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An object is placed beyond the centre of curvature \( C \) of the given concave mirror. If the distance of the object is \( d_1 \) from \( C \) and the distance of the image formed is \( d_2 \) from \( C \), the radius of curvature of this mirror is:

If the distance between object and its two times magnified virtual image produced by a curved mirror is \( 15 \, \text{cm} \), the focal length of the mirror must be:

An object is placed at a distance of \( 12 \, \text{cm} \) in front of a plane mirror. The virtual and erect image is formed by the mirror. Now the mirror is moved by \( 4 \, \text{cm} \) towards the stationary object. The distance by which the position of image would be shifted, will be:

Refer the figure given below, \( \mu_{1} \) and \( \mu_{2} \) are refractive indices of air and lens material. The height of image will be ____ \( \text{cm} \).

A bi-convex lens has radius of curvature of both the surfaces same as \( \frac{1}{6} \, \text{cm} \). If this lens is required to be replaced by another convex lens having different radii of curvatures on both sides \( (R_1 \neq R_2) \), without any change in lens power then possible combination of \( R_1 \) and \( R_2 \) is:

An unpolarized light is incident at an interface of two dielectric media having refractive indices of \( 2 \) (incident medium) and \( 2\sqrt{3} \) (medium) respectively. To satisfy the condition that reflected and refracted rays are perpendicular to each other, the angle of incidence is ______.

The X-Y plane be taken as the boundary between two transparent media \( M_1 \) and \( M_2 \). \( M_1 \) in \( Z \ge 0 \) has a refractive index of \( \sqrt{2} \) and \( M_2 \) with \( Z < 0 \) has a refractive index of \( \sqrt{3} \). A ray of light travelling in \( M_1 \) along the direction given by the vector \( \vec{P}=4\sqrt{3}\,\hat{i}-3\sqrt{3}\,\hat{j}-5\hat{k} \), is incident on the plane of separation. The value of difference between the angle of incident in \( M_1 \) and the angle of refraction in \( M_2 \) will be __________ degree.

A spherical interface lens of radius \( R \) separates two media of refractive indices \( 1 \) and \( 1.4 \) respectively as shown in the figure below. A point source is placed at a distance of \( 4R \) in front of spherical interface. The magnitude of the magnification of point source image is _____.

The position of the image formed by the combination of lenses is

Region I and II are separated by a spherical surface of radius \( 25 \, \text{cm} \). An object is kept in region I at a distance of \( 40 \, \text{cm} \) from the surface. The distance of the image from the surface is:

Distance between an object and three times magnified real image is \( 40 \, \text{cm} \). The focal length of the mirror used is _______________ cm.

Consider a light ray travelling in air is incident into a medium of refractive index \( \sqrt{2n} \). The incident angle is twice that of refracting angle. Then, the angle of incidence will be:

In a medium the speed of light wave decrease to \( 0.2 \) times to its speed in free space the ratio of relative permittivity to the refractive index of the medium is \( x:1 \). The value of \( x \) is_____ :
(Given speed of light in free space = \( 3 \times 10^8 \, \text{m s}^{-1} \) and for the given medium \( \mu_r = 1 \) )

A plano-convex lens having radius of curvature of first surface \( 2 \, \text{cm} \) exhibits focal length of \( f_{1} \) in air. Another plano-convex lens with first surface radius of curvature \( 3 \, \text{cm} \) has focal length of \( f_{2} \) when it is immersed in a liquid of refractive index \( 1.2 \). If both the lenses are made of same glass of refractive index \( 1.5 \), the ratio of \( f_{1} \) and \( f_{2} \) will be:

A rod of length \( 10 \, \text{cm} \) lies along the principle axis of a concave mirror of focal length \( 10 \, \text{cm} \) as shown in figure. The length of the image is ______ \( \text{cm} \).

Two concave refracting surfaces of equal radii of curvature and refractive index \(1.5\) face each other in air as shown in figure. A point object \(O\) is place midway, between \(P\) and \(B\). The separation between the images of \(O\), formed by each refracting surface is:

As shown in the diagram, when the incident ray is parallel to base of the prism, the emergent ray grazes along the second surface.

If refractive index of the material of prism is \( \sqrt{2} \), the angle \( \theta \) of prism is.

The driver sitting inside a parked car is watching vehicles approaching from behind with the help of his side view mirror, which is a convex mirror with radius of curvature \( R = 2 \, \text{m} \). Another car approaches him from behind with a uniform speed of \( 90 \, \text{km} / \text{hr} \). When the car is at a distance of \( 24 \, \text{m} \) from his, the magnitude of the acceleration of the image of the car in the side view mirror is \(' a' \). The value of \( 100a \) is ______ \( \text{m} / \text{s}^2 \).

A parallel beam of light travelling in air (refractive index \(1.0\)) is incident on a convex spherical glass surface of radius of curvature \(50 \, \text{cm}\). Refractive index of glass is \(1.5\). The rays converge to a point at a distance \( x \, \text{cm} \) from the centre of the curvature of the spherical surface. The value of \( x \) is __________ \( \text{cm} \).

Given a thin convex lens (refractive index \( \mu_2 \)), kept in a liquid (refractive index \( \mu_1 \), \( \mu_1 < \mu_2 \)) having radii of curvatures \( |R_1| \) and \( |R_2| \). Its second surface is silver polished. Where should an object be placed on the optic axis so that a real and inverted image is formed at the same place?

