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CBSE Class 12 Maths Exam 2024 Important Case Study Based Questions

Case study questions for class 12 maths: check here the important case study based questions of section e in the cbse class 12 maths exam 2024 for last minute preparation..

CBSE Class 12 Maths Exam 2024 Important Questions: The Central Board of Secondary Education is the largest and one of the most famed school boards in India, and lakhs of students are currently enrolled in it. The CBSE conducts the Class 12 board exams annually. The next paper is arguably the most important for science and commerce stream students: Maths on 9 March. Maths is essential for non-medical science and commerce aspirants and is also required in subjects like physics, statistics and accounts. CBSE Class 12 Maths requires extensive practice, especially important topics like Calculus and Algebra. There will be five sections in the 2024 CBSE Class 12 Maths exam, and the last section E will comprise two case study-based questions of 4 marks. These questions are quite important from the exam point of view, and you can check and practice the solved versions here.

CBSE Class 12   Maths   Unit Wise Marks Distribution 2024

CBSE Maths Previous Year Question Paper Class 12

CBSE Class 12 Maths Case Study Questions 2023

Q uestion 1:  Ramesh is   elder brother of Suresh. Ramesh wants to help his younger brother Suresh to solve the following problems of integrals. Write the suitable substitution by which Ramesh can help him.

Question 2: Mr Shashi, who is an architect, designs a building for a small company. The design of window on the ground floor is proposed to be different than other floors. The window   is in the shape of a rectangle which is surmounted by a semi-circular opening. This window is having a perimeter of 10 m as shown below :

Based on the above information answer the following :

(i) If 2x and 2y represents the length and breadth of the rectangular portion of the windows, then the relation between the variables is:      

(ii) The combined area (A) of the rectangular region and semi-circular region of the window expressed as a function of x is:

(iii) The maximum value of area A, of the whole window is

The owner of this small company is interested in maximizing the area of the whole window so that maximum light input is possible.

For this to happen, the length of rectangular portion of the window should be

(i) 4y = 10 - (2 + π)x

(ii) A = 10x - (2 + 12π)x 2

(iii) 50/4 + π

20/4 + π  

Question 3:  Read the following and answer the questions given below 

The front gate of a building is in the shape of a trapezium as shown below. Its three sides other than base are of 10 m each. The height of the gate is h meter. On the basis of below figure, answer the following questions:

(i)  Write the Area (A) of the gate in terms of   .

(ii) Write the value of  when Area (A)  is maximum.

(iii) Write the value of h when Area (A) is maximum .

      Write the Maximum value of Area (A) .

(i) (10 + x)√100 - x 2

(iii) 5√3m OR 75√3/m.m2

Question 4 : Read the following and answer the questions given below 

Given three identical boxes 1 st,  2 nd  and 3 rd  each containing two coins. In 1 st  box both coins are gold coins, in 2 nd  box both are silver coins and in 3 rd  box there is one gold and one silver coin. A person chooses a box at random and takes out a coin.

On the basis of above information, answer the following questions:    

(i)   What is the probability of choosing 1 st  box  ?

(ii)  What is the probability of getting gold coin from 3 rd  box ?

(iii)  What is the total probability of drawing gold coin ?

       If drawn coin is of gold the probability that other coin in the box is also of gold?

(iii) 1/2  Or  2/3            

Question 5:  Read the following and answer the questions given below

Sand is pouring from a pipe at the rate of 12 cm 3 / second the falling sand forms a cone on the ground in such a way that the height of the cone is always 1/6 th of the radius of the base. Based on above information answer the following:

(i)  Write the expression for volume in terms of height only.

(ii)   What is the rate of Change of height, when height is 4 cm?

i)  12πh 3

(ii) 1/48 cm/s

Question 6: There are two antiaircraft guns, named as A and B. The probabilities that the shell fired from them hits an airplane are 0.3 and 0.2 respectively. Both of them fired one shell at an airplane at the same time.

(i) What is the probability that the shell fired from exactly one of them hit the plane?

(ii) If it is known that the shell fired from exactly one of them hit the plane, then what is the probability that it was fired from B?

• CBSE Class 12 Maths Deleted Syllabus 2023-24
• NCERT Solutions 12th Maths PDF

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CBSE Expert

Case Study Questions for Class 12 Maths PDF Download

We have provided here Case Study questions for the Class 12 Maths for board exams. You can read these chapter-wise Case Study questions. These questions are prepared by subject experts and experienced teachers. The answer key is also provided so that you can check the correct answer for each question. Practice these questions to score well in your final exams.

CBSE 12th Standard CBSE Maths question papers, important notes, study materials, Previous Year questions, Syllabus, and exam patterns. Free 12th Standard CBSE Maths books and syllabus online. Important keywords, Case Study Questions, and Solutions.

Class 12 Maths Case Study Questions

CBSE Class 12 Maths question paper will have case study questions too. These case-based questions will be objective type in nature. So, Class 12 Maths students must prepare themselves for such questions. First of all, you should study NCERT Textbooks line by line, and then you should practice as many questions as possible.

Chapter-wise Solved Case Study Questions for Class 12 Maths

Class 12 students should go through important Case Study problems for Maths before the exams. This will help them to understand the type of Case Study questions that can be asked in Grade 12 Maths examinations. Our expert faculty for standard 12 Maths have designed these questions based on the trend of questions that have been asked in last year’s exams. The solutions have been designed in a manner to help the grade 12 students understand the concepts and also easy to learn solutions.

Books for Class 12 Maths

Strictly as per the new term-wise syllabus for Board Examinations to be held in the academic session 2022-23 for class 12 Multiple Choice Questions based on new typologies introduced by the board- Stand-Alone MCQs, MCQs based on Assertion-Reason Case-based MCQs. Include Questions from CBSE official Question Bank released in April 2022 Answer key with Explanations What are the updates in the book: Strictly as per the Term wise syllabus for Board Examinations to be held in the academic session 2022-23. Chapter-wise -Topic-wise Multiple choice questions based on the special scheme of assessment for Board Examination for Class 12th.

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Class 12 Maths Case Study Questions

Table of Contents

myCBSEguide App

Download the app to get CBSE Sample Papers 2023-24, NCERT Solutions (Revised), Most Important Questions, Previous Year Question Bank, Mock Tests, and Detailed Notes.

Class 12 Maths question paper will have 1-2 Case Study Questions. These questions will carry 5 MCQs and students will attempt any four of them. As all of these are only MCQs, it is easy to score good marks with a little practice. Class 12 Maths Case Study Questions are available on the myCBSEguide App and Student Dashboard .

Why Case Studies in CBSE Syllabus?

CBSE has introduced case study questions in the CBSE curriculum recently. The purpose was to make students ready to face real-life challenges with the knowledge acquired in their classrooms. It means, there was a need to connect theories with practicals. Whatsoever the students are learning, they must know how to apply it in their day-to-day life. That’s why CBSE is emphasizing case studies and competency-based education .

Case Study Questions in Maths

Let’s have a look over the class 12 Mathematics sample question paper issued by CBSE, New Delhi. Question numbers 17 and 18 are case study questions.

Focus on concepts

If you go through each MCQ there, you will find that the theme/case study is common but the questions are based on different concepts related to the theme. It means, that if you have done ample practice on the various concepts, you can solve all these MCQs in minutes.

Easy Questions with a Practical Approach

The difficulty level of the questions is average or say easy in some cases. On the other hand, you get four options to choose from. So, you get two levels of support to get full marks with very little effort.

