$n$-Propyl alcohol and isopropyl alcohol can be chemically distinguished by which of the following reagents?

Observe the following statements:
$(I)$ In $\triangle ABC, b \cos^2 \frac{C}{2} + c \cos^2 \frac{B}{2} = s$
$(II)$ In $\triangle ABC, \cot \frac{A}{2} = \frac{b+c}{a} \Rightarrow B = 90^{\circ}$
Which of the following is correct?

$A$ uniform beam of weight $W$ is attached to a vertical wall by a hinge $H$. The beam is held horizontal by a rope as shown below. Which one of the following best shows the direction of the reaction force $R$ at the hinge?

Which one of the following has the smallest diameter?

The refractive index of a glass convex lens is $1.5$. The radius of curvature of each of the two surfaces of the lens is $20 ~cm$. The ratio of the power of the lens when immersed in a liquid of refractive index $1.25$ to that when placed in air is:

The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is $1000$ at $t = 0$. The number of bacteria is increased by $20 \%$ in $2 \text{ hours}$. If the population of bacteria is $2000$ after $\frac{k}{\log \left(\frac{6}{5}\right)} \text{ hours}$,then $\left(\frac{k}{\log 2}\right)^2$ is