Which of the following inequalities is/are $TRUE$?
$(A)$ $\int_0^1 x \cos x \, dx \geq \frac{3}{8}$
$(B)$ $\int_0^1 x \sin x \, dx \geq \frac{3}{10}$
$(C)$ $\int_0^1 x^2 \cos x \, dx \geq \frac{1}{2}$
$(D)$ $\int_0^1 x^2 \sin x \, dx \geq \frac{2}{9}$

The number of continuous functions $f:[0,1] \rightarrow(-\infty, \infty)$ satisfying the condition $\int_0^1 (f(x))^2 dx = 2 \int_0^1 f(x) dx$ is

List $I$	List $II$
$P.$ The number of polynomials $f(x)$ with non-negative integer coefficients of degree $\leq 2$,satisfying $f(0)=0$ and $\int_0^1 f(x) dx=1$,is	$1.$ $8$
$Q.$ The number of points in the interval $(-\sqrt{13}, \sqrt{13})$ at which $f(x)=\sin(x^2)+\cos(x^2)$ attains its maximum value,is	$2.$ $2$
$R.$ $\int_{-2}^2 \frac{3x^2}{1+e^x} dx$ equals	$3.$ $4$
$S.$ $\frac{\int_{-1/2}^{1/2} \cos 2x \log(\frac{1+x}{1-x}) dx}{\int_0^{1/2} \cos 2x \log(\frac{1+x}{1-x}) dx}$ equals	$4.$ $0$

Codes: $P \quad Q \quad R \quad S$

Let $f(x)=(1-x)^2 \sin ^2 x+x^2$ for all $x \in \mathbb{R}$ and let $g(x)=\int_1^x \left(\frac{2(t-1)}{t+1}-\ln t\right) f(t) dt$ for all $x \in (1, \infty)$.
$1.$ 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 $x \in \mathbb{R}$ such that $f(x)+2x=2(1+x^2)$
$Q$ : There exists some $x \in \mathbb{R}$ such that $2f(x)+1=2x(1+x)$
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 for question $1$ and $2$.

$A$ continuous and differentiable function $f$ satisfies the condition $\int_{0}^{x} f(t) dt = f^2(x) - 1$ for all real $x$. Then:

If $f(x) = A \sin \left( \frac{\pi x}{2} \right) + B$,$f'(1/2) = \sqrt{2}$ and $\int_{0}^{1} f(x) dx = \frac{2A}{\pi}$,then the constants $A$ and $B$ are respectively.