SSC CGL 2024 Tier-I Official Paper (Held On: 10 Sept, 2024 Shift 3) (Previous Year Paper)

The logic followed here is:

(121, 93, 214)

121 + 93 = 214 

214 = 214 (LHS = RHS)

And,

(115, 87, 202)

115 + 87 = 202 

202 = 202 (LHS = RHS)

So, checking each option one by one:

Option 1) (135, 117, 259)

135 + 117 = 259 

252 ≠ 259 (LHS ≠ RHS)

Option 2) (110, 94, 224)

110 + 94 = 224 

204 ≠ 224 (LHS ≠ RHS)

Option 3) (84, 99, 189)

84 + 99 = 189 

183 ≠ 189 (LHS ≠ RHS)

Option 4) (97, 112, 209)

97 + 112 = 209 

209 = 209 (LHS = RHS)

Thus, out of all the options, '(97, 112, 209)' follows the same logic as given.

Hence, "Option 4" is the correct answer.

The logic followed here is:

1. Alternately one '

' element is replaced by '

' starting from the first box.

2) Alternately one '

' element is replaced by '

' starting from the second box.

Thus, the final series is:

Hence, "Option 4" is the correct answer.

The logic followed here is :-

From the given two different cubes the adjacent sides are as shown below :

As 8 is common on both the dices, it is clear that 3, 7, 2, and 9 are adjacent to it, and the remaining i.e. 4 will be opposite to it.

Opposite pairs:

3 → 2

7 → 9

8 → 4

So, the opposite side of "4" will be side "8".

Hence, the correct answer is "Option 1".

The number of squares in the given figure is shown below:

Thus, there are total 13 squares.

Hence, "Option 4" is the correct answer.

The logic followed here is:

Logic: Degree of intensity/quantity.

Here,

Presence : Abundance → Abundance is a higher degree of Presence.

Similarly,

Infinite : Cosmic → Cosmic is a higher degree of Infinite.

Hence, the correct answer is "Option 2".

The logic followed here is :-

From the given three different cubes the adjacent sides are as shown below :

As K is common on both the dices I and III, it is clear that M, O, N, and L are adjacent to it, and the remaining i.e. P will be opposite to it.

Opposite pairs:

M → N

O → L

K → P

So, the opposite side of "M" will be side "N".

Hence, the correct answer is "Option 2".

Given: 15 × 9 + 7 ÷ 26 – 13 = ?

According to the question, after ‘÷’ and ‘–‘ are interchanged and ‘×’ and ‘+’ are interchanged:

15 × 9 + 7 ÷ 26 – 13 = ? 

⇒ 15 + 9 × 7 - 26 ÷ 13 = ?

⇒ 15 + 9 × 7 - 26 ÷ 13 = ?

⇒ 15 + 9 × 7 - 2 = ?

⇒ 15 + 63 - 2 = ?

⇒ 78 - 2 = ?

⇒ 76 = ?

Hence, "Option 3" is the correct answer.

Decoding symbols:

Given: 5 − 2 × 4 ÷ 3

5 − 2 → 5 is the brother of 2.

2 × 4 → 2 is the wife of 4.

4 ÷ 3 → 4 is the father of 3.

Thus, 5 is 3's mother’s brother or maternal uncle.

Hence, the correct answer is "Option 2"

The logic followed here is:

Option 1) EJM

Option 2) DIM

Option 3) FKN

Option 4) CHK

Thus, out of all the options, 'DIM' is the odd one out.

Hence, "Option 2" is the correct answer.

The logic followed here is:

1. All the elements shift clockwise including the center element.

Thus, the final series is:

Hence, "Option 4" is the correct answer.

The logic followed here is:

62 is related to 79

62 + 2 = 64 → Perfect square of 8

79 + 2 = 81 → Perfect square of 9

And,

98 is related to 119

98 + 2 = 100 → Perfect square of 10

119 + 2 = 121 → Perfect square of 11

Similarly,

14 is related to ?

14 + 2 = 16 → Perfect square of 4

? + 2 = 25 → Perfect square of 5

? = 25 - 2

? = 23

Hence, "Option 2" is the correct answer.

The logic followed here is:

Hence, "Option 4" is the correct answer.

The logic followed here is:

Hence, "Option 2" is the correct answer.

Given word: WISDOM

According to the question,

Thus, the new formed letter cluster → D I M O S W

Now, the number of letters are there in the English alphabetical series between the letter which is third from the left end and the one which is second from the right end:

(Left end) D I M O S W (Right end)

Thus, five letters i.e N, O, P, Q, and R are between M and S.

Hence, "Option 3" is the correct answer.

The correct mirror image of the given figure is:

​​

Hence, the correct answer is "Option 2".

Given: 11 ÷ 12 – 78 × 6 + 57 = ?

According to the question, after ‘+’ and ‘ – ‘ are interchanged and ‘×’ and ‘÷’ are interchanged:

11 ÷ 12 – 78 × 6 + 57 = ? ⇒ 11 × 12 + 78 ÷ 6 - 57 = ?

⇒ 11 × 12 + 78 ÷ 6 - 57 = ?

⇒ 11 × 12 + 13 - 57 = ?

⇒ 132 + 13 - 57 = ?

⇒ 145 - 57 = ?

⇒ 88 = ?

Hence, "Option 4" is the correct answer.

The logic followed here is:

Option 1) 81 – 118 – 91

81 ÷ 3 + 91 = 118

27 + 91 = 118

118 = 118 (LHS = RHS)

Option 2)  52 – 68 – 51

52 ÷ 3 + 51 = 68

17.33 + 51 = 68

68.33 ≠ 68 (LHS ≠ RHS)

Option 3) 99 – 104 – 71

99 ÷ 3 + 71 = 104

33 + 71 = 104

104 = 104 (LHS = RHS)

Option 4) 21 – 38 – 31

21 ÷ 3 + 31 = 38

7 + 31 = 38

38 = 38 (LHS = RHS)

Thus, out of all the options, '52 – 68 – 51' is the odd one out.

Hence, "Option 2" is the correct answer.

The figure which will look like the paper when unfolded is shown below:

Thus, option Figure 4 will look like the paper when unfolded.

Hence, "Option 4" is the correct answer.

Given series: P Q _ S _ P P Q P _ A P P Q _ S A P _ Q

Option 1) QPASP→ P Q Q S P P P Q P A A P P Q S S A P P Q → (It doesn't follow any pattern)

Option 2) PQSAP → P Q P S Q P P Q P S A P P Q A S A P P Q → (It doesn't follow any pattern)

Option 3) PASPP → P Q P S A P P Q P S A P P Q P S A P P Q → (It follows the pattern of: PQPSAP - PQPSAP - PQPSAP - PQPSAP - PQ)

Option 4) QSPAP → P Q Q S S P P Q P P A P P Q A S A P P Q → (It doesn't follow any pattern)

Hence, "Option 3" is the correct answer.

The logic followed here is:

SWEET : WSETE

And,

SALTY : ASLYT

Similarly,

DRINK : ?

Hence, "Option 1" is the correct answer.

According to the given Venn diagram:

The shaded region represents the total number of women entrepreneurs who  are NOT Indian.

So, the total number of women entrepreneurs who  are NOT Indian = 18

Hence, the correct answer is "Option 4".

According to the given information,

Thus, according to the given codes the code 'gq' is for the word 'more'.

Hence, the correct answer is "Option 3".

The logic followed here is:

(173, 246)

173 × 2 - 100 = 246 

346 - 100 = 246 

246 = 246 (LHS = RHS)

And,

(148, 196)

148 × 2 - 100 = 196 

296 - 100 = 196

196 = 196 (LHS = RHS)

So, checking each option one by one:

Option 1) (142, 176)

142 × 2 - 100 = 176 

284 - 100 = 176

184 ≠ 176 (LHS ≠ RHS)

Option 2) (169, 247)

169 × 2 - 100 = 247 

338 - 100 = 247

238 ≠ 247 (LHS ≠ RHS)

Option 3) (127, 194)

127 × 2 - 100 = 194 

254 - 100 = 194

154 ≠ 194 (LHS ≠ RHS)

Option 4) (154, 208)

154 × 2 - 100 = 208 

308 - 100 = 208

208 = 208 (LHS = RHS)

Thus, out of all the options, '(154, 208)' follows the same logic as given.

Hence, "Option 4" is the correct answer.

The logic followed here is:

‘VERY’ is written as ‘6314’

‘VARY’ is written as ‘1438’

Thus, 'A' is written as '8'.

Hence, "Option 2" is the correct answer.

According to the given Venn diagram:

The shaded region represents the total number of smart people who are polite.

So, the total number of smart people who are polite = (12 + 7) = 19

Hence, the correct answer is "Option 1".

The correct answer is Vakataka

Key Points

• Prabhavati Gupta married Rudrasena II of the Vakataka dynasty:
• Prabhavati Gupta
• The daughter of Chandragupta II, the ruler of the Gupta Empire, and queen Kuberanaga.
• She was a powerful monarch in early Indian history.
• Rudrasena II
• The ruler of the Pravarapura-Nandivardhana branch of the Vakataka dynasty.
• He ruled in the Deccan region to the south of the Gupta empire.
• Rudrasena II's reign was short, from around 385–390 CE.
• He died early, leaving his sons Divakarasena, Damodarasena, and Pravarasena II minors.
• Prabhavati Gupta ruled as regent for an extended period of time.
• Rudrasena became a devotee of Chakrapani or Vishnu, unlike his ancestors who were all devout Shaivites.
• His change of faith was probably encouraged by his father-in-law and his queen, who were both staunch Vaishnavites.

