Relations and Functions — Maths STD 12 Science — Question

$1.$ 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 $2.$

$1.$ The correct statement$(s)$ is(are)

$(A)$ $f^{\prime}(1) < 0$

$(B)$ $f(2) < 0$

$(C)$ $f^{\prime}(x) \neq 0$ for any $x \in(1,3)$

$(D)$ $f^{\prime}(x)=0$ for some $x \in(1,3)$

$2.$ If $\int_1^3 x^2 F^{\prime}(x) d x=-12$ and $\int_1^3 x^3 F^{\prime \prime}(x) d x=40$, then the correct expression$(s)$ is(are)

$(A)$ $9 f^{\prime}(3)+f^{\prime}(1)-32=0$

$(B)$ $\int_1^3 f(x) d x=12$

$(C)$ $9 f^{\prime}(3)-f^{\prime}(1)+32=0$

$(D)$ $\int_1^3 f(x) d x=-12$

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

Statement $-1$ : The probability that system of equations has a solution is $1$ .

Statement $-2$ : The probability that the system of equations has a unique solution is $\frac {3}{8}$ .