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question_answer1)
The equation of the curve which passes through the point (1, 1) and whose slope is given by \[\frac{2y}{x}\], is [Roorkee 1987]
A)
\[y={{x}^{2}}\] done
clear
B)
\[{{x}^{2}}-{{y}^{2}}=0\] done
clear
C)
\[2{{x}^{2}}+{{y}^{2}}=3\] done
clear
D)
None of these done
clear
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question_answer2)
The equation of the curve that passes through the point \[(1,\,2)\] and satisfies the differential equation \[\frac{dy}{dx}=\frac{-2xy}{({{x}^{2}}+1)}\]is
A)
\[y({{x}^{2}}+1)=4\] done
clear
B)
\[y({{x}^{2}}+1)+4=0\] done
clear
C)
\[y({{x}^{2}}-1)=4\] done
clear
D)
None of these done
clear
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question_answer3)
Equation of curve through point \[(1,\,0)\]which satisfies the differential equation \[(1+{{y}^{2}})dx-xydy=0\], is [WB JEE 1986]
A)
\[{{x}^{2}}+{{y}^{2}}=1\] done
clear
B)
\[{{x}^{2}}-{{y}^{2}}=1\] done
clear
C)
\[2{{x}^{2}}+{{y}^{2}}=2\] done
clear
D)
None of these done
clear
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question_answer4)
Equation of curve passing through (3, 9) which satisfies the differential equation \[\frac{dy}{dx}=x+\frac{1}{{{x}^{2}}}\], is [WB JEE 1986]
A)
\[6xy=3{{x}^{2}}-6x+29\] done
clear
B)
\[6xy=3{{x}^{3}}-29x+6\] done
clear
C)
\[6xy=3{{x}^{3}}+29x-6\] done
clear
D)
None of these done
clear
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question_answer5)
The differential equation \[y\frac{dy}{dx}+x=a\](a is any constant) represents
A)
A set of circles having centre on the y-axis done
clear
B)
A set of circles centre on the x-axis done
clear
C)
A set of ellipses done
clear
D)
None of these done
clear
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question_answer6)
The equation of a curve passing through \[\left( 2,\frac{7}{2} \right)\] and having gradient \[1-\frac{1}{{{x}^{2}}}\]at\[(x,\,y)\]is
A)
\[y={{x}^{2}}+x+1\] done
clear
B)
\[xy={{x}^{2}}+x+1\] done
clear
C)
\[xy=x+1\] done
clear
D)
None of these done
clear
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question_answer7)
The equation of the curve through the point (1,0) and whose slope is \[\frac{y-1}{{{x}^{2}}+x}\]is
A)
\[(y-1)(x+1)+2x=0\] done
clear
B)
\[2x(y-1)+x+1=0\] done
clear
C)
\[x(y-1)(x+1)+2=0\] done
clear
D)
None of these done
clear
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question_answer8)
The slope of a curve at any point is the reciprocal of twice the ordinate at the point and it passes though the point (4, 3). The equation of the curve is
A)
\[{{x}^{2}}=y+5\] done
clear
B)
\[{{y}^{2}}=x-5\] done
clear
C)
\[{{y}^{2}}=x+5\] done
clear
D)
\[{{x}^{2}}=y-5\] done
clear
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question_answer9)
A particle moves in a straight line with a velocity given by \[\frac{dx}{dt}=x+1\](x is the distance described). The time taken by a particle to traverse a distance of 99 metre is
A)
\[{{\log }_{10}}e\] done
clear
B)
\[2{{\log }_{e}}10\] done
clear
C)
\[2{{\log }_{10}}e\] done
clear
D)
\[\frac{1}{2}{{\log }_{10}}e\] done
clear
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question_answer10)
Solution of differential equation \[x\,dy-y\,dx=0\] represents [MP PET 1996]
A)
Rectangular hyperbola done
clear
B)
Straight line passing through origin done
clear
C)
Parabola whose vertex is at origin done
clear
D)
Circle whose centre is at origin done
clear
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question_answer11)
Integral curve satisfying \[y'=\frac{{{x}^{2}}+{{y}^{2}}}{{{x}^{2}}-{{y}^{2}}},\ y(1)=2\] has the slope at the point (1, 0) of the curve, equal to [MP PET 2000]
A)
? 5/3 done
clear
B)
? 1 done
clear
C)
1 done
clear
D)
5/3 done
clear
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question_answer12)
A particle starts at the origin and moves along the x?axis in such a way that its velocity at the point (x, 0) is given by the formula \[\frac{dx}{dt}={{\cos }^{2}}\pi x.\] Then the particle never reaches the point on [AMU 2000]
A)
\[x=\frac{1}{4}\] done
clear
B)
\[x=\frac{3}{4}\] done
clear
C)
\[x=\frac{1}{2}\] done
clear
D)
x = 1 done
clear
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question_answer13)
The slope of the tangent at (x, y) to a curve passing through a point (2, 1) is \[\frac{{{x}^{2}}+{{y}^{2}}}{2xy}\], then the equation of the curve is [MP PET 2002]
A)
\[2({{x}^{2}}-{{y}^{2}})=3x\] done
clear
B)
\[2({{x}^{2}}-{{y}^{2}})=6y\] done
clear
C)
\[x({{x}^{2}}-{{y}^{2}})=6\] done
clear
D)
\[x({{x}^{2}}+{{y}^{2}})=10\] done
clear
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question_answer14)
A function \[y=f(x)\] has a second order derivatives \[{{f}'}'(x)=6(x-1)\]. If its graph passes through the point (2, 1) and at that point the tangent to the graph is \[y=3x-5\], then the function is [AIEEE 2004]
A)
\[{{(x+1)}^{3}}\] done
clear
B)
\[{{(x-1)}^{3}}\] done
clear
C)
\[{{(x+1)}^{2}}\] done
clear
D)
\[{{(x-1)}^{2}}\] done
clear
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