Additional Information

• Pushyabhuti
• Also known as the Vardhana dynasty, this dynasty ruled in northern India during the 6^th and 7^th centuries.
• The dynasty's last ruler, Harsha Vardhana, ruled from Kanyakubja (modern-day Kannauj, Uttar Pradesh) and extended his empire to Kamarupa in the east and Narmada River in the south.
• Chalukya
• This dynasty ruled in central and southern India, consolidating its power after defeating the Pallavas and the Gangas.
• The Chalukyas expanded their power over the Deccan Plateau after Pulakesin I.
• Pallava
• This dynasty ruled in northern Tamil Nadu and was a major power for a brief time.
• The Pallavas' art and architecture include the Vaikuntha Perumal Temple at Kanchipuram, the Shore Temple, and the Pancha Rathas of Mahabalipuram

The correct answer is Uttarakhand

Key Points

• Madhuri Barthwal is a folk singer from Uttarakhand, India.
• She was awarded the Padma Shri in 2022 for her contributions to the preservation and promotion of folk music in the region.
• She was the first woman to become a music composer for All India Radio.
• She is also said to be the first female musician from Uttarakhand to become a music teacher.
• She was awarded the Nari Shakti Puraskar in 2019 by Ram Nath Kovind.
• She has also received the Uttarakhand Ratna and Uttarakhand Bhushan awards.
• She has documented and taught folk music traditions, and has also written five books on Garhwali folk music.
• Her research work, Raag Raganiya in Garhwali folk songs, explains how Indian classical ragas have been incorporated into Garhwali folk song

The correct answer is Parliament

Key Points

• The legislative organ of the Union government is Parliament.
• Parliament is made up of the President, the Rajya Sabha, and the Lok Sabha:
• Rajya Sabha
• Also known as the Council of States, this is the upper house of Parliament.
• It has 245 members, with 233 representing states and union territories and 12 nominated by the President.
• The Rajya Sabha is permanent and cannot be dissolved.
• Lok Sabha
• Also known as the House of the People, this is the lower house of Parliament.
• It is a temporary house that can be dissolved if the party in power loses the support of the majority of the house.
• President
• The President has the power to summon and prorogue either House of Parliament or to dissolve Lok Sabha.
• Parliament is the supreme legislative body of India and the highest representative authority of the people.
• However, the Supreme Court can review Parliament's laws, and the President and Vice President are part of the Executive branch.

Additional Information

• Judiciary
• Interprets laws, settles disputes, and administers justice.
• For example, the Supreme Court interprets laws and can declare them unconstitutional.
• Legislative assembly
• Makes laws and procedures, and examines the administration and its resolutions. In a state with one house, it is known as the Legislative Assembly (Vidhan Sabha).
• In a state with two houses, the lower house is known as the Legislative Assembly (Vidhan Sabha) and the upper house is known as the Legislative Council (Vidhan Parishad).
• Executive
• Implements the laws enacted by the legislature and enforces the will of the state. For example, the President enforces laws and has the power to veto legislation. 
• Separation of Powers in India: Executive, Legislature ...
• The separation of powers is the division of these functions among separate and independent bodies.
• The purpose of separation of powers is to limit the possibility of arbitrary excesses by the government

The correct answer is Silicon

Key Points

• Silicon is the metalloid among the given elements:
• Silicon:
• A metalloid is an element that has properties of both metals and nonmetals.
• Silicon has a metallic luster but is not a good conductor of electricity.
• Silicon is used to make alloys with iron and aluminum in the steel and automotive industries.

Additional Information

• Iron:
• It is a metal
• Symbol: Fe; Atomic number: 26.
• It is a strong, malleable, and ductile metal widely used in construction and manufacturing.
• It is essential for human health; a key component of haemoglobin in red blood cells.
• Oxygen (O)
• Symbol: O; Atomic number: 8.
• It is a colourless, odourless gas making up 21% of Earth's atmosphere.
• It is  Vital for respiration in living organisms; and supports combustion.
• Sodium (Na)
• Symbol: Na; Atomic number: 11.
• A soft, silvery metal that reacts vigorously with water.
• It is essential for nerve function and maintaining fluid balance

The correct answer is the French Constitution

Key Points

• The Republic feature of the Indian Constitution was borrowed from the French Constitution:
• The ideals of liberty, equality, and fraternity in the Indian Constitution's Preamble are from the French Constitution.
• These ideals emerged during the French Revolution.
• Democratic principles
• The Indian Constitution's emphasis on democratic principles, republicanism, and the sovereignty of the people can be traced back to the French Constitution.

Additional Information

• The Indian Constitution has also borrowed features from other countries, including:
• South African Constitution:
• The procedure for amending the Constitution and the election of members of Rajya Sabha
• Japanese Constitution:
• The procedure established by law
• Australian Constitution:
• The concurrent list, freedom of trade, commerce, and intercourse, and joint sitting of the two Houses of Parliament
• Canadian Constitution:
• Federation with a strong center, vesting of residuary powers in the center, the appointment of state governors by the center, and advisory jurisdiction of the Supreme Court
• Irish Constitution:
• The Directive Principles of State Policy, nomination of members to Rajya Sabha, and the method of election of the president
• Weimar Constitution of Germany:
• The suspension of fundamental rights during emergencies

The correct answer is Odisha.

Key Points

• The government of Odisha has announced a social security scheme specifically for registered newspaper hawkers in the state.
• This initiative was taken considering the financial losses that newspaper hawkers faced during the COVID-19 pandemic.
• Newspaper hawkers are part of the unorganized sector, which often lacks the social security benefits available to organized workers.
• The scheme aims to provide financial assistance and security to newspaper hawkers to help them cope with their economic challenges.

Additional Information

• Social Security Schemes:
• Social security schemes are government initiatives aimed at providing financial support and security to individuals, especially those in the unorganized sector.
• These schemes can include health insurance, pension plans, and financial aid during unemployment or economic hardships.
• Such initiatives are crucial for ensuring the well-being of workers who do not have access to formal employment benefits.
• Unorganized Sector:
• The unorganized sector refers to employment that is not regulated by the government and does not provide standard employment benefits like health insurance, retirement plans, or job security.
• Workers in this sector include daily wage laborers, agricultural workers, domestic workers, and small-scale vendors, among others.
• Due to the lack of formal employment benefits, workers in the unorganized sector are often vulnerable to economic instability.
• COVID-19 Pandemic Impact:
• The COVID-19 pandemic caused significant economic disruptions worldwide, affecting both organized and unorganized sectors.
• Many workers in the unorganized sector faced job losses, reduced income, and financial hardships due to lockdowns and restrictions.
• Governments around the world introduced various relief measures and social security schemes to support affected individuals and stabilize the economy.
• Newspaper Hawkers:
• Newspaper hawkers are individuals who distribute newspapers to households, offices, and other establishments.
• They play a vital role in ensuring the timely delivery of news and information to the public.
• Being part of the unorganized sector, newspaper hawkers often lack formal employment benefits and face economic challenges during crises like the pandemic.

The correct answer is Deforestation due to mining

Key Points

• The main cause of land degradation in Jharkhand and Chhattisgarh is deforestation due to mining:
• Mining activities in these states have caused extensive damage to the land and environment.
• This is because mining involves deforestation, removal of topsoil, and excavation, which results in soil erosion, loss of fertility, and land degradation.
• Other factors
• Other factors that contribute to land degradation in these states include overgrazing, over-irrigation, and mineral processing.
• Sustainable practices
• To mitigate the negative impact of mining, sustainable mining practices, reforestation, and restoration of degraded lands are crucial.

The correct answer is below 30%.

Key Points

• The Accelerated Female Literacy Programme was initiated in districts where the female literacy rate was below 30% based on the 2001 census.
• This initiative aimed to address the significant gender gap in literacy rates.
• The program focused on empowering women through education, which is crucial for social and economic development.
• It included various measures such as adult education centers, literacy campaigns, and skill development programs to improve female literacy rates.

Additional Information

• Female Literacy Rate:
• The female literacy rate is the percentage of women who can read and write at a specified age, typically 15 years and older.
• It is a crucial indicator of gender equality and women’s empowerment.
• Higher literacy rates among women are associated with better health outcomes, lower fertility rates, and improved economic opportunities.
• 2001 Census:
• The 2001 Census of India was the 14^th census conducted in the country.
• It provided comprehensive data on various demographic parameters, including literacy rates, population distribution, and socio-economic indicators.
• The data from the census is used for planning and implementing various government policies and programs.
• Educational Initiatives for Women in India:
• Beti Bachao Beti Padhao (BBBP): Launched in 2015, this campaign aims to address the declining child sex ratio and promote the education and empowerment of girls.
• Saakshar Bharat Mission: Aimed at increasing adult literacy rates, particularly among women, by providing basic education and skill development.
• National Programme for Education of Girls at Elementary Level (NPEGEL): Focuses on enhancing the education of girls in educationally backward blocks.
• Importance of Female Education:
• Educated women are more likely to participate in the workforce, contributing to economic growth.
• They are better equipped to make informed decisions about health, nutrition, and family planning.
• Female education contributes to overall social development and reduces gender disparities.

The correct answer is Amazon Basin

Key Points

• The Amazon River Basin is the world's largest drainage basin, covering an area of about 7,000,000 square kilometers:
• The Amazon basin is located in the center and eastern parts of South America and includes parts of Brazil, Peru, Colombia, Ecuador, Bolivia, Suriname, French Guiana, and Venezuela.
• The Amazon basin is more than twice the size of the Congo River basin in Africa.
• It covers about 35.5% of the South American continent.
• The Amazon basin is home to the world's largest rainforest, which covers most of the basin and is also known as Amazonia.
• The Amazon rainforest is home to a huge variety of plant and animal species, including more than 40,000 plant species and 1,300 bird species.
• About one-fifth of all freshwater runoff on Earth passes through the Amazon basin.

