Plz answed the question marked with tick

Asked by lovemaan5500 | 7th Feb, 2020, 09:51: AM

Expert Answer:

f is conntinuous at x = pi/2
Therefor, LHL = f(pi/2) = RHL
rightwards double arrow limit as x rightwards arrow straight pi over 2 to the power of minus of space f left parenthesis x right parenthesis space equals space f open parentheses straight pi over 2 close parentheses equals limit as x rightwards arrow straight pi over 2 to the power of plus of space f left parenthesis x right parenthesis
rightwards double arrow limit as x rightwards arrow straight pi over 2 of space fraction numerator 1 minus sin cubed x over denominator 3 cos squared x end fraction equals p
rightwards double arrow p equals limit as x rightwards arrow straight pi over 2 of space fraction numerator open parentheses 1 minus sin x close parentheses open parentheses 1 plus sin squared x plus sin x close parentheses over denominator 3 open parentheses 1 minus sin squared x close parentheses end fraction
rightwards double arrow p equals limit as x rightwards arrow straight pi over 2 of space fraction numerator open parentheses 1 minus sin x close parentheses open parentheses 1 plus sin squared x plus sin x close parentheses over denominator 3 open parentheses 1 minus sin x close parentheses open parentheses 1 plus sin x close parentheses end fraction equals limit as x rightwards arrow straight pi over 2 of space fraction numerator open parentheses 1 plus sin squared x plus sin x close parentheses over denominator 3 open parentheses 1 plus sin x close parentheses end fraction equals fraction numerator 1 plus 1 plus 1 over denominator 3 cross times 2 end fraction equals 1 half
rightwards double arrow p equals 1 half
A l s o comma space space f open parentheses straight pi over 2 close parentheses equals limit as x rightwards arrow straight pi over 2 to the power of plus of space f left parenthesis x right parenthesis rightwards double arrow 1 half equals limit as x rightwards arrow straight pi over 2 of fraction numerator q open parentheses 1 minus sin x close parentheses over denominator open parentheses straight pi minus 2 straight x close parentheses squared end fraction
T a k e space straight pi over 2 minus x equals theta rightwards double arrow straight pi minus 2 straight x equals 2 straight theta
rightwards double arrow 1 half equals limit as straight theta rightwards arrow 0 of fraction numerator straight q open parentheses 1 minus sin open parentheses begin display style straight pi over 2 end style minus straight theta close parentheses close parentheses over denominator 4 straight theta squared end fraction equals limit as straight theta rightwards arrow 0 of fraction numerator straight q open parentheses 1 minus cosθ close parentheses over denominator 4 straight theta squared end fraction equals limit as straight theta rightwards arrow 0 of fraction numerator straight q times 2 sin squared begin display style straight theta over 2 end style over denominator 4 straight theta squared end fraction equals limit as straight theta rightwards arrow 0 of fraction numerator straight q times 2 sin squared begin display style straight theta over 2 end style over denominator 16 open parentheses begin display style straight theta over 2 end style close parentheses squared end fraction
rightwards double arrow 1 half equals fraction numerator 2 straight q over denominator 16 end fraction limit as straight theta rightwards arrow 0 of open parentheses fraction numerator sin begin display style straight theta over 2 end style over denominator straight theta over 2 end fraction close parentheses squared
rightwards double arrow straight q equals 4rightwards double arrow limit as x rightwards arrow straight pi over 2 to the power of minus of space f left parenthesis x right parenthesis space equals space f open parentheses straight pi over 2 close parentheses equals limit as x rightwards arrow straight pi over 2 to the power of plus of space f left parenthesis x right parenthesis
rightwards double arrow limit as x rightwards arrow straight pi over 2 of space fraction numerator 1 minus sin cubed x over denominator 3 cos squared x end fraction equals p
rightwards double arrow p equals limit as x rightwards arrow straight pi over 2 of space fraction numerator open parentheses 1 minus sin x close parentheses open parentheses 1 plus sin squared x plus sin x close parentheses over denominator 3 open parentheses 1 minus sin squared x close parentheses end fraction
rightwards double arrow p equals limit as x rightwards arrow straight pi over 2 of space fraction numerator open parentheses 1 minus sin x close parentheses open parentheses 1 plus sin squared x plus sin x close parentheses over denominator 3 open parentheses 1 minus sin x close parentheses open parentheses 1 plus sin x close parentheses end fraction equals limit as x rightwards arrow straight pi over 2 of space fraction numerator open parentheses 1 plus sin squared x plus sin x close parentheses over denominator 3 open parentheses 1 plus sin x close parentheses end fraction equals fraction numerator 1 plus 1 plus 1 over denominator 3 cross times 2 end fraction equals 1 half
rightwards double arrow p equals 1 half
A l s o comma space space f open parentheses straight pi over 2 close parentheses equals limit as x rightwards arrow straight pi over 2 to the power of plus of space f left parenthesis x right parenthesis rightwards double arrow 1 half equals limit as x rightwards arrow straight pi over 2 of fraction numerator q open parentheses 1 minus sin x close parentheses over denominator open parentheses straight pi minus 2 straight x close parentheses squared end fraction
T a k e space straight pi over 2 minus x equals theta rightwards double arrow straight pi minus 2 straight x equals 2 straight theta
rightwards double arrow 1 half equals limit as straight theta rightwards arrow 0 of fraction numerator straight q open parentheses 1 minus sin open parentheses begin display style straight pi over 2 end style minus straight theta close parentheses close parentheses over denominator 4 straight theta squared end fraction equals limit as straight theta rightwards arrow 0 of fraction numerator straight q open parentheses 1 minus cosθ close parentheses over denominator 4 straight theta squared end fraction equals limit as straight theta rightwards arrow 0 of fraction numerator straight q times 2 sin squared begin display style straight theta over 2 end style over denominator 4 straight theta squared end fraction equals limit as straight theta rightwards arrow 0 of fraction numerator straight q times 2 sin squared begin display style straight theta over 2 end style over denominator 16 open parentheses begin display style straight theta over 2 end style close parentheses squared end fraction
rightwards double arrow 1 half equals fraction numerator 2 straight q over denominator 16 end fraction limit as straight theta rightwards arrow 0 of open parentheses fraction numerator sin begin display style straight theta over 2 end style over denominator straight theta over 2 end fraction close parentheses squared
rightwards double arrow straight q equals 4

Answered by Renu Varma | 7th Feb, 2020, 10:42: AM