Question

The values of $m$ for which $y=\frac{m{x}^2+3x-4}{-4x^2+3x+m}$ has range $R$ is

• A. $[2,5]$
• B. $(2,5)$
• C. $(1,7)$
• D. $[1,7]$

Answer 1

The correct option is C $(1,7)$

Given expression is $y=\frac{m{x}^2+3x-4}{-4x^2+3x+m}$

Finding the common roots of numerator and denominator
$m{x}^2+3x-4=0-4x^2+3x+m=0\Rightarrow (m+4)x^2-(m+4)=0$
When $m=-4$, then numerator and denominator becomes same, so $m\ne -4$
$\Rightarrow x^2=1\Rightarrow x=\pm 1$

Now, $f\left(x\right)=m{x}^2+3x-4$
We know that,
$f(-1)f\left(1\right)<0\Rightarrow (m-7)(m-1)<0\Rightarrow m\in (1,7)$