$A$ concave mirror has a radius of curvature of $40\, cm$. It is at the bottom of a glass that has water filled up to $5\, cm$ (see figure). If a small particle is floating on the surface of water,its image as seen from directly above the glass is at a distance $d$ from the surface of water. The value of $d$ is close to ......$cm$ (Refractive index of water $= 1.33$)

The $45^{\circ}-45^{\circ}-90^{\circ}$ prism of height $10 \,cm$ (see image) has a refractive index $\mu = 2$,with a silvered hypotenuse surface. $A$ convex lens of focal length $10 \,cm$ is placed $15 \,cm$ in front of a wall. The system produces a sharp image of point $P$ on the wall. The value of $h$ (in $cm$) is closest to:

$A$ 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:

$A$ $60 \text{ W}$ bulb is hung over the center of a table of size $4 \text{ m} \times 4 \text{ m}$ at a height of $3 \text{ m}$. The ratio of the intensities of illumination at a point on the center of the edge $(A)$ and at the corner of the table $(B)$ is:

$A$ point source of $100$ candela is held $5\,m$ above a sheet of blotting paper which reflects $75\%$ of light incident upon it. The illuminance of the blotting paper is

$A$ 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 $20 \ cm$