Question

For $p>2$ and $x\in R$ , if the number of natural numbers in the range of $f\left(x\right)=\frac{x^{2} + 2 x + p}{x^{2} + 2 x + 2}$ is $3,$ then the value of $p$ is equal to

• A. 3
• B. 4
• C. 5
• D. 6

Answer 1

Answer: (C)

Here, $f\left(x\right)=1+\frac{p - 2}{\left(x + 1\right)^{2} + 1}\in \left(1 , p - 1\right]$
If there are $3$ natural numbers in the range
$\Rightarrow p-1=4\Rightarrow p=5$