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NCERT Solutions for Exercise 6.1 Class 12 Maths Chapter 6 Application of Derivatives are discussed here. These NCERT solutions are created by subject matter expert at Careers360 considering the latest syllabus and pattern of CBSE 2023-24. NCERT solutions for exercise 6.1 Class 12 Maths chapter 6 surrounds by questions related to the topic rate of change quantities. The Class 12th Maths chapter 6 exercise 6.1 are curated by Mathematics faculties according to the CBSE pattern. All these NCERT solutions given in Class 12 Maths chapter 6 exercise 6.1 give a more deep knowledge on the concept of rate of change of quantities that is discussed in the NCERT book. There are 6 exercises in Class 12 NCERT chapter applications of derivatives. Other than Exercise 6.1 Class 12 Maths, the following exercise, are also present in the NCERT syllabus of this chapter.
12th class Maths exercise 6.1 answers are designed as per the students demand covering comprehensive, step by step solutions of every problem. Practice these questions and answers to command the concepts, boost confidence and in depth understanding of concepts. Students can find all exercise together using the link provided below.
Question:1 b) Find the rate of change of the area of a circle with respect to its radius r when
r = 4 cm
Answer:
Area of the circle (A) =
Rate of change of the area of a circle with respect to its radius r = = =
So, when r = 4, Rate of change of the area of a circle = =
Hence, Rate of change of the area of a circle with respect to its radius r when r = 4 is
Answer:
The volume of the cube(V) = where x is the edge length of the cube
It is given that the volume of a cube is increasing at the rate of
we can write ( By chain rule)
- (i)
Now, we know that the surface area of the cube(A) is
- (ii)
from equation (i) we know that
put this value in equation (i)
We get,
It is given in the question that the value of edge length(x) = 12cm
So,
Answer:
Radius of a circle is increasing uniformly at the rate = 3 cm/s
Area of circle(A) =
(by chain rule)
It is given that the value of r = 10 cm
So,
Hence, the rate at which the area of the circle is increasing when the radius is 10 cm is
Answer:
It is given that the rate at which edge of cube increase = 3 cm/s
The volume of cube =
(By chain rule)
It is given that the value of x is 10 cm
So,
Hence, the rate at which the volume of the cube increasing when the edge is 10 cm long is
Answer:
Given =
To find = at r = 8 cm
Area of the circle (A) =
(by chain rule)
Hence, the rate at which the area increases when the radius of the circular wave is 8 cm is
Answer:
Given =
To find = , where C is circumference
Solution :-
we know that the circumference of the circle (C) =
(by chain rule)
Hence, the rate of increase of its circumference is
the perimeter of rectangle
Answer:
Given = Length x of a rectangle is decreasing at the rate = -5 cm/minute (-ve sign indicates decrease in rate)
the width y is increasing at the rate = 4 cm/minute
To find = and at x = 8 cm and y = 6 cm , where P is perimeter
Solution:-
Perimeter of rectangle(P) = 2(x+y)
Hence, Perimeter decreases at the rate of
Question:7(b) The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x = 8cm and y = 6cm, find the rates of change of the area of the rectangle.
Answer:
Given same as previous question
Solution:-
Area of rectangle = xy
Hence, the rate of change of area is
Answer:
Given =
To find = at r = 15 cm
Solution:-
Volume of sphere(V) =
Hence, the rate at which the radius of the balloon increases when the radius is 15 cm is
Answer:
We need to find the value of at r =10 cm
The volume of the sphere (V) =
Hence, the rate at which its volume is increasing with the radius when the later is 10 cm is
Answer:
Let h be the height of the ladder and x be the distance of the foot of the ladder from the wall
It is given that
We need to find the rate at which the height of the ladder decreases
length of ladder(L) = 5m and x = 4m (given)
By Pythagoras theorem, we can say that
Differentiate on both sides w.r.t. t
at x = 4
Hence, the rate at which the height of ladder decreases is
Answer:
We need to find the point at which
Given the equation of curve =
Differentiate both sides w.r.t. t
(required condition)
when x = 4 ,
and
when x = -4 ,
So , the coordinates are
Answer:
It is given that
We know that the shape of the air bubble is spherical
So, volume(V) =
Hence, the rate of change in volume is
Answer:
Volume of sphere(V) =
Diameter =
So, radius(r) =
Answer:
Given = and
To find = at h = 4 cm
Solution:-
Volume of cone(V) =
Question:15 The total cost C(x) in Rupees associated with the production of x units of an
item is given by
Find the marginal cost when 17 units are produced.
