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CBSE Class 12 Maths Case Study Questions With Solutions

CBSE Class 12 Mathematics Case Study Questions are introduced this year in the updated CBSE Board Exam Pattern. According to that this year the candidates need to prepare for the case study problems along with the questions that are legacy of CBSE Board.

In Class XIIth Mathematics Case Study Questions there are problems based on Objective types of questions, Assertions and Reason, and cae based problems. These problems have the intention to examine the students' overall understanding of the subject. If you are preparing for the Maths board exam then this is the right place for you. 

Here, we have provided the complete set of CBSE class 12 maths case study questions With Solutions. These are developed by the subject matter expert. If you want to score good marks in the board papers then you should practice these Case based problems. It is absolutely free of cost. 

Download PDF CBSE Class 12 Maths Case Study From the Given links. It is in chapter wise format.

Class 12th Mathematics Important Formulas, MCQ, Case-Based, Assertion and Reason

Class 12th maths case study.

In class 12th Maths Case Study the questions are based on the real world scenarios. A passage filled with information or data is provided to the students. On the basis of that paragraph upto 5 questions are developed which should be answered by the students. To answer them they need to read the passage carefully and then pay attention to the given data to solve such questions.

These types of problems are generally known as case based questions which can be only solved by referring to the given paragraph.

Class 12 Maths Important Formulas

Knowing about the Maths Important Formulas is a crucial part of solving the Maths questions. Formulas help in solving the problems more efficiently and faster. Therefore the PDF file that we have provided here consists of all the basics and Important formulas of maths. It will help in revisions and solving the questions more accurately and easily.

Every chapter has its own topics and formulas so the PDF has been divided into chapter wise format. And if you access them from this place then you will be able to get all the formulas and other questions separately.

Class 12 Maths Assertion and Reason MCQs

Class 12th Maths Assertion and Reason MCQs PDF with Solutions are also given here. It is the most important part of Case Study Questions because scoring good marks in this section is not that much hard. If your basic concepts are clear. Because most of the time such types of questions are directly prepared by referring to the concepts.

Furthermore, the assertion and reason MCQs are solved by applying the distinct approaches. For instance, to answer these questions first candidates need to verify the Assertion (Statement) and then the reason if both are correct, then learners need to verify whether both statement and reason support each other or not.

There are a total of 5 sections in CBSE Class 12 Maths Questions Term 1. Section A contains 1 mark, Section B contains 2 marks, Section C contains 3 marks, Section D contains 4 marks and the last Section E contains 5 marks. A total of 5 questions are given in these sections.

No, In Term 1 exam there will be a total of 5 questions in each case study of class 12 maths, out of which 4 are compulsory to solve.

CBSE Board Class 12 Computer Science Answer Key 2024 and Question Papers, Download PDF All SETs

CBSE Class 12 Computer Science Exam 2024 : Important MCQs with Answers For Last Minute Revision

CBSE Board Class 12 History Answer Key 2024 and Question Papers, Download PDF All SETs

CBSE Board 12th History Exam 2024 : Chapterwise Most Important Question with Answers

CBSE Board Class 12 Business Studies Answer Key 2024 and Question Papers, Download PDF All SETs

CBSE Class 12th Exam 2024: Business Studies Important MCQs & Assertion-Reason Questions

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Unit 7: Integrals

Antiderivatives.

• Antiderivatives and indefinite integrals (Opens a modal)
• Antiderivatives and indefinite integrals Get 3 of 4 questions to level up!

Indefinite integrals intro

• Reverse power rule (Opens a modal)
• Rewriting before integrating: challenge problem (Opens a modal)
• Visually determining antiderivative (Opens a modal)
• Graphs of indefinite integrals (Opens a modal)
• Reverse power rule review (Opens a modal)
• Reverse power rule Get 3 of 4 questions to level up!
• Reverse power rule: negative and fractional powers Get 3 of 4 questions to level up!
• Reverse power rule: sums & multiples Get 3 of 4 questions to level up!

Indefinite integrals of common functions

• Indefinite integrals of sin(x), cos(x), and eˣ (Opens a modal)
• Indefinite integral of 1/x (Opens a modal)
• Particular solutions to differential equations: rational function (Opens a modal)
• Particular solutions to differential equations: exponential function (Opens a modal)
• Antiderivatives and indefinite integrals review (Opens a modal)
• Common integrals review (Opens a modal)
• Indefinite integrals: eˣ & 1/x Get 3 of 4 questions to level up!
• Particular solutions to differential equations Get 3 of 4 questions to level up!
• Indefinite integrals: sin & cos Get 3 of 4 questions to level up!
• Integrating trig functions Get 5 of 7 questions to level up!
• Indefinite integrals & antiderivatives challenge Get 5 of 7 questions to level up!

Integration by parts

• Integration by parts intro (Opens a modal)
• Integration by parts: ∫x⋅cos(x)dx (Opens a modal)
• Integration by parts: ∫ln(x)dx (Opens a modal)
• Integration by parts: ∫x²⋅𝑒ˣdx (Opens a modal)
• Integration by parts: ∫𝑒ˣ⋅cos(x)dx (Opens a modal)
• Integration by parts challenge (Opens a modal)
• Integration by parts review (Opens a modal)
• Integration by parts Get 3 of 4 questions to level up!

u-substitution

• 𝘶-substitution intro (Opens a modal)
• 𝘶-substitution: rational function (Opens a modal)
• 𝘶-substitution: multiplying by a constant (Opens a modal)
• 𝘶-substitution: logarithmic function (Opens a modal)
• 𝘶-substitution: challenging application (Opens a modal)
• 𝘶-substitution: special application (Opens a modal)
• 𝘶-substitution warmup (Opens a modal)
• 𝘶-substitution: definite integral of exponential function (Opens a modal)
• 𝘶-substitution: double substitution (Opens a modal)
• 𝘶-substitution: indefinite integrals Get 3 of 4 questions to level up!
• u-substitution challenge Get 3 of 4 questions to level up!

Reverse chain rule

• Reverse chain rule introduction (Opens a modal)
• Reverse chain rule example (Opens a modal)
• Integral of tan x (Opens a modal)
• Reverse chain rule Get 3 of 4 questions to level up!

Partial fraction expansion

• Partial fraction decomposition to evaluate integral (Opens a modal)
• Integration using long division (Opens a modal)
• Integration with partial fractions Get 3 of 4 questions to level up!

Integration using trigonometric identities

• Integral of cos^3(x) (Opens a modal)
• Integral of sin^2(x) cos^3(x) (Opens a modal)
• Integral of sin^4(x) (Opens a modal)
• Integration using trigonometric identities Get 3 of 4 questions to level up!

Trigonometric substitution

• Introduction to trigonometric substitution (Opens a modal)
• Substitution with x=sin(theta) (Opens a modal)
• More trig sub practice (Opens a modal)
• Trig and u substitution together (part 1) (Opens a modal)
• Trig and u substitution together (part 2) (Opens a modal)
• Trig substitution with tangent (Opens a modal)
• More trig substitution with tangent (Opens a modal)
• Long trig sub problem (Opens a modal)
• Trigonometric substitution Get 3 of 4 questions to level up!

Functions defined by integrals

• Worked example: Breaking up the integral's interval (Opens a modal)
• Functions defined by integrals: switched interval (Opens a modal)
• Functions defined by integrals: challenge problem (Opens a modal)
• Functions defined by definite integrals (accumulation functions) Get 3 of 4 questions to level up!
• Functions defined by integrals challenge Get 3 of 4 questions to level up!

Fundamental theorem of calculus

• The fundamental theorem of calculus and accumulation functions (Opens a modal)
• Finding derivative with fundamental theorem of calculus (Opens a modal)
• Finding derivative with fundamental theorem of calculus: x is on both bounds (Opens a modal)
• Proof of fundamental theorem of calculus (Opens a modal)
• Finding derivative with fundamental theorem of calculus Get 3 of 4 questions to level up!

Fundamental theorem of calculus: chain rule

• Finding derivative with fundamental theorem of calculus: x is on lower bound (Opens a modal)
• Fundamental theorem of calculus review (Opens a modal)
• Finding derivative with fundamental theorem of calculus: chain rule Get 3 of 4 questions to level up!

Definite integral as area

• Warmup: Definite integrals intro (Opens a modal)
• Worked example: Definite integral by thinking about the function's graph (Opens a modal)
• Finding definite integrals using area formulas Get 3 of 4 questions to level up!
• Definite integral by thinking about the function's graph Get 3 of 4 questions to level up!

Definite integral properties

• Integrating scaled version of function (Opens a modal)
• Integrating sums of functions (Opens a modal)
• Definite integral over a single point (Opens a modal)
• Definite integrals on adjacent intervals (Opens a modal)
• Definite integral of shifted function (Opens a modal)
• Switching bounds of definite integral (Opens a modal)
• Worked examples: Finding definite integrals using algebraic properties (Opens a modal)
• Definite integral properties (no graph): function combination (Opens a modal)
• Worked examples: Definite integral properties 2 (Opens a modal)
• Definite integral properties (no graph): breaking interval (Opens a modal)
• Warmup: Definite integral properties (no graph) (Opens a modal)
• Examples leveraging integration properties (Opens a modal)
• Definite integrals properties review (Opens a modal)
• Using multiple properties of definite integrals Get 3 of 4 questions to level up!
• Finding definite integrals using algebraic properties Get 3 of 4 questions to level up!

Definite integral evaluation

• Area between a curve and the x-axis (Opens a modal)
• Area between a curve and the x-axis: negative area (Opens a modal)
• Definite integral of rational function (Opens a modal)
• Definite integral of radical function (Opens a modal)
• Definite integral of trig function (Opens a modal)
• Definite integral involving natural log (Opens a modal)
• Definite integrals: reverse power rule Get 3 of 4 questions to level up!
• Definite integrals: common functions Get 3 of 4 questions to level up!
• Area using definite integrals Get 3 of 4 questions to level up!

Definite integrals of piecewise functions

• Definite integral of piecewise function (Opens a modal)
• Definite integral of absolute value function (Opens a modal)
• Definite integrals of piecewise functions Get 3 of 4 questions to level up!

Challenging definite integrals

• Challenging definite integration (Opens a modal)
• Integration by parts: definite integrals Get 3 of 4 questions to level up!
• 𝘶-substitution: definite integrals Get 3 of 4 questions to level up!

Improper integrals

• Introduction to improper integrals (Opens a modal)
• Improper integral with two infinite bounds (Opens a modal)
• Divergent improper integral (Opens a modal)
• Improper integrals review (Opens a modal)
• Improper integrals Get 3 of 4 questions to level up!
• Improper integrals challenge Get 3 of 4 questions to level up!

myCBSEguide

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• Class 12 Maths Case...

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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Class 12 Maths: Case Study Based Questions PDF Download

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You must practice some good Case Study questions of Class 12 Maths to boost your preparation to score 95+% on Boards. In this post, you will get Case Study Questions of All Chapters which will come in CBSE Class 12 Maths Board Exams.

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

We have provided here Case Study questions for the Class 12 Maths exams. You can read these chapter-wise Case Study questions. 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 Board Final exams.

We are providing Case Study questions for class 12 Biology based on the latest syllabi. There is a total of 13 chapters included in CBSE class 12 Maths exams. Students can practice these questions for concept clarity and score better marks in their exams.

Table of Contents

CBSE Class 12th – MATHS : Chapterwise Case Study Question & Solution

CBSE will ask two Case Study Questions in the CBSE class 12 maths questions paper. Question numbers 15 and 16 are case-based questions where 5 MCQs will be asked based on a paragraph. Each theme will have five questions and students will have a choice to attempt any four of them.

Case Study Based Questions for Class 12 Maths

Class 12 Physics Case Study Questions Class 12 Chemistry Case Study Questions Class 12 Biology Case Study Questions Class 12 Maths Case Study Questions

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.

Class 12 Maths Syllabus 2022-23

Unit-i: relations and functions.

1. Relations and Functions (15 Periods)

Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto functions.

2. Inverse Trigonometric Functions (15 Periods)

Definition, range, domain, principal value branch. Graphs of inverse trigonometric functions.

Unit-II: Algebra

1. Matrices (25 Periods)

Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew symmetric matrices. Operation on matrices: Addition and multiplication and multiplication with a scalar. Simple properties of addition, multiplication and scalar multiplication. Oncommutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order 2). Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).

2. Determinants 25 Periods

Determinant of a square matrix (up to 3 x 3 matrices), minors, co-factors 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.

Unit-III: Calculus

1. Continuity and Differentiability (20 Periods)

Continuity and differentiability, chain rule, derivative of inverse trigonometric functions, 𝑙𝑖𝑘𝑒 sin −1  𝑥 , cos −1  𝑥 and tan −1  𝑥, derivative of implicit functions. Concept of exponential and logarithmic functions. Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives.

2. Applications of Derivatives (10 Periods)

Applications of derivatives: rate of change of bodies, increasing/decreasing functions, 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 reallife situations).

3. Integrals (20 Periods)

Integration as inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, Evaluation of simple integrals of the following types and problems based on them.

Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.

4. Applications of the Integrals (15 Periods)

Applications in finding the area under simple curves, especially lines, circles/ parabolas/ellipses (in standard form only)

5. Differential Equations (15 Periods)

Definition, order and degree, general and particular solutions of a differential equation. Solution of differential equations by method of separation of variables, solutions of homogeneous differential equations of first order and first degree. Solutions of linear differential equation of the type:

Unit-IV: Vectors and Three-Dimensional Geometry

1. Vectors (15 Periods)

Vectors and scalars, magnitude and direction of a vector. Direction cosines and direction ratios of a vector. 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. Definition, Geometrical Interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors.

2. Three – dimensional Geometry (15 Periods)

Direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation of a line, skew lines, shortest distance between two lines. Angle between two lines.

Unit-V: Linear Programming

1. Linear Programming (20 Periods)

Introduction, related terminology such as constraints, objective function, optimization, graphical method of solution for problems in two variables, feasible and infeasible regions (bounded or unbounded), feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints).

Unit-VI: Probability

1. Probability 30 (Periods)

Conditional probability, multiplication theorem on probability, independent events, total probability, Bayes’ theorem, Random variable and its probability distribution, mean of random variable.

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Chapter 7 Class 12 Integrals

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Get NCERT Solutions of Class 12 Integration, Chapter 7 of the NCERT book . Solutions of all questions, examples and supplementary questions explained here. Download formulas and practice questions as well.

Topics include

• Integration as anti-derivative - Basic definition of integration. Using derivative rules, finding integration
• Integration using Trigonometry Formulas - where we use trigonometry formulas like cos 2x, sin 2 x, sin 3x, Inverse formulas and make the function easier to integrate
• Integration by substitution - Where we substitute functions as some other functions and integrate using the formulas we know - x n , lnx, e x to find the integration
• Integration by parts - We do integration by using by parts formula
• By parts integration of e x - We use integration formula of e x (f(x) + f'(x)) to solve questions
• Integration by partial fractions - We use partial fractions to solve the integration. Like in this question .
• Integration by special formulas - We use special formulas mentioned in our Integral Table to solve questions
• Integration as limit as a sum - We use basic definition of integration , Integration = Area to form limit as a sum formula and then solve its questions
• Definite Integration - Solving definite integration using methods of indefinite integration, and using properties of definite integration

To check all formulas of Integrals used in this chapter, check Integration Formulas

Important Questions are marked as Important , you can also check all Important Questions for Class 12 Maths . As well as Sample papers .

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Class 12th Maths - Application of Integrals Case Study Questions and Answers 2022 - 2023

By QB365 on 08 Sep, 2022

QB365 provides a detailed and simple solution for every Possible Case Study Questions in Class 12 Maths Subject - Application of Integrals, CBSE. It will help Students to get more practice questions, Students can Practice these question papers in addition to score best marks.

QB365 - Question Bank Software

Application of integrals case study questions with answer key.

12th Standard CBSE

Final Semester - June 2015

Consider the following equations of curves : x ? - = y and y = x. On the basis of above information, answer the following questions. (i) The point(s) of intersection of both the curves is (are)

(iv) The value of the integral  \(\int_{0}^{1} x^{2} d x\)  

(v) The value of area bounded by the curves x ? - = y and x = y is

Consider the curve x 2 +y 2 = 16 and line y = x in the first quadrant. Based on the above information, answer the following questions. (i) Point of intersection of both the given curves is

(iv) The value of the integral  \(\int_{2 \sqrt{2}}^{4} \sqrt{16-x^{2}} d x\)  is

(v) Area bounded by the two given curves is

(iv) Area of each slice of pizza when child cut the pizza into 4 equal pieces is

(v) Area of whole pizza is

Consider the following equation of curve I' = 4x and straight line x + y = 3. Based on the above information, answer the following questions. (i) The line x + y = 3 cuts the x-axis and y-axis respectively at

(ii) Point(s) of intersection of two given curves is (are)

(v) Value of area bounded by given curves is

(ii) Area of curve explained in the passage from 0 to  \(\frac{\pi}{2}\)  is

(iii) Area of curve discussed in classroom from  \(\frac{\pi}{2} \text { to } \frac{3 \pi}{2}\)  is

(iv) Area of curve discussed in classroom from  \(\frac{3 \pi}{2} \text { to } 2 \pi\)  is

(v) Area of explained curve from 0 to  \(2 \pi\)  is

(ii) Value of  \(\int_{0}^{\pi / 4} \sin x d x\)  is

(iii) Value of  \(\int_{\pi / 4}^{\pi / 2} \cos x d x\)  is

(iv) Value of  \(\int_{0}^{\pi} \sin x d x\)  is

(v) Value of  \(\int_{0}^{\pi / 2} \sin x d x\)  is

(ii) Equation of line BC is

(iii) Area of region ABCD is

(iv) Area of  \(\Delta A D C\)  is

(iv) Area of \(\Delta A B C\)  is

(iv) Value of  \(\int_{1 / 2}^{1} \sqrt{1-x^{2}} d x\)  is

(v) Area of hidden portion of lower circle is

(iv) The value of  \(2 \int_{0}^{3}\left(1-\frac{x}{3}\right) d x\)  is 

(v) Area of the smaller region bounded by the mirror and scratch is

Consider the following equations of curves y = cos x, y = x + 1 and y = 0. On the basis of above information, answer the following questions. (i) The curves y = cos x and y = x + 1 meet at

(ii) y = cos x meet the x-axis at

(iii) Value of the integral  \(\int_{-1}^{0}(x+1) d x\)  is

(iv) Value of the integral  \(\int_{0}^{\pi / 2} \cos x d x\)  is

(v) Area bounded by the given curves is

*****************************************

Application of integrals case study questions with answer key answer keys.

(i) (a) : Here, teacher explained about cosine curve. (ii) (c) : Required area  \(\int_{0}^{\pi / 2} \cos x d x\)   \(=[\sin x]_{0}^{\pi / 2}=\sin \frac{\pi}{2}-\sin 0=1-0=1 \text { sq. unit }\)   (iii) (b) : Required area =  \(\left|\int_{\pi / 2}^{3 \pi / 2} \cos x d x\right|=\left|[\sin x]_{\pi / 2}^{3 \pi / 2}\right|\)   \(=\left|\sin \frac{3 \pi}{2}-\sin \frac{\pi}{2}\right|=|-1-1|=|-2|\)   = 2 sq. units [Since, area can't be negative] (iv) (a) : Required area =  \(\int_{3 \pi / 2}^{2 \pi} \cos x d x=[\sin x]_{3 \pi / 2}^{2 \pi}\)   \(=\sin 2 \pi-\sin \frac{3 \pi}{2}=0-(-1)=1 \text { sq. unit }\)   (v) (d) : Required area \(=\int_{0}^{\pi / 2} \cos x d x+\left|\int_{\pi / 2}^{3 \pi / 2} \cos x d x\right|+\int_{3 \pi / 2}^{2 \pi} \cos x d x\)   = 1 + 2 + 1 = 4 sq. units.

(i) (c) : For point of intersection, we have sin x = cos x \(\Rightarrow \frac{\sin x}{\cos x}=1 \Rightarrow \tan x=1 \Rightarrow x=\frac{\pi}{4}\)   (ii) (a) :  \(\int_{0}^{\pi / 4} \sin x d x=[-\cos x]_{0}^{\pi / 4}=-\cos \frac{\pi}{4}+\cos 0\)   \(=1-\frac{1}{\sqrt{2}}\)   (iii) (b) :  \(\int_{\pi / 4}^{\pi / 2} \cos x d x=[\sin x]_{\pi / 4}^{\pi / 2}=\sin \frac{\pi}{2}-\sin \frac{\pi}{4}\)   \(=1-\frac{1}{\sqrt{2}}\)   (iv) (c) :   \(\int_{0}^{\pi} \sin x d x=[-\cos x]_{0}^{\pi}=[-\cos \pi+\cos 0]=2\)   (v) (b) :   \(\int_{0}^{\pi / 2} \sin x d x=[-\cos x]_{0}^{\pi / 2}=\left[-\cos \frac{\pi}{2}+\cos 0\right]\)  = 0+1=1

(i) (a) : Equation of line AB is \(y-0=\frac{3-0}{1+1}(x+1) \Rightarrow y=\frac{3}{2}(x+1)\)   (ii) (c) : Equation of line BC is  \(y-3=\frac{2-3}{3-1}(x-1)\)   \(\Rightarrow y=-\frac{1}{2} x+\frac{1}{2}+3 \Rightarrow y=\frac{-1}{2} x+\frac{7}{2}\)   (iii) (d) : Area of region ABCD = Area of \(\triangle A B E\) + Area of region BCDE \(=\int_{-1}^{1} \frac{3}{2}(x+1) d x+\int_{1}^{3}\left(\frac{-1}{2} x+\frac{7}{2}\right) d x\)   \(=\frac{3}{2}\left[\frac{x^{2}}{2}+x\right]_{-1}^{1}+\left[\frac{-x^{2}}{4}+\frac{7}{2} x\right]_{1}^{3}\)   \(=\frac{3}{2}\left[\frac{1}{2}+1-\frac{1}{2}+1\right]+\left[\frac{-9}{4}+\frac{21}{2}+\frac{1}{4}-\frac{7}{2}\right]\)   = 3 + 5 = 8 sq. units (iv) (a) : Equation of line AC is  \(y-0=\frac{2-0}{3+1}(x+1)\)   \(\Rightarrow y=\frac{1}{2}(x+1)\)   \(\therefore \text { Area of } \Delta A D C=\int_{-1}^{3} \frac{1}{2}(x+1) d x=\left[\frac{x^{2}}{4}+\frac{1}{2} x\right]_{-1}^{3}\)   \(=\frac{9}{4}+\frac{3}{2}-\frac{1}{4}+\frac{1}{2}=4 \text { sq. units }\)   (v) (b) : Area of  \(\Delta A B C\) = Area of region ABCD - Area of  \(\Delta A C D=8-4=4 \mathrm{sq} . \text { units }\)

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• Application Of Integrals For Class 12

Application of Integrals for Class 12

From the previous classes, we are already aware of the various method of integration in Maths.  In the class 12 maths chapter 2, students will learn a specific application of integrals to find the area under simple curves, area between lines and arcs of standard forms of different curves such as circles, parabolas and ellipses. We shall also deal with finding the area bounded by these curves along with the combination of lines. In this article, let us look at the various application of integrals Class 12 Maths Chapter 8 .

Application of Integrals Class 12 Concepts

Chapter 8 – Application of Integrals class 12  CBSE covers the following concepts:

• Introduction
• Areas and Simple curves
• The area of a region bounded by a curve and a line
• The area between the two curves

From the previous classes, we are already aware of the various method of integration in maths.  In this article, let us look at the various application of integrals class 12 Maths .

One of the major application of integrals is in determining the area under the curves.

Consider a function y = f(x), then the area is given as

Consider the two curves having equation of f(x) and g(x), the area between the region a,b of the two curves is given as-

dA = f(x) – g(x)]dx, and the total area A can be taken as-

Application of Integrals Examples

Example 1: 

Determine the area enclosed by the circle x 2 + y 2 = a 2

Given, circle equation is x 2 + y 2 = a 2

From the given figure, we can say that the whole area enclosed by the given circle is as

= 4(Area of the region AOBA bounded by the curve, coordinates x=0 and x=a, and the x-axis)

As the circle is symmetric about both x-axis and y-axis, the equation can be written as 

= 4 0 ∫ a y dx (By taking the vertical strips) ….(1)

From the given circle equation, y can be written as

y = ±√(a 2 -x 2 )

As the region, AOBA lies in the first quadrant of the circle, we can take y as positive, so the value of y becomes √(a 2 -x 2 )

Now, substitute y = √(a 2 -x 2 ) in equation (1), we get

= 4 0 ∫ a √(a 2 -x 2 ) dx 

Integrate the above function, we get

= 4 [(x/2)√(a 2 -x 2 ) +(a 2 /2)sin -1 (x/a)] 0 a

Now, substitute the upper and lower limit, we get

= 4[{(a/2)(0)+(a 2 /2)sin -1 1}-{0}]

= 4(a 2 /2)(π/2)

Hence, the area enclosed by the circle x 2 +y 2 =a 2  is πa 2 .

Let’s have a look at the example to understand how to find the area of the region bounded by a curve and a line.

Find the area of the region bounded between the line x = 2 and the parabola y 2 = 8x.

Given equation of parabola is y 2 = 8x.

Equation of line is x = 2.

y 2 = 8x has only even power of y and is symmetrical about x-axis.

So, the required area = Area of OAC + Area of OAB

= 2 (Area of OAB)

= 2 ∫ 0 2 y dx

Substituting the value of y, i.e. y2 = 8x and y = √(8x) = 2 √2 √x, we get;

= 2 ∫ 0 2 (2 √2 √x) dx

= 4√2 ∫ 0 2 (√x) dx

= 4√2 [x 3/2 / (3/2)] 0 2

By applying the limits,

= 4√2 {[2 3/2 / (3/2)] – 0}

= (8√2/3)  × 2√2

= (16 × √2 × √2)

Go through the example given below to learn how to find the area between two curves.

Determine the area which lies above the x-axis and included between the circle and parabola, where the circle equation is given as x 2 +y 2 = 8x, and parabola equation is y 2 = 4x.

The circle equation x 2 +y 2 = 8x can be written as (x-4) 2 +y 2 =16. Hence, the centre of the circle is (4, 0), and the radius is 4 units. The intersection of the circle with the parabola y 2 = 4x is as follows:

Now, substitute y 2 = 4x in the given circle equation,

x 2 +4x = 8x

x 2 – 4x = 0

On solving the above equation, we get

x=0 and x=4

Therefore, the point of intersection of the circle and the parabola above the x-axis is obtained as O(0,0) and P(4,4).

Hence, from the above figure, the area of the region OPQCO included between these two curves above the x-axis is written as

= Area of OCPO + Area of PCQP

= 0 ∫ 4 y dx + 4 ∫ 8 y dx

= 2 0 ∫ 4 √x dx + 4 ∫ 8  √[4 2 – (x-4) 2 ]dx

Now take x-4 = t, then the above equation is written in the form 

= 2 0 ∫ 4 √x dx + 0 ∫ 4  √[4 2 – t 2 ]dx …. (1)

Now, integrate the functions.

2 0 ∫ 4 √x dx = (2)(⅔) (x 3 /2 ) 0 4

2 0 ∫ 4  √x dx = 32/3  …..(2)

0 ∫ 4  √[4 2 – t 2 ]dx = [(t/2)(√[4 2 -t 2 ] + (½)(4 2 )(sin -1 (t/4)] 0 4

0 ∫ 4 √[4 2 – t 2 ]dx  = 4π …..(3)

Now, substitute (2) and (3) in (1), we get

= (32/3) +   4π 

= (4/3) (8+3π) 

Therefore, the area of the region that lies above the x-axis, and included between the circle and parabola is (4/3) (8+3π).

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• Sep 1, 2023

Integrals Made Easy-A Class 12 Student's Perspective

Updated: Apr 5

How to learn Integral Calculus for Class 12 Mathematics

I am a mathematics educator and teacher having taught Maths for three decades to High school and college students. You may be a little wary of Integration, but if you follow the techniques outlined here, I assure you that you'll sail through Integration. ​ ​ So, let's learn integral Calculus for Class 12 Maths.

How to start learning Integration for Class 12?

Basically, integration and differentiation are a pair of inverse operations. Integral of a function is also called the anti derivative of the function. You'll start with the study of Indefinite integrals. There are a few basic formulas of Integration which you need to start with.

Basic Integration Formulas for a Quick Revision

MCQ-Integrals

Lesson on Basic Integration

Evaluate this Integral CUET Question 1

Evaluate this algebraic integral ​ ​What is the Substitution method in Integration?

Note that for indefinite integrals, we add the constant c, which is just the integration constant. The most basic method of Integration is using the Substitution method. Once you master this method, then you can basically solve any problem on Integration.

MCQ-Substitution method in Integration

MCQ-Substitution method

Substitution method in Integration lesson

MCQ-Integration- More problems on the Substitution method.

​ What happens when the degree of the Numerator is greater than or equal to degree of the denominator? ​

Find k, Integrals CUET

Next, what you'll need to know is how to integrate quotient functions. There are two cases here. Firstly when the degree of the numerator is greater than or equal to the degree of the denominator. ​ ​ In this case, you divide the numerator by the denominator and write the quotient and remainder. You will then proceed to integrate using the formula given. The methods are illustrated in my video links below.

How to integrate if degree of the Numerator is greater than or equal to the Denominator

What are Partial Fractions in Integration?

In case, you still need help, you can contact me for an online session on Integration. When the degree of the numerator is lesser than the degree of the denominator, you'll use the method of partial fractions. Again, keep in mind, that there are at least 4 cases which you need to know, that is, when the roots are distinct, repeated or complex.

MCQ-Evaluating Integrals

IMPROVE YOUR RESULTS -LEARN PARTIAL FRACTIONS IN INTEGRATION

Tips on Integral Calculus

​ ​ You'll then move on to learn four standard integrals, namely tan, cot, cosec and sec and solve problems based on these.You also have special cases of the form linear/quadratic or linear/sqrt(quadratic). These two are more or less similar in solving.

MCQ-Special Integrals

MCQ-Trigonometric Integrals

MCQ-Special Integrals 2

Special Integrals Video

​ How will you integrate Trigonometric Functions? ​

Next, you'll learn how to integrate rational functions of sinx and cosx . As a special type you'll also learn how to integrate functions of the form (asinx+bcosx)/(csinx+dcosx). There's another special type of integral where only even powers of x occur both in the numerator and denominator. Basically, you have degree two in the numerator and degree 4 in the denominator.

Integration Question ​ ​

What is Integration by parts?

The next important topic that you'll learn is Integration by parts. Note that you need to remember the order of the functions while integrating by parts. ILATE OR LIATE. There's an exception to this rule and that's discussed in the links below.There's also a special form of integration by parts, involving the exponential function which is equally important.

Application of Integration by parts ​ ​

How to use limits to calculate integral as a sum?

There are three more standard integrals which are useful while solving problems involving area of bounded regions. We now move to Definite Integrals. You'll learn how to integrate definite integral as the limit of a sum.

Tips on how to understand Integral as the limit of a Sum ​ ​ As of now, this topic has been excluded from the syllabus for 2024. You'll learn the Fundamental Theorem of Integral Calculus, using which you'll learn to evaluate Definite Integrals. You'll see how the Substitution method is applied for definite Integrals.

​ ​ Keep in mind that when you use the Substitution method in definite integrals, the limit also changes. Last but not the least, you'll learn Properties of Definite Integrals. These properties are extensively used in problem solving.

MCQ on integral of an odd function

Question on Definite Integrals ​ ​ These are the basic topics that you'll need to learn in Integration. I have provided a number of free resources, in the form of videos and multiple choice questions below. Feel free to make use of these. ​ ​

What if you still need that little extra help?

In case you need more help, you can contact me for online tutoring. You can choose specific topics or the entire syllabus. Again, if you have any fellow students from your school, you can bring them along. ​ ​ I can help prepare you for the conceptual and case study based questions . My students have always excelled in Maths. So, let's start learning!

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Case study chapter 8 (application of integral ).

Case study 3: Read the following and answer the question(Case study application of integral 3) A student designs an open air Honeybee nest on the branch of a tree, whose plane figure is parabolic and the branch of tree is given by a straight line.

(i) Point of intersection of the parabola and straight line are

(a) (4, 0) and (-4, 0)           (b) (4, -4) and (4, 0)

(c) (-4, 4) and (4, 4)            (d) (2, 4) and (-2, 4)

(ii) Length of each horizontal strip of the bounded region is given by

(iii) Length of each vertical strip is given by

(c) 4                                           (d) None of these

(c) 64/3                                   (d) 128/3

(v) Area of each vertical strip is given by

Solution: (i) Answer (c)

Given equation of parabola is

x² = 4 y  —–(i)

And equation of straight line y = 4

∴ From (i), we get

x² = 4×4 = 16

∴ Point of intersection are (4, 4) and (-4, 4)

(ii) Answer (c)

x² = 4 y  —(i)

Length of horizontal strip be = 2×2√y = 4√y

(iii) Answer (a)

(iv) Answer (c)

Area of required bounded region

(v) Answer (b)

Area of each(one) vertical strip

Some Other Case study problem

Case study 1: Read the following and answer the question.(Case study application of integral 1)

Nowadays, almost every boat has a triangular sail. By using a triangular sail design it has become possible to travel against the wind using a technique known as tacking. Tacking allows the boat to travel forward with r triangular sail on the walls and three edges(lines) at the triangular sail are given by the equation x = 0, y = 0 and y + 2x – 4 = 0 respectively.

Solution: For solution click here

Case study 2: Read the following and answer the question(Case study application of integral 2)

Case study 4: Read the following and answer the question

A boy design a pizza by cutting it with a knife on a card board. If pizza is circular in shape which is represented by the

Solution: for solution click here

Case study 5:-A farmer has a triangular shaped field. His, son a science student observes the triangular field has three edges and can be drawn on a plain paper with three lines given by its equations.(Case study application of integral 5)

Based on the above information answer the following question:

(i) Find the area of the shaped region in the figure shown below.

(ii) Find the area of the triangle  ΔABC.

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Class 12 CBSE Applied Maths ML AGGARWAL Integrals Exercise 7.1

Please select, determine the order and the degree (when defined) of each of the following (1 to 9) differential equations:.

View Solution