Differentiate each of the following w.r.t
Question: Differentiate each of the following w.r.t $x$ : $\sin ^{-1}\left(\frac{1-\tan ^{2} x}{1+\tan ^{2} x}\right)$ Solution:...
Read More →A small object is placed at the center of the bottom of
Question: A small object is placed at the center of the bottom of a cylindrical vessel of radius $3 \mathrm{~cm}$ and height $4 \mathrm{~cm}$ filled completely with water. Consider the ray leaving the vessel through a corner. Suppose this ray and the ray along the axis of the vessel are used to trace the image. Find the apparent depth under the assumption taken. Refractive index of water is $-1.33 .$ Solution:...
Read More →Consider the situation in figure.
Question: Consider the situation in figure. The bottom of the pot is a reflecting plane mirror, $S$ is a small fish and $T$ is a human eye. Refractive index of the water is $\mu$ (a) At what distance from itself will the fish see the image of the eye? (b) At what distance from itself will the eye see the image of the fish. Solution:...
Read More →A cylindrical vessel of diameter
Question: A cylindrical vessel of diameter $12 \mathrm{~cm}$ contains $800 \pi \mathrm{cm}$ of water. A cylindrical glass piece of diameter $8.0 \mathrm{~cm}$ and height $8 \mathrm{~cm}$ is placed in the vessel. If the bottom of the vessel under the glass piece is seen by the paraxial rays fig. locate its image. The index of refractive of glass is $1.50$ and that of water is $1.33$. Solution:...
Read More →k transparent slabs are arranged one over another.
Question: $k$ transparent slabs are arranged one over another. The refractive index of the slabs are $\mu_{1}, \mu_{2}, \mu_{3}, \ldots \ldots \ldots \ldots \ldots \mu_{\mathrm{k}}$ and the thickness are $t_{1}, t_{2}, t_{3} \ldots \ldots \ldots \ldots t_{k}$. An object is seen through this combination with nearly perpendicular light. Find the equivalent refractive index of the system which will allow the image to be formed at the same place. Solution:...
Read More →Differentiate each of the following w.r.t
Question: Differentiate each of the following w.r.t $x$ : $\sec ^{-1}\left(\frac{1+\tan ^{2} x}{1-\tan ^{2} x}\right)$ Solution:...
Read More →Locate the image of the point P
Question: Locate the image of the point $P$ as seen $y$ the eye in the fig Solution:...
Read More →A vessel contains water up to a height of
Question: A vessel contains water up to a height of $20 \mathrm{~cm}$ and above it oil up to another $20 \mathrm{~cm}$. The refractive index of the water and the oil are $1.33$ and $1.30$ resp. Find the apparent depth of the vessel when viewed from above. Solution:...
Read More →Differentiate each of the following w.r.t
Question: Differentiate each of the following w.r.t $x$ : $\cot ^{-1}\left(\sqrt{\frac{1+\cos 3 x}{1-\cos 3 x}}\right)$ Solution:...
Read More →An object P is focused by a microscope M.
Question: An object $\mathrm{P}$ is focused by a microscope M. A glass slab of thickness $2.1 \mathrm{~cm}$ is introduced between $\mathrm{P}$ and $\mathrm{M}$. If the refractive index of the slab is $1.5$, by what distance should the microscope be shifted to focus the object again? Solution:...
Read More →A small piece of wood is floating on the surface of
Question: A small piece of wood is floating on the surface of a $2.5 \mathrm{~m}$ deep lake. Where does the shadow from on the bottom when the sum is just setting? Refractive index of water $=4 / 3$. Solution:...
Read More →A pole of length 1m stands half dipped in a swimming pool with
Question: A pole of length $1 \mathrm{~m}$ stands half dipped in a swimming pool with water level $50.0 \mathrm{~cm}$ higher than the bed. The refractive index of water is $1.33$ and sunlight is coming at an angle of $45^{\circ}$ with the vertical. Find the length of the shadow of the pole on the bed. Solution:...
