Total no. of ways $=5!=120$
Since all numbers are $>20,000$
$\therefore $ all numbers $2,3,5,7,9$ can come at first place.
$q\,:\,\begin{array}{*{20}{c}}
0&0&0&0&0 \\
3&4&3&2&1
\end{array}\,\,\,\begin{array}{*{20}{c}}
{place} \\
{ways}
\end{array}$
Total no. of ways $=3 \times 4!=72$
( $\therefore $ $2$ and $9$ can not be put at first place)
So, $p :q=120 : 72=5 :3$
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
$1.$ Which of the following is true?
$(A)$ $g$ is increasing on $(1, \infty)$
$(B)$ $g$ is decreasing on $(1, \infty)$
$(C)$ $g$ is increasing on $(1,2)$ and decreasing on $(2, \infty)$
$(D)$ $g$ is decreasing on $(1,2)$ and increasing on $(2, \infty)$
$2.$ Consider the statements :
$P$ : There exists some $x \in \operatorname{IR}$ such that $f(x)+2 x=2\left(1+x^2\right)$
$Q$ : There exists some $x \in \operatorname{IR}$ such that $2 f(x)+1=2 x(1+x)$ Then
$(A)$ both $P$ and $Q$ are true
$(B)$ $P$ is true and $Q$ is false
$(C)$ $P$ is false and $Q$ is true
$(D)$ both $P$ and $Q$ are false
Give the answer question $1$ and $2.$
$x+y+z=5, x+2 y+\lambda^2 z=9$
$x+3 y+\lambda z=\mu$, where $\lambda, \mu \in R$. Then, which of the following statement is NOT correct?