Area bounded by the lines y = x, , ,x = - 1, , ,x = 2 and x - axi…

$1.$ One of the two boxes, box $I$ and box $II$, was selected at random and a ball was drawn randomly out of this box. The ball was found to be red. If the probability that this red ball was drawn from box $II$ is $\frac{1}{3}$, then the correct option$(s)$ with the possible values of $n_1, n_2, n_3$ and $n_4$ is(are)

$(A)$ $n_1=3, n_2=3, n_3=5, n_4=15$

$(B)$ $n_1=3, n_2=6, n_3=10, n_4=50$

$(C)$ $n_1=8, n_2=6, n_3=5, n_4=20$

$(D)$ $n_1=6, n_2=12, n_3=5, n_4=20$

$2.$ A ball is drawn at random from box $I$ and transferred to box $II$. If the probability of drawing a red ball from box $I$, after this transfer, is $\frac{1}{3}$, then the correct option$(s)$ with the possible values of $n_1$ and $n_2$ is(are)

$(A)$ $n_1=4, n_2=6$ $(B)$ $n_1=2, n_2=3$

$(C)$ $n_1=10, n_2=20$ $(D)$ $n_1=3, n_2=6$

Give the answer question $1$ and $2.$

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Let $f$ be a real-valued function defined on the interval $(0, \infty)$ by $f(x)=\ln x+\int_0^x \sqrt{1+\sin t} d t$. Then which of the following statement(s) is (are) true?

$(A)$ $f^{\prime \prime}(x)$ exists for all $x \in(0, \infty)$

$(B)$ $f^{\prime}(x)$ exists for all $x \in(0, \infty)$ and $f^{\prime}$ is continuous on $(0, \infty)$, but not differentiable on $(0, \infty)$

$(C)$ there exists $\alpha>1$ such that $\left|f^{\prime}(x)\right|<|f(x)|$ for all $x \in(\alpha, \infty)$

$(D)$ there exists $\beta>0$ such that $|f(x)|+\left|f^{\prime}(x)\right| \leq \beta$ for all $x \in(0, \infty)$

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$\left[ {\sum\limits_{n = 1}^{10} {\int_{ - 2n - 1}^{2n} {{{\sin }^{27}}x\,dx} } } \right] + \left[ {\sum\limits_{n = 1}^{10} {\int_{2n}^{2n + 1} {{{\sin }^{27}}x\,dx} } } \right]$ equals

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