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NCERT Solutions for Exercise 6.3 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 6.3 Class 12 Maths chapter 6 is related to the topic normal and tangents. The equations of normals and tangent to a curve and questions related to these are discussed in exercise 6.3 Class 12 Maths. There are twenty-seven questions presented in the Class 12 Maths chapter 6 exercise 6.3. These questions in Class 12th Maths chapter 6 exercise 6.3 are solved by our Mathematics subject matter experts and are reliable and according to the CBSE pattern. The NCERT solutions for Class 12 Maths chapter 6 exercise 6.3 along with all other exercises of the NCERT chapter applications of derivatives gives a good idea of concepts discussed in the NCERT Book.
12th class Maths exercise 6.3 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 . Find the slope of the tangent to the curve
Answer:
Given curve is,
Now, the slope of the tangent at point x =4 is given by
Question:2 . Find the slope of the tangent to the curve
Answer:
Given curve is,
The slope of the tangent at x = 10 is given by
at x = 10
hence, slope of tangent at x = 10 is
Question:3 Find the slope of the tangent to curve at the point whose x-coordinate is 2.
Answer:
Given curve is,
The slope of the tangent at x = 2 is given by
Hence, the slope of the tangent at point x = 2 is 11
Question:4 Find the slope of the tangent to the curve at the point whose x-coordinate is 3.
Answer:
Given curve is,
The slope of the tangent at x = 3 is given by
Hence, the slope of tangent at point x = 3 is 24
Question:5 Find the slope of the normal to the curve
Answer:
The slope of the tangent at a point on a given curve is given by
Now,
Similarly,
Hence, the slope of the tangent at is -1
Now,
Slope of normal = =
Hence, the slope of normal at is 1
Question:6 Find the slope of the normal to the curve
Answer:
The slope of the tangent at a point on given curves is given by
Now,
Similarly,
Hence, the slope of the tangent at is
Now,
Slope of normal = =
Hence, the slope of normal at is
Question:7 Find points at which the tangent to the curve is parallel to the x-axis.
Answer:
We are given :
Differentiating the equation with respect to x, we get :
or
or
It is given that tangent is parallel to the x-axis, so the slope of the tangent is equal to 0.
So,
or
Thus, Either x = -1 or x = 3
When x = -1 we get y = 12 and if x =3 we get y = -20
So the required points are (-1, 12) and (3, -20).
(4, 4).
Answer:
Points joining the chord is (2,0) and (4,4)
Now, we know that the slope of the curve with given two points is
As it is given that the tangent is parallel to the chord, so their slopes are equal
i.e. slope of the tangent = slope of the chord
Given the equation of the curve is
Now, when
Hence, the coordinates are (3, 1)
Question:9 Find the point on the curve at which the tangent is
Answer:
We know that the equation of a line is y = mx + c
Know the given equation of tangent is
y = x - 11
So, by comparing with the standard equation we can say that the slope of the tangent (m) = 1 and value of c if -11
As we know that slope of the tangent at a point on the given curve is given by
Given the equation of curve is
When x = 2 ,
and
When x = -2 ,
Hence, the coordinates are (2,-9) and (-2,19), here (-2,19) does not satisfy the equation y=x-11
Hence, the coordinate is (2,-9) at which the tangent is
Question:10 Find the equation of all lines having slope –1 that are tangents to the curve
Answer:
We know that the slope of the tangent of at the point of the given curve is given by
Given the equation of curve is
It is given thta slope is -1
So,
Now, when x = 0 ,
and
when x = 2 ,
Hence, the coordinates are (0,-1) and (2,1)
Equation of line passing through (0,-1) and having slope = -1 is
y = mx + c
-1 = 0 X -1 + c
c = -1
Now equation of line is
y = -x -1
y + x + 1 = 0
Similarly, Equation of line passing through (2,1) and having slope = -1 is
y = mx + c
1 = -1 X 2 + c
c = 3
Now equation of line is
y = -x + 3
y + x - 3 = 0
Question:11 Find the equation of all lines having slope 2 which are tangents to the curve
Answer:
We know that the slope of the tangent of at the point of the given curve is given by
Given the equation of curve is
It is given that slope is 2
So,
So, this is not possible as our coordinates are imaginary numbers
Hence, no tangent is possible with slope 2 to the curve
Question:12 Find the equations of all lines having slope 0 which are tangent to the curve
Answer:
We know that the slope of the tangent at a point on the given curve is given by