The exit surface of a prism with refractive index \( n \) is coated with a material having refractive index \( \dfrac{n}{2} \). When this prism is set for minimum angle of deviation, it exactly meets the condition of critical angle. The prism angle is _______.

In an experiment with a convex lens. The plot of the image distance \( (v') \) against the object distance \( (u') \) measured from the focus gives a curve \( v' u' = 225 \). If all the distances are measured in \( \text{cm} \). The magnitude of the focal length of the lens is __________ \( \text{cm} \).

Two beams of light having intensities \( I \) and \( 4I \) interfere to produce a fringe pattern on a screen. The phase difference between the two beams are \( \pi/2 \) and \( \pi/3 \) at points \( A \) and \( B \) respectively. The difference between the resultant intensities at the two points is \( xI \). The value of \( x \) will be ________.

A small bulb is placed at the bottom of a tank containing water to a depth of \( \sqrt{7} \, \text{m} \). The refractive index of water is \( \frac{4}{3} \). The area of the surface of water through which light from the bulb can emerge out is \( x \pi \, \text{m}^2 \). The value of \( x \) is ________.

An object is placed at a distance of \( 12 \, \text{cm} \) from a convex lens. A convex mirror of focal length \( 15 \, \text{cm} \) is placed on other side of lens at \( 8 \, \text{cm} \) as shown in the figure. Image of object coincides with the object.

When the convex mirror is removed, a real and inverted image is formed at a position. The distance of the image from the object will be ________ \( (\text{cm}) \).

Two identical objects are placed in front of convex mirror and concave mirror having same radii of curvature of \( 12 \, \text{cm} \), at same distance of \( 18 \, \text{cm} \) from the respective mirrors. The ratio of sizes of the images formed by convex mirror and by concave mirror is:

The refracting angle of a prism is \( A \) and refractive index of the material of the prism is \( \cot(A/2) \). Then the angle of minimum deviation will be:

A concave mirror of focal length \( f \) in air is dipped in a liquid of refractive index \( \mu \). Its focal length in the liquid will be:

The speed of light in media ‘A’ and ‘B’ are \( 2.0 \times 10^{10} \, \text{cm/ s}^{} \) and \( 1.5 \times 10^{10} \, \text{cm/ s}^{} \) respectively. A ray of light enters from the medium \( B \) to \( A \) at an incident angle \( \theta \). If the ray suffers total internal reflection, then

Given is a thin convex lens of glass (refractive index \( \mu \)) and each side having radius of curvature \( R \). One side is polished for complete reflection. At what distance form the lens, an object be placed on the optic axis so that the image gets formed on the object itself?

An object is placed on the principal axis of convex lens of focal length \( 10 \, \text{cm} \) as shown. A plane mirror is placed on the other side of lens at a distance of \( 20 \, \text{cm} \). The image produced by the plane mirror is \( 5 \, \text{cm} \) inside the mirror. The distance of the object from the lens is ______ \( \text{cm} \).

A glass tumbler having inner depth of \( 17.5 \, \text{cm} \) is kept on a table. A student starts pouring water \( (\mu = \tfrac{4}{3}) \) into it while looking at the surface of water from the above. When he feels that the tumbler is half filled, he stops pouring water. Up to what height, the tumbler is actually filled?

Time taken by light to travel in two different materials A and B of refractive indices \( \mu_A \) and \( \mu_B \) of same thickness is \( t_1 \) & \( t_2 \) respectively. If \( t_2-t_1=5\times 10^{-10}\,s \) and the ratio of \( \mu_A \) to \( \mu_B \) is \( 1:2 \). Then, the thickness of material, in meter is: (Given \( v_A \) & \( v_B \) are velocities of light in A and B materials respectively.)

A convex lens is made from glass material having refractive index of \( 1.4 \) with same radius of curvature on both sides. The ratio of its focal length and radius of curvature is __________.

If the refractive index of the material of a prism is \( \cot\!\left(\dfrac{A}{2}\right) \) where \( A \) is the angle of prism then the angle of minimum deviation will be :

If sunlight is focused on a paper using convex lens, it starts burning the paper in shortest time when the lens is kept at \( 30\ \text{cm} \) above the paper. If the radius of curvature of the lens is \( 60\ \text{cm} \) then the refractive index of the lens material is \( \frac{\alpha}{10} \). The value of \( \alpha \) is __________.

Given below are two statements: one labelled as Assertion (A) and the other is labelled as Reason (R)
Assertion (A): Refractive index of glass is higher than that of air.
Reason (R): Optical density of a medium is directly proportionate to its mass density which results in a proportionate refractive index.
In the light of the above statements, choose most appropriate answer from the options given below:

A slanted object \( AB \) is placed on one side of convex lens as shown in the diagram. The image is formed on the opposite side. Angle made by the image with principal axis is:

A spherical surface separates two media of refractive indices \( 1 \) and \( 1.5 \) as shown in figure. Distance of the image of an object \( 'O' \), is: (\( C \) is the center of curvature of the spherical surface and \( R \) is the radius of curvature)

Light from a point source in air falls on a convex curved surface of radius \( 20 \, \text{cm} \) and refractive index \( 1.5 \). If the source is located at \( 100 \, \text{cm} \) from the convex surface, the image will be formed at _______ \( \text{cm} \) from the object.