Additional Information

• Congo Basin: The world's second-largest river basin, located in west-central Africa: 
• It Covers over 1.3 million square miles (3.4 million square km)
• It is the Home to the world's second-largest tropical rainforest, which is home to many endangered species, including gorillas, elephants, and chimpanzees
• It Provides water for agriculture and energy generation
• The Congo Basin is home to eight World Heritage Sites, including five that are in danger
• Amur Basin:
• It Drains an area of about 716,200 square miles (1,855,000 square km)
• Nile Basin:
• It Drains an area of about 1,293,000 square miles (3,349,000 square kilometers)
• It Includes parts of Tanzania, Burundi, Rwanda, the Democratic Republic of the Congo, Kenya, Uganda, South Sudan, Ethiopia, Sudan, and Egypt
• The longest river in the world, flowing through northeastern Africa and draining into the Mediterranean Sea

The correct answer is Manipur

Key Points

• The National Sports University (NSU) is located in Imphal, Manipur:
• The university is currently operating from its temporary campus at the Khuman Lampak Sports Complex in Imphal.
• It is about 10 kilometres from the Imphal airport.
• Prime Minister Narendra Modi laid the foundation stone for the university on March 16, 2018.
• It also has a new campus under construction in the West District of Manipur that will include sports infrastructure, academic and administrative blocks, a library, auditorium, conference hall, and hostels

The correct answer is They give birth to live young ones with some exceptions those lay eggs

Key Points

• The characteristic that is not of Aves is giving birth to live young ones with some exceptions those lay eggs:
• Aves are oviparous, meaning they lay eggs, while mammals are viviparous, meaning they give birth to live young.
• The only mammals that lay eggs are the platypus and the echidnas.
• Some other characteristics of Aves include:
• They are warm-blooded animals
• They have a four-chambered heart
• They have feathers covering their bodies
• Their forelimbs are modified into wings
• They breathe through lungs
• They have no teeth
• They have sharp eyesight
• They have a modified beak made from their upper and lower jaws

The correct answer is Sikkim

Key Points

• Sikkim has limited rail transport development due to its challenging geographical conditions, such as:
• Mountainous terrain and steep slopes.
• Dense forests and fragile ecosystems.
• Lack of large plains suitable for laying railway tracks.
• Sikkim is a small Himalayan state with mountainous terrain.
• This makes it difficult and expensive to construct and maintain railway tracks.
• The challenging topography limits the potential for rail transport development in the state.
• While Sikkim is working on developing its first railway line, the Sivok-Rangpo railway project, the state's geographical conditions pose significant challenges to rail transport development.

The correct answer is The Discovery of India- Subhash Chandra Bose

Key Points

• The incorrect pairing is: 
• The Discovery of India – Subhash Chandra Bose
• The correct author of "The Discovery of India" is Jawaharlal Nehru.
• He wrote this book while imprisoned in the Ahmednagar Fort during the Quit India Movement.
• It is a reflection on Indian history and culture.

Additional Information

• Hind Swaraj – Mahatma Gandhi
• This pairing is correct.
• Mahatma Gandhi wrote "Hind Swaraj" in 1909, where he outlined his views on Swaraj (self-rule), modern civilization, and the path to Indian independence.
• It's a significant work in understanding Gandhi's philosophy and political thought.
• Gitanjali – Rabindranath Tagore
• This pairing is also correct. "Gitanjali" is a collection of poems by Rabindranath Tagore, for which he was awarded the Nobel Prize in Literature in 1913.
• It's a beautiful collection of poems expressing spiritual and philosophical ideas.
• Why I am an Atheist – Bhagat Singh
• This pairing is correct as well.
• Bhagat Singh, a prominent Indian revolutionary, wrote this essay expressing his views on atheism and the need for rational thinking.
• It's a significant text in understanding his intellectual and philosophical beliefs.

The correct answer is Rabindranath Tagore

Key Points

• Rabindranath Tagore was the poet who won the Nobel Prize in Literature in 1913:
• Tagore was a poet, writer, artist, and lyricist from Bengal.
• His work was rooted in both Indian and Western learning traditions.
• He wrote poetry, songs, stories, dramas, literary criticism, philosophy, and social issues.
• Tagore won the Nobel Prize for his "profoundly sensitive, fresh and beautiful verse".
• The prize was awarded for his book of song offerings, Gitanjali.
• Tagore was the first person from India and Asia to win the Nobel Prize in Literature.
• He was also the first non-European to receive the award.

Additional Information 

• Aurobindo Ghose (Sri Aurobindo):
• Aurobindo Ghose, commonly known as Sri Aurobindo, was a prominent Indian philosopher, poet, and nationalist.
• He was a key figure in the Indian independence movement and later became a spiritual leader.
• Sri Aurobindo played an essential role in the early years of India's struggle for freedom from British rule and is known for his writings on spiritual evolution, the nature of the universe, and the role of human beings in achieving higher states of consciousness.
• His major works include The Life Divine and Savitri, an epic poem.
• Aurobindo also founded the Sri Aurobindo Ashram in Pondicherry, which became a center for spiritual learning.
• Rudyard Kipling:
• Rudyard Kipling was a British author and poet, best known for his works related to British India.
• He was born in 1865 and is famous for books like The Jungle Book, Kim, and poems such as If— and Gunga Din.
• Kipling's writings often dealt with themes of British imperialism, colonialism, and the relationship between the British and the Indian people.
• His work earned him the Nobel Prize for Literature in 1907.
• Bankim Chandra Chattopadhyay:
• Bankim Chandra Chattopadhyay was a Bengali writer, poet, and journalist, known for his contributions to literature and his role in the Indian nationalist movement.
• He is best known for writing the national song Vande Mataram, which became a rallying cry for India's independence movement.
• His novel Anandamath (1882) is also notable for its portrayal of Bengali culture and history.

The correct answer is 1964

Key Points

• National Institute of Kathak Dance, also known as Kathak Kendra.
• It was Established in 1964 (the institution became a part of the Sangeet Natak Akademi).
• It is the premier dance institution for the Indian classical dance form of Kathak, and a unit of the Sangeet Natak Akademi, India's National Academy of Music, Dance and Drama, situated in New Delhi.
• The institution is primarily dedicated to Kathak, though it also offers courses in Hindustani classical music (Vocal) and Pakhawaj and Tabla.
• Kathak Kendra was originally established as the Kathak wing of the Shriram Bharatiya Kala Kendra, in 1955, through the efforts of Nirmala Joshi, the first secretary of the Sangeet Natak Akademi.
• Here Pandit Shambhu Maharaj, celebrated Guru of the Lucknow Gharana (school) of Kathak, was head of the department.
• Shambhu Maharaj received the highest award of the Sangeet Natak Akademi, the Sangeet Natak Akademi Fellowship in 1967.
• Birju Maharaj a Kathak dancer was his nephew.

Additional Information

Sangeet Natak Akademi

• The Sangeet Natak Akademi is constituted of three main units. The two of these units are dance teaching Institutes, as given below.
• Jawaharlal Nehru Manipur Dance Academy, called JNMDA located in Imphal.
• Kathak Kendra is located in Delhi.
• The President of India appoints the chairman of this Akademi.
• The term of office for the chairman is of 5 years.
• The duties, responsibilities, and functions of Sangeet Natak Akademi have been described in the Academy’s Memorandum of Association.
• This memoran

The correct answer is Madhya Pradesh

Key Points

• Madhya Pradesh had the lowest unemployment rate among people aged 15 years and above in 2020-21.

Additional Information

• Gujarat: The unemployment rate in Gujarat for the year 2020-21 was 3.1%.
• Jharkhand: In 2020-21, Jharkhand had an unemployment rate of 3.6%.
• Delhi: The unemployment rate in Delhi during 2020-21 stood at 4.6%.
• Chhattisgarh: Chhattisgarh has the lowest unemployment rate in India, with 14 unemployed individuals per 1,000 people.
• Youth Unemployment: The unemployment rate among youth aged 15-29 was 12.9% in 2020-21.

The Correct Answer is 1925.

Key Points

• Indian Hockey Federation (IHF) was the apex governing body of field hockey in India formed on 7 November 1925 in Gwalior.
• It is the first Non-European Country which was a part of International Hockey Federation
• As a member of the International Hockey Federation, it represented India in all international matches under the former leadership of KPS Gill & the secretary of the federation, K.Jyothikumaran.
• The women's team was directed by the Indian Women's Hockey Federation.
• R. K. Shetty was elected as a new IHF president in 2010.

The correct answer is Peroxisomes.

Key Points

• Peroxisomes -
• Peroxisomes are small, membrane-bound organelles that contain enzymes involved in various metabolic reactions, including the oxidation of fatty acids and amino acids.
• These organelles play a crucial role in detoxifying harmful substances and in lipid metabolism.
• They produce hydrogen peroxide as a byproduct of these reactions, which is then broken down by the enzyme catalase.

Additional Information

• Centrosome
• Centrosome was discovered by T. Boberi in 1888.
• In almost all types of colonocytes except the nerve cell, a star-like structure is visible in the cytoplasm near the nucleus, which is called the centrosome.
• It helps in cell division,
• Forms cilia and flagellum and makes the tail of spermatozoa.
• Vacuole
• Vacuole is surrounded by the tonoplast, a type of cytoplasmic membrane that can stretch and fills itself with a solution known as cell sap
• Vacuoles are found mainly in plant cells and are filled with water, containing inorganic and organic molecules.
• Nucleus
• The nucleus is a pivotal organelle responsible to regulate almost all forms of cellular activities.
• Mostly, every type of cell that exists is categorized on the basis of the absence or presence of the nucleus within its cell (categorized either as a prokaryotic or eukaryotic cell.)

The correct answer is transfer of electrons between two reactants.

Key Points

Redox Reactions

• Redox reactions are chemical reactions in which the oxidation states of atoms are changed.
• These reactions involve the transfer of electrons from one substance to another.
• One substance is oxidized (losing electrons) and another is reduced (gaining electrons).

Explanation:-

1. Zn + CuSO4 → ZnSO4 + Cu

• Zn goes from 0 to +2 oxidation state (oxidation).
• Cu goes from +2 to 0 oxidation state (reduction).
• This is a redox reaction.

2. 2KClO3 + I2 → 2KIO3 + Cl2

• Cl in KClO3 goes from +5 to 0 oxidation state (reduction).
• I in I2 goes from 0 to +5 oxidation state in KIO3 (oxidation).
• This is a redox reaction.

3. H2 + Cl2 → 2HCl

• H goes from 0 to +1 oxidation state (oxidation).
• Cl goes from 0 to -1 oxidation state (reduction).
• This is a redox reaction.

The correct answer is Buddhism.

Key Points

• Buddhism
• Gautama Buddha, variously termed Sakyamuni or Tathagata, is considered as the founder of Buddhism.
• He was born as Siddhartha to Suddhodhana, the ruler of the Sakyan republic, and his wife Maya, on Vaisakha Purnima in the Lumbini gardens near Kapilavasthu (in Nepal) in the 6^th century BC.

Additional Information

• Jainism
• Jainism is one of the world’s oldest living religions.
• Jainism grounds itself in 24 Tirthankaras.
• A ‘Tirthankara’, is the one who revealed religious truth at different times.
• The first Tirthankara was Rishabhadev.
• The last and 24^th tirthankara of the Jainas was Mahaveer Swami or Vardhamana Mahavira.
• Jainism gained prominence under the aegis of Mahavira, during the sixth century BC.
• He was a kshatriya prince of the Lichchhavis, a group that was part of the Vajji sangha.
• At the age of thirty, he left home and went to live in a forest.
• For twelve years he led a hard and lonely life, at the end of which he attained enlightenment.
• The followers of Mahavira were known as Jainas.
• The teachings of Mahavira and his followers were transmitted orally for several centuries.
• They were written down in the form in which they are presently available at a place called Valabhi, in Gujarat, about 1500 years ago.

