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Given as f(x) = x³ – 2x² – x + 3 on [0, 1]

The every polynomial function is continuous everywhere on (−∞, ∞) and differentiable for all arguments. Here, f(x) is a polynomial function. Therefore it is continuous in [0, 1] and differentiable in (0, 1). Therefore both the necessary conditions of Lagrange’s mean value theorem is satisfied.

Therefore, there exist a point c ∈ (0, 1) such that:

f'(c) = (f(1) - f(0))/1 - 0

f'(c) = (f(1) - f(0))/1

f (x) = x³ – 2x² – x + 3

Differentiate with respect to x

f’(x) = 3x² – 2(2x) – 1

= 3x² – 4x – 1

For the f’(c), put the value of x = c in f’(x)

f’(c)= 3c² – 4c – 1

For the f(1), put the value of x = 1 in f(x)

f (1)= (1)³ – 2(1)² – (1) + 3

= 1 – 2 – 1 + 3

= 1

For the f(0), put the value of x = 0 in f(x)

f (0)= (0)³ – 2(0)² – (0) + 3

= 0 – 0 – 0 + 3

= 3

∴ f’(c) = f(1) – f(0)

⇒ 3c² – 4c – 1 = 1 – 3

⇒ 3c² – 4c = 1 + 1 – 3

⇒ 3c² – 4c = – 1

⇒ 3c² – 4c + 1 = 0

⇒ 3c² – 3c – c + 1 = 0

⇒ 3c(c – 1) – 1(c – 1) = 0

⇒ (3c – 1) (c – 1) = 0

c = 1/3, 1

c = (1/3) ∈ (0,1)

Thus, lagrange's mean value theorem is verified.

Given as the function is y = 16 – x², x ϵ [– 1, 1]

As we know that polynomial function is continuous and differentiable over R.

Let's check the values of ‘y’ at extremes

⇒ y (– 1) = 16 – (– 1)²

⇒ y (– 1) = 16 – 1

⇒ y (– 1) = 15

⇒ y (1) = 16 – (1)²

⇒ y (1) = 16 – 1

⇒ y (1) = 15

y(– 1) = y(1). Therefore, there exists a c ϵ (– 1, 1) such that f’(c) = 0.

As we know that for a curve g, the value of the slope of the tangent at a point r is given by g’(r).

Let us find the derivative of curve y

y' = d(16 - x²)/dx

⇒ y’ = – 2x

y’(c) = 0

⇒ – 2c = 0

⇒ c = 0 ϵ (– 1, 1)

Value of y at x = 1 is

⇒ y = 16 – 0²

⇒ y = 16

Hence, the point at which the curve y has a tangent parallel to x – axis (since the slope of x – axis is 0) is (0, 16).

Given as f(x) = (1/x) on [-1,1]

It is clear, x ≠ 0

⇒ f (x) exists for all the values of x except 0

⇒ f (x) is the discontinuous at x = 0

So, f (x) is not continuous in [– 1, 1]

Thus, the Lagrange’s mean value theorem is not applicable for the function f(x) = 1/x on [-1, 1]

It is done to accommodate the linear expansion of railway line during summer. Fi the gap is not left, in summer, the lines will bend causing a threat of derailment.

Yes. For example, ice contracts on heating.

Given function is 

f(x) = x sin (1/x), x ∈ [-1,1]

f(x) = 0

Now, lim_{x → 0} f(x) = lim_{x → 0} x sin (1/x) = (0 x a) a number between 1 and -1

= 0 = f'(0)

∴ f(x) is continuous at x = 0 

Again, left hand derivative L f'(0) = lim_{h → 0} (f(0 - h) - f(0))/-h

= lim_{h → 0} (-h sin(-1/h) - 0)/-h = lim_{h → 0} (-sin(1/h))

Which does not exist.

∴ The given function is not derivable at x = 0

So, Rolle's theorem is not applicable. 

The boiling point of a substance at one standard atmospheric pressure is called its normal boiling point.

Example: Normal boiling point of water is 99.97 °C,

Charle's law of pressure.

98.4° F or 37°C.

Correct Answer is: C) Both a and b

Whetstone’s bridge is an instrument which is used to determine the unknown resistance R other given resistances.

for example:

R4 is unknown resistance then it can be find from following equation:

R₄=R₃ x (R₂/R₁)

So it can find both low and high resistances.

Correct Answer is: (A)

The relation between peak current I_o and rms current I_Rms is I0 = √2 I_rms

Given relation aRb is defined only when, both a & b are not even or odd at a time.

i.e., if a is even & b is odd then aRb is well defined.

(i) Let a and be odd and even then aRb is defined and aRb ⇒ bRa

Hence, R is symmetric.

(ii) Let a be an odd then aRa is not defined and a be an even then also aRa is not defined.

So, R is not reflexive.

(iii) Let a and b be odd and even respectively. Then if R is transitive, then aRb, bRa ⇒ aRa but, then aRa is not defined as a is odd. Hence, R is not transitive.

(a) Mahatma Gandhi

Electric charges has three basic properties :

(a) quantisation 

(b) additivity. 

(c) conservation

• My Experiments with Truth 
• Hind Swaraj.

(d) Dr. Bhimrao Ambedkar

(c) Bhimrao Ambedkar

(a) Dr. Bhimrao Ambedkar

Vedas, Upanishads, Ramayana and Shrimad Bhagwad Gita etc.

Be literate, remain united and struggle.

Dr. Bhimrao Ambedkar.

An Introduction of Bhimrao Ambedkar:

Dr. Bhimrao Ambedkar was born at Mhow cantonment in Madhya Pradesh on April 14, 1891. He was deeply influenced by the thoughts of Gautama Buddha, Saint Kabir and Mahatma Jyotiba Phule,etc. He passed his high school examination in 1907 from Elphinston High School, Bombay and B.A. examination in 1912, from the same Elphinston College.

During his study period, he got ₹ 25 per month as scholarship from Maharaja of Baroda, Sayaji Rao Gayekwad. He was given a scholarship for higher education in America. The diligent and intelligent young man Ambedkar got his M.A. degree from Columbia University in 1915, Pn.D. degree from London in 1917, M.Sc. degree from London in 1922 and law graduation and D.C.C. in 1923. In this way, Dr. Bhimrao Ambekar was the most highly educated leader during national independence movement. Dr Ambedkar’s concepts are still greatly relevant in present time.

His concepts are given in the following points: 

1. Opposition to Untouchability:

Dr. Bhimrao Ambedkar belonged to Mahar caste, ,which was considered untouchable in those days. He was discriminated against due to his caste. He felt that the inhuman customs like untouchability and inequality were social evils that had divided the society into fragments and so they were a curse for the nation. Due to these social and religious evils, a large group of population was unable to get the opportunity of progress and development. Being disappointed by all this he started a movement for the upliftment of the daligts.

2. Education: 

Dr. Bhimrao Ambedkar considered that a society can progress only through proper education. So, he gave the slogan, “Be literate, be united and then struggle”. His thoughts on education are relevant even today. He considered that it is education which opens the doors of success. These days, the Government as well as other social institutions are trying hard to spread the light of education all over India. 

3. Support to inter – caste marriage: 

Dr Bhimrao Ambedkar supported inter-caste marriage. He considered that intercaste marriage is a medium of mixing the blood of different groups. Today, the government is also promoting inter-caste marriages. An incentive is provided to the parties involved in inter-caste marriage by the government.

4. Opposition to Communal-ism and Evangelists:

Dr. Ambedkar always opposed communism and evangelists. He opposed the demand of separating sindh from Bombay made by the Muslim League before the Simon commission in 1928. In his famous book – “Pakistan and the Partition of India”, he practically suggested the exchange of population between two new nations regarding unity of India, communal peace, economic development and security of border. He was very unhappy on the incidents of state – sponsored conversion of schedule castes in Pakistan.

5. Lover of Indian Culture:

Dr. Ambedkar had great affection for India and its culture. He considered Sanskrit language as a root source of Indian knowledge, philosophy and culture. He wanted to make Hindi the national language for the unity and integrity of India. On the basis of Vedic literature, he proved that Aryans were the natives of India and the Shudras were the scholars of Vedas, and they practiced thread ceremony and were rulers too. He further said that if we love our country and our culture, then it is our duty to accept Hindi as our national language.

6. Emphasis on the importance of state and democracy:

Dr. Ambedkar considered democracy as a medium of social and economic revolutionary change. He acknowledged the importance of freedom of individual, but opposed his selflessness. He disapproved of the totalitarian nature of dividing mankind. He was of the opinion that the state must be for public welfare and its role of governance should moderate. The single and only objective of a state must be to provide social justice to all and to make its people prosperous.

7. Supporter of Parliamentary System:

Dr. Ambedkar supported the Parliamentary System in India and considered it in favor of India. The unitary ruling system is most suitable for the unity and integrity of this country. There should be a control of legislature over the executive so that if might not avoid the directions and directives of state policy as laid down in the constitution of India. There should be a balance and adjustment among all departments of Government, so that they might be free, and have control over each other at the same time. The interest of minorities must not be compromised. In parliamentary democracy, party – based democracy should be promoted instead of specific individual.

8. Adult franchise or Voting: 

To establish a responsible government in the country, every adult must have the right to vote without any discrimination. The government has adopted this policy, so even a backward class adult could get the right to vote, and they could become an important link in the chain of democracy. 

9. Opposition to the Alliance of Communism and Islamic countries:

Dr. Bhimrao described communism as a danger for democratic countries. China, being a communist, can attack India any moment. His forecast was proved correct in 1962, when China attacked India suddenly and secretly. On the other hand, Dr, Ambedkar warned India to be aware of the alliance of Islamic countries. The increasing Islamic terrorism at the global level has proved his warning true. 

10. His thoughts about the Constitution:

Dr. Ambedkar was the chairperson of the Drafting Committee in the Constituent Assembly. His speeches and statements in the Constituent Assembly present a wonderful confluence of politics, law, history, economics and philosophy. What he said in his speeches and through his statements, was a valuable treasure for the law experts. He said, during the framing of the constitution, that our constitution must be practical and flexible. It must be capable in ensuring peace and unity during war disasters and other crises. 

11. Opposition to the formation of Language-based States: 

Dr. Ambedkar opposed the formation of states on the basis of language. This, he believed, could take the shape of communal-ism in the long rum. It is therefore necessary, that the government must take wise decisions for national interest and integrity.

(c) Physically weak and frail

• Castes in India 
• Who were the Shudras.

Truth and Non – Violence.

Physical education is helpful to make a inactive person active and energetic.

Physical education is essential for complete development of a human’s personality. Though its uses has been made since the ancient times. But, today, it has become very important to apply in our daily life. Because, lack of physical education leads to many problems. Physically as well as mentally. It makes a complete impact on the life of a person. Thus, its need and application in one’s life cannot be avoided

(d) Social Development objectives

Introduction of Mahatma Gandhi:

Mohandas Karamchand Gandhi was born in Porbanda in Gujarat on 2th October, 1869. His father was Karamchand Uttamchand Gandhi. He went to England for higher education of law in 1888 and returned to India in 1891. In 1893, he went to South Africa to prosecute a case and remain there till 1914. There,he adopted the policy of truth and non-violence to oppose the British Government for its racial discrimination and oppressive rule through his practice of ‘Satyagraha’.

In 1914, Gandhiji came back to India from South Africa. In India, he led many movements and satyagrahas in Champaran, Ahmedabad and Kheda. In the beginning till 1918, Gandhiji was a co – operator of British Government. He co – operated with the British Government during First World War, so he was honoured and awarded the title ‘Kesar – e – Hind’. But the events of Rowlatt Act 1919, Jallianwala Bagh massacre (1919),and Khilafat Movement 1920, made Gandhiji a great opponent of the British government.

The concept of Ram – Rajya by Gandhiji is attached with the concept of classless and stateless society. According to Gandhiji, Ram – rajya could be established only when the governance would be decentralized, there would be cottage industries in every village everyone would be self sufficient and be able to cultivate as much as needed. In the society, the most inferior person would get social acceptance and benefits, also economic and political justice.

Gandhiji considered that politics is deeply related to religion and to morality. According to him, without religion, the politics in a state is like death. Gandhiji himself was a karmyogi. He was inspired deeply by the ‘Geeta’ and considered that the other name of religion is to perform one’s duty and only a dutiful man can be called a perfect man.

According to Gandhiji, the form of state is:

Religious opinion: A godly state.

Economic opinion: 

The decentralized arrangement of simple and easy lifestyle on the basis of minimum human necessities. The satisfaction of human is the major basis instead of profit motive.

Social opinion:

A society in the form of an extended family without any discrimination.

Political opinion:

All bindings in this decentralized governance are moral and self-controlled. There is no inequality of caste, religion, language, region, color, wealth etc. Justice is always available for all. All the citizens use their freedom in a pious way.

The root and basis of entire thoughts of Dayanand Saraswati are the Vedas. He put an intellectual and moral foundation stone for independence. At the very outset, he gave a message that good governance could not replace self governance. He proved that Indian intellectual culture and the Vedas are the best in the world. He called upon the Arya to make India undivided and independent.

He considered that domestic goods and patriotic thoughts are the spiritual duty of an Indian. He said that a foreign government might be very good, but it cannot take the place of self governance. In this way, Swamiji created the feeling of self respect, pride and patriotism among Indian youths. The Arya Samaj was established by Swami Dayanand.

He always opposed evil social customs such as child marriage, unmatched marriage and Sati practice. He also opposed religious hypocrisy and promoted women’s education. The consciousness that he awakened on the basis of four pillars of Swadeshi, Swadharma, Swaraj and Swabhasha is relevant even today.

According to J.R. Sherman, “The fundamental goal of Physical Education is to affect the experiences of the individual to the extent, that each individual can help to adjust successfully in changing conditions, which will result in an increase in his desires and develop the ability to fulfill his wants”,

According to J.F. Williams, “Physical Education should aim to provide skilled leadership, adequate facilities and ample time for the individual and the groups to participate in activities that are physically wholesome, mentally stimulating and socially sound”. 

Physical education helps in the:

1. All round development of the child. 

2. Physical fitness and wellness 

3. Cultivation of Leadership qualities. 

4. Healthy physical growth and development

Physical education is essential for a person’s complete development-physically, mentally, emotionally as well as socially. Physical education will help humans

• to develop an understanding of the role of physical activity in promoting health 
• to develop social skills. 
• to improve self confidence and problem solving skills. 
• to learn discipline, and to increase their creativity.
• to develop a sense of cooperation and team work.

Thus, physical education is very necessary for the overall development of a human being.

Physical Development objectives: The objectives included in this category are those by which man develops physically by making his body strong, healthy and attractive. He does physical activities and takes active part in games which strengthen him and accelerate his activities like running, weight lifting, catching, jumping, climbing up and down etc.

According to J.F. Williams, “Physical Education should aim to provide skilled leadership, adequate facilities and ample time for the individual and the groups to participate in activities that are physically wholesome, mentally stimulating and socially sound”.

• Satyagraha and 
• The concept of Sarvodaya as a whole.

• Marshal Tito (Yugoslavia) 
• Colonel Nasser (Egypt).

Swami Vivekananda never played an active role in Indian politics. He was a messenger of humanism and Advait Vedanta. Being a saint, he was attached with religion. He himself said, “Neither I am a politician nor a political movement”. But what he provided to the Indians was the sense of established supremacy of Indian culture over meaning – less foreign rule. He tried his best to attract the Indian youth towards Indian culture.

Moral strength, fearlessness and motivation of deeds were, in his view, the most valuable for the growth of Indian nationalism. Swami Vivekananda was an internationalist. He addressed the people present in Chicago Conference “My own brothers and sisters ”, which was a symbol of world brotherhood and the feeling of oneness.

He was a supporter to provide opportunities to all. He imagined an ideal nation. He considered that political redemption could be possible only by the collective progress of all. He presented the spiritual principle of nationalism. He believed in the theory of global fraternity. He considered morality as only basis of national unity.

Vinayak Damodar Savarkar (1883 – 1966) was born on 28th May,1883 at a small village Bhayur in Nashik district of Maharashtra. His family belonged to the clan of Chitpawan Brahmin caste. His father was Damodarpant Savarkar. His elder brother was Ganesh Savarkar and younger brother was Narayan Savarkar. These three brothers of Savarkar family played important role in Indian revolutionary movement in the form of the forerunners of Indian independence movement.

Due to his bravery in revolutionary movements, he was called as Veer Savarkar by people. He was an undefeated revolutionary. In 1904,he founded a revolutionary institution ‘Abhinav Bharat’. He lived in England during 1906 to 1910. while livening in England, he wrote a research-based book, ‘Bharatiya Swatantrata Samar’ which was seized before its publication. He was arrested in March 1910 in London.

After arrest, when he was being brought to India in a ship, he tried to abscond and jumped into the sea near the Marseilles Sea Port. This event gave him international fame. He was again arrested and sent to Andaman Jail where he spent very torturous 10 long years (1911-21). He was released in May 1937. After that, he played an important and active role in Indian politics. He was the chairperson of Hindu Mahasabha from 193 7 to 1949. His only goal of life was an independent and undivided India.

Nationalism of Veer Savarkar: 

Veer Savarkar was a first – grade nationalist. He never – bowed down before the British Government. His nationalist thoughts are the central point of Indian political philosophy. According to him, the geographic expression of nationalism is the reflection of his Hindu nationalist culture. The man who considers the land from River Indus to coastal Bengal as his motherland is a Hindu and it is the only original parameter of nationalism. These thoughts must be infused in the heart of every Indian.

It does not matter to what caste or creed or belief he belongs to. It is the basis of our faith. Veer Savarkar said that the majority of Hindus would decide the future of India and its form. So,other groups and castes and religions should assimilate themselves in the social, economic and political life of India.They should mingle themselves in this country as sugar in milk and make the Indian culture the best.

To check the evangelists, he called upon the Hindus to avoid the caste system. He considered the caste system always inimical to the unity and strength of a country. He emphasis-ed to end the dividing nature of such type of evils. He supported intercaste marriage. According to him, Jains, Sikhs, Aryas and Brahma Samaj is are all Hindus.

He called upon the people who had converted their religion to come back in their original religion after Vedic purification. He warned them that if they did not do so, then there was no room in India for them. Hindus should play a deciding role in the fight for freedom, on the question of Islamic and Christian equality in India, he said,they might be equal to Hindus if they consider this country as their motherland and as their holy land.

He called upon the Indian youth and said that the foreign clothes that they were wearing with pride, must be burnt by them at once without any hesitation. They must take an oath before the holy fire to adopt everything that is made in their country and to dedicate themselves in the service of their motherland.

Veer Savarkar made an important contribution to India by successfully persuading the learned and prosperous youth to join the freedom movement. He encouraged them to follow Shivaji to weaken and overcome the British oppression. He also emphasis-ed on Hindu-Muslim unity. His sacrifices are the footprints of service to motherland for the modern Indian youth.

In the mechanical age of today man has become totally dependent upon machines. He has lost muscular energy and has become the victim of tension and worries leading to many physical and mental ailments. 

It is very essential in such a situation to understand the significance of Physical Education, as underlined below :

1. All round development of the child. 

2. Physical fitness and wellness 

3. Cultivation of Leadership qualities. 

4. Healthy physical growth and development 5. Formation of sound conduct. 

6. Development of Neuro-muscular coordination. 

7. Cultural development. 

8. Development of good and healthy habits. 

9. Development of healthy emotional expressions.

10.Proper use of the leisure time.

11.National unity and integrity.

12.Inter-state and international understanding and amity. 

13.Spirit of socialisation. 

14.Development of self expression qualities in children. 

15.Development of spirit of creativity and expression. 

16.Financial gains. 

17.Promotion of sense of discipline, tolerance and communal amity. 

18.Relaxation of mind and body.

1. Able Leadership 

2. Ample Facilities 

3. Likelihood of participation in physical activities by each individual and group. 

4. Creation of situations for physical fitness, mental alertness and social stability or soundness.

The ultimate and the final aim of the Physical Education is the wholesome development of an individual and the group of individuals. The able leadership, ample time, adequate facilities etc. are the means or objectives to attain the aim, and the physical exercises and the activities are the fields of Physical Education.

Pt. Jawaharlal Nehru was born at Allahabad on November 14, 1889.

Pt. Jawaharlal Nehru.

“The objectives of Physical Education are characterised by fitness of different organs of the body, Neuro-muscular coordination, development of skill, behaviour and personality”. According to it, the aim of Physical Education must be to make every child physically, mentally and emotionally fit, and also to develop in him such personal and social qualities as will help him to live happily with others and build him up as good citizens.

Aims of Physical Education according to Charles A Butcher are :

1. Physical Development Aims. 

2. Movement and Motor development aims. 

3. Cognitive and Mental Development aims. 

4. Social Development aims.

Just as the ultimate aim of a Basket ball player is to basket the ball, the same way the utmost aim of Physical Education is to prepare the wholesome citizens through various physical activities and games, and to make them energetic, stimulating, enthusiast and disciplined so that they may face all the adverse situations and find an effective solution to them.

Physical Development objectives: The objectives included in this category are those by which man develops physically by making his body strong, healthy and attractive. He does physical activities and takes active part in games which strengthen him and accelerate his activities like running, weightlifting, catching, jumping, climbing up and down etc.

Mental Development objectives: They are such objectives which relieve a child of his mental tensions and stress. Children are trained to think properly and they are enabled to solve many intricate problems. Thus they are helped to achieve their aim. They learn to face many difficult situations, develop confidence and take independent decisions.

Social Development objectives: In this category those objectives are included, which develop qualities like leadership, tolerance, cooperation, fair play, boldness, selfdiscipline, sportsmanship, self-expression etc. A society can prosper in a healthy social environment and such type of environment is possible through the programmes of Physical Education.

• Panchjanya 
• Rashtradharma 
• Swadesh Patrika.

Pt. Deendayal Upadhyaya was born at Nagla Chandrabhan in Mathura district, on 25thSeptember, 1916.

Life and thoughts of Pt. Deendayal:

Pt. Deendayal Upadhyaya was born on 25 September, 1916, in Nagla Chandrabhan village in Mathura district. His parents died after few years of his birth. So, he was supported and brought up by the maternal side of his family. He left his home for the sake of the service of the nation and became a propagator of Rashtriya Swayam Sewak Sangh. He had deep faith in eternal culture of India.

The brief description of his thoughts is given further: 

1. Integral Humanism:

Integral Humanism is the basic concept of Pt. Deendayal Upadhyaya’s thoughts. The dimensions of his concept include all the aspects of human life.

Indian customs and traditions: 

The human being is integral, it means, the human being cannot be divided into parts. He is a body of society in the form of human. According to Pt. Deendayal’s concept, human beings need both freedom as well as equality.

They are not considered as rival,but complementary to each other. When mankind wishes to overcome nature and start an uncontrolled consumption of nature, a danger arises upon the existence of human culture. The earthly resources are the means of human satisfaction and happiness but not an end. The bad customs,such as untouchability, caste difference, death feast, dowry system and oppression of women are the social evils spread in the society and in Indian culture,and great men have struggled against these.

Everybody in the world believes in capitalism, communism and socialism, while our culture opposes these, as the strong as well as the weak both live in the same society happily The basic need of life and society and the country is cooperation instead of struggle. The nature has its cycle. All the living beings, creatures and vegetation are made for one another. We need oxygen, while trees need carbon dioxide. The fact is that both are complementary to each other.

2. Western Dualism and Indian Monoism:

Clash is not suitable for creation. Unity in diversity has been our glorious heritage. A couple is different from each other before marriage, but after marriage, there is a oneness. In western culture, it is considered an agreement, but we take oath to redeem the relation during this life, and even after life.

We have adopted the concept of eternity in life, while the western thought does not believe in it. According to saint Ramanand, “Jati-pati poochey nahin koi, Hari ko bhaje so Hari ka hoi” Means, in India the cast or creed is not asked for, and he who worships the almighty, belongs to God.

3. Selfishness and Altruism: 

We know nothing about capitalism or socialism, but believe in integral philosophy. We believe in unity and union. We have a spiritual inside ourselves. Taking this godly feeling, we make progress West has faith in selfishness, but we give importance to welfare of all.

All the social sciences, political science, sociology, economics stand on the basis of selfishness, as they all alone are helpless, and so, due to their selfishness, they become united for vested interest, but we promote welfare of other, an activity, that we do not as work but for pleasure.

It is a godly imagination. We go ahead with a viewpoint of oneness and perfection. Duty is the basis of life. If spirit is not happy, body will also not happy. Thus, spiritual pleasure is also necessary. The synthesis of body, mind, wisdom and soul is the individual. So, the pleasure must be of body, mind, intellect and soul.

(b) Books and directions

Integral Humanism.