A biconvex lens of refractive index \( 1.5 \) has a focal length of \( 20 \, \text{cm} \) in air. Its focal length when immersed in a liquid of refractive index \( 1.6 \) will be:

A convex lens of refractive index \( 1.5 \) and focal length \( 18 \, \text{cm} \) in air is immersed in water. The change in focal length of the lens will be ________ \( \text{cm} \). (Given refractive index of water = \( \frac{4}{3} \))

A telescope with objective diameter \( R \) is used to observe a distant star emitting light of wavelength \( 500 \, \text{nm} \), at a resolution of \( 5 \times 10^{-7} \, \text{radian} \). The value of \( R \) is ______ \( \text{cm} \).

Two convex lenses of focal length \( 20\,\text{cm} \) each are placed coaxially with a separation of \( 60\,\text{cm} \) between them. The image of the distant object formed by the combination is at _____ \( \text{cm} \) from the first lens.

Consider light travelling from a medium A to medium B separated by a plane interface. If the light undergoes total internal reflection during its travel from medium A to B and the speed of light in media A and B are \( 2.4 \times 10^8 \, \text{m/s} \) and \( 2.7 \times 10^8 \, \text{m/s} \), respectively, then the value of critical angle is:

Consider an equilateral prism (refractive index $\sqrt{2}$). A ray of light is incident on its one surface at a certain angle $i$. If the emergent ray is found to graze along the other surface then the angle of refraction at the incident surface is close to _________.

The difference of speed of light in the two media A and \( (v_A - v_B) \) is \( 2.6 \times 10^7 \, \text{m /s}^{} \). If the refractive index of medium B is \( 1.47 \), then the ratio of refractive index of medium B to medium A is: (Given : speed of light in vacuum \( c = 3 \times 10^8 \, \text{m s}^{-1} \) )

In an experiment with a closed organ pipe, it is filled with water by \( \left( \dfrac{1}{5} \right) \)th of its volume. The frequency of the fundamental node will change by:

A biconvex lens of focal length \( 10 \, \text{cm} \) is cut in two identical parts along a plane perpendicular to the principal axis. The power of each lens after cut is _________ \( \text{D} \).

Two thin convex lenses of focal lengths \( 30 \, \text{cm} \) and \( 10 \, \text{cm} \) are placed coaxially, \( 10 \, \text{cm} \) apart. The power of this combination is:

The distance between object and its two times magnified real image as produced by a convex lens is \( 45 \, \text{cm} \). The focal length of the lens used is _____ \( \text{cm} \).

Light travels in two media \( M_1 \) and \( M_2 \) with speeds \( 1.5 \times 10^8 \, \text{ms}^{-1} \) and \( 2.0 \times 10^8 \, \text{ms}^{-1} \) respectively. The critical angle between them is:

A thin prism of angle \( 6^\circ \) and refractive index for yellow light \( (n_\gamma) \, 1.5 \) is combined with another prism of angle \( 5^\circ \) and \( n_\gamma = 1.55 \). The combination produces no dispersion. The net average deviation \( (\delta) \) produced by the combination is \( \left(\dfrac{1}{x}\right)^\circ \). The value of \( x \) is ______. 

A convex lens of focal length \( 20 \, \text{cm} \) is placed in front of a convex mirror with principal axis coinciding each other. The distance between the lens and mirror is \( 10 \, \text{cm} \). A point object is placed on principal axis at a distance of \( 60 \, \text{cm} \) from the convex lens. The image formed by combination coincides the object itself. The focal length of the convex mirror is ______________ \( \text{cm} \).

A container contains a liquid with refractive index of \( 1.2 \) up to a height of \( 60 \, \text{cm} \) and another liquid having refractive index \( 1.6 \) is added to height \( H \) above first liquid. If viewed from above, the apparent shift in the position of bottom of container is \( 40 \, \text{cm} \). The value of \( H \) is __________ \( \text{cm} \). (Consider liquids are immiscible)

A spherical surface of radius of curvature \( R \), separates air from glass (refractive index \( = 1.5 \)). The centre of curvature is in the glass medium. A point object ‘O’ placed in air on the optic axis of the surface, so that its real image is formed at ‘I’ inside glass. The line \( OI \) intersects the spherical surface at \( P \) and \( PO = PI \). The distance \( PO \) equals to:

Critical angle of incidence for a pair of optical media is \( 45^\circ \). The refractive indices of first and second media are in the ratio:

The radii of curvature for a thin convex lens are \( 10 \, \text{cm} \) and \( 15 \, \text{cm} \) respectively. The focal length of the lens is \( 12 \, \text{cm} \). The refractive index of the lens material is:

A convex mirror of radius of curvature \( 30 \, \text{cm} \) forms an image that is half the size of the object. The object distance is:

Light emerges out of a convex lens when a source of light kept at its focus. The shape of wavefront of the light is:

The refractive index of a prism with apex angle \( A \) is \( \cot \frac{A}{2} \). The angle of minimum deviation is:

A ray light passing through an equilateral prism is having velocity \( 2.12 \times 10^{8} \, \text{m/s} \) in the prism material, then the minimum angle of deviation is ______ degrees.

As shown in the figure, a combination of thin plano concave lens and a thin plano convex lens is used to image an object at infinity. The radius of curvature of both the lenses is \( 30 \, \mathrm{cm} \) and refraction index of the material for both the lenses is \( 1.75 \). Both the lenses are placed at distance of \( 40 \, \mathrm{cm} \) from each other. Due to the combination, the image of the object is formed at distance \( x = \) ______ \( \mathrm{cm} \) from concave lens.

Given below are two statements:
Statement (I): When an object is placed at the centre of curvature of a concave lens, image is formed at the centre of curvature of the lens on the other side.
Statement (II): Concave lens always forms a virtual and erect image.
In the light of the above statements, choose the correct answer from the options given below:

A thin prism with angle \( 5^\circ \) of refractive index \( 1.72 \) is combined with another prism of refractive index \( 1.9 \) to produce dispersion without deviation. The angle of second prism is ______.

