$A$ six-faced unbiased die is thrown twice and the sum of the numbers appearing on the upper face is observed to be $7$. The probability that the number $3$ has appeared at least once is:

$A$ body cools from $62^{\circ}C$ to $50^{\circ}C$ in $10$ minutes and to $42^{\circ}C$ in the next $10$ minutes. The temperature of the surrounding is ........... $^{\circ}C$.

In a saturated solution of $Mg(OH)_2$,the degree of dissociation of $Mg(OH)_2$ is $\alpha$. Find the concentration $(C)$ of $Mg(OH)_2$ if the concentration of $[OH^{-}]$ is $2$.

An electron moves with speed $2 \times 10^5 \ m/s$ along the positive $x$-direction in the presence of a magnetic field of induction $B = (\hat{i} + 4\hat{j} - 3\hat{k}) \ T$. The magnitude of the force experienced by the electron in newtons is (Charge on the electron $= 1.6 \times 10^{-19} \ C$)

Two tangent galvanometers $A$ and $B$ have coils of radii $8 \, cm$ and $16 \, cm$ respectively and resistance $8 \, \Omega$ each. They are connected in parallel with a cell of emf $4 \, V$ and negligible internal resistance. The deflections produced in the tangent galvanometers $A$ and $B$ are $30°$ and $60°$ respectively. If $A$ has $2$ turns, then $B$ must have: (in $turns$)

An increase in pressure required to decrease the $200\,L$ volume of a liquid by $0.004\%$ in a container is ........... $kPa$. (Bulk modulus of the liquid $= 2100\,MPa$)