An object is placed $20 \, cm$ in front of a convex mirror. $A$ plane mirror is also placed such that the virtual images formed by both mirrors coincide. If the plane mirror is placed $15 \, cm$ from the object,find the focal length of the convex mirror in $cm$.

$A$ point source of light moves in a straight line parallel to a plane table. Consider a small portion of the table directly below the line of movement of the source. The illuminance $E$ at this portion varies with its distance $r$ from the source as:

Assertion : $A$ concave mirror and convex lens both have the same focal length in air. When they are submerged in water,they will have same focal length.
Reason : The refractive index of water is smaller than the refractive index of air.

$A$ double convex lens $(R_1 = R_2 = 10\,cm)$ with refractive index $(\mu = 1.5)$ has a focal length equal to the focal length of a concave mirror. The radius of curvature of the concave mirror is.......$cm$.

Write the names of instruments developed by the use of properties of reflection and refraction of mirrors,lenses,and prisms.

$A$ plano-convex lens is made of a material of refractive index $n$. When a small object is placed $30 \ cm$ away in front of the curved surface of the lens,an image of double the size of the object is produced. Due to reflection from the curved surface of the lens,another faint image is observed at a distance of $10 \ cm$ away from the lens. Which of the following statement$(s)$ is(are) true?
$(A)$ The refractive index of the lens is $2.5$
$(B)$ The radius of curvature of the convex surface is $45 \ cm$
$(C)$ The faint image is erect and real
$(D)$ The focal length of the lens is $20 \ cm$