Physics-
General
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A Ray of light goes from A in a medium where the speed of light is V1 to a point B in a medium where the speed of light is V2 as shown in figure The path of the rays as shown in figure Answer the following questions,based on the above paragraph The time taken for the light to go from the point A to the point B in the figure

  1. fraction numerator a s i n invisible function application i over denominator V subscript 1 end subscript end fraction    
  2. fraction numerator a s e c invisible function application i over denominator v subscript 1 end subscript end fraction plus fraction numerator b s e c invisible function application r over denominator v subscript 2 end subscript end fraction    
  3. fraction numerator b s i n invisible function application r over denominator v subscript 2 end subscript end fraction    
  4. fraction numerator v subscript 2 end subscript a s i n to the power of 2 end exponent over denominator v subscript 1 end subscript b s i n invisible function application r end fraction    

The correct answer is: fraction numerator a s e c invisible function application i over denominator v subscript 1 end subscript end fraction plus fraction numerator b s e c invisible function application r over denominator v subscript 2 end subscript end fraction

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General
physics-

Spherical aberration in spherical mirrors is a defect which is due to dependence of focal length ‘f’ on angle of incidence ‘ q ’ as shown in figure is given by f equals R minus fraction numerator K over denominator 2 end fraction s e c invisible function application theta where R is radius of curvature of mirror and q is the angle of incidence The rays which are closed to principal axis are called paraxial rays and the rays far away from principal axis are called marginal rays As a result of above dependence different rays are brought to focus at different points and the image of a point object is on a point Which of the following statements are correct regarding spherical aberration :

Spherical aberration in spherical mirrors is a defect which is due to dependence of focal length ‘f’ on angle of incidence ‘ q ’ as shown in figure is given by f equals R minus fraction numerator K over denominator 2 end fraction s e c invisible function application theta where R is radius of curvature of mirror and q is the angle of incidence The rays which are closed to principal axis are called paraxial rays and the rays far away from principal axis are called marginal rays As a result of above dependence different rays are brought to focus at different points and the image of a point object is on a point Which of the following statements are correct regarding spherical aberration :

physics-General
General
physics-

Spherical aberration in spherical mirrors is a defect which is due to dependence of focal length ‘f’ on angle of incidence ‘ q ’ as shown in figure is given by f equals R minus fraction numerator K over denominator 2 end fraction s e c invisible function application theta where R is radius of curvature of mirror and q is the angle of incidence The rays which are closed to principal axis are called paraxial rays and the rays far away from principal axis are called marginal rays As a result of above dependence different rays are brought to focus at different points and the image of a point object is on a point For paraxial rays, focal length approximately is

Spherical aberration in spherical mirrors is a defect which is due to dependence of focal length ‘f’ on angle of incidence ‘ q ’ as shown in figure is given by f equals R minus fraction numerator K over denominator 2 end fraction s e c invisible function application theta where R is radius of curvature of mirror and q is the angle of incidence The rays which are closed to principal axis are called paraxial rays and the rays far away from principal axis are called marginal rays As a result of above dependence different rays are brought to focus at different points and the image of a point object is on a point For paraxial rays, focal length approximately is

physics-General
General
physics-

Spherical aberration in spherical mirrors is a defect which is due to dependence of focal length ‘f’ on angle of incidence ‘ q ’ as shown in figure is given by f equals R minus fraction numerator K over denominator 2 end fraction s e c invisible function application theta where R is radius of curvature of mirror and q is the angle of incidence The rays which are closed to principal axis are called paraxial rays and the rays far away from principal axis are called marginal rays As a result of above dependence different rays are brought to focus at different points and the image of a point object is on a point The total deviation suffered by the ray falling on mirror at an angle of incidence equal to 60° is

Spherical aberration in spherical mirrors is a defect which is due to dependence of focal length ‘f’ on angle of incidence ‘ q ’ as shown in figure is given by f equals R minus fraction numerator K over denominator 2 end fraction s e c invisible function application theta where R is radius of curvature of mirror and q is the angle of incidence The rays which are closed to principal axis are called paraxial rays and the rays far away from principal axis are called marginal rays As a result of above dependence different rays are brought to focus at different points and the image of a point object is on a point The total deviation suffered by the ray falling on mirror at an angle of incidence equal to 60° is

physics-General
General
physics-

Spherical aberration in spherical mirrors is a defect which is due to dependence of focal length ‘f’ on angle of incidence ‘ q ’ as shown in figure is given by f equals R minus fraction numerator K over denominator 2 end fraction s e c invisible function application theta where R is radius of curvature of mirror and q is the angle of incidence The rays which are closed to principal axis are called paraxial rays and the rays far away from principal axis are called marginal rays As a result of above dependence different rays are brought to focus at different points and the image of a point object is on a point If fp and fm represent the focal length of paraxial and marginal rays respectively, then correct relationship is :

