Question

Set of all real value of a such that $f\left(x\right)=\frac{(2a-1)x^2(a+1)x+(2a-1)}{x^{2}2x+40}$ always negative is

• A. $(-\infty ,0)$
• B. $(0,\infty )$
• C. $(-\infty ,\frac{1}{2})$
• D. $(2,\infty )$

Answer 1

The correct option is A $(-\infty ,0)$

Set of all real value........................
$\frac{(2a-1)x^2+2(a+1)x+(2a-1)}{x^2-2x+40}<0$, for all x R

$(2a-1)x^2+2(a+1)x+(2a-1)<0,$ for all x R

$2a-1<0$ and $4(a+1{)}^{2}4(2a-1{)}^2<0$

$a<\frac{1}{2}$ and $3a(-a+2)<0$

$a<\frac{1}{2}$ and $a(a-2)>0$

$\iff a<\frac{1}{2}$ and {a < 0 or a > 2}

Thus a < 0