A finite size object is placed normal to the principal axis at a distance of \( 30 \, \text{cm} \) from a convex mirror of focal length \( 30 \, \text{cm} \). A plane mirror is now placed in such a way that the image produced by both the mirrors coincide with each other. The distance between the two mirrors is:

A parallel beam of light is allowed to fall on a transparent spherical globe of diameter \( 30 \, \text{cm} \) and refractive index \( 1.5 \). The distance from the centre of the globe at which the beam of light can converge is __________ \( \text{mm} \).

As shown in the figure, a plane mirror is fixed at a height of \( 50 \, \text{cm} \) from the bottom of tank containing water \( \left( \mu = \dfrac{4}{3} \right) .\) The height of water in the tank is \( 8 \, \text{cm} .\) A small bulb is placed at the bottom of the water tank. The distance of image of the bulb formed by mirror form the bottom of the tank is _________ \( \text{cm} .\)

The magnitudes of power of a biconvex lens (refractive index \( 1.5 \)) and that of a plano-concave lens (refractive index \( = 1.7 \)) are same. If the curvature of plano-concave lens exactly matches with the curvature of back surface of the biconvex lens, then ratio of radius of curvature of front and back surface of the biconvex lens is ____________.

A pole is vertically submerged in swimming pool, such that it gives a length of shadow \( 2.15 \, \text{m} \) within water when sunlight is incident at an angle of \( 30^\circ \) with the surface of water. If swimming pool is filled to a height of \( 1.5 \, \text{m} \), then the height of the pole above the water surface in centimeters is: \( (n_w = {4}/{3}) \)

A thin convex lens and a thin concave lens are kept in contact and are co-axial. Which of the following statements is correct for this combination of two lenses?

An object viewed from a near point distance of \( 25 \, \text{cm} \), using a microscopic lens with magnification '\( 6 \)', gives an unresolved image. A resolved image is observed at infinite distance with a total magnification double the earlier using an eyepiece along with the given lens and a tube of length \( 0.6 \, \text{m} \), if the focal length of the eyepiece is equal to __________ \( \text{cm} \).

Your friend is having eye sight problem. She is not able to see clearly a distant uniform window mesh and it appears to her as non-uniform and distorted. The doctor diagnosed the problem as:

A person has been using spectacles of power \( -1.0 \) dioptre for distant vision and a separate reading glass of power \( 2.0 \) dioptres. What is the least distance of distinct vision for this person?

The radius of curvature of each surface of a convex lens having refractive index \( 1.8 \) is \( 20 \, \text{cm} \). The lens is now immersed in a liquid of refractive index \( 1.5 \). The ratio of power of lens in air to its power in the liquid will be \( x : 1 \). The value of \( x \) is __________.

In the given figure, the face \( AC \) of the equilateral prism is immersed in a liquid of refractive index '\( n \)'. For incident angle \( 60^\circ \) at the side \( AC \), the refracted light beam just grazes along face \( AC \). The refractive index of the liquid \( n = \dfrac{\sqrt{x}}{4} \). The value of \( x \) is ________. (Given refractive index of glass \( = 1.5 \)). 

Light enters from air into a given medium at an angle of \( 45^\circ \) with interface of the air-medium surface. After refraction, the light ray is deviated through an angle of \( 15^\circ \) from its original direction. The refractive index of the medium is :

A fish rising vertically upward with a uniform velocity of \( 8 \, \text{m s}^{-1} \), observes that a bird is diving vertically downward towards the fish with the velocity of \( 12 \, \text{m s}^{-1} \). If the refractive index of water is \( \dfrac{4}{3} \), then the actual velocity of the diving bird to pick the fish, will be ________ \( \text{m s}^{-1} \).

The expected graphical representation of the variation of angle of deviation '\( \delta \)' with angle of incidence '\( i \)' in a prism is:

A thin cylindrical rod of length \( 10\,\text{cm} \) is placed horizontally on the principle axis of a concave mirror of focal length \( 20\,\text{cm} \). The rod is placed in a such a way that mid point of the rod is at \( 40\,\text{cm} \) from the pole of mirror. The length of the image formed by the mirror will be \( \frac{x}{3}\,\text{cm} \). The value of \( x \) is ______.

Which of the following statement is correct?

In parallax method for the determination of focal length of a concave mirror, the object should always be placed:

The thickness at the centre of a plano convex lens is \( 3 \, \text{mm} \) and the diameter is \( 6 \, \text{cm} \). If the speed of light in the material of the lens is \( 2 \times 10^{8} \, \text{m s}^{-1} \). The focal length of the lens is __________.

A thin biconvex lens is prepared from the glass \( (\mu=1.5) \) both curved surfaces of which have equal radii of \( 20 \,\text{cm} \) each. Left side surface of the lens is silvered from outside to make it reflecting. To have the position of image and object at the same place, the object should be placed, from the lens at a distance of ______ \( \text{cm} \).

Car \( B \) overtakes another car \( A \) at a relative speed of \( 40 \, \text{ms}^{-1} \). How fast will the image of car \( B \) appear to move in the mirror of focal length \( 10 \, \text{cm} \) fitted in car \( A \), when the car \( B \) is \( 1.9 \, \text{m} \) away from the car \( A \)?

