Question

Let $\mathrm{f}\left(\mathrm{x}\right)=\left[\mathrm{x}\right]$, where $\left[\mathrm{x}\right]$ denotes the greatest integer less than or equal to $\mathrm{x}$. If $\mathrm{a}=\sqrt{{\left(2011\right)}^2+2012}$, then the value of $\mathrm{f}\left(\mathrm{a}\right)$ is equal to

• A. $2010$
• B. $2011$
• C. $2012$
• D. $2013$

Answer 1

The correct option is B

$2011$

Determine the value of $\mathrm{f}\left(\mathrm{a}\right)$

Here, $\left[\mathrm{x}\right]$ denotes the greatest integer less than or equal to $\mathrm{x}$.

Let $\mathrm{x}=3.4$, then by the definition of the greatest integer function we have:

$$\begin{gathered} \Rightarrow \mathrm{f}\left(\mathrm{x}\right)=\left[\mathrm{x}\right] \\ \Rightarrow \mathrm{f}(3.4)=\left[3.4\right] \\ \Rightarrow \mathrm{f}(3.4)=3 \end{gathered}$$

Now we have,

$$\begin{gathered} \mathrm{a}=\sqrt{{\left(2011\right)}^2+2012} \\ \Rightarrow \mathrm{a}=2011.5 \end{gathered}$$

Therefore, using the greatest integer function we get,

$$\begin{gathered} \mathrm{f}\left(\mathrm{a}\right)=\left[\mathrm{a}\right] \\ \Rightarrow \mathrm{f}(2011.5)=\left[2011.5\right] \\ \Rightarrow \mathrm{f}(2011.5)=2011 \end{gathered}$$

Hence, option (B) is the correct answer.