Practice Questions Regularly

Most of the time we feel that it’s easy and neglect it. But in the end, we have to pay for this negligence. This may happen here too. Although it’s easy to score good marks on the case study questions if you don’t practice such questions, you may lose your marks. So, we suggest students should practice at least 30-40 such questions before writing the board exam.

12 Maths Case-Based Questions

We are giving you some examples of case study questions here. We have arranged hundreds of such questions chapter-wise on the myCBSEguide App. It is the complete guide for CBSE students. You can download the myCBSEguide App and get more case study questions there.

Case Study Question – 1

• A is a diagonal matrix
• A is a scalar matrix
• A is a zero matrix
• A is a square matrix
• If A and B are two matrices such that AB = B and BA = A, then B 2 is equal to

Case Study Question – 2

• 4(x 3  – 24x 2   + 144x)
• 4(x 3 – 34x 2   + 244x)
• x 3  – 24x 2   + 144x
• 4x 3  – 24x 2   + 144x
• Local maxima at x = c 1
• Local minima at x = c 1
• Neither maxima nor minima at x = c 1
• None of these

Case Study Questions Matrices -1

Answer Key:

Case Study Questions Matrices – 2

Read the case study carefully and answer any four out of the following questions: Once a mathematics teacher drew a triangle ABC on the blackboard. Now he asked Jose,” If I increase AB by 11 cm and decrease the side BC by 11 cm, then what type of triangle it would be?” Jose said, “It will become an equilateral triangle.”

Again teacher asked Suraj,” If I multiply the side AB by 4 then what will be the relation of this with side AC?” Suraj said it will be 10 cm more than the three times AC.

Find the sides of the triangle using the matrix method and  answer the following questions:

• (a) 3  ×  3

Case Study Questions Determinants – 01

DETERMINANTS:  A determinant is a square array of numbers (written within a pair of vertical lines) that represents a certain sum of products. We can solve a system of equations using determinants, but it becomes very tedious for large systems. We will only do 2 × 2 and 3 × 3 systems using determinants. Using the properties of determinants solve the problem given below and answer the questions that follow:

Three shopkeepers Ram Lal, Shyam Lal, and Ghansham are using polythene bags, handmade bags (prepared by prisoners), and newspaper envelopes as carrying bags. It is found that the shopkeepers Ram Lal, Shyam Lal, and Ghansham are using (20,30,40), (30,40,20), and (40,20,30) polythene bags, handmade bags, and newspapers envelopes respectively. The shopkeeper’s Ram Lal, Shyam Lal, and Ghansham spent ₹250, ₹270, and ₹200 on these carry bags respectively.

• (b) Shyam Lal
• (a) Ram Lal

Case Study Questions Determinants – 02

Case study questions application of derivatives.

• R(x) = -x 2  + 200x + 150000
• R(x) = x 2  – 200x – 140000
• R(x) = 200x 2  + x + 150000
• R(x) = -x 2  + 100 x + 100000
• R'(x) > 0
• R'(x) < 0
• R”(x) = 0
• (a) -x 2  + 200x + 150000
• (a) R'(x) = 0
• (c) 257, -63

Case Study Questions Vector Algebra

• tan−1⁡(5/12)
• tan−1⁡(12/3)
• (b) 130 m/s
• (a)  tan−1⁡(5/12)
• (b) 170 m/s

More Case Study Questions

These are only some samples. If you wish to get more case study questions for CBSE class 12 maths, install the myCBSEguide App. It has class 12 Maths chapter-wise case studies with solutions.

12 Maths Exam pattern

Question Paper Design of CBSE class 12 maths is as below. It clearly shows that 20% weightage will be given to HOTS questions. Whereas 55% of questions will be easy to solve.

• No. chapter-wise weightage. Care to be taken to cover all the chapters
• Suitable internal variations may be made for generating various templates keeping the overall weightage to different forms of questions and typology of questions the same.

Choice(s): There will be no overall choice in the question paper. However, 33% of internal choices will be given in all the sections

12 Maths Prescribed Books

• Mathematics Part I – Textbook for Class XII, NCERT Publication
• Mathematics Part II – Textbook for Class XII, NCERT Publication
• Mathematics Exemplar Problem for Class XII, Published by NCERT
• Mathematics Lab Manual class XII, published by NCERT

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CBSE NCERT Solutions

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CBSE Class 12 Maths Case Study Problems

In Coming Exams, CBSE will ask two Case Study Questions in the CBSE class 12 maths questions paper. Each theme will have five questions and students will have a choice to attempt any four of them.

Here are some example questions Based On Case Study Problems:

Question-1 Read the case study carefully and answer any four out of the following questions:

Three friends Ganesh, Dinesh, and Ramesh went for playing a Tug of war game. Team A, B, and C belong to Ganesh, Dinesh, and Ramesh respectively. Teams A, B, C have attached a rope to a metal ring and are trying to pull the ring into their own area (team areas shown below). Team A pulls with F1 = 4i + 0j KN Team B → F2 = -2i+ 4j KN Team C → F3 = -3i – 3j KN

1. Which team will win the game? (a). Team B (b). Team A (c). Team C (d). No one

2. What is the magnitude of the team’s combine Force? (a). 7 KN (b). 1.4 KN (c). 1.5 KN (d). 2 KN

3. In with direction approx the ring getting pulls: (a). 2.0 radian (b). 2.5 radian (c). 2.4 radian (d). 3 radian

4. What is the magnitude of the force of Team B? (a). 25–√25 KN (b). 6 KN (c). 2 KN (d). 6–√6 KN

5. How many KN Force is applied by Team A? (a). 5 KN (b). 4 KN (c). 2 KN (d). 16 KN

Question-2 Read the case study carefully and answer any four out of the following questions:

Three shopkeepers Ram Lal, Shyam Lal, and Ghansham are using polythene bags, handmade bags (prepared by prisoners), and newspaper’s envelope as carry bags. It is found that the shopkeepers Ram Lal, Shyam Lal, and Ghansham are using (20,30,40), (30,40,20), and (40,20,30) polythene bags, handmade bags, and newspapers envelopes respectively. The shopkeeper’s Ram Lal, Shyam Lal, and Ghansham spent ₹250, ₹270, and ₹200 on these carry bags respectively.

1. What is the cost of one polythene bag? (a). ₹ 1 (b). ₹ 2 (c). ₹ 3 (d). ₹ 5

2. What is the cost of one handmade bag? (a). ₹ 1 (b). ₹ 2 (c). ₹ 3 (d). ₹ 5

3. What is the cost of one newspaper bag? (a). ₹ 1 (b). ₹ 2 (c). ₹ 3 (d). ₹ 5

4. Keeping in mind the social conditions, which shopkeeper is better? (a). Ram Lal (b). Shyam Lal (c). Ghansham (d). None of these

5. Keeping in mind the environmental conditions, which shopkeeper is better? (a). Ram Lal (b). Shyam Lal (c). Ghansham (d). None of these

Question-3 Read the case study carefully and answer any four out of the following questions:

Three friends Ravi, Raju, and Rohit were doing buying and selling stationery items in a market. The price per dozen of Pen, Notebook, and toys are Rupees x, y, and z respectively. Ravi purchases 4 dozens of notebooks and sells 2 dozens of pens and 5 dozens of toys. Raju purchases 2 dozens of toys and sells 3 dozens of pens and 1 dozen of notebooks. Rohit purchases one dozen of pens and sells 3 dozens of notebooks and one dozen toys. In the process, Ravi, Raju, and Rohit earn ₹ 1500, ₹ 100, and ₹400 respectively.

