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∵ The sum of all three angles of a triangle is always 180°.

∴ ∠P + ∠Q +∠R = 180°
⇒ ∠P + 25° + 65° = 180°
⇒ ∠P + 90° = 180°
⇒ ∠P = 180° – 90° = 90°

∴ ∆PQR is a right angled triangle where ∠P = 90°

∴ From Pythagorous Theorem PR² + PQ² = QR²
= QR²

∴ (ii) is true.

Let ABCD is a rectangle where AB = 4.5 cm and AC = 5.5 cm

In right angled triangle ∆ABC, ∠B = 90°.

∴ From Pythagorous Theorem

BC² = AC² – AB² = 5.5² – 4.5²

= (5.5 + 4.5) (5.5 – 4.5) = 10 x 1

BC² = 10

BC= 3.16 m

∴ Perimeter of rectangle = 2(AB + BC)

= 2(4.5 + 3.16) cm

= 2 x 7.66

= 15.32 cm

(i) Acute angle

(ii) Obtuse angle

(iii) greater

(iv) isosceles triangle

(v) median

(vi) centroid

(vii) altitudes

We have the relation, 

R = {(T₁, T₂) : T₁ ≅ T₂}

Reflexivity : As Each triangle is congruent to itself,

i.e., T₁ ≅ T₂ ∀ T₁ ∈ T

Thus, R is reflexive.

Symmetry : Let T₁, T₂ ∈ T, such that

(T_1, T₂₎ ∈ R

⇒ T₁ ≅ T₂

T₂ ≅ T₁ 

⇒ (T_2, T₁) ∈ R

i.e., R is symmetric.

Transitivity : Let T₁, T_2, T₃ ∈ T, such that (T₁, T₂) ∈ R and (T₂, T₃) ∈ R

⇒ T₁ ≅ T₂ and T₂ ≅ T₃

⇒ T₁ ≅ T₃ ⇒ (T₁, T₃) ∈ R

i.e., R is transitive.

Hence, R is an equivalence relation.

(T₁, T₁) ∈ R ⇒ T₁ is similar to T₁ which is true. 

Hence R is reflexive 

(T₁, T₂)∈ R ⇒ T₁ is similar to T₂ 

⇒ T₂ is similar to T₁ 

⇒ (T₂, T₁) ∈ 2 

hence R is symmetric 

(T₁, T₂) ∈ R, (T₂, T₃) ∈ R 

⇒ T is similar to T₂, T₂ is similar to T₃ 

⇒ T₁ is similar to T₃ ⇒(T₁,T₃) ∈ R 

hence R is transitive 

∴ R is an equivalence Relation. 

T₁ is similar to T₃ as hence T₁ and T₃ are related.

(i) Reflexive 

R is reflexive if T₁ R T₁ 

Since T₁ ≅ T₁ 

∴ R is reflexive. 

(ii) Symmetric 

R is symmetric if T₁ R T₂ ⇒ T₂ R T₁ 

Since T₁ ≅ T₂ ⇒ T₂ ≅ T₁ 

∴ R is symmetric. 

(iii) Transitive 

R is transitive if T₁ R T₂ and T₂ R T₃ ⇒ T₁R T₃ 

Since T₁ ≅ T₂ and T₂ ≅ T₃ ⇒ T₁ ≅ T₃ 

∴ R is transitive From (i), (ii) and (iii), we get R is an equivalence relation.

Answer is (C) x + y = 0

Answer is (B) a₁b₁ + a₂b₂ + a₃b₃

Answer is (A) I² = 1

Correct option is C) eight

Lencho was a simple and a poor farmer who had a firm faith in God. When he asked God to send him a hundred pesos, he received only seventy pesos in reply. Due to his faith in God, he blamed the post office employees for stealing his money, as he could not believe that got could send him any less money. He doubted those employees. But he was not right to call them ''a bunch of crooks'', as they helped him in his problem. 

His reaction towards the post office employees was not right justified but it was just because of his innocence that he could not believe that God has done such a mistake. They were the post office employees who had stolen the money according to him.

In 1951 the population of Delhi was about 20,00,000.

1. In 1951, the population of Delhi was only 20,00,000. Today it is more than 1,60,00,000. During the last 60 years it has grown 8 times. 

2. Over decades its attracted people from all parts of India as they migrated in search of jobs and livelihood, etc.

3. As the capital of the country, with the parliament and central government offices, there are people from all parts living in the city. 

4. Every city usually has a master plan to design and allocate different types of areas and Delhi had three such plans. 

5. But they have not been implemented. 

6. On one hand there is growing population in the city and on the other hand there is a long delay In planning and announcing how the areas are to be used. 

