from Mathematics Continuity and Differentiability
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    Continuity and Differentiability

    Multiple Choice QuestionsShort Answer Type

    361. If space straight x equals straight a left parenthesis straight theta plus sinθ right parenthesis comma space straight y equals straight a left parenthesis 1 minus cosθ right parenthesis comma space fimd space dy over dx at space straight theta equals straight pi over 2.
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    362. If x=a sin 2t(1+os2t) and y=b cos2t(1-cos2t), show that
    open parentheses dy over dx close parentheses subscript straight t equals straight pi over 2 end subscript equals straight b over straight a.
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    363. FInd space dy over dx x equals straight a open parentheses cos space straight t plus log space tan straight t over 2 close parentheses comma space straight y equals straight a space sin space straight t.

    space space space space space space space space space space space straight x equals straight a open parentheses cos space straight t plus tog space tan straight t over 2 close parentheses therefore space dx over dt equals straight a open square brackets negative sin space straight t plus fraction numerator 1 over denominator tan straight t over 2 end fraction. straight d over dt open parentheses tan straight t over 2 close parentheses close square brackets equals straight a open square brackets negative sin space straight t plus fraction numerator 1 over denominator tan straight t over 2 end fraction cross times sec squared straight t over 2 cross times 1 half close square brackets space space space space space space space space space space space space space equals straight a open square brackets negative sin space straight t plus fraction numerator cos begin display style straight t over 2 end style over denominator sin begin display style straight t over 2 end style end fraction cross times fraction numerator 1 over denominator cos squared begin display style straight t over 2 end style end fraction cross times 1 half close square brackets space space space space space space space space space space space space space equals straight a open square brackets negative sin space straight t plus fraction numerator 1 over denominator 2 space sin begin display style straight t over 2 end style cos begin display style straight t over 2 end style end fraction close square brackets equals straight a open square brackets negative sin space straight t plus fraction numerator 1 over denominator sin space straight t end fraction close square brackets equals straight a open square brackets fraction numerator negative sin squared space straight t plus 1 over denominator sin space straight t end fraction close square brackets equals fraction numerator straight a space cos squared straight t over denominator sin space straight t end fraction space space space space space space space space space space space straight y equals straight a space sin space straight t therefore space dy over dx equals straight a space cos space straight t Now space dy over dx equals fraction numerator begin display style dy over dt end style over denominator begin display style dx over dt end style end fraction equals fraction numerator straight a space cos space straight t cross times sin space straight t over denominator straight a space cos squared straight t end fraction equals fraction numerator sin space straight t over denominator cos space straight t end fraction equals tan space straight t
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    364. Find space dy over dx space when space straight x space equals straight a space cosө comma space straight y space equals straight b space cosө space
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    365. Find space dy over dx space when space straight x equals sin space straight t comma space straight y space equals cos space 2 straight t
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    366. Find space dy over dx when space straight x equals straight a space secө comma space straight y equals straight b space tanө space
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    367. Find space dy over dx when space straight x equals cosө minus cos 2 straight ө comma space straight y equals sinө minus sin 2 straight ө
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    368. Find space dy over dx when space straight x equals straight a left parenthesis straight ө minus sinө right parenthesis space comma straight y equals straight a left parenthesis 1 plus cosө right parenthesis
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    369. Find space dy over dx when space straight x space equals straight a space left parenthesis cosө plus straight ө space sinө right parenthesis comma space straight y space equals space straight a space left parenthesis sinө space minus space straight ө space cosө right parenthesis
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    Multiple Choice QuestionsLong Answer Type

    370. Find space dy over dx when space straight x equals fraction numerator sin cubed straight t over denominator square root of cos space 2 straight t end root end fraction comma space straight y equals fraction numerator cos cubed straight t over denominator square root of cos space 2 straight t end root end fraction
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