Answer the following questions using the matrix method:

1. What is the price of one dozen of pens? (a). ₹ 100 (b). ₹ 200 (c). ₹ 300 (d). ₹ 400

2. What is the total price of one dozen of pens and one dozen of notebooks? (a). ₹ 100 (b). ₹ 200 (c). ₹ 300 (d). ₹ 400

3. What is the sale amount of Ravi? (a). ₹ 1000 (b). ₹ 1100 (c). ₹ 1300 (d). ₹ 1200

4. What is the amount of purchases made by all three friends? (a). ₹ 1200 (b). ₹ 1500 (c). ₹ 1300 (d). ₹ 1400

5. What is the price of sales made by all three friends? (a). ₹ 3000 (b). ₹ 2500 (c). ₹ 2700 (d). ₹ 2400

Question-4 Read the case study carefully and answer any four out of the following questions:

Dinesh is having a jewelry shop at Green Park, Normally he does not sit in the shop as he remains busy in political meetings. The manager Lisa takes care of jewelry shop where she sells earrings and necklaces. She sells earrings for ₹ 30 and necklaces for 40. It takes 30 minutes to make a pair of earrings and 1 hour to make a necklace, and there are 10 hours a week to make jewelry. In addition, there are only enough materials to make 15 total jewelry items per week. The profit of 15 and 20 is made on each pair of earrings and a necklace respectively. The graph of the equations formed is shown here:

Answer the following questions:

1. What is the first inequality formed? (a). 0.5x + y ≤ 10 (b). x +0.5 y ≤ 10 (c). 0.5x + y ≤ 15 (d). x + 0.5 y ≤ 20

2. What is the second inequality formed? (a). 0.5x + y ≤ 10 (b). x + y ≤ 10 (c). x + y ≤ 15 (d). x + 0.5 y ≤ 10

3. In order to maximize the profits, per week how many pair of earrings and necklaces should be made? (a). 10 and 5 (b). 5 and 10 (c). 15 and 5 (d). 5 and 15

4. What would be the profit if 5 pairs of earrings and 5 necklaces are made? (a). 150 (b). 300 (c). 175 (d). 250

5. What would be the maximum possible profit? (a). 150 (b). 300 (c). 175 (d). 250

Question-5 Read the case study carefully and answer any four out of the following questions:

A telephone company in a town has 500 subscribers on its list and collects fixed charges of 300 per subscriber per year. The company proposes to increases the annual subscription and it is believed that for every increase of 1 one subscriber will discontinue the service.

1. If x be the annual subscription then the total revenue of the company after increment will be: (a). R(x) = -x2 + 200x + 150000 (b). R(x) = x2 – 200x – 140000 (c). R(x) = 200×2 + x + 150000 (d). R(x) = -x2 + 100 x + 100000

2. To find maximum profit we put (a). R'(x) = 0 (b). R'(x) > 0 (c). R'(x) < 0 (d). R”(x) = 0

3. How much fee the company should increase to have maximum profit? (a). Rs. 150 (b). Rs. 100 (c). Rs. 200 (d). Rs. 250

4. Find the maximum profit that the company can make if the profit function is given by P(x) = 41 + 24x – 18×2. (a). 25 (b). 44 (c). 45 (d). 49

5. Find both the maximum and minimum value respectively of 3×4 – 8×3 + 48x + 1 on the interval [1, 4]. (a). -63, 257 (b). 258, -63 (c). 257, -63 (d). -63, -257

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Class 12 Maths: Case Study of Chapter 1 Relations and Functions PDF Download

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In Class 12 Boards there will be Case studies and Passage Based Questions will be asked, So practice these types of questions. Study Rate is always there to help you. Free PDF Download of CBSE Class 12 Mathematics Chapter 1 Relations and Functions Case Study and Passage Based Questions with Answers were Prepared Based on Latest Exam Pattern. Students can solve NCERT Class 12 Maths Relations and Functions  to know their preparation level.

Join our Telegram Channel, there you will get various e-books for CBSE 2024 Boards exams for Class 9th, 10th, 11th, and 12th.

In CBSE Class 12 Maths Paper, There will be a few questions based on case studies and passage-based as well. In that, a paragraph will be given, and then the MCQ questions based on it will be asked.

Relations and Functions Case Study Questions With answers

Here, we have provided case-based/passage-based questions for Class 12 Mathematics  Chapter 1 Relations and Functions

Case Study/Passage-Based Questions

Case Study 1:

A general election of the Lok Sabha is a gigantic exercise. About 911 million people were eligible to vote and voter turnout was about 67%, the highest ever.

Let I be the set of all citizens of India who were eligible to exercise their voting right in the general election held in 2019. A relation ‘R’ is defined on I as follows: R = {(𝑉1, 𝑉2) ∶ 𝑉1, 𝑉2 ∈ 𝐼 and both use their voting right in the general election – 2019}

• Two neighbors X and Y∈ I. X exercised his voting right while Y did not cast her vote in general election – 2019. Which of the following is true? a. (X,Y) ∈R b. (Y,X) ∈R c. (X,X) ∉R d. (X,Y) ∉R
• Mr.’𝑋’ and his wife ‘𝑊’both exercised their voting right in general election -2019, Which of the following is true? a. both (X,W) and (W,X) ∈ R b. (X,W) ∈ R but (W,X) ∉ R c. both (X,W) and (W,X) ∉ R d. (W,X) ∈ R but (X,W) ∉ R
• Three friends F1, F2, and F3 exercised their voting right in the general election 2019, then which of the following is true? a. (F1,F2 ) ∈R, (F2,F3) ∈ R and (F1,F3) ∈ R b. (F1,F2 ) ∈ R, (F2,F3) ∈ R and (F1,F3) ∉ R c. (F1,F2 ) ∈ R, (F2,F2) ∈R but (F3,F3) ∉ R d. (F1,F2 ) ∉ R, (F2,F3) ∉ R and (F1,F3) ∉ R
• The above-defined relation R is __ a. Symmetric and transitive but not reflexive b. Universal relation c. Equivalence relation d. Reflexive but not symmetric and transitive
• Mr. Shyam exercised his voting right in General Election – 2019, then Mr. Shyam is related to which of the following? a. All those eligible voters who cast their votes b. Family members of Mr.Shyam c. All citizens of India d. Eligible voters of India

Answer: 1. (d) (X,Y) ∉R 2. (a) both (X,W) and (W,X) ∈ R 3. (a) (F1,F2 ) ∈R, (F2,F3) ∈ R and (F1,F3) ∈ R 4. (c) Equivalence relation 5. (a) All those eligible voters who cast their votes

Case Study 2:

Sherlin and Danju are playing Ludo at home during Covid-19. While rolling the dice, Sherlin’s sister Raji observed and noted the possible outcomes of the throw every time belonging to set {1, 2, 3, 4, 5, 6}. Let A be the set of players while B be the set of all possible outcomes. A = {S, D}, B = {1, 2, 3, 4, 5, 6}

(i) Let R : B –> B be defined by R = {(x, y) : y is divisible by x} is (a) Reflexive and transitive but not symmetric (b) Reflexive and symmetric but not transitive (c) Not reflexive but symmetric and transitive (d) Equivalence

Answer: (a) Reflexive and transitive but not symmetric

(ii) Raji wants to know the number of functions from A to B. How many number of functions are possible? (a) 6 2 (b) 2 6 (c) 6! (d) 2 12

