The figure shows a convex lens cut symmetrically into two equal halves and separated laterally by a distance $h$. $A$ point object placed at a distance $30\, cm$ from the lens halves forms two real images separated by a distance $d$. $A$ plot of $d$ versus $h$ is shown in the figure. The focal length of the lens is ......$cm$.

In an experiment to measure the focal length $(f)$ of a convex lens,the least counts of the measuring scales for the position of the object $(u)$ and for the position of the image $(v)$ are $\Delta u$ and $\Delta v$,respectively. The error in the measurement of the focal length of the convex lens will be:

$A$ convex lens has its radii of curvature equal. The focal length of the lens is $f$. If it is divided vertically into two identical plano-convex lenses by cutting it, then the focal length of the plano-convex lens is ($\mu =$ the refractive index of the material of the lens).

$A$ biconvex lens of focal length $f$ and radii of curvature of both surfaces $R$ is made of a material of refractive index $n_{1}$. This lens is placed in a liquid of refractive index $n_{2}$. How will this lens behave?

Consider an equi-convex lens of glass $\left( \mu = \frac{3}{2} \right)$. If the temperature of the lens is increased by $40\, ^\circ C$,its focal length remains the same. The coefficient of linear expansion of glass is $2.5 \times 10^{-4} / ^\circ C$. Calculate the change in the refractive index of the glass on increasing the temperature by $40\, ^\circ C$.

In an optical bench experiment, the index error is $+1 \, cm$ and $-1 \, cm$ between the object needle and lens, and between the lens and image needle, respectively. The observed values of $u$ and $v$ are $9 \, cm$ and $17 \, cm$. The focal length of the lens is ..... $cm$.