(2a + 3b) (5a + 7b) =

$1.$ 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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When the tangent to the curve $\text{y}=\text{x}\log\text{x}$ is parallel to the chord joining the points $(1, 0)$ and $(e, e),$ the value of $x$ is :

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Choose the correct answer from the given four options.The value of $\sin^{-1}\bigg[\cos\Big(\frac{33\pi}{5}\Big)\bigg]$ is:

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$\int_{}^{} {\frac{{\sin x}}{{\sin x - \cos x}}} \;dx = $

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