When sunlight falls normally on Earth,a luminous flux density (illuminance) of $1.57 \times 10^5 \; lm/m^2$ is produced on Earth. If the distance of Earth from the Sun is $1.5 \times 10^8 \; km$,what is the total luminous flux emitted by the Sun?

$1\%$ of light of a source with luminous intensity $50 \,cd$ is incident on a circular surface of radius $10 \,cm$. The average illuminance of the surface is: (in $,lux$)

$A$ reflecting surface is represented by the equation $y = \frac{2L}{\pi} \sin \left( \frac{\pi x}{L} \right)$,$0 \leq x \leq L$. $A$ ray travelling horizontally becomes vertical after reflection. The coordinates of the point$(s)$ where this ray is incident is?

An object is placed in front of a convex mirror at a distance of $50\, cm$. $A$ plane mirror is introduced covering the lower half of the convex mirror. If the distance between the object and the plane mirror is $30\, cm$,it is found that there is no parallax between the images formed by the two mirrors. The magnification of the convex mirror will be......

$A$ lamp is hanging at a height of $4\,m$ above a table. The lamp is lowered by $1\,m$. The percentage increase in illuminance will be.....$\%$

$A$ beaker of radius $r$ is filled with water (refractive index $\frac{4}{3}$) up to a height $H$ as shown in the figure. The beaker is kept on a horizontal table rotating with angular speed $\omega$. This makes the water surface curved so that the difference in the height of the water level at the center and at the circumference of the beaker is $h$ $(h \ll H, h \ll r)$,as shown in the figure. Take this surface to be approximately spherical with a radius of curvature $R$. Which of the following is/are correct? ($g$ is the acceleration due to gravity)
$(A)$ $R=\frac{h^2+r^2}{2 h}$
$(B)$ $R=\frac{r^2}{2 h}$
$(C)$ Apparent depth of the bottom of the beaker is close to $\frac{3 H}{4}\left(1+\frac{\omega^2 H}{4 g}\right)^{-1}$
$(D)$ Apparent depth of the bottom of the beaker is close to $\frac{3 H}{2}\left(1+\frac{\omega^2 H}{2 g}\right)^{-1}$