Determinants — Maths STD 12 Science — Question

$1.$ 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 $2.$ 

Consider the two statements :

($I$) $\mathrm{R}$ is reflexive but not symmetric.

($II$) $\mathrm{R}$ is transitive

Then which one of the following is true?