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# NCERT Solutions for Class 12 maths chapter 6

NCERT solutions for class 12 Maths chapter 6 exercise 6.5, 6.4, 6.3, 6.2, 6.1 Applications of Derivatives (rate of change, increasing decreasing, approximation, tangent normal and maxima minima) in PDF form to free download. 12th NCERT solutions of other subjects, NCERT books, revisions books, assignments, chapter tests based on applications of derivatives class xii, previous year’s board papers questions are in PDF. ## NCERT solutions for class 12 Maths chapter 6

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## Mixed Chapter Tests

Chapter 5 & 6

#### Previous Years Questions

1. The sides of an equilateral triangle are increasing at the rate of 2 cm/sec. Find the rate at which its area increases, when side is 10 cm long. [CBSE Sample Paper 2017]
2. 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]
3. The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm? [Delhi 2015]
4. Determine for what values of x, the function f(x) = x^3 + 1/x^3, where x ≠ 0, is strictly increasing or strictly decreasing. [CBSE Sample Paper 2017]
5. Show that the function f(x) = 4x^3 – 18x^2 + 27x – 7 is always increasing on R. [Delhi 2017]
6. Find the interval in which f(x) = sin 3x – cos 3x, 0 < x < π, is strictly increasing or strictly decreasing. [Delhi 2016]
7. Find the point on the curve y = x^3 – 11x + 5 at which the tangent is y = x – 11. [CBSE Sample Paper 2017]
8. Find the equation of tangents to the curve y = cos(x + y), where x lies in [- 2π, 2π], that are parallel to the line x + 2y = 0. [Foreign 2016]
9. Find the shortest distance between the line x – y + 1 = 0 and the curve y^2 = x. [CBSE Sample Paper 2017]
10. If the sum of lengths of the hypotenuse and a side of a right angled triangle is given, show that the area of the triangle is maximum, when the angle between them is π/3. [Delhi 2017]
11. Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is 4r/3. Also find maximum volume in terms of volume of the sphere. [Delhi 2016]
12. The sum of the surface areas of a cuboid with sides x, 2x and x/3 and a sphere is given to be constant. Prove that the sum of their volumes is minimum, if x is equal to three times the radius of the sphere. Also find the minimum value of the sum of their volumes. [Foreign 2016]
13. A tank with rectangular base and rectangular sides open at the top is to be constructed so that its depth is 3 m and volume is 75 cubic meter. If building of tank costs ₹ 100 per square metre for the base and ₹ 50 per square meters for the sides, find the cost of least expensive tank. [Delhi 2015C]
14. A point on the hypotenuse of a right triangle is at distances ‘a’ and ‘b’ from the sides of the triangle.  Show that the minimum length of the hypotenuse is (a^2/3 + b^2/3)^3/2. [Delhi 2015C]
15. Find the local maxima and local minima, of the function f(x) = sin x – cos x, 0 < x < 2π. Also find the local maximum and local minimum values. [Delhi 2015]