case study based questions class 12 maths pdf

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.

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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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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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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 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?

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CBSE Case Study Class 12

While preparing for the board exams, students are being judged on different levels of skills, such as writing, reading, etc. CBSE Case Study Class 12 questions are one of them that helps in assessing critical thinking.

The Central Board of Secondary Education will be asking the case study questions in the Class 12 board examination. Therefore, here on this page, we have provided the CBSE Case Study Class 12 for Maths, Physics, Chemistry, Biology and other subjects. Our subject matter experts have prepared Case Study questions so that Apart from the basic standard questions, students can have a variety of problems to solve. 

Just like MCQs, and other written types questions CBSE Case Study Class 12 questions will impact the overall performance of a student. Therefore for the convenience of the students we have provided the download links here, so that they can easily access the Class 12 Case Study.

Download Subject Wise CBSE Case Study Class 12 Question and Answers PDF 

There are three different streams in class 12th: Science, Arts & Commerce. And all these streams have a variety of subjects therefore, here our subject experts have crafted the Subject Wise Case Study For Class 12. Download Subject Wise CBSE Case Study Class 12 Question and Answers PDF from the below given links.

CBSE Case Study Questions Class 12 for Maths

To learn how to answer CBSE Case Study Class 12 maths questions, it would be a good choice to refer to the separate set of questions in PDF. CBSE Case Study Questions for Class 12 Maths can be quite useful and helpful in understanding how the board will prepare the case based questions for maths. 

The Maths Case studies are distributed into different chapters as per the prescribed syllabus of maths. Each and every chapter contains several number of case studies problems which enable the learners to gain in-depth knowledge in the subject.

The maths case study questions of class 12 are usually based on the formula. If any student wants to get a good score in maths case studies then they have to be well versed with a set of formulas as per the chapters and topics. Case Study Class 12 Maths can be downloaded here on this website for free of cost.

Case Study Questions Class 12 Chapter 1 Relations And Functions

Case Study Questions Class 12 Chapter  3 Matrices

Case Study Questions Class 12 Chapter  4 Determinants

Case Study Questions Class 12 Chapter  5 Continuity And Differentiability

Case Study Questions Class 12 Chapter  6 Application Of Derivatives

Case Study Questions Class 12 Chapter  8 Application Of Integrals

Case Study Questions Class 12 Chapter  9 Differential Equations

Case Study Questions Class 12 Chapter  10 Vector Algebra

Case Study Questions Class 12 Chapter  11 Three Dimensional Geometry

Case Study Questions Class 12 Chapter  12 Linear Programming

Case Study Questions Class 12 Chapter  13 Probability

CBSE Case Study Questions Class 12 for Physics

CBSE Class 12 Physics Case Study for different chapters can be interesting to solve. Class 12th physics case studies generally contain descriptive paragraphs and 4 to 5 different questions are given based on the paragraph. Students are required to read the paragraph thoroughly and understand them as well as observe the given data and information to answer the Class 12 Case Study Questions for Physics. 

In order to solve those questions students are required to have a better understanding of the basic level of physics concepts too, so make sure to use Class 12 Revision Notes of Physics. 

Case Study Questions Class 12 Chapter 1 Electric Charges and Fields

Case Study Questions Class 12 Chapter 2 Electrostatic Potential And Capacitance

Case Study Questions Class 12 Chapter 3 Current Electricity

Case Study Questions Class 12 Chapter 4 Moving Charges And Magnetism

Case Study Questions Class 12 Chapter 5 Magnetism And Matter

Case Study Questions Class 12 Chapter 6 Electromagnetic Induction

Case Study Questions Class 12 Chapter 7 Alternating Current

Case Study Questions Class 12 Chapter 8 Electromagnetic Waves

Case Study Questions Class 12 Chapter 9 Ray Optics & Optical Instruments

Case Study Questions Class 12 Chapter 10 Wave Optics

Case Study Questions Class 12 Chapter 11 Dual Nature Radiation & Matter

Case Study Questions Class 12 Chapter 12 Atoms

Case Study Questions Class 12 Chapter 13 Nuclei

Case Study Questions Class 12 Chapter 14 Semiconductor Electronics - Materials, Devices & Simple Circuits

CBSE Case Study Questions Class 12 for Chemistry

Students opted for Science stream will have to study CBSE Case Study Class 12 Chemistry. The given PDF of CBSE Class 12 Case Study Questions for Chemistry will enable the students to practice tons of questions on a regular basis. It will help them to revise the chemistry questions and its syllabus frequently. The Case study questions allow students to think more creative and find the answers in the quickest way possible. Links to download the PDF file of Class 12 Chemistry Case Study are given here.

Case Study Questions Class 12 Chapter 1 The Solid State

Case Study Questions Class 12 Chapter  2 Solutions

Case Study Questions Class 12 Chapter  3 Electrochemistry

Case Study Questions Class 12 Chapter  4 Chemical Kinetics

Case Study Questions Class 12 Chapter  5 Surface Chemistry  

Case Study Questions Class 12 Chapter  7 The p-Block Elements

Case Study Questions Class 12 Chapter  8 The d- and f-Block Elements

Case Study Questions Class 12 Chapter  9 Coordination Compounds

Case Study Questions Class 12 Chapter  10 Haloalkanes And Haloarenes

Case Study Questions Class 12 Chapter 11 Alcohols, Phenols And Ethers

Case Study Questions Class 12 Chapter  12 Aldehydes. Ketones & Carboxylic Acids

Case Study Questions Class 12 Chapter  13 Amines

Case Study Questions Class 12 Chapter 14 Biomolecules

CBSE Case Study Questions Class 12 for Biology

There are lots of chapters in class 12 Biology from which Class 12 Case Study Questions can be framed. Going through such types of questions help the students to assess their understanding level in the topics discussed in NCERT Class 12 Biology Books. By practicing the Class 12 Case Study Questions for Biology students will be very confident to ace the board exam. Also the Biology case study will be very useful for the NEET exam preparation.

Doing a regular practice of Class 12 Biology Case Study questions is a great way to score higher marks in the board exams as it will help students to develop a grip on the concepts. 

Case Study Questions Class 12 Chapter 2 Sexual Reproduction in Flowering Plants

Case Study Questions Class 12 Chapter 3 Human Reproduction

Case Study Questions Class 12 Chapter 4 Reproductive Health

Case Study Questions Class 12 Chapter 5 Principles of Inheritance & Variation

Case Study Questions Class 12 Chapter 6 Molecular Basic of Inheritance

Case Study Questions Class 12 Chapter 8 Human Health And Diseases

Case Study Questions Class 12 Chapter 10 Microbes in Human Welfare

Case Study Questions Class 12 Chapter 11 Biotechnology - Principles & Processes

Case Study Questions Class 12 Chapter 12 Biotechnology And Its Applications

Case Study Questions Class 12 Chapter 13 Organisms And Populations

Case Study Questions Class 12 Chapter 15 Biodiversity And Conversation

Case study types of questions are generally descriptive that helps to gather more information easily so, it is kinda easy to answer. However, our subject matter experts have given the solutions of all the CBSE Case Study Class 12 Biology questions.

Passage Based Class 12 Case Study Questions in PDF

CBSE Class 12 Case studies are known as Passage Based Questions. These types of problems usually contain a short/long paragraph with 4 to 5 questions. 

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CBSE 12th Standard Maths Subject Differential Equations 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 differential equations case study questions 2021.

12th Standard CBSE

Final Semester - June 2015

Case Study Questions

(ii) fP o be the initial principal, then find the solution of differential equation formed in given situation.

(iii) If the interest is compounded continuously at 5% per annum, in how many years will Rs. 100 double itself?

(iv) At what interest rate will Rs.100 double itself in 10 years? (log e 2 = 0.6931).

(v) How much will Rs. 1000 be worth at 5% interest after 10 years? (e 0.5 = 1.648).

(ii)  \(\frac{d y}{d t}\)  is proptional to 

(iii) The value of y(0) is

(iv) The value of y(2) is

(v) The value of y at any time t is given by

Order: The order of a differential equation is the order of the highest order derivative appearing in the differential equation. Degree : The degree of differential equation is the power of the highest order derivative, when differential coefficients are made free from radicals and fractions. Also, differential equation must be a polynomial equation in derivatives for the degree to be defined. Based on the above information, answer the following questions. (i) Find the degree of the differential equation  \(2 \frac{d^{2} y}{d x^{2}}+3 \sqrt{1-\left(\frac{d y}{d x}\right)^{2}-y}=0\)  

(ii) Order and degree of the differential equation  \(y \frac{d y}{d x}=\frac{x}{\frac{d y}{d x}+\left(\frac{d y}{d x}\right)^{3}}\)  are respectively

(iii) Find order and degree of the equation  \(y^{\prime \prime \prime}+y^{2}+e^{y^{\prime}}=0\)  

(iv) Determine degree of the differential equation  \((\sqrt{a+x}) \cdot\left(\frac{d y}{d x}\right)+x=0\)  

(v) Order and degree of the differential equation  \(\left(1+\left(\frac{d y}{d x}\right)^{3}\right)^{\frac{7}{3}}=7 \frac{d^{2} y}{d x^{2}}\)  are respectively

If the equation is of the form  \(\frac{d y}{d x}=\frac{f(x, y)}{g(x, y)} \text { or } \frac{d y}{d x}=F\left(\frac{y}{x}\right)\)  ,wheref (x, y), g(x, y) are homogeneous functions of the same degree in x and y, then put y = vx and  \(\frac{d y}{d x}=v+x \frac{d v}{d x}\) , so that the dependent variable y is changed to another variable v and then apply variable separable method. Based on the above information, answer the following questions. (i) The general solution of  \(x^{2} \frac{d y}{d x}=x^{2}+x y+y^{2}\)  is

(ii) Solution of the differential equation  \(2 x y \frac{d y}{d x}=x^{2}+3 y^{2} \)   is

(iii) Solution of the differential equation  \(\left(x^{2}+3 x y+y^{2}\right) d x-x^{2} d y=0\)   is 

(iv) General solution ofthe differential equation   \(\frac{d y}{d x}=\frac{y}{x}\left\{\log \left(\frac{y}{x}\right)+1\right\}\)  is

(v) Solution ofthe differential equation  \(\left(x \frac{d y}{d x}-y\right) e^{\frac{y}{x}}=x^{2} \cos x\)   is

If the equation is of the form  \(\frac{d y}{d x}+P y=Q\)   , where P, Q are functions of x, then the solution of the differential  equation is given by  \(y e^{\int P d x}=\int Q e^{\int P d x} d x+c\) , where  \(e^{\int P d x}\)  is called the integrating factor (I.F.). Based on the above information, answer the following questions. (i) The integrating factor of the differential equation \(\sin x \frac{d y}{d x}+2 y \cos x=1 \text { is }(\sin x)^{\lambda}, \text { where } \lambda=\)

(ii) Integrating  factor of the differential equation  \(\left(1-x^{2}\right) \frac{d y}{d x}-x y=1 \)  is

(iii) The solution of  \(\frac{d y}{d x}+y=e^{-x}, y(0)=0\)   is

(iv) General solution of  \(\frac{d y}{d x}+y \tan x=\sec x\)  is

(v) The integrating factor of differential equation  \(\frac{d y}{d x}-3 y=\sin 2 x\)   is

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Cbse 12th standard maths differential equations case study questions 2021 answer keys.

(i) (b) : Here, P denotes the principal at any time t and the rate of interest be r% per annum compounded continuously, then according to the law given in the problem, we get \(\frac{d P}{d t}=\frac{P r}{100}\)   (ii) (a) : We have, \(\frac{d P}{d t}=\frac{\operatorname{Pr}}{100}\)   \(\Rightarrow \frac{d P}{P}=\frac{r}{100} d t \Rightarrow \int \frac{1}{P} d P=\frac{r}{100} \int d t\)   \(\Rightarrow \log P=\frac{r t}{100}+C\)   At t = 0, P = P o \(\therefore \quad C=\log P_{0}\)   So,  \(\log P=\frac{r t}{100}+\log P_{0}\)   \(\Rightarrow \log \left(\frac{P}{P_{0}}\right)=\frac{r t}{100}\)   (iii) (c) : We have, r = 5, P o = Rs. 100 and P = Rs. 200 = 2P o Substituting these values in (2), we get \(\log 2=\frac{5}{100} t\)   \(\Rightarrow\)  t = 20 log e 2 = 20 x 0.6931 years = 13.862 years (iv) (d) : We have P o = Rs. 100, P = Rs. Rs. 200 = 2P o and t = 10 years Substituting these values in (2), we get \(\log 2=\frac{10 r}{100} \Rightarrow r=10 \log 2=10 \times 0.6931=6.931\)   (v) (a) : We have P o = Rs. 1000, r = 5 and t = 10 Substituting these values in (2), we get \(\log \left(\frac{P}{1000}\right)=\frac{5 \times 10}{100}=\frac{1}{2}=0.5 \Rightarrow \frac{P}{1000}=e^{0.5}\)   \(\Rightarrow\)  P = 1000 x 1.648 = Rs. 1648

(i) (c) : Since, size of population is 5000. \(\therefore\)  Maximum value of y(t) is 5000. (ii) (d) : Clearly, according to given information \(\frac{d y}{d t}=k y(5000-y)\)  ,where k is the constant of proportionality. (iii) (a): Since, rumour is initiated with 100 people. \(\therefore\)  When t = 0, then y = 100 Thus y(O) = 100 (iv) (b) : Since, rumour is spread in 500 people, after 2 days. \(\therefore\)  When t = 2, then y = 500. Thus, y(2) = 500 (v) (c) : We know that, when t = 0, then y = 100 This condition is satisfied by option (c) only.

(i) (c) : We have,  \(2 \frac{d^{2} y}{d x^{2}}+3 \sqrt{1-\left(\frac{d y}{d x}\right)^{2}-y}=0\)   \(\therefore \quad 2 \frac{d^{2} y}{d x^{2}}=-3 \sqrt{1-\left(\frac{d y}{d x}\right)^{2}-y}\)   Squaring both sides, we get \(4\left(\frac{d^{2} y}{d x^{2}}\right)^{2}=9\left[1-\left(\frac{d y}{d x}\right)^{2}-y\right]\)   Here, highest order derivative is  \(\frac{d^{2} y}{d x^{2}}\)  and its power is 2. So, its degree is 2. (ii) (d) : We have,  \(y \frac{d y}{d x}=\frac{x}{\frac{d y}{d x}+\left(\frac{d y}{d x}\right)^{3}}\)   \(\Rightarrow y\left(\frac{d y}{d x}\right)^{2}+y\left(\frac{d y}{d x}\right)^{4}=x\)   \(\Rightarrow\)  Here, highest order derivative is  \(\frac{d y}{d x}\)  is So , its order is 1 and degree is 4. (iii) (a) : We have, \(y^{\prime \prime \prime}+y^{2}+e^{y^{\prime}}=0\)   \(\frac{d^{3} y}{d x^{3}}+y^{2}+e^{(d y / d x)}=0\)   Highest order derivative is  \(\frac{d^{3} y}{d x^{3}}\)  .So, its order is 3. Also, the given differential cannot be expressed as a polynomial. So, its degree is not defined. (iv) (c) : The given differential equation is, \(\sqrt{a+x} \cdot\left(\frac{d y}{d x}\right)+x=0 \Rightarrow \frac{d y}{d x}=\frac{-x}{\sqrt{a+x}}\)   Clearly, degree = 1 (v) (b) :  We have  \(y \frac{d y}{d x}=\frac{x}{\frac{d y}{d x}+\left(\frac{d y}{d x}\right)^{3}}\)   \(\Rightarrow y\left(\frac{d y}{d x}\right)^{2}+y\left(\frac{d y}{d x}\right)^{4}=x\)   \(\Rightarrow\)  Here, highest order derivative is  \(\frac{d y}{d x}\)  ,So , its order is 1 and degree is 4. (iii) (a) : We have, y'" +y 2 + e y = 0 \(\frac{d^{3} y}{d x^{3}}+y^{2}+e^{(d y / d x)}=0\)   Highest order derivative is  \(\frac{d^{3} y}{d x^{3}}\)  So, its order is 3. Also, the given differential cannot be expressed as a polynomial. So, its degree is not defined (iv) (c) : The given differential equation is, \(\sqrt{a+x} \cdot\left(\frac{d y}{d x}\right)+x=0 \Rightarrow \frac{d y}{d x}=\frac{-x}{\sqrt{a+x}}\)   Clearly, degree = 1. (v) (b) : We have  \(\left(1+\left(\frac{d y}{d x} \mid\right)^{3}\right)^{\frac{1}{3}}=7 \frac{d^{2} y}{d x^{2}}\)   \(\therefore\)  Order is 2 and degree is 3. 

(i) (b): We have,  \(\frac{d y}{d x}=\frac{x^{2}+x y+y^{2}}{x^{2}}\) Put y = vx and  \(\frac{d y}{d x}=v+x \frac{d v}{d x}\) \(\therefore v+x \frac{d v}{d x}=\frac{x^{2}+x \cdot v x+v^{2} x^{2}}{x^{2}}=1+v+v^{2}\) \(\Rightarrow x \frac{d v}{d x}=1+v^{2} \Rightarrow \int \frac{d v}{1+v^{2}}=\int \frac{d x}{x}+c\) \(\Rightarrow \tan ^{-1} v=\log |x|+c \Rightarrow \tan ^{-1} \frac{y}{x}=\log |x|+c\) (ii) (d): We have ,  \( 2 x y \frac{d y}{d x}=x^{2}+3 y^{2} \Rightarrow \frac{d y}{d x}=\frac{x^{2}+3 y^{2}}{2 x y}\) Put y = vx and  \(\frac{d y}{d x}=v+x \frac{d v}{d x}\) \(\therefore v+x \frac{d v}{d x}=\frac{x^{2}+3 v^{2} x^{2}}{2 v x^{2}} \Rightarrow x \frac{d v}{d x}=\frac{1+3 v^{2}}{2 v}-v\) \(\Rightarrow x \frac{d v}{d x}=\frac{1+v^{2}}{2 v_{*}} \Rightarrow \int \frac{2 v}{1+v^{2}} d v=\int \frac{d x}{x}+\log c\) \(\Rightarrow \log \left|1+v^{2}\right|=\log |x|+\log |c| \Rightarrow \log \left|v^{2}+1\right|=\log |x c|\) \(\Rightarrow \quad v^{2}+1=x c \Rightarrow \frac{y^{2}}{2}+1=x c \Rightarrow x^{2}+y^{2}=x^{3} c\) (iii) (d): We have,  \(\left(x^{2}+3 x y+y^{2}\right) d x-x^{2} d y=0\) \(\Rightarrow \frac{x^{2}+3 x y+y^{2}}{x^{2}}=\frac{d y}{d x}\) Put y = vx and  \(\frac{d y}{d x}=v+x \frac{d v}{d x}\) \(\therefore \frac{x^{2}+3 x^{2} v+x^{2} v^{2}}{x^{2}}=\left(v+x \frac{d v}{d x}\right)\) \(\Rightarrow 1+3 v+v^{2}=v+x \frac{d v}{d x} \Rightarrow 1+2 v+v^{2}=x \frac{d v}{d x}\) \(\Rightarrow \int \frac{d x}{x}-\int(v+1)^{-2} d v=c \Rightarrow \log x+\frac{1}{v+1}=c\) \(\Rightarrow \log x+\frac{x}{x+y}=c\) (iv) (c): We have,  \(\frac{d y}{d x}=\frac{y}{x}\left\{\log \left(\frac{y}{x}\right)+1\right\}\) Put y = vx and  \(\frac{d y}{d x}=v+x \frac{d v}{d x}\) \(\therefore v+x \frac{d v}{d x}=v\{\log (v)+1\} \Rightarrow x \frac{d v}{d x}=v \log v\) \(\Rightarrow \int \frac{d v}{v \log v}=\int \frac{d x}{x} \Rightarrow \log |\log v|=\log |x|+\log |c|\) \(\Rightarrow \log \left(\frac{y}{x}\right)=c x\)  (v) (a):   We have, \(\left(x \frac{d y}{d x}-y\right) e^{\frac{y}{x}}=x^{2} \cos x\) \(\Rightarrow\left(\frac{d y}{d x}-\frac{y}{x}\right) e^{\frac{y}{x}}=x \cos x\) \(\text { Put } y=v x \text { and } \frac{d y}{d x}=v+x \frac{d v}{d x}\) \(\Rightarrow\left(v+x \frac{d v}{d x}-v\right) e^{v}=x \cos x \Rightarrow x e^{v} \frac{d v}{d x}=x \cos x\) \(\Rightarrow \int e^{v} d v=\int \cos x d x \Rightarrow e^{v}=\sin x+c\) \(\Rightarrow e^{\frac{y}{x}}-\sin x=c\)

  (i) (c) :  The given differential equation can be written as  \( \frac{d y}{d x}+2 y \cot x=\operatorname{cosec} x\) \(\therefore \text { I.F. }=e^{\int 2 \cot x d x}=e^{2 \log |\sin x|}=(\sin x)^{2} \) \(\therefore \quad \lambda=2\) (ii) (c) : We have,  \(\left(1-x^{2}\right) \frac{d y}{d x}-x y=1\) \(\Rightarrow \frac{d y}{d x}-\frac{x}{1-x^{2}} \cdot y=\frac{1}{1-x^{2}} \) \(\therefore \text { I.F. }=e^{-\int \frac{x}{1-x^{2}} d x}=e^{\frac{1}{2} \int \frac{-2 x}{1-x^{2}} d x}\) \(=e^{\frac{1}{2} \log \left(1-x^{2}\right)}=e^{\log \left(1-x^{2}\right)^{\frac{1}{2}}}=\sqrt{1-x^{2}}\) (iii) (b) : We have,  \(\frac{d y}{d x}+y=e^{-x} \)  It is a linear differential equation with I.F. =  \(e^{\int d x}=e^{x}\) Now, solution is \(y \cdot e^{x}=\int e^{x} \cdot e^{-x} d x+c\) \(\Rightarrow y e^{x}=\int d x+c \Rightarrow y e^{x}=x+c \Rightarrow y=x e^{-x}+c e^{-x}\) \(\because y(0)=0 \Rightarrow c=0 \quad \therefore y=x e^{-x}\) (iv) (a) : We have, \( \frac{d y}{d x}+y \tan x=\sec x\) It is a linear differential equation with I.F. =  \(e^{\int \tan x d x}=e^{\log |\sec x|}=\sec x\) Now, solution is  \(y \sec x=\int \sec ^{2} x d x+c\) \(\Rightarrow y \sec x=\tan x+c\) (v) (c) : We have,  \(\frac{d y}{d x}-3 y=\sin 2 x\) It is a linear differential equation with  \(\text { I.F. }=e^{\int-3 d x}=e^{-3 x}\)

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