Given below are two statements:
Statement I: When the white light passed through a prism, the red light bends lesser than yellow and violet.
Statement II: The refractive indices are different for different wavelengths in dispersive medium.
In the light of the above statements, chose the correct answer from the options given below:

A prism of refractive index \( \mu \) and angle of prism \( A \) is placed in the position of minimum angle of deviation. If minimum angle of deviation is also \( A \). then in terms of refractive index value of \( A \) is:

The power of a lens (biconvex) is \( 1.25 \, \text{m}^{-1} \) in particular medium. Refractive index of the lens is \( 1.5 \) and radii of curvature are \( 20 \, \text{cm} \) and \( 40 \, \text{cm} \) respectively. The refractive index of surrounding medium:

A prism of angle \( 75^\circ \) and refractive index \( \sqrt{3} \) is coated with thin film of refractive index \( 1.5 \) only at the back exit surface. To have total internal reflection at the back exit surface the incident angle must be ______.
\( (\sin 15^\circ = 0.25 \text{ and } \sin 25^\circ = 0.43) \)

A ray of light passes from a denser medium to a rarer medium at an angle of incidence \( i \). The reflected and refracted rays make an angle of \( 90^\circ \) with each other. The angle of reflection and refraction are respectively \( r \) and \( r' \). The critical angle is given by:

In the diagram given below, there are three lenses formed. Considering negligible thickness of each of them as compared to \( |R_1| \) and \( |R_2| \), i.e., the radii of curvature for upper and lower surfaces of the glass lens, the power of the combination is:

What is the relative decrease in focal length of a lens for an increase in optical power by \( 0.1 \, \text{D} \) from \( 2.5 \, \text{D} \)? (\( \text{‘D’} \) stands for dioptre)

A convex lens of focal length \( 30 \, \text{cm} \) is placed in contact with a concave lens of focal length \( 20 \, \text{cm} \). An object is placed at \( 20 \, \text{cm} \) to the left of this lens system. The distance of the image from the lens in \( \text{cm} \) is ______.

An object is placed at the focus of concave lens having focal length \( f \). What is the magnification and distance of the image from the optical centre of the lens?

One side of an equilateral prism is painted by a transparent material of refractive index \( n_{2} \). The refractive index of prism is \( 1.6 \). The minimum value of \( n_{2} \) required for total internal reflection from painted face is ______.

The refractive index of prism is \( \mu = \sqrt{3} \) and the ratio of the angle of minimum deviation to the angle of prism is one. The value of angle of prism is ________.

Two objects A and B are placed at \( 15 \, \text{cm} \) and \( 25 \, \text{cm} \) from the pole in front of a concave mirror having radius of curvature \( 40 \, \text{cm} \). The distance between images formed by the mirror is:

A compound microscope is designed with two symmetric biconvex lenses. The objective lens is cut vertically, creating two identical plano-convex lenses. One of them is used in place of original objective lens. To retain same magnification keeping the object distance unchanged, the tube length has to be

An object ‘\( o \)’ is placed at a distance of \( 100 \, \text{cm} \) in front of a concave mirror of radius of curvature \( 200 \, \text{cm} \) as shown in the figure. The object starts moving towards the mirror at a speed \( 2 \, \text{cm/ s}\). The position of the image from the mirror after \( 10 \, \text{s} \) will be at __________ \( \text{cm} \).

The following figure represents two biconvex lenses \( L_1 \) and \( L_2 \) having focal length \( 10 \, \text{cm} \) and \( 15 \, \text{cm} \) respectively. The distance between \( L_1 \) and \( L_2 \) is:

Two light beams fall on a transparent material block at point \( 1 \) and \( 2 \) with angle \( \theta_{1} \) and \( \theta_{2} \), respectively, as shown in figure. After refraction, the beams intersect at point \( 3 \) which is exactly on the interface at other end of the block.
Given: the distance between \( 1 \) and \( 2 \), \( d = 4\sqrt{3} \, \text{cm} \) and \( \theta_{1} = \theta_{2} = \cos^{-1}\!\left(\frac{n_{2}}{2 n_{1}}\right) \), where refractive index of the block \( n_{2} > \) refractive index of the outside medium \( n_{1} \), then the thickness of the block is __________ cm.

Light from a point source in air falls on a spherical glass surface (refractive index, \( \mu = 1.5 \) and radius of curvature = \( 50 \, \text{cm} \)). The image is formed at a distance of \( 200 \, \text{cm} \) from the glass surface inside the glass. The magnitude of distance of the light source from the glass surface is ________ \( \text{m} \),

For the thin convex lens, the radii of curvature are at \( 15 \, \text{cm} \) and \( 30 \, \text{cm} \) respectively. The focal length the lens is \( 20 \, \text{cm} \). The refractive index of the material is:

Curved surfaces of a plano-convex lens of refractive index \( \mu_1 \) and a plano-concave lens of refractive index \( \mu_2 \) have equal radius of curvature as shown in figure. Find the ratio of radius of curvature to the focal length of the combined lenses.

A ray of light suffers minimum deviation when incident on a prism having angle of the prism equal to \( 60^\circ \). The refractive index of the prism material is \( \sqrt{2} \). The angle of incidence (in degrees) is __________.

