The value of the limit $\lim _{n \rightarrow \infty} \int _{0}^{1} x^{10} \sin (n x) d x$ equals

Let $f: \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \rightarrow \mathbb{R}$ be a continuous function such that $f(0)=1$ and $\int_0^{\frac{\pi}{3}} f(t) dt = 0$. Then which of the following statements is (are) $TRUE$?
$(A)$ The equation $f(x) - 3 \cos 3x = 0$ has at least one solution in $\left(0, \frac{\pi}{3}\right)$
$(B)$ The equation $f(x) - 3 \sin 3x = -\frac{6}{\pi}$ has at least one solution in $\left(0, \frac{\pi}{3}\right)$
$(C)$ $\lim_{x \rightarrow 0} \frac{x \int_0^x f(t) dt}{1 - e^{x^2}} = -1$
$(D)$ $\lim_{x \rightarrow 0} \frac{\sin x \int_0^x f(t) dt}{x^2} = -1$

The numbers $P, Q$ and $R$ for which the function $f(x) = P{e^{2x}} + Q{e^x} + Rx$ satisfies the conditions $f(0) = -1$,$f'(\log 2) = 31$ and $\int_0^{\log 4} [f(x) - Rx] \, dx = \frac{39}{2}$ are given by

Let $f(x) = \min \{[x-1], [x-2], \ldots, [x-10]\}$ where $[t]$ denotes the greatest integer $\leq t$. Then $\int_{0}^{10} f(x) \, dx + \int_{0}^{10} (f(x))^2 \, dx + \int_{0}^{10} |f(x)| \, dx$ is equal to

If $A_n = \int_{\frac{\pi}{2}}^{\infty} e^{-x} \cos^n x \, dx$,then $\frac{A_4 - A_6}{A_4} = $

If $m \in Z^{+}$,$n=2m$ and $\int_0^{\frac{\pi}{2}} \sin ^{m} x \cos ^{n} x \, dx = K(m) \int_0^{\frac{\pi}{2}} \sin ^m x \, dx$,then $\frac{2^{m-1}(m-1)!}{(2m-1)!} K(m) =$