SSC CGL 2024 Tier-I Official Paper (Held On: 17 Sept, 2024 Shift 1) (Previous Year Paper)

Given series: _DRG_ZDR_BZD_GBZ_RGB

Option 1) ZBDRD → ZDRGBZDRGBZDRGBZDRGB → (It doesn't follow any pattern)

Option 2) BBGRD → ZDRGBZDRGBZDRGBZDRGB → (It doesn't follow any pattern)

Option 3) ZBGRB → ZDRGBZDRGBZDRGBZDRGB → (It doesn't follow any pattern)

Option 4) ZBGRD → ZDRGBZDRGBZDRGBZDRGB → (It follows the pattern of: ZDRGB - ZDRGB - ZDRGB - ZDRGB)

Hence, "Option 4" is the correct answer.

The logic followed here is:

Logic: 1^st number + 85 = 2^nd number.

Now,

For, 429 : 514

⇒ 429 + 85 = 514.

For, 688 : 773

⇒ 688 + 85 = 773.

Similarly, for 546 : ?

⇒ 546 + 85 = 631.

Hence, the correct answer is "Option 2".

The logic followed here is:

Hence, "Option 2" is the correct answer.

The logic followed here is:

'BIRTHDAY' is written as 'ADHTRIBY'

And,

'CONGRESS' is written as 'SERGNOCS'

Similarly,

'DIVISION' is written as

Hence, "Option 4" 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.

The logic followed here is:

Option 1) BEH

Option 2) SVY

Option 3) ILN

Option 4) KNQ

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

Hence, "Option 3" is the correct answer.

The logic followed here is:

Logic: 1^st number ÷ 9 = 2^nd number.

Now,

For, 189 : 21

⇒ 189 ÷ 9 = 21.

For, 234 : 26

⇒ 234 ÷ 9 = 26.

Similarly, for 468 : ?

⇒ 468 ÷ 9 = 52.

Hence, the correct answer is "Option 3".

The correct mirror image of the given figure is:

Hence, the correct answer is "Option 1".

The logic followed here is:

1. The circle element shifts clockwise.

2. The number if lines is increasing in the rhombus shaped element subsequently.

Thus, the final series is:

Hence, "Option 2" is the correct answer.

Family Chart:

Given: G is the brother of F. E is the father of K. C is the daughter of A. F is the mother of E. C is the sister of E.

So, G is the "Wife's brother" of A.

Hence, the correct answer is "Wife's brother" .

The logic followed here is: Sum of place value of letters × 4 = Number code.

'FLBS' is coded as '156'

'MXT' is coded as '228'

Similarly, 'VDZN' will be coded as:

Hence, the correct answer is "264".

BODMAS Table: 

After decoding signs:

 

Given equation: 101 ÷ 48 − 13013 × 13 + 12 = ?

After interchanging the signs then:

⇒ 101 × 48 + 13013 ÷ 13 - 12 = ?

⇒ ? = 101 × 48 + 1001 - 12

⇒ ? = 4848 + 1001 - 12

⇒ ? = 5849 - 12

⇒ ? = 5837

Hence, the correct answer is "5837".

For find the opposite surface of '6', we need to compare the common surface in the given dices,

So, From Dice 1 and Dice 2,

From both dices, 6 is common and on the same surface. So, all remaining surface will be opposite to each other.

5 is opposite to 2.

3 is opposite to 1.

So, 8 will be opposite to 6.

Hence, the correct answer is "8".

Statements:

Some balls are toys.

All circles are balls.

All balls are pits.

The least possible Venn diagram is as follows:

Conclusions:

(I) Some circles are toys. - False (because there is no direct relation between circles and toys so it can be possible but not definite)

(II) All pits are toys. - False (because All balls are pits and Some balls are toys. So, All pits are toys it is not possible)

Hence, the correct answer is "Neither conclusion (I) nor (II) follow".

The logic followed here is:

Hence, the correct answer is "HQYD".

The logic followed here is: (2^nd number / 5) + (3^rd number / 3) = 1^st number.

Option 1) 26 – 95 – 21 → 95 / 5 + 21 / 3 = 19 + 7 = 26

Option 2) 22 – 67 – 22 → 67 / 5 + 22 / 3 = 13.4 + 7.3 = 20.7 (not equal to 22)

Option 3) 20 – 15 – 51 → 15 / 5 + 51 / 3 = 3 + 17 = 20

Option 4) 16 – 45 – 21 → 45 / 5 + 21 / 3 = 9 + 7 = 16

Hence, the correct answer is "22 – 67 – 22".

The logic followed here is:

Logic: Synonyms

Here,

Magnify : Expand → Magnify is a synonym of Expand.

Similarly,

Curtail : Restrict → Curtail is a synonym of Restrict.

Hence, the correct answer is "Option 3".

Given word: 'CERTAIN'

After arranged in alphabetical order:

 

Now,

there are Zero alphabets that will remain unchanged if each of the letters in the word 'CERTAIN' is arranged in English alphabetical order.

Hence, the correct answer is "Option 2".

Statements:

All cars are birds.

Some birds are plants.

All plants are leaves.

The least possible Venn diagram is as follows:

Conclusions:

I. Some birds are cars. - True (because All cars are birds, so some birds are cards it is true)

II. Some plants are birds. - True (because Some birds are plants. So, Some plants are birds it is true)

III. Some leaves are plants. - True (because All plants are leaves. So, Some leaves are plants it is true)

Hence, the correct answer is "All the conclusions follow".

The sheet will appear when the given transparent sheet is folded at the middle horizontal line:

Hence, the correct answer is "Option 4".

The logic followed here is:

Numbers are shifting their position starting from middle to bottom-left corner number.

Bottom-left number is removed in each figure and a new number is added in middle of the position.

Hence, the correct answer is "Option 3".

BODMAS Table: 

After decoding signs:

 

Given equation: 4 + 9 × 48 – 6 ÷ 7 = ?

After interchanging the signs then:

⇒ 4 × 9 + 48 ÷ 6 - 7 = ?

⇒ ? = 4 × 9 + 8 - 7

⇒ ? = 36 + 8 - 7

⇒ ? = 44 - 7

⇒ ? = 37

Hence, the correct answer is "37".

Counting of triangles:

Thus triangles are there in the given figure = 6

Hence, the correct answer is "6".

The logic follows here is:

Logic: n² + 5

Now, 

30 is related to 41

→ 30 = 5² + 5

→ 41 = 6² + 5

14 is related to 21

→ 14 = 3² + 5

→ 21 = 4² + 5

Similarly, for 54 and 69

→ 54 = 7² + 5

→ 69 = 8² + 5

Hence, the correct answer is "69".

The logic followed here is:

Hence, the correct answer is "29".

The correct answer is Rohini Bhate.

Key Points

• Rohini Bhate was an eminent Indian classical dancer renowned for her expertise in the Kathak dance form.
• She was a disciple of Guru Lachhu Maharaj and Shambhu Maharaj, both legendary exponents of Kathak.
• Her dance style was characterized by its technical precision and expressive storytelling.
• She was also a respected teacher and choreographer, contributing significantly to the propagation and evolution of Kathak.

Additional Information

• Kathak
• Kathak is one of the eight major forms of Indian classical dance.
• It originated from the nomadic bards of ancient northern India, known as Kathakars or storytellers.
• The dance form emphasizes intricate footwork, spins, and expressive gestures to convey stories, often from epics like the Mahabharata and Ramayana.
• Kathak has three main gharanas (schools) - Lucknow, Jaipur, and Banaras, each with its unique style and emphasis.
• Other Indian Classical Dance Forms
• Bharatanatyam - Originating from Tamil Nadu, it is known for its fixed upper torso, bent legs, and intricate footwork combined with expressive hand gestures and facial expressions.
• Kathakali - A dance-drama from Kerala, characterized by elaborate costumes, detailed gestures, and well-defined body movements presented in the form of a story-play.
• Kuchipudi - Hailing from Andhra Pradesh, it blends fast rhythms and fluid movements with dramatic storytelling.
• Odissi - From Odisha, it is known for its grace, fluid movements, and sculpturesque poses.
• Manipuri - A dance form from Manipur, characterized by smooth and graceful movements, often depicting themes from the life of Lord Krishna.
• Mohiniyattam - A classical dance from Kerala, it combines elements of Bharatanatyam and Kathakali, performed by women with slow, graceful movements.
• Sattriya - Originating from Assam, it is a dance-drama that combines dance, drama, and music.

The correct answer is 1861.

Key Points

• The Indian High Courts Act was introduced by the British Parliament in the year 1861.
• This act led to the establishment of High Courts in India for the first time, replacing the older Supreme Courts at Calcutta, Bombay, and Madras.
• The High Courts were given extensive original and appellate jurisdiction, marking a significant shift in the Indian judicial system.
• These High Courts were modeled after the English judicial system and were intended to provide a higher level of judicial administration in the major cities of India.

Additional Information

• Indian High Courts Act 1861
• The Indian High Courts Act 1861 was an act of the British Parliament that led to the creation of High Courts in India in the three Presidency towns of Calcutta, Bombay, and Madras.
• Before this act, the judicial system in these cities was managed by Supreme Courts, which were established under the Charter Act of 1774.
• The new High Courts were granted original civil and criminal jurisdiction, appellate jurisdiction, and supervisory jurisdiction over the lower courts.
• This act was significant as it brought uniformity and improvement in the judicial administration of British India.
• Supreme Courts of Calcutta, Bombay, and Madras
• These courts were established under the Regulating Act of 1773 and subsequent acts, and they functioned until the High Courts were established in 1861.
• They were the highest courts in British India and had both original and appellate jurisdiction.
• The Supreme Courts were replaced by the High Courts to streamline and improve the judicial system.
• Judicial Administration in British India
• The British introduced several reforms to the judicial system in India to make it more efficient and aligned with British legal principles.
• The High Courts Act was a part of these reforms, which aimed to enhance the administration of justice in the colonies.
• These reforms were crucial in shaping the modern Indian judicial system.

The correct answer is Tata Iron and Steel Company.

Key Points

• Tata Iron and Steel Company (TISCO), now known as Tata Steel Ltd, is one of the largest private sector steel manufacturing companies in India.
• Tata Steel was established by Jamsetji Tata in 1907 and is headquartered in Mumbai, Maharashtra, India.
• It is part of the Tata Group, which is a prominent Indian multinational conglomerate.
• Tata Steel is known for its significant contribution to the industrialization of India and remains a leading player in the global steel industry.