The correct answer is Orphans, specially abled children, destitute women and senior citizens.

Key Points

• The Mukhya Mantri Sukh Aashray Kosh is a new initiative of the government, constituted on the auspicious occasion of new year by the state government.
• Taking an innovative step towards humanity to shoulder his social responsibility, Himachal Pradesh Chief Minister Sukhvinder Singh has contributed his first salary to the Mukhya Mantri Sukh Aashray Kosh.
• The Hon'ble Chief Minister of Himachal Pradesh also announced on 1^st January that all designated ministers and Congress MLAs have volunteered to donate their first salary as MLAs to the fund.
• The outlay budget of the scheme is 101 crores.
• The fund has been started with a vision to help and assist needy students and destitute women to get higher education.
• This would also help them to lead a life of dignity.
• A specific amount would be given to these students and women which would help them in acquiring desired higher education.

The correct answer is 10 million LPG connections will be provided to the beneficiaries.

Key Points

• Prime Minister Narendra Modi launched Ujjwala 2.0 (Pradhan Mantri Ujjwala Yojana - PMUY) by handing over LPG connections, at Mahoba, Uttar Pradesh on 10^th August 2021.

 Additional Information

• Ujjwala 1.0 was launched in 2016 and the target was set to provide LPG connections to five crore women members of BPL households.
• Subsequently, the scheme was expanded in April 2018 to include women beneficiaries from seven more categories: SC/ST, PMAY, AAY, Most backward classes, tea garden, forest dwellers, Islands.
• In the Union budget for FY 21-22, provision for an additional one crore LPG connection under the PMUY scheme was announced.
• These one crore additional PMUY connections (under Ujjwala 2.0) aim to provide deposit-free LPG connections to those low-income families who could not be covered under the earlier phase of PMUY.
• Along with a deposit free LPG connection, Ujjwala 2.0 will provide the first refill and hotplate free of cost to the beneficiaries. Also, the enrollment procedure will require minimal paperwork.

The correct answer is Nagaland.

Key Points

• The MG Music Awards is an annual music awards show presented by the Musicians Guild Nagaland.
• The first edition was held virtually in 2021, while the second edition took place offline in 2022 at the RCEMPA Hall in Jotsoma, Nagaland.
• The awards aim to recognize and celebrate musical talent from the northeastern states of India.
• The 2022 edition featured 12 categories, including Best Rock Artist, Best Indie Artist, Best Rap/Hiphop Artist, Best Gospel Artist, Best Folk Fusion Artist, Best Electronic Artist, Best Pop Solo Artist, Best Pop Duo/Group, Best Sound Engineer, Most Promising Artist,and Song of the Year.
• The awards ceremony also included performances by various artists from the region.
• The MG Music Awards have become a significant platform for showcasing and promoting the diverse musical talent of the northeastern states.

The correct answer is net worth.

Key Points

• Net worth is a financial metric that represents the overall value of an individual, company, or entity.
• It's calculated by subtracting total liabilities from total assets.
• A positive net worth indicates financial health, while a negative one signifies debt exceeding assets.

 Additional Information

• Assets:
• These are items of value owned by an individual or entity, such as real estate, vehicles, investments, bank accounts, and personal belongings.
• Liabilities:
• These are financial obligations or debts, including mortgages, loans, credit card debt, and other outstanding payments.

The correct answer is Diwan-i-Arz

Key Points

• Balban created the department of Diwan-i-Arz in Delhi Sultanate.
• Diwan-i-Arz
• Diwan-i-Arz was essentially the Department of Military managed by Ariz-i-Mamalik.
• He was accountable for the regulation and preservation of the royal army.

Additional Information

• Administration under the Sultanate:
• It was also influenced by the Iranian system of administration.
• The situation in India and Indian traditions was kept in mind during these systems.
• Organs of the Government:
• Diwan-i-Wizarat:
• Department of revenue and finance headed by the Wazir.
• Diwan-i-Arz:
• Department of Military headed by Ariz-i-Mamalik.
• Diwan-i- Insha:
• Department of royal correspondence headed by Dabir-i-lnsha.
• Diwan-i-Risalat:
• Department of religious affairs headed by the chief Qazi.
• Diwan-i-bandagan:
• ​Diwan-i-bandagan (department of slaves) and Diwan-i-Khairat (charity department) was created by Firuz Shah Tughluq.
• Diwan-i-mustakharaj:
• Diwan-i-mustakharaj  (to realise arrears) was created by Alauddin Khiiji.
• Diwan- i-kohi:
• Diwan- i-kohi (department of agriculture) was created by Muhammad bin Tughluq.

Given:

45 ÷ 3 + 4 × A – 48 ÷ 24 + 4 = 1

Formula Used:

Perform arithmetic operations following BODMAS/BIDMAS rules (Brackets, Orders (i.e. powers and square roots, etc.), Division and Multiplication, Addition and Subtraction)

Calculation:

First simplify the given equation:

45 ÷ 3 + 4 × A – 48 ÷ 24 + 4 = 1

⇒ 15 + 4A – 2 + 4 = 1

⇒ 17 + 4A = 1

⇒ 4A = 1 – 17

⇒ 4A = -16

⇒ A = -16 ÷ 4

⇒ A = -4

The value of A is -4.

Given:

Bar graph shows the foreign direct investment (in million dollars) in Country A from all over the world over the years.

Formula used:

Ratio of investment in 2018 to that in 2015 = (Investment in 2018) / (Investment in 2015)

Calculation:

By the given bar graph,

Investment in 2018 = 65 million dollars

Investment in 2015 = 60 million dollars

Ratio of investment in 2018 to that in 2015 = $\frac{65}{60}$

Simplify the ration,

⇒$\frac{65}{60} = \frac{13}{12}$ = 13 : 12

The ratio of investment in 2018 to that in 2015 was 13 : 12.

Shortcut Trick

There is a direct algebraic relationship between the direct common tangent (DCT), the transverse common tangent (TCT), and the product of the radii of two circles:

4 × R × r = DCT² − TCT²

⇒ 4 × Product of radii = 9² − 5²

⇒ 4 × Product of radii = 81 − 25 = 56

⇒ Product of radii = 56 ÷ 4 = 14

∴ The correct answer is 14.

Alternate Method

             C1 C2  DCT = 9 TCT = 5 d R r  4Rr = DCT² − TCT²

Given:

Length of direct common tangent (DCT) = 9

Length of transverse common tangent (TCT) = 5

Let the radii of the two circles be R and r, and the distance between their centers be d.

Formula Used:

Length of DCT = √(d² − (R − r)²)

Length of TCT = √(d² − (R + r)²)

Calculations:

Squaring the formulas for DCT and TCT, we get:

⇒ d² − (R − r)² = 9² = 81     −−− (Equation 1)

⇒ d² − (R + r)² = 5² = 25     −−− (Equation 2)

Subtracting Equation 2 from Equation 1:

⇒ (d² − (R − r)²) − (d² − (R + r)²) = 81 − 25

⇒ (R + r)² − (R − r)² = 56

Using the basic algebraic identity (A + B)² − (A − B)² = 4AB :

⇒ 4 × R × r = 56

⇒ R × r = 56 ÷ 4

⇒ R × r = 14

∴ The correct answer is 14.

Additional Information

Direct Common Tangent (DCT)

A common tangent is called a direct common tangent if both circles lie on the same side of the tangent. Its length is always greater than or equal to the transverse common tangent. Formula: DCT = √(d² − (R − r)²)

Transverse Common Tangent (TCT)

A tangent is called a transverse common tangent if it intersects the line segment joining the centers of the two circles. The two circles lie on opposite sides of this tangent. Formula: TCT = √(d² − (R + r)²)

Number of Common Tangents

When two circles do not intersect and lie outside each other, there are exactly 4 common tangents (2 DCT + 2 TCT). When they touch externally, there are 3 common tangents. When they intersect at two points, there are 2 common tangents.

Shortcut Trick

In congruent triangles, corresponding parts are identical (CPCTC).

From ΔPQR ≅ ΔMON, the corresponding vertices give ∠Q = ∠O and QR = ON.

Equating the angles: 82 = 5y − 8 ⇒ 5y = 90 ⇒ y = 18

Equating the sides: 24 = 3x + y ⇒ 24 = 3x + 18 ⇒ 3x = 6 ⇒ x = 2

We need to find (x + y): 2 + 18 = 20

∴ The correct answer is 20.

Alternate Method

Given:

ΔPQR ≅ ΔMON

∠PQR = 82°, ∠QRP = 47°, ∠RPQ = 51°, ∠MON = (5y − 8)°

QR = 24, ON = 3x + y

Concept Used:

When two triangles are congruent, their corresponding angles and corresponding sides are equal (CPCTC).

P Q R

M O N

82° 47° 51°

(5y − 8)°

24 3x + y

Calculations:

Since ΔPQR ≅ ΔMON, the exact mapping of corresponding vertices is P ↔ M, Q ↔ O, and R ↔ N.

⇒ ∠PQR = ∠MON

⇒ 82 = 5y − 8

⇒ 5y = 82 + 8 = 90

⇒ y = 18

Similarly, for the corresponding sides mapping to each other:

⇒ QR = ON

⇒ 24 = 3x + y

Substitute the obtained value of y = 18 in the equation above:

⇒ 24 = 3x + 18

⇒ 3x = 24 − 18

⇒ 3x = 6

⇒ x = 2

Finally, find the value of (x + y):

⇒ x + y = 2 + 18 = 20

∴ The correct answer is 20.

Additional Information

Congruence of Triangles

Two triangles are congruent if all three corresponding sides are equal in length and all three corresponding angles are equal in measure, meaning they can perfectly overlap each other.

CPCTC Rule

CPCTC stands for "Corresponding Parts of Congruent Triangles are Congruent". It serves as a theorem to establish that matching sides or angles are mathematically identical once triangle congruence is proved.

