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question_answer1)
The points on the curve \[y=12x-{{x}^{3}}\]at which the gradient is zero are [MP PET 1999]
A)
(0, 2), (2,16) done
clear
B)
(0, ? 2), (2, ? 16) done
clear
C)
(2, ?16), (? 2, 16) done
clear
D)
(2, 16), (? 2, ? 16) done
clear
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question_answer2)
The area of the triangle formed by the coordinate axes and a tangent to the curve \[xy={{a}^{2}}\]at the point \[({{x}_{1}},{{y}_{1}})\]on it is [DCE 2001]
A)
\[\frac{{{a}^{2}}{{x}_{1}}}{{{y}_{1}}}\] done
clear
B)
\[\frac{{{a}^{2}}{{y}_{1}}}{{{x}_{1}}}\] done
clear
C)
\[2{{a}^{2}}\] done
clear
D)
\[4{{a}^{2}}\] done
clear
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question_answer3)
The slope of tangent to the curve\[x={{t}^{2}}+3t-8\], \[y=2{{t}^{2}}-2t-5\] at the point (2, ?1) is [MNR 1994]
A)
\[\frac{22}{7}\] done
clear
B)
\[\frac{6}{7}\] done
clear
C)
? 6 done
clear
D)
None of these done
clear
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question_answer4)
The point of the curve \[{{y}^{2}}=2(x-3)\] at which the normal is parallel to the line\[y-2x+1=0\]is [MP PET 1998]
A)
(5,2) done
clear
B)
\[\left( -\frac{1}{2},-2 \right)\] done
clear
C)
(5, ?2) done
clear
D)
\[\left( \frac{3}{2},\,2 \right)\] done
clear
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question_answer5)
The line \[x+y=2\]is tangent to the curve \[{{x}^{2}}=3-2y\] at its point [MP PET 1998]
A)
(1, 1) done
clear
B)
(?1, 1) done
clear
C)
(\[\sqrt{3}\], 0) done
clear
D)
(3, ?3) done
clear
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question_answer6)
If \[x={{t}^{2}}\]and \[y=2t\], then equation of the normal at \[t=1\]is [RPET 1996]
A)
\[x+y-3=0\] done
clear
B)
\[x+y-1=0\] done
clear
C)
\[x+y+1=0\] done
clear
D)
\[x+y+3=0\] done
clear
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question_answer7)
The equation of the normal to the curve \[y=\sin \frac{\pi x}{2}\]at (1, 1) is [AMU 1999]
A)
\[y=1\] done
clear
B)
\[x=1\] done
clear
C)
\[y=x\] done
clear
D)
\[y-1=\frac{-2}{\pi }(x-1)\] done
clear
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question_answer8)
The equation of tangent to the curve \[y=2\cos x\] at \[x=\frac{\pi }{4}\] is [RPET 1997]
A)
\[y-\sqrt{2}=2\sqrt{2}\left( x-\frac{\pi }{4} \right)\] done
clear
B)
\[y+\sqrt{2}=\sqrt{2}\left( x+\frac{\pi }{4} \right)\] done
clear
C)
\[y-\sqrt{2}=-\sqrt{2}\left( x-\frac{\pi }{4} \right)\] done
clear
D)
\[y-\sqrt{2}=\sqrt{2}\left( x-\frac{\pi }{4} \right)\] done
clear
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question_answer9)
At which point the line \[\frac{x}{a}+\frac{y}{b}=1\], touches the curve \[y=b{{e}^{-x/a}}\] [RPET 1999]
A)
(0, 0) done
clear
B)
(0, a) done
clear
C)
(0, b) done
clear
D)
(b, 0) done
clear
View Solution play_arrow
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question_answer10)
The angle between curves \[{{y}^{2}}=4x\] and \[{{x}^{2}}+{{y}^{2}}=5\]at (1, 2) is [Karnataka CET 1999]
A)
\[{{\tan }^{-1}}(3)\] done
clear
B)
\[{{\tan }^{-1}}(2)\] done
clear
C)
\[\frac{\pi }{2}\] done
clear
D)
\[\frac{\pi }{4}\] done