Answer: (a) 62

(iii) Let R be a relation on B defined by R = {(1, 2), (2, 2), (1, 3), (3, 4), (3, 1), (4, 3), (5, 5)}. Then R is (a) Symmetric (b) Reflexive (c) Transitive (d) None of these three

Answer: (d) None of these three

(iv) Raji wants to know the number of relations possible from A to B. How many numbers of relations are possible? (a) 6 2 (b) 2 6 (c) 6! (d) 2 12

Answer: (d) 212

(v) Let R : B –> B be defined by R = {(1, 1), (1, 2), (2, 2)(3, 3), (4, 4), (5, 5), (6, 6)}, then R is (a) Symmetric (b) Reflexive and Transitive (c) Transitive and symmetric (d) Equivalence

Answer: (b) Reflexive and Transitive

Hope the information shed above regarding Case Study and Passage Based Questions for Class 12 Maths Chapter 1 Relations and Functions with Answers Pdf free download has been useful to an extent. If you have any other queries of CBSE Class 12 Mathematics Relations and Functions Case Study and Passage Based Questions with Answers, feel free to comment below so that we can revert back to us at the earliest possible. By Team Study Rate

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CBSE Case Study Questions for Class 12 Maths Matrices Free PDF

Mere Bacchon, you must practice the CBSE Case Study Questions Class 12 Maths Matrices  in order to fully complete your preparation . They are very very important from exam point of view. These tricky Case Study Based Questions can act as a villain in your heroic exams!

I have made sure the questions (along with the solutions) prepare you fully for the upcoming exams. To download the latest CBSE Case Study Questions , just click ‘ Download PDF ’.

CBSE Case Study Questions for Class 12 Maths Matrices PDF

Mcq set 1 -, mcq set 2 -, checkout our case study questions for other chapters.

• Chapter 1 Relations and Functions Case Study Questions
• Chapter 2 Inverse Trigonometric Functions Case Study Questions
• Chapter 4 Determinants Case Study Questions
• Chapter 5 Continuity and Differentiability Case Study Questions

How should I study for my upcoming exams?

First, learn to sit for at least 2 hours at a stretch

Solve every question of NCERT by hand, without looking at the solution.

Solve NCERT Exemplar (if available)

Sit through chapter wise FULLY INVIGILATED TESTS

Practice MCQ Questions (Very Important)

Practice Assertion Reason & Case Study Based Questions

Sit through FULLY INVIGILATED TESTS involving MCQs. Assertion reason & Case Study Based Questions

After Completing everything mentioned above, Sit for atleast 6 full syllabus TESTS.

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CBSE Class 12 Maths Study Material

Learn cbse class xii maths with our easy-to-follow ncert-based study material. we've covered everything a student needs to learn in maths, equipping you with the core concepts to top your exams. we'll guide you through the essential concepts in a clear, engaging, and easy-to-understand way, so you can understand every chapter of class 12 maths and unlock your potential in this field to score 95+ marks in your class 12th board exam..

R D Sharma Solutions

Discover a smoother learning journey through our effortless roadmap

Chapter 1: Relations and Functions

What are Relations?

Types of Functions

Composition of Functions

Invertible Functions

Chapter 2: Inverse Trigonometric Functions

Inverse Trigonometric Functions

Graphs of Inverse Trigonometric Functions - Trigonometry | Class 12 Maths

Properties of Inverse Trigonometric Functions

Inverse Trigonometric Identities

Chapter 3: Matrices

19 articles

Order of Matrix

Types of Matrices

Column Matrix

Rectangular Matrix

Square Matrix

Zero Matrix

Diagonal Matrix

Scalar Matrix

Identity Matrix

Triangular Matrix

How to solve Equality of Matrices?

Matrix Operations

Matrix Addition

Transpose of a Matrix

Symmetric and Skew Symmetric Matrices

Invertible Matrix

Inverse of a Matrix by Elementary Operations - Matrices | Class 12 Maths

Chapter 4: Determinants

Determinant of a Matrix with Solved Examples

Properties of Determinants

Area of a Triangle using Determinants

Minors and Cofactors

Adjoint of a Matrix

Inverse of a Matrix

Applications of Matrices and Determinants

System of Linear Equations

Chapter 5: Continuity and Differentiability

15 articles

Continuity of Functions

Algebra of Continuous Functions - Continuity and Differentiability | Class 12 Maths

Derivatives of Composite Functions

Chain Rule Derivative - Theorem, Proof, Examples

Quotient Rule | Formula, Proof and Examples

Derivatives of Implicit Functions - Continuity and Differentiability | Class 12 Maths

Implicit Differentiation

Implicit differentiation - Advanced Examples

Derivatives of Inverse Functions

Derivatives of Inverse Trigonometric Functions | Class 12 Maths

Exponential Functions| Definition, Formula, Examples

Logarithmic Differentiation - Continuity and Differentiability

Derivative of Functions in Parametric Forms

Second Order Derivatives in Continuity and Differentiability | Class 12 Maths

Higher Order Derivatives

Chapter 6: Application of Derivatives

Derivatives as Rate of Change

Increasing and Decreasing Functions

Increasing and Decreasing Intervals

Maxima and Minima in Calculus

Relative Minima and Maxima

Absolute Minima and Maxima

Concave Function

Inflection Point

Chapter 7: Integrals

Differentiation and Integration Formula

Methods of Integration

Integration by Substitution

Integration by U-substitution

Trigonometric Substitution

Partial Fraction Expansion

Integration of Trigonometric Functions

Integration by Partial Fractions

Integration by Parts

Definite Integral

Computing Definite Integrals

Fundamental Theorem of Calculus

Evaluating Definite Integrals

Properties of Definite Integrals

Definite Integrals of Piecewise Functions

Chapter 8: Application of Integrals

Area under Simple Curves

Chapter 9: Differential Equations

Differential Equations

Order and Degree of Differential Equations

Particular Solutions to Differential Equations

First Order Differential Equation: Type, Solution and Examples

Homogeneous Differential Equations

Separable Differential Equations

Linear Differential Equations

Exact Equations and Integrating Factors

Chapter 10: Vector Algebra

Vector Algebra

Zero Vector

Unit Vector

Triangle Law of Vector Addition

Parallelogram Law of Vector Addition

Commutative Property - Definition | Commutative Law and Examples

Associative Property

Additive Identity Property Definition and Examples

Additive Inverse

Section Formula - Vector Algebra

Dot and Cross Products on Vectors

How to calculate Dot Product of Two Vectors?

Cross Product of two Vectors

Chapter 11: Three Dimensional Geometry

Direction Cosines and Direction Ratios

Equation of a Line in 3D

Angles Between two Lines in 3D Space

Shortest Distance Between Two Lines in 3D Space | Class 12 Maths

Distance between two parallel lines

Chapter 12: Linear Programming

Linear Programming

Graphical Solution of Linear Programming Problems

Chapter 13: Probability

Conditional Probability and Independence - Probability | Class 12 Maths

Multiplication Theorem

Dependent and Independent Events

Bayes' Theorem

Mathematics - Law of Total Probability

Class 12th Maths Paper Set

CBSE Class 12 Previous Year Question Papers

Class 12th Maths R D Sharma Solution

RD Sharma Class 12 Solutions for Maths

This CBSE Class 12 Maths Study Material is made by expert teachers to help you learn Theory and Applications. Our Class 12 Maths study material provides a comprehensive journey through all the essential chapters you'll encounter in your exams. We'll start by building a strong foundation in relations and functions, including their different types, compositions, and inverses. In addition, this study material by GeeksforGeeks also provides detailed solutions to previous year's questions, so students can learn how to answer them correctly. This helps students to understand the concepts behind the questions and to improve their problem-solving skills.