Additional Information

• Steel Authority of India Limited (SAIL)
• It is a public sector undertaking and one of the largest steel-making companies in India.
• SAIL is owned and operated by the Government of India.
• It was founded in 1973 and is headquartered in New Delhi.
• SAIL operates and owns 5 integrated steel plants and 3 special steel plants in India.
• Oil India Limited (OIL)
• It is the second-largest hydrocarbon exploration and production public sector company in India.
• OIL is owned by the Government of India.
• It was founded in 1959 and is headquartered in Duliajan, Assam, India.
• OIL is involved in the exploration, development, and production of crude oil and natural gas.
• Bharat Heavy Electricals Limited (BHEL)
• It is a public sector engineering and manufacturing company in India.
• BHEL is owned by the Government of India.
• It was established in 1964 and is headquartered in New Delhi.
• BHEL is engaged in the design, engineering, manufacturing, construction, testing, commissioning, and servicing of a wide range of products and services for the core sectors of the economy, including power, transmission, industry, transportation, renewable energy, oil & gas, and defense.

The correct answer is 1950.

Key Points

• The Election Commission of India (ECI) was established on 25^th January 1950.
• It is an autonomous constitutional authority responsible for administering election processes in India at both the national and state levels.
• The ECI ensures the conduct of free and fair elections in the country, which is a fundamental aspect of the democratic framework of India.
• The Election Commission operates under the authority of the Constitution per Article 324, and subsequently enacted Representation of the People Acts.

Additional Information

• Election Commission of India (ECI)
• The ECI is responsible for the superintendence, direction, and control of the preparation of the electoral rolls for, and the conduct of, all elections to Parliament and to the Legislature of every State and of elections to the offices of President and Vice President held under the Constitution.
• The body administers elections to the Lok Sabha, Rajya Sabha, and State Legislative Assemblies in India, and the offices of the President and Vice President in the country.
• It was originally a one-member body but was made a three-member body in 1989.
• The Chief Election Commissioner (CEC) heads the Election Commission of India.
• Article 324 of the Indian Constitution
• Article 324 vests the power of superintendence, direction, and control of elections in an independent body, which is the Election Commission.
• The Election Commission has the power to take all necessary steps to ensure free and fair elections.
• Representation of the People Acts
• The Representation of the People Act, 1950 provides for the allocation of seats in the House of the People and in the Legislative Assemblies, the delimitation of constituencies for elections, the qualifications of voters at such elections, and other related matters.
• The Representation of the People Act, 1951 provides for the actual conduct of elections in India, the qualifications and disqualifications for membership of Parliament and State Legislatures, and the resolution of election disputes.

The correct answer is ₹6,000.

Key Points

• The Pradhan Mantri Kisan Samman Nidhi Yojana (PM-Kisan) is a central sector scheme launched by the Government of India.
• Under this scheme, income support of ₹6,000 per year is provided to all farmer families across the country.
• This amount is paid in three equal installments of ₹2,000 each, which are directly transferred to the bank accounts of the beneficiaries.
• The scheme aims to support the financial needs of farmers in procuring various inputs to ensure proper crop health and appropriate yields.

Additional Information

• Pradhan Mantri Kisan Samman Nidhi (PM-Kisan)
• PM-Kisan is a central sector scheme with 100% funding from the Government of India.
• It was launched on 24^th February 2019, and the first installment was paid out in December 2018.
• The scheme is implemented by the Ministry of Agriculture and Farmers' Welfare.
• All landholding farmer families are eligible for this scheme, provided they meet the criteria set by the government.
• Objectives
• The main objective of PM-Kisan is to provide financial support to farmers, helping them to procure various inputs for agriculture and allied activities, as well as domestic needs.
• The scheme aims to supplement the financial needs of the farmers in procuring inputs for crop production, ensuring proper crop health and yields.
• It also aims to protect farmers from falling into the clutches of moneylenders for meeting such expenses and ensure their continuance in farming activities.
• Eligibility Criteria
• All landholding farmer families, which have cultivable landholding in their names, are eligible for this scheme.
• However, certain exclusions apply, such as institutional landholders, farmer families holding constitutional posts, serving or retired officers and employees of state/central government as well as public sector undertakings and government autonomous bodies, and professionals like doctors, engineers, and lawyers.
• Implementation
• The scheme is implemented through an electronic database containing the details of all the beneficiaries.
• The funds are directly transferred to the bank accounts of the beneficiaries through Direct Benefit Transfer (DBT) mode.
• The states and union territories are responsible for identifying the farmer families which are eligible for support as per scheme guidelines.

The correct answer is 2014.

Key Points

• The Pro Kabaddi League was inaugurated for the first time in India in 2014.
• It was established with the aim to popularize the sport of kabaddi and provide a professional platform for players.
• The league was inspired by the success of the kabaddi tournament at the 2006 Asian Games.
• The inaugural season of the Pro Kabaddi League featured eight teams, which competed in a double round-robin format followed by playoffs.

Additional Information

• Pro Kabaddi League (PKL)
• The Pro Kabaddi League is a professional-level kabaddi league in India, established in 2014.
• It is organized by Mashal Sports and STAR India, with the backing of the Amateur Kabaddi Federation of India (AKFI), the Asian Kabaddi Federation (AKF), and the International Kabaddi Federation (IKF).
• The league follows a format similar to that of the Indian Premier League (IPL), with franchises representing various cities across India.
• Players from different countries, including Iran, South Korea, and Kenya, participate in the league, making it a global event.
• The league has significantly contributed to the growth and popularity of kabaddi, both in India and internationally.
• Kabaddi
• Kabaddi is a contact team sport that originated in ancient India.
• The game involves two teams of seven players each, who take turns sending a "raider" into the opposing team's half of the court to tag as many defenders as possible and return to their half of the court without being tackled.
• Points are scored by the raider for each defender tagged, while the opposing team earns points for stopping the raider.
• Kabaddi is known for its fast-paced and physically demanding nature, requiring strength, agility, and strategic thinking.

The correct answer is Education.

Key Points

• Wood's Dispatch of 1854 is considered the Magna Carta of English Education in India.
• The Dispatch recommended a system of education from the primary level to the university level.
• It proposed the establishment of universities in major cities like Bombay, Calcutta, and Madras.
• Wood's Dispatch emphasized the need for vocational and female education.

Additional Information

• Wood's Dispatch
• Wood's Dispatch was a comprehensive plan for the education system in India, formulated by Sir Charles Wood, the President of the Board of Control of the British East India Company, in 1854.
• It laid the foundation for a systematic and organized education policy in India under British rule.
• The Dispatch recommended the establishment of a Department of Public Instruction in each province to oversee education matters.
• It also suggested the use of vernacular languages at the primary level, while English was to be the medium of instruction for higher education.
• Universities in Major Cities
• Following the recommendations of Wood's Dispatch, universities were established in Bombay, Calcutta, and Madras in 1857.
• These universities played a crucial role in shaping modern education in India and produced many notable scholars and leaders.
• Vocational and Female Education
• Wood's Dispatch emphasized the importance of vocational education to improve the skills of the workforce.
• It also highlighted the need for female education to uplift the status of women in society.
• Special grants were recommended to promote female education.

The correct answer is Lovlina Borgohain.

Key Points

• Lovlina Borgohain is an Indian amateur boxer who won a bronze medal at the 2020 Tokyo Olympics in the women's welterweight event.
• In December 2022, Lovlina was elected as the Chairperson and a voting member of the International Boxing Association (IBA) Athletes' Committee.
• She has also won a bronze medal in the 2018 and 2019 AIBA Women's World Boxing Championships.
• Lovlina Borgohain hails from Assam, India, and is known for her significant contributions to Indian boxing on an international platform.

Additional Information

• International Boxing Association (IBA)
• The International Boxing Association (IBA), formerly known as AIBA, is a worldwide governing body for the sport of boxing, both amateur and professional.
• It was established in 1946 and aims to promote and develop boxing globally while ensuring fair and equal opportunities for all boxers.
• The IBA organizes major international boxing events, including the World Boxing Championships and the Women's World Boxing Championships.
• It also works to implement rules and regulations for the sport, ensuring the safety and integrity of boxing competitions.
• AIBA Women's World Boxing Championships
• The AIBA Women's World Boxing Championships is an international event organized by the International Boxing Association (IBA) for female boxers.
• The competition features multiple weight categories, allowing women boxers from different countries to compete for world titles.
• The championship is held biennially and has been instrumental in promoting women's boxing on a global scale.
• It provides a platform for female boxers to showcase their skills and gain international recognition.
• Tokyo 2020 Olympics
• The Tokyo 2020 Olympics was an international multi-sport event held in Tokyo, Japan, from July 23 to August 8, 2021, after being postponed due to the COVID-19 pandemic.
• It featured competitions in 33 sports and 339 events, with athletes from over 200 countries participating.
• The Tokyo 2020 Olympics was notable for its strict COVID-19 protocols and the absence of spectators in most venues.
• India achieved its best-ever medal tally at the Tokyo 2020 Olympics, winning a total of 7 medals, including 1 gold.

The correct answer is 1993.

Key Points

• Members of Parliament Local Area Development Scheme (MPLADS) was launched in December 1993.
• The scheme enables Members of Parliament (MPs) to recommend developmental work in their constituencies with emphasis on creating durable community assets based on locally felt needs.
• Initially, the scheme allowed MPs to recommend works for Rs. 5 lakh per annum, which has now been increased to Rs. 5 crore per annum.
• The MPLADS funds can be used for various development activities including the construction of roads, schools, sanitation facilities, and drinking water facilities.

Additional Information

• Objective of MPLADS:
• The primary objective of MPLADS is to enable MPs to identify and fund small-scale developmental projects in their constituencies, addressing specific local needs and promoting the welfare of the people.
• Implementation:
• The MPLADS is implemented by the Ministry of Statistics and Programme Implementation (MoSPI).
• The funds released under MPLADS are non-lapsable, meaning they can be carried forward to subsequent years if not utilized within the financial year.
• Monitoring and Accountability:
• District authorities are responsible for the execution of projects recommended by MPs and ensuring that the funds are used appropriately.
• Regular audits and reviews are conducted to ensure transparency and accountability in the utilization of MPLADS funds.
• Recent Amendments:
• In recent years, the MPLADS guidelines have been amended to include provisions for funding projects related to COVID-19 relief and other national priorities.

The correct answer is willy-willies.

Key Points

• Willy-willies is the term used to describe tropical cyclones in Australia.
• Tropical cyclones are intense low-pressure systems that form over warm ocean waters and can cause significant damage to coastal regions.
• These cyclones are characterized by strong winds, heavy rainfall, and can lead to flooding and storm surges.
• The term willy-willies is specifically used in the Australian region, whereas other regions have different terms for similar weather phenomena.

Additional Information

• Hurricanes
• Hurricanes are tropical cyclones that occur in the North Atlantic Ocean and northeastern Pacific Ocean.
• The term is used primarily in the Caribbean Sea, Gulf of Mexico, and along the eastern coastlines of the United States and Mexico.
• They are known for their destructive power, which includes strong winds, heavy rains, and storm surges.
• Typhoons
• Typhoons are tropical cyclones that occur in the northwestern Pacific Ocean, primarily affecting Southeast Asia, Japan, and eastern China.
• They are similar in structure and impact to hurricanes and willy-willies, with strong winds and heavy rainfall.
• The term "typhoon" is derived from the Chinese term "tai fung," meaning "great wind."
• Cyclones
• The term "cyclone" is a general term used to describe a rotating, organized system of clouds and thunderstorms that originate over tropical or subtropical waters.
• In the Indian Ocean and South Pacific, these systems are referred to as cyclones.
• Cyclones can cause extensive damage, including high winds, heavy rainfall, and coastal flooding.

