The electrode potentials for $Cu^{2+}_{(aq)} + e^- \to Cu^{+}_{(aq)}$ and $Cu^{+}_{(aq)} + e^- \to Cu_{(s)}$ are $+0.15 \, V$ and $+0.50 \, V$ respectively. The value of $E^o_{Cu^{2+}/Cu}$ will be ............ $V$.

Sodium metal crystallizes in a body-centred cubic $(bcc)$ lattice with a unit cell edge of $4.29 \ \mathring{A}$. The radius of the sodium atom is approximately: .............. $\mathring{A}$

Two particles $A$ and $B$ move with constant velocities $\vec{v}_1$ and $\vec{v}_2$. At the initial moment,their position vectors are $\vec{r}_1$ and $\vec{r}_2$ respectively. The condition for their collision is

In a compound microscope,the focal lengths of two lenses are $1.5 \ cm$ and $6.25 \ cm$. An object is placed at $2 \ cm$ from the objective lens and the final image is formed at $25 \ cm$ from the eye lens. The distance between the two lenses is $.... \ cm$.

In transgenics,the expression of a transgene in the target tissue is determined by:

$\int e^x \left( \frac{2+\sin 2x}{1+\cos 2x} \right) dx$ is equal to