Given the equation of the curve as
It is given thta slope is 0
So,
Now, when x = 1 ,
Hence, the coordinates are
Equation of line passing through and having slope = 0 is
y = mx + c
1/2 = 0 X 1 + c
c = 1/2
Now equation of line is
Question:13(i) Find points on the curve at which the tangents are parallel to x-axis
Answer:
Parallel to x-axis means slope of tangent is 0
We know that slope of tangent at a given point on the given curve is given by
Given the equation of the curve is
From this, we can say that
Now. when ,
Hence, the coordinates are (0,4) and (0,-4)
Question:13(ii) Find points on the curve at which the tangents are parallel to y-axis
Answer:
Parallel to y-axis means the slope of the tangent is , means the slope of normal is 0
We know that slope of the tangent at a given point on the given curve is given by
Given the equation of the curve is
Slope of normal =
From this we can say that y = 0
Now. when y = 0,
Hence, the coordinates are (3,0) and (-3,0)
Question:14(i) Find the equations of the tangent and normal to the given curves at the indicated
points:
Answer:
We know that Slope of the tangent at a point on the given curve is given by
Given the equation of the curve
at point (0,5)
Hence slope of tangent is -10
Now we know that,
Now, equation of tangent at point (0,5) with slope = -10 is
equation of tangent is
Similarly, the equation of normal at point (0,5) with slope = 1/10 is
equation of normal is
Question:14(ii) Find the equations of the tangent and normal to the given curves at the indicated
points:
Answer:
We know that Slope of tangent at a point on given curve is given by
Given equation of curve
at point (1,3)
Hence slope of tangent is 2
Now we know that,
Now, equation of tangent at point (1,3) with slope = 2 is
y = 2x + 1
y -2x = 1
Similarly, equation of normal at point (1,3) with slope = -1/2 is
y = mx + c
equation of normal is
Question:14(iii) Find the equations of the tangent and normal to the given curves at the indicated
points:
Answer:
We know that Slope of the tangent at a point on the given curve is given by
Given the equation of the curve
at point (1,1)
Hence slope of tangent is 3
Now we know that,
Now, equation of tangent at point (1,1) with slope = 3 is
equation of tangent is
Similarly, equation of normal at point (1,1) with slope = -1/3 is
y = mx + c
equation of normal is
Question:14(iv) Find the equations of the tangent and normal to the given curves at the indicated points
Answer:
We know that Slope of the tangent at a point on the given curve is given by
Given the equation of the curve
at point (0,0)
Hence slope of tangent is 0
Now we know that,
Now, equation of tangent at point (0,0) with slope = 0 is
y = 0
Similarly, equation of normal at point (0,0) with slope = is
Question:14(v) Find the equations of the tangent and normal to the given curves at the indicated points:
Answer:
We know that Slope of the tangent at a point on the given curve is given by
Given the equation of the curve
Now,
and
Now,
Hence slope of the tangent is -1
Now we know that,
Now, the equation of the tangent at the point with slope = -1 is
and
equation of the tangent at
i.e. is
Similarly, the equation of normal at with slope = 1 is
and
equation of the tangent at
i.e. is
Question:15(a) Find the equation of the tangent line to the curve which is parallel to the line
Answer:
Parellel to line means slope of tangent and slope of line is equal
We know that the equation of line is
y = mx + c
on comparing with the given equation we get slope of line m = 2 and c = 9
Now, we know that the slope of tangent at a given point to given curve is given by
Given equation of curve is
Now, when x = 2 ,
Hence, the coordinates are (2,7)
Now, equation of tangent paasing through (2,7) and with slope m = 2 is
y = mx + c
7 = 2 X 2 + c
c = 7 - 4 = 3
So,
y = 2 X x+ 3
y = 2x + 3
So, the equation of tangent is y - 2x = 3
Question:15(b) Find the equation of the tangent line to the curve which is perpendicular to the line
Answer:
Perpendicular to line means
We know that the equation of the line is
y = mx + c
on comparing with the given equation we get the slope of line m = 3 and c = 13/5
Now, we know that the slope of the tangent at a given point to given curve is given by
Given the equation of curve is
Now, when ,
Hence, the coordinates are
Now, the equation of tangent passing through (2,7) and with slope is
So,
Hence, equation of tangent is 36y + 12x = 227
Question:16 Show that the tangents to the curve at the points where x = 2 and x = – 2 are parallel .
