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JAMIA MILLIA ISLAMIA Jamia Millia Islamia Solved Paper-2012

  • question_answer
        The order and degree of the differential equation representing the family of curves \[{{y}^{2}}=2k(x+\sqrt{k})\](where,  k is  positive parameter) are respectively,

    A)  1 and 2                

    B)  2 and 4

    C)  1 and 4                

    D)  1 and 3

    Correct Answer: D

    Solution :

                    We have the equation of curve \[y=-{{x}^{3}}+3{{x}^{2}}+2x-27\] \[\Rightarrow \]               \[\frac{dy}{dx}=-3{{x}^{2}}+6x+2\] which is the slope of the curve Let \[Z=-3{{x}^{2}}+6x+2\] For maximum and minimum, put \[\frac{dZ}{dx}=0\] \[\Rightarrow \]               \[-6x+6=0\] \[\Rightarrow \]               \[x=1\] Now,    \[\frac{{{d}^{2}}Z}{d{{x}^{2}}}=-6<0\] Thus, Z is maximum for\[x=1\]and maximum value of Z \[=-3{{(1)}^{2}}+6(1)+2\] \[=-3+8=5\] Hence, maximum slope of the given curve is 5.


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