The correct answer is Starch: Egg yolk.

Key Points

• Starch is a carbohydrate consisting of a large number of glucose units joined by glycosidic bonds.
• It is produced by most green plants as an energy store.
• Egg yolk does not contain starch; it is rich in proteins and fats.
• Common sources of starch include potatoes, rice, wheat, and corn.

Additional Information

• Lactose
• Lactose is a disaccharide sugar derived from galactose and glucose that is found in milk.
• It makes up around 2-8% of milk (by weight).
• Lactose is important for the proper digestion of milk in mammals.
• Fructose
• Fructose is a simple sugar found in many plants, where it is often bonded to glucose to form the disaccharide sucrose.
• It is one of the three dietary monosaccharides, along with glucose and galactose, that are absorbed directly into the bloodstream during digestion.
• Fructose is commonly found in fruits such as grapes, apples, and honey.
• Maltose
• Maltose, also known as malt sugar, is a disaccharide formed from two units of glucose joined with an α(1→4) bond.
• It is found in germinating grains like wheat, cornmeal, and barley.
• Maltose is used in the brewing of beer and is a component of malt.

The correct answer is i (b), ii (c), iii (d), iv (a).

Key Points

• Global System of Trade Preferences (GSTP) was established in 1988 to increase trade among developing countries.
• The India-Gulf Cooperation Council Free Trade Agreement negotiations began in 2004 to enhance economic cooperation between India and the GCC states.
• The India-Republic of Korea Comprehensive Economic Partnership Agreement (CEPA) was signed in 2010 to promote greater economic collaboration and trade between India and South Korea.
• The 7^th Trade Policy Review of India was conducted in 2021 by the World Trade Organization (WTO) to review India's trade policies and practices.

Additional Information

• Global System of Trade Preferences (GSTP)
• It was established in 1988 during the UN Conference on Trade and Development (UNCTAD).
• It aims to promote trade among developing countries through the exchange of tariff concessions.
• India-Gulf Cooperation Council (GCC) Free Trade Agreement
• The GCC consists of Bahrain, Kuwait, Oman, Qatar, Saudi Arabia, and the UAE.
• FTA negotiations started in 2004 to facilitate trade and economic cooperation between India and GCC countries.
• India-Republic of Korea Comprehensive Economic Partnership Agreement (CEPA)
• CEPA was signed in 2010 to foster trade and economic relations between India and South Korea.
• The agreement covers trade in goods, services, investment, and intellectual property rights.
• Trade Policy Review Mechanism (TPRM)
• It was established in 1989 under the General Agreement on Tariffs and Trade (GATT) and is now an integral part of the WTO.
• The 7^th Trade Policy Review of India in 2021 aimed to evaluate and improve the transparency of India's trade policies.

The correct answer is Kirana gharana.

Key Points

• Prabha Atre is a renowned Indian classical vocalist who belongs to the Kirana gharana.
• The Kirana gharana is known for its emphasis on perfect pitch (sur) and intricate note patterns (taan).
• Prabha Atre has been a significant figure in Indian classical music and has contributed extensively to the Kirana gharana's repertoire.
• She has received numerous awards and accolades, including the Padma Vibhushan in 2022, which is India's second-highest civilian award.

Additional Information

• Gharana
• A gharana is a system of social organization linking musicians or dancers by lineage or apprenticeship, and by adherence to a particular musical style.
• The term "gharana" comes from the Hindi word "ghar," which means "house" and refers to a family tradition.
• Each gharana has its own distinct features and styles that differentiate it from others.
• The concept of gharana is pivotal in the Indian classical music tradition, playing a crucial role in preserving the unique characteristics of different musical styles.
• Kirana Gharana
• The Kirana gharana is one of the most prominent gharanas in Hindustani classical music.
• It was founded in the 19^th century by Ustad Abdul Karim Khan and Ustad Abdul Wahid Khan.
• The gharana is named after the town of Kirana in Uttar Pradesh, India.
• The Kirana gharana is renowned for its emphasis on the development and elaboration of notes and the emotional expression of ragas.
• Padma Vibhushan
• The Padma Vibhushan is the second-highest civilian award in India, after the Bharat Ratna.
• It is awarded to recognize exceptional and distinguished service in any field, including government service, arts, literature, science, and public affairs.
• The award was instituted in 1954 and is conferred by the President of India.
• Recipients of the Padma Vibhushan are announced annually on Republic Day (26^th January).

The correct answer is Parthians.

Key Points

• The Parthian Empire ruled over a significant part of ancient Iran and extended their domain into the northwestern regions of the Indian subcontinent, including parts of present-day Pakistan.
• The Parthian period is known for its cultural syncretism, blending Greek, Persian, and regional influences, which is reflected in their art, architecture, and inscriptions.
• Takht-i-Bahi, a historical site near Mardan in present-day Pakistan, is famous for its Buddhist monastery ruins, which flourished during the Parthian period.
• The Parthian rulers in this region were known for their patronage of Buddhism, which contributed to the spread and development of Buddhist culture and architecture.

Additional Information

• Takht-i-Bahi
• Takht-i-Bahi is an archaeological site of an ancient Buddhist monastery situated in Mardan, Khyber Pakhtunkhwa, Pakistan.
• The site is renowned for its well-preserved ruins, which date back to the 1^st century CE and were in use until the 7^th century CE.
• The name "Takht-i-Bahi" means "Throne of the Water Spring" in Persian, indicating the presence of a spring near the monastery.
• The monastery complex includes stupas, courtyards, meditation cells, and assembly halls, reflecting the architectural style and cultural influences of the Parthian period.
• Parthian Empire
• The Parthian Empire, also known as the Arsacid Empire, was a major Iranian political and cultural power in ancient Persia, which lasted from approximately 247 BCE to 224 CE.
• The Parthians were known for their unique style of governance, which combined elements of Greek, Persian, and regional traditions.
• Their empire stretched from the Euphrates River in the west to the Indus River in the east, including regions of modern-day Iran, Iraq, Armenia, and parts of Pakistan and Afghanistan.
• Parthian art, architecture, and inscriptions, including those found at sites like Takht-i-Bahi, reflect a blend of Hellenistic and Persian influences.
• Sakas
• The Sakas, also known as the Scythians, were a group of ancient Iranian nomadic tribes who migrated into the Indian subcontinent during the 1^st century BCE.
• They established several kingdoms in the northwestern regions of India, including the Indo-Scythian kingdom.
• The Sakas were known for their contributions to the cultural and political landscape of ancient India, including the development of the Saka era calendar.
• Indo-Greek Kingdom
• The Indo-Greek Kingdom was established by Greek rulers who invaded and settled in the northwestern regions of the Indian subcontinent during the 2^nd century BCE.
• The Indo-Greek rulers are known for their coinage, which often featured a blend of Greek and Indian artistic styles and inscriptions.
• Their rule contributed to the spread of Hellenistic culture and the fusion of Greek and Indian cultural elements in art, architecture, and religion.

The correct answer is Delhi.

Key Points

• According to the Census of India 2011, Delhi has the highest slum population among the Union Territories.
• The slum population in Delhi contributes significantly to the overall urban population of the Union Territory.
• Delhi's slums are characterized by inadequate housing and poor living conditions.
• Efforts to improve living conditions in slums include various government schemes and initiatives aimed at urban development and poverty alleviation.

Additional Information

• Census of India
• The Census of India is a comprehensive enumeration of the Indian population conducted every ten years by the Office of the Registrar General and Census Commissioner, under the Ministry of Home Affairs, Government of India.
• The census collects a wide range of demographic, social, and economic data, which is used for planning and policy-making at various levels of government.
• The most recent census was conducted in 2011, and the next one is scheduled for 2021.
• Slum
• A slum is a heavily populated urban residential area characterized by substandard housing and squalor.
• Slums often lack basic services such as sanitation, clean water, reliable electricity, and adequate housing.
• Slum dwellers typically live in conditions of extreme poverty, with limited access to education and healthcare.
• Union Territory
• A Union Territory is a type of administrative division in India, which is governed directly by the Central Government.
• Unlike states, Union Territories have more limited self-governance and are administered by an appointed Lieutenant Governor or Administrator.
• As of 2021, India has eight Union Territories.
• Urban Development Schemes
• Various urban development schemes have been launched by the Indian government to improve the living conditions in urban areas, including slums.
• Examples include the Pradhan Mantri Awas Yojana (PMAY) for affordable housing, the Atal Mission for Rejuvenation and Urban Transformation (AMRUT), and the Smart Cities Mission.
• These schemes aim to provide better infrastructure, housing, and basic services to urban residents, including those living in slums.

The correct answer is 0.07.

Key Points

• The Gross National Product (GNP) is a broad measure of a nation's total economic activity.
• The GNP is 16% higher than the last year, indicating overall economic growth.
• Inflation rate is 9%, which represents the rate at which the general level of prices for goods and services is rising.
• The production growth rate can be determined by adjusting the GNP growth rate for inflation.

Additional Information

• Gross National Product (GNP)
• GNP measures the market value of all the products and services produced in one year by labor and property supplied by the residents of a country.
• It includes the value of goods and services produced by a country's residents, regardless of the location of the actual production.
• Inflation
• Inflation is the rate at which the general level of prices for goods and services is rising and subsequently eroding purchasing power.
• Central banks attempt to limit inflation, and avoid deflation, in order to keep the economy running smoothly.
• Economic Growth
• Economic growth is an increase in the production of economic goods and services, compared from one period of time to another.
• It can be measured in nominal or real terms, the latter of which is adjusted for inflation.
• Calculation of Production Growth Rate
• To find the production growth rate, we adjust the GNP growth rate for inflation.
• Formula: Production Growth Rate = (1 + GNP Growth Rate) / (1 + Inflation Rate) - 1
• Using the given values: GNP Growth Rate = 16% or 0.16, Inflation Rate = 9% or 0.09
• Production Growth Rate = (1 + 0.16) / (1 + 0.09) - 1 = 1.16 / 1.09 - 1 = 1.0642 - 1 ≈ 0.07

The correct answer is Assam.

Key Points

• Bhupen Hazarika was a legendary figure from Assam, known for his contribution to music, cinema, and literature.
• He was a playback singer, lyricist, musician, poet, and filmmaker, renowned for his works in Assamese cinema.
• Bhupen Hazarika is often referred to as the 'Bard of Brahmaputra' because of his deep connection with the river and his impactful songs about it.
• His songs are known for their socio-political themes, often addressing issues of humanity and universal brotherhood.

