- Home
- Standard 12
- Mathematics
1.Relation and Function
hard
Let $[t]$ be the greatest integer less than or equal to $t$. Let $A$ be the set of al prime factors of $2310$ and $f: A \rightarrow \mathbb{Z}$ be the function $f(x)=\left[\log _2\left(x^2+\left[\frac{x^3}{5}\right]\right)\right]$. The number of one-to-one functions from $A$ to the range of $f$ is :
A
$20$
B
$120$
C
$25$
D
$24$
(JEE MAIN-2024)
Solution
$\mathrm{N}=2310 $$ =231 \times 10 $
$= 3 \times 11 \times 7 \times 2 \times 5$
$ A=\{2,3,5,7,11\} $
$ f(x)=\left[\log _2\left(x^2+\left[\frac{x^3}{5}\right]\right)\right]$
$ f(2)=\left[\log _2(5)\right]=2 $
$ f(3)=\left[\log _2(14)\right]=3 $
$ f(5)=\left[\log _2(25+25)\right]=5 $
$ f(7)=\left[\log _2(117)\right]=6 $
$ f(11)=\left[\log _2 387\right]=8$
Range of $f: B=\{2,3,5,6,8\}$
No. of one-one functions $=5 !=120$
Standard 12
Mathematics
