$A$ conductor of resistance $3 \Omega$ is stretched uniformly until its length is doubled. The wire is now bent in the form of an equilateral triangle. The effective resistance between the ends of any side of the triangle in ohms is

If $\alpha$ is a non-real root of $x^6=1$,then $\frac{\alpha^5+\alpha^3+\alpha+1}{\alpha^2+1}$ is equal to

Two coins $A$ and $B$ are kept in an urn. When coin $A$ is flipped,the probability of getting a Head is $1/4$,while for coin $B$ it is $3/4$. One coin is randomly chosen from this bag and tossed twice,and it falls heads on both occasions. The probability that it is coin $A$ is:

$A$ ray of light is incident on the hypotenuse of a right-angled prism after travelling parallel to the base inside the prism. If $\mu$ is the refractive index of the material of the prism,the maximum value of the base angle for which light is totally reflected from the hypotenuse is

$A$ thermodynamic system is taken through the cycle $PQRSP$ as shown in the $PV$ diagram. The net work done by the system is...... $J$. (Note: $1 \text{ kp} = 10^3 \text{ Pa}$,$1 \text{ cc} = 10^{-6} \text{ m}^3$)

If $\alpha+\beta+\gamma=2 \theta$,then $\cos \theta+\cos (\theta-\alpha)+\cos (\theta-\beta)+\cos (\theta-\gamma)$ is equal to