Question

Assertion :Consider the function
$f\left(x\right)={(}^{x+1}{C}_{2x-8}\left){(}^{2x-8}{C}_{x+1}\right)$
Statement-1: Domain of $f\left(x\right)$ is singleton. Reason: Statement-2: Range of $f\left(x\right)$ is singleton.

• A. Statement-1 is true, Statement-2 is true and Statement-2 is correct explanation for Statement-1.
• B. Statement-1 is true, Statement-2 is true and Statement-2 is NOT the correct explanation for Statement-1.
• C. Statement-1 is true, Statement-2 is false.
• D. Statement-1 is false, Statement-2 is true.

Answer 1

The correct option is B Statement-1 is true, Statement-2 is true and Statement-2 is NOT the correct explanation for Statement-1.

$f\left(x\right)={(}^{x+1}{C}_{2x-8}\left){(}^{2x-8}{C}_{x+1}\right)$
For $f\left(x\right)$ to be defined we must have $x+1\ge 2x-8$ and $2x-8\ge x+1$
Thus $2x-8=x+1\Rightarrow x=9$
Hence domain of $f\left(x\right)$ is singleton set Also range is a singleton set corresponding to $x=9$ which is $1.$
Clearly both the statement are correct but they are not related.