Class 12 – Page 37

CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k)

February 18, 2025February 17, 2025 by Bhagya

Odisha State Board Elements of Mathematics Class 12 Solutions CHSE Odisha Chapter 9 Integration Ex 9(k) Textbook Exercise questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Exercise 9(k)

Evaluate the following Integrals:
Question 1.
(i) $\int_0^{\frac{\pi}{2}} \frac{d x}{1+\tan x}$dx
Solution:
$CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.1(1)$

(ii) $\int_0^{\frac{\pi}{2}} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}$dx
Solution:
$CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.1(2)$

(iii) $\int_0^1 \frac{\ln (1+x)}{2+x^2}$dx (x = tan θ)
Solution:
$CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.1(3)$

(iv) $\int_0^\pi \frac{x d x}{1+\sin x}$
Solution:
$CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.1(4)$

Question 2.
(i) $\int_{-a}^a$x⁴ dx
Solution:
$CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.2(1)$

(ii) $\int_{-a}^a$(x⁵ + 2x² + x) dx
Solution:
$CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.2(2)$

(iii) $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}$cos² x dx
Solution:
$CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.2(3)$

(iv) $\int_{-\frac{\pi}{6}}^{\frac{\pi}{6}}$sin⁵ x dx
Solution:
Let f(x) = sin⁵ x
Then f(-x) = sin⁵ (-x)
= -sin⁵ x = -f(x)
So f(x) is an odd function.
Thus $\int_{-a}^a$f(x) dx = 0
$\int_{-\frac{\pi}{6}}^{\frac{\pi}{6}}$sin⁵ x dx = 0

Question 3.
(i) $\int_0^\pi$cos³ x dx
Solution:
$CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.3(1)$

(ii) $\int_0^\pi$cos² x dx
Solution:
$CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.3(2)$

(iii) $\int_0^\pi$sin³ x cos x dx
Solution:
$\int_0^\pi$sin³ x cos x dx
[Put sin x = t, then cos x dx = dt
When x = 0, t = 0, when x = π, t = 0
$\int_0^\pi$t³ dt = 0

(iv) $\int_0^\pi$sin x cos² x dx
Solution:
$CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.3(4)$

Question 4.
Show that
(i) $\int_0^1 \frac{\ln x}{\sqrt{1-x^2}}$ dx = $\frac{\pi}{2} \ln \frac{1}{2}$
Solution:
$CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.4(1)$

(ii) $\int_0^{\frac{\pi}{2}} \frac{\cos x-\sin x}{1+\sin x \cos x}$ dx = 0
Solution:
$CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.4(2)$

(iii) $\int_0^\pi$x ln sin x dx = $\frac{\pi^2}{2} \ln \frac{1}{2}$
Solution:
$CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.4(3)$

Question 5.
(i) $\int_0^{\pi / 2}$ln (tan x + cot x) dx
Solution:
$CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.5(1)$

(ii) $\int_0^\pi \frac{x \tan x-\sin x}{1+\sin x \cos x}$ dx
Solution:
$CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.5(2)$

(iii) $\int_1^3 \frac{\sqrt{x} d x}{\sqrt{4-x}+\sqrt{x}}$
Solution:
$CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.5(3)$

(iv) $\int_0^\pi \frac{x \sin x d x}{1+\cos ^2 x}$
Solution:
$CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.5(4)$

(v) $\int_0^1$x (1 – x)¹⁰⁰ dx
Solution:
$CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.5(5)$

(vi) $\int_{\pi / 6}^{\pi / 3} \frac{d x}{1+\sqrt{\cot x}}$
Solution:
$CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.5(6)$

(vii) $\int_0^{50}$e^x-[x] dx
Solution:
$CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Ex 9(k) Q.5(7)$

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c)

February 18, 2025February 17, 2025 by Bhagya

Odisha State Board Elements of Mathematics Class 12 Solutions CHSE Odisha Chapter 11 Differential Equations Ex 11(c) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Exercise 11(c)

Find the solutions of the following differential equations:
Question 1.
(x + y) dy + (x – y) dx = 0
Solution:
Given equation can be written as
$CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.1$

Question 2.
$\frac{d y}{d x}$ = $\frac{1}{2}\left(\frac{y}{x}+\frac{y^2}{x^2}\right)$
Solution:
$CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.2$

Question 3.
(x² – y²) dx + 2xy dy = 0
Solution:
Given equation can be written as
$CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.3$

Question 4.
x$\frac{d y}{d x}$ + $\sqrt{x^2+y^2}$ = y
Solution:
Given equation can be written as
$CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.4$

Question 5.
x (x + y) dy = (x² + y²) dx
Solution:
$CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.5$
$CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.5.1$
This is the required solution.

Question 6.
y² + x² $\frac{d y}{d x}$ = xy $\frac{d y}{d x}$
Solution:
Given equation can be written as
$CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.6$
This is the required solution.

Question 7.
x sin$\frac{y}{x}$ dy = $\left(y \sin \frac{y}{x}-x\right)$dx
Solution:
Given equation can be written as
$CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.7$

Question 8.
x dy – y dx= $\sqrt{x^2+y^2}$ dx
Solution:
$CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.8$
This is the required solution.

Question 9.
$\frac{d y}{d x}$ = $\frac{y-x+1}{y+x+5}$
Solution:
Given equation is
$CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.9$
$CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.9.1$
This is the required solution.

Question 10.
(x – y) dy = (x + y + 1) dx
Solution:
Given equation can be written as
$CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.10$
$CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.10.1$
This is the required solution.

Question 11.
(x – y – 2) dx + (x – 2y – 3) dy = 0
Solution:
Given equation can be written as
$CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.11$
$CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.11.1$
This is the required solution.

Question 12.
$\frac{d y}{d x}$ = $\frac{3 x-7 y+7}{3 y-7 x-3}$
Solution:
Given equation can be written as
$CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.12$
$CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.12.1$
This is the required solution.

Question 13.
(2x + y + 1) dx + (4x + 2y – 1) dy = 0
Solution:
Given equation can be written as
$CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.13$
⇒ 2z + ln (z – 1) = 3x + C
⇒ 2 (2x + y) + ln (2x + y – 1 ) = 3x + C
⇒ (x + 2y) + ln (2x + y – 1 ) = C
This is the required solution.

Question 14.
(2x + 3y – 5)$\frac{d y}{d x}$ + 3x + 2y – 5 = 0
Solution:
Given equation can be written as
$CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.14$
$CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.14.1$

Question 15.
(4x + 6y + 5) dx – (2x + 3y + 4) dy = 0
Solution:
Given equation can be written as
$CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.15$
$CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Ex 11(c) Q.15.1$
This is the required solution.

CHSE Odisha Class 12 Political Science Book Solutions (+2 2nd Year)

February 17, 2025February 16, 2025 by Bhagya

CHSE Odisha 12th Class Political Science Book Solutions (+ 2 2nd Year)

CHSE Odisha Class 12 Political Science Book Solutions in English Medium

Unit 1 Democracy in India

Unit 2 Democratic Process in India-I

Unit 3 Democratic Process in India-II

Unit 4 India in World Politics

Unit 5 Issues in International Politics

CHSE Odisha Class 12 Political Science Book Solutions in Odia Medium

Chapter 1 ଗଣତନ୍ତ୍ର

Chapter 2 ଭାରତରେ ରାଜନୈତିକ ଦଳୀୟ ବ୍ୟବସ୍ଥା

Chapter 3 ଭାରତରେ ସଂଘୀୟବାଦ

Chapter 4 ଭାରତରେ ଗ୍ରାମାଞ୍ଚଳ ଓ ସହରାଞ୍ଚଳ ସ୍ୱାୟତ୍ତ ଶାସନ ବ୍ୟବସ୍ଥା

Chapter 5 ଜାତି ଗଠନରେ ପ୍ରତିବନ୍ଧକ

Chapter 6 ଭାରତୀୟ ରାଜନୀତିରେ ସାଂପ୍ରତିକ ପ୍ରସଙ୍ଗ-ଜନପ୍ରିୟ ଆନ୍ଦୋଳନ

Chapter 7 ଭାରତର ବୈଦେଶିକ ନୀତି

Chapter 8 ଆନ୍ତର୍ଜାତୀୟ ସଙ୍ଗଠନ

Chapter 9 ସାଂପ୍ରତିକ ବିଶ୍ଵରେ ନିରାପତ୍ତା ପ୍ରସଙ୍ଗର ପରିବର୍ତ୍ତିତ ପୃଷ୍ଠଭୂମି

Chapter 10 ପରିବେଶ ଓ ପ୍ରାକୃତିକ ସମ୍ପଦ

CHSE Odisha Class 12 Political Science Syllabus (+2 2nd Year)

There shall be two papers in Political Science modeled on the Syllabi of CBSE. Paper-I: TitleFoundation of Political Theory and Indian Government at work (For First Year). Paper-II: Title-Democracy and Nation Building in India and International affairs (For Second Year).

The subject of Political Science modeled on the Syllabi of CBSE consists of two papers as mentioned above. Paper-I is to be covered in the +2 First Year class and Paper-ll is to be covered in the +2 Second Year class. Each paper is divided into two sections and each section is further subdivided into two/three units. Thus there are five units in both Paper-I and Paper II. Periods have been allocated for the respective units approximately. Teachers are advised to take at least those numbers of periods to cover the particular unit. The major concepts and principles should be taught in such a manner as to stimulate higher mental abilities among students like application, logical thinking, analysis, etc., and not factual information. Paper-setters and Examiners are requested to keep the above in mind while setting questions and examining, respectively. Questions should of short (one word/ multiple-choice/ one sentence), medium (50/100 words/ five sentences), and long (500 words or thereabout). Also, Questions of the final/AHS Examination shall cover all five units of Paper -II.

Objectives of the course/syllabus are, as briefly mentioned above are:

To enable the students to acquire knowledge about the important concepts, theories, principles, provisions, processes and Institutions of the Indian constitution, and some rudimentary knowledge about International affairs;
To acquaint the students with the changing dimension of politics and political theory both in the national and international knowledge domain;
To develop an interest among the students regarding problems of the political domain and to find out the possible solution to those problems.

CHSE Odisha Class 12 Text Book Solutions

CHSE Odisha Class 12 Psychology Book Solutions (+2 2nd Year)

February 17, 2025February 16, 2025 by Bhagya

CHSE Odisha 12th Class Psychology Book Solutions (+ 2 2nd Year)

CHSE Odisha 12th Class Psychology Book Solutions in English Medium

Unit 1 Life Span Development & Self and Personality

Unit 2 Stress: Meeting Life Challenges & Physical Environment and Behaviour

Unit 3 Group Processess and Leadership & Counselling Process

Unit 4 Psychological Disorder & Therapeutic Approaches

Unit 5 Statistics in Psychology

Unit 5 Objective Questions and Answers

CHSE Odisha 12th Class Psychology Book Solutions in Odia Medium

Unit 1 ଜୀବନବ୍ୟାପୀ ବିକାଶ & ସ୍ଵୟଂ ତଥା ବ୍ୟକ୍ତିତ୍ଵ

Unit 2 ମନୋ-ସାମାଜିକ ଚାପ & ଭୌତିକ ପରିବେଶ ଓ ବ୍ୟବହାର

Unit 3 ସମୂହ ପ୍ରକ୍ରିୟା ଓ ନେତୃତ୍ଵ & ପରାମର୍ଶ ପ୍ରକ୍ରିୟା

Unit 4 ମାନସିକ ବିକାର & ମାନସିକ ବିକାର ଚିକିତ୍ସା

Unit 5 ମନୋବିଜ୍ଞାନରେ ପରିସଂଖ୍ୟାନର ବ୍ୟବହାର

CHSE Odisha Class 12 Psychology Syllabus (+2 2nd Year)

Psychology is introduced as an elective subject at the higher secondary stage of education. The course deals with Psychological knowledge and practices which are contextually rooted, It emphasizes the complexity of cognitive and behavioural processes of human beings within a socio-cultural context. It encourages critical reasoning, allowing students to appreciate the role of social factors in behaviour and illustrates how biology and experience shape behaviour.

In the second year, there will be a council examination and the theory paper carries 70 marks and the practical paper carries 30 marks. The question paper pattern for both papers is same.

QUESTION PAPER PATTERN

Theory Paper:
Group-A: Objective type (Compulsory)
Q.No.1: Multiple choice (Fill up the blanks from all units) 1 mark each × 10 = 10 marks
Q.No.2: Statements “True” or “False” 1 mark each × 10 = 10 marks

Group-B: Short Type
Q.No.3: Short type answer (Answer within two /three sentences) and one has to answer 10 bits out of 12 bits.
2 marks each × 10 = 20 marks
Q.No.4: Short type answer (Answer within six sentences) and one has to answer 3 bits out of 5 bits.
3 mark each × 3 = 09 marks

Group-C: Long Type
Q.No.5: Question No-5 to Q.No-10 (Questions will be from all the units and one has to answer any three questions).
7 marks each × 3 = 21 marks

PRACTICAL
There will be 4 number of questions and the examinee is to choose/draw 2 number of questions through the lottery and is to conduct any one question out of the two questions.
Distribution of marks :
Record – 03
Viva -Voce – 07
Conduction & Report writing – 20
Total marks – 30

PSYCHOLOGY IN APPLICATION
SECOND YEAR
Total Marks – 100
Theory – 70 marks
Practical – 30 marks

Theory
UNIT-I
1. Life span development [10 Periods]
This chapter deals with variations in development and developmental tasks across the life span.
a) Meaning of development – Life span perspective
b) Principles of development
c) Stages of development: Pre-natal stage, Infancy, childhood stage, Adolescence, Adulthood and old age.

2. Self and Personality [9 Periods]
This chapter focuses on the study of self and personality in the context of different approaches in an effort to appraise the person. The assessment of personality will also be discussed.
a) Concept of self and personality
b) Personality types and traits
c) Assessment of Personality
This chapter deals with the nature of stress and strategies to cope with stress.
a) Meaning, Nature and causes of stress.
b) Coping strategies to deal with stress.

UNIT-II
3. Stress: Meeting life challenges [6 Periods]
This chapter deals with the nature of stress and strategies to cope with stress.
a) Meaning, Nature and causes of stress.
b) Coping strategies to deal with stress.

4. Physical environment and behaviour. [6 Periods]
This chapter focuses on the application of psychological understanding of human-environment relationship.
a) Human impact on the environment: Noise Pollution, Crowding, Natural disasters.
b) Impact of environment on human behaviour.

UNIT-III
5. Group Processes and leadership. [7 Periods]
This chapter deals with the concept of a group and the role of the leader in a group.
a) Groups: Nature, types and formation.
b) Leadership: Nature, functions and styles of leadership.

6. Counselling Processes [6 Periods]
This chapter focuses on helping the client in living a meaningful and fulfilling life.
a) Meaning and concept of counselling; Goals of counselling.
b) Characteristics of an effective counsellor.

UNIT-IV
7. Psychological disorder [10 Periods]
This chapter discusses the concept of normality and abnormality and the major psychological disorders.
a) Concept of normality and abnormality, criteria of studying abnormal behaviour
b) Causal factors associated with abnormal behaviour.
c) Major Psychological disorders: Anxiety disorders, somatoform disorders and mood disorders.

8. Therapeutic Approaches [6 Periods]
This chapter discusses the purpose and processes to treat Psychological disorders:
a) Nature and Processes of therapy
b) Types of therapy: Psychotherapy, Behaviour therapy, Cognitive therapy and Biomedical therapy.

UNIT-V
9. Statistics in Psychology [10 Periods]
This chapter deals with some basic statistical methods to be used in psychological studies.
a) Frequency distribution
b) Measures of Central Tendency: Computation and uses of mean, median and mode.

PRACTICALS

RCPM (Children) / RPM (Adults)
Case History Method (Preparation of at least one case profile)
Personality Test (Type A/B)
Piagetian Task (Conservation of Liquid Quantity)

Books Recommended:
1. Bureau’s Higher Secondary +2 Psychology Part-II Published by Odisha State Bureau of Test Book Preparation and Production, Bhubaneswar.
2. Psychology Part-II, NCERT.

CHSE Odisha Class 12 Text Book Solutions

CHSE Odisha Class 12 Sociology Book Solutions (+2 2nd Year)

February 17, 2025February 16, 2025 by Bhagya

CHSE Odisha 12th Class Sociology Book Solutions (+ 2 2nd Year)

CHSE Odisha 12th Class Sociology Book Solutions in English Medium

Unit 1 Introducing Indian Society

Unit 2 Indian Social Structure

Unit 3 The Challenges of Cultural Diversity

Unit 4 Social Inequality, Exclusion and Movement

Unit 5 Change and Development in India

CHSE Odisha 12th Class Sociology Book Solutions in Odia Medium

Unit 1 ଭାରତୀୟ ସମାଜର ପରିଚୟ

Unit 2 ଭାରତୀୟ ସାମାଜିକ ସଂରଚନା

Unit 3 ସାଂସ୍କୃତିକ ବିବିଧତାର ଆହ୍ଵାନ

Unit 4 ସାମାଜିକ ବୈଷମ୍ୟ, ବହିଷ୍କରଣ ଓ ଆନ୍ଦୋଳନ

Unit 5 ଭାରତରେ ପରିବର୍ତ୍ତନ ଓ ବିକାଶ

CHSE Odisha Class 12 Sociology Syllabus (+2 2nd Year)

Sociology is introduced as an elective subject at Higher Secondary Education. This subject studies the processes and patterns of individual and group interaction, the forms of organization of social groups, the relationships among them, and group influences on individual behaviour. lt has focused on the understanding of group or other collective factors in human behavior. This syllabus is designed to help the learners to know about society and develop a constructive attitude towards different facets of life.

Objectives:

To enable learners to relate classroom teaching to their outside environment,
To introduce them to the basic concepts of sociology that would enable them to observe and interpret social life,
To make them aware of the complexity of social processes.
The overall objective of this syllabus is to provide learners with knowledge and understanding of the main sociological theories, concepts, and methods.

Distribution of Marks will be as follows: (Total Marks-100)

GROUP-A: Objective type (Compulsory)
Q.1 Multiple Choice (From all Units)
1 mark each × 15 marks = 15 marks
Q.2 One word answer/ (From all Units)
Very short answer/ Correct the sentence/ Filling in the blanks
1 mark each × 15 marks = 15 marks

GROUP-B: Short Answer Type
Q.3 Answer within Two/Three sentences (Out of 14 bits, one has to answer 11 bits)
2 marks each × 11 = 22 marks
Q.4 Answer within six sentences (Out of 8 bits, one has to answer 6 bits)
3 mark each × 6 = 18 marks

GROUP-C: Long Answer Type
Q.5 (out of 6 Questions from all units, one has to answer 4 questions)
7.5 marks each × 4 = 30 marks

SOCIOLOGY
Paper-II
Indian Society

Unit-I Introducing Indian Society
Composition of Indian Society – Demographic, Geographical, Racial, Linguistic, Religious, Tribal
Unity & Diversity – Factors of Unity & Diversity 4

Unit-II Indian Social Structure
Caste System – Meaning, Characteristics, Functions, Dysfunctions, Recent Changes, Caste and Class
Hindu Joint family – Meaning, Characteristics Merits, Demerits, Recent Changes
Village Community – Meaning and Characteristics/ Rural-Urban Linkages and Divisions

Unit-III The Challenges of Cultural Diversity
National Integration: Concept and obstacles – Communalism, Regionalism, Casteism, Terrorism

Unit-IV Social Inequality, Exclusion and Movement
Caste and Social Inequality, Class and Social Inequality, Marginalized Classes: Scheduled Castes, Scheduled Tribes, Constitutional Safeguards, Tribal Movements, Peasant Movements.

Unit-V Change and Development in India
Industrialization – Meaning, Characteristics, Impact
Urbanization – Meaning, Characteristics, Impact
Modernization – Meaning, Characteristics, Impact
Globalization – Meaning, Characteristics, Impact

BOOK PRESCRIBED:
1. Bureau’s Higher Secondary (+2) Sociology, Part-II Published by Odisha State Bureau of Textbook Preparation & Production, Bhubaneswar.
2. Sociology, Part-II, NCERT.

CHSE Odisha Class 12 Text Book Solutions

CHSE Odisha Class 12 Text Book Solutions | +2 2nd Year Science Arts Commerce Book Solutions Pdf Download

July 4, 2025February 16, 2025 by Veerendra

CHSE Odisha 12th Class Text Book Solutions | + 2 2nd Year Science Arts Commerce Solutions Book Pdf Download

Elements of Mathematics Class 12 CHSE Odisha Solutions (Maths)
CHSE Odisha Class 12 Physics Book Solutions
CHSE Odisha Class 12 Chemistry Book Solutions
CHSE Odisha Class 12 Biology Book Solutions
CHSE Odisha Class 12 Odia Book Solutions (MIL Odia Sahitya Jyoti)
CHSE Odisha Class 12 Optional Odia Solutions (Odia Sahitya Dipti)
CHSE Odisha Class 12 Alternative English Book Solutions
CHSE Odisha Class 12 Invitation to English Book Solutions
CHSE Odisha Class 12 Sanskrit Book Solutions
CHSE Odisha Class 12 Sanskrit Optional Book Solutions (Sanskrit Elective)
CHSE Odisha Class 12 Hindi Book Solutions
CHSE Odisha Class 12 History Book Solutions
CHSE Odisha Class 12 Political Science Book Solutions
CHSE Odisha Class 12 Economics Book Solutions
CHSE Odisha Class 12 Accountancy Book Solutions
CHSE Odisha Class 12 Business Studies Book Solutions
CHSE Odisha Class 12 Education Book Solutions
CHSE Odisha Class 12 Logic Book Solutions
CHSE Odisha Class 12 Sociology Book Solutions
CHSE Odisha Class 12 Psychology Book Solutions

BSE Odisha Solutions

CHSE Odisha Class 12 Alternative English Approaches to English Book 1, 2 Solutions

February 17, 2025February 16, 2025 by Bhagya

CHSE Odisha Class 12 Approaches to English Book 1, 2 Solutions (+2 2nd Year)

CHSE Odisha Class 12 Approaches to English Book 1 Solutions

Prose

Unit 1 THE WONDER WORLD OF SCIENCE

Unit 2 OUR ENVIRONMENT

Unit 3 THE WORLD OF BUSINESS

Unit 4 THE CHANGING WORLD

CHSE Odisha Class 12 Approaches to English Book 2 Solutions

Poems

Short Stories

One-Act Plays

Chapter 1 The Hour of Truth

Grammar & Usage

CHSE Odisha Class 12 Alternative English Syllabus

CHSE Odisha Class 12 Alternative English Question Pattern and Distribution of Marks

Alternative English +2 2nd Year
Full Marks – 100, Time – 3 Hours

1. Reading Comprehension.
(a) A prescribed prose piece or extract (10 Marks)
(5 questions including inferential questions are to be answered)
(b) A prescribed poem/extract (10 Marks)
(5 questions including inferential questions and those on poetic devices, figures of speech, mode, tone, and style, etc.)
(c) A prescribed story/one-act play or its extract (10 Marks)
(5 questions including Inferential questions and those on literary devices, tone, etc.)
(d) An unseen passage of at least 200 words. (10 Marks)
(5 questions including inferential ones)

2. Reading-related skills.
(a) Un Guided note-making based on the passage 1(d) (10 Marks)

3. Writing skills.
(a) Designing and writing a brochure (10 Marks)
(b) Writing dialogues of a face-to-face/telephonic conversation. (10 Marks)
(c) Rewriting a poem/short story as a different form of discourse i.e. a page of a diary, a newspaper report/article, or a script for a play, etc. (10 Marks)
(d) Adding a suitable beginning/ending/title to a given poem/story. (5 Marks)

4. Grammar and usage (in context) (15 Marks)
(3 questions on the prescribed grammar units including modified close tests.

CHSE Odisha Class 12 Text Book Solutions

CHSE Odisha Class 12 Math Notes – Elements of Mathematics Class 12 Notes

February 17, 2025February 16, 2025 by Veerendra

CHSE Odisha 12th Class Math Notes | Elements of Mathematics Class 12 Notes CHSE Odisha

CHSE Odisha Class 12 Math Book Solutions | Elements of Mathematics Class 12 CHSE Odisha Solutions Pdf Download

February 17, 2025February 16, 2025 by Veerendra

CHSE Odisha 12th Class Math Book Solutions | Elements of Mathematics Class 12 Solutions CHSE Odisha Pdf Download

Elements of Mathematics CHSE Solutions Class 12 Chapter 1 Relation and Function

CHSE Math Solution Class 12 Pdf Chapter 2 Inverse Trigonometric Functions

Chapter 2 Inverse Trigonometric Functions Ex 2

Elements of Mathematics Class 12 Volume 2 Chapter 3 Linear Programming

Elements of Mathematics Class 12 Book Solutions Chapter 4 Matrices

CHSE Odisha Class 12 Math Book Pdf Download Chapter 5 Determinants

Elements of Mathematics Class 12 CHSE Odisha Pdf Download Chapter 6 Probability

Elements of Mathematics Class 12 CHSE Odisha Solutions Chapter 7 Continuity and Differentiability

Plus 2 2nd Year Science Math Book Pdf Chapter 8 Application of Derivatives

Elements of Mathematics Class 12 Solutions CHSE Odisha Pdf Download Chapter 9 Integration

Class 12 Elements of Mathematics Book Pdf Chapter 10 Area Under Plane Curves

Chapter 10 Area Under Plane Curves Ex 10

CHSE Odisha Class 12 Math Book Pdf Chapter 11 Differential Equations

Elements of Mathematics Class 12 Odisha Pdf Download Chapter 12 Vectors

Elements of Mathematics Class 12 Solutions CHSE Odisha Chapter 13 Three Dimensional Geometry

CHSE Odisha Class 12 Maths Syllabus (+2 2nd year)

Mathematics (+2 2nd year)
Course Structure

Unit	Topic	Marks	No. of Periods
I	Relations and Functions & Linear Programming	20	45
II	Algebra and Probability	20	45
III	Differential Calculus	20	45
IV	Integral Calculus	20	45
V	Vector 3-D Geometry	20	45
Total		100	220

General Instructions:

All questions are compulsory in Group A, which are very short answer-type questions. All questions in the group are to be answered in one word, one sentence, or as per the exact requirement of the question. (1 × 10 = 10 Marks)
Group B contains 5(five) questions and each question has 5 bits, out of which only 3 bits are to be answered (Each bit carries 4 Marks) (4 × 15 = 60 Marks)
Group-C contains 5(five) questions and each question contains 2/3 bits, out of which only 1(one) bit is to be answered. Each bit caries 6(six) Mark (6 × 5 = 30 Marks)

Unit I Relations and Functions

Chapter 1 Relations and Functions
Types of relations, reflexive, symmetric, transitive, and equivalence relations. One-to-one and onto functions, composite functions, the inverse of a function. Binary operations.

Chapter 2 Inverse Trigonometric Functions
Definition, range, domain, principal value branch. Graphs of inverse trigonometric functions. Elementary properties of inverse trigonometric functions.

Chapter 3 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 (bounded and unbounded), feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints).

Unit II Algebra

Chapter 4 Matrices
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. Noncommutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrices (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).

Chapter 5 Determinants
Determinant of a square matrix (up to 3 x 3 matrices), properties of determinants, minors, co-factors and applications of determinants in finding the area of a triangle, and Adjoint and inverse of a square matrix. Consistency, inconsistency, and a number of solutions of a system of linear equations by examples, solving a system of linear equations in two or three variables (having unique solution) using the inverse of a matrix.

Chapter 6 Probability
Conditional probability, multiplication theorem on probability. Independent events, total probability, Baye’s theorem, Random variable, and its probability distribution, mean and variance of a random variable. Independent (Bernoulli) trials and Binomial distribution.

Unit III Differential Calculus

Chapter 7 Continuity and Differentiability
Continuity and differentiability, a derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, 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. Rolle’s and Lagrange’s Mean Value Theorems (without proof) and their geometric interpretation.

Chapter 8 Applications of Derivatives
Applications of derivatives: rate of change of bodies, increasing and decreasing functions, tangents, and normals, use of derivatives in approximation, maxima, and minima (first derivative test motivates 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).

Unit IV Integral Calculus

Chapter 9 Integration
Integration as the 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.
$\int \frac{d x}{x^2 \pm a^2}, \int \frac{d x}{x^2 \pm a^2}, \int \frac{d x}{a^2-x^2}, \int \frac{d x}{a x^2+b x+c}$
$\int \frac{d x}{a x^2+b x+c}, \int \frac{p x+q}{a x^2+b x+c} d x$
$\int \frac{p x+q}{a x^2+b x+c} d x, \int \sqrt{a^2 \pm x^2} d x$
$\int \sqrt{x^2-a^2} d x$
$\int \sqrt{a x^2+b x+c} d x, \int(p x+q) \sqrt{a x^2+b x+c} d x$
Definite integrals as a limit of a sum, Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.

Chapter 10 Applications of the Integrals
Applications in finding the area under simple curves, especially lines, circles/parabolas/ ellipses (in standard form only). The area between any of the two above-said curves (the region should be clearly identifiable).

Chapter 11 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 the method of separation of variables, solutions of homogeneous differential equations of the first order and first degree. Solutions of linear differential equation of the type:
$\frac{dy}{dx}$ + py = q, wherep and q are functions of x or constants.
$\frac{dx}{dy}$ + px = q, where p and q are functions of y or constants.

Unit V Vectors and Three-Dimensional Geometry

Chapter 12 Vectors
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, the 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, a scalar triple product of vectors, and Coplanarity of three vectors.

Chapter 13 Three-dimensional Geometry
Direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation of a line, coplanar and skew lines, the shortest distance between two lines. Cartesian and vector equation of a plane. The angle between (i) two lines, (ii) two planes, (iii) a line and a plane. Distance of a point from a plane.

Books Recommended:
Bureau’s Higher Secondary (+2) Elements of Mathematics, Part-II, Published by Odisha State Bureau of Text Book Preparation and Production, Bhubaneswar.

CHSE Odisha Class 12 Text Book Solutions

Plus Two Second Year Optional Odia Question Answer – Sahitya Dipti Question Answer (+2 2nd Year)

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Class 12 Odia Optional Book Solutions – Sahitya Dipti Part 2 Question Answer ସାହିତ୍ୟ ଦୀପ୍ତି

Plus Two Second Year Optional Odia Question Answer | ସାହିତ୍ୟ ଦୀପ୍ତି Sahitya Dipti Part 2 Question Answer

CHSE Odisha Class 12 Text Book Solutions

CHSE Odisha Class 12 Math Solutions Chapter 10 Area Under Plane Curves Ex 10

February 17, 2025February 16, 2025 by Bhagya

Odisha State Board Elements of Mathematics Class 12 CHSE Odisha Solutions Chapter 10 Area Under Plane Curves Ex 10 Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 10 Area Under Plane Curves Exercise 10

Question 1.
Find the area bounded by
(i) y = e^x, y = 0, x = 4, x = 2
Solution:
Area = $\int_2^4$e^x dx
= $\left[e^x\right]_2^4$
= e⁴ – e²

(ii) y = x², y = 0, x = 1
Solution:
Area = $\int_0^1$x² dx
= $\left[\frac{x^3}{3}\right]_0^1$
= $\frac{1}{3}$

(iii) xy = a², y = 0, x = α, x = β (β > α > 0)
Solution:
Area = $\int_\alpha^\beta y$y dx
= $\int_\alpha^\beta \frac{a^2}{x}$ dx
= a²$[\ln x]_\alpha^\beta$
= a² ln (β/α)

(iv) y = sin x, y = 0, x = $\frac{\pi}{2}$
Solution:
Area = $\int_0^{\frac{\pi}{2}}$sin x dx
= $[-\cos x]_0^{\frac{\pi}{2}}$
= -cos$\frac{\pi}{2}$ + cos θ = 1

Question 2.
Find the area enclosed by
(i) y = e^x, x = 0, y = 2, y = 3
Solution:
$CHSE Odisha Class 12 Math Solutions Chapter 10 Area Under Plane Curves Ex 10 Q.2$

(ii) y² = x, x = 0, y = 1
Solution:
Area = $\int_0^1$x dy
= $\int_0^1$y² dy
= $\left[\frac{y^3}{3}\right]_0^1$
= $\frac{1}{3}$

(iii) xy = a², x = 0, y = α, y = β (β > α > 0)
Solution:
Area = $\int_\alpha^\beta$x dy
= $\int_\alpha^\beta \frac{a^2}{y}$dy
= a²$[\ln y]_\alpha^\beta$
= a² ln (β/α)

(iv) y² = x³, x = 0, y = 1
Solution:
Given curve is y² = x³
⇒ x = y^2/3
It passes through the origin. So the required area
= $\int_0^1$x dy
= $\int_0^1 y^{\frac{2}{3}}$ dy
= $\left[\frac{y^{\frac{5}{3}}}{5 / 3}\right]_0^1$
= $\frac{3}{5}$

Question 3.
(i) Determineellipse the area with in the ellipse
$\frac{x^2}{a^2}$ + $\frac{y^2}{b^2}$ = 1
Solution:
$CHSE Odisha Class 12 Math Solutions Chapter 10 Area Under Plane Curves Ex 10 Q.3(1)$
The ellipse is symmetrical about x-axis and y-axis.It is divided into 4 equal parts by the coordinate axes. So required area
$CHSE Odisha Class 12 Math Solutions Chapter 10 Area Under Plane Curves Ex 10 Q.3(1.1)$

(ii) Find the area of the circle x² + y² = 2ax.
Solution:
Given circle is x² + y² = 2ax
⇒ x² – 2ax + a² + y² = a²
⇒ (x – a)² + y² = a² … (1)
The centre of the circle is (a, 0) and radius is a.
$CHSE Odisha Class 12 Math Solutions Chapter 10 Area Under Plane Curves Ex 10 Q.3(2)$

(iii) Find the area of the portion of the parabola y² = 4x bounded by the double ordinate through (3, 0).
Solution:
Given parabola is y² = 4x
$CHSE Odisha Class 12 Math Solutions Chapter 10 Area Under Plane Curves Ex 10 Q.3(3)$

(iv) Determine the area of the region bounded by y² = x³ and the double ordinate through (2, 0)
Solution:
Given curve is y² = x³
⇒ y = ±x^3/2 … (1)
The curve passes through the origin and symmetrical about x-axis because the power of y is even.
$CHSE Odisha Class 12 Math Solutions Chapter 10 Area Under Plane Curves Ex 10 Q.3(4)$

Question 4.
(i) Find the area of the regions into which the circle x² + y² = 4 is divided by the line x + √3y = 2.
Solution:
Given circle and the straight line are x²+ y² = 4 and x+ √3y = 2
The circle has the centre at (0, 0) and radius ‘2’.
The eqn. (2) can be written as
y = –$\frac{1}{\sqrt{3}}$x + $\frac{2}{\sqrt{3}}$
Slope of the strainght line = –$\frac{1}{\sqrt{3}}$
The line makes and angle of 150° with x-axis making intercept $\frac{2}{\sqrt{3}}$ from y-axis.
It intersects x-axis at (2, 0).
$CHSE Odisha Class 12 Math Solutions Chapter 10 Area Under Plane Curves Ex 10 Q.4(1)$
Solveing (1) and (2),
ge wet (2 – 3√y)² + y² = 4
4 + 3√y² – 4√3y + y² = 4
4y² – 4√3y = 0
y(y – √3) = 0
y = 0 or y = √3
When y = 0, x = 2
When y = √3, x = -1
Thus the straight line intersects the circle at (2, 0) and (-1, √3).
Area of the portion ACBA.
$CHSE Odisha Class 12 Math Solutions Chapter 10 Area Under Plane Curves Ex 10 Q.4(1.1)$

(ii) Determine the area of the region between the curves y = cos x and y = sin x, bounded by x = 0.
Solution:
$CHSE Odisha Class 12 Math Solutions Chapter 10 Area Under Plane Curves Ex 10 Q.4(2)$
The curves y = cos x and y = sin x are shown in the above figure. The region included between these two curves in [0, $\frac{\pi}{4}$] is OABO.
$CHSE Odisha Class 12 Math Solutions Chapter 10 Area Under Plane Curves Ex 10 Q.4(2.1)$

(iii) Find the area enclosed by the two parabolas y² = 4 ax and x² = 4ay.
Solution:
The given parabolas are y² = 4ax and x² = 4ay.
The graphs of the two parabolas are shown in the figure.
$CHSE Odisha Class 12 Math Solutions Chapter 10 Area Under Plane Curves Ex 10 Q.4(3)$
⇒ x⁴ = 64 a⁴
⇒ x⁴ – 64 a³x = 0
⇒ x (x³ – (4a)³) = 0
⇒ x (x – 4a) (x² + 4ax + 16a²) = 0
⇒ x = 0, 4a
When x = 0, y = 0 and
when x = 4a, y = 4a
Thus the two parabolas intersect at (0, 0) and (4a, 4a).
Area between two parabolas
$CHSE Odisha Class 12 Math Solutions Chapter 10 Area Under Plane Curves Ex 10 Q.4(3.1)$

(iv) Determine the area common to the parabola y² = x and the circle x²+ y² = 2x.
Solution:
Gien parabola is y² = x
Given circle is
x² + y² = 2x ⇒ (x – 1)² + y² = 1
The centre is at (1, 0) and radius is 1.
$CHSE Odisha Class 12 Math Solutions Chapter 10 Area Under Plane Curves Ex 10 Q.4(4)$
Solving (1) and (2) we get
x² + x = 2x ⇒ x² – x = 0 ⇒ x(x – 1) = 0
⇒ x = 0, x = 1
When x = 0, y = 0 and when x = 1, y = 1.
Thus both the parabola and circle intersect at (0, 0) and (1, 1).
Required Area
$CHSE Odisha Class 12 Math Solutions Chapter 10 Area Under Plane Curves Ex 10 Q.4(4.1)$