What do we offer?

GeeksforGeeks has designed this study material for class 12 students to make learning Maths an interactive and enjoyable experience. Our expert teachers provide clear and concise explanations of the concepts to keep you motivated and focused. We've also incorporated practice problems and quizzes to help you apply your knowledge to score 95+ Marks in your class 12 Maths board exam.

Key Highlights of this Study Material:

• Clear & Engaging Explanations: Break down complex concepts with easy-to-understand language and step-by-step breakdowns.
• Self-Paced: Learn at your own pace with visual aids, diagrams, and animations that bring concepts to life.
• Practice questions: Here we provide selected practice questions to test your skills and make sure you're ready for the exam.
• Highlights All formulas: Important Maths formulas are covered throughout the lessons, equipping you with the necessary formulas to score 95+ marks in your class 12 Maths exam with confidence.
• Solved Previous Year Questions: Gain an edge with previous year's questions integrated throughout the material. Analyze past exam trends and question styles to score well in the CBSE Class 12 Maths board exam.

Your resources are top-notch! They've been instrumental in enhancing my understanding of complex concepts. Kudos to your team for creating such insightful content!

I'm thoroughly impressed by the quality of your materials. They've significantly enriched my learning experience.

Your Class 12 content is exceptional! It's clear, comprehensive, and immensely helpful. The engaging presentation and practical examples make learning enjoyable.

Niraj Chopra

Thank you for providing such valuable resources that greatly enhance students' understanding of Maths.

Q1. How to get 95 percent in Maths class 12?

Learn from the past. Gain insights into the exam pattern and question types by solving previous years' question papers. Additionally, familiarise yourself with the Sample Question Papers 23-24 released by CBSE to grasp the expected pattern comprehensively.

Q2. Is CBSE 12th maths tough?

Not really, Class 12 maths is considered to be easier than Class 11 maths.CBSE 12th Maths Difficulty is moderate for prepared students. Practice and mock tests are key to success.

Q3. How many chapters are present in Class 12 Maths?

Class 12 Mathematics textbook comprises a total of 13 chapters. All the chapters are significant from the board examination perspective.

Q4. What are the Crucial Chapters in the Mathematics Syllabus for Class 12?

The latest syllabus for class 12 Mathematics 2024-25 covers some crucial chapters and that is: Relations and Functions Algebra Calculus Vectors and Three-Dimensional Geometry Linear Programming Probability

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NCERT Solutions For Class 12 Maths

Ncert solutions for class 12 maths – free 2023-24 cbse pdf download.

NCERT Solutions for Class 12 Maths provides a solid conceptual base for all the topics included in the CBSE Class 12 Maths Syllabus. It covers all the important theorems and formulae with detailed explanations to ensure better conceptual understanding in students. NCERT Solutions for Class 12 Maths will help students in preparation for undergraduate-level Engineering entrance exams like JEE Mains, BITSAT, VITEEE, etc., to be conducted for the year 2023-2024.

BYJU’S brings you NCERT Solutions for Class 12 Maths, designed by our subject experts to facilitate a smooth and precise understanding of concepts. These Class 12 Solutions of NCERT have detailed step-by-step explanations of problems given in the textbooks. Apart from solutions for each chapter of CBSE Class 12 Maths, BYJU’S also provides Class 12 Maths notes, solved exemplar problems, and previous year question papers, along with worksheets as study materials for students who are preparing for their Class 12 board exams.

NCERT Solutions for Class 12 Maths Chapter-Wise PDF

Chapter 1 relations and functions, chapter 2 inverse trigonometric functions, chapter 3 matrices.

• Chapter 4 Determinants

Chapter 5 Continuity and Differentiability

Chapter 6 applications of derivatives, chapter 7 integrals, chapter 8 applications of integrals, chapter 9 differential equations, chapter 10 vector algebra.

• Chapter 11 Three dimensional Geometry

Chapter 12 Linear Programming

Chapter 13 probability, class 12 maths ncert solutions exercises.

Students can download NCERT Solution of Maths of Class 12 , chapter-wise from the links given below.

Chapter 1 of NCERT textbook deals with relations and functions , types of relation, types of functions, the composition of functions and invertible function, binary operations and miscellaneous examples. Here, you can find the exercise solution links for the topics covered in this chapter.

Topics Covered in Class 12 Maths Chapter 1 Relations and Functions: Types of relations: Reflexive, symmetric, transitive and equivalence relations. One to one and onto functions, composite functions, inverse of a function. Binary operations.

Also, access the following resources for Class 12 Chapter 1 Relations and Functions at BYJU’S:

• Relations And Functions Class 12 Maths Notes Chapter 1
• Important Questions For Class 12 Maths Chapter 1 – Relations and Functions
• Maths Revision Notes for Class 12 Chapter 1 Relations and Functions
• NCERT Exemplar Solutions for Class 12 Maths Chapter 1 Relations and Functions

Chapter 2, Inverse Trigonometric Functions of NCERT Class 12 Maths, gives an account of various topics such as the remarks based on basic concepts of inverse trigonometric functions, properties of inverse trigonometric functions and miscellaneous examples. These concepts are explained very well through examples. Solutions to the exercises from the chapter can be accessed from here.

Topics Covered in Class 12 Maths Chapter 2 Inverse Trigonometric Functions: Definition, range, domain, principal value branches. Graphs of inverse trigonometric functions. Elementary properties of inverse trigonometric functions.

Also access the following resources for Class 12 Chapter 2 Inverse Trigonometric Functions at BYJU’S:

• Inverse Trigonometric Functions Class 12 Notes – Chapter 2
• Important Questions for Class 12 Maths Chapter 2 – Inverse Trigonometric Functions
• Maths Revision Notes for Class 12 Chapter 2 Inverse Trigonometric Functions
• NCERT Exemplar Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions

In Chapter 3 of NCERT textbook, we shall see the definition of a matrix, types of matrices, equality of matrices, operations on matrices such as the addition of matrices and multiplication of a matrix by a scalar, properties of matrix addition, properties of scalar multiplication, multiplication of matrices, properties of multiplication of matrices, transpose of a matrix, properties of the transpose of the matrix, symmetric and skew-symmetric matrices, elementary operation or transformation of a matrix, the inverse of a matrix by elementary operations and miscellaneous examples. Here, you can find the exercise solution links for the topics covered in this chapter.

Topics Covered in Class 12 Maths Chapter 3 Matrices: Concept, notation, order, equality, types of matrices, zero matrix, transpose of a matrix, symmetric and skew symmetric matrices. Addition, multiplication and scalar multiplication of matrices, simple properties of addition, multiplication and scalar multiplication. Noncommutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order 2). Concept of elementary row and column operations. Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).

Also access the following resources for Class 12 Chapter 3 Matrices at BYJU’S:

• CBSE Class 12 Maths Notes Chapter 3 Matrices
• Important Questions for Class 12 Maths Chapter 3 – Matrices
• Maths Revision Notes for Class 12 Chapter 3 Matrices
• NCERT Exemplar Solutions for Class 12 Maths Chapter 3 Matrices

Chapter 4 Determinants –

Chapter 4 of 12 Class NCERT Maths Solution discusses the topic of determinants. Students will get to learn about the definition and meaning of determinants, remarks based on order of determinants, properties of determinants, finding area of triangle using determinants, minors and cofactors of determinants, adjoint of a matrix, inverse of matrix , applications of determinants and matrices, and miscellaneous examples. Below, we have links provided to each exercise solution covered in this chapter.