Answer:
Slope of tangent =
When x = 2
When x = -2
Slope is equal when x= 2 and x = - 2
Hence, we can say that both the tangents to curve is parallel
Question:17 Find the points on the curve at which the slope of the tangent is equal to the y-coordinate of the point.
Answer:
Given equation of curve is
Slope of tangent =
it is given that the slope of the tangent is equal to the y-coordinate of the point
We have
So, when x = 0 , y = 0
and when x = 3 ,
Hence, the coordinates are (3,27) and (0,0)
Question:18 For the curve , find all the points at which the tangent passes
through the origin.
Answer:
Tangent passes through origin so, (x,y) = (0,0)
Given equtaion of curve is
Slope of tangent =
Now, equation of tangent is
at (0,0) Y = 0 and X = 0
and we have
Now, when x = 0,
when x = 1 ,
when x= -1 ,
Hence, the coordinates are (0,0) , (1,2) and (-1,-2)
Question:19 Find the points on the curve at which the tangents are parallel
to the x-axis.
Answer:
parellel to x-axis means slope is 0
Given equation of curve is
Slope of tangent =
When x = 1 ,
Hence, the coordinates are (1,2) and (1,-2)
Question:20 Find the equation of the normal at the point for the curve
Answer:
Given equation of curve is
Slope of tangent
at point
Now, we know that
equation of normal at point and with slope
Hence, the equation of normal is
Question:21 Find the equation of the normals to the curve which are parallel
to the line
Answer:
Equation of given curve is
Parellel to line means slope of normal and line is equal
We know that, equation of line
y= mx + c
on comparing it with our given equation. we get,
Slope of tangent =
We know that
Now, when x = 2,
and
When x = -2 ,
Hence, the coordinates are (2,18) and (-2,-6)
Now, the equation of at point (2,18) with slope
Similarly, the equation of at point (-2,-6) with slope
Hence, the equation of the normals to the curve which are parallel
to the line
are x +14y - 254 = 0 and x + 14y +86 = 0
Question:22 Find the equations of the tangent and normal to the parabola at the point
Answer:
Equation of the given curve is
Slope of tangent =
at point
Now, the equation of tangent with point and slope is
We know that
Now, the equation of at point with slope -t
Hence, the equations of the tangent and normal to the parabola
at the point are
Question:23 Prove that the curves and xy = k cut at right angles*
Answer:
Let suppose, Curve and xy = k cut at the right angle
then the slope of their tangent also cut at the right angle
means,
-(i)
Now these values in equation (i)
Hence proved
Question:24 Find the equations of the tangent and normal to the hyperbola
at the point
Answer:
Given equation is
Now ,we know that
slope of tangent =
at point
equation of tangent at point with slope
Now, divide both sides by
Hence, the equation of tangent is
We know that
equation of normal at the point with slope
Question:25 Find the equation of the tangent to the curve which is parallel to the line
Answer:
Parellel to line means the slope of tangent and slope of line is equal
We know that the equation of line is
y = mx + c
on comparing with the given equation we get the slope of line m = 2 and c = 5/2
Now, we know that the slope of the tangent at a given point to given curve is given by
Given the equation of curve is
Now, when
,
but y cannot be -ve so we take only positive value
Hence, the coordinates are
Now, equation of tangent paasing through
and with slope m = 2 is
Hence, equation of tangent paasing through and with slope m = 2 is 48x - 24y = 23
Question:26 The slope of the normal to the curve is
(A) 3 (B) 1/3 (C) –3 (D) -1/3
Answer:
Equation of the given curve is
Slope of tangent =
at x = 0
Now, we know that
Hence, (D) is the correct option
Question:27 The line is a tangent to the curve at the point
(A) (1, 2) (B) (2, 1) (C) (1, – 2) (D) (– 1, 2)
Answer:
The slope of the given line is 1
given curve equation is
If the line is tangent to the given curve than the slope of the tangent is equal to the slope of the curve
The slope of tangent =
Now, when y = 2,
Hence, the coordinates are (1,2)
Hence, (A) is the correct answer
As the questions in the Class 12 Maths chapter 6 exercise 6.3 deal with an application of derivatives, it is better for the students to revise the basic derivatives of trigonometric functions, exponential functions and some other special functions and rules and properties related to the derivatives. Out of the 27 problems in the Class 12th Maths chapter, 6 exercise 6. 3 question 26 is to find the slope of the normal to a given curve and question 27 asks to find the points for which a line is a tangent to the given curve.
