In a $\triangle ABC$,$\cos \left(\frac{B+2C+3A}{2}\right) + \cos \left(\frac{A-B}{2}\right)$ is equal to

$A$ $100\%$ efficient transformer has $100$ turns in the primary and $25$ turns in its secondary coil. If the current in the secondary coil is $4\, A$,then the current in the primary coil is.....$A$.

For the electrochemical cell, $Mg ( s )\left| Mg ^{2+}( aq , 1 M ) \| Cu ^{2+}( aq , 1 M )\right| Cu ( s )$ the standard emf of the cell is $2.70 V$ at $300 K$. When the concentration of $Mg ^{2+}$ is changed to $x M$, the cell potential changes to $2.67 V$ at $300 K$. The value of $x$ is.

(given, $\frac{ F }{ R }=11500 K V ^{-1}$, where $F$ is the Faraday constant and $R$ is the gas constant, The value of $\ln (10)=2.30$ )

$A$ man fires a bullet standing between two cliffs. The first echo is heard after $3 \ s$ and the second echo is heard after $5 \ s$. If the velocity of sound is $330 \ m/s$,then the distance between the cliffs is .... $m$.

$A$ conducting rod $AC$ of length $4l$ is rotated about a point $O$ in a uniform magnetic field $\vec{B}$ directed into the paper. $AO = l$ and $OC = 3l$. Then:

The diagram shows a venturimeter through which water is flowing. The speed of water at $X$ is $2\,cm/s$. The speed of water at $Y$ (taking $g = 1000\,cm/s^2$) is ........ $cm/s$.