Physics-
General
Easy

Question

A ray of light travelling in air is incident at a grazing angle on a large transparent slab of thickness t equals 2.0 m. The point of incidence is the origin. The medium has a variable refractive index(y) given by mu left parenthesis y right parenthesis equals square root of k y plus 1 end root Where y is in m and k equals 0.25 m to the power of negative 1 end exponent

a) Express a relation between the angle of incidence and the slope of the trajectory m, in terms of the refractive index at that point m ( y)</span

  1. fraction numerator square root of mu to the power of 2 end exponent minus 1 end root over denominator sin to the power of negative 1 end exponent invisible function application open parentheses fraction numerator 1 over denominator mu end fraction close parentheses end fraction    
  2. fraction numerator square root of mu to the power of 2 end exponent plus 1 end root over denominator sin to the power of negative 1 end exponent invisible function application open parentheses fraction numerator 1 over denominator mu end fraction close parentheses end fraction    
  3. fraction numerator square root of mu to the power of 2 end exponent plus 1 end root over denominator sin to the power of negative 1 end exponent invisible function application open parentheses fraction numerator 1 over denominator mu end fraction close parentheses end fraction    
  4. fraction numerator mu over denominator sin to the power of negative 1 end exponent invisible function application open parentheses fraction numerator 1 over denominator mu end fraction close parentheses end fraction    

The correct answer is: fraction numerator square root of mu to the power of 2 end exponent minus 1 end root over denominator sin to the power of negative 1 end exponent invisible function application open parentheses fraction numerator 1 over denominator mu end fraction close parentheses end fraction

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Find the variation of refractive index assuming it to be a function of y such that a ray entering origin at grazing incidence follows a parabolic path y = x2 as shown in fig

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Maths-

If 3 Tan to the power of 4 space alpha minus 10 Tan squared space alpha plus 3 equals 0 then principal values of ' alpha ' are

space space space space 3 t an to the power of 4 space alpha minus 10 t an squared space alpha plus 3 equals 0
rightwards double arrow 3 t an to the power of 4 space alpha minus 9 tan squared alpha minus t an squared space alpha plus 3 equals 0
rightwards double arrow 3 space tan squared alpha left parenthesis tan squared alpha minus 3 right parenthesis minus 1 left parenthesis tan squared alpha minus 3 right parenthesis equals 0
rightwards double arrow left parenthesis 3 space tan squared alpha minus 1 right parenthesis left parenthesis tan squared alpha minus 3 right parenthesis equals 0
rightwards double arrow tan squared alpha equals 1 third o r space tan squared alpha equals 3
rightwards double arrow tan alpha equals fraction numerator 1 over denominator square root of 3 end fraction o r space tan space alpha equals square root of 3
rightwards double arrow tan space alpha equals tan space left parenthesis plus-or-minus 30 degree right parenthesis o r space tan space left parenthesis plus-or-minus 60 degree right parenthesis
rightwards double arrow alpha space equals space plus-or-minus 30 degree space o r space plus-or-minus 60 degree

If 3 Tan to the power of 4 space alpha minus 10 Tan squared space alpha plus 3 equals 0 then principal values of ' alpha ' are

Maths-General
space space space space 3 t an to the power of 4 space alpha minus 10 t an squared space alpha plus 3 equals 0
rightwards double arrow 3 t an to the power of 4 space alpha minus 9 tan squared alpha minus t an squared space alpha plus 3 equals 0
rightwards double arrow 3 space tan squared alpha left parenthesis tan squared alpha minus 3 right parenthesis minus 1 left parenthesis tan squared alpha minus 3 right parenthesis equals 0
rightwards double arrow left parenthesis 3 space tan squared alpha minus 1 right parenthesis left parenthesis tan squared alpha minus 3 right parenthesis equals 0
rightwards double arrow tan squared alpha equals 1 third o r space tan squared alpha equals 3
rightwards double arrow tan alpha equals fraction numerator 1 over denominator square root of 3 end fraction o r space tan space alpha equals square root of 3
rightwards double arrow tan space alpha equals tan space left parenthesis plus-or-minus 30 degree right parenthesis o r space tan space left parenthesis plus-or-minus 60 degree right parenthesis
rightwards double arrow alpha space equals space plus-or-minus 30 degree space o r space plus-or-minus 60 degree
General
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If Sin space open parentheses fraction numerator pi Cot space theta over denominator 4 end fraction close parentheses equals Cos space open parentheses fraction numerator pi Tan space theta over denominator 4 end fraction close parentheses and theta is in the first quadrant then theta =....