Additional Information

• Assam
• Assam is a state in northeastern India known for its rich culture, tea plantations, and biodiversity.
• The state is home to the mighty Brahmaputra River, which plays a crucial role in the lives of the people and the state's economy.
• Assam has a diverse population with numerous ethnic groups and languages, and its capital is Dispur.
• The state celebrates various traditional festivals, including Bihu, which marks the Assamese New Year and is celebrated with much fervor.
• Bhupen Hazarika's Contributions
• He was awarded several prestigious awards, including the Dadasaheb Phalke Award, Padma Shri, and Padma Bhushan.
• Hazarika's music and films often highlighted the culture and issues of Assam and the northeastern region of India.
• He was also a prominent politician and served as a member of the Assam Legislative Assembly.
• His legacy continues to inspire and influence artists and activists in India and beyond.

The correct answer is copper.

Key Points

• Copper is known for its excellent thermal conductivity, making it ideal for applications where heat transfer is crucial.
• In a calorimeter, quick and accurate measurement of heat is essential, and copper’s high thermal conductivity ensures this efficiency.
• Copper is also malleable and ductile, allowing it to be shaped into thin sheets needed for the construction of calorimeters.
• Copper has good corrosion resistance, which is necessary for the durability and longevity of the calorimeter.

Additional Information

• Calorimeter
• A calorimeter is an instrument used to measure the amount of heat involved in a chemical or physical process.
• It works on the principle of heat transfer and is used in various scientific and industrial applications to determine heat capacity, specific heat, and other thermal properties.
• Thermal Conductivity
• Thermal conductivity is a physical property of materials that indicates their ability to conduct heat.
• Materials with high thermal conductivity, like copper, transfer heat more efficiently than materials with low thermal conductivity.
• This property is critical in designing devices like calorimeters where accurate heat measurements are required.
• Other Metals Mentioned
• Silver - While silver has higher thermal conductivity than copper, it is more expensive and less commonly used in calorimeters.
• Platinum - Platinum is also a good conductor of heat but is expensive and used in more specialized applications.
• Cesium - Cesium is a metal with lower thermal conductivity and is not suitable for use in calorimeters.

The correct answer is H2.

Key Points

• When dilute sulphuric acid (H2SO4) reacts with magnesium (Mg), hydrogen gas (H2) is released.
• The chemical equation for the reaction is: Mg + H2SO4 → MgSO4 + H2.
• Magnesium (Mg) is a highly reactive metal and reacts vigorously with dilute acids.
• The reaction of a metal with an acid typically produces a salt and hydrogen gas.

Additional Information

• Sulphuric Acid (H2SO4)
• It is a strong mineral acid with various industrial applications.
• It is used in the production of fertilizers, in petroleum refining, and in wastewater processing.
• H2SO4 is highly corrosive and can cause severe burns.
• Hydrogen Gas (H2)
• Hydrogen is the lightest and most abundant element in the universe.
• It is commonly used in the chemical industry and in fuel cells for energy production.
• Hydrogen gas is colorless, odorless, and highly flammable.
• Magnesium (Mg)
• Magnesium is a lightweight metal with a silvery-white appearance.
• It is used in the production of alloys, in the automotive industry, and in the manufacture of fireworks and flares.
• Magnesium is essential for all living cells and is the 11^th most abundant element by mass in the human body.
• Chemical Reactions of Metals with Acids
• Metals react with acids to produce a salt and hydrogen gas.
• The general reaction can be represented as: Metal + Acid → Salt + Hydrogen gas.
• The reactivity of the metal determines the vigor of the reaction.

The correct answer is 2) 2 and 3 both.

Key Points

• The Indian Constitution prescribes certain duties to its citizens known as Fundamental Duties.
• Among the given options, "To pay income tax" and "To cast the vote in election" are not listed as Fundamental Duties in the Indian Constitution.
• The Fundamental Duties are listed in Article 51A of the Indian Constitution.
• There are a total of 11 Fundamental Duties for Indian citizens, introduced by the 42^nd Amendment in 1976, and one more added by the 86^th Amendment in 2002.

Additional Information

• Fundamental Duties
• The concept of Fundamental Duties in India is inspired by the Constitution of the erstwhile USSR.
• These duties are intended to serve as a constant reminder to every citizen that they should not only be conscious of their rights but also of their duties.
• Some of the key Fundamental Duties include respecting the national symbols, cherishing the noble ideals of the freedom struggle, and protecting the sovereignty and integrity of India.
• Fundamental Duties are non-justiciable, meaning they are not enforceable by the courts for their violation.
• Amendments Related to Fundamental Duties
• The 42^nd Amendment Act of 1976 introduced Fundamental Duties in the Indian Constitution.
• The 86^th Amendment Act of 2002 added the 11^th duty, which mandates that parents or guardians provide opportunities for education to their children between the age of six and fourteen years.
• List of Fundamental Duties
• To abide by the Constitution and respect its ideals and institutions, the National Flag, and the National Anthem.
• To cherish and follow the noble ideals which inspired our national struggle for freedom.
• To uphold and protect the sovereignty, unity, and integrity of India.
• To defend the country and render national service when called upon to do so.
• To promote harmony and the spirit of common brotherhood amongst all the people of India transcending religious, linguistic, and regional or sectional diversities; to renounce practices derogatory to the dignity of women.
• To value and preserve the rich heritage of our composite culture.
• To protect and improve the natural environment including forests, lakes, rivers, and wildlife, and to have compassion for living creatures.
• To develop the scientific temper, humanism, and the spirit of inquiry and reform.
• To safeguard public property and to abjure violence.
• To strive towards excellence in all spheres of individual and collective activity so that the nation constantly rises to higher levels of endeavor and achievement.
• To provide opportunities for education to children between the ages of six and fourteen years (added by the 86^th Amendment in 2002).

The correct answer is Humayun.

Key Points

• Humayun was the son of Babur, the founder of the Mughal Empire in India.
• Humayun succeeded his father Babur after his death in 1530.
• Humayun's reign was marked by a series of setbacks and recoveries, including being ousted from his kingdom by Sher Shah Suri.
• He eventually regained his throne with the help of the Persian Empire and re-established Mughal rule in India before his accidental death in 1556.

Additional Information

• Babur
• Babur was the founder of the Mughal Empire in India, establishing it after his victory at the First Battle of Panipat in 1526.
• He was a descendant of Timur on his father's side and Genghis Khan on his mother's side.
• Babur's contributions laid the foundation for a powerful and enduring empire in the Indian subcontinent.
• Humayun
• Humayun was the eldest son of Babur and succeeded him as the second Mughal Emperor.
• His full name was Nasir-ud-Din Muḥammad Humayun.
• Humayun faced challenges from Afghan and Rajput rulers and was defeated by Sher Shah Suri, leading to his temporary exile.
• With Persian assistance, Humayun was able to reclaim his throne in 1555.
• Sher Shah Suri
• Sher Shah Suri was an Afghan ruler who defeated Humayun and established the Suri Empire in India.
• He is known for his administrative reforms, including the introduction of the Rupiya and the improvement of infrastructure.
• Sher Shah's rule was short-lived, but his policies greatly influenced the Mughal administration.
• Mughal Empire
• The Mughal Empire was a prominent empire in the Indian subcontinent that lasted from the early 16^th to the mid-18^th century.
• It was known for its rich cultural contributions, monumental architecture, and the amalgamation of Persian and Indian cultures.
• The empire reached its zenith under rulers like Akbar, Jahangir, Shah Jahan, and Aurangzeb.

The correct answer is Shevaroy hill.

Key Points

• The Shevaroy Hills are part of the Eastern Ghats in Tamil Nadu, India.
• They are located in the Salem district and are known for their coffee plantations and fruit orchards.
• The Shevaroy Hills are home to the popular hill station of Yercaud.
• The highest point in the Shevaroy Hills is Solaikaradu which stands at an elevation of 1620 meters.

Additional Information

• Eastern Ghats
• The Eastern Ghats are a discontinuous range of mountains along India's eastern coast, extending from West Bengal through Odisha and Andhra Pradesh to Tamil Nadu.
• The Eastern Ghats are older than the Western Ghats and are characterized by a series of low hills, plateaus, and valleys.
• These mountains are rich in minerals and biodiversity, with several protected areas and wildlife sanctuaries.
• Mizo Hills
• The Mizo Hills, also known as Lushai Hills, are located in the state of Mizoram in northeastern India.
• These hills are part of the Patkai range and are characterized by steep, rugged terrain and dense forests.
• Khasi Hills
• The Khasi Hills are located in the central part of Meghalaya in northeastern India.
• They are known for their unique matrilineal culture and the picturesque hill station of Shillong.
• Anai Mudi
• Anai Mudi is the highest peak in the Western Ghats and South India, located in the state of Kerala.
• It stands at an elevation of 2,695 meters and is part of the Anaimalai Hills range.

The correct answer is Typhoid, Cholera, Tuberculosis.

Key Points

• Typhoid fever is caused by the bacterium Salmonella typhi and is typically spread through contaminated food and water.
• Cholera is caused by the bacterium Vibrio cholerae and is also usually spread through contaminated water and food, leading to severe diarrhea and dehydration.
• Tuberculosis (TB) is caused by the bacterium Mycobacterium tuberculosis and primarily affects the lungs, but can also affect other parts of the body.
• These diseases are treatable with antibiotics, which target and kill the bacterial pathogens responsible for the infections.

Additional Information

• Influenza
• Influenza, also known as the flu, is caused by influenza viruses, not bacteria.
• It is a contagious respiratory illness with symptoms like fever, cough, and body aches.
• Vaccines are available to help prevent influenza.
• Dengue
• Dengue is caused by the dengue virus, which is transmitted by Aedes mosquitoes.
• It leads to symptoms such as high fever, severe headache, pain behind the eyes, joint and muscle pain, and rash.
• There is no specific treatment for dengue, but supportive care can help manage symptoms.
• Malaria
• Malaria is caused by Plasmodium parasites, which are transmitted through the bites of infected Anopheles mosquitoes.
• Symptoms include fever, chills, and flu-like illness; if not treated, it can lead to severe complications and death.
• Antimalarial drugs are used to treat and prevent malaria.
• Common Cold
• The common cold is caused by a variety of viruses, most commonly rhinoviruses.
• It is characterized by symptoms such as sneezing, runny nose, and sore throat.
• There is no cure for the common cold, but symptoms can be managed with over-the-counter medications.

The correct answer is 16.

• In chess, each player starts with 16 pieces.
• The 16 pieces for each side include: 1 King, 1 Queen, 2 Rooks, 2 Knights, 2 Bishops, and 8 Pawns.
• White and Black both have identical sets of pieces at the start of the game.
• The objective of chess is to checkmate the opponent's King while strategically using all pieces.
• King:
• The King is the most important piece, but it is not very powerful. It can only move one square in any direction.
• The game ends when the King is checkmated.
• Queen:
• The Queen is the most powerful piece. It can move any number of squares in any direction.
• Rook:
• The Rook can move any number of squares along a row or column.
• Bishop:
• The Bishop moves diagonally any number of squares.
• Knight:
• The Knight moves in an L-shape: two squares in one direction and then one square perpendicular.
• Knights are the only pieces that can jump over other pieces.
• Pawn:
• Pawns move forward one square, but capture diagonally.
• Pawns have the option to move two squares forward on their first move.
• If a pawn reaches the opposite end of the board, it can be promoted to any other piece (except a King).

The correct answer is Jain.

Key Points

• Paryushana is one of the most important religious festivals celebrated by the Jain community.
• The festival lasts for 8 days for the Svetambara sect and 10 days for the Digambara sect.
• The word "Paryushana" means "to stay in one place," signifying a time of increased religious observance.
• It is marked by fasting, prayer, meditation, and introspection to purify oneself and seek forgiveness for past wrongs.

Additional Information

• Jainism:
• Jainism is one of the oldest religions in the world, originating in India around the 6^th century BCE.
• The religion emphasizes non-violence (ahimsa), truth (satya), and asceticism as key principles.
• Jains believe in the cycle of reincarnation and the ultimate goal of achieving liberation (moksha) from the cycle of birth and death.
• The two major sects within Jainism are Svetambara and Digambara, each with its own unique practices and traditions.
• Paryushana Festival:
• Paryushana is considered a time for spiritual rejuvenation, involving rigorous fasting and self-discipline.
• During Paryushana, Jains engage in various religious activities such as reading scriptures, attending sermons, and participating in communal prayers.
• The festival concludes with the celebration of Kshamavani, or "Forgiveness Day," where Jains seek forgiveness from others for any harm or hurt caused intentionally or unintentionally.
• It is a time for Jains to reflect on their actions, repent for their sins, and make a renewed commitment to living a virtuous life.

Given:

E×pression: $\left[ \left( \frac{1}{6} \div \frac{1}{9} × \frac{2}{3} \right) × 18 + 5 \right]$

Formula used:

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} × \frac{d}{c}$

For simplifying the entire e×pression, follow BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction).

Calculations:

Step 1: Simplify $\frac{1}{6} \div \frac{1}{9}$:

$\frac{1}{6} \div \frac{1}{9} = \frac{1}{6} × \frac{9}{1} = \frac{9}{6} = \frac{3}{2}$

Step 2: Multiply the result by $\frac{2}{3}$:

$\frac{3}{2} × \frac{2}{3} = \frac{6}{6} = 1$

Step 3: Multiply by 18:

1 × 18 = 18

Step 4: Add 5:

18 + 5 = 23

The value of the e×pression is 23.

Shortcut Trick

The average of the squares of the first n natural numbers can be calculated directly using the formula: (n + 1)(2n + 1) ÷ 6

Here, the total number of terms, n = 47.

Average = (47 + 1)(2 × 47 + 1) ÷ 6

Average = (48 × 95) ÷ 6 = 8 × 95 = 760

∴ The correct answer is 760.

Alternate Method

Given:

Total natural numbers, n = 47

Formula Used:

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

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

Total Sum of Squares n(n+1)(2n+1) ÷ 6

Total Numbers n = 47

Average = (n+1)(2n+1) ÷ 6 Avg = (48 × 95) ÷ 6 = 760

Calculations:

⇒ Average = (47 + 1)(2 × 47 + 1) ÷ 6

⇒ Average = (48 × 95) ÷ 6

⇒ Average = 8 × 95

⇒ Average = 760

∴ The correct answer is 760.

Additional Information

Sum of First 'n' Natural Numbers

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

Sum of Cubes of First 'n' Natural Numbers

The sum of the cubes of the first n natural numbers is [n(n + 1) ÷ 2]². The average is 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 two persons start together from a point, and one reaches the destination and returns to meet the other, the total distance covered by both is 2 × Distance between the points.

Total distance covered = 2 × 72 = 144 km.

Ratio of distances covered = Ratio of their speeds = 8 : 10 = 4 : 5.

Distance covered by Akshara (PR) = (4 ÷ 9) × 144 = 64 km.

∴ The correct answer is 64 km.

Alternate Method

Given:

Distance between P and Q = 72 km

Speed of Akshara = 8 km/h

Speed of Siddharth = 10 km/h

Formula Used:

Distance = Speed × Time

Total distance covered by both = 2 × Distance between two points

Calculations:

⇒ Let the time taken for them to meet at point R be 't' hours.

⇒ Distance covered by Akshara in t hours = PR = 8t km

⇒ Distance covered by Siddharth in t hours = PQ + QR = 10t km

⇒ The sum of the distances covered by both = PR + PQ + QR

⇒ Since PR + QR = PQ, the total distance covered = 2 × PQ = 2 × 72 = 144 km.

⇒ 8t + 10t = 144

⇒ 18t = 144

⇒ t = 8 hours

⇒ Distance from P to R = 8t = 8 × 8 = 64 km

∴ The correct answer is 64 km.

Additional Information

Meeting Point Concept

When two individuals start simultaneously from point A towards B, and the faster one returns immediately from B to meet the slower one at C, the total distance covered by both is exactly twice the distance between A and B (2D).

Direct Distance Ratio

Since the time of travel until they meet is constant for both, the distance covered is directly proportional to their respective speeds. Thus, Distance Ratio = S₁ : S₂.

Shortcut Trick

Initial Ratio (Gold : Non-Gold) = 1 : 4 = 3 : 12

New Ratio (Gold : Non-Gold) = 2 : 3 = 8 : 12 (Non-Gold parts made equal)

Increase in Gold parts = 8 − 3 = 5 parts

Given 5 parts = 20 coins ⇒ 1 part = 4 coins

Total new parts = 8 + 12 = 20 parts ⇒ New total coins = 20 × 4 = 80 coins

∴ The correct answer is 80 coins.

Alternate Method

Given:

Initial ratio of gold to non-gold coins = 1 : 4

Number of gold coins added = 20

New ratio of gold to non-gold coins = 2 : 3

Formula / Logic Used:

Since the number of non-gold coins remains unchanged, we can equate the number of non-gold coins in both cases to simplify the calculation.

Calculations:

⇒ Let the initial number of gold coins be x and non-gold coins be 4x.

⇒ After adding 20 gold coins, new number of gold coins = x + 20

⇒ Number of non-gold coins remains 4x.

⇒ New ratio = (x + 20) ÷ 4x = 2 ÷ 3

⇒ Cross-multiplying: 3(x + 20) = 2 × 4x

⇒ 3x + 60 = 8x

⇒ 8x − 3x = 60

⇒ 5x = 60

⇒ x = 12

⇒ New total number of coins = (x + 20) + 4x = 5x + 20

⇒ Total coins = 5 × 12 + 20 = 60 + 20 = 80

∴ The correct answer is 80 coins.

Additional Information

Making Parts Equal

When a mixture or collection changes due to the addition of only one component, making the ratio parts of the unchanged component equal across both ratios provides a direct unit-based solution without requiring basic algebra equations.

Cross-Multiplication Method

If an initial ratio a : b transforms to c : d by adding x and y respectively, it can be directly solved using the algebraic relationship: (a + x) ÷ (b + y) = c ÷ d.

Compound Ratio

The compound ratio of two or more given ratios is the product of their corresponding antecedents (first terms) and consequents (second terms). For example, the compound ratio of a : b and c : d is ac : bd.

Shortcut Trick

For a number to be divisible by 6, it must be divisible by both 2 and 3.

The unit digit is 6, so it is already divisible by 2.

For divisibility by 3, the sum of digits must be a multiple of 3.

Sum of digits (ignoring multiples of 3) → 2 + 4 + 9 + k + 8 + 7 + 6 = 36 + k.

Since 36 is divisible by 3, the least possible single-digit value for k is 0.

∴ The correct answer is 0.

Alternate Method

Given:

A 7-digit number: 249k876

Formula Used:

Divisibility by 6: A number must be completely divisible by both 2 and 3.

Divisibility by 3: The sum of all the digits of the number must be a multiple of 3.

Calculations:

⇒ The unit digit of 249k876 is 6, which is an even number. Thus, the given number is always divisible by 2 irrespective of the value of k.

⇒ Now, check for the divisibility by 3.

⇒ Sum of the digits = 2 + 4 + 9 + k + 8 + 7 + 6

⇒ Sum of the digits = 36 + k

⇒ We need the expression (36 + k) to be a multiple of 3.

⇒ Since 36 is exactly divisible by 3, the possible single-digit values of k are 0, 3, 6, and 9.

⇒ The question asks for the least value of k.

⇒ The least possible value among the options is 0.

∴ The correct answer is 0.

Additional Information

Divisibility Rule of 3

A number is divisible by 3 if the sum of all its digits is completely divisible by 3.

Divisibility Rule of 4

A number is divisible by 4 if the number formed by its last two digits is divisible by 4.

Divisibility Rule of 8

A number is divisible by 8 if the number formed by its last three digits is divisible by 8.

Divisibility Rule of 9

A number is divisible by 9 if the sum of all its digits is completely divisible by 9.

Divisibility Rule of 11

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

Shortcut Trick

Since the simple interest is equal in both cases, the products of their Principal, Rate, and Time must be equal.

P₁ × R₁ × T₁ = P₂ × R₂ × T₂

7200 × 16 × 3 = 9600 × x × 4

345600 = 38400 × x ⇒ x = 9

∴ The correct answer is 9%.

Alternate Method

Given:

Principal 1 (P₁) = ₹7,200, Rate 1 (R₁) = 16%, Time 1 (T₁) = 3 years

Principal 2 (P₂) = ₹9,600, Rate 2 (R₂) = x%, Time 2 (T₂) = 4 years

Formula Used:

Simple Interest (SI) = (P × R × T) ÷ 100

Calculations:

⇒ SI₁ = (7200 × 16 × 3) ÷ 100

⇒ SI₂ = (9600 × x × 4) ÷ 100

According to the question, SI₁ = SI₂

⇒ (7200 × 16 × 3) ÷ 100 = (9600 × x × 4) ÷ 100

⇒ 72 × 48 = 96 × 4x

⇒ 3456 = 384x

⇒ x = 3456 ÷ 384

⇒ x = 9

∴ The correct answer is 9%.

Additional Information

Simple Interest (SI)

It is the interest calculated strictly on the initial principal amount throughout the entire period. The formula used is SI = (P × R × T) ÷ 100.

Constant Interest Concept

When the simple interest obtained from two different investments is completely equal, their respective (P × R × T) products must balance out equally.

Amount

The amount refers to the total final sum of the principal and the simple interest earned over the given duration. Formula: A = P + SI = P(1 + (R × T) ÷ 100).

Shortcut Trick

Cost Price (CP) per kg = Selling Price ÷ (1 + Profit%) = 44.8 ÷ 1.12 = ₹40.

To gain a 5% overall profit, the target revenue per kg of initial stock must be 40 × 1.05 = ₹42.

Since 1/5 of the sugar is damaged, only 4/5 of the quantity remains to be sold.

The entire target revenue must be generated by the remaining sugar.

New Selling Price per kg = Target Revenue ÷ Remaining Fraction = 42 ÷ (4/5) = 42 × 5 ÷ 4 = ₹52.5.

∴ The correct answer is ₹52.5.

Alternate Method

Given:

Selling Price (SP) of sugar = ₹44.8 per kg

Initial profit = 12%

Damaged portion = 1/5 of the total sugar

Target overall profit = 5%

Formula Used:

Cost Price = Selling Price ÷ (1 + Profit/100)

Total Selling Price = Total Cost Price × (1 + Desired Profit/100)

Calculations:

⇒ Let the total quantity of sugar be 5 kg.

⇒ Cost Price (CP) per kg = 44.8 ÷ 1.12 = ₹40

⇒ Total CP for 5 kg = 5 × 40 = ₹200

⇒ Since 5% overall profit is required, target Total SP = 200 × 1.05 = ₹210

⇒ Quantity of damaged sugar = 1/5 of 5 kg = 1 kg (yields ₹0 revenue)

⇒ Remaining quantity to be sold = 5 − 1 = 4 kg

⇒ New Selling Price per kg = Target Total SP ÷ Remaining Quantity

⇒ New SP per kg = 210 ÷ 4 = ₹52.5

∴ The correct answer is ₹52.5.

Additional Information

Cost Price and Selling Price Relation

When profit is made, SP = CP × (100 + Profit%) ÷ 100. For loss, SP = CP × (100 − Loss%) ÷ 100.

Impact of Damaged Goods on Profit

Goods that are completely damaged yield zero revenue. This effectively acts as a 100% loss on that specific portion, meaning the remaining goods must be sold at a higher markup to achieve the target overall profit.

Weighted Average in Profit & Loss

If quantity is split into parts (e.g., Q₁ and Q₂) sold at profit % P₁ and P₂, overall profit % = (Q₁P₁ + Q₂P₂) ÷ (Q₁ + Q₂). For completely damaged goods, P = −100%.

Shortcut Trick

For a system of linear equations to have no solution, the ratio of x-coefficients must equal the ratio of y-coefficients.

a₁ / a₂ = b₁ / b₂

k / 7 = −15 / −21

k = 7 × (5 / 7) = 5

∴ The correct answer is 5.

Alternate Method

Given:

Equation 1: kx − 15y + 7 = 0

Equation 2: 7x − 21y − 19 = 0

Formula Used:

For two linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 to have NO solution, the condition is:

a₁ / a₂ = b₁ / b₂ ≠ c₁ / c₂

Calculations:

⇒ Comparing the given equations with the standard form, we get:

⇒ a₁ = k, b₁ = −15, c₁ = 7

⇒ a₂ = 7, b₂ = −21, c₂ = −19

⇒ Applying the condition for no solution:

⇒ k / 7 = (−15) / (−21)

⇒ k / 7 = 15 / 21

⇒ k / 7 = 5 / 7

⇒ k = 5

⇒ Checking the second part of the condition:

⇒ b₁ / b₂ = 5 / 7

⇒ c₁ / c₂ = 7 / (−19)

⇒ 5 / 7 ≠ −7 / 19 (Condition satisfied)

∴ The correct answer is 5.

Additional Information

Condition for Unique Solution (Intersecting Lines)

The system of equations has exactly one solution if a₁ / a₂ ≠ b₁ / b₂.

Condition for Infinitely Many Solutions (Coincident Lines)

The system has infinitely many solutions if all ratios are equal: a₁ / a₂ = b₁ / b₂ = c₁ / c₂.

Condition for No Solution (Parallel Lines)

The system has no solution if the lines are strictly parallel: a₁ / a₂ = b₁ / b₂ ≠ c₁ / c₂.

Given:

The cost price per unit and profit percentage for each item:

Formula used:

Profit = (Profit Percentage / 100) × Cost Price

Total Profit = Sum of profits for all items

Calculations:

Step 1: Calculate profit for each item:

Profit on Mobile = (10 / 100) × 20,000 = 2,000

Profit on Laptop = (20 / 100) × 72,000 = 14,400

Profit on Mobile Covers = (12 / 100) × 300 = 36

Step 2: Add up the profits:

Total Profit = 2,000 + 14,400 + 36 = 16,436

The total profit earned by the shopkeeper is ₹16,436.

Given:

Total agricultural production = 8600 tonnes

Percentage of rice = 25%

Formula used:

Production of rice = (Percentage of rice / 100) × Total production

Calculations:

Production of rice = (25 / 100) × 8600

= 0.25 × 8600

= 2150 tonnes

The annual production of rice in the region is 2150 tonnes.

Given:

The number of cars manufactured over three years is as follows:

Formula used:

Average = (Total cars manufactured in 2024) / (Number of car types)

Calculations:

Total cars manufactured in 2024 = Cars of type X + Cars of type Y + Cars of type Z

= 3551 + 3020 + 2600

= 9171

Number of car types = 3

Average = 9171 / 3 = 3057

The average number of cars manufactured by the company in the year 2024 is 3057.

Shortcut Trick

Total area of the circle = π × r² = π × 8² = 64π

Area of the sector is given as 16π.

Ratio of sector area to total area = 16π ÷ 64π = 1/4

Central angle = (1/4) × 360° = 90°

∴ The correct answer is 90°.

Alternate Method

Given:

Area of the sector = 16π

Radius (r) = 8

Formula Used:

Area of a sector = (θ ÷ 360°) × πr²

Calculations:

⇒ 16π = (θ ÷ 360°) × π × (8)²

⇒ 16π = (θ ÷ 360°) × 64π

⇒ 16 ÷ 64 = θ ÷ 360°

⇒ 1/4 = θ ÷ 360°

⇒ θ = 360° ÷ 4 = 90°

∴ The correct answer is 90°.

Additional Information

Length of an Arc

The length of an arc of a sector with central angle θ is given by l = (θ ÷ 360°) × 2πr.

Perimeter of a Sector

The perimeter of a sector includes the arc length and the two radii: P = l + 2r.

Area of a Segment

The area of the segment enclosed by a chord and its corresponding arc is determined by subtracting the area of the triangle formed by the radii and the chord from the area of the sector.

Shortcut Trick

Area of ΔABC = (10 ÷ 4) × √(4 × 20² − 10²) = 2.5 × √1500 = 25√15 cm²

⇒ Area of ΔABC = 25 × 3.87 = 96.75 cm²

Base of right ΔPQR = √(13² − 12²) = 5 cm

⇒ Area of ΔPQR = &frac{1}{2} × 12 × 5 = 30 cm²

⇒ Difference in area = 96.75 − 30 = 66.75 cm²

∴ The correct answer is 66.75 cm².

Alternate Method

Given:

Sides of ΔABC are 10 cm, 20 cm, and 20 cm.

In ΔPQR, hypotenuse = 13 cm, one perpendicular side = 12 cm.

Formula Used:

Heron's Formula for Area = √(s(s − a)(s − b)(s − c))

Pythagorean Theorem: Hypotenuse² = Perpendicular² + Base²

Area of right triangle = &frac{1}{2} × Base × Height

Calculations:

⇒ Semi-perimeter of ΔABC, s = (10 + 20 + 20) ÷ 2 = 50 ÷ 2 = 25 cm

⇒ Area of ΔABC = √(25 × (25 − 10) × (25 − 20) × (25 − 20))

⇒ Area of ΔABC = √(25 × 15 × 5 × 5) = √(9375) = 25√15 cm²

⇒ Given √15 ≈ 3.87, Area = 25 × 3.87 = 96.75 cm²

⇒ In right ΔPQR, Base = √(13² − 12²) = √(169 − 144) = √25 = 5 cm

⇒ Area of ΔPQR = &frac{1}{2} × 12 × 5 = 30 cm²

⇒ Difference in area = 96.75 − 30 = 66.75 cm²

∴ The correct answer is 66.75 cm².

Additional Information

Heron's Formula

Used to find the area of a general triangle when all three sides are known: Area = √(s(s−a)(s−b)(s−c)), where s = (a+b+c) ÷ 2.

Pythagorean Theorem

For any right-angled triangle, the square of the hypotenuse equals the sum of squares of perpendicular and base: H² = P² + B².

Area of an Isosceles Triangle

Can also be calculated using the direct formula: Area = (b ÷ 4) × √(4a² − b²), where 'a' is the length of equal sides and 'b' is the base.

Shortcut Trick

The total perimeter (circumference) of a circle corresponds to a complete central angle of 360°.

Since the arc length is exactly half of the total perimeter, the corresponding angle will also be exactly half.

⇒ Required Angle = 360° ÷ 2

⇒ Required Angle = 180°

∴ The correct answer is 180°.

Alternate Method

Given:

Arc length = 1/2 × Perimeter of the circle

Formula Used:

Perimeter of a circle = 2πr

Arc length = (θ ÷ 360°) × 2πr

Calculation:

⇒ Arc length = 1/2 × 2πr

⇒ Arc length = πr

Equating this to the arc length formula:

⇒ (θ ÷ 360°) × 2πr = πr

⇒ θ ÷ 360° = πr ÷ 2πr

⇒ θ ÷ 360° = 1/2

⇒ θ = 1/2 × 360°

⇒ θ = 180°

∴ The correct answer is 180°.

Additional Information

Area of a Sector

The area of a sector with central angle θ is directly proportional to its angle and is given by (θ ÷ 360°) × πr².

Perimeter of a Semicircular Region

While the curved arc length of a semicircle is πr, the total perimeter of a closed semicircular region includes the straight diameter as well, making it πr + 2r.

Quadrant of a Circle

A quadrant represents one-fourth of a circle with a central angle of 90°. Its curved arc length is πr ÷ 2.

Shortcut Trick

When a value is a product of two variables, the net percentage change can be found using the successive percentage formula: x + y + (x × y)/100

Here, the number of employees increased by 20% (x = 20) and wages decreased by 18% (y = −18).

Net Change = 20 − 18 + [(20 × −18) / 100]

Net Change = 2 − 3.6 = −1.6%

The negative sign clearly indicates a total decrease in wages.

∴ The correct answer is 1.6%.

Alternate Method

Given:

Increase in number of employees = 20%

Decrease in wages per head = 18%

Formula Used:

Total Wages = Number of Employees × Wages per head

Percentage Change = [(Final Value − Initial Value) / Initial Value] × 100

Calculations:

Let the initial number of employees be 100 and the wage per head be 100.

⇒ Initial Total Wages = 100 × 100 = 10000

⇒ New number of employees = 100 + 20% of 100 = 120

⇒ New wage per head = 100 − 18% of 100 = 82

⇒ Final Total Wages = 120 × 82 = 9840

⇒ Decrease in Total Wages = 10000 − 9840 = 160

⇒ Percentage Decrease = (160 / 10000) × 100 = 1.6%

∴ The correct answer is 1.6%.

Additional Information

Successive Percentage Formula

When a quantity undergoes two consecutive percentage changes x and y, the net change is given by x + y + (xy/100). Use positive signs for increments and negative signs for decrements.

Product Rule in Percentages

If a final quantity is determined by the product of two variables (Z = A × B), the percentage change in Z equals the successive percentage changes applied to A and B.

Base Value Assumption

Assuming initial values as 100 makes calculating percentage increases or decreases highly straightforward, allowing avoidance of cumbersome variables and fractions.

Shortcut Trick

We can use the successive percentage change formula: Net % Change = a + b + (a × b) / 100

Here, the markup percentage (a) = +20% and the discount percentage (b) = −30%.

Net % Change = 20 − 30 + (20 × −30) / 100

Net % Change = −10 − 6 = −16%

The negative sign indicates an overall loss of 16%.

∴ The correct answer is 16%.

Alternate Method

Given:

⇒ Markup Percentage = 20%

⇒ Discount Percentage = 30%

Calculations:

⇒ Let the Cost Price (CP) of the article be 100.

⇒ Marked Price (MP) = CP + 20% of CP = 100 + 20 = 120

⇒ Discount is always calculated on the Marked Price.

⇒ Selling Price (SP) = MP − 30% of MP

⇒ SP = 120 − (30 / 100 × 120) = 120 − 36 = 84

⇒ Since Selling Price is less than Cost Price, there is a loss.

⇒ Loss = CP − SP = 100 − 84 = 16

⇒ Loss Percentage = (Loss / CP) × 100 = (16 / 100) × 100 = 16%

∴ The correct answer is 16%.

Additional Information

Successive Discounts

If two successive discounts of a% and b% are given on an article, the single equivalent discount is calculated as: D = a + b − (a × b) / 100.

CP and MP Relationship

The direct ratio between Cost Price and Marked Price is given by the formula: CP / MP = (100 − Discount%) / (100 + Profit%). If there is a loss, use (100 − Loss%).

Dishonest Dealer Rule

When a shopkeeper claims to sell goods at Cost Price but uses a false weight, the actual profit percent is: Profit% = (Error / True Weight − Error) × 100.

Shortcut Trick

Total capacity (LCM of 15, 25, 40) = 600 units.

Efficiencies: A = 40, B = 24, C = −15 units/hr.

Work done in first 5 hours = 5 × (40 + 24 − 15) = 245 units.

Remaining work = 600 − 245 = 355 units.

Time to fill remaining part by B and C = 355 ÷ (24 − 15) = 355 ÷ 9 = 39 ⁴/₉ hours.

∴ The correct answer is 39 ⁴/₉ hours.

Alternate Method

Given:

⇒ Time taken by pipe A to fill the tank = 15 hours

⇒ Time taken by pipe B to fill the tank = 25 hours

⇒ Time taken by pipe C to empty the tank = 40 hours

⇒ All three pipes are opened together for 5 hours before A is closed.

Calculations:

⇒ Combined efficiency of A, B, and C = 40 + 24 − 15 = 49 units/hr

⇒ Total work done by all three pipes in 5 hours = 49 × 5 = 245 units

⇒ Remaining work to be completed = 600 − 245 = 355 units

⇒ After 5 hours, pipe A is closed. Thus, the remaining work is done by pipes B and C.

⇒ Combined efficiency of pipes B and C = 24 − 15 = 9 units/hr

⇒ Time required by B and C to fill the remaining tank = 355 ÷ 9 = 39 ⁴/₉ hours

∴ The correct answer is 39 ⁴/₉ hours.

Additional Information

Pipes and Cisterns Concept

If a pipe can fill a tank in x hours, its efficiency is 1/x of the tank per hour. If a pipe empties the tank in y hours, its efficiency is −1/y per hour.

Total Work using LCM

Taking the Least Common Multiple (LCM) of the given times as the total capacity of the tank avoids complex fraction calculations and directly gives individual efficiencies in integer units.

Negative Efficiency

Any pipe or leak that empties a cistern performs negative work. Its efficiency is assigned a minus sign and must be algebraically added to the efficiencies of the filling pipes to find the net work rate.

Shortcut Trick

Given, tanA = 1 ⇒ ∠A = 45°.

We know the double angle formula: 2 sinA cosA = sin2A

⇒ 4 sinA cosA = 2 × (2 sinA cosA) = 2 sin2A

⇒ 2 sin(2 × 45°) = 2 sin90°

⇒ 2 × 1 = 2

∴ The correct answer is 2.

Alternate Method

Given:

In a right angle ΔABC, ∠B = 90°.

tanA = 1

Formula Used:

tanA = Perpendicular ÷ Base

sinA = Perpendicular ÷ Hypotenuse

cosA = Base ÷ Hypotenuse

Hypotenuse² = Perpendicular² + Base²

Calculations:

Let Perpendicular (BC) = x and Base (AB) = x, since tanA = BC ÷ AB = x ÷ x = 1.

⇒ Hypotenuse (AC) = √(x² + x²) = √(2x²) = x√2

⇒ sinA = BC ÷ AC = x ÷ x√2 = 1 ÷ √2

⇒ cosA = AB ÷ AC = x ÷ x√2 = 1 ÷ √2

⇒ 4 sinA cosA = 4 × (1 ÷ √2) × (1 ÷ √2)

⇒ 4 × (1 ÷ 2) = 2

∴ The correct answer is 2.

Additional Information

Trigonometric Ratios

sinθ = Perpendicular ÷ Hypotenuse, cosθ = Base ÷ Hypotenuse, tanθ = Perpendicular ÷ Base.

Double Angle Formulas

sin2θ = 2 sinθ cosθ and cos2θ = cos²θ − sin²θ.

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.

Trigonometric Values at 45°

sin45° = 1 ÷ √2, cos45° = 1 ÷ √2, and tan45° = 1.

Shortcut Trick If x2 + 1/x2 = k, then x − 1/x = √(k − 2). Here, k = 83, so x − 1/x = √(83 − 2) = √81 = 9. If x − 1/x = a, then x3 − 1/x3 = a3 + 3a. So, x3 − 1/x3 = 93 + 3 × 9 = 729 + 27 = 756. ∴ The correct answer is 756.

Alternate Method Given: x2 + 1/x2 = 83, where x > 1 Formula Used: (a − b)2 = a2 + b2 − 2ab a3 − b3 = (a − b)(a2 + b2 + ab) Calculations: ⇒ (x − 1/x)2 = x2 + 1/x2 − 2(x)(1/x) ⇒ (x − 1/x)2 = 83 − 2 ⇒ (x − 1/x)2 = 81 ⇒ x − 1/x = ±9 ⇒ Since x > 1, x − 1/x must be positive, so x − 1/x = 9 ⇒ x3 − 1/x3 = (x − 1/x)(x2 + 1/x2 + 1) ⇒ x3 − 1/x3 = 9 × (83 + 1) ⇒ x3 − 1/x3 = 9 × 84 ⇒ x3 − 1/x3 = 756 ∴ The correct answer is 756.

Additional Information Reciprocal Identities If x + 1/x = p, then x2 + 1/x2 = p2 − 2 and x3 + 1/x3 = p3 − 3p. Negative Reciprocal Identities If x − 1/x = p, then x2 + 1/x2 = p2 + 2 and x3 − 1/x3 = p3 + 3p. Cube Factorization The difference of cubes formula is a3 − b3 = (a − b)(a2 + b2 + ab). Substituting 1/x for b gives the identity x3 − 1/x3 = (x − 1/x)(x2 + 1/x2 + 1).

Shortcut Trick In a right-angled triangle, if secθ = 29/20, then Hypotenuse = 29 and Base = 20. Using the standard Pythagorean triplet (20, 21, 29), we directly get Perpendicular = 21. We need the value of: 3cosecθ + 3cotθ = 3(Hypotenuse/Perpendicular) + 3(Base/Perpendicular) ⇒ 3 × (29/21) + 3 × (20/21) = 3 × (49/21) = 147/21 = 7 ∴ The correct answer is 7.

Alternate Method Given: secθ = 29/20, where 0 < θ < 90°

Formula Used: By Pythagoras Theorem: Perpendicular2 + Base2 = Hypotenuse2 cosecθ = Hypotenuse / Perpendicular cotθ = Base / Perpendicular

Calculations: ⇒ In a right-angled triangle, secθ = Hypotenuse / Base = 29 / 20 ⇒ Hypotenuse = 29, Base = 20 ⇒ Perpendicular = √(Hypotenuse2 − Base2) ⇒ Perpendicular = √(292 − 202) ⇒ Perpendicular = √(841 − 400) = √441 = 21 ⇒ cosecθ = 29/21 ⇒ cotθ = 20/21 Now, substitute the values into the expression (3cosecθ + 3cotθ): ⇒ 3 × (29/21) + 3 × (20/21) ⇒ (87/21) + (60/21) ⇒ (87 + 60) / 21 ⇒ 147 / 21 = 7 ∴ The correct answer is 7.

Additional Information Pythagorean Triplets Common sets of right-angle triangle sides that appear frequently in exams are (3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), and (20, 21, 29). Trigonometric Ratios sinθ = Perpendicular/Hypotenuse | cosθ = Base/Hypotenuse | tanθ = Perpendicular/Base. The reciprocals are cosecθ, secθ, and cotθ respectively. Fundamental Trigonometric Identities sin2θ + cos2θ = 1 | sec2θ − tan2θ = 1 | cosec2θ − cot2θ = 1.

Shortcut Trick

The angle subtended by the two tangents at an external point and the angle subtended by their respective radii at the center are supplementary.

⇒ ∠POQ + ∠PAQ = 180°

⇒ ∠POQ + 40° = 180°

⇒ ∠POQ = 180° − 40° = 140°

∴ The correct answer is 140°.

Alternate Method

A P Q O 40° ?

∠PAQ + ∠POQ = 180°

Given:

∠PAQ = 40°

AP and AQ are tangents to the circle with center O.

Formula Used:

The radius drawn to the point of tangency is perpendicular to the tangent at that point.

The sum of the interior angles of a quadrilateral is 360°.

Calculations:

⇒ OP ⊥ AP and OQ ⊥ AQ

⇒ ∠OPA = 90° and ∠OQA = 90°

⇒ In quadrilateral APOQ,

⇒ ∠PAQ + ∠OPA + ∠OQA + ∠POQ = 360°

⇒ 40° + 90° + 90° + ∠POQ = 360°

⇒ 220° + ∠POQ = 360°

⇒ ∠POQ = 360° − 220°

⇒ ∠POQ = 140°

∴ The correct answer is 140°.

Additional Information

Equal Tangents Property

The lengths of tangents drawn from an external point to a circle are always equal. Hence, AP = AQ.

Angle Bisector Property

The line segment joining the external point to the center of the circle bisects the angle between the tangents. Hence, ∠OAP = ∠OAQ.