Rules for Triangle Congruence

The five primary criteria to verify if triangles are congruent are SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle), AAS (Angle-Angle-Side), and RHS (Right angle-Hypotenuse-Side).

Shortcut Trick

For divisibility by 5, the last digit (#) must be 0 or 5.

For divisibility by 8, the number must be even, so # cannot be 5. Thus, # = 0.

The last three digits form 4*0, which must be a multiple of 8.

Checking the options, if * = 4, then 440 ÷ 8 = 55 (perfectly divisible).

∴ The correct answer is 4 and 0.

Alternate Method

Given:

The 7-digit number is 62684*#

It is perfectly divisible by both 8 and 5.

Formula Used:

Divisibility by 5: The unit digit must be 0 or 5.

Divisibility by 8: The number formed by the last three digits must be divisible by 8 (and the overall number must be even).

Calculation:

⇒ For the number to be divisible by 8, it must be an even number. So, the unit digit # cannot be 5.

⇒ From the divisibility rule of 5, the possible unit digits are 0 or 5. Since it must be even, we get # = 0.

⇒ Now, the last three digits form the number 4*0.

⇒ We need 4*0 to be perfectly divisible by 8. Let us evaluate the given options where # = 0:

⇒ Option 1: If * = 1, 410 ÷ 8 = 51.25 (Not perfectly divisible)

⇒ Option 2: If * = 4, 440 ÷ 8 = 55 (Perfectly divisible)

⇒ Option 3: If * = 5, 450 ÷ 8 = 56.25 (Not perfectly divisible)

⇒ Option 4 is directly eliminated because # = 5 is not possible for an even number.

⇒ Therefore, the correct digits are * = 4 and # = 0.

∴ The correct answer is 4 and 0.

Additional Information

Divisibility by 3 and 9

A number is divisible by 3 or 9 if the sum of all its digits is a multiple of 3 or 9, respectively.

Divisibility by 4 and 16

A number is divisible by 4 if the number formed by its last two digits is divisible by 4. It is divisible by 16 if its last four digits are divisible by 16.

Divisibility by 11

A number is divisible by 11 if the difference between the sum of its digits at odd positions and the sum of its digits at even positions is either 0 or a multiple of 11.

Shortcut Trick

Area of square = 16x² + 40x + 25 = (4x)² + 2(4x)(5) + 5² = (4x + 5)².

Side of the square = √(4x + 5)² = 4x + 5.

Perimeter = 4 × Side = 4 × (4x + 5) = 16x + 20.

Taking 2 as a common factor, Perimeter = 2(8x + 10).

∴ The correct answer is 2(8x + 10).

Alternate Method

Given:

Area of the square = 16x² + 40x + 25

Formula Used:

Area of a square = (Side)²

Perimeter of a square = 4 × Side

Algebraic Identity: (a + b)² = a² + 2ab + b²

Calculation:

⇒ (Side)² = 16x² + 40x + 25

⇒ (Side)² = (4x)² + 2 × (4x) × 5 + (5)²

⇒ (Side)² = (4x + 5)²

⇒ Side = 4x + 5

Now, calculate the perimeter:

⇒ Perimeter = 4 × (4x + 5)

⇒ Perimeter = 16x + 20

⇒ Perimeter = 2(8x + 10)

∴ The correct answer is 2(8x + 10).

Additional Information

Algebraic Identities

Standard identities include (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b². These are frequently used to form perfect squares in geometry problems.

Properties of a Square

All four sides are equal, and each interior angle is 90°. The diagonals are equal and bisect each other at right angles.

Diagonal of a Square

The diagonal is given by the formula d = Side × √2. The area can also be determined directly from the diagonal using the relation Area = d² ÷ 2.

Shortcut Trick

The average of the squares of the first n natural numbers is given by the direct formula: (n + 1)(2n + 1) ÷ 6.

Here, n = 45.

Average = (45 + 1)(2 × 45 + 1) ÷ 6

Average = (46 × 91) ÷ 6 = (23 × 91) ÷ 3

Average = 2093 ÷ 3 ≈ 697.67

∴ The correct answer is 697.67.

Alternate Method

Given:

Number of terms (n) = 45 natural numbers.

Formula Used:

Sum of squares of first n natural numbers = n(n + 1)(2n + 1) ÷ 6

Average = Sum ÷ n = (n + 1)(2n + 1) ÷ 6

1²

2²

3²

. . .

44²

45²

Average = (n + 1)(2n + 1) ÷ 6 Avg for n=45 → 697.67

Calculations:

⇒ Average = (45 + 1)(2 × 45 + 1) ÷ 6

⇒ Average = (46 × 91) ÷ 6

⇒ Average = (23 × 91) ÷ 3

⇒ Average = 2093 ÷ 3

⇒ Average = 697.666...

Rounding off to two decimal places,

⇒ Average ≈ 697.67

∴ The correct answer is 697.67.

Additional Information

Average of first n natural numbers

The average of the first n natural numbers is given by the formula (n + 1) ÷ 2.

Average of cubes of first n natural numbers

The average of the cubes of the first n natural numbers is given by n(n + 1)² ÷ 4.

Average of first n even and odd numbers

The average of the first n even numbers is (n + 1), and the average of the first n odd numbers is n.

Shortcut Trick

When rolled along its breadth, Breadth (B) = Circumference, and Length (L) = Height.

Direct formula for Volume = (B² × L) ÷ 4π

Volume = (22 × 22 × 50 × 7) ÷ (4 × 22)

Volume = 7700 ÷ 4 = 1925 cm³

∴ The correct answer is 1925 cm³.

Alternate Method

Given:

Length of paper (l) = 50 cm

Breadth of paper (b) = 22 cm

When the rectangular sheet is rolled along its breadth:

Circumference of the cylinder's base = 22 cm

Height of the cylinder (h) = 50 cm

Formula Used:

Circumference = 2πr

Volume of Cylinder = πr²h

Calculations:

⇒ 2 × (22/7) × r = 22

⇒ r = (22 × 7) ÷ (2 × 22)

⇒ r = 3.5 cm

⇒ Volume = (22/7) × 3.5 × 3.5 × 50

⇒ Volume = (22/7) × (7/2) × (7/2) × 50

⇒ Volume = 11 × 7 × 25

⇒ Volume = 1925 cm³

∴ The correct answer is 1925 cm³.

Additional Information

Rolling a Rectangular Sheet

When rolled along its length, the length forms the circumference and the breadth becomes the height. When rolled along its breadth, the breadth forms the circumference and the length becomes the height.

Surface Areas of a Cylinder

The Curved Surface Area (CSA) is given by 2πrh, and the Total Surface Area (TSA) is 2πr(h + r), where r is the radius and h is the height.

Relationship with Cone

A cylinder and a cone with the same base radius and height have volumes in the ratio 3:1. The volume of the cone is 1/3 × πr²h.

Shortcut Trick

We know that ((x − y) / (x + y))² = (x² + y² − 2xy) / (x² + y² + 2xy)

⇒ ((x − y) / (x + y))² = (280 − 2 × 120) / (280 + 2 × 120)

⇒ ((x − y) / (x + y))² = (280 − 240) / (280 + 240) = 40 / 520 = 1 / 13

⇒ (x − y) / (x + y) = 1 / √13

∴ The correct answer is 1/√13.

Alternate Method

Given:

x² + y² = 280

xy = 120

Formula Used:

(x + y)² = x² + y² + 2xy

(x − y)² = x² + y² − 2xy

Calculations:

⇒ (x + y)² = 280 + 2 × 120

⇒ (x + y)² = 280 + 240 = 520

⇒ (x − y)² = 280 − 2 × 120

⇒ (x − y)² = 280 − 240 = 40

We need to find the value of the ratio (x − y) / (x + y).

Squaring the required expression:

⇒ [(x − y) / (x + y)]² = (x − y)² / (x + y)²

⇒ [(x − y) / (x + y)]² = 40 / 520

⇒ [(x − y) / (x + y)]² = 1 / 13

Taking the square root on both sides:

⇒ (x − y) / (x + y) = 1 / √13

∴ The correct answer is 1/√13.

Additional Information

Sum and Difference of Squares

The identities (x + y)² = x² + y² + 2xy and (x − y)² = x² + y² − 2xy are fundamentally used to relate the sum and difference of variables to the sum of their squares and their product.

Relation between Squares

Subtracting the difference squared from the sum squared yields: (x + y)² − (x − y)² = 4xy. Adding them gives: (x + y)² + (x − y)² = 2(x² + y²).

Difference of Squares

Another key algebraic identity is x² − y² = (x − y)(x + y), which is widely used in solving factorization and simplification problems.

Shortcut Trick

Total Time = Distance ÷ (Speed of Boat + Speed of Stream) + Distance ÷ (Speed of Boat − Speed of Stream)

Total Time = 105 ÷ (18 + 3) + 105 ÷ (18 − 3)

Total Time = 105 ÷ 21 + 105 ÷ 15

Total Time = 5 + 7 = 12 hours.

∴ The correct answer is 12 hours.

Alternate Method

Given:

Speed of boat in still water (u) = 18 km/h

Speed of stream (v) = 3 km/h

Distance covered each way (D) = 105 km

Formula Used:

Downstream Speed (D_s) = u + v

Upstream Speed (U_s) = u − v

Time = Distance ÷ Speed

Calculations:

⇒ Downstream Speed (D_s) = 18 + 3 = 21 km/h

⇒ Upstream Speed (U_s) = 18 − 3 = 15 km/h

⇒ Time taken downstream = 105 ÷ 21 = 5 hours

⇒ Time taken upstream = 105 ÷ 15 = 7 hours

⇒ Total Time = 5 + 7 = 12 hours

∴ The correct answer is 12 hours.

Additional Information

Speed of Boat and Stream from Downstream / Upstream

If downstream speed is D_s and upstream speed is U_s, then Speed of Boat = (D_s + U_s) ÷ 2, and Speed of Stream = (D_s − U_s) ÷ 2.

Average Speed for a Round Trip

For a journey traveling the same distance downstream and upstream, Average Speed = (D_s × U_s) ÷ Speed of Boat in still water.

Total Time Formula

If total time for a round trip is T, then Total Distance = (T × (u² − v²)) ÷ (2 × u), where u is boat speed and v is stream speed.

Shortcut Trick

Since Simple Interest (SI) is 1/5 of the principal, total interest earned is 20%.

