(x^(2)-2x+4)lt3,AAxepsilonR`
Let `y=(9.3^(2x)+6.3^(x)+4)/(9.3^(2x)-6.3^(x)+4)=((3^(x+1))^(2)+2.3^(x+1)+4)/((e^(x+1))^(2)-2.3^(x+1)+4)`
`(t^(2)+2t+4)/(t^(2)-2t+4)` where `t=3^(x+1)`
`(y-1)t^(2)-2(y+1)t+4(y-1)=0`
By the given condition for every `t epsilonR`
`1/2ltylt3`.....i
But`t=3^(x+1)gt0`
We have product of the roots `=4gt0` which is true.
And sum of the roots `=(2(y+1))/((y-1)ge0`
`implies(y+1)/(y-1)gt0`
`:.yepsilon (-oo,-1)uu(1,o)`...ii
From Eqs (i) and (ii) we get
`1ltylt3`
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