Question

If $f\left(x\right)=\frac{x^2+14x+9}{x^2+2x+3},g\left(x\right)=x^2-4x+9$ and $h\left(x\right)=-3x^2-4x+p$,

then which of the following option(s) is/are correct?

• A. Number of integers in the range of $f\left(x\right)$ is $8$
• B. Maximum value of $g\left(f\right(x\left)\right)$ is $54$
  
• C. Number of integral values of $p$ for which $h\left(f\right(x\left)\right)>0$ and $h\left(g\right(x\left)\right)<0$ is $28$
• D. Number of integral values of $p$ for which $h\left(f\right(x\left)\right)>0$ and $h\left(g\right(x\left)\right)<0$ is $30$

Answer 1

Answer: (B), (D)

The correct options are
B Maximum value of $g\left(f\right(x\left)\right)$ is $54$

D Number of integral values of $p$ for which $h\left(f\right(x\left)\right)>0$ and $h\left(g\right(x\left)\right)<0$ is $30$

$y=\frac{x^2+14x+9}{x^2+2x+3}$
$\Rightarrow x^2(y-1)+(2y-14)x+3y-9=0$
$(2y-14{)}^2-4(y-1\left)\right(3y-9)\ge 0$
$\Rightarrow y\in [-5,4]$

$g\left(x\right)=x^2-4x+9=(x-2{)}^2+5$
Maximum value of $g\left(f\right(x\left)\right)$ will occur at $f\left(x\right)=-5$ and $g(-5)=54$

Now, $h(-5)>0$
$\Rightarrow -3(-5{)}^2-4(-5)+p>0\Rightarrow p>55$

Also,$h\left(4\right)>0\Rightarrow -3(4{)}^2-4(4)+p>0\Rightarrow p>64$
and $h\left(5\right)<0\Rightarrow -3(5{)}^2-4(5)+p<0\Rightarrow p<95$