An object is placed at a distance of $10 \, cm$ from a lens. The image is formed at a distance of $20 \, cm$ behind the lens. The power of the lens is ......

$(i)$ If $f=0.5 \,m$ for a glass lens,what is the power of the lens?
$(ii)$ The radii of curvature of the faces of a double convex lens are $10 \,cm$ and $15 \,cm$. Its focal length is $12 \,cm$. What is the refractive index of glass?
$(iii)$ $A$ convex lens has $20 \,cm$ focal length in air. What is its focal length in water? (Refractive index of water $= 1.33$,refractive index of glass $= 1.5$)

$A$ magnifying glass is to be used at a fixed object distance of $1 \text{ inch}$. If it is to produce an erect image magnified $5$ times,its focal length should be ........ $\text{inch}$.

An object is placed at a distance $x$ from the focus of a convex lens,and its image is formed at a distance $x'$ from the other focus,as shown in the figure. Which relation do the distances $x$ and $x'$ satisfy?

The focal lengths for violet,green,and red light rays are ${f_V}$,${f_G}$,and ${f_R}$ respectively. Which of the following is the true relationship?

$A$ convex lens has a power of $+5.0 \, D$ in air,where the refractive index of the lens material is $_a\mu_g = 1.5$. In a liquid of what refractive index should it be immersed so that it acts as a concave lens of focal length $100 \, cm$?