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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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