Given Rate (R) = 5 × Time (T), the total interest percentage = R × T = 5T² = 20%.

⇒ T² = 4 ⇒ T = 2 years ⇒ Initial Rate (R) = 5 × 2 = 10% p.a.

New Rate = 10% + 3% = 13% p.a.

New SI for 6 years = 13% × 6 = 78% of ₹5,600 = ₹4,368.

∴ The correct answer is ₹4368.

Alternate Method

Given:

Simple interest (SI) = 1/5 of the Principal (P)

Rate of interest (R) = 5 × Time in years (T)

New Principal = ₹5,600, New Time = 6 years

Formula Used:

SI = (P × R × T) ÷ 100

P = ₹5,600

R = 13% p.a. T = 6 years

SI = 78% of P

SI = ₹4,368

Calculations:

⇒ P ÷ 5 = (P × 5T × T) ÷ 100

⇒ 1 ÷ 5 = 5T² ÷ 100

⇒ 1 ÷ 5 = T² ÷ 20

⇒ T² = 20 ÷ 5 = 4

⇒ T = √4 = 2 years

⇒ R = 5 × 2 = 10%

New rate of interest = 10% + 3% = 13%

Now, finding the SI on ₹5,600 for 6 years at the new rate of 13% p.a.:

⇒ New SI = (5600 × 13 × 6) ÷ 100

⇒ New SI = 56 × 78 = 4368

∴ The correct answer is ₹4368.

Additional Information

Simple Interest (SI) Formula

SI = (P × R × T) ÷ 100, where P is Principal, R is Rate of interest, and T is Time period.

Amount Formula

Total Amount = Principal + Simple Interest = P × (1 + (R × T) ÷ 100).

Effective Interest Rate

In simple interest, the effective percentage of interest earned over T years is directly R × T%. E.g., 13% p.a. for 6 years yields exactly 78% of the principal as interest.

Shortcut Trick

Successive discounts act as successive multiplying factors on the marked price.

SP = MP × (1 − 0.20) × (1 − 0.30) × (1 − 0.10)

SP = 1000 × 0.8 × 0.7 × 0.9

SP = 1000 × 0.504 = 504

∴ The correct answer is ₹504.

Alternate Method

Given:

Marked Price (MP) = ₹1,000

Successive discounts = 20%, 30%, and 10%

Formula Used:

SP = MP × (100 − D₁)/100 × (100 − D₂)/100 × (100 − D₃)/100

Calculations:

⇒ Discount 1 = 20% of 1000 = ₹200

⇒ Price after 1^st discount = 1000 − 200 = ₹800

⇒ Discount 2 = 30% of 800 = ₹240

⇒ Price after 2^nd discount = 800 − 240 = ₹560

⇒ Discount 3 = 10% of 560 = ₹56

⇒ Final Selling Price = 560 − 56 = ₹504

∴ The correct answer is ₹504.

Additional Information

Equivalent Discount Formula

For two successive discounts of x% and y%, the equivalent single discount is given by (x + y − xy/100)%.

Base for Discount

Discount is always calculated on the Marked Price (MP), never on the Cost Price (CP). Selling Price is calculated as MP − Discount.

Successive Discounts Concept

When multiple discounts are offered, each subsequent discount is calculated on the reduced price from the previous step, not on the original Marked Price.

Shortcut Trick

Total capacity = LCM of 10, 15, and 12 = 60 units.

Efficiency of P = +6, Q = +4, R = −5.

Work done by P and Q from 6 a.m. to 9 a.m. (3 hours) = 3 × (6 + 4) = 30 units.

Remaining capacity to fill = 60 − 30 = 30 units.

Time to fill remaining capacity = 30 ÷ (6 + 4 − 5) = 6 hours after 9 a.m. = 3 p.m.

∴ The correct answer is 3 p.m.

Alternate Method

Given:

Pipe P completely fills the tank in 10 hours.

Pipe Q completely fills the tank in 15 hours.

Pipe R empties the filled tank in 12 hours.

Formula Used:

Total Work = Time × Efficiency

Calculations:

⇒ Pipes P and Q are initially opened at 6 a.m. and work until 9 a.m. (for 3 hours).

⇒ Work done by P and Q in 3 hours = (6 + 4) × 3 = 10 × 3 = 30 units

⇒ Remaining work to be done = Total Work − Work Done = 60 − 30 = 30 units

⇒ At 9 a.m., pipe R is also opened.

⇒ Combined efficiency of P, Q, and R = 6 + 4 − 5 = +5 units/hour

⇒ Time required to complete the remaining work = 30 ÷ 5 = 6 hours

⇒ The tank will be completely filled 6 hours after 9 a.m.

⇒ 9 a.m. + 6 hours = 3 p.m.

∴ The correct answer is 3 p.m.

Additional Information

Inlet and Outlet Pipes

An inlet pipe fills the tank and is assigned a positive efficiency, while an outlet pipe empties the tank and gets a negative efficiency.

Total Capacity Approach

The total capacity of a tank is commonly assumed to be the Lowest Common Multiple (LCM) of the times taken by the individual pipes.

Alternate Working

When pipes operate on an alternate hour basis, calculate the net filling per cycle. Extra care must be taken near completion to prevent overfilling before an emptying cycle begins.

Shortcut Trick

For a system of linear equations to have no solution, the lines must be parallel.

Ratio of their x-coefficients = Ratio of their y-coefficients

⇒ 2 / 1 = 7k / 1

⇒ 7k = 2

⇒ k = 2/7

∴ The correct answer is 2/7.

Alternate Method

XYx + y - 9 = 02x + 7ky - 8 = 0For 'No Solution', lines must be parallelCondition: a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Given:

Equation 1: 2x + 7ky − 8 = 0

Equation 2: x + y − 9 = 0

Formula Used:

For two linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 to have no solution, they must represent parallel lines.

Condition: a₁ / a₂ = b₁ / b₂ ≠ c₁ / c₂

Calculations:

⇒ Comparing given equations with standard form:

⇒ a₁ = 2, b₁ = 7k, c₁ = −8

⇒ a₂ = 1, b₂ = 1, c₂ = −9

⇒ Substituting these values into the parallel lines condition:

⇒ 2 / 1 = 7k / 1

⇒ 7k = 2

⇒ k = 2/7

⇒ Checking the constant terms ratio:

⇒ c₁ / c₂ = −8 / −9 = 8/9

⇒ Since 2/1 ≠ 8/9, the condition for no solution is satisfied.

∴ The correct answer is 2/7.

Additional Information

Unique Solution (Intersecting Lines)

The lines intersect at exactly one point, meaning they have different slopes. Condition: a₁ / a₂ ≠ b₁ / b₂.

Infinite Solutions (Coincident Lines)

The lines are overlapping. Every point on one line is a solution for both equations. Condition: a₁ / a₂ = b₁ / b₂ = c₁ / c₂.

No Solution (Parallel Lines)

The lines are parallel and never intersect. They have the same slope but different intercepts. Condition: a₁ / a₂ = b₁ / b₂ ≠ c₁ / c₂.

Shortcut Trick

Length of Transverse Common Tangent = √(d² − (R + r)²)

⇒ TCT = √(17² − (10 + 5)²)

⇒ TCT = √(289 − 225)

⇒ TCT = √64 = 8 cm

∴ The correct answer is 8 cm.

Alternate Method

Given:

Radius of larger circle (R) = 10 cm

Radius of smaller circle (r) = 5 cm

Distance between centres (d) = 17 cm

d = 17 cm

O₁ O₂ 10 cm 5 cm TCT

A B

Formula Used:

The length of the Transverse Common Tangent to two circles is given by:

TCT = √(d² − (R + r)²)

Calculations:

⇒ TCT = √(17² − (10 + 5)²)

⇒ TCT = √(17² − 15²)

⇒ TCT = √(289 − 225)

⇒ TCT = √64

⇒ TCT = 8 cm

∴ The correct answer is 8 cm.

Additional Information

Direct Common Tangent (DCT)

The tangent which does not intersect the line segment joining the centres of the circles. Its length is given by: DCT = √(d² − (R − r)²).

Transverse Common Tangent (TCT)

The tangent which intersects the line segment joining the centres. The point of intersection divides the distance between the centres in the ratio of their radii (R : r).

Circles Touching Externally

If two circles touch externally, the distance between centres is d = R + r. The transverse tangent length becomes 0, and the length of the direct common tangent simplifies to 2√(Rr).

Given: 

The pie chart of the number of hours spent by a student for different activities on a working day.

Formula used:

% = {(required value) / (Total value)} × 100

Calculation:

From the given pie chart,

Number of hours spent by the student at school = 6 hours

Total numbers of hours spent by the student for all the activities on a working day = 4 + 6 + 8 + 4 + 2 = 24 hours

The percentage of hours spent at school in a day = $\frac{6}{24} \times100$

⇒$\frac{100}{4}$

⇒25 %

The percentage of hours spent at school in a day is 25%

Shortcut Trick

Total voters can be found directly by applying the reverse multiplying factors for each percentage reduction.

Total Voters = Sunita's Votes × (100 ÷ 85) × (100 ÷ 98) × (100 ÷ 78)

Total Voters = 64974 × (20 ÷ 17) × (50 ÷ 49) × (50 ÷ 39)

Total Voters = 64974 × (50000 ÷ 32487)

Total Voters = 2 × 50000 = 1,00,000

∴ The correct answer is 1,00,000.

Alternate Method

Given:

→ Percentage of voters who cast their votes = 85%

→ Percentage of invalid votes = 2% of cast votes

→ Sunita's share = 78% of valid votes

→ Sunita's total votes = 64,974

Calculations:

⇒ Let the total number of enrolled voters in the election be 100x.

⇒ Number of voters who cast their votes = 85% of 100x = 85x

⇒ Number of invalid votes = 2% of 85x = 1.7x

⇒ Number of valid votes = 85x − 1.7x = 83.3x

⇒ Sunita got 78% of the valid votes.

⇒ Sunita's votes = 78% of 83.3x = 0.78 × 83.3x = 64.974x

⇒ 64.974x = 64974

⇒ x = 1000

⇒ Total number of enrolled voters = 100x = 100 × 1000 = 1,00,000

∴ The correct answer is 1,00,000.

Additional Information

Successive Percentage Change

When a value undergoes continuous percentage changes, the final value is found by multiplying the successive multiplying factors: Final = Initial × (1 ± P₁/100) × (1 ± P₂/100).

Valid Votes Concept

Valid Votes = Total Enrolled Voters − Votes Not Cast − Invalid Votes. Winner's margin over the loser is always evaluated as (Winner's % − Loser's %) × Valid Votes.

Base Change Rule

Always identify the correct base for each percentage applied. In this scenario, invalid votes are computed specifically on "cast votes" (85%), and not on the "total votes" (100%).

Given:

Data regarding the population of a city at the end of the years mentioned.

Formula used:

Rate of increase = (population in current year - population in previous year) / (population in previous year) × 100

Calculation:

The rate of increase for a given year compared to the previous year can be calculated by using above formula.

Lets calculate the rate of increase for a given year compared to the previous year, 

Rate of increase in 2013 = $\frac{68.4-64.5}{64.5}\times 100$ = 6.04

Rate of increase in 2014 = $\frac{72.6-68.4}{68.4}\times 100$ = 10.93

Rate of increase in 2015 = $\frac{83.8-72.6}{72.6}\times 100$ = 15.42

Rate of increase in 2016 = $\frac{100.6-83.8}{83.8}\times 100$ = 20.4

Rate of increase in 2017 = $\frac{112.8-100.6}{100.6}\times 100$ = 12.12

Rate of increase in 2018 = $\frac{128.4-112.8}{112.8}\times 100$ = 13.8

Rate of increase in 2019 = $\frac{134.6-128.4}{128.4}\times 100$ = 4.82

Rate of increase in 2020 = $\frac{156.8-134.6}{134.6}\times 100$ = 16.49

Rate of increase in 2021 = $\frac{176.4-156.8}{156.8}\times 100$ = 12.5

2016 has the highest rate of increase compared to the previous year which is 20.4%

Shortcut Trick

If A + B = 90°, then sinA = cosB and cosA = sinB.

Since 58° + 32° = 90° and 62° + 28° = 90°, their respective sine and cosine ratios are identical.

⇒ sin58° / cos32° = 1 and cos62° / sin28° = 1

⇒ Expression = (3 × 1) / 5 + (2 × 1) / 5 = 3/5 + 2/5 = 1

∴ The correct answer is 1.

Alternate Method

Given:

Expression: (3sin58°) / (5cos32°) + (2cos62°) / (5sin28°)

Formula Used:

sin(90° − θ) = cosθ

cos(90° − θ) = sinθ

Calculations:

⇒ sin58° = sin(90° − 32°) = cos32°

⇒ cos62° = cos(90° − 28°) = sin28°

⇒ Substituting these equivalent values in the given expression:

⇒ (3 × cos32°) / (5 × cos32°) + (2 × sin28°) / (5 × sin28°)

⇒ 3/5 + 2/5

⇒ (3 + 2) / 5

⇒ 5 / 5 = 1

∴ The correct answer is 1.

Additional Information

Complementary Angles

Two angles are complementary if their sum is 90°. The key trigonometric conversions are: sin(90° − θ) = cosθ, tan(90° − θ) = cotθ, and sec(90° − θ) = cosecθ.

Supplementary Angles

Two angles are supplementary if their sum is 180°. Important rules include: sin(180° − θ) = sinθ and cos(180° − θ) = −cosθ.

Basic Trigonometric Identities

Fundamental identities include: sin²θ + cos²θ = 1, sec²θ − tan²θ = 1, and cosec²θ − cot²θ = 1.

Shortcut Trick

Given 7 tan θ = 3 ⇒ tan θ = 3/7.

Divide the numerator and denominator of the expression by cos θ to convert it into tan θ terms.

Expression = (5 tan θ − 1) / (5 tan θ + 2)

Value = (5 × 3/7 − 1) / (5 × 3/7 + 2) = (15/7 − 7/7) / (15/7 + 14/7) = 8 / 29

∴ The correct answer is 8/29.

Alternate Method

Given:

7 tan θ = 3 ⇒ tan θ = 3/7

Formula Used:

By Pythagoras Theorem, Hypotenuse² = Perpendicular² + Base²

sin θ = Perpendicular / Hypotenuse

cos θ = Base / Hypotenuse

Calculations:

⇒ Let Perpendicular (AB) = 3 and Base (BC) = 7

⇒ Hypotenuse (AC) = √(3² + 7²) = √(9 + 49) = √58

⇒ sin θ = 3 / √58

⇒ cos θ = 7 / √58

⇒ Expression = (5 sin θ − cos θ) / (5 sin θ + 2 cos θ)

⇒ Substitute the values of sin θ and cos θ

⇒ Numerator = 5 × (3 / √58) − (7 / √58) = (15 − 7) / √58 = 8 / √58

⇒ Denominator = 5 × (3 / √58) + 2 × (7 / √58) = (15 + 14) / √58 = 29 / √58

⇒ Expression Value = (8 / √58) ÷ (29 / √58) = 8 / 29

∴ The correct answer is 8/29.

Additional Information

Trigonometric Ratios

sin θ = Perpendicular / Hypotenuse, cos θ = Base / Hypotenuse, and tan θ = Perpendicular / Base.

Pythagoras Theorem

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides: H² = P² + B².

Homogeneous Expressions

When calculating a ratio where both the numerator and denominator are of the same degree in sine and cosine, dividing by a suitable power of cosine directly yields the result in terms of tangent.

Shortcut Trick Substitute basic angles to satisfy the given equation 7 cos2θ + 5 sin2θ − 6 = 0. ⇒ For θ = 45°: 7(1/2) + 5(1/2) − 6 = 12/2 − 6 = 0 (Condition satisfied) Substitute θ = 45° in the given expression 1 + √((sec θ + tan θ) / (sec θ − tan θ)) ⇒ 1 + √((√2 + 1) / (√2 − 1)) ⇒ 1 + √((√2 + 1)2 / (2 − 1)) = 1 + √2 + 1 = √2 + 2 ∴ The correct answer is √2 + 2. Alternate Method Given: 7 cos2θ + 5 sin2θ − 6 = 0 Formula Used: sin2θ + cos2θ = 1 sec θ = 1 / cos θ tan θ = sin θ / cos θ Calculations: ⇒ 7(1 − sin2θ) + 5 sin2θ − 6 = 0 ⇒ 7 − 7 sin2θ + 5 sin2θ − 6 = 0 ⇒ 1 − 2 sin2θ = 0 ⇒ 2 sin2θ = 1 ⇒ sin θ = 1/√2 ⇒ θ = 45° (since 0° < θ < 90°) Now, simplify the given expression: ⇒ 1 + √((sec 45° + tan 45°) / (sec 45° − tan 45°)) ⇒ 1 + √((√2 + 1) / (√2 − 1)) Rationalising the denominator inside the square root: ⇒ 1 + √((√2 + 1)(√2 + 1) / ((√2 − 1)(√2 + 1))) ⇒ 1 + √((√2 + 1)2 / (2 − 1)) ⇒ 1 + √2 + 1 ⇒ √2 + 2 ∴ The correct answer is √2 + 2. Additional Information Trigonometric Identities Fundamental identities: sin2θ + cos2θ = 1, sec2θ − tan2θ = 1, and cosec2θ − cot2θ = 1. Values at 45° sin 45° = cos 45° = 1/√2, tan 45° = cot 45° = 1, sec 45° = cosec 45° = √2. Rationalisation To simplify an expression with surds in the denominator, multiply the numerator and denominator by the conjugate of the denominator.

Shortcut Trick

Proportion of milk in the 1^st mixture = 10 ÷ (10 + 6) = 10 ÷ 16 = 5 / 8

Proportion of milk in the 2^nd mixture = 8 ÷ (8 + 4) = 8 ÷ 12 = 2 / 3

Ratio of milk proportion in both mixtures = (5 / 8) : (2 / 3)

Cross multiplying the denominators gives 15 : 16.

∴ The correct answer is 15 : 16.

Alternate Method

Given:

Mixture 1: Quantity of milk = 10 L, Quantity of water = 6 L

Mixture 2: Quantity of milk = 8 L, Quantity of water = 4 L

Formula Used:

Proportion of Milk = Quantity of Milk ÷ Total Mixture Volume

⇒ Total volume of 1^st mixture = 10 + 6 = 16 L

⇒ Proportion of milk in 1^st mixture = 10 ÷ 16 = 5 / 8

⇒ Total volume of 2^nd mixture = 8 + 4 = 12 L

⇒ Proportion of milk in 2^nd mixture = 8 ÷ 12 = 2 / 3

⇒ Ratio of milk in the two mixtures = (5 / 8) : (2 / 3)

⇒ Required ratio = (5 × 3) : (8 × 2)

⇒ Required ratio = 15 : 16

∴ The correct answer is 15 : 16.

Additional Information

Alligation Method

Used to find the ratio in which two ingredients of different concentrations or prices are mixed to produce a desired combination.

Successive Replacement

If x litres are replaced with water n times from V litres of pure liquid, the remaining pure liquid quantity is V × (1 − x ÷ V)ⁿ.

Comparing Fractions & Ratios

To easily compare or divide two fraction ratios like (a/b) ÷ (c/d), we multiply by the reciprocal of the second fraction resulting in (a × d) : (b × c).

Shortcut Trick

For problems relating Cost Price (CP) and Marked Price (MP) directly with profit and discount percentages, use the standard ratio formula:

Ratio of CP to MP = (100 − Discount%) : (100 + Profit%)

Substitute the given values (Discount = 20%, Profit = 10%):

CP : MP = (100 − 20) : (100 + 10)

CP : MP = 80 : 110 = 8 : 11

∴ The correct answer is 8:11.

Alternate Method

Given:

Discount (D) = 20%

Profit (P) = 10%

Formula Used:

Selling Price (SP) = MP × (100 − D%) ÷ 100

Selling Price (SP) = CP × (100 + P%) ÷ 100

Equating both equations for Selling Price (SP):

MP × (100 − D%) = CP × (100 + P%)

CP / MP = (100 − D%) / (100 + P%)

Calculations:

⇒ CP / MP = (100 − 20) / (100 + 10)

⇒ CP / MP = 80 / 110

⇒ CP / MP = 8 / 11

∴ The correct answer is 8:11.

Additional Information

Marked Price (MP)

It is the price printed or tagged on an article. Discount is always calculated on the Marked Price, not on the Cost Price.