Cyclic Quadrilateral Formation

The quadrilateral formed by the two tangents and the two corresponding radii (APOQ) is always a cyclic quadrilateral because the sum of its opposite angles (∠OPA and ∠OQA) is 180°.

Shortcut Trick

Simplify given ratios to lowest terms: A : B = 6 : 8 = 3 : 4

Apply the inverted 'N' multiplication method for A : B = 3 : 4 and B : C = 7 : 10

A = 3 × 7 = 21

B = 4 × 7 = 28

C = 4 × 10 = 40

∴ The correct answer is 21:28:40.

Alternate Method

Given:

A : B = 6 : 8

B : C = 7 : 10

Calculations:

⇒ First simplify A : B to its lowest terms.

⇒ A : B = 3 : 4

⇒ To combine both ratios, the common term 'B' must be made equal.

⇒ The LCM of 4 and 7 is 28.

⇒ Multiply the first ratio by 7 ⇒ A : B = (3 × 7) : (4 × 7) = 21 : 28

⇒ Multiply the second ratio by 4 ⇒ B : C = (7 × 4) : (10 × 4) = 28 : 40

⇒ Now, B is equal in both equations, allowing direct combination.

⇒ A : B : C = 21 : 28 : 40

∴ The correct answer is 21:28:40.

Additional Information

Compound Ratio

The compound ratio of two different ratios a : b and c : d is calculated by multiplying the antecedents together and consequents together as ac : bd.

Duplicate & Triplicate Ratio

The duplicate ratio of a : b is given by a² : b², and the triplicate ratio is given by a³ : b³.

Invertendo

If a : b = c : d, then according to the invertendo property, the inverse ratios are also equal, meaning b : a = d : c.

Shortcut Trick

Given, cos 24° = m / n

Using complementary angles, cos 66° = cos(90° − 24°) = sin 24°

Expression: cosec 24° − cos 66° = 1 / sin 24° − sin 24° = (1 − sin² 24°) / sin 24° = cos² 24° / sin 24°

Since cos 24° = m / n, we get sin 24° = √(1 − m²/n²) = √(n² − m²) / n

⇒ Value = (m / n)² × [n / √(n² − m²)] = m² / [n√(n² − m²)]

∴ The correct answer is m² / [n√(n² − m²)].

Alternate Method

Given:

⇒ cos 24° = Base / Hypotenuse = m / n

Formula Used:

⇒ Perpendicular = √(Hypotenuse² − Base²)

⇒ cosec θ = Hypotenuse / Perpendicular

Calculations:

⇒ Let Base for angle 24° be m and Hypotenuse be n.

⇒ Perpendicular = √(n² − m²)

⇒ cosec 24° = Hypotenuse / Perpendicular = n / √(n² − m²)

⇒ The other acute angle in the right-angled triangle is 90° − 24° = 66°.

⇒ For angle 66°, the side adjacent to it is the Perpendicular of 24°.

⇒ Base for 66° = √(n² − m²) and Hypotenuse = n.

⇒ cos 66° = Base / Hypotenuse = √(n² − m²) / n

⇒ Required Expression = cosec 24° − cos 66°

⇒ Expression = [n / √(n² − m²)] − [√(n² − m²) / n]

⇒ Taking LCM as n√(n² − m²):

⇒ Expression = [n² − (√(n² − m²))²] / [n√(n² − m²)]

⇒ Expression = [n² − (n² − m²)] / [n√(n² − m²)]

⇒ Expression = m² / [n√(n² − m²)]

∴ The correct answer is m² / [n√(n² − m²)].

Additional Information

Complementary Angles

Two angles are complementary if their sum is 90°. Key relations: sin(90° − θ) = cos θ and sec(90° − θ) = cosec θ.

Pythagorean Trigonometric Identities

The fundamental identities are: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ.

Right-Angled Triangle Ratios

For any acute angle θ, sin θ = Perpendicular / Hypotenuse and cos θ = Base / Hypotenuse.

Shortcut Trick

Radius (r) = Circumference ÷ 2π = 44 × (7/44) = 7 cm.

Volume in cm³ = π × r² × h = (22/7) × 7² × 25 = 3850 cm³.

Convert to liters (1 L = 1000 cm³): 3850 ÷ 1000 = 3.85 liters.

∴ The correct answer is 3.85 liters.

Alternate Method

Given:

Circumference of the base = 44 cm

Height of the cylinder (h) = 25 cm

Formula Used:

Circumference of base = 2πr

Volume of cylinder = πr²h

1 liter = 1000 cm³

Calculations:

⇒ 2 × (22/7) × r = 44

⇒ (44/7) × r = 44

⇒ r = 44 × (7/44)

⇒ r = 7 cm

⇒ Volume of cylinder = (22/7) × (7)² × 25

⇒ Volume = (22/7) × 49 × 25

⇒ Volume = 22 × 7 × 25

⇒ Volume = 154 × 25

⇒ Volume = 3850 cm³

⇒ Capacity in liters = 3850 ÷ 1000 = 3.85 liters

∴ The correct answer is 3.85 liters.

Additional Information

Total Surface Area of Cylinder

TSA = 2πr(r + h), where r is the radius of the base and h is the height.

Curved Surface Area of Cylinder

CSA = 2πrh. This represents the outer wall surface area, excluding the top and bottom circular faces.

Metric Volume Conversion

1 m³ = 1000 liters. Conversely, 1 liter = 1000 cm³ or 1000 ml.

Given:

Sales of products P, Q, and R by companies A and C are as follows:

Formula used:

Percentage = (Total production by company C / Total production by company A) × 100

Calculations:

Step 1: Calculate total production by company A:

Total production by company A = Sales of R + Sales of Q + Sales of P

Total production by company A = 9 + 6 + 8 = 23

Step 2: Calculate total production by company C:

Total production by company C = Sales of R + Sales of Q + Sales of P

Total production by company C = 8 + 9 + 6 = 23

Step 3: Calculate percentage:

Percentage = (Total production by company C / Total production by company A) × 100

Percentage = (23 / 23) × 100 = 100%

The total production by company C is 100% of the total production by company A.

The correct answer is Option 1.

 

Key Points

• The primary issue in the sentence is the choice of the preposition "of" after the term "necessities."
• The preposition "to" is more appropriate to indicate the direction of providing something to someone.
• The correct phrase should be "the necessities to her family", ensuring grammatical correctness and clarity.
• Thus, the error is located in the underlined segment of the sentence.

Therefore, the correct answer is Option 1.

 

Correct Sentence: "It is very hard for her to provide the necessities to her family."

 

Additional Information

• Option 2: "the necessities by her family" – This implies the family provides the necessities, which is incorrect in this context.
• Option 3: "the necessity of her family" – This changes the meaning to a single necessity, which is not the intended meaning.
• Option 4: "the necessities from her families" – The plural form "families" and the preposition "from" are incorrect in this context.

The correct answer is Option 1.

 

Key Points

• The primary issue in the sentence is the incorrect use of the article "an" before "intellectual."
• The correct article to use before "intellectual" is "the" because it refers to a specific type of solidarity.
• The phrase should be "strengthen the intellectual" to ensure grammatical correctness and clarity.
• Thus, the error is located in option 1 of the sentence.

Therefore, the correct answer is Option 1.

 

Correct Sentence: "UNESCO works to strengthen the intellectual and moral solidarity of humankind."

The correct answer is 'Option 1'.

Key Points

• The term "Numismatics" refers to the study of coins, which directly matches the meaning of the given phrase.
• Other options like "Choreography", "Informatics", and "Cartography" do not have meaning for the study of coins.

Hence, the correct answer is 'Option 1'.

Correct Sentence*: The study of coins is known as Numismatics.*

Additional Information

• Option 2: Choreography refers to the art of designing dance movements, not the study of coins.
• Option 3: Informatics involves the study of information science, not coin study.
• Option 4: Cartography refers to the study of maps, which is unrelated to the study of coins.

The correct answer is option 1.

Key Points

The misspelled word is 'Eficient'.

• The correct spelling would be 'Efficient'.
• The word 'Efficient' is an adjective which means 'achieving maximum productivity with minimum wasted effort or expense'. (कुशल)
• For Example -
• The new system is highly efficient and saves us a lot of time.
• Therefore, the correct answer would be 'Eficient'.

 

Additional Information

Let's understand the meanings of the other marked words - 

• Contribution - a gift or payment to a common fund or collection (योगदान)
• Miscalculation - an error or mistake in calculation (गणना में त्रुटि)
• Mediation - intervention in a dispute in order to resolve it (मध्यस्थता)

The correct answer is 'Option 1'.

Key Points

• The term "Pessimist" refers to a person who tends to see the worst aspect of things, which perfectly matches the given description.
• Other options like "Philanthropist", "Optimist", and "Altruist" refer to people with positive tendencies, not negative ones.

Hence, the correct answer is 'Option 1'.

Correct Sentence*: A pessimist is a person who is inclined to see the worst aspect of things.*

Additional Information

• Option 2: A philanthropist is someone who seeks to promote the welfare of others, which does not match the description provided.
• Option 3: An optimist has a positive outlook, which is the opposite of what is described in the question.
• Option 4: An altruist is a person who is selflessly concerned for the well-being of others, which does not fit the given definition either.

The correct answer is Option 3.

 

Key Points

• Passive voice structure: The object of the active voice sentence becomes the subject of the passive voice sentence.
• In the active voice, the sentence is: "He will give a lecture on personality development."
• To convert to passive voice: Place the object "a lecture on personality development" at the beginning.
• Change the active verb form "will give" to "will be given" (future simple passive).
• Add the preposition "by" followed by the original subject "him."

Therefore, the correct answer is Option 3.

 

Correct Sentence: "A lecture on personality development will be given by him."

 

Additional Information

• Option 1: "A lecture on personality development can be given by him." – Incorrect modality (can instead of will).
• Option 2: "A lecture will be given on personality development." – Incorrect structure, missing the agent (by him).
• Option 4: "A lecture on personality development will be giving by him." – Incorrect verb form (giving instead of given).

The correct answer is Option 3.

 

Key Points

• The paragraph begins by introducing the unique qualities of the ice cream.
• Sentence C is the first sentence as it sets the introduction:
• "Silky thick double ice cream has a unique, fresh taste and texture."
• Sentence B follows as it describes the specific characteristics of the ice cream:
• "Beautifully firm, nice, silky rather than thin and watery, it stays in shape and stands on top of fruits or pastries without soaking in."
• Sentence A is the third sentence as it concludes with why the ice cream is different:
• "That's why Silky is deliciously different and exceptionally versatile, natural, thick and durable ice cream."
• The correct sequence is CBA.

 

Coherent paragraph after rearrangement:

"Silky thick double ice cream has a unique, fresh taste and texture. Beautifully firm, nice, silky rather than thin and watery, it stays in shape and stands on top of fruits or pastries without soaking in. That's why Silky is deliciously different and exceptionally versatile, natural, thick and durable ice cream."