Physics-
General
Easy

Question

A ray of light incident normally on an isosceles right angled prism travels as shown in the figure. The refractive index of the prism must be greater than

  1. square root of 2    
  2. square root of 3    
  3. 1.5    
  4. 2    

The correct answer is: square root of 2

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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
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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
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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
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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
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rightwards double arrow c o t theta equals 2 minus t a n theta
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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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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
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rightwards double arrow fraction numerator cos squared theta plus sin squared theta over denominator sin theta cos theta end fraction equals 2
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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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