NCERT Class 12-Mathematics: Chapter – 6 Application of Derivatives Part 3 (For CBSE, ICSE, IAS, NET, NRA 2022)

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Question 8:

Find the condition for the curves o intersect orthogonally.

Let the curves intersect at . Therefore,

Slope of tangent at the point of intersection

Again

For orthoganality,

Question 9:

Find all the points of local maxima and local minima of the function

. Therefore, is point of local maxima

. Therefore, is point of local minima

. Therefore is point of local maxima.

Question 10:

Show that the local maximum value of is less than local minimum value.

Let

Hence local maximum value of is at and the local maximum value .

Local minimum value of is at and local minimum value .

Therefore, local maximum value is less than local minimum value .

Question 11:

Water is dripping out at a steady rate of rough a tiny hole at the vertex of the conical vessel, whose axis is vertical. When the slant height of water in the vessel is , find the rate of decrease of slant height, where the vertical angle of the conical vessel is

Given that , where v is the volume of water in the conical vessel.

From the Fig 6.2,

Therefore,

Therefore,

Therefore, the rate of decrease of slant height

Question 12:

Find the equation of all the tangents to the curve , that are parallel to the line .

Given that

Or

Since tangent is parallel to , therefore slope of tangent

Therefore,

Since .

Therefore,

Therefore,

Thus, satisfy equation (ii)

Hence, the points are

Therefore, equation of tangent at and equation of tangent at

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