Also Read| Application of Derivatives Class 12 Notes
Solving exercise 6.3 Class 12 Maths will be beneficial for both the CBSE board exam and JEE Main exam.
One question from NCERT solutions for Class 12 Maths chapter 6 exercise 6.3 can be expected for the CBSE Class 12 Maths board paper.
The applications of tangents will be used in Class 12 Physics and Chemistry also.
Also see-
y-y0=f’(x0)(x-x0)
(y-y0)f’(x0)+(x-x0)=0
The slope =0. Therefore the equation of tangent is y=y0
The equation of tangent with infinite slope is x=x0
Normal is perpendicular to the tangent
The slope of normal is the negative of the inverse of the slope of the tangent.
27 questions are answered in the NCERT Solutions for Class 12 Maths chapter 6 exercise 6.3. For more questions refer to NCERT exemplar. Following NCERT syllabus will be useful for CBSE board exams.
There are 7 solved examples explained before the NCERT book Class 12 Maths chapter 6 exercise 6.3.
hello,
Yes you can appear for the compartment paper again since CBSE gives three chances to a candidate to clear his/her exams so you still have two more attempts. However, you can appear for your improvement paper for all subjects but you cannot appear for the ones in which you have failed.
I hope this was helpful!
Good Luck
Hello dear,
If you was not able to clear 1st compartment and now you giving second compartment so YES, you can go for your improvement exam next year but if a student receives an improvement, they are given the opportunity to retake the boards as a private candidate the following year, but there are some requirements. First, the student must pass all of their subjects; if they received a compartment in any subject, they must then pass the compartment exam before being eligible for the improvement.
As you can registered yourself as private candidate for giving your improvement exam of 12 standard CBSE(Central Board of Secondary Education).For that you have to wait for a whole year which is bit difficult for you.
Positive side of waiting for whole year is you have a whole year to preparing yourself for your examination. You have no distraction or something which may causes your failure in the exams. In whole year you have to stay focused on your 12 standard examination for doing well in it. By this you get a highest marks as a comparison of others.
Believe in Yourself! You can make anything happen
All the very best.
Hello Student,
I appreciate your Interest in education. See the improvement is not restricted to one subject or multiple subjects and we cannot say if improvement in one subject in one year leads to improvement in more subjects in coming year.
You just need to have a revision of all subjects what you have completed in the school. have a revision and practice of subjects and concepts helps you better.
All the best.
If you'll do hard work then by hard work of 6 months you can achieve your goal but you have to start studying for it dont waste your time its a very important year so please dont waste it otherwise you'll regret.
Yes, you can take admission in class 12th privately there are many colleges in which you can give 12th privately.
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Depending on the video game designer job description and experience they may also have to lead a team and do the early testing of the game in order to suggest changes and find loopholes.
Radio Jockey is an exciting, promising career and a great challenge for music lovers. If you are really interested in a career as radio jockey, then it is very important for an RJ to have an automatic, fun, and friendly personality. If you want to get a job done in this field, a strong command of the language and a good voice are always good things. Apart from this, in order to be a good radio jockey, you will also listen to good radio jockeys so that you can understand their style and later make your own by practicing.
A career as radio jockey has a lot to offer to deserving candidates. If you want to know more about a career as radio jockey, and how to become a radio jockey then continue reading the article.
The word “choreography" actually comes from Greek words that mean “dance writing." Individuals who opt for a career as a choreographer create and direct original dances, in addition to developing interpretations of existing dances. A Choreographer dances and utilises his or her creativity in other aspects of dance performance. For example, he or she may work with the music director to select music or collaborate with other famous choreographers to enhance such performance elements as lighting, costume and set design.
