CBSE Class 12th Mathematics Chapter 6 - Applications of Derivatives Important Questions with Answers

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The volume of a cube is increasing at the rate of 8 cm³/s. How fast is the surface area increasing when the length of its edge is 12 cm? (All India 2019)

Question 2.

Find the equation of tangent to the curve y =

\sqrt{3 x - 2}

which is parallel to the line 4x – 2y + 5 = 0. Also, write the equation of normal to the curve at the point of contact. (Delhi 2019)

Question 3.

Find the equation of the normal to the curve x² = 4y, which passes through the point (-1, 4). (All India 2019)

Question 4.

A tank with rectangular base and rectangular sides, open at the top is to be constructed, so that its depth is 2 m and volume is 8 m³. If building of tank cost ? 70 per sq m for the base and ? 45 per sq m for sides. What is the cost of least expensive tank? (Delhi 2019)

Question 5.

Prove that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is

\frac{2 R}{\sqrt{3}}

. Also, find the maximum volume. (All India 2019,2014,2012C, 2011: Delhi 2013)

Question 6.

Find the point on the curve y² = 4x, which is nearest to the point (2, – 8). (All India 2019)

Question 7.

Show that the altitude of the right circular cone of maximum volume that can he inscribed in a sphere of radius r is

\frac{4 r}{3}

Also, find the maximum volume in terms of volume of the sphere. (Delhi 2019, 2016)
Or
Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is

\frac{4 r}{3}

. Also, show that the maximum volume of the cone is

\frac{8}{27}

of the volume of the sphere. (All India 2014)

Question 8.

The total cost C(x) associated with the production of x units of an item is given by C(x) = 0.005x³ – 0.02x² + 30x + 5000. Find the marginal cost when 3 units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output. (CBSE 2018)

Question 9.

The total revenue received from the sale of x units of a product is given by R(x) = 3x² + 36x + 5 in rupees. Find the marginal revenue when x = 5, where by marginal revenue we mean the rate of change of total revenue with respect to the number of items sold at an instant. (CBSE 2018 C)

Question 10.

Find the intervals in which the function
f(x) =

\frac{x^{4}}{4}

– x³ – 5x² + 24x + 12 is
(i) strictly increasing
(ii) strictly decreasing. (CBSE 2018)

Question 1.

Find the equations of the tangent and the normal to the curve 16x² + 9y² = 145 at the point (x₁, y₁), where x₁ = 2 and y₁ > 0. (CBSE 2018)

Question 2.

Find the angle of intersection of the curves x² + y² = 4 and (x – 2)² + y² = 4, at the point in the first quadrant. (CBSE 2018C)

Question 3.

An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when depth of the tank is half of its width. If the cost is to he borne by nearby settled lower income families, for whom water will be provided. (CBSE 2018; All India 2010 C)

Question 4.

A window is of the form of a semi-circle with a rectangle on its diameter. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening. (CBSE 2018 C; All India 2017,2011; Foreign 2014)
Or
A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. ,Find the dimensions of the window to ‘ admit maximum light through the whole j opening. (All India 2017)

Question 5.

The volume of a sphere is increasing at the rate of 8 cm³/s. Find the rate at which its surface area is increasing when the radius of the sphere is 12 cm. (All India 2017)

Question 6.

Show that the function f(x) = x³ – 3x² + 6x – 100 is increasing on R. (All India 2017)

Question 7.

The volume of a sphere is increasing at the rate of 3 cubic centimeter per second. Find the rate of increase of its surface area, when the radius is 2 cm. (Delhi 2017)

Question 8.

Show that the function f(x) = 4x³ – 18x² + 27x – 7 is always increasing on R. (Delhi 2017)

Question 9.

The length x of a rectangle is decreasing at the rate of 5 cm/min and the width y is increasing at the rate of 4 cm/min. When x = 8 cm and y = 6 cm, find the rate of change of
(i) the perimeter.
(ii) area of rectangle. (All India 2017)