$A$ convex lens of focal length $30 \ cm$ is placed in contact with a concave lens of focal length $20 \ cm$. An object is placed at $20 \ cm$ to the left of this lens system. The distance of the image from the lens in $cm$ is . . . . . .

$A$ lens of power $+2$ diopters is placed in contact with a lens of power $-1$ diopter. The combination will behave like

An optical device is constructed by fixing three identical convex lenses of focal lengths $10 \,cm$ each inside a hollow tube at equal spacing of $30 \,cm$ each. One end of the device is placed $10 \,cm$ away from a point source. What is the image shift in $cm$ when the device is moved away from the source by another $10 \,cm$?

$A$ convex lens has $9 \ cm$ focal length and a concave lens has $-18 \ cm$ focal length. The focal length of the combination in contact will be.....$cm$

$A$ parallel beam of light is incident on a system of two convex lenses of focal lengths $f_1 = 20\, cm$ and $f_2 = 10\, cm$. What should be the distance between the two lenses so that the rays after refraction from both the lenses emerge as a parallel beam (in $, cm$)? (Note: The condition for the final rays to emerge parallel is that the focal point of the first lens must coincide with the focal point of the second lens).

Find the resultant focal length for the following system where the common radius of curvature is $15 \ cm$. The glass has a refractive index of $1.5$ and water has a refractive index of $\frac{4}{3}$.