In normal adjustment, for a refracting telescope, the distance between objective and eye piece is \( 30 \, \text{cm} \). The focal length of the objective, when the angular magnification of the telescope is \( 2 \), will be:

A ray of light entering from air into a denser medium of refractive index \( \dfrac{4}{3} \) as shown in figure.
The light ray suffers total internal reflection at the adjacent surface as shown. The maximum value of angle \( \theta \) should be equal to:

As shown in the figure, three identical polaroids \( P_{1} \), \( P_{2} \) and \( P_{3} \) are placed one after another. The pass axis of \( P_{2} \) and \( P_{3} \) are inclined at angle of \( 60^\circ \) and \( 90^\circ \) with respect to axis of \( P_{1} \). The source \( S \) has an intensity of \( 256 \, \mathrm{W/m^{2}} \). The intensity of light at point \( O \) is ________ \( \mathrm{W/m^{2}} \).

For an object placed at a distance \( 2.4 \, \text{m} \) from a lens, a sharp focused image is observed on a screen placed at a distance \( 12 \, \text{cm} \) form the lens. A glass plate of refractive index \( 1.5 \) and thickness \( 1 \, \text{cm} \) is introduced between lens and screen such that the glass plate plane faces parallel to the screen. By what distance should the object be shifted so that a sharp focused image is observed again on the screen?

Find the distance of the image from object \( O \), formed by the combination of lenses in the figure:

A mirror is used to produce an image with magnification of \( \frac{1}{4} \). If the distance between object and its image is \( 40 \, \text{cm} \), then the focal length of the mirror is ______.

A vessel of depth ‘\( d \)’ is half filled with oil of refractive index \( n_{1} \) and the other half is filled with water of refractive index \( n_{2} \). The apparent depth of this vessel when viewed from above will be:

A thin plano convex lens made of glass of refractive index \( 1.5 \) is immersed in a liquid of refractive index \( 1.2 \). When the plane side of the lens is silver coated for complete reflection, the lens immersed in the liquid behaves like a concave mirror of focal length \( 0.2 \, \text{m} \). The radius of curvature of the curved surface of the lens is:

For a thin symmetric prism made of glass (refractive index \( 1.5 \)), the ratio of incident angle and minimum deviation will be ________.

Two immiscible liquids of refractive indices \( \frac{8}{5} \) and \( \frac{3}{2} \) respectively are put in a beaker as shown in the figure. The height of each column is \( 6 \, \text{cm} \). A coin is placed at the bottom of the beaker. For near normal vision, the apparent depth of the coin is \( \frac{\alpha}{4} \, \text{cm} \). The value of \( \alpha \) is ________.

The refractive index of a transparent liquid filled in an equilateral hollow prism is \( \sqrt{2} \). The angle of minimum deviation for the liquid will be _________\( ^\circ \).

The refractive index of the material of a glass prism is \( \sqrt{3} \). The angle of minimum deviation is equal to the angle of the prism. What is the angle of the prism?

A light ray is incident, at an incident angle \( \theta_1 \), on the system of two plane mirrors \( M_1 \) and \( M_2 \) having an inclination angle \( 75^\circ \) between them (as shown in figure). After reflecting from mirror \( M_1 \) it gets reflected back by the mirror \( M_2 \) with an angle of reflection \( 30^\circ \). The total deviation of the ray will be ________ degree.

A convex lens of refractive index \( 1.5 \) and focal length \( f = 18 \, \text{cm} \) is immersed in water. The difference in focal lengths of the given lens when it is in water and in air is \( \alpha \times f \). The value of \( \alpha \) is __________. (refractive index of water \( = \frac{4}{3} \))

The wavelength of light, while it is passing through water is \( 540 \,\text{nm} \). The refractive index of water is \( \frac{4}{3} \). The wavelength of the same light when it is passing through a transparent medium having refractive index of \( \frac{3}{2} \) is ________ \( \text{nm} \).

Two identical symmetric double convex lenses of focal length \( f \) are cut into two equal parts \( L_{1}\!, L_{2} \) by \( AB \) plane and \( L_{3}\!, L_{4} \) by \( XY \) plane as shown in figure respectively. The ratio of focal lengths of lenses \( L_{1} \) and \( L_{3} \) is:

Which of the following are true?
A. Speed of light in vacuum is dependent on the direction of propagation.
B. Speed of light in a medium is independent of the wavelength of light.
C. The speed of light is independent of the motion of the source.
D. The speed of light in a medium is independent of intensity.
Choose the correct answer from the options given below:

A ray of light is incident at an angle of incidence \( 60^\circ \) on the glass slab of refractive index \( \sqrt{3} \). After refraction, the light ray emerges out from other parallel faces and lateral shift between incident ray and emergent ray is \( 4\sqrt{3} \, \text{cm} \). The thickness of the glass slab is ________ \( \text{cm} \).

As shown in the figure, in Young’s double slit experiment, a thin plate of thickness \( t = 10 \, \mu \text{m} \) and refractive index \( \mu = 1.2 \) is inserted infront of slit \( S_{1} \). The experiment is conducted in air \( (\mu = 1) \) and uses a monochromatic light of wavelength \( \lambda = 500 \, \text{nm} \). Due to the insertion of the plate, central maxima is shifted by a distance of \( x \beta_{0} \), \( \beta_{0} \) is the fringe-width before the insertion of plate. The value of the \( x \) is _______.

A ray of light passing through a prism \( (\mu = \sqrt{3} )\) suffers minimum deviation. It is found that the angle of incidence is double the angle of refraction within the prism. Then, the angle of prism is ______ (in degrees).

What is the lateral shift of a ray refracted through a parallel-sided glass slab of thickness \( 'h' \) in terms of the angle of incidence \( 'i' \) and angle of refraction \(' r '\), if the glass slab is placed in air medium?

