Let $f: R \rightarrow R$ be a function defined as $f(x) = a \sin \left(\frac{\pi[x]}{2}\right) + [2-x]$,$a \in R$,where $[t]$ is the greatest integer less than or equal to $t$. If $\lim_{x \rightarrow -1} f(x)$ exists,then the value of $\int_{0}^{4} f(x) dx$ is equal to.

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

If ${I_n} = \int_{ - n}^n {{{\tan }^2}\{x\}dx} $ then (where $\{.\}$ denotes the fractional part function and $n \in N$):

Let $F: R \rightarrow R$ be a thrice differentiable function. Suppose that $F(1)=0, F(3)=-4$ and $F'(x) < 0$ for all $x \in (1/2, 3)$. Let $f(x)=x F(x)$ for all $x \in R$.
$1.$ The correct statement$(s)$ is(are):
$(A) f'(1) < 0$
$(B) f(2) < 0$
$(C) f'(x) \neq 0$ for any $x \in (1, 3)$
$(D) f'(x)=0$ for some $x \in (1, 3)$
$2.$ If $\int_1^3 x^2 F '(x) dx = -12$ and $\int_1^3 x^3 F''(x) dx = 40$,then the correct expression$(s)$ is(are):
$(A) 9 f'(3)+f'(1)-32=0$
$(B) \int_1^3 f(x) dx = 12$
$(C) 9 f'(3)-f'(1)+32=0$
$(D) \int_1^3 f(x) dx = -12$
Give the answer for question $1$ and $2$.

Let $\operatorname{Max} \limits _{0 \leq x \leq 2}\left\{\frac{9-x^{2}}{5-x}\right\}=\alpha$ and $\operatorname{Min} \limits _ {0 \leq x \leq 2}\left\{\frac{9-x^{2}}{5-x}\right\}=\beta$. If $\int\limits_{\beta-\frac{8}{3}}^{2 \alpha-1} \operatorname{Max}\left\{\frac{9- x ^{2}}{5- x }, x \right\} dx =\alpha_{1}+\alpha_{2} \log _{e}\left(\frac{8}{15}\right)$,then $\alpha_{1}+\alpha_{2}$ is equal to

$A$ polynomial function $f(x)$ satisfying the conditions $f(x) = [f'(x)]^2$ and $\int_{0}^{1} f(x) dx = \frac{19}{12}$ can be: