The specific rotation of the equilibrium mixture of $\alpha-D$-glucose and $\beta-D$-glucose is ....... $^\circ$.

The direction ratios of the normal to the plane passing through $(0,0,1)$, $(0,1,2)$, and $(1,0,3)$ are

Which one of the following elements,when present as an impurity in silicon,makes it a $p$-type semiconductor?

$A$ gas is compressed at a constant pressure of $50 \ N/m^2$ from a volume of $10 \ m^3$ to a volume of $4 \ m^3$. Energy of $100 \ J$ is then added to the gas by heating. Its internal energy is:

Which of the following statements is correct?

If $\alpha+\beta=-2$ and $\alpha^3+\beta^3=-56$,then the quadratic equation whose roots are $\alpha$ and $\beta$ is