MCQ
$\frac{d}{{dx}}\log \left( {\frac{1}{x}} \right) = ............\left( {x > 0} \right)$
- A$ - \frac{1}{{{x^2}}}$
- B$\frac{1}{{{x^2}}}$
- ✓$ - \frac{1}{x}$
- D$ - \frac{2}{{{x^2}}}$
$\frac{d}{{dx}}\log \left( {\frac{1}{x}} \right) = \frac{d}{{dx}}\left[ { - \log x} \right] = - \frac{1}{x}$
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$1.$ 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 $2.$