Topics Covered in Class 12 Maths Chapter 4 Determinants: Determinant of a square matrix (up to 3 × 3 matrices), properties of determinants, minors, cofactors and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix. Consistency, inconsistency and number of solutions of system of linear equations by examples, solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix.

Also access the following resources for Class 12 Chapter 4 Determinants at BYJU’S:

• Determinants Class 12 Notes – Chapter 4
• Important Question for Class 12 Maths Chapter 4 – Determinants
• Maths Revision Notes for Class 12 Chapter 4 Determinants
• NCERT Exemplar Solutions for Class 12 Maths Chapter 4 Determinants

Chapter 5 of NCERT textbook starts with the definition of continuity. Students go on to learn about continuity, algebra of continuous function, definition and meaning of differentiability, derivatives of composite functions, derivative of implicit functions, derivative of inverse trigonometric functions, exponential and logarithmic functions, logarithmic differentiation, derivatives of functions in parametric forms, second order derivatives , mean value theorem via miscellaneous examples. Here, students can find the exercises explaining these concepts properly with solutions.

Topics Covered in Class 12 Maths Chapter 5 Continuity and Differentiability: Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit function. Concept of exponential and logarithmic functions and their derivatives. Logarithmic differentiation. Derivative of functions expressed in parametric forms. Second order derivatives. Rolle’s and Lagrange’s Mean Value Theorems (without proof) and their geometric interpretations.

Also access the following resources for Class 12 Chapter 5 Continuity and Differentiability at BYJU’S:

• Continuity And Differentiability Class 12 Notes Chapter 5
• Important Questions for Class 12 Maths Chapter 5 – Continuity and Differentiability
• Maths Revision Notes for Class 12 Chapter 5 Continuity and Differentiability
• NCERT Exemplar Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability

Chapter 6 of NCERT textbook provides definition of derivatives, rate of change of quantities, increasing and decreasing functions, tangents and normals, approximations, maxima and minima , first derivative test, maximum and minimum values of a function in a closed interval, and miscellaneous examples. Here, students can find the exercises explaining these concepts properly.

Topics Covered in Class 12 Maths Chapter 6 Applications of Derivatives: Applications of derivatives: Rate of change, increasing/decreasing functions, tangents and normals, approximation, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool). Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations).

Also access the following resources for Class 12 Chapter 6 Applications of Derivatives at BYJU’S:

• Application of Derivatives Class 12 Notes Chapter 6
• Important Questions for Class 12 Maths Chapter 6 – Applications of Derivatives
• Maths Revision Notes for Class 12 Chapter 6 Application of Derivatives
• NCERT Exemplar Solutions for Class 12 Maths Chapter 6 Application of Derivatives

In Chapter 7, Integrals, we shall see the definition of indefinite integral, integration as an inverse process of differentiation, geometrical interpretation of indefinite integral, properties of indefinite integral, comparison between differentiation and integration, methods of integration such as integration by substitution, integration using partial fractions, integration by parts , integration using trigonometric identities, integration of some integral functions, definition and meaning of definite integrals, definite integral as the limit of a sum, fundamental theorem of calculus, evaluation of definite integrals by substitution, some properties of definite integrals and miscellaneous examples. Here, we have given exercises with solutions based on these topics from the chapter.

Topics Covered in Class 12 Maths Chapter 7 Integrals: Integration as inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, only simple integrals of the type

Also access the following resources for Class 12 Chapter 7 Integrals at BYJU’S:

• Integrals Class 12 Notes Chapter 7
• Important Questions for Class 12 Maths Chapter 7 – Integrals
• Maths Revision Notes for Class 12 Chapter 7 Integrals
• NCERT Exemplar Solutions for Class 12 Maths Chapter 7 Integrals

In Chapter 8 of NCERT textbook, we continue the discussions about integrals starting with definition, area under simple curves, area of the region bounded by a curve and a line , area between two curves and miscellaneous examples. From the below-given links, students can access the exercises with solutions explaining the concepts from this chapter.

Topics Covered in Class 12 Maths Chapter 8 Applications of Integrals: Applications in finding the area under simple curves, especially lines, arcs of circles/parabolas/ ellipses (in standard form only), area between the two above-said curves (the region should be clearly identifiable).

Also, access the following resources for Class 12 Chapter 8 Applications of Integrals at BYJU’S:

• Application of Integrals Class 12 Notes Chapter 8
• Important Questions Class 12 Maths Chapter 8-Applications of Integrals
• Maths Revision Notes for Class 12 Chapter 8 Application of Integrals
• NCERT Exemplar Solutions for Class 12 Maths Chapter 8 Application of Integrals

In this chapter, students concentrate on the definition of differential equations, basic concepts related to differential equations, degree of a differential equation , order of a differential equation, general and particular solutions of a differential equation, formation of a differential equation whose general solution is given, procedure to form a differential equation that will represent a given family of curves, methods of solving first order, first degree differential equations, differential equations with variable separable, homogeneous differential equations, linear differential equations, procedure involved to solve first order linear differential equations, and miscellaneous examples. Solutions to the exercises from the chapter can be accessed from here.

Topics Covered in Class 12 Maths Chapter 9 Differential Equations: Definition, order and degree, general and particular solutions of a differential equation. Formation of differential equation whose general solution is given. Solution of differential equations by method of separation of variables, homogeneous differential equations of first order and first degree. Solutions of linear differential equation of the type: dy/dx + Py = Q, where P and Q are functions of x 

Also access the following resources for Class 12 Chapter 9 Differential Equations at BYJU’S:

• Differential Equation Class 12 Notes Chapter 9
• Important Questions Class 12 Maths Chapter 9 – Differential Equations
• Maths Revision Notes for Class 12 Chapter 9 Differential Equations
• NCERT Exemplar Solutions for Class 12 Maths Chapter 9 Differential Equations

In this chapter, students are introduced to vector algebra. The concepts in this chapter include how to find position vector, some basic concepts related to vector algebra, direction cosines, types of vectors such as zero vector, unit vector, collinear vector, equal vector, negative of a vector, addition of vectors, properties of vector addition, multiplication of a vector by a scalar, components of a vector, vector joining two points, section formula, product of two vectors, scalar or dot product of two vectors, properties of scalar product, projection of a vector on a line, vector or cross product of two vectors and miscellaneous examples. From the below-given links, students can access the exercises with solutions explaining the concepts from this chapter.

Topics Covered in Class 12 Maths Chapter 10 Vector Algebra: Vectors and scalars, magnitude and direction of a vector. Direction cosines/ratios of vectors. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Scalar (dot) product of vectors, projection of a vector on a line. Vector (cross) product of vectors.

Also, access the following resources for Class 12 Chapter 10 Vector Algebra at BYJU’S:

• Vector Algebra Class 12 Notes – Chapter 10
• Important Questions Class 12 Maths Chapter 10 – Vectors
• Maths Revision Notes for Class 12 Chapter 10 Vector Algebra
• NCERT Exemplar Solutions for Class 12 Maths Chapter 10 Vector Algebra

Chapter 11 Three Dimensional Geometry

Chapter 11 of NCERT Class 12 Maths constitutes topics related to three dimensional geometry, direction cosines and direction ratios of a line, relation between the direction cosines of a line, direction cosines of a line passing through two points, equation of a line in space, equation of a line through a given point and parallel to a given vector, derivation of cartesian form from vector form, equation of a line passing through two given points, angle between two lines, shortest distance between two lines, distance between two skew lines, distance between two parallel lines, plane in cartesian form, equation of a plane in a normal form, equation of a plane perpendicular to a given vector and passing through a given point, equation of a plane passing through three non collinear points, intercept form of the equation of a plane, plane passing through the intersection of two given planes, coplanarity of two lines , angle between two planes, distance of a point from a plane, angle between a line and a plane and miscellaneous examples. Below, we have listed links to the solutions for the exercises from this chapter.

Topics Covered in Class 12 Maths Chapter 11 Three Dimensional Geometry: Direction cosines/ratios of a line joining two points. Cartesian and vector equation of a line, coplanar and skew lines, shortest distance between two lines. Cartesian and vector equation of a plane. Angle between (i) two lines, (ii) two planes, (iii) a line and a plane. Distance of a point from a plane.

Also access the following resources for Class 12 Chapter 11 Three Dimensional Geometry at BYJU’S:

• Three Dimensional Geometry Class 12 Notes Chapter 11
• Important Questions Class 12 Maths Chapter 11-Three Dimensional Geometry
• Maths Revision Notes for Class 12 Chapter 11 Three Dimensional Geometry
• NCERT Exemplar Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry

In this chapter, students are introduced to linear programming. The concepts covered in this chapter are linear programming problem and its mathematical formulation, mathematical formulation of the problem, graphical method of solving linear programming problems, different types of linear programming problems with miscellaneous examples. Students can find the exercises explaining these concepts properly with solutions.

Topics Covered in Class 12 Maths Chapter 12 Linear Programming: Introduction, related terminology such as constraints, objective function, optimization, different types of linear programming (L.P.) problems, mathematical formulation of L.P. problems, graphical method of solution for problems in two variables, feasible and infeasible regions, feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constrains).

Also access the following resources for Class 12 Chapter 12 Linear Programming at BYJU’S:

• Linear Programming Class 12 Notes Chapter 12
• Important questions Class 12 Maths Chapter 12 Linear Programming
• Maths Revision Notes for Class 12 Chapter 12 Linear Programming
• NCERT Exemplar Solutions for Class 12 Maths Chapter 12 Linear Programming

This chapter deals with probability. By reviewing the basic facts and formulae studied earlier in Class 11, we shall implement those formulas again in this chapter along with the advance concepts mentioned, i.e., conditional probability, properties of conditional probability , multiplication theorem on probability, multiplication rule of probability for more than two events, independent events, Bayes’ theorem, partition of a sample space, theorem on total probability, random variables and its probability distributions, probability distribution of random variable, mean of random variable, variance of random variable, Bernoulli trials and binomial distribution with miscellaneous examples. We have compiled here the solutions to exercises from this chapter that explain the concept thoroughly.

Topics Covered in Class 12 Maths Chapter 13 Probability: Multiplication theorem on probability. Conditional probability, independent events, total probability, Baye’s theorem. Random variable and its probability distribution, mean and variance of haphazard variable. Repeated independent (Bernoulli) trials and Binomial distribution.

Also access the following resources for Class 12 Chapter 13 Probability at BYJU’S:

• Probability Class 12 Notes Chapter 13
• Important Questions for Class 12 Maths chapter 13 Probability
• Maths Revision Notes for Class 12 Chapter 13 Probability
• NCERT Exemplar Solutions for Class 12 Maths Chapter 13 Probability

CBSE Class 12 Maths 2023-24 Unit-Wise Weightage

Internal assessment class 12, ncert solutions class 12 maths chapter 1 to 13.

NCERT Solutions PDF Class 12 Maths acts as a vital tool for students in their exam preparation and home assignments. Maths plays a vital role in the success of students in Class 12 boards and competitive exams. Solutions PDF of Class 12 can be your best support for board exam preparations. It’s difficult to score in Mathematics simply by reading and memorizing the formulae. Instead, practice is the perfect method to succeed.

Learning from BYJU’S NCERT Solutions will help students of Class 12 to grasp the conceptual meaning of every problem quite effectively. CBSE Class 12th NCERT Maths Solutions provides in-depth answers to all the problems covered in the syllabus. All the above-mentioned chapters are equally important and can be mastered with consistent practice and hard work. The topics like probability, calculus, coordinate geometry, and trigonometry are easy, and with proper practice, students can easily score good marks for questions from these units. We have also provided solved examples at the end of every chapter to make concepts clear to our readers.

Key Features of CBSE Class 12 Maths NCERT Solutions

• Students can easily practise and revise concepts by referring to these solutions.
• It saves your time by providing all the necessary concepts and formulae in a proper sequence.
• The CBSE Class 12th Maths NCERT Solutions   PDF Download option promotes among students the logical approach, which is crucial to ace the exam.
• These solutions are extremely helpful for quick revisions during examinations.

Students are advised to follow the NCERT Class 12 books and solutions on a regular basis to get in-depth knowledge about all the concepts covered in the syllabus.

Frequently Asked Questions on NCERT Solutions for Class 12 Maths

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Get NCERT solutions for Class 12 Maths free with videos. Solutions of all exercise questions, examples, miscellaneous exercise, supplementary exercise are given in an easy to understand way

The chapters and the topics in them are

• Chapter 1 Relation and Functions – Types of Relation - Reflexive, Symmetric, Transitive & Equivalence relation, Type of functions - One-one (injective), Onto (surjective) & Bijective, Composition of functions and Inverse of a function, Binary Operation - Commutative, Associative, Identity and Inverse.
• Chapter 2 Inverse Trigonometric Functions –  Finding principle value, solving identities using properties
• Chapter 3 Matrices – Order of matrix, Types of matrices - Column matrix, Row matrix, Square matrix, Diagnol matrix, Scalar Matrix, Identity matrix, Equal matrices, Addition, Subtraction and Multiplication of matrices, Transpose of a Matrix, Symmetric, Skew Symmetric Matrices, Finding inverse of matrix by elementary operations
• Chapter 4 Determinants – Evaluating 2 x 2, 3 x 3 Determinant, Properties of Determinant, Area of triangle, Minors and Cofactors, Adjoint and inverse of matrix, Solution of system of linear equations using matrix method
• Chapter 5 Continuity and Differentiability – Finding whether function is continuous, composition of continuous functions is continuous, Finding Derivative using chain rules, logarithmic and exponential differentiation, Mean value of theorem, Rolle's Theorem.
• Chapter 6 Application of Derivatives – Finding rate of change, Increasing & decreasing functions, Point of Inflexion, Finding slope of tangent  and normal, Finding approximate of functions and numbers, Finding maxima & minima using first derivative & second derivative test and Finding absolute maxima and minima
• Chapter 7 Integrals – Finding integration by substitution, trigonometric identities, Partial Fractions & using special formulas. Finding definite integrals as Limit as a Sum, Finding definite integral using indefinite integral and finding Definite Integral using properties
• Chapter 8 Application of Integrals –  Finding area of region of simple curve, curve and line. Finding area between two curves.
• Chapter 9 Differential Equations – Degree and order of differential equations, General & Particular Solutions of Differential Equation, Forming differential equation when general solution is equation, Solving by variable separating, solving homogeneous differential equations, Linear Differential Equation
• Chapter 10 Vector Algebra – What is vector, Difference between vector and scalar, Types of vector - Unit vector, Equal vector, Addition of vectors, Components of vector, Scalar(Dot) product of vectors, Projection of vector on line, Vector (Cross) product of vectors, Finding area of parallelogram
• Chapter 11 Three Dimensional Geometry – Direction Ratio & Direction cosines of line, Finding equation of line - given point & parallel vector, though two points, Angle between two lines, Distance between two skew lines, Distance between two parallel lines, Finding equation of plane - normal form, perpendicular to given vector & passing through point, through 3 non-collinear points, intercept form, through two planes. Coplanarity of two planes, Angle between two planes, Distance between point and plane, Angle between Line & Plane
• Chapter 12 Linear Programming  – Finding maximum or minimum value of Z using bounded or unbounded region of graphical inequality.  Solving Diet, Manufacturing & Transport problems.
• Chapter 13 Probability – Conditional Probability, Multiplication Theorem of Probability, Independent events, Bayes Theorem, Random Variable & its Probability Distributions, Mean & Variance of Random Variable, Bernoulli Trials & Binomial Distribution

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Chapter 1 Class 12 Relation and Functions

Chapter 2 class 12 inverse trigonometric functions, chapter 3 class 12 matrices, chapter 4 class 12 determinants, chapter 5 class 12 continuity and differentiability, chapter 6 class 12 application of derivatives, chapter 7 class 12 integrals, chapter 8 class 12 application of integrals, chapter 9 class 12 differential equations, chapter 10 class 12 vector algebra, chapter 11 class 12 three dimensional geometry, chapter 12 class 12 linear programming, chapter 13 class 12 probability, important questions for exams class 12, solutions of sample papers and past year papers - for class 12 boards.

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CBSE 12th Standard Maths Subject Linear Programming Case Study Questions 2021

By QB365 on 21 May, 2021

QB365 Provides the updated CASE Study Questions for Class 12 Maths, and also provide the detail solution for each and every case study questions . Case study questions are latest updated question pattern from NCERT, QB365 will helps to get  more marks in Exams

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Cbse 12th standard maths linear programming case study questions 2021.

12th Standard CBSE

Final Semester - June 2015

Case Study Questions

Linear programming is a method for finding the optimal values (maximum or minimum) of quantities subject to the constraints when relationship is expressed as linear equations or inequations. Based on the above information, answer the following questions. (i) The optimal value of the objective function is attained at the points

(ii) The graph of the inequality 3x + 4y < 12 is

(iv) The corner points of the feasible region determined by the system of linear constraints are (0, 10), (5, 5), (15, 15), (0, 20). Let Z = px + qy, where p, q > 0. Condition on p and q so that the maximum of Z occurs at both the points (15, 15) and (0, 20) is

(v)  The corner points of the feasible region determined by the system of linear constraints are (0, 0), (0, 40), (20,40), (60,20), (60, 0). The objective function is Z = 4x + 3y. Compare the quantity in Column A and Column B

(ii) The corner points of the feasible region shown in above graph are

(iii) If Z = x + y be the objective function and max Z = 30. The maximum value occurs at point

(iv) If Z = 6x - 9y be the objective function, then maximum value of Z is

(v) If Z = 6x + 3y be the objective function, then what is the minimum value of Z?

Corner points of the feasible region for an LPP are (0, 3), (5, 0), (6, 8), (0, 8). Let Z = 4x - 6y be the objective function. Based on the above information, answer the following questions. (i) The minimum value of Z occurs at

(ii) Maximum value of Z occurs at

(iii) Maximum of Z - Minimumof Z =

(v) The feasible solution of LPP belongs to

(ii) Let the constraints in the given problem is represented by the following inequalities x + y ≤ 20 360x + 240y ≤ 5760 x, y ≥ 0 Then which of the following point lie in its feasible region.

(iii) If the objective function of the given problem is maximise z = 22x + 18y, then its optimal value occur at

(iv) Suppose the following shaded region APDO, represent the feasible region corresponding to mathematical formulation. of given problem. Then which of the following represent the coordinates of one of its corner points.

Let R be the feasible region (convex polygon) for a linear programming problem and let Z = ax + by be the objective function. When Z has an optimal value (maximum or minimum), where the variables x and y are subject to, constraints described by linear inequalities, this optimal value must occur at a corner point (vertex) of the feasible region. Based on the above information, answer the following questions. (i) Objective function of a L.P.P. is

(ii) Which of the following statement is correct?

(iii) In solving the LPP : "minimize f = 6x + 10y subject to constraints x ≥  6, Y ≥ 2, 2x + y  ≥ 10,x  ≥ 0,y ≥ 0" redundant constraints are

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Cbse 12th standard maths linear programming case study questions 2021 answer keys.

(iv) (d): Value of Z = px + qy at (15, 15) = 15p + 15q and that at (0, 20) = 20 q. According to given condition, we have 15p + 15q = 20q \(\Rightarrow\) 15p = 5q \(\Rightarrow\) q = 3p (v) (b): Construct the following table of values of the objective function:

(i) (b):  Let B(x, y) be the point of intersection of the given lines 2x + 5y = 100 ....(i) and  \(\frac{x}{25}+\frac{y}{40}=1 \Rightarrow 8 x+5 y=20\) ...(ii) Solving (i) and (ii), we get \(x=\frac{50}{3}, y=\frac{40}{3}\) ஃ The point of intersection \(B(x, y)=\left(\frac{50}{3}, \frac{40}{3}\right)\) (ii) (d): The corner points of the feasible region shown in the given graph are \((0,0), A(25,0), B\left(\frac{50}{3}, \frac{40}{3}\right), C(0,20)\) (iii) (a): Here Z = x + y

Thus, max Z = 30 occurs at point  \(\left(\frac{50}{3}, \frac{40}{3}\right)\) (iv) (b): 

Construct the following table of values of objective function

(i) (d): Minimum value of Z is -48 which occurs at (0,8). (ii) (a): Maximumvalue of Z is 20, which occurs at (5,0). (iii) (b): Maximum of Z - Minimum of Z = 20 - (-48) = 20 + 48 = 68 (iv) (c): The corner points of the feasible region are O(0,0), A(3, 0), B(3, 2), C(2, 3), D(0, 3). (v) (d)

(i) (c) (ii) (b): Since (8, 12) satisfy all the inequalities therefore (8, .1.2) is the point in its feasible region. (iii) (c) : At (0, 0), z = 0 At (16, 0), z = 352 At (8, 12), z = 392 At (0, 20), z = 360 It can be observed that max z occur at (8, 12). Thus, z will attain its optimal value at (8, 12). (iv) (c) : We have, x + y = 20 ... (i) and 3x + 2y = 48.. (ii) On solving (i) and (ii), we get X = 8, y = 12. Thus, the coordinates of P are (8,12) and hence (8, 12) is one of its corner points. (v) (b): The optimal solution occurs at every point on the line joining these two points.

(i) (b): Objective function is a linear function (involve variable) whose maximum or minimum value is to be found. (ii) (c): If optimal solution is obtained at two distinct points A and B (corners of the feasible region), then optimal solution is obtained at every point of segment [AB]. (iii) (b): When x ≥ 6 and y ≥ 2, then 2x + y ≥ 2 x 6 + 2, i.e., 2x + y ≥ 14 Hence, x ≥ 0, Y ≥ 0 and 2x + y ≥ 10 are automatically satisfied by every point of the region. \(\{(x, y): x \geq 6\} \cap\{(x, y): y \geq 2\}\) (iv) (b): Construct the following table of values of the objective function:

Minimum of Z = - 32 at (0, 8) (v) (a): Construct the following table of values of the objective function F

Hence, maximum of F = 0

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