Physics-
General
Easy

Question

The frequency which is not part of AM broadcast

  1. 100 kHz    
  2. 700 kHz    
  3. 600 kHz    
  4. 1500 kHz    

The correct answer is: 100 kHz

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Let f left parenthesis x right parenthesis equals fraction numerator c o s e c to the power of 4 end exponent invisible function application x minus 2 c o s e c to the power of 2 end exponent invisible function application x plus 1 over denominator c o s e c invisible function application x left parenthesis c o s e c invisible function application x minus s i n invisible function application x right parenthesis plus fraction numerator s i n invisible function application x minus c o s invisible function application x over denominator sin invisible function application x end fraction plus c o t invisible function application x end fraction The sum of all the solutions of f (x) = 0 in [0, 100p] is

table row cell table row blank cell f left parenthesis x right parenthesis equals fraction numerator open parentheses c o s e c to the power of 2 end exponent invisible function application x minus 1 close parentheses to the power of 2 end exponent over denominator open parentheses c o s e c to the power of 2 end exponent invisible function application x minus 1 close parentheses plus 1 minus c o t invisible function application x plus c o t invisible function application x end fraction end cell row blank cell text  defined for  end text R minus n pi comma n element of I end cell row blank cell f left parenthesis x right parenthesis equals fraction numerator left parenthesis c o t invisible function application x right parenthesis to the power of 4 end exponent over denominator 1 plus c o t to the power of 2 end exponent invisible function application x end fraction end cell row blank cell f left parenthesis x right parenthesis equals 0 rightwards double arrow c o t invisible function application x equals 0 rightwards double arrow x equals left parenthesis 2 n minus 1 right parenthesis fraction numerator pi over denominator 2 end fraction end cell row blank cell therefore x equals fraction numerator pi over denominator 2 end fraction comma fraction numerator 3 pi over denominator 2 end fraction comma fraction numerator 5 pi over denominator 2 end fraction comma horizontal ellipsis horizontal ellipsis comma fraction numerator 199 pi over denominator 2 end fraction end cell row blank cell s u m equals fraction numerator pi over denominator 2 end fraction left square bracket 1 plus 3 plus 5 plus horizontal ellipsis horizontal ellipsis plus 199 right square bracket end cell end table end cell row cell text end text text ( end text text 100 end text text end text text s end text text o end text text l end text text u end text text t end text text i end text text o end text text n end text text s end text text ) end text text end text end cell row cell equals fraction numerator pi over denominator 2 end fraction times fraction numerator 100 over denominator 2 end fraction times 200 equals 5000 pi end cell end table

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table row cell table row blank cell f left parenthesis x right parenthesis equals fraction numerator open parentheses c o s e c to the power of 2 end exponent invisible function application x minus 1 close parentheses to the power of 2 end exponent over denominator open parentheses c o s e c to the power of 2 end exponent invisible function application x minus 1 close parentheses plus 1 minus c o t invisible function application x plus c o t invisible function application x end fraction end cell row blank cell text  defined for  end text R minus n pi comma n element of I end cell row blank cell f left parenthesis x right parenthesis equals fraction numerator left parenthesis c o t invisible function application x right parenthesis to the power of 4 end exponent over denominator 1 plus c o t to the power of 2 end exponent invisible function application x end fraction end cell row blank cell f left parenthesis x right parenthesis equals 0 rightwards double arrow c o t invisible function application x equals 0 rightwards double arrow x equals left parenthesis 2 n minus 1 right parenthesis fraction numerator pi over denominator 2 end fraction end cell row blank cell therefore x equals fraction numerator pi over denominator 2 end fraction comma fraction numerator 3 pi over denominator 2 end fraction comma fraction numerator 5 pi over denominator 2 end fraction comma horizontal ellipsis horizontal ellipsis comma fraction numerator 199 pi over denominator 2 end fraction end cell row blank cell s u m equals fraction numerator pi over denominator 2 end fraction left square bracket 1 plus 3 plus 5 plus horizontal ellipsis horizontal ellipsis plus 199 right square bracket end cell end table end cell row cell text end text text ( end text text 100 end text text end text text s end text text o end text text l end text text u end text text t end text text i end text text o end text text n end text text s end text text ) end text text end text end cell row cell equals fraction numerator pi over denominator 2 end fraction times fraction numerator 100 over denominator 2 end fraction times 200 equals 5000 pi end cell end table