Successive Discounts

If two successive discounts of a% and b% are given, the equivalent single discount percentage is (a + b − ab/100)%.

Dishonest Dealer

When a dealer sells goods at cost price but uses a false or lesser weight, his profit percentage is given by: Profit % = [Error ÷ (True Value − Error)] × 100.

Given:

Incomes (in rupees) earned by five employees of a company during the month of March 2019 and the income sources.

Formula used:

The percentage of the total amount of arrears relative to the total amount of bonus = (total arrears) / (total bonus) × 100

Calculation:

Total arrears = Sum of arrears for all employees = 4500 + 4800 + 4900 + 5300 + 5600 = 25100

Total bonus = Sum of bonuses for all employees = 2300 + 2100 + 2500 + 2700 + 2800 = 12400

The percentage of the total amount of arrears relative to the total amount of bonus:

⇒$\left( \frac{25100}{12400} \right) \times 100$ = 202.419 %

Thus, the total amount of arrears paid by the company during the given period is approximately 202.42 %.

Shortcut Trick

For a sector, radius (r), perimeter (P), and area (A) share the relation: 2r² − Pr + 2A = 0.

Substitute P = 63 and A = 209: 2r² − 63r + 418 = 0.

Solving the quadratic equation: (r − 22)(2r − 19) = 0 ⇒ r = 22 cm (since r must be an integer).

Arc length, l = P − 2r = 63 − 44 = 19 cm.

Central angle θ = (l ÷ r) × (180º ÷ π) = (19 ÷ 22) × (180º × 7 ÷ 22) ≈ 49.46º ≈ 49.5º.

∴ The correct answer is 49.5º.

Alternate Method

Given:

Area of the sector (A) = 209 cm²

Perimeter of the sector (P) = 63 cm

Formula Used:

Area of a sector = ½ × l × r

Perimeter of a sector = l + 2r

Arc length (l) = (θ ÷ 360º) × 2πr

Calculations:

⇒ Area = ½ × l × r = 209

⇒ l × r = 418

⇒ Perimeter = l + 2r = 63

⇒ l = 63 − 2r

Substitute the value of l into the area equation:

⇒ (63 − 2r) × r = 418

⇒ 63r − 2r² = 418

⇒ 2r² − 63r + 418 = 0

⇒ 2r² − 44r − 19r + 418 = 0

⇒ 2r(r − 22) − 19(r − 22) = 0

⇒ (2r − 19)(r − 22) = 0

⇒ r = 9.5 or r = 22

Since the radius is given as a non-negative integer, we take r = 22 cm.

⇒ Arc length l = 63 − 2(22) = 19 cm

Now, using the arc length formula with π ≈ 22/7:

⇒ 19 = (θ ÷ 360º) × 2 × (22 ÷ 7) × 22

⇒ θ = (19 × 360º × 7) ÷ (2 × 22 × 22)

⇒ θ = 47880º ÷ 968

⇒ θ = 49.4628...º

Rounding to one decimal place:

⇒ θ ≈ 49.5º

∴ The correct answer is 49.5º.

Additional Information

Sector of a Circle

A sector is a portion of a circle enclosed by two radii and an arc. The angle formed at the center of the circle is known as the central angle.

Area of a Sector

The area can be calculated as Area = (θ ÷ 360º) × πr² or Area = ½ × l × r, where l is the arc length.

Arc Length & Perimeter

The arc length is given by l = (θ ÷ 360º) × 2πr. The total perimeter of the sector includes both radii, so P = l + 2r.

The correct answer is: Option 4.

Key Points

• The word "Claustrophobia" means an extreme or irrational fear of confined places. (संकीर्ण स्थानों का भय)
• Example: She refused to get into the elevator due to her claustrophobia.
• "Claustrophobia" refers to the fear of closed or confined spaces. (संकीर्ण स्थानों का भय)
• Example: His claustrophobia made it difficult for him to travel in the crowded subway.
• Hence, we can infer that the correct one-word substitute for "Fear of closed spaces" is 'Claustrophobia'.

Therefore, the correct answer is: Option 4.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Photophobia (प्रकाश का भय): An abnormal fear or intolerance of light.
• Example: His migraine was accompanied by photophobia, making it difficult for him to be in bright environments.
• Optophobia (आंखें खोलने का भय): Fear of opening one's eyes.
• Example: After the traumatic event, he developed optophobia and often kept his eyes closed.
• Agoraphobia (खुले स्थान का भय): Fear of open or crowded spaces.
• Example: She rarely left her house due to her agoraphobia.

The correct answer is Option 3.

Key Points

• The primary task is to convert the given sentence into passive voice while retaining its original meaning.
• The given sentence is: "Historians tell us how this country was divided into three."
• In passive voice, the object of the active sentence becomes the subject of the passive sentence.
• The correct passive sentence is: "We are told by historians how this country was divided into three."

Therefore, the correct answer is Option 3.

Correct Sentence: "We are told by historians how this country was divided into three."

Additional Information

• Option 1: "We are told by historians how this country has been divided into three." - This changes the tense of the original sentence.
• Option 2: "We are told by historians how this country divided into three." - This is grammatically incorrect.
• Option 4: "We have been told by historians how this country was divided into three." - This changes the tense of the original sentence.

The correct answer is: Option 1.

Key Points

• The word "components" refers to parts or elements that make up a larger system or structure. (घटक)
• Example: The engine is one of the most important components of a car.
• "Components" are essential parts that are used to build or assemble something larger. (घटक)
• Example: The transistors are used as the main components, which are built into computer chips.

Therefore, the correct answer is: Option 1.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Cumponents (कंपोनेंट्स): This is an incorrect spelling of "components".
• Example: Incorrect usage.
• Kamponents (काम्पोनेंट्स): This is an incorrect spelling of "components".
• Example: Incorrect usage.
• Kumponents (कंपोनेंट्स): This is an incorrect spelling of "components".
• Example: Incorrect usage.

The correct answer is Option (3)

Key Points

• To form a meaningful sentence, it is essential to begin with the subject and then logically follow with the statement that provides a coherent idea.
• Part (P) "technology has revolutionised" serves as the beginning of the sentence, introducing the subject 'technology' and stating its impact.
• Part (O) "every aspect of our lives" follows next, describing what has been revolutionised by technology.
• Part (Q) "and it has transformed the way" continues by explaining the specific transformation caused by technology.
• Part (R) "we communicate and connect" completes the sentence by specifying the areas that have been transformed by technology.
• Therefore, the correct sequence is POQR, forming the sentence: "Technology has revolutionised every aspect of our lives and it has transformed the way we communicate and connect."

Additional Information

• Option 1 (RPQO): This option starts with a phrase that does not introduce the subject correctly and leads to a sentence that lacks coherence.
• Option 2 (RPOQ): This option misplaces parts of the sentence, creating a grammatically incorrect and unclear structure.
• Option 4 (ORQP): Starting with "every aspect of our lives" disrupts the logical flow of the sentence, making it awkward and unclear.

The correct answer is: Option 3.

Key Points

• The word "exit" means a way out of a place or a departure from a location. (निर्गम)
• Example: We made a quick exit from the crowded room.
• "Entry" refers to an act of going or coming in, or a way in. (प्रवेश)
• Example: The entry to the museum was grand and welcoming.
• Hence, we can infer that the opposite of 'exit' is 'entry'.

Therefore, the correct answer is: Option 3.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• House (घर): A building for human habitation.
• Example: We visited her house for the weekend.
• Road (सड़क): A wide way leading from one place to another.
• Example: The road was under construction.
• Ceremony (समारोह): A formal event held on special occasions.
• Example: The award ceremony was held in the evening.

The correct answer is: Option 4.

Key Points

• The idiom "care a hang" means to show no interest or to not care at all. (कोई दिलचस्पी नहीं दिखाना या बिल्कुल भी परवाह नहीं करना)
• Example: He doesn't care a hang about what people think of him.
• "Showing no interest" means to be indifferent or not concerned about something. (बिना किसी दिलचस्पी के होना या किसी चीज की परवाह न करना)
• Example: Despite all the excitement, she showed no interest in the event.

Therefore, the correct answer is: Option 4.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Hanging clothes to dry (कपड़े सुखाने के लिए लटकाना): The act of suspending clothes on a line or rack to dry.
• Example: She was hanging clothes to dry in the backyard.
• Putting an outfit on a hanger (कपड़ों को हैंगर पर लगाना): The act of placing clothes on a hanger for storage.
• Example: He carefully put his suit on a hanger to keep it wrinkle-free.
• Caring too much (बहुत ज्यादा परवाह करना): Being excessively concerned about something.
• Example: She is always caring too much about her grades.

The correct answer is Option 1.

Key Points

• The correct passive voice construction involves changing the object of the active sentence into the subject of the passive sentence.
• The new subject (the new product) is followed by the appropriate form of the verb "to be" (will be) and the past participle of the main verb (launched).
• The second part of the sentence also needs to be in passive voice: "it is already being promoted."

Therefore, the correct answer is Option 1.

Correct Sentence: "The new product will be launched by the company next month, and it is already being promoted through various marketing channels."

Additional Information

• Option 2: Contains incorrect verb forms "were been" and "was promoting," making it grammatically incorrect.
• Option 3: Uses incorrect future tense "was launched" and does not use passive voice correctly in the second part.
• Option 4: Incorrect future perfect tense "will have been launched" and incorrect past continuous "was already being promoted."

The correct answer is: Option 4.

Key Points

• The word "Mechanical" means something that is operated by machinery or done without thought or spontaneity; it is automatic and lacks emotion or originality. (यांत्रिक)
• Example: His responses were so mechanical that it felt like he wasn't even listening.
• "Manual" refers to something that is done with the hands or by a person rather than a machine. (हाथ से किया हुआ)
• Example: Manual labor can be very exhausting but also rewarding.
• Hence, we can infer that the opposite of 'mechanical' is 'manual'.

Therefore, the correct answer is: Option 4.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Surrogate (प्रतिनिधि): A substitute, especially a person deputizing for another in a specific role or office.
• Example: She acted as a surrogate mother for the couple.
• Automatised (स्वचालित): Something that is made to operate automatically or without human intervention.
• Example: The factory has fully automatised its production line.
• Yielding (झुकने वाला): Giving way under pressure; not stiff or rigid.
• Example: The material is soft and yielding underfoot.