Read More →Differentiate each of the following w.r.t
Question: Differentiate each of the following w.r.t $x$ : $\cot ^{-1}\left(\frac{\cos x-\sin x}{\cos x+\sin x}\right)$ Solution:...
Read More →A light ray falling at an angle of
Question: A light ray falling at an angle of $45^{\circ}$ with the surface of a clean slab of ice of thickness $1 \mathrm{~m}$ is refracted into it at an angle of $30^{\circ}$. Calculate the time taken by the light rays to cross the slab. Speed of light in vacuum $=3^{*} 10^{6} \mathrm{~m} / \mathrm{s}$. Solution:...
Read More →A converging mirror M1, a point source S and a diverging mirror M2
Question: A converging mirror $\mathrm{M}_{1}$, a point source $\mathrm{S}$ and a diverging mirror $\mathrm{M}_{2}$ are arranged as shown in figure The source is placed at a distance of $30 \mathrm{~cm}$ from $\mathrm{M}_{1}$. The focal length of each of the mirror is $20 \mathrm{~cm}$. Consider only the images formed by a maximum of two reflections. It is found that one image is formed on the source itself. (a) Find the distance between the two mirrors. (b) Find the location of the images forme...
Read More →Differentiate each of the following w.r.t
Question: Differentiate each of the following w.r.t $x$ : $\tan ^{-1}\left(\frac{\cos x+\sin x}{\cos x-\sin x}\right)$ Solution:...
Read More →A point source S is placed midway between two converging
Question: A point source $S$ is placed midway between two converging mirror having equal focal length $f$ as shown in fig. Find the values od d for which only one image is formed. Solution: Both mirror produces one image in following two cases: (A) When Source is at distance $=2 f$ or at center of curvature then image is produce at $v=2 f+2 f=4 f$ (B) When source is at distance $f$ then rays become parallel after reflecting through one mirror, then it is reflected through other mirror and only o...
Read More →Differentiate each of the following w.r.t
Question: Differentiate each of the following w.r.t $x$ : $\cot ^{-1}\left(\sqrt{\frac{1+\cos x}{1-\cos x}}\right)$ Solution: Ans) $\frac{1}{2}$...
Read More →A concave mirror of radius
Question: A concave mirror of radius $\mathrm{R}$ is kept on a horizontal table. Water (refractive index $=\mu$ ) is poured into it up to a height $h$. Where an object should be placed so that its image is formed on itself? Solution:...
Read More →A particle goes in a circle of radius
Question: A particle goes in a circle of radius $2.0 \mathrm{~cm}$. A concave mirror of focal length $20 \mathrm{~cm}$ is placed with its principal axis passing through the centre of the circle and perpendicular to its plane. The distance between the pole of the mirror and the center of the circle is $30 \mathrm{~cm}$. Calculate the radius of the circle formed by the image. Solution:...
Read More →Differentiate each of the following w.r.t
Question: Differentiate each of the following w.r.t $x$ : $\cot ^{-1}\left(\frac{1+\cos x}{\sin x}\right)$ Solution:...
Read More →Find the diameter of the images of the moon formed
Question: Find the diameter of the images of the moon formed by a spherical concave mirror of focal length $7.6 \mathrm{~m}$. The diameter of the moon is $3450 \mathrm{~km}$ and the distance between the earth and the moon is $3.8 \times 10^{5} \mathrm{~km}$. Solution:...
Read More →Differentiate each of the following w.r.t
Question: Differentiate each of the following w.r.t $x$ : $\tan ^{-1}\left(\frac{\sin x}{1+\cos x}\right)$ Solution:...
Read More →A man uses a concave mirror for shaving.
Question: A man uses a concave mirror for shaving. He keeps his face at a distance of $25 \mathrm{~cm}$ from the mirror and gets an image which is $1.4$ times enlarged. Find the focal length of the mirror. Solution:...
Read More →A U-shaped wire is placed before a concave mirror having radius
Question: A U-shaped wire is placed before a concave mirror having radius of curvature $20 \mathrm{~cm}$ as shown in figure. Find the total length of the image. Solution:...
Read More →