A multimedia specialist is a media professional who creates, audio, videos, graphic image files, computer animations for multimedia applications. He or she is responsible for planning, producing, and maintaining websites and applications.
A career as social media manager involves implementing the company’s or brand’s marketing plan across all social media channels. Social media managers help in building or improving a brand’s or a company’s website traffic, build brand awareness, create and implement marketing and brand strategy. Social media managers are key to important social communication as well.
In a career as a copywriter, one has to consult with the client and understand the brief well. A career as a copywriter has a lot to offer to deserving candidates. Several new mediums of advertising are opening therefore making it a lucrative career choice. Students can pursue various copywriter courses such as Journalism, Advertising, Marketing Management. Here, we have discussed how to become a freelance copywriter, copywriter career path, how to become a copywriter in India, and copywriting career outlook.
Careers in journalism are filled with excitement as well as responsibilities. One cannot afford to miss out on the details. As it is the small details that provide insights into a story. Depending on those insights a journalist goes about writing a news article. A journalism career can be stressful at times but if you are someone who is passionate about it then it is the right choice for you. If you want to know more about the media field and journalist career then continue reading this article.
For publishing books, newspapers, magazines and digital material, editorial and commercial strategies are set by publishers. Individuals in publishing career paths make choices about the markets their businesses will reach and the type of content that their audience will be served. Individuals in book publisher careers collaborate with editorial staff, designers, authors, and freelance contributors who develop and manage the creation of content.
In a career as a vlogger, one generally works for himself or herself. However, once an individual has gained viewership there are several brands and companies that approach them for paid collaboration. It is one of those fields where an individual can earn well while following his or her passion.
Ever since internet costs got reduced the viewership for these types of content has increased on a large scale. Therefore, a career as a vlogger has a lot to offer. If you want to know more about the Vlogger eligibility, roles and responsibilities then continue reading the article.
Individuals in the editor career path is an unsung hero of the news industry who polishes the language of the news stories provided by stringers, reporters, copywriters and content writers and also news agencies. Individuals who opt for a career as an editor make it more persuasive, concise and clear for readers. In this article, we will discuss the details of the editor's career path such as how to become an editor in India, editor salary in India and editor skills and qualities.
Linguistic meaning is related to language or Linguistics which is the study of languages. A career as a linguistic meaning, a profession that is based on the scientific study of language, and it's a very broad field with many specialities. Famous linguists work in academia, researching and teaching different areas of language, such as phonetics (sounds), syntax (word order) and semantics (meaning).
Other researchers focus on specialities like computational linguistics, which seeks to better match human and computer language capacities, or applied linguistics, which is concerned with improving language education. Still, others work as language experts for the government, advertising companies, dictionary publishers and various other private enterprises. Some might work from home as freelance linguists. Philologist, phonologist, and dialectician are some of Linguist synonym. Linguists can study French, German, Italian.
The career of a travel journalist is full of passion, excitement and responsibility. Journalism as a career could be challenging at times, but if you're someone who has been genuinely enthusiastic about all this, then it is the best decision for you. Travel journalism jobs are all about insightful, artfully written, informative narratives designed to cover the travel industry. Travel Journalist is someone who explores, gathers and presents information as a news article.
Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues.
A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product.
A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
A QA Lead is in charge of the QA Team. The role of QA Lead comes with the responsibility of assessing services and products in order to determine that he or she meets the quality standards. He or she develops, implements and manages test plans.
A metallurgical engineer is a professional who studies and produces materials that bring power to our world. He or she extracts metals from ores and rocks and transforms them into alloys, high-purity metals and other materials used in developing infrastructure, transportation and healthcare equipment.
An Azure Administrator is a professional responsible for implementing, monitoring, and maintaining Azure Solutions. He or she manages cloud infrastructure service instances and various cloud servers as well as sets up public and private cloud systems.
An AWS Solution Architect is someone who specializes in developing and implementing cloud computing systems. He or she has a good understanding of the various aspects of cloud computing and can confidently deploy and manage their systems. He or she troubleshoots the issues and evaluates the risk from the third party.
Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack