If $\overline{a}=\frac{1}{\sqrt{10}}(3 \hat{i}+\hat{k})$ and $\overline{b}=\frac{1}{7}(2 \hat{i}+3 \hat{j}-6 \hat{k})$,then the value of $(2 \overline{a}-\overline{b}) \cdot [(\overline{a} \times \overline{b}) \times (\overline{a}+2 \overline{b})]$ is

Assertion $(A)$: The boiling points of noble gases increase from $He$ to $Xe$.
Reason $(R)$: The interatomic van der Waals' attractive forces increase from $He$ to $Xe$.
The correct answer is:

In the circuit shown below,the resistances are given in ohm and the battery is assumed ideal with $emf$ equal to $3\, V$. The voltage across the resistance $R_4$ is ................ $V$.

$A$ thin prism $P_1$ with angle $4^o$ and made from a glass of refractive index $1.54$ is combined with another thin prism $P_2$ made from glass of refractive index $1.72$ to produce dispersion without deviation. The angle of the prism $P_2$ is.....$^o$

Match the following columns and select the correct option:

$(a)$ Placenta	$(i)$ Androgens
$(b)$ Zona pellucida	$(ii)$ Human Chorionic Gonadotropin $(hCG)$
$(c)$ Bulbourethral glands	$(iii)$ Layer of the ovum
$(d)$ Leydig cells	$(iv)$ Lubrication of the penis

$(a)\quad (b)\quad (c)\quad (d)$

Two rigid boxes containing different ideal gases are placed on a table. Box $A$ contains one mole of nitrogen at temperature $T_0$,while box $B$ contains one mole of helium at temperature $\left( \frac{7}{3} \right) T_0$. The boxes are then put into thermal contact with each other,and heat flows between them until the gases reach a common final temperature (ignore the heat capacity of the boxes). The final temperature of the gases,$T_f$,in terms of $T_0$ is: