JEE Main & Advanced Mathematics Differentiation Question Bank

done Partial Differentiation Total Questions - 21

• question_answer1)
  If \[z=\frac{{{({{x}^{4}}+{{y}^{4}})}^{1/3}}}{{{({{x}^{3}}+{{y}^{3}})}^{1/4}}}\],  then \[x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y}=\]
  A)
             \[\frac{1}{12}z\] done clear
  B)
             \[\frac{1}{4}z\] done clear
  C)
             \[\frac{1}{3}z\] done clear
  D)
             \[\frac{7}{12}z\] done clear
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• question_answer2)
  If \[z={{\tan }^{-1}}\left( \frac{x}{y} \right)\], then \[{{z}_{x}}:{{z}_{y}}=\]
  A)
             \[y:x\] done clear
  B)
             \[x:y\] done clear
  C)
             \[-y:x\] done clear
  D)
             \[-x:y\] done clear
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• question_answer3)
  If \[u=\frac{x+y}{x-y}\], then \[\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=\]                                         [EAMCET 1991]
  A)
             \[\frac{1}{x-y}\] done clear
  B)
             \[\frac{2}{x-y}\] done clear
  C)
             \[\frac{1}{{{(x-y)}^{2}}}\] done clear
  D)
             \[\frac{2}{{{(x-y)}^{2}}}\] done clear
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• question_answer4)
  If \[u=\log ({{x}^{2}}+{{y}^{2}}),\] then \[\frac{{{\partial }^{2}}u}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}u}{\partial {{y}^{2}}}=\]                                     [EAMCET 1994]
  A)
             \[\frac{1}{{{x}^{2}}+{{y}^{2}}}\] done clear
  B)
             0 done clear
  C)
             \[\frac{{{x}^{2}}-{{y}^{2}}}{{{({{x}^{2}}+{{y}^{2}})}^{2}}}\] done clear
  D)
             \[\frac{{{y}^{2}}-{{x}^{2}}}{{{({{x}^{2}}+{{y}^{2}})}^{2}}}\] done clear
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• question_answer5)
  If \[u={{\sin }^{-1}}\left( \frac{y}{x} \right),\] then \[\frac{\partial u}{\partial x}\] is equal to [Tamilnadu (Engg.) 1992]
  A)
             \[-\frac{y}{{{x}^{2}}+{{y}^{2}}}\] done clear
  B)
             \[\frac{x}{\sqrt{1-{{y}^{2}}}}\] done clear
  C)
             \[\frac{-y}{\sqrt{{{x}^{2}}-{{y}^{2}}}}\] done clear
  D)
             \[\frac{-y}{x\sqrt{{{x}^{2}}-{{y}^{2}}}}\] done clear
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• question_answer6)
  If \[u={{\tan }^{-1}}\frac{y}{x}\], then by Euler?s Theorem the value of x \[\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=\]                                           [Tamilnadu (Engg.) 1993]
  A)
             \[\tan u\] done clear
  B)
             \[\sin u\] done clear
  C)
             \[0\] done clear
  D)
             \[\cos 2u\] done clear
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• question_answer7)
  If \[u={{\tan }^{-1}}\left( \frac{{{x}^{3}}+{{y}^{3}}}{x-y} \right)\], then \[x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=\]                                                                                                                [EAMCET 1999]
  A)
             \[\sin 2u\] done clear
  B)
             \[\cos 2u\] done clear
  C)
             \[\tan 2u\] done clear
  D)
             \[\sec 2u\] done clear
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• question_answer8)
  If \[F(u)=f(x,\,y,\,z)\] be a homogeneous function of degree \[n\] in \[x,\,y,\,z\] then \[x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}+z\frac{\partial u}{\partial z}=\]
  A)
             \[nu\] done clear
  B)
             \[n\,F(u)\] done clear
  C)
             \[\frac{n\,F(u)}{{F}'(u)}\] done clear
  D)
             None of these done clear
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• question_answer9)
  If \[u=\log ({{x}^{3}}+{{y}^{3}}+{{z}^{3}}-3xyz)\], then \[\left( \frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}+\frac{\partial u}{\partial z} \right)\] \[(x+y+z)\] =                                        [EAMCET 1996]
  A)
             0 done clear
  B)
             1 done clear
  C)
             2 done clear
  D)
             3 done clear
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• question_answer10)
  If \[z={{\sin }^{-1}}\left( \frac{x+y}{\sqrt{x}+\sqrt{y}} \right)\], then \[x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y}\] is equal to [EAMCET 1998; Orissa JEE 2000]
  A)
             \[\frac{1}{2}\sin z\] done clear
  B)
             \[\frac{1}{2}\tan z\] done clear
  C)
             \[0\] done clear
  D)
             None of these done clear
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• question_answer11)
  If \[u={{\log }_{e}}({{x}^{2}}+{{y}^{2}})+{{\tan }^{-1}}\left( \frac{y}{x} \right)\], then \[\frac{{{\partial }^{2}}u}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}u}{\partial {{y}^{2}}}=\]                                                               [EAMCET 2000]
  A)
             0 done clear
  B)
             2u done clear
  C)
             1/u done clear
  D)
             u done clear
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• question_answer12)
  If \[{{x}^{x}}{{y}^{y}}{{z}^{z}}=c\], then \[\frac{\partial z}{\partial x}=\]                                  [EAMCET 1999]
  A)
             \[\frac{1+\log x}{1+\log z}\] done clear
  B)
             \[-\frac{1+\log x}{1+\log z}\] done clear
  C)
             \[-\frac{1+\log y}{1+\log z}\] done clear
  D)
             None of these done clear
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• question_answer13)
  If \[u=x{{y}^{2}}{{\tan }^{-1}}\left( \frac{y}{x} \right)\], then \[x{{u}_{x}}+y{{u}_{y}}=\]                 [EAMCET 2001]
  A)
             2u done clear
  B)
             u done clear
  C)
             3u done clear
  D)
             u/3 done clear
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• question_answer14)
  If \[{{z}^{2}}=\frac{{{x}^{1/2}}+{{y}^{1/2}}}{{{x}^{1/3}}+{{y}^{1/3}}}\] then \[x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y}=\]   [EAMCET 1999]
  A)
             \[\frac{z}{6}\] done clear
  B)
             \[\frac{z}{3}\] done clear
  C)
             \[\frac{z}{2}\] done clear
  D)
             \[\frac{z}{12}\] done clear
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• question_answer15)
  If \[u={{\tan }^{-1}}(x+y),\] then \[x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=\]   [EAMCET 1996]
  A)
             \[\sin 2u\] done clear
  B)
             \[\frac{1}{2}\sin 2u\] done clear
  C)
             \[2\tan u\] done clear
  D)
             \[{{\sec }^{2}}u\] done clear
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• question_answer16)
  If \[u={{({{x}^{2}}+{{y}^{2}}+{{z}^{2}})}^{3/2}}\], then \[{{\left( \frac{\partial u}{\partial x} \right)}^{2}}+{{\left( \frac{\partial u}{\partial y} \right)}^{2}}+{{\left( \frac{\partial u}{\partial z} \right)}^{2}}=\]                               [EAMCET 1996]
  A)
             9u done clear
  B)
             \[9{{u}^{4/3}}\] done clear
  C)
             \[9{{u}^{2}}\] done clear
  D)
             \[{{u}^{4/3}}\] done clear
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• question_answer17)
  If \[u={{x}^{2}}{{\tan }^{-1}}\frac{y}{x}-{{y}^{2}}{{\tan }^{-1}}\frac{x}{y}\], then \[\frac{{{\partial }^{2}}u}{\partial x\,\partial \,y}=\] [Tamilnadu (Engg.) 2002]
  A)
             \[\frac{{{x}^{2}}+{{y}^{2}}}{{{x}^{2}}-{{y}^{2}}}\] done clear
  B)
             \[\frac{{{x}^{2}}-{{y}^{2}}}{{{x}^{2}}+{{y}^{2}}}\] done clear
  C)
             \[\frac{{{x}^{2}}+{{y}^{2}}}{{{x}^{2}}-{{y}^{2}}}\] done clear
  D)
             \[-\frac{{{x}^{2}}{{y}^{2}}}{{{x}^{2}}+{{y}^{2}}}\] done clear
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• question_answer18)
  If \[{{u}^{2}}={{(x-a)}^{2}}+{{(y-b)}^{2}}+{{(z-c)}^{2}}\], then \[\sum \frac{{{\partial }^{2}}u}{\partial {{x}^{2}}}=\]                    [Tamilnadu (Engg.) 2002]
  A)
             \[\frac{2}{u}\] done clear
  B)
             \[\frac{3}{u}\] done clear
  C)
             0 done clear
  D)
             \[\frac{1}{u}\] done clear
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• question_answer19)
  If \[z=\sec \,(y-ax)+\tan (y+ax),\] then \[\frac{{{\partial }^{2}}z}{\partial {{x}^{2}}}-{{a}^{2}}\frac{{{\partial }^{2}}z}{\partial {{y}^{2}}}=\]                    [EAMCET 2002]
  A)
             z done clear
  B)
             2z done clear
  C)
             0 done clear
  D)
             ?z done clear
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• question_answer20)
  If \[z=\frac{y}{x}\left[ \sin \frac{x}{y}+\cos \left( 1+\frac{y}{x} \right) \right]\], then \[x\frac{\partial z}{\partial x}=\]                                                                                          [EAMCET 2002]
  A)
             \[y\frac{\partial z}{\partial y}\] done clear
  B)
             \[-y\frac{\partial z}{\partial y}\] done clear
  C)
             \[2y\frac{\partial z}{\partial y}\] done clear
  D)
             \[2y\frac{\partial z}{\partial x}\] done clear
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• question_answer21)
  If \[u={{e}^{-{{x}^{2}}-{{y}^{2}}}}\], then                                                              [EAMCET 2001]
  A)
             \[x{{u}_{x}}=y{{y}_{y}}\] done clear
  B)
             \[y{{u}_{x}}=x{{u}_{y}}\] done clear
  C)
             \[y{{u}_{x}}+x{{u}_{y}}=0\] done clear
  D)
             \[{{x}^{2}}{{u}_{y}}+{{y}^{2}}{{u}_{x}}=0\] done clear
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