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

Options

A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.

Answer

A

Explanation

Let `y = (x^(2)-2x+4)/(x^(2)+2x+4)`. Then, `x^(2)(y-1) + 2x(y+1) + 4(y-1) = 0`
As x is real.
`therefore" "4(y+1)^(2)-16(y-1)^(2) gt 0 rArr (3y -1)(y-3) le 0 rArr (1)/(3) le y le 3`
Som statement-2 is true.
Let `alpha = 3^(x+1)`. Then, `(9xx3^(2x)+6xx3^(x)+4)/(9xx3^(2x)-6xx3^(x)+4)=((3^(x+1))^(2)+2xx3^(x+1)+4)/((3^(x+1))^(2)-2xx3^(x+1)+4)`
`rArr" "(9xx3^(2x)+6xx3^(x)+4)/(9xx3^(2x)-6xx3^(x)+4)=(alpha^(2)+2alpha+4)/(alpha^(2)-2alpha+4)`
From Statement-2, we have `(1)/(3) le (alpha^(2)-2alpha+4)/(alpha^(2)+2alpha+4) le 3 "for all" alpha in R`
`rArr" "(1)/(3)le(alpha^(2)+2alpha+4)/(alpha^(2)-2a+4)le 3 "for all" alpha in R`
`rArr" "(1)/(3)le(9xx3^(2x)+6xx3^(x)+4)/(9xx3^(2x)-6xx3^(x)+4) le 3 "for all x in R`
Statement-1 is true and statement-2 is a correct explanation for statement-1.