# NCERT Solutions for Class 12 Maths Chapter 6

NCERT solutions for class 12 Maths chapter 6 Applications of Derivatives exercise 6.5, 6.4, 6.3, 6.2, 6.1 (rate of change, increasing decreasing, approximation, tangent normal and maxima minima) in PDF format for new academic session 2021-22.

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 format. UP Board Intermediate students 2021-2022 can take help from these solutions. Download UP Board Solution for Class 12 Math chapter 6 in Hindi Medium or English Medium or in Videos format.

## NCERT solutions for class 12 Maths chapter 6

• ### Class 12 Maths Chapter 6 Solutions in English Medium

 Class: 12 Mathematics Chapter 6: Applications of Derivatives

### 12th Maths Chapter 6 Solutions

NCERT solutions for class 12 Maths chapter 6 AOD is given below to free download in PDF form. Download NCERT Books 2021-22 and Offline Apps based on latest CBSE Syllabus. Join the discussion forum to ask your doubts of NIOS and CBSE Board with our experts and other users.

• ### Class 12 Maths Chapter 6 Solutions in PDF

#### Class 12 Maths Chapter 6 Solutions in Videos

Class 12 Maths Chapter 6 Exercise 6.1 Solution
Class 12 Maths Chapter 6 Exercise 6.2 Solution
Class 12 Maths Chapter 6 Exercise 6.3 Solution
Class 12 Maths Chapter 6 Exercise 6.4 Solution
Class 12 Maths Chapter 6 Exercise 6.5 Solution
Class 12 Maths Chapter 6 Miscellaneous Exercise Solution

#### Class 12 Maths Chapter 6 Solution in Hindi Medium Videos

Class 12 Maths Exercise 6.1 Solution in Hindi
Class 12 Maths Exercise 6.2 Solution in Hindi

Class 12 Maths Exercise 6.3 Solution in Hindi
Class 12 Maths Exercise 6.4 Solution in Hindi
Class 12 Maths Exercise 6.5 Solution in Hindi
Class 12 Maths Miscellaneous Exercise Solution

#### Previous Years CBSE 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) = x³ + 1/x³, where x ≠ 0, is strictly increasing or strictly decreasing. [CBSE Sample Paper 2017]
5. Show that the function f(x) = 4x³ – 18x² + 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 = x³ – 11x + 5 at which the tangent is y = x – 11. [CBSE Sample Paper 2017]

##### Questions from Board Papers

1. 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]
2. Find the shortest distance between the line x – y + 1 = 0 and the curve y² = x. [CBSE Sample Paper 2017]
3. 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]
4. 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]
5. 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]
6. 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]
7. 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 (a²/³ + b²/³)³/². [Delhi 2015C]
8. 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]

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##### How interesting the chapter 6 Class 12th Maths is?

Yes, applications of derivatives (chapter 6 grade 12th Maths) is an interesting chapter. Some real-life applications of derivatives (chapter 6 grade 12th Maths) are:

1. To find the profit and loss in business using graphs.
2. To check the variations in temperature.
3. To determine the speed and distance, such as miles per hour, kilometer per hour, etc.
4. Derivatives are used in physics to derive many equations.
5. In the study of Seismology, like, to find the range of magnitudes of the earthquake.
6. Also, students can study applications of the derivatives in various fields like engineering, science, social science, etc.
7. Students can use derivatives in economics also.
8. In daily life, the derivative can help you predict fluctuations in the stock market.
9. Derivatives are important for function optimization in machine learning.
##### What is the basic motive of chapter 6 of 12th Maths NCERT?

The basic motive of chapter 6 of grade 12th Maths is to make the following things clear to the students:

• 1. Rate of change of quantities.
• 2. Increasing and Decreasing functions.
• 3. Tangents and Normals.
• 4. Approximations.
• 5. Maxima and Minima.
##### Which exercise of chapter 6 of 12th Class Maths has the highest number of problems?

Chapter 6 of grade 12th Maths has 6 exercises.
In the first exercise (Ex 6.1), there are 24 problems (6 examples and 18 questions).
In the second exercise (Ex 6.2), there are 26 problems (7 examples and 19 questions).
The third exercise (Ex 6.3) contains 34 problems (7 examples and 27 questions).
The fourth exercise (Ex 6.4) has 14 problems (5 examples and 9 questions).
There are 45 problems (16 examples and 29 questions) in the fifth exercise (Ex 6.5).
In the last (Miscellaneous) exercise, there are 34 problems (10 examples and 24 questions).
So, the fifth exercise (Ex 6.5) has the highest number of problems.

##### Is there any chapter that students should revise before starting chapter 6 Class 12th standard Maths?

Before starting chapter 6 (Applications of Derivatives) of 12th standard Maths, students should revise chapter 5 (Continuity and Differentiability) of grade 12th Maths. Chapter 5 of class 12th Maths works as a base for chapter 6 of class 12th Maths.

##### Does chapter 6 of grade 12th Maths has any miscellaneous exercise?

Yes, chapter 6 of grade 12th Maths has a miscellaneous exercise. There are 6 exercises in chapter 6 of class 12th Maths, and the last exercise is the miscellaneous exercise of chapter 6 of grade 12th Maths.