clear
View Solution play_arrow
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question_answer11)
For the curve \[b{{y}^{2}}={{(x+a)}^{3}}\]the square of subtangent is proportional to [RPET 1999]
A)
\[{{\text{(Subnormal)}}^{1/2}}\] done
clear
B)
Subnormal done
clear
C)
\[{{\text{(Subnormal)}}^{\text{3/2}}}\] done
clear
D)
None of these done
clear
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question_answer12)
The tangent to the curve \[y=a{{x}^{2}}+bx\] at \[(2,\,-8)\] is parallel to x-axis. Then [AMU 1999]
A)
\[a=2,\,b=-2\] done
clear
B)
\[a=2,\,\,b=-4\] done
clear
C)
\[a=2\,\,b=-8\] done
clear
D)
\[a=4,\,b=-4\] done
clear
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question_answer13)
The sum of intercepts on co-ordinate axes made by tangent to the curve \[\sqrt{x}+\sqrt{y}=\sqrt{a}\]is [RPET 1999]
A)
\[a\] done
clear
B)
\[2a\] done
clear
C)
\[2\sqrt{a}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer14)
Co-ordinates of a point on the curve \[y=x\log x\] at which the normal is parallel to the line \[2x-2y=3\] are [RPET 2000]
A)
(0,0) done
clear
B)
\[(e,\,\,e)\] done
clear
C)
\[({{e}^{2}},\,2{{e}^{2}})\] done
clear
D)
\[({{e}^{-2}}-2{{e}^{-2}})\] done
clear
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question_answer15)
If normal to the curve \[y=f(x)\] is parallel to x-axis, then correct statement is [RPET 2000]
A)
\[\frac{dy}{dx}=0\] done
clear
B)
\[\frac{dy}{dx}=1\] done
clear
C)
\[\frac{dx}{dy}=0\] done
clear
D)
None of these done
clear
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question_answer16)
The length of normal to the curve \[x=a\,(\theta +\sin \theta ),\] \[y=a(1-\cos \theta )\] at the point \[\theta =\pi /2\]is [RPET 1999]
A)
\[2a\] done
clear
B)
\[a/2\] done
clear
C)
\[\sqrt{2}\,a\] done
clear
D)
\[a/\sqrt{2}\] done
clear
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question_answer17)
The normal of the curve \[x=a(\cos \theta +\theta \sin \theta )\] \[y=a(\sin \theta -\theta \cos \theta )\] at any \[\theta \] is such that [DCE 2000; AIEEE 2005]
A)
It makes a constant angle with x-axis done
clear
B)
It passes through the origin done
clear
C)
It is at a constant distance from the origin done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer18)
The slope of the tangent to the curve \[x=3{{t}^{2}}+1,y={{t}^{3}}-1\] at \[x=1\] is [Karnataka CET 2003]
A)
0 done
clear
B)
\[\frac{1}{2}\] done
clear
C)
\[\infty \] done
clear
D)
\[-2\] done
clear
View Solution play_arrow
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question_answer19)
An equation of the tangent to the curve \[y={{x}^{4}}\] from the point (2, 0) not on the curve is [RPET 2000]
A)
\[y=0\] done
clear
B)
\[x=0\] done
clear
C)
\[x+y=0\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer20)
The angle of intersection of the curves \[y={{x}^{2}}\] and \[x={{y}^{2}}\] at (1, 1) is [Roorkee 2000; Karnataka CET 2001]
A)
\[{{\tan }^{-1}}\left( \frac{4}{3} \right)\] done
clear
B)
\[{{\tan }^{-1}}(1)\] done
clear
C)
\[{{90}^{o}}\] done
clear
D)
\[{{\tan }^{-1}}\left( \frac{3}{4} \right)\] done
clear
View Solution play_arrow
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question_answer21)
The abscissae of the points, where the tangent to curve \[y={{x}^{3}}-3{{x}^{2}}-9x+5\]is parallel to x-axis, are [Karnataka CET 2001]
A)
0 and 0 done
clear
B)
x = 1 and ? 1 done
clear
C)
x = 1 and ? 3 done
clear
D)
x = ? 1 and 3 done
clear
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question_answer22)
If the curve \[y={{a}^{x}}\] and \[y={{b}^{x}}\] intersect at angle \[\alpha \] then,\[\tan \alpha =\] [MP PET 2001]
A)
\[\frac{a-b}{1+ab}\] done
clear
B)
\[\frac{\log a-\log b}{1+\log a\log b}\] done
clear
C)
\[\frac{a+b}{1-ab}\] done
clear
D)
\[\frac{\log a+\log b}{1-\log a\log b}\] done
clear
View Solution play_arrow
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question_answer23)
The equation of tangent at \[(-4,\,-4)\] on the curve \[{{x}^{2}}=-4y\] is [Karnataka CET 2001: Pb. CET 2000]
A)
\[2x+y+4=0\] done
clear
B)
\[2x-y-12=0\] done
clear
C)
\[2x+y-4=0\] done
clear
D)
\[2x-y+4=0\] done
clear
View Solution play_arrow
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question_answer24)
The point at which the tangent to the curve \[y=2{{x}^{2}}-x+1\] is parallel to \[y\text{ }=\text{ 3}x+\text{9 }\]will be [Karnataka CET 2001]
A)
(2, 1) done
clear
B)
(1, 2) done
clear
C)
(3, 9) done
clear
D)
(?2, 1) done
clear
View Solution play_arrow
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question_answer25)
At what point on the curve \[{{x}^{3}}-8{{a}^{2}}y=0\], the slope of the normal is \[\frac{-2}{3}\] [RPET 2002]
A)
\[(a,\,a)\] done
clear
B)
\[(2a,\,-a)\] done
clear
C)
\[(2a,\,a)\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer26)
The length of the normal at point ?t? of the curve \[x=a(t+\sin t),\] \[y=a(1-\cos t)\] is [RPET 2001]
A)
\[a\sin t\] done
clear
B)
\[2a{{\sin }^{3}}(t/2)\sec (t/2)\] done
clear
C)
\[2a\sin (t/2)\,\,\tan \,(t/2)\] done
clear
D)
\[2a\sin (t/2)\] done
clear
View Solution play_arrow
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question_answer27)
The tangent drawn at the point (0, 1) on the curve \[y={{e}^{2x}}\] meets x-axis at the point [RPET 2002]
A)
\[(1/2,\,0)\] done
clear
B)
\[(-1/2,\,\,0)\] done
clear
C)
(2, 0) done
clear
D)
\[(0,\,\,0)\] done
clear
View Solution play_arrow
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question_answer28)
The equation of the tangent to the curve \[(1+{{x}^{2}})y=2-x,\] where it crosses the x-axis, is [Kerala (Engg.) 2002]
A)
\[x+5y=2\] done
clear
B)
\[x-5y=2\] done
clear
C)
\[5x-y=2\] done
clear
D)
\[5x+y-2=0\] done
clear
View Solution play_arrow
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question_answer29)
The equation of the tangent to curve \[y=b{{e}^{-x/a}}\] at the point where it crosses y-axis is [Karnataka CET 2002]
A)
\[ax+by=1\] done
clear
B)
\[ax-by=1\] done
clear
C)
\[\frac{x}{a}-\frac{y}{b}=1\] done
clear
D)
\[\frac{x}{a}+\frac{y}{b}=1\] done
clear
View Solution play_arrow
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question_answer30)
The angle of intersection of curves \[y={{x}^{2}},\] \[6y=7-{{x}^{3}}\] at (1, 1) is [Kurukshetra CEE 2002]
A)
\[\pi /4\] done
clear
B)
\[\pi /3\] done
clear
C)
\[\pi /2\] done
clear
D)
\[\pi \] done
clear
View Solution play_arrow
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question_answer31)
The tangent to the curve \[y=2{{x}^{2}}-x+1\] at a point P is parallel to \[y=3x+4,\]the co-ordinates of P are [RPET 2003]
A)
(2, 1) done
clear
B)
(1, 2) done
clear
C)
(? 1, 2) done
clear
D)
(2, ? 1) done
clear
View Solution play_arrow
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question_answer32)
For the curve \[xy={{c}^{2}}\] the subnormal at any point varies as [Karnataka CET 2003]
A)
\[{{x}^{2}}\] done
clear
B)
\[{{x}^{3}}\] done
clear
C)
\[{{y}^{2}}\] done
clear
D)
\[{{y}^{3}}\] done
clear
View Solution play_arrow
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question_answer33)
The angle between the curves \[y=\sin x\] and \[y=\cos x\] is [EAMCET 2003]
A)
\[{{\tan }^{-1}}(2\sqrt{2})\] done
clear
B)
\[{{\tan }^{-1}}(3\sqrt{2})\] done
clear
C)
\[{{\tan }^{-1}}(3\sqrt{3})\] done
clear
D)
\[{{\tan }^{-1}}(5\sqrt{2})\] done
clear
View Solution play_arrow
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question_answer34)
If the normal to the curve \[{{y}^{2}}=5x-1\], at the point (1, ?2) is of the form \[ax-5y+b=0\], then a and b are [Pb. CET 2001]
A)
4, ? 14 done
clear
B)
4, 14 done
clear
C)
?4, 14 done
clear
D)
?4, ?14 done
clear
View Solution play_arrow
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question_answer35)
If a tangent to the curve \[y=6x-{{x}^{2}}\]is parallel to the line \[4x-2y-1=0\], then the point of tangency on the curve is [Karnataka CET 2004]
A)
(2, 8) done
clear
B)
(8, 2) done
clear
C)
(6, 1) done
clear
D)
(4, 2) done
clear
View Solution play_arrow
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question_answer36)
The normal to the curve \[x=a\text{ }(1+\cos \theta ),\,y=a\sin \theta \]at \['\theta '\] always passes through the fixed point [AIEEE 2004]
A)
(a, a) done
clear
B)
(0, a) done
clear
C)
(0, 0) done
clear
D)
(a, 0) done
clear
View Solution play_arrow
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question_answer37)
. If ST and SN are the lengths of the subtangent and the subnormal at the point \[\theta =\frac{\pi }{2}\]on the curve \[x=a(\theta +\sin \theta ),y=a(1-\cos \theta ),a\ne 1\], then [Karnataka CET 2005]
A)
\[ST=SN\] done
clear
B)
\[ST=2\,SN\] done
clear
C)
\[S{{T}^{2}}=a\,S{{N}^{3}}\] done
clear
D)
\[S{{T}^{3}}=a\,SN\] done
clear
View Solution play_arrow
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question_answer38)
The equation of the tangent to the curve \[x=2{{\cos }^{3}}\theta \] and \[y=3{{\sin }^{3}}\theta \] at the point \[\theta =\pi /4\] is [J & K 2005]
A)
\[2x+3y=3\sqrt{2}\] done
clear
B)
\[2x-3y=3\sqrt{2}\] done
clear
C)
\[3x+2y=3\sqrt{2}\] done
clear
D)
\[3x-2y=3\sqrt{2}\] done
clear
View Solution play_arrow
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question_answer39)
The curve given by \[x+y={{e}^{xy}}\] has a tangent parallel to the y-axis at the point [AMU 2005]
A)
(0, 1) done
clear
B)
(1, 0) done
clear
C)
(1, 1) done
clear
D)
(?1, ?1) done
clear
View Solution play_arrow