If $a, b, c$ are in $AP$,and $b-a, c-b, a$ are in $GP$,then $a: b: c$ is

An air column in a tube $32 ~cm$ long,closed at one end,is in resonance with a tuning fork. The air column in another tube,open at both ends,of length $66 ~cm$ is in resonance with another tuning fork. When these two tuning forks are sounded together,they produce $8$ beats per second. Then the frequencies of the two tuning forks are (Consider fundamental frequencies only):

If $|\vec{A} \times \vec{B}| = \sqrt{3} \vec{A} \cdot \vec{B}$,then the value of $|\vec{A} + \vec{B}|$ is

$A$ bullet of mass $0.02 \ kg$ travelling horizontally with velocity $250 \ ms^{-1}$ strikes a block of wood of mass $0.23 \ kg$ which rests on a rough horizontal surface. After the impact,the block and bullet move together and come to rest after travelling a distance of $40 \ m$. Find the coefficient of sliding friction of the rough surface. (Take $g = 9.8 \ ms^{-2}$)

Which of the following structures corresponds to the product expected when an excess of $C_6H_6$ reacts with $CH_2Cl_2$ in the presence of anhydrous $AlCl_3$?

$A \rightarrow D$ is an endothermic reaction occurring in $3$ steps (elementary).
$(i)$ $A \rightarrow B$ $\Delta H_{i} = +ve$
$(ii)$ $B \rightarrow C$ $\Delta H_{ii} = -ve$
$(iii)$ $C \rightarrow D$ $\Delta H_{iii} = -ve$
Which of the following graphs between potential energy ($y$-axis) vs reaction coordinate ($x$-axis) correctly represents the reaction profile of $A \rightarrow D$?