Cross-section view of a prism is the equilateral triangle \( ABC \) shown in the figure. The minimum deviation is observed using this prism when the angle of incidence is equal to the prism angle. The time taken by light to travel from \( P \) (midpoint of \( BC \)) to \( A \) is _______ \( \times 10^{-10} \, \text{s} \). (Given, speed of light in vacuum \( = 3 \times 10^{8} \, \text{m/s} \) and \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \))

Two plane mirrors \( M_1 \) and \( M_2 \) are at right angle to each other, shown. A point source ‘P’ is placed at \( a \) and \( 2a \) meter away from \( M_1 \) and \( M_2 \) respectively. The shortest distance between the images thus formed is: (Take \( \sqrt{5} = 2.3 \).)

When one light ray is reflected from a plane mirror with \( 30^\circ \) angle of reflection, the angle of deviation of the ray after reflection is:

$7.9\text{ MeV}$ $\alpha$-particle scatters from a target material of atomic number $79$. From the given data the estimated diameter of nuclei of the target material is (approximately) _________ $\text{m}$.
$\left[ \frac{1}{4\pi\epsilon_0} = 9 \times 10^9\text{ Nm}^2/\text{C}^2 \text{ and electron charge} = 1.6 \times 10^{-19}\text{ C} \right]$

Consider following statements for refraction of light through prism, when angle of deviation is minimum.
(A) The refracted ray inside prism becomes parallel to the base.
(B) Larger angle prisms provide smaller angle of minimum deviation.
(C) Angle of incidence and angle of emergence becomes equal.
(D) There are always two sets of angle of incidence for which deviation will be same except at minimum deviation setting.
(E) Angle of refraction becomes double of prism angle.
Choose the correct answer from the options given below:

The light rays from an object have been reflected towards an observer from a standard flat mirror, the image observed by the observer are:
A. Real
B. Erect
C. Smaller in size than object
D. Laterally inverted
Choose the most appropriate answer from the options given below:

A thin convex lens of focal length \( 5 \, \text{cm} \) and a thin concave lens of focal length \( 4 \, \text{cm} \) are combined together (without any gap) and this combination has magnification \( m_1 \) when an object is placed \( 10 \, \text{cm} \) before the convex lens. Keeping the positions of convex lens and object undisturbed a gap of \( 1 \, \text{cm} \) is introduced between the lenses by moving the concave lens away, which lead to a change in magnification of total lens system to \( m_2 \). The value of \( \left| \dfrac{m_1}{m_2} \right| \) is ________.

An effective power of a combination of \( 5 \) identical convex lenses which are kept in contact along the principal axis is \( 25 \, \text{D} \). Focal length of each of the convex lens is:

A hemispherical vessel is completely filled with a liquid of refractive index \( \mu \). A small coin is kept at the lowest point \( (O )\) of the vessel as shown in figure. The minimum value of the refractive index of the liquid so that a person can see  the coin from point \( E \) (at the level of the vessel) is ______.

A scientist is observing a bacteria through a compound microscope. For better analysis and to improve its resolving power he should. (Select the best option)

A lens having refractive index \( 1.6 \) has focal length of \( 12 \, \text{cm} \), when it is in air. Find the focal length of the lens when it is placed in water: (Take refractive index of water as \( 1.28 \))

The image of an object placed in air formed by a convex refracting surface is at a distance of \( 10\,m \) behind the surface. The image is real and is at \( \dfrac{2^{rd}}{3} \) of the distance of the object from the surface. The wavelength of light inside the surface is \( \dfrac{2}{3} \) times the wavelength in air. The radius of the curved surface is \( \dfrac{x}{13}\,m \). The value of 'x' is ________.

A convex lens has power \( P \). It is cut into two halves, along its principal axis. Further one piece (out of the two halves) is cut into two halves perpendicular to the principal axis (as shown in figures). Choose the incorrect option for the reported pieces.

The size of the image of an object, formed by a thin lens are equal when the object is placed at two different positions \( 8 \; \mathrm{cm} \) and \( 24 \; \mathrm{cm} \) from the lens. The focal length of the lens is ______ \( \mathrm{cm} \).

A symmetric thin biconvex lens is cut into four equal parts by two planes AB and CD as shown in figure. If the power of original lens is \( 4D \) then the power of a part of the divided lens is:

When a beam of while light is allowed to pass through convex lens parallel to principal axis, the different colours of light converge at different point on the principle axis after refraction. This is called:

Light ray incident along a vector \( \overrightarrow{AO} \, ( \overrightarrow{AO} = 2\hat{i} - 3\hat{j} ) \) emerges out along vector \( \overrightarrow{OB} \, ( \overrightarrow{OB} = C\hat{i} - 4\hat{j} ) \) as shown in the figure below. The value of \( C \) is _______.

The critical angle for a denser-rarer interface is \( 45^\circ \). The speed of light in rarer medium is \( 3 \times 10^8 \, \text{m/s} \). The speed of light in the denser medium is:

A thin prism \( P_1 \) with an angle \( 6^\circ \) and made of glass of refractive index \( 1.54 \) is combined with another prism \( P_2 \) made from glass of refractive index \( 1.72 \) to produce dispersion without average deviation. The angle of prism \( P_2 \) is:

Distance between object and its image (magnified by \( -\frac{1}{3} \)) is \( 30 \, \text{cm} \). The focal length of the mirror used is \( \left( \frac{x}{4} \right) \, \text{cm} \), where magnitude of value of \( x \) is ______.

A concave-convex lens of refractive index \( 1.5 \) and the radii of curvature of its surfaces are \( 30 \, \text{cm} \) and \( 20 \, \text{cm} \), respectively. The concave surface is upwards and is filled with a liquid of refractive index \( 1.3 \). The focal length of the liquid-glass combination will be:

A collimated beam of light of diameter \( 2 \, \text{mm} \) is propagating along \( x \)-axis. The beam is required to be expanded in a collimated beam of diameter \( 14 \, \text{mm} \) using a system of two convex lenses. If first lens has focal length \( 40 \, \text{mm} \), then the focal length of second lens is __________ \( \text{mm} \).

The refractive index of a converging lens is \( 1.4 \). What will be the focal length of this lens if it is placed in a medium of same refractive index? Assume the radii of curvature of the faces of lens are \( R_{1} \) and \( R_{2} \) respectively.

The distance between object and its \( 3 \) times magnified virtual image as produced by a convex lens is \( 20 \, \text{cm} \). The focal length of the lens used is ______ \( \text{cm} \).

In a microscope of tube length \( 10 \, \text{cm} \) two convex lenses are arranged with focal length of \( 2 \, \text{cm} \) and \( 5 \, \text{cm} \). Total magnification obtained with this system for normal adjustment is \( (5)^k \). The value of \( k \) is ______.

A concave mirror of focal length \( 10 \, \text{cm} \) forms an image which is double the size of object when the object is placed at two different positions. The distance between the two positions of the object is ______ \( \text{cm} \).

A light wave travelling linearly in a medium of dielectric constant \( 4 \), incidents on the horizontal interface separating medium with air. The angle of incidence for which the total intensity of incident wave will be reflected back into the same medium will be:

At the interface between two materials having refractive indices \( n_1 \) and \( n_2 \), the critical angle for reflection of an em wave is \( \theta_{1C} \). The \( n_2 \) material is replaced by another material having refractive index \( n_3 \) such that the critical angle at the interface between \( n_1 \) and \( n_3 \) materials is \( \theta_{2C} \). If \( n_3 > n_2 > n_1 \); \( \frac{n_2}{n_3} = \frac{2}{5} \) and \( \sin \theta_{2C} - \sin \theta_{1C} = \frac{1}{2} \), then \( \theta_{1C} \) is:

A photograph of a landscape is captured by a drone camera at a height of \( 18 \,\text{km} \). The size of the camera film is \( 2 \,\text{cm} \times 2 \,\text{cm} \) and the area of the landscape photographed is \( 400 \,\text{km}^{2} \). The focal length of the lens in the drone camera is:

Let \( u \) and \( v \) be the distances of the object and the image from a lens of focal length \( f \). The correct graphical representation of \( u \) and \( v \) for a convex lens when \( |u| > f \), is:

A concave mirror produces an image of an object such that the distance between the object and image is \( 20 \, \text{cm} \). If the magnification of the image is \( '-3' \), then the magnitude of the radius of curvature of the mirror is:

When an object is placed \( 40 \, \text{cm} \) away from a spherical mirror an image of magnification \( \dfrac{1}{2} \) is produced. To obtain an image with magnification of \( \dfrac{1}{3} \), the object is to be moved:

A convex lens made of glass (refractive index \( = 1.5 \)) has focal length \( 24 \, \text{cm} \) in air. When it is totally immersed in water (refractive index \( = 1.33 \)), its focal length changes to:

Two identical thin biconvex lenses of focal length \( 15 \, \text{cm} \) and refractive index \( 1.5 \) are in contact with each other. The space between the lenses is filled with a liquid of refractive index \( 1.25 \). The focal length of the combination is ________ \( \text{cm} \).

A ray of laser of a wavelength \( 630 \ \text{nm} \) is incident at an angle of \( 30^\circ \) at the diamond–air interface. It is going from diamond to air. The refractive index of diamond is \( 2.42 \) and that of air is \( 1 \). Choose the correct option.

A point object, ‘O’ is placed in front of two thin symmetrical coaxial convex lenses \( L_1 \) and \( L_2 \) with focal length \( 24 \, \text{cm} \) and \( 9 \, \text{cm} \) respectively. The distance between two lenses is \( 10 \, \text{cm} \) and the object is placed \( 6 \, \text{cm} \) away from lens \( L_1 \) as shown in the figure. The distance between the object and the image formed by the system of two lenses is ______ \( \text{cm} \).

A ray of light is incident from air on a glass plate having thickness \( \sqrt{3} \, \text{cm} \) and refractive index \( \sqrt{2} \). The angle of incidence of a ray is equal to the critical angle for glass-air interface. The lateral displacement of the ray when it passes through the plate is ________ \( \times 10^{-2} \, \text{cm} \). (Given \( \sin 15^\circ = 0.26 \))

A microscope is focused on an object at the bottom of a bucket. If liquid with refractive index \( \frac{5}{3} \) is poured inside the bucket, the microscope have to be raised by \( 30 \, \text{cm} \) to focus the object again. The height of the liquid in the bucket is:

A prism of refractive index \( n_1 \) and another prism of refractive index \( n_2 \) are stuck together (as shown in the figure). \( n_1 \) and \( n_2 \) depend on \( \lambda \), the wavelength of light, according to the relation \( n_1 = 1.2 + \frac{10.8 \times 10^{-14}}{\lambda^{2}} \) and \( n_2 = 1.45 + \frac{1.8 \times 10^{-14}}{\lambda^{2}} \). The wavelength for which rays incident at any angle on the interface BC pass through without bending at that interface will be ________ \( \text{nm} \).