sin open parentheses πcotθ over 4 close parentheses equals cos open parentheses πtanθ over 4 close parentheses
rightwards double arrow sin open parentheses πcotθ over 4 close parentheses equals sin open parentheses straight pi over 2 minus πtanθ over 4 close parentheses
rightwards double arrow πcotθ over 4 equals straight pi over 4 minus πtanθ over 4
rightwards double arrow πcotθ over 4 equals straight pi over 4 open parentheses 2 minus tan theta close parentheses
rightwards double arrow c o t theta equals 2 minus t a n theta
rightwards double arrow fraction numerator cos theta over denominator sin theta end fraction plus fraction numerator sin theta over denominator cos theta end fraction equals 2
rightwards double arrow fraction numerator cos squared theta plus sin squared theta over denominator sin theta cos theta end fraction equals 2
rightwards double arrow 1 equals 2 sin theta cos theta
rightwards double arrow sin straight pi over 2 equals sin 2 theta
rightwards double arrow straight pi over 2 equals 2 theta
rightwards double arrow theta equals straight pi over 4

If Sin space open parentheses fraction numerator pi Cot space theta over denominator 4 end fraction close parentheses equals Cos space open parentheses fraction numerator pi Tan space theta over denominator 4 end fraction close parentheses and theta is in the first quadrant then theta =....

Maths-General
sin open parentheses πcotθ over 4 close parentheses equals cos open parentheses πtanθ over 4 close parentheses
rightwards double arrow sin open parentheses πcotθ over 4 close parentheses equals sin open parentheses straight pi over 2 minus πtanθ over 4 close parentheses
rightwards double arrow πcotθ over 4 equals straight pi over 4 minus πtanθ over 4
rightwards double arrow πcotθ over 4 equals straight pi over 4 open parentheses 2 minus tan theta close parentheses
rightwards double arrow c o t theta equals 2 minus t a n theta
rightwards double arrow fraction numerator cos theta over denominator sin theta end fraction plus fraction numerator sin theta over denominator cos theta end fraction equals 2
rightwards double arrow fraction numerator cos squared theta plus sin squared theta over denominator sin theta cos theta end fraction equals 2
rightwards double arrow 1 equals 2 sin theta cos theta
rightwards double arrow sin straight pi over 2 equals sin 2 theta
rightwards double arrow straight pi over 2 equals 2 theta
rightwards double arrow theta equals straight pi over 4
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If Tan squared space theta equals square root of 3 plus left parenthesis square root of 3 minus 1 right parenthesis Tan space theta and theta lies in open parentheses negative pi over 2 comma pi over 2 close parentheses then theta =

space space space space t an squared space theta equals square root of 3 plus left parenthesis square root of 3 minus 1 right parenthesis t an space theta
rightwards double arrow tan squared theta minus square root of 3 tan space theta minus tan space theta minus square root of 3 equals 0
rightwards double arrow tan space theta left parenthesis tan space theta minus square root of 3 right parenthesis space plus space left parenthesis tan space theta space minus square root of 3 right parenthesis equals 0
rightwards double arrow open parentheses tan space theta plus 1 close parentheses open parentheses tan space theta minus square root of 3 close parentheses equals 0
rightwards double arrow tan space theta equals equals negative 1 space o r space tan space theta space equals square root of 3
rightwards double arrow tan space theta space equals tan space open parentheses fraction numerator negative pi over denominator 4 end fraction close parentheses space o r space tan space theta space equals tan pi over 3 space space space space space space space space space space open square brackets a s space theta space l i e s space i n space open parentheses fraction numerator negative pi over denominator 2 end fraction comma pi over 2 close parentheses close square brackets
rightwards double arrow theta equals fraction numerator negative pi over denominator 4 end fraction o r pi over 3
S o comma space open curly brackets fraction numerator negative pi over denominator 4 end fraction comma pi over 3 close curly brackets

If Tan squared space theta equals square root of 3 plus left parenthesis square root of 3 minus 1 right parenthesis Tan space theta and theta lies in open parentheses negative pi over 2 comma pi over 2 close parentheses then theta =

Maths-General
space space space space t an squared space theta equals square root of 3 plus left parenthesis square root of 3 minus 1 right parenthesis t an space theta
rightwards double arrow tan squared theta minus square root of 3 tan space theta minus tan space theta minus square root of 3 equals 0
rightwards double arrow tan space theta left parenthesis tan space theta minus square root of 3 right parenthesis space plus space left parenthesis tan space theta space minus square root of 3 right parenthesis equals 0
rightwards double arrow open parentheses tan space theta plus 1 close parentheses open parentheses tan space theta minus square root of 3 close parentheses equals 0
rightwards double arrow tan space theta equals equals negative 1 space o r space tan space theta space equals square root of 3
rightwards double arrow tan space theta space equals tan space open parentheses fraction numerator negative pi over denominator 4 end fraction close parentheses space o r space tan space theta space equals tan pi over 3 space space space space space space space space space space open square brackets a s space theta space l i e s space i n space open parentheses fraction numerator negative pi over denominator 2 end fraction comma pi over 2 close parentheses close square brackets
rightwards double arrow theta equals fraction numerator negative pi over denominator 4 end fraction o r pi over 3
S o comma space open curly brackets fraction numerator negative pi over denominator 4 end fraction comma pi over 3 close curly brackets
parallel
General
Maths-

If , thenTan space theta plus Sec space theta equals square root of 3 he principal value of open parentheses theta plus pi over 6 close parentheses is

Tan space theta plus Sec space theta equals square root of 3
rightwards double arrow fraction numerator s i n theta over denominator cos theta end fraction plus fraction numerator 1 over denominator cos theta end fraction equals square root of 3
rightwards double arrow sin theta plus 1 equals square root of 3 cos theta
rightwards double arrow square root of 3 cos theta minus sin theta equals 1
rightwards double arrow fraction numerator square root of 3 over denominator 2 end fraction cos theta minus 1 half sin theta equals 1 half
rightwards double arrow cos straight pi over 6 cos theta minus sin straight pi over 6 sin theta equals 1 half
rightwards double arrow cos open parentheses straight pi over 6 plus theta close parentheses equals cos straight pi over 3
rightwards double arrow theta plus straight pi over 6 equals straight pi over 3

If , thenTan space theta plus Sec space theta equals square root of 3 he principal value of open parentheses theta plus pi over 6 close parentheses is

Maths-General
Tan space theta plus Sec space theta equals square root of 3
rightwards double arrow fraction numerator s i n theta over denominator cos theta end fraction plus fraction numerator 1 over denominator cos theta end fraction equals square root of 3
rightwards double arrow sin theta plus 1 equals square root of 3 cos theta
rightwards double arrow square root of 3 cos theta minus sin theta equals 1
rightwards double arrow fraction numerator square root of 3 over denominator 2 end fraction cos theta minus 1 half sin theta equals 1 half
rightwards double arrow cos straight pi over 6 cos theta minus sin straight pi over 6 sin theta equals 1 half
rightwards double arrow cos open parentheses straight pi over 6 plus theta close parentheses equals cos straight pi over 3
rightwards double arrow theta plus straight pi over 6 equals straight pi over 3
General
maths-

In a straight triangle A B C comma sin to the power of 4 invisible function application A plus sin to the power of 4 invisible function application B plus sin to the power of 4 invisible function application C equals

In a straight triangle A B C comma sin to the power of 4 invisible function application A plus sin to the power of 4 invisible function application B plus sin to the power of 4 invisible function application C equals

maths-General
General
Maths-

If 4 cos squared space theta equals 3 text  then  end text theta equals negative midline horizontal ellipsis minus negative

4 space cos squared theta equals 3
rightwards double arrow cos squared theta equals 3 over 4
rightwards double arrow cos space theta space equals fraction numerator square root of 3 over denominator 2 end fraction
rightwards double arrow cos space theta equals cos space pi over 6 space o r space cos space fraction numerator 5 pi over denominator 6 end fraction
rightwards double arrow theta equals pi over 6 comma fraction numerator 5 pi over denominator 6 end fraction

If 4 cos squared space theta equals 3 text  then  end text theta equals negative midline horizontal ellipsis minus negative

Maths-General
4 space cos squared theta equals 3
rightwards double arrow cos squared theta equals 3 over 4
rightwards double arrow cos space theta space equals fraction numerator square root of 3 over denominator 2 end fraction
rightwards double arrow cos space theta equals cos space pi over 6 space o r space cos space fraction numerator 5 pi over denominator 6 end fraction
rightwards double arrow theta equals pi over 6 comma fraction numerator 5 pi over denominator 6 end fraction
parallel

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