Answer:
Marginal cost (MC) =
Now, at x = 17
MC
Hence, marginal cost when 17 units are produced is 20.967
Question:16 The total revenue in Rupees received from the sale of x units of a product is
given by
Find the marginal revenue when x = 7
Answer:
Marginal revenue =
at x = 7
Hence, marginal revenue when x = 7 is 208
Answer:
Area of circle(A) =
Now, at r = 6cm
Hence, the rate of change of the area of a circle with respect to its radius r at r = 6 cm is
Hence, the correct answer is B
Answer:
Marginal revenue =
at x = 15
Hence, marginal revenue when x = 15 is 126
Hence, the correct answer is D
18 questions and their solutions are discussed in exercise 6.1 Class 12 Maths solutions. Out of these eighteen questions, 2 are multiple-choice questions. NCERT book gives 6 solved examples before Class 12th Maths chapter 6 exercise 6.1. Solving these examples helps in understanding the questions in NCERT solutions given in Class 12 Maths chapter 6 exercise 6.1. After going through the examples try to solve the exercise alone and if any confusion arises refer to the solutions given here. The solutions given here are in detail and is beneficial for the CBSE Class 12 Mathematics board exam.
Also Read| Application of Derivatives Class 12 Notes
Including miscellaneous, there are 6 exercises in the unit Applications of Derivatives
Two multiple-choice questions are discussed in the NCERT exercise 6.1
There are six solved examples
dy/dx means the rate of change of y with respect to x
dy/dx is positive if y increases as x increases
The topic number 6.3 increasing and decreasing function is coming after exercise 6.1.
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Hello there! Thanks for reaching out to us at Careers360.
Ah, you're looking for CBSE quarterly question papers for mathematics, right? Those can be super helpful for exam prep.
Unfortunately, CBSE doesn't officially release quarterly papers - they mainly put out sample papers and previous years' board exam papers. But don't worry, there are still some good options to help you practice!
Have you checked out the CBSE sample papers on their official website? Those are usually pretty close to the actual exam format. You could also look into previous years' board exam papers - they're great for getting a feel for the types of questions that might come up.
If you're after more practice material, some textbook publishers release their own mock papers which can be useful too.
Let me know if you need any other tips for your math prep. Good luck with your studies!
It's understandable to feel disheartened after facing a compartment exam, especially when you've invested significant effort. However, it's important to remember that setbacks are a part of life, and they can be opportunities for growth.
Possible steps:
Re-evaluate Your Study Strategies:
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I hope this information helps you.
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hello mahima,
If you have uploaded screenshot of your 12th board result taken from CBSE official website,there won,t be a problem with that.If the screenshot that you have uploaded is clear and legible. It should display your name, roll number, marks obtained, and any other relevant details in a readable forma.ALSO, the screenshot clearly show it is from the official CBSE results portal.
hope this helps.
Hello Akash,
If you are looking for important questions of class 12th then I would like to suggest you to go with previous year questions of that particular board. You can go with last 5-10 years of PYQs so and after going through all the questions you will have a clear idea about the type and level of questions that are being asked and it will help you to boost your class 12th board preparation.
You can get the Previous Year Questions (PYQs) on the official website of the respective board.
I hope this answer helps you. If you have more queries then feel free to share your questions with us we will be happy to assist you.
Thank you and wishing you all the best for your bright future.
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