SSC CGL 2024 Tier-I Official Paper (Held On: 18 Sept, 2024 Shift 2) (Previous Year Paper)

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

Thus, option Figure 2 will look like the paper when unfolded.

Hence, "Option 2" is the correct answer.

The logic followed here is:

7 – 14 – 28 – 38

7 × 2 + 14 = 28

14 + 14 = 28

28 = 28 (LHS = RHS);

28 + 10 = 38

38 = 38 (LHS = RHS)

And, 

4 – 8 – 16 – 26

4 × 2 + 8 = 16

8 + 8 = 16

16 = 16 (LHS = RHS);

16 + 10 = 26

26 = 26 (LHS = RHS)

So, checking each option one by one:

Option 1) 13 – 26 – 52 – 62

13 × 2 + 26 = 52

26 + 26 = 52

52 = 52 (LHS = RHS);

52 + 10 = 62

62 = 62 (LHS = RHS)

Option 2) 8 – 16 – 32 – 40

8 × 2 + 16 = 32

16 + 16 = 32

32 = 32 (LHS = RHS);

32 + 10 = 40

42 ≠ 40 (LHS ≠ RHS)

Option 3) 6 – 12 – 24 – 36

6 × 2 + 12 = 24

12 + 12 = 24

24 = 24 (LHS = RHS);

24 + 10 = 36

34 ≠ 36 (LHS ≠ RHS)

Option 4) 15 – 25 – 50 – 60

15 × 2 + 25 = 50

30 + 25 = 50

55 ≠ 50 (LHS ≠ RHS);

50 + 10 = 60

60 = 60 (LHS = RHS)

Thus, out of all the options '13 – 26 – 52 – 62' follows the same logic as given.

Hence, "Option 1" is the correct answer.

BODMAS Table: 

Given equation: 272 × 16 + 97 − 23 ÷ 4 = ?

Now, if ‘+’ and ‘-‘ and ‘×' and '÷’ are interchanged, then:

⇒ 272 ÷ 16 - 97 + 23 × 4 = ?

⇒ 17 - 97 + 92 = ?

⇒ 109 - 97 = 12

Hence, the correct answer is "Option 4".

The logic followed here is:

Logic: 1st number × 8 = 2nd number.

Now,

For, 28 : 224

⇒ (28 × 8) = 224.

For, 47 : 376

⇒ (47 × 8) = 376.

Similarly, for 31 ∶ ?

⇒ (31 × 8) = 248.

Hence, the correct answer is "Option 2".

The logic followed here is:

Option 1) KJI

Option 2) VUT

Option 3) AAY

Option 4) HGF

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

Hence, "Option 3" is the correct answer.

Given word: 'SLANTED'

After arranged in alphabetical order:

After arranged in alphabetical order,

Fourth alphabet from the left end is: L,

Second alphabet from the right end is: S,

Now, there are 6 letters between L and S.

Hence, the correct answer is "Option 1".

The logic followed here is:

1. The top element shifts clockwise to immediate right section subsequently.

2. The bottom element shifts clockwise to immediate left section subsequently.

3) The bottom element is replaced by new element subsequently.

Thus, the final series is:

Hence, "Option 4" is the correct answer.

According to the given information,

Thus, according to the given codes the code either 'sh' or 'ko' is for the word 'sunshine'.

Hence, the correct answer is "Option 4".

The image embedded in the given figure is as shown below :

∴ Here, given figure is embedded in option figure 4.

Hence, the correct answer is "Option 4".

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

As marked there are total 9 triangles.

Hence, "Option 4" is the correct answer.

The logic followed here is:

1. One new element is added in each box subsequently.

Thus, the final series is:

Hence, "Option 4" is the correct answer.

The logic followed here is:

'TRIED' is coded as '54123'

'TREAD' is coded as '12365'

Thus, code of 'I' is '4'.

Hence, "Option 1" is the correct answer.

The logic followed here is:

Hence, "Option 1" is the correct answer.

BODMAS Table: 

Given equation: 121 × 11 – 28 ÷ 3 + 17 = ?

Now, if ‘+’ and ‘–‘ and ‘×’ and ‘÷’ are interchanged, then:

⇒ 121 ÷ 11 + 28  × 3 - 17 = ?

⇒ 11 + 84 – 17 = ?

⇒ 95 - 17 = 78

Hence, the correct answer is "Option 3".

The logic followed here is:

Logic: Synonyms

Persist : Endure  → Persist is a synonym of Endure.

Similarly,

Stall : Delay  → Stall is a synonym of Delay.

Hence, the correct answer is "Delay".

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

Thus, option Figure 1 will look like the paper when unfolded.

Hence, "Option 1" is the correct answer.

Given series: L M N _ O L L M _ Q O L L _ N Q O _ L M N Q _ L L M

Option 1) QNMLO → L M N Q O L L M N Q O L L M N Q O L L M N Q O L L M → (It follows the pattern of: LMNQOL - LMNQOL - LMNQOL - LMNQOL - LM)

Option 2) QMNLO → L M N Q O L L M M Q O L L N N Q O L L M N Q O L L M → (It doesn't follow any pattern)

Option 3) QLMNO → L M N Q O L L M L Q O L L M N Q O N L M N Q O L L M → (It doesn't follow any pattern)

Option 4) NQLMO → L M N N O L L M Q Q O L L L N Q O M L M N Q O L L M → (It doesn't follow any pattern)

Hence, "Option 1" is the correct answer.

The logic followed here is:​

Hence, "Option 4" is the correct answer.

Decoding symbols:

Given: A × B ÷ C + D

A × B → A is the wife of B.

B ÷ C → B is the father of C.

C + D → C is the mother of D.

Thus, A is D's mother's mother or maternal grandmother.

Hence, the correct answer is "Option 1".

The logic followed here is:

Logic: (1st number)³ - 10 = 2nd number.

Now,

For, 7 : 333

⇒ 7³ - 10 = 343 - 10 = 333.

For, 5 : 115

⇒ 5³ - 10 = 125 - 10 = 115.

Similarly, for other options:

1. (4, 54)

⇒ 4³ - 10 = 64 - 10 = 54.

2. (3, 24)

⇒ 3³ - 10 = 27 - 10 = 17 (Not 24).

3. (8, 492)

⇒ 8³ - 10 = 512 - 10  = 502 (Not 492).

4. (6, 226)

⇒ 6³ - 10 = 216 - 10 = 206 (Not 226).

Hence, the correct answer is "Option 1".

The logic followed here is:

CARBON : ACIYNO

And,

DAMAGE : ADNZEG

Similarly,

MARGIN : ?

Hence, "Option 3" is the correct answer.

The correct mirror image of the given figure is:

Hence, the correct answer is "Option 4".

The logic followed here is: (First number × 3) × First number = Second number

Option 1) 11 – 363

(11 × 3) × 11 = 363

33 × 11 = 363

363 = 363 (LHS = RHS)

Option 2) 13 – 507

(13 × 3) × 13 = 507

39 × 13 = 507

507 = 507 (LHS = RHS)

Option 3) 15 – 450

(15 × 3) × 15 = 450

45 × 15 = 450

675 ≠ 450 (LHS ≠ RHS)

Option 4) 9 – 243

(9 × 3) × 9 = 243

27 × 9 = 243

243 = 243 (LHS = RHS)

Thus, out of all the options '15 – 450' is the odd one out.

Hence, "Option 3" is the correct answer.

The logic followed here is:

'HIT' is coded as '132'

And,

'SBYZ' is coded as '144'

Similarly,

'YRF' is coded as

Hence, "Option 2" is the correct answer.

The logic followed here is:

Option 1) WUQ

Option 2) YWT

Option 3) ZXU

Option 4) XVS

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

Hence, "Option 1" is the correct answer.

The correct answer is Fourth Five-Year Plan.

Key Points

• The Drought Prone Area Programme (DPAP) was launched by the Government of India during the Fourth Five-Year Plan (1969-1974).
• The primary objective of DPAP was to provide employment to people in drought-prone areas and to ensure economic development by stabilizing the income of the people living in these areas.
• The program aimed at reducing the adverse effects of drought on the productivity of land, water resources, and overall livelihood of the people.
• DPAP was part of the broader strategy of the Fourth Five-Year Plan to tackle regional disparities and promote balanced regional development in India.

Additional Information

• Five-Year Plans
• The Five-Year Plans of India were centralized and integrated national economic programs. The first plan was launched in 1951 and subsequent plans were formulated for five-year periods.
• The Fourth Five-Year Plan (1969-1974) aimed at growth with stability and progressive achievement of self-reliance.
• It focused on agriculture, including the Green Revolution, and improvements in industrial production.
• Drought Prone Area Programme (DPAP)
• DPAP is one of the earliest area development programs initiated in India to address the issues of drought and desertification.
• The program focuses on the development of natural resources, including water conservation, afforestation, and soil conservation to reduce the impact of drought.
• It also aims to improve the socio-economic conditions of the people living in drought-prone areas through employment generation and infrastructure development.
• Regional Disparities
• Regional disparities refer to the differences in income, wealth, and living standards across different regions within a country.
• Addressing regional disparities is crucial for balanced and inclusive growth, ensuring that all regions benefit from economic development.
• Programs like DPAP are aimed at reducing these disparities by focusing on the development of less developed and drought-prone regions.

The correct answer is Kerala.

Key Points

• Padayani is a traditional folk dance and ritualistic art form that originates from Kerala, India.
• It is primarily performed in the central part of Kerala, especially in the Pathanamthitta district.
• The dance is performed to please Goddess Bhadrakali, a fierce form of the Goddess Kali.
• Padayani is known for its elaborate and colorful costumes, masks, and energetic performances.

Additional Information

• Padayani
• Padayani, also spelled Padeni, is a ceremonial dance performed in Bhadrakali temples in Kerala.
• It is a combination of music, dance, theatre, satire, facial masks, and paintings.
• The term Padayani literally means "rows of warriors" and represents the martial traditions of the region.
• Padayani is performed during the festival season, particularly in the Malayalam months of Kumbham and Meenam (February to April).
• The performers wear huge masks known as "Kolams" which are traditionally made of areca nut tree sheaths, and they depict various characters including gods, goddesses, and spirits.
• The dance is accompanied by the rhythmic beats of the "Chenda" (drum) and "Ilathalam" (cymbals).
• Goddess Bhadrakali
• Bhadrakali is a Hindu goddess popular in Southern India, particularly in Kerala and Tamil Nadu.
• She is considered a fierce and powerful form of the goddess Kali, associated with the destruction of evil and protection of the good.
• Bhadrakali is worshipped for her protective and benevolent qualities, and is often depicted with multiple arms, holding various weapons, and riding a lion.
• Various rituals and ceremonies, including Padayani, are performed to honor and seek blessings from Goddess Bhadrakali.

The correct answer is (a) and (d).

Key Points

• In the given reaction, nitrogen (N) in NH₃ goes from -3 oxidation state to +2 oxidation state in NO, which indicates oxidation (loss of electrons).
• Oxygen (O) in O₂ goes from 0 oxidation state to -2 oxidation state in H₂O, which indicates reduction (gain of electrons).
• Since nitrogen is being oxidized and oxygen is being reduced, statements (a) and (d) are true.
• The other statements (b) and (c) are incorrect as they contradict the changes in oxidation states observed in the reaction.

Additional Information

• Oxidation:
• Oxidation is the process where an atom, ion, or molecule loses electrons, leading to an increase in its oxidation state.
• In the given reaction, nitrogen in NH₃ (oxidation state -3) is oxidized to NO (oxidation state +2).
• Reduction:
• Reduction is the process where an atom, ion, or molecule gains electrons, leading to a decrease in its oxidation state.
• In the given reaction, oxygen in O₂ (oxidation state 0) is reduced to H₂O (oxidation state -2).
• Oxidation State:
• The oxidation state, or oxidation number, is a measure of the degree of oxidation of an atom in a substance.
• It is a hypothetical charge that an atom would have if all bonds to atoms of different elements were 100% ionic.
• Redox Reaction:
• A redox (reduction-oxidation) reaction is a type of chemical reaction that involves a transfer of electrons between two species.
• It consists of two half-reactions: oxidation (loss of electrons) and reduction (gain of electrons).

The correct answer is Zakir Hussain.

Key Points

• Zakir Hussain is a renowned Indian tabla virtuoso, composer, percussionist, music producer, and film actor.
• He was born on 9 March 1951 in Mumbai, Maharashtra, India.
• Zakir Hussain has received several prestigious awards including the Padma Shri in 1988 and the Padma Bhushan in 2002, making him one of the most celebrated percussionists in India.
• In 2023, he was awarded the Padma Vibhushan, the second-highest civilian award in India, for his exceptional contribution to the field of arts.

Additional Information

• Padma Awards
• The Padma Awards are one of the highest civilian honors in India, instituted in 1954.
• The awards are given in three categories: Padma Vibhushan (for exceptional and distinguished service), Padma Bhushan (for distinguished service of a high order), and Padma Shri (for distinguished service).
• The awards are announced on Republic Day (26th January) and are conferred by the President of India at a ceremonial function held at Rashtrapati Bhavan.
• These awards recognize achievements in all fields of activities or disciplines where an element of public service is involved.
• Zakir Hussain
• He is the son of legendary tabla player Ustad Allah Rakha.
• Zakir Hussain started his career at a young age and was a child prodigy in the field of Indian classical music.
• He has collaborated with many international artists and has been part of several cross-genre musical projects, contributing to the global recognition of Indian classical music.
• In addition to his musical career, Zakir Hussain has also been involved in teaching and has held positions as a visiting professor at various prestigious institutions worldwide.
• Maharashtra
• Maharashtra is a state in the western region of India, and it is the second-most populous state and third-largest by area.
• Mumbai, the capital city of Maharashtra, is the financial, commercial, and entertainment capital of India.
• The state has a rich cultural heritage and is known for its contributions to music, dance, and theater.
• Several renowned artists, musicians, and performers hail from Maharashtra, contributing significantly to the Indian cultural landscape.

The correct answer is Chota Nagpur region.

Key Points

• The Chota Nagpur region is one of the most important industrial regions in India primarily due to its rich mineral resources especially coal and iron ore.
• This region includes the eastern part of India, covering Jharkhand, northern Odisha, and parts of West Bengal, making it a significant area for industrial activities.
• Major industries in this region include steel, iron, and other metallurgical industries, alongside various other manufacturing industries.
• Prominent industrial cities in the Chota Nagpur region include Jamshedpur, Bokaro, Dhanbad, and Ranchi, which are well-known for their large-scale industrial establishments.

Additional Information

• Industrial Regions in India
• India is divided into several industrial regions based on the concentration of industries and the availability of resources.
• Some other significant industrial regions include the Mumbai-Pune region, the Gujarat region, and the Chennai-Bengaluru region.
• Each region has its unique advantages, such as proximity to ports, availability of raw materials, and skilled labor.
• Chota Nagpur Plateau
• The Chota Nagpur Plateau is a mineral-rich region in eastern India, known for its deposits of coal, iron ore, manganese, and limestone.
• This plateau plays a critical role in the industrial development of India, particularly in the steel and metallurgical sectors.
• The region's mining activities contribute significantly to the local and national economy.
• Jamshedpur
• Jamshedpur, located in Jharkhand, is one of India's first planned industrial cities.
• It is home to Tata Steel, one of the largest steel manufacturing companies in the world.
• The city has a well-developed infrastructure, including educational institutions, healthcare facilities, and recreational areas, making it a key industrial hub.
• Bokaro Steel City
• Bokaro Steel City is another major industrial city in the Chota Nagpur region, known for its large steel plant, Bokaro Steel Plant, operated by the Steel Authority of India Limited (SAIL).
• The city has a significant influence on the region's economy, providing employment opportunities and contributing to the industrial output.

The correct answer is National Games.

Key Points

• The Raja Bhalindra Singh Trophy is awarded in the context of the National Games of India.
• This trophy is named after Raja Bhalindra Singh, who was the president of the Indian Olympic Association.
• It is given to the state or union territory that achieves the highest points tally in the National Games.
• The National Games are a significant multi-sport event held in India, involving athletes from all Indian states and union territories.

Additional Information

• Khelo India Youth Games
• Khelo India Youth Games is an initiative by the Government of India to promote sports and fitness among youth.
• It includes various disciplines and is held annually with participation from schools and colleges across the country.
• Khelo India Winter Games
• Khelo India Winter Games focuses on winter sports and aims to nurture talent in sports like skiing, ice hockey, and snowboarding.
• It is held in regions with suitable winter conditions, such as Jammu and Kashmir.
• Khelo India University Games
• Khelo India University Games aims to bring together the best university athletes in India to compete in various sports.
• This event is a platform to identify and nurture young talent at the university level.
• National Games
• The National Games of India is a multi-sport event, similar to the Olympics, but held at the national level.
• Athletes from different states and union territories compete in a variety of sports.
• The first modern National Games were held in 1924 in Lahore, undivided India.
• The event is organized by the Indian Olympic Association (IOA).

The correct answer is Uttar Pradesh.

Key Points

• Girija Devi was a renowned Indian classical and ghazal singer of the Seniya and Banaras gharanas.
• She was born on May 8, 1929, in Varanasi, which is located in the state of Uttar Pradesh.
• Girija Devi is known for her contributions to the Thumri genre of Indian classical music.
• She was awarded several prestigious awards, including the Padma Shri and Padma Bhushan, in recognition of her contributions to Indian classical music.

Additional Information

• Ghazal
• A ghazal is a form of amatory poem or ode, originating in Arabic poetry.
• It is a poetic form consisting of rhyming couplets and a refrain, with each line sharing the same meter.
• Ghazals often express the pain of loss and separation and the beauty of love in spite of that pain.
• Its roots are in Arabic poetry and it has been adapted into many languages, including Urdu and Hindi.
• Banaras Gharana
• The Banaras Gharana, or Varanasi Gharana, is one of the most prominent styles of Indian classical music.
• This Gharana is known for its rich traditions in vocal music, particularly the Thumri, Dadra, Chaiti, and Kajri forms.
• The Banaras Gharana is characterized by its emotive expressiveness and the incorporation of folk music elements.
• Musicians from this Gharana have made significant contributions to the development and popularization of Indian classical music.
• Thumri
• Thumri is a common genre of semi-classical Indian music.
• The text is romantic and devotional in nature, and usually revolves around a girl's love for Krishna.
• Thumri is characterized by its sensuality, and the compositions are designed to bring out the subtleties of the Raga.
• It is closely associated with dance, gestures, and dramatic representation.

The correct answer is India has a multi-party system classified as national, state or regional level..

Key Points

• India follows a multi-party system where multiple political parties can exist and participate in elections.
• Political parties in India are classified into national, state or regional level based on their performance and influence.
• National parties are those that have influence over a larger part of India, while state or regional parties have influence limited to specific states or regions.
• This classification helps in managing the democratic process and ensuring representation at both the national and state levels.

Additional Information

• Election Commission of India (ECI)
• The Election Commission of India is an autonomous constitutional authority responsible for administering election processes in India at both the national and state levels.
• It administers elections to the Lok Sabha, Rajya Sabha, and State Legislative Assemblies, as well as the offices of the President and Vice President in the country.
• The ECI ensures free and fair elections in India.
• Registration of Political Parties
• Political parties in India are registered with the Election Commission of India under the Representation of the People Act, 1951.
• Registration with the ECI is a precondition for a political party to be recognized as a national or state party.
• Recognition of Political Parties
• The status of a political party, whether national or state, is determined based on its performance in elections.
• Criteria for recognition include the number of seats won in elections and the percentage of votes obtained.
• Symbols of Political Parties
• The Election Commission of India assigns symbols to political parties to help voters identify them during elections.
• Symbols play a significant role in the election process, especially for illiterate voters who may not recognize party names.

The correct answer is Disproportionate.

Key Points

• The Green Revolution led to a significant increase in the production of food grains, but the growth was not uniform across all types of grains.
• While crops like wheat and rice saw substantial yield improvements due to the introduction of high-yielding varieties (HYVs), other grains like pulses and coarse cereals did not benefit as much.
• The growth in food grain production was therefore termed as disproportionate because it heavily favored certain crops over others.
• This uneven growth also led to regional disparities, with states like Punjab and Haryana benefiting more compared to other regions.

Additional Information

• Green Revolution:
• The Green Revolution refers to a period during the 1960s and 1970s when agriculture in many developing countries was transformed by the introduction of high-yielding varieties of grains, particularly wheat and rice.
• This revolution was spearheaded by the use of chemical fertilizers, pesticides, and improved irrigation techniques.
• Norman Borlaug, an American agronomist, is often called the "Father of the Green Revolution" for his work in developing high-yielding crop varieties.
• The Green Revolution is credited with preventing famines in countries like India and significantly increasing food security.
• High-Yielding Varieties (HYVs):
• These are strains of crops that have been scientifically bred to produce more grain per acre compared to traditional varieties.
• HYVs often require specific conditions such as high levels of chemical fertilizers and controlled irrigation.
• In India, the introduction of HYVs of wheat and rice during the Green Revolution led to a substantial increase in production.
• Impact on Food Security:
• The Green Revolution played a key role in making countries like India self-sufficient in food grains.
• It helped in reducing the reliance on food imports and improved the overall food security of the nation.
• However, the focus on a few high-yielding crops also led to a reduction in the cultivation of traditional crops, affecting the agricultural biodiversity.
• Regional Disparities:
• The benefits of the Green Revolution were not evenly distributed across all regions.
• States with better irrigation facilities and infrastructure, like Punjab and Haryana, saw greater gains compared to states with less developed agricultural systems.
• This led to economic and developmental disparities among different regions within the country.

The correct answer is Dual Citizenship.

Key Points

• The Constitution of India does not provide for dual citizenship.
• India follows a single citizenship policy, which means that a person is a citizen of India only and not of any state individually.
• Dual citizenship could lead to complications in matters of loyalty and allegiance, thus the Indian Constitution avoids it.
• Other countries may allow dual citizenship, but India mandates renouncing Indian citizenship upon acquiring foreign nationality.

Additional Information

• Universal Adult Franchise
• It is a fundamental principle of democracy where all adult citizens have the right to vote without any discrimination.
• In India, it was adopted with the commencement of the Constitution on January 26, 1950.
• It ensures equality and inclusiveness in the electoral process.
• Every citizen above the age of 18 has the right to vote in elections to the Lok Sabha and the Legislative Assemblies.
• Independent Judiciary
• It refers to a judiciary that is free from any influence or control by the executive or the legislature.
• The Indian Constitution ensures the independence of the judiciary through various provisions such as the security of tenure, fixed service conditions, and restrictions on practice after retirement.
• Judicial independence is crucial for the fair administration of justice and upholding the rule of law.
• The Supreme Court and High Courts in India are the guardians of the Constitution and are empowered to review laws and executive actions.
• Blend of Rigidity & Flexibility
• The Indian Constitution is a unique blend of rigidity and flexibility.
• Some parts of the Constitution can be amended by a simple majority, while others require a special majority or ratification by half of the state legislatures.
• This blend allows the Constitution to adapt to changing needs while preserving its core principles.
• Article 368 of the Constitution provides for the amendment process.

The correct answer is The Epidemic Diseases Act, 1897.

Key Points

• The Epidemic Diseases Act, 1897 was enacted to provide for better prevention of the spread of dangerous epidemic diseases.
• The Act gives special powers to state and central governments to take measures necessary to prevent the outbreak or spread of an epidemic.
• The Epidemic Diseases (Amendment) Ordinance, 2020 was introduced to include protections for healthcare personnel combatting epidemic diseases.
• The Ordinance also provides for stricter penalties for acts of violence against healthcare service personnel and damage to property during epidemics.

Additional Information

• Epidemic Diseases Act, 1897
• The Act was passed by the British colonial government in India in 1897 in response to the bubonic plague outbreak in Mumbai (then Bombay).
• The Act allows the central and state governments to take special measures and prescribe regulations to prevent the spread of epidemic diseases.
• The Act also provides the legal framework for the implementation of containment measures such as quarantine, isolation, and restrictions on the movement of people.
• The Act has been invoked several times in India, including during the outbreaks of H1N1 influenza (swine flu), dengue, and COVID-19.
• The Epidemic Diseases (Amendment) Ordinance, 2020
• The Ordinance was promulgated by the President of India on April 22, 2020, in response to the COVID-19 pandemic.
• It amends the Epidemic Diseases Act, 1897 to include protections for healthcare service personnel combating epidemic diseases.
• The Ordinance defines "healthcare service personnel" to include healthcare professionals, nurses, paramedical workers, and any other person empowered under the Act to take measures to prevent the outbreak or spread of an epidemic disease.
• It provides for imprisonment and fines for acts of violence against healthcare service personnel and damage to property during epidemics.
• Other Important Terminologies
• Quarantine: A restriction on the movement of people and goods intended to prevent the spread of disease or pests.
• Isolation: The separation of people who are infected with a contagious disease from those who are healthy to prevent the spread of the disease.
• Containment Measures: Strategies and actions taken to prevent the spread of a disease within a community, including quarantine, isolation, and restrictions on movement.

The correct answer is 100.

Key Points

• The 'Mero Rukh Mero Santati' initiative was launched by the Sikkim Chief Minister Prem Singh Tamang in 2023.
• Under this initiative, 100 trees will be planted for every child born in Sikkim.
• This initiative aims to promote environmental conservation and create a sustainable environment for future generations.
• The program encourages parental participation in the plantation and nurturing of these trees, fostering a sense of responsibility towards the environment.

Additional Information

• Sikkim
• Sikkim is a state in northeastern India, bordered by Bhutan, Tibet, and Nepal.
• It is known for its biodiversity, including alpine and subtropical climates, as well as being home to Kangchenjunga, the highest peak in India and third highest on Earth.
• Sikkim is India's least populous and second smallest state, with a population of around 600,000 people.
• The state capital is Gangtok, and it is famous for its rich cultural heritage and beautiful landscapes.
• Environmental Conservation
• Environmental conservation is the practice of protecting the natural environment for current and future generations.
• It involves sustainable management of natural resources and the prevention of pollution and environmental degradation.
• Planting trees is a key aspect of conservation as it helps to combat climate change, restore habitats, and maintain biodiversity.
• Initiatives like 'Mero Rukh Mero Santati' are critical in promoting environmental awareness and action at the community level.
• Sustainable Development
• Sustainable development is development that meets the needs of the present without compromising the ability of future generations to meet their own needs.
• It encompasses three main pillars: economic growth, social inclusion, and environmental protection.
• Programs that involve tree planting and environmental conservation contribute significantly to sustainable development.
• Such initiatives ensure a balance between human progress and environmental well-being.

The correct answer is nephrons.

Key Points

• Nephrons are the basic structural and functional units of the kidney.
• Each kidney contains around 1 to 1.5 million nephrons.
• Nephrons are responsible for the removal of waste products, excess substances, and toxins from the blood.
• They play a crucial role in the regulation of water, electrolytes, and maintaining the body's acid-base balance.

Additional Information

• Bowman's Capsule
• The Bowman's capsule is a cup-like sac at the beginning of the nephron in the kidney.
• It encases the glomerulus and is involved in the first step of the filtration of blood to form urine.
• It captures the filtrate produced from the blood and passes it into the renal tubule.
• Capillaries
• Capillaries are the smallest blood vessels in the body.
• They facilitate the exchange of oxygen, carbon dioxide, nutrients, and waste products between blood and tissues.
• In the kidneys, capillaries are part of the glomerulus and play a critical role in the filtration process.
• Alveoli
• Alveoli are small air sacs in the lungs where gas exchange takes place.
• Oxygen from the inhaled air passes through the walls of the alveoli and into the blood in the capillaries, while carbon dioxide from the blood passes into the alveoli to be exhaled.
• They are crucial for efficient respiratory function but are not related to kidney function.

The correct answer is 6.022 × 10²³.

Key Points

• Avogadro's constant, also known as Avogadro's number, is named after the Italian scientist Amedeo Avogadro.
• It is defined as the number of constituent particles (usually atoms or molecules) that are contained in one mole of a substance.
• The value of Avogadro's constant is 6.022 × 10²³ mol^-1.
• Avogadro's constant is a fundamental constant in chemistry and physics that allows chemists to count the number of atoms, ions, or molecules in a given sample.

Additional Information

• Avogadro's Law:
• Avogadro's law states that equal volumes of all gases, at the same temperature and pressure, have the same number of molecules.
• This law provides the relationship between the volume of a gas and the amount of substance of the gas present.
• The Mole Concept:
• The mole is a unit of measurement for the amount of substance in the International System of Units (SI).
• One mole contains exactly 6.02214076 × 10²³ elementary entities (Avogadro's constant).
• This concept is used to convert between the number of particles and the amount of substance.
• Importance in Chemistry:
• Avogadro's constant is crucial for understanding and working with chemical reactions, as it allows chemists to determine the number of particles involved.
• It is essential for calculations involving the ideal gas law, stoichiometry, and other chemical principles.
• Historical Context:
• Amedeo Avogadro (1776-1856) was an Italian scientist who made significant contributions to molecular theory.
• Avogadro's hypothesis, proposed in 1811, led to the eventual development of the concept of the mole and Avogadro's constant.

The correct answer is Paleoecology.

Key Points

• Paleoecology is the branch of ecology that deals with the study of the characteristics of ancient environments and their relationships with ancient plants and animals.
• It involves the reconstruction of past ecosystems based on fossil records and other geological and biological evidence.
• Paleoecologists study various aspects such as climate, geography, and the interactions between ancient organisms.
• By understanding ancient ecosystems, paleoecologists can provide insights into the evolutionary history of life on Earth and the processes that have shaped current biodiversity.

Additional Information

• Demecology
• It is also known as population ecology.
• Demecology focuses on the study of populations of organisms, especially with regard to population size, density, and structure, as well as how these populations interact with their environment.
• Ichthyology
• It is the branch of zoology that deals with the study of fish.
• Ichthyologists study various aspects of fish biology, including their anatomy, behavior, ecology, and evolution.
• Mycology
• It is the branch of biology concerned with the study of fungi.
• Mycologists study fungi in all their forms, including their genetics, ecology, and use in medicine and industry.

The correct answer is Dead Sea.

Key Points

• The Dead Sea is a landlocked salt lake located in the Middle East, bordered by Jordan to the east and Israel and Palestine to the west.
• It is one of the world's saltiest bodies of water, with a salinity of around 34.2%, which is almost ten times saltier than the ocean.
• The Dead Sea is also the lowest point on the surface of the Earth, lying more than 430 meters below sea level.
• Due to its high salinity, the Dead Sea is inhospitable to most life forms, hence its name.

Additional Information

• Don Juan Pond
• Don Juan Pond is located in Antarctica and is one of the saltiest bodies of water on Earth.
• It is not landlocked in South-West Asia and does not lie between Israel and Jordan.
• Aral Sea
• The Aral Sea is situated between Kazakhstan and Uzbekistan in Central Asia.
• It was once one of the world's largest lakes, but has drastically shrunk due to water diversion projects.
• Caspian Sea
• The Caspian Sea is the world's largest enclosed inland body of water, located between Europe and Asia.
• It is bordered by Kazakhstan, Russia, Turkmenistan, Iran, and Azerbaijan.

The correct answer is Godavari river.

Key Points

• The Godavari is the second longest river in India after the Ganges.
• It flows eastwards across the Deccan Plateau and drains into the Bay of Bengal.
• The Godavari river is also known as 'Dakshina Ganga' or 'Ganga of the South'.
• It originates near Trimbak in the Nashik District of Maharashtra.

Additional Information

• Mahi River
• The Mahi is one of the major west-flowing rivers in India.
• It originates in Madhya Pradesh and flows through Rajasthan and Gujarat before emptying into the Arabian Sea.
• It is notable for its drainage basin which is shared by multiple states.
• Narmada River
• The Narmada is also a west-flowing river.
• It originates from the Amarkantak Plateau in Madhya Pradesh and flows through Gujarat, eventually draining into the Arabian Sea.
• The Narmada is known for the picturesque Marble Rocks and the Dhuandhar Falls in Jabalpur.
• Tapi River
• The Tapi is another major river that flows westwards.
• It originates in the Satpura range of Madhya Pradesh and flows through Maharashtra and Gujarat, emptying into the Arabian Sea.
• It is one of the major rivers that do not drain into the Bay of Bengal.

The correct answer is Lord Cornwallis.

Key Points

• Lord Cornwallis, who served as Governor-General of Bengal from 1786 to 1793, is often associated with significant administrative reforms in India.
• He introduced the Daroga system in 1792 as part of the police reforms to establish a more structured and efficient law enforcement system.
• The Daroga system involved the appointment of Darogas (police officers) in different regions to maintain law and order.
• This system laid the foundation for the modern police administration in India.

Additional Information

• Lord Cornwallis
• Lord Cornwallis is also known for the Cornwallis Code, which included judicial and administrative reforms.
• The Cornwallis Code aimed at improving the efficiency and fairness of the judicial system, including the separation of revenue and judicial functions.
• His tenure is marked by significant changes in the governance of British India, including the introduction of the Permanent Settlement of Bengal in 1793.
• The Permanent Settlement established a system of land revenue collection that aimed at stabilizing revenue for the British administration and creating a class of loyal landowners.
• Lord Minto I:
• Lord Minto served as Governor-General of India from 1807 to 1813.
• His tenure is notable for the Charter Act of 1813, which renewed the East India Company's charter but also allowed missionaries to come to India.
• Lord Minto II:
• He served as Viceroy (1905-1910).
• He is also remembered for the Minto-Morley Reforms, which were introduced in 1909 and were aimed at increasing Indian participation in the governance.
• Minto II's tenure was marked by the Morley-Minto Reforms, the formation of the All India Muslim League, and the Surat Split of the Indian National Congress.
• Lord Mayo
• Lord Mayo served as Viceroy of India from 1869 to 1872.
• He is known for his efforts in administrative and financial reforms, including the establishment of the Department of Agriculture and Commerce.
• Lord Mayo was assassinated in 1872 by a convict in the Andaman Islands.
• Lord William Bentinck
• Lord William Bentinck served as Governor-General of India from 1828 to 1835.
• He is renowned for his social reforms, including the abolition of Sati (the practice of widow immolation) and the suppression of Thuggee (banditry).
• His tenure also saw the introduction of English education in India, with Thomas Babington Macaulay's Minute on Indian Education.

The correct answer is Balabhadra.

Key Points

• The Chariot Festival, also known as Rath Yatra, is primarily celebrated in the city of Puri in Odisha, India.
• It is dedicated to Lord Jagannath, who is considered a form of Lord Krishna, along with his siblings Subhadra and Balabhadra (also known as Balarama).
• Balabhadra is the elder brother of Lord Jagannath and Subhadra, and he is often depicted with a plough in his hand, symbolizing agriculture and strength.
• The festival involves the procession of deities in chariots, pulled by devotees through the streets, symbolizing the journey of Lord Jagannath, Balabhadra, and Subhadra from their temple to their garden temple in the countryside.

Additional Information

• Rath Yatra:
• Rath Yatra is one of the oldest and largest chariot festivals in the world, attracting millions of devotees and tourists from all over the globe.
• It is celebrated annually on the second day of the bright fortnight of the month of Ashadha (June-July).
• The three deities, Lord Jagannath, Balabhadra, and Subhadra, are taken out in a grand procession from their temple in Puri to the Gundicha Temple, which is considered their aunt's house.
• The return journey of the deities to the Jagannath Temple is known as Bahuda Yatra.
• Lord Jagannath:
• Lord Jagannath is a form of Lord Vishnu and is primarily worshipped in the state of Odisha.
• He is known for his large, round eyes and is often depicted without limbs.
• The Jagannath Temple in Puri is one of the four major Hindu pilgrimage sites in India, known as Char Dham.
• Balabhadra (Balarama):
• Balabhadra, also known as Balarama, is the elder brother of Lord Krishna and is worshipped as an incarnation of Sheshnag, the serpent on whom Lord Vishnu rests.
• He is depicted with a fair complexion and is often shown carrying a plough and a mace.
• Balarama is considered a deity of agriculture and strength and is revered for his role in various Hindu epics and scriptures.
• Subhadra:
• Subhadra is the younger sister of Lord Jagannath and Balabhadra and is often depicted riding in the middle chariot during the Rath Yatra.
• She is worshipped as a manifestation of Goddess Durga and is believed to protect and bless her devotees.

The correct answer is Production of the firm during the year – Sale of the firm during the year.

Key Points

• Change in inventories refers to the difference between the production and the sale of goods in a given period.
• It helps in determining the amount of goods that are produced but not sold during the period.
• This metric is crucial for understanding the stock levels and inventory management of a firm.
• It impacts the working capital and liquidity position of the firm since unsold inventory ties up capital.

Additional Information

• Inventory:
• Inventory refers to the goods and materials that a business holds for the ultimate goal of resale.
• It includes raw materials, work-in-progress, and finished goods.
• Production:
• Production is the process of creating, expanding, manufacturing, or improving goods and services.
• It involves various stages of transformation of raw materials into finished products.
• Sales:
• Sales refer to the exchange of a good or service for money.
• It is the process of selling the produced goods to the end consumer or another business.
• Working Capital:
• Working capital is the difference between a company's current assets and current liabilities.
• It is a measure of a company's operational efficiency and short-term financial health.

The correct answer is Ministry of Youth Affairs and Sports.

Key Points

• The Sports Authority of India (SAI) operates under the Ministry of Youth Affairs and Sports, Government of India.
• SAI was established in 1984 to promote sports and oversee the development of sports facilities and training programs in India.
• It is responsible for the preparation of athletes for major international events such as the Olympics, Asian Games, and Commonwealth Games.
• SAI also manages various sports training centers and academies across the country, providing infrastructure, coaching, and other support to budding sportspersons.

Additional Information

• Ministry of Education:
• The Ministry of Education is responsible for the development and implementation of educational policies and programs in India.
• It oversees various institutions such as the University Grants Commission (UGC), National Council of Educational Research and Training (NCERT), and others.
• The Ministry's focus is on improving the quality of education and ensuring access to education for all.
• Ministry of Social Justice and Empowerment:
• This ministry focuses on the welfare, social justice, and empowerment of disadvantaged and marginalized sections of society.
• It implements schemes and policies for the upliftment of Scheduled Castes, Scheduled Tribes, Other Backward Classes, Divyangjan (Persons with Disabilities), and other vulnerable groups.
• Ministry of Health and Family Welfare:
• The Ministry of Health and Family Welfare is responsible for the formulation and implementation of health policies and programs in India.
• It aims to ensure the accessibility and affordability of quality healthcare for all citizens.
• It manages various national health schemes like Ayushman Bharat, National Health Mission, and others.
• Ministry of Youth Affairs and Sports:
• The Ministry of Youth Affairs and Sports is dedicated to the development and promotion of sports and youth welfare in India.
• It formulates policies and programs to nurture young talent in sports and provides necessary infrastructure and training facilities.
• It also organizes national and international sports events and collaborates with various sports bodies and organizations.

The correct answer is Satavahana.

Key Points

• The Satavahana dynasty was an ancient Indian dynasty based in the Deccan region.
• The kingdom was divided into subdivisions called Aharas or Rashtras, which were essentially districts.
• These districts were governed by officers known as Amatyas and Mahamatras.
• The Satavahana administration was highly organized, with efficient management of its various subdivisions.

Additional Information

• Satavahana Dynasty
• The Satavahana dynasty ruled over the Deccan region of India from the 1st century BCE to the 3rd century CE.
• Their rule was marked by the establishment of significant trade routes, both inland and maritime.
• The Satavahanas are known for their contribution to Indian culture, including the promotion of Prakrit and Sanskrit literature.
• They played a crucial role in the cultural and political unification of the Deccan region.
• Administrative Division
• The Satavahana kingdom was divided into various administrative units for effective governance.
• Aharas or Rashtras were the primary administrative divisions, equivalent to modern districts.
• Each Ahara was managed by a set of officers who were responsible for collecting taxes and maintaining law and order.
• This administrative structure helped in the efficient management of the vast territories under Satavahana control.
• Amatyas and Mahamatras
• Amatyas were high-ranking officials who played a crucial role in the administration of the kingdom.
• Mahamatras were officers responsible for various administrative functions, including justice, revenue collection, and public works.
• These officials were appointed based on their abilities and were accountable to the central authority.
• Their roles were critical in maintaining the administrative efficiency and stability of the Satavahana dynasty.

The correct answer is Tolkappiyam.

Key Points

• Tolkappiyam is considered the earliest extant Tamil literary work.
• It was authored by the ancient Tamil scholar Tolkappiyar.
• This work is a comprehensive treatise on Tamil grammar, phonology, morphology, and syntax.
• Tolkappiyam also provides valuable insights into the social and cultural life of ancient Tamil society.

Additional Information

• Ettutogai
• Ettutogai, also known as the Eight Anthologies, is a classical Tamil poetic work.
• It is a collection of eight different anthologies of Tamil poems.
• These poems focus on various themes such as love, valor, and ethics, reflecting the Tamil culture and traditions.
• Kalittogai
• Kalittogai is a classical Tamil poetic work and one of the Eight Anthologies (Ettutogai).
• It comprises 150 poems focusing on the theme of love.
• The poems in Kalittogai are unique in their meter and style, known as Kali meter.
• Narrinai
• Narrinai is also part of the Eight Anthologies (Ettutogai).
• This work consists of 400 poems, primarily focusing on the theme of love.
• Narrinai poems are known for their simplicity and emotional depth.

The correct answer is Jai Prakash Narayan.

Key Points

• Jai Prakash Narayan, also known as JP, was an Indian independence activist, social reformer, and political leader.
• He is famously known as "Lok Nayak," which translates to "The People's Leader" for his leadership and contribution to Indian society.
• He played a key role in the Quit India Movement against British rule in 1942.
• In the 1970s, he led a movement against Prime Minister Indira Gandhi's government, which eventually led to the imposition of Emergency in India.

Additional Information

• Ram Manohar Lohia
• He was an activist in the Indian independence movement and a socialist political leader.
• Lohia was known for his advocacy of non-violent struggle and his opposition to the caste system.
• He co-founded the Congress Socialist Party in 1934.
• Post-independence, he was a prominent member of the Socialist Party.
• Bal Gangadhar Tilak
• He was an Indian nationalist, teacher, and an independence activist.
• Tilak is often referred to as "Lokmanya Tilak," meaning "Beloved leader of the people."
• He was one of the first leaders of the Indian Independence Movement.
• Tilak was a strong advocate for Swaraj (self-rule) and is famously quoted saying, "Swaraj is my birthright and I shall have it."
• Subhash Chandra Bose
• He was a prominent leader in the Indian independence movement against British rule.
• Bose was known for his militant approach and leadership of the Indian National Army (INA).
• He earned the title "Netaji," meaning "Respected Leader."
• Bose's famous slogan was "Give me blood, and I shall give you freedom."

The correct answer is Amitabh Bachchan.

Key Points

• Narendra Modi, the Prime Minister of India, was included in the Time Magazine's list of '100 most influential people of 2021'.
• Mamta Banerjee, the Chief Minister of West Bengal, was also featured in the list for her influential political presence.
• Adar Poonawalla, CEO of Serum Institute of India, was recognized for his contributions to vaccine production during the COVID-19 pandemic.
• Amitabh Bachchan, the iconic Bollywood actor, was not included in the list of '100 most influential people of 2021'.

Additional Information

• Time Magazine's '100 most influential people'
• This annual list recognizes individuals who have made significant impacts globally in various fields such as politics, entertainment, technology, and more.
• The list is highly regarded and features people from diverse backgrounds and professions.
• Influence is measured by the ability of these individuals to shape the world around them, whether through policy, innovation, or cultural impact.
• In 2021, the list included leaders, creators, advocates, and pioneers who were at the forefront of their respective fields.
• Narendra Modi
• The Prime Minister of India, known for his leadership and policy reforms in the country.
• He has been a recurring figure on the list due to his significant political influence both within India and internationally.
• Mamta Banerjee
• The Chief Minister of West Bengal, recognized for her strong political stance and leadership in her state.
• Her inclusion highlights her impactful role in Indian politics.
• Adar Poonawalla
• CEO of Serum Institute of India, the world's largest vaccine manufacturer.
• He played a critical role in the global fight against COVID-19 through the production and distribution of vaccines.

Shortcut Trick

Since the chord lengths (7 units) equal the radius (7 units), ΔPOQ and ΔQOR are equilateral triangles.

Thus, each chord subtends a 60° angle at the center, making the total central angle for chord PR = 60° + 60° = 120°.

Using the direct chord length formula: PR = 2 × R × sin(θ/2) = 2 × 7 × sin(60°).

PR = 14 × (√3 / 2) = 7√3 units.

To match the given options, multiply the numerator and denominator by √3: 7√3 × (√3 / √3) = 21/√3 units.

∴ The correct answer is 21/√3 units.

Alternate Method

Given:

Radius of the circle (R) = 7 units

Length of chords PQ = QR = 7 units

     O  P  Q  R   60° 60° 7 7 7 7 PR = ?

Calculations:

⇒ Let O be the center of the circle. The radii OP, OQ, and OR all measure 7 units.

⇒ In ΔPOQ, we have OP = OQ = PQ = 7 units. Thus, ΔPOQ is an equilateral triangle, meaning ∠POQ = 60°.

⇒ Similarly, in ΔQOR, OQ = OR = QR = 7 units. Thus, ΔQOR is also an equilateral triangle, meaning ∠QOR = 60°.

⇒ The total central angle subtended by chord PR is ∠POR = ∠POQ + ∠QOR = 60° + 60° = 120°.

⇒ Applying the cosine rule in ΔPOR:

⇒ PR² = OP² + OR² − 2 × OP × OR × cos(120°)

⇒ PR² = 7² + 7² − 2 × 7 × 7 × (−1/2)

⇒ PR² = 49 + 49 + 49 = 147

⇒ PR = √147 = 7√3 units

⇒ To match the fraction format given in the options, multiply the numerator and denominator by √3:

⇒ PR = (7√3 × √3) ÷ √3 = 21/√3 units

∴ The correct answer is 21/√3 units.

Additional Information

Central Angle Theorem for Chords

The length of a chord L in a circle of radius R subtending a central angle θ is directly given by the formula L = 2 × R × sin(θ/2).

Cosine Rule

In any triangle, the square of a side is equal to the sum of the squares of the other two sides minus twice their product multiplied by the cosine of the included angle: c² = a² + b² − 2ab × cos(C).

Equilateral Triangle Properties

When a chord length is perfectly equal to the radius of the circle, the triangle formed by the chord and the two radii is perfectly equilateral, with all internal angles naturally measuring 60°.

Shortcut Trick

Relative speed in opposite direction = 36 + 54 = 90 km/h = 90 × (5/18) = 25 m/s.

Time to meet for the first time = Total Distance ÷ Relative Speed

Time = 4800 ÷ 25 = 192 seconds.

∴ The correct answer is 192 second.

Alternate Method

Given:

Length of the circular track = 4800 m

Speed of A = 36 km/h

Speed of B = 54 km/h

Formula Used:

Relative Speed (Opposite Direction) = Speed of A + Speed of B

Time taken to meet = Distance ÷ Relative Speed

1 km/h = 5/18 m/s

Calculations:

⇒ Speed of A in m/s = 36 × (5/18) = 10 m/s

⇒ Speed of B in m/s = 54 × (5/18) = 15 m/s

⇒ Relative Speed = 10 + 15 = 25 m/s

⇒ Time to meet = Total Distance ÷ Relative Speed

⇒ Time = 4800 ÷ 25

⇒ Time = 192 seconds

∴ The correct answer is 192 second.

Additional Information

Meeting for the first time anywhere on the track

Time = Length of track ÷ Relative Speed. The relative speed is the sum of their speeds if moving in opposite directions, and difference of their speeds if moving in the same direction.

Meeting for the first time at the starting point

Time = LCM of individual times taken to complete one full round (LCM of T₁, T₂, ...). This applies irrespective of the direction of motion.

Speed Unit Conversion

To convert km/h to m/s directly, multiply the given speed by 5/18. Conversely, to convert m/s to km/h, multiply by 18/5.

Given:

Discipline: Medical

Boys in Medical = 80

Girls in Medical = 120

Formula Used:

Percentage of boys in Medical = (Number of boys in Medical / Total number of students in Medical) × 100

Calculation:

Total number of students in Medical = Boys in Medical + Girls in Medical

Total number of students in Medical = 80 + 120 = 200

Percentage of boys in Medical = (80 / 200) × 100

Percentage of boys in Medical = 0.4 × 100 = 40%

The percentage of students who chose medical and are boys is 40%.

Shortcut Trick

Total increase in weight = Number of students × Increase in average weight

Total weight increase = 10 × 0.5 kg = 5 kg

Weight of new student = Weight of replaced student + Total weight increase

Weight of new student = 51 kg + 5 kg = 56 kg

∴ The correct answer is 56 kg.

Alternate Method

Given:

Initial number of students = 10

Weight of the student replaced = 51 kg

Increase in average weight = 0.5 kg

Formula Used:

Average = Total Sum of Observations ÷ Number of Observations

Group of 10 Students Average increases by 0.5 kg

51 kg Leaves

New Student Joins

Net Weight Increase = 10 × 0.5 kg = +5 kg

Calculations:

Let the initial average weight of the 10 students be A kg.

⇒ Total initial weight of the 10 students = 10 × A = 10A

Let the weight of the new student be W kg.

When the 51 kg student is replaced by the new student W, the new total weight becomes:

⇒ New total weight = 10A − 51 + W

The new average weight increases by 0.5 kg, so it becomes A + 0.5 kg.

⇒ New average = (10A − 51 + W) ÷ 10

⇒ A + 0.5 = (10A − 51 + W) ÷ 10

⇒ 10 × (A + 0.5) = 10A − 51 + W

⇒ 10A + 5 = 10A − 51 + W

Subtracting 10A from both sides, we get:

⇒ 5 = W − 51

⇒ W = 51 + 5

⇒ W = 56 kg

∴ The correct answer is 56 kg.

Additional Information

Replacement Method in Averages

When a person is replaced by a new person, use the formula: New Value = Replaced Value + (Total Number of Items × Change in Average). If the average decreases, the sign before the change becomes negative.

Combined Average

When two groups of sizes n₁ and n₂ have averages A₁ and A₂ respectively, their combined average is given by: (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Inclusion & Exclusion Principle

If a new person joins a group and the average increases, their value is strictly greater than the old average. Conversely, if someone leaves the group and the average increases, their value was less than the old average.

Shortcut Trick

Total capacity = LCM(6, 8) = 24 units.

Combined efficiency of A and B = (24 ÷ 6) + (24 ÷ 8) = 4 + 3 = 7 units/hr.

Work done in first 3 hours = 3 × 7 = 21 units, leaving 24 − 21 = 3 units.

Time taken by B to complete remaining 3 units = 3 ÷ 3 = 1 hour.

Total time taken = 3 (together) + 1 (B alone) = 4 hours.

∴ The correct answer is 4 hours.

Alternate Method

Given:

Time taken by Pipe A to fill the tank = 6 hours

Time taken by Pipe B to fill the tank = 8 hours

Both pipes operate together for the first 3 hours.

Calculations:

⇒ Combined efficiency of A and B = 4 + 3 = 7 units/hr

⇒ Work completed in 3 hours = 7 × 3 = 21 units

⇒ Remaining work = 24 − 21 = 3 units

⇒ Time taken by B to complete the rest = 3 ÷ 3 = 1 hour

⇒ Total time required to fill the tank = 3 + 1 = 4 hours

∴ The correct answer is 4 hours.

Additional Information

Pipes and Cisterns Trick

Always assume Total Capacity as the LCM of the given times to avoid fractional calculations.

Efficiency Concept

Efficiency is inversely proportional to the time taken. Total Work = Efficiency × Time.

Emptying Pipes

If a pipe empties the tank, its efficiency is taken as negative during calculations.

Shortcut Trick

Let the Selling Price (SP) be 5 units.

Given, Gain = 1/5 of SP ⇒ Gain = 1 unit.

Cost Price (CP) = SP − Gain = 5 − 1 = 4 units.

Gain % = (Gain ÷ CP) × 100 = (1 ÷ 4) × 100 = 25%.

∴ The correct answer is 25%.

Alternate Method

Given:

Selling Price (SP) of the scooter = ₹71785

Gain = 1/5 of the Selling Price

Formula Used:

Cost Price (CP) = SP − Gain

Gain % = (Gain ÷ CP) × 100

Calculations:

⇒ Gain = (1 ÷ 5) × 71785 = ₹14357

⇒ CP = 71785 − 14357 = ₹57428

⇒ Gain % = (14357 ÷ 57428) × 100

⇒ Gain % = (1 ÷ 4) × 100 = 25%

∴ The correct answer is 25%.

Additional Information

Base of Profit/Loss Calculation

Profit and Loss percentages are always calculated on the Cost Price (CP) unless specifically stated to be calculated on the Selling Price.

Marked Price & Discount

Discount is always offered on the Marked Price (MP). Formula: Selling Price = Marked Price × (100 − Discount%) ÷ 100.

Successive Discount

If two successive discounts of x% and y% are given, the effective single discount is (x + y − xy ÷ 100)%.

Shortcut Trick

If A + B = 90°, then tan A = cot B.

Here, the sum of the angles is 55° + 35° = 90°.

Therefore, we can directly say tan 55° = cot 35°.

Substituting this in the given expression, we get: cot 35° ÷ cot 35° = 1.

∴ The correct answer is 1.

Alternate Method

Given:

