Find the points on the curve y = x3 – 2x2 – x at which th

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 Multiple Choice QuestionsShort Answer Type

101. Find the point on the curve y = x3 – 11 x + 5 at which the tangent has the equation y = x – 11.
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102.

Find the point on the curve y = x3 – 2x2 – 2x at which the tangent lines are parallel to the line y = 2x – 3.

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103.

Find the points on the curve y = x3 – 2x2 – x at which the tangent lines are parallel to the line y = 3x – 2.


The equation of curve is   straight y equals straight x cubed minus 2 straight x squared minus straight x                                  ...(1)
therefore space space space space dy over dx space equals space 3 straight x squared minus 4 straight x minus 1
At space left parenthesis straight x comma space straight y right parenthesis comma space space slope space of space tangent space equals 3 straight x squared minus 4 straight x minus 1
therefore space space space space space straight m subscript 1 space equals space 3 straight x squared minus 4 straight x minus 1 comma space space space where space straight m subscript 1 space is space slope space of space tangent
Let space space straight m subscript 2 space be space slope space of space line space straight y equals space space 3 straight x minus 2
therefore space space space space straight m subscript 2 space equals space 3
because space space tangent space is space parallel space to space line space left parenthesis 2 right parenthesis
therefore space space space space space straight m subscript 1 space equals space straight m subscript 2 space space space space rightwards double arrow space space space space space 3 straight x squared minus 4 straight x minus 1 space space space space space rightwards double arrow space space space space space 3 straight x squared minus 4 straight x minus 4 space equals space 0
rightwards double arrow space space space space space straight x space equals space fraction numerator 4 plus-or-minus square root of 16 plus 48 end root over denominator 6 end fraction space equals space fraction numerator 4 plus-or-minus 8 over denominator 6 end fraction space equals space 2 comma space space space space minus 2 over 3
When space straight x space equals space 2 comma space space space from space left parenthesis 1 right parenthesis comma space space space straight y space equals space 8 minus 8 minus 2 space equals space minus 2
When space straight x space equals space minus 2 over 3 comma space space from space left parenthesis 1 right parenthesis comma space space space straight y space equals space minus 8 over 27 minus 8 over 9 plus 2 over 3 space equals space fraction numerator negative 8 minus 24 plus 18 over denominator 27 end fraction space equals space minus 14 over 27
therefore space space space space space required space points space are space left parenthesis 2 comma space minus 2 right parenthesis comma space space space open parentheses negative 2 over 3 comma space space minus 14 over 27 close parentheses

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104.

Find the equation of the tangent to the curve x2 + 3y = 3, which is parallel to the line y – 4x + 5 = 0. 

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105.

Find the equations of all lines having slope -1 that are tangents to the curve straight y equals fraction numerator 1 over denominator straight x minus 1 end fraction. straight x not equal to 1.

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106.

Find the equation of all lines having slope 2 and being tangent to the curve straight y plus fraction numerator 2 over denominator straight x minus 3 end fraction space equals space 0.

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 Multiple Choice QuestionsLong Answer Type

107.

Find the equation of all lines having 0 slope and that are tangent to the curve straight y equals fraction numerator 1 over denominator straight x squared minus 2 straight x plus 2 end fraction.

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 Multiple Choice QuestionsShort Answer Type

108.

Find the equation of the tangent to the curve straight y equals square root of 3 straight x minus 2 end root which is parallel to the line 4x - 2y + 5 = 0

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 Multiple Choice QuestionsLong Answer Type

109.

Find the equation of tangents to the curve
y = cos (x + y), – 2 straight pi ≤ x ≤ 2 straight pi
that are parallel to the line x + 2y = 0.

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110.

Find the point on curve 4x2 + 9y2 = 1, where the tangents are perpendicular to the line 2y + x = 0.

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