7. In fact, Delhi has grown in an unplanned manner.

Lencho was a poor farmer. Once the hailstorm destroyed his harvest. He had firm faith in God. So he wrote a letter to God, asking Him to send him money. The postmaster saw this letter. He decided to help Lencho. He collected some money with the help of his employees. He put it inside an envelope. When Lencho came to the post-office to check his mail, the envelope was delivered to him. Lencho opened it and took out the money. He found that it was less than he had asked for. He became angry. He wrote another letter to God to send him the rest of money, but not through mail because the post-office employees were a bunch of crooks.

The population of Delhi increased mostly in the decades of 1981 and 1991. 

I think the reasons are:

1.  It is our national capital. 
2.  Its development.

At the beginning the rulers encourage the crafts persons to settle in urban area. Slowly the facilities are increasing In urban areas. People who are not having land they are migrating to towns & cities and improve their living conditions. After some years industrials are established they left their occupations and joined in industries for better life. 

Slowly town areas are increasing, population also increasing and the people who depends on agriculture Is also slowly decreasing. The share of agriculture in GDP also reducing. But In urban areas the people are facing so many problems.

1. The city of Delhi can claim to have been central to many empires that rilled India. 

2. When India gained independence, the city remained its capital. 

3. Over the decades, the city has attracted people from all parts of India as they migrated in search of livelihood, jobs, etc. 

4. As the capital of the country, with the Parliament and Central Government offices there are people from all parts living in the city. 

5. Surrounding areas of Delhi are developed.

6. Establishment of industries in surrounding areas of Delhi.

Visakhapatnam has a long history. It was ruled by different dynasties during pre-colonial times. During the 19th century, the British and French fought a naval battle over this city. Coastal places were of immense significance for colonial powers because they could build ports there. These ports would enable export of raw materials to the colonizing country.

As settlements became larger, they also became more complex because there were more functions to be performed – food production was not the quility concem. Within each settlement, people specialised in certain skills. Many goods were produced for the market and sold to traders who carried them to far off places.

1. Due to his confidence Lencho did not show any surprise on seeing the money.He became angry because the money was short.

2. Lencho asked for paper and ink to write a letter to God. He licked the stamp and affixed it to the envelope with a blow of fist.

3. Lencho got only seventy pesos.

4. Lencho requested God not to send money through the mail because post office employees were a bunch of crooks.

Let A be an event of always 4 exhibits on second dice

i.e., A = {(1,4),(2,4),(3,4),(4,4),(5,4),(6,4)}

n(A) = 6

and B be an event of the numbers appeared has a sum of 8 

i;e., B = {(4,4)}

∴ A ∩ B = {(4,4)}
∴ n(A ∩ B) = 1

Hence, P(B/A) = n(A ∩ B)/n(A) = 1/6

As population increased, there was more specialization – weavers, potters, metal workers and other professions emerged. The number and variety of goods produced increased and so did the trade in them. Rulers began to encourage craft persons to settle in urban areas. Urban settlements, i.e., towns where people did not work in agriculture expanded.

Urbanisation: 

The migration of people from rural areas to urban areas is called urbanisation. 

Causes of urbanisation:

1. Natural growth 
2.  Inclusion of rural areas and 
3.  Migrations 

Challenges of urbanisation: 

1. The rapid inflows of rural population to urban places give rise to housing problem and thus slums are developed in these places.
2. The decrease in rural population effects the agricultural production due to shortage of workers in rural areas. 
3.  Due to use of more vehicles, air, water, sound pollutions are increased. 
4.  Traffic problem will arise. 
5.  Proper drainage facilities are not expanded. 
6.  Using of plastic covers are increased. 
7.  Unemployment increases in urban areas. 
8.  Due to this, various criminal activities, corruption, etc. increase affecting the law and order system.

The cities having more than 10 million persons are Mega cities.

Correct option is A) Berhampur

1.The postmaster was collecting the hundred pesos to send to Lencho.

2. The postmaster put a little more than fifty pesos in the envelope. The envelope was addressed to Lencho.

3. The next day Lencho went to the post-office if there was a letter for him. The postman handed him a letter.

4. The postmaster decided to answer the letter addressed to God not to shake the writer’s faith in God.

5. The postmaster collected the money to be sent to Lencho from his employees and friends. He himself gave a part of his salary.

6. Postman himself handed the letter to Lencho. The postmaster was experiencing the contentment of a man who had performed a good deed.

7. Lencho was angry because the money was short.

Given that, f(x) = tan^-1(sin x + cos x), x > 0

∴ f(x) = 1/(1 + (sin x + cos x)²).(cos x - sin x)

= cos x/(1 + (sin x + cos x)²).(1 - tan x) > 0, ∀ x ∈ (0,π/4)

Hence, f(x) > 0 in (0,π/4) 

Therefore f(x) is an increasing function is (0,π/4)

An aerotropolis is an urban plan in which the layout, infrastructure and economy is centered on an airport, existing as an airport city. 

(or) 

The settlements which centered around large airports are called aerotropolis. 

Advantages:

1. People can fly to conduct their business with their counterparts right there, and fly out – with all the comfort of a city. 

2. Many facilities like hotels, shopping, entertainment, food, business, conferencing, etc, are provided right there. People and Settlement

C) Kempegouda international airport

People can fly in, conduct their business with their counterparts right there, and fly out- with ail ‘ the comfort of a city, without the traffic and other problems.

Correct option is D) Kolkata

A) Suyarna Bhoomi International Airport

Aerotropolis: An Aerotropolis is an urban plan in which the layout, infrastructure, and economy Is centered on an airport, existing as an airport city

Hotels, shopping, entertainment, food, business conferencing, etc.

tan^-1 2x + tan^-1 3x = π/4

tan^-1(2x + 3x)/(1 - 6x²) = π/4

tan^-1 5x/(1 - 6x²) = 1

5x = 1 - 6x²

6x² + 5x -1 = 0

6x² + 6x - x - 1 = 0

(x + 1)(6x - 1) = 0

x = -1, x = 1/6

D) Suvarna Bhoomi international airport

Correct option is (c) statement 2 is the result of statement 1

(iv) Both (A) and (R) are true and (R) is not the correct explanation of (A).

Wrong Pair: 

Improvement in quality of life – 12th five-year plan.

(iii) Both (A) and (R) are true and (R) is the correct explanation of (A).

(iii) Both (A) and (R) are true and (R) is the correct explanation of (A).

(iii) Both (A) and (R) are true and (R) is the correct explanation of (A).

1. Faster sustainable growth

2. 12th Five year plan

3. To reduce fertility rate to 2.1%

4. Ex-Officio Members

5. Mobilisation of Resources

6. Annual Plans.

Humid Farming.

(a) Mixed Cropping

(b) Inter Cropping

Targets for infrastructure are:

• To connect all villages of a country with all-weather roads. 
• To increase rural television and telephone density to 70%. 
• To increase infrastructure investment to 9% of G.D.P. 
• To upgrade national and state highways to a minimum two-lane standard. 
• To achieve real GDP growth rate at 8%, agriculture growth rate at 4%, and manufacturing and industrial growth rate at 10 %.
•  To reduce the headcount ratio of poverty by 10%. 
• To create 50 million (5 crores) new work opportunities in the non-farm sector.
• To increase average years of schooling to 7 years. To eliminate gender and social gap in school enrollment. To reduce the total fertility rate to 2.1%.

Targets for infrastructure are: 

• To connect all villages of a country with all-weather roads.
• To increase rural television and telephone density to 70%. 
• To increase infrastructure investment to 9% of G.D.P. 
• To upgrade national and state highways to a minimum two-lane standard. 
• To achieve real GDP growth rate at 8%, agriculture growth rate at 4%, and manufacturing and industrial growth rate at 10%. 
• To reduce the headcount ratio of poverty by 10%.
• To create 50 million (5 crores) new work opportunities in the non-farm sector. 
• To increase average years of schooling to 7 years. 
• To eliminate gender and social gap in school enrollment.
• To reduce the total fertility rate to 2.1%.

In Jodhpur district.

Agriculture holds a significant position in the state’s economy where half of the state income comes from agriculture and allied activities. Irrigation is very important in agricultural production. 

There is a lack of groundwater sources in the state.

Place of Rajasthan from the Point of view of agriculture in India : Rajasthan is an agriculture based state where 70% of the population is dependent upon farming. Agriculture holds a significant position in the state’s economy where half of the state income comes from agriculture and related activities. In 2009-10 in Rajasthan, 11.3% of total agricultural area of India was there, whereas Uttar Pradesh holds the first position in India and the state holds the third. Rajasthan has become a leading state in the production of mustard and black mustard seeds and half of their total production in the country takes place in Rajasthan. In 2013-14, from the point of view of food grain production, Rajasthan’s total production was 7.5% of India.

Place of Rajasthan from the point of view of Industry in India : The economy of Rajasthan is slowly developing, and with industrial point of view, it is underdeveloped. But the state government is putting a lot of efforts in creating a conductive environment for industries. Rajasthan is a backward state from the point of view of factory area but the state’s status is mentionable in handloom and handicrafts. The number of factories in Gujarat and Maharashtra is about 2.5 times of Rajasthan.

Place of Rajasthan from the view of surface Area in India : Total surface area of Rajasthan is 3,42,239 square kilometers, which is about 10.41% of India. Whereas the area of Gujarat is approx. 6% and area of Uttar Pradesh is approx 7.3%. Around 61% part of Rajasthan is desert. Therefore, it is clear that Rajasthan is the largest state in terms of surface area.