The vapour pressure of water at $23^{\circ} C$ is $19.8 \ mm$. $0.1 \ mole$ of glucose is dissolved in $178.2 \ g$ of water. What is the vapour pressure (in $mm$) of the resultant solution?

The origin is translated to $(1,2)$. The point $(7,5)$ in the old system undergoes the following transformations successively.
$I$. Moves to the new point under the given translation of origin.
$II$. Translated through $2$ units along the negative direction of the new $X$-axis.
$III$. Rotated through an angle $\frac{\pi}{4}$ about the origin of the new system in the clockwise direction. The final position of the point $(7,5)$ is

If $\overrightarrow{a} = 2\widehat{i} - \widehat{j} + \widehat{k}$,$\overrightarrow{b} = \widehat{i} + \widehat{j} - 2\widehat{k}$ and $\overrightarrow{c} = \widehat{i} + 3\widehat{j} - (\lambda^2 + 3\lambda)\widehat{k}$ (where $\lambda$ is a constant) and $\overrightarrow{a}$ is perpendicular to $\overrightarrow{c} - \lambda\overrightarrow{b}$,then the sum of different values of $\lambda$ is:

$A$ father with $8$ children takes them $3$ at a time to the Zoological gardens,as often as he can without taking the same $3$ children together more than once. The number of times he will go to the garden is

The wavelengths of electron waves in two orbits is $3: 5$. The ratio of kinetic energy of electrons will be

The sensitivity of a galvanometer is $60 \text{ div/A}$. When a shunt is used,its sensitivity becomes $10 \text{ div/A}$. If the resistance of the galvanometer is $20 \ \Omega$,then the value of the shunt used is: (in $\Omega$)