Spherical aberration in spherical mirrors is a defect which is due to dependence of focal length ‘f’ on angle of incidence ‘ q ’ as shown in figure is given by f equals R minus fraction numerator K over denominator 2 end fraction s e c invisible function application theta where R is radius of curvature of mirror and q is the angle of incidence The rays which are closed to principal axis are called paraxial rays and the rays far away from principal axis are called marginal rays As a result of above dependence different rays are brought to focus at different points and the image of a point object is on a point If fp and fm represent the focal length of paraxial and marginal rays respectively, then correct relationship is :

physics-General
General
physics-

Most materials have the refractive index, n > 1 So, when a light ray from air enters a naturally occurring material, then by Snell’s text law,  end text fraction numerator sin invisible function application theta subscript 1 end subscript over denominator sin invisible function application theta subscript 2 end subscript end fraction equals fraction numerator n subscript 1 end subscript over denominator n subscript 2 end subscript end fraction comma it is understood that the refracted ray bends towards the normal But it never emerges on the same side of the normal as the incident ray According to electromagnetism, the refractive index of the medium is given by the relation, n equals open parentheses fraction numerator c over denominator V end fraction close parentheses equals plus-or-minus square root of epsilon subscript r end subscript mu subscript r end subscript end root, where c is the speed of the electromagnetic waves in vacuum, v its speed in the medium, er and mr are negative, one must choose the negative root of n Such negative refractive index materials can now be artificially prepared and are called metamaterials They exhibit signficantly different optical behaviour, without violating any physical laws Since n is negative, it results in a change in the direction of propagation of the refracted light However, similar to normal materials, the frequency of light remains unchanged upon refraction even in metamaterials For light incident from air on a meta-material, the appropriate ray diagrams

Most materials have the refractive index, n > 1 So, when a light ray from air enters a naturally occurring material, then by Snell’s text law,  end text fraction numerator sin invisible function application theta subscript 1 end subscript over denominator sin invisible function application theta subscript 2 end subscript end fraction equals fraction numerator n subscript 1 end subscript over denominator n subscript 2 end subscript end fraction comma it is understood that the refracted ray bends towards the normal But it never emerges on the same side of the normal as the incident ray According to electromagnetism, the refractive index of the medium is given by the relation, n equals open parentheses fraction numerator c over denominator V end fraction close parentheses equals plus-or-minus square root of epsilon subscript r end subscript mu subscript r end subscript end root, where c is the speed of the electromagnetic waves in vacuum, v its speed in the medium, er and mr are negative, one must choose the negative root of n Such negative refractive index materials can now be artificially prepared and are called metamaterials They exhibit signficantly different optical behaviour, without violating any physical laws Since n is negative, it results in a change in the direction of propagation of the refracted light However, similar to normal materials, the frequency of light remains unchanged upon refraction even in metamaterials For light incident from air on a meta-material, the appropriate ray diagrams

physics-General
General
maths-

The element in the first row and third column of the inverse of the matrix open square brackets table row 1 2 cell negative 3 end cell row 0 1 2 row 0 0 1 end table close square bracketsis

Let A equals open square brackets table row 1 2 cell negative 3 end cell row 0 1 2 row 0 0 1 end table close square brackets Þ |A| = 1
a d j left parenthesis A right parenthesis equals open square brackets table row 1 2 cell negative 1 end cell row cell negative 2 end cell 1 1 row 7 cell negative 2 end cell 1 end table close square brackets to the power of T end exponent.
Hence, A to the power of negative 1 end exponent equals fraction numerator a d j left parenthesis A right parenthesis over denominator vertical line A vertical line end fraction
Þ A to the power of negative 1 end exponent equals open square brackets table row 1 cell negative 2 end cell 7 row 2 1 cell negative 2 end cell row cell negative 1 end cell 1 1 end table close square brackets . Hence, element z equals 3.

The element in the first row and third column of the inverse of the matrix open square brackets table row 1 2 cell negative 3 end cell row 0 1 2 row 0 0 1 end table close square bracketsis

maths-General
Let A equals open square brackets table row 1 2 cell negative 3 end cell row 0 1 2 row 0 0 1 end table close square brackets Þ |A| = 1
a d j left parenthesis A right parenthesis equals open square brackets table row 1 2 cell negative 1 end cell row cell negative 2 end cell 1 1 row 7 cell negative 2 end cell 1 end table close square brackets to the power of T end exponent.
Hence, A to the power of negative 1 end exponent equals fraction numerator a d j left parenthesis A right parenthesis over denominator vertical line A vertical line end fraction
Þ A to the power of negative 1 end exponent equals open square brackets table row 1 cell negative 2 end cell 7 row 2 1 cell negative 2 end cell row cell negative 1 end cell 1 1 end table close square brackets . Hence, element z equals 3.
General
maths-

If A and B are non-singular square matrices of same order, then a d j left parenthesis A B right parenthesis is equal to

It is obvious.

