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A sequence is called an arithmetic progression if its terms continually increase or decrease by the same number. The fixed number by which they increase or decrease is called the common difference. 

Three quantities a, b, c will be in A.P. if b – a = c – b, i.e., 2b = a + c.

After inserting x A.M's between 2 and 38, we get an A.P. of (x + 2) terms with first term as 2 and last term as 38. 

Now, sum of n terms of an A.P. = \(\frac{n}{2}\) (a + l), where a = first term, l = last term

∴ Here, 200 = \(\frac{(x+2)}{2}\)(2+38) ⇒ 200 = 20 (x + 2)

⇒ 20x + 40 = 200 

⇒ 20x = 160 

⇒ x = 8

Given, \(\frac{a_7}{a_{10}}\) = \(\frac{5}{7}\)

Let the first term and common difference of AP be ′A’ and ′D’, respectively.

∴ \(\frac{A\, +\, 6D}{A\, +\, 9D}\) = \(\frac{5}{7}\)

⇒ 7A + 42D = 5A + 45D

⇒ 7A = 3D

⇒ A \(\frac{3}{2}\) D ….. (i)

Now, \(\frac{a_4}{a_7}\) = \(\frac{A\,+\,3D}{A\,+\,6D}\)

= \(\frac{\frac{3}{2}D}{\frac{3}{2}D}\) = \(\frac{3D}{6D}\) … . . (using(i))

= \(\frac{9D}{15D}\)

= \(\frac{3}{5}\)

Answer : (c) - 90

 T_n = a + (n – 1)d 

⇒ T₂₆ = a + 25d 

Here a = 10, d = – 4 

∴ T₂₆ = 10 + 25 × (–4) 

= 10 – 100 

= – 90.

Given A.P. is

a, b … c

First term (A) = a

Common difference (d) = b − a

Last term i.e., n^th term (A_n) = c

Using formula, A_n = A + (n − 1)d, we get

c = a + (n − 1)(b − a)

⇒ c − a + (n − 1)(b − a)

⇒ n − 1 = \(\frac{c−a}{b−a}\)

⇒ n = \(\frac{c−a}{b−a}\) + 1

⇒ n = \(\frac{b+c−2a}{b−a}\) …...(i)

Now, using formula S_n = \(\frac{n}{2}\)(A + A_n) for sum of n terms, we get

S_n = \(\frac{n}{2}\)[a + c]

= \(\frac{(b+c−2a)(a+c)}{2}\)          using (i)

= \(\frac{(a+c)(b+c−2a)}{2}\)

Answer : (d) 59

Let the first term, common difference and number of terms of the A.P. be a, d and n respectively. 

Then, 

a + d = \(7\frac{3}{4}\) ... (i)

a + 30d = \(\frac{1}{2}\) ...(ii) 

and a + (n – 1)d = − \(6\frac{1}{2}\)  ...(iii)

Eqn (ii) – Eqn (i)

⇒ 29d = \(\frac{1}{2}\) - \(\frac{31}{4}\) = - \(\frac{29}{4}\) 

⇒ d = - \(\frac{1}{4}\) 

Putting d = - \(\frac{1}{4}\)  in (i), we get

a - \(\frac{1}{4}\) = \(7\frac{3}{4}\) + \(\frac{1}{4}\)

⇒ a = 8 

∴ Putting the values of a and d in (iii), we have

8 + (n – 1) ( - \(\frac{1}{4}\) ) = - \(\frac{13}{2}\)  

⇒ 8 + \(\frac{1}{4}\)- \(\frac{1}{4}\)n = - \(\frac{13}{2}\)

⇒ −\(\frac{1}{4}\)n  = − \(\frac{13}{2}\) − \(\frac{33}{4}\) 

⇒ − \(\frac{1}{4}\)n = − \(\frac{59}{4}\)

⇒ n = 59

The reason for this is the leakage of Methyl isocynate into the air from an insecticide factory managed by union carbide.

Air pollution causes certain diseases including shortness of breath, sore throat, chest pain, nausea, asthma, bronchitis and lung cancer.

Diseases of the respiratory system like lung cancer, Asthma etc.

The white pine trees are found at the height of 2000 – 3000 m in the western Himalayan region.

(a) Jammu – Kashmir

The trees of sal (oak) are found in the Terai region on the lower slopes of Himalayas. They are also found in Uttaranchal, Uttar Pradesh, Madhya Pradesh, Chhattisgarh, Jharkhand, Bihar and Orissa.

The trees of sagwan (Teak) are found in southern Rajasthan, Madhya Pradesh, Chhattisgarh, Maharashtra, Kerala, Tamil Nadu and Odisha.

n India teak is distributed naturally in the Peninsular region below 24oN latitude. The most important teak forests are found in Madhya Pradesh, Maharashtra, Tamil Nadu, Karnataka, and Kerala besides Uttar Pradesh (small extent), Gujarat, Orissa, Rajasthan, Andhra Pradesh, and Manipur.

• Tribal people are slow to change because. 
• Tribal practice simple occupations based on primitive technology. Most of the occupations fall into the primary categories such as hunting, gathering, and agriculture. There is no profit nor surplus making in such an economy. 
• Illiteracy among tribal is a major hindrance to their development.

The trees of cedar are mainly found in Kashmir, hilly areas of Punjab and Garhwal region.

Belief in black and white magic, ancestor worship, benevolent and malevolent spirits. Sacred Groves are an integral part of tribal society, a tract of forest land dedicated to local deities who are believed by the tribal people to be fiercely protective about the grove and the village.

• Hunting and fishing. 
• Simple agriculture and shifting cultivation. 
• Food gathering of forest products.

The woods obtained from the Himalayan Region are mainly divided into cedar wood, pine wood and white pine wood.

On administrative basis, Indian forests have been divided into three categories, that is, Protected forests, Conserved forests, and Unclassified forests.

The forests in which wood – cutting and animal grazing is prohibited are called as Protected Forests.

(c) Threcimax

These woods are divided as woods of monsoon forests, woods of Himalayan region and woods of dry forests.

• Animism: The custom of worshipping the soul or ancestors. 
• Manaism: The custom of worshipping force which is believed to move swiftly across the world and enter into objects and people, giving them powers that they previously lacked. 
• Naturism: The custom of worshipping elements of nature like river, stream, Sun, Moon, forest, etc.

(b) Anarchists

Jacques is the writer of ‘History of politics’.

The Female Infanticide (Prevention) Act was enacted in 1870

(a) Right of franchise

Locke considers man as a naturally intelligent and thoughtful creature. According to him, man naturally has the virtues of love, compassion, sympathy, co – operation, kindness, etc. According to Locke, state of nature was peaceful and ideal because of kind and logical nature of man. All the men were equal and free in-the state of nature. They used to protect their life and property themselves and lived life as per their own wish.

At that time, there was no king, no subjects, no ruler and no ruled. This was a state of freedom but there were no chaos because people followed natural laws and laws of morality, but they explained the natural laws as per their own wish and reason.

Man used to live in simplicity and freedom in the state of nature as is explained by Rousseau. He did not have the feeling of selfishness. His life was that of a ‘Noble Savage’. He was happy and content. But, this heavenly state of nature could not last long. The will to create property and family arose in man. Natural equality and freedom of‘ Noble Savage’ came to an end along with the advent of personal property. It became impossible to live together. Jealousy and conflicts arose and all the comforts and peace of state of nature came to an end. As a result, man felt the need to organize themselves into a state.

Swami Dayanand Saraswati strived for the progress of women and eradication of illiteracy.

(a) Inalienability

Human nature: 

According to Thomas Hobbes, Man basically was an unsocial, lonely, selfish and quarrelsome creature. He used to indulge in continuous strife for his selfishness. To get success in his endeavor, he resorted to lies, corruption and violence.

State of Nature: 

According to Hobbes, due to brutal tendencies of human nature, his state of nature was extremely painful and full of struggles. This was the state of continuous war and conflicts, in which, every person was in a state of undeclared war with the other. Thus, human life in the state of nature was lonely, unsafe and brutal.

(d)  Jean Bodin

John Locke is the propounder of social contract theory which gave two contracts.

• According to Hobbes, contract was made among individuals mutually. 
• King was not a part but a result of that contract.

Propounder of General Will theory is Rousseau.

(d) All the above

(c) Peace and co – operation

There are four elements of state: 

• Population 
• Fixed Territory 
• Government
• Sovereignty.

This is written by John Locke.

According to Prof Gettel, main features of evolution of state are as under: 

1.  State has developed from simple to complex. Initially, the structure of state was very simple, which became complex with the development of human life. 

2.  Initially, number of states was very less and area was limited, but gradually jaulation and area increased. 

3.  With the development of political awareness among people along with the development of state, (systems like) representative democracy and feudal government originated. 

4.  There were autocratic monarchies in the beginning of the development of states where individual freedom had no place. Gradually, importance of individual freedom increased, and compromise between the sovereignty of state and individual freedom was accepted.

5.  In the beginning, state and religion were connected, but presently, they are separated from each other. Today the concept of secular state is prevalent.

When jealousy and conflicts started in the state of nature, then men felt the need to organize themselves into the organization of state to get rid of that state of nature. For this, they entered into a mutual contract. According to the contract, every man surrenders his freedom, rights and power to society because society is group of people and it is made by the people.Thus, man alienates his freedom and rights and surrenders them to society and finally gets*it back as being part of society.

According to Rousseau, the contract which was made to end the state of nature, was done by two parties. One party was man as an individual and second party was men as a collective group. According to Rousseau, men do not surrender their freedom and rights after organizing themselves into state. The keep it with themselves, but not individually, rather collectively. Being a part of society, chaos and insecurity were removed from the society due to this contract and the state originated, which is representative of general will.

Movement of the whole organism from place to place is called locomotion.

Locomotion is helpful to animals and enables them to: 

(i) shift from unfavourable environment to favourable conditions. 

(ii) go in search of food, water and shelter. 

(iii) find its partner for reproduction and proper place for laying eggs or giving birth to young ones. 

(iv) Prevents from getting captured by predators.

The connective tissue covering around each fascicle is the perimysium.

The connective tissue surrounding the muscle fibre is called the endomysium.

The combination of transportation and communication is known as circulation.

(d) Afghanistan, Pakistan and India