If A and B are non-singular square matrices of same order, then a d j left parenthesis A B right parenthesis is equal to

maths-General
It is obvious.
General
maths-

If d is the determinant of a square matrix A of order n, then the determinant of its adjoint is

vertical line A d j A vertical line equals vertical line A vertical line to the power of n minus 1 end exponent equals d to the power of n minus 1 end exponent.

If d is the determinant of a square matrix A of order n, then the determinant of its adjoint is

maths-General
vertical line A d j A vertical line equals vertical line A vertical line to the power of n minus 1 end exponent equals d to the power of n minus 1 end exponent.
General
chemistry-

C subscript 6 H subscript 5 minus C H equals C H C H O not stretchy rightwards arrow with X on top C subscript 6 H subscript 5 C H equals C H C H subscript 2 O H  In the above sequence X can be

N a B H subscript 4 space a n d space L i A l H subscript 4 attacks only carbonyl group and reduce it into alcohol group. They do not attack on double bond. 

C subscript 6 H subscript 5 minus C H equals C H C H O not stretchy rightwards arrow with X on top C subscript 6 H subscript 5 C H equals C H C H subscript 2 O H  In the above sequence X can be

chemistry-General
N a B H subscript 4 space a n d space L i A l H subscript 4 attacks only carbonyl group and reduce it into alcohol group. They do not attack on double bond. 
General
physics-

A ray of light traveling in air is incident at grazing angle (Ð >i 90º) on a long rectangular slab of a transparent medium of thickness t = 1.0 m The point of incidence is the medium A (0, 0) The medium has a variable index of refraction n(y) given by n left parenthesis y right parenthesis equals open square brackets k y to the power of 3 divided by 2 end exponent plus 1 close square brackets to the power of 1 divided by 2 end exponent where k equals 1.0 left parenthesis m right parenthesis to the power of negative 3 divided by 2 end exponent The refractive index of air is 1 The coordinates (x1, y(A) of the point P where the ray intersects the upper surface of the slabair boundary are

A ray of light traveling in air is incident at grazing angle (Ð >i 90º) on a long rectangular slab of a transparent medium of thickness t = 1.0 m The point of incidence is the medium A (0, 0) The medium has a variable index of refraction n(y) given by n left parenthesis y right parenthesis equals open square brackets k y to the power of 3 divided by 2 end exponent plus 1 close square brackets to the power of 1 divided by 2 end exponent where k equals 1.0 left parenthesis m right parenthesis to the power of negative 3 divided by 2 end exponent The refractive index of air is 1 The coordinates (x1, y(A) of the point P where the ray intersects the upper surface of the slabair boundary are

physics-General
General
physics-

A ray of light traveling in air is incident at grazing angle (Ð >i 90º) on a long rectangular slab of a transparent medium of thickness t = 1.0 m The point of incidence is the medium A (0, 0) The medium has a variable index of refraction n(y) given by n left parenthesis y right parenthesis equals open square brackets k y to the power of 3 divided by 2 end exponent plus 1 close square brackets to the power of 1 divided by 2 end exponent where k equals 1.0 left parenthesis m right parenthesis to the power of negative 3 divided by 2 end exponent The refractive index of air is 1 Equation for the trajectory y(x) of the ray in the medium is

A ray of light traveling in air is incident at grazing angle (Ð >i 90º) on a long rectangular slab of a transparent medium of thickness t = 1.0 m The point of incidence is the medium A (0, 0) The medium has a variable index of refraction n(y) given by n left parenthesis y right parenthesis equals open square brackets k y to the power of 3 divided by 2 end exponent plus 1 close square brackets to the power of 1 divided by 2 end exponent where k equals 1.0 left parenthesis m right parenthesis to the power of negative 3 divided by 2 end exponent The refractive index of air is 1 Equation for the trajectory y(x) of the ray in the medium is

physics-General
General
physics-

A ray of light traveling in air is incident at grazing angle (Ð >i 90º) on a long rectangular slab of a transparent medium of thickness t = 1.0 m The point of incidence is the medium A (0, 0) The medium has a variable index of refraction n(y) given by n left parenthesis y right parenthesis equals open square brackets k y to the power of 3 divided by 2 end exponent plus 1 close square brackets to the power of 1 divided by 2 end exponent where k equals 1.0 left parenthesis m right parenthesis to the power of negative 3 divided by 2 end exponent The refractive index of air is 1 The incident angle at B(x, y) in the medium and the slope at B are related by the formula

A ray of light traveling in air is incident at grazing angle (Ð >i 90º) on a long rectangular slab of a transparent medium of thickness t = 1.0 m The point of incidence is the medium A (0, 0) The medium has a variable index of refraction n(y) given by n left parenthesis y right parenthesis equals open square brackets k y to the power of 3 divided by 2 end exponent plus 1 close square brackets to the power of 1 divided by 2 end exponent where k equals 1.0 left parenthesis m right parenthesis to the power of negative 3 divided by 2 end exponent The refractive index of air is 1 The incident angle at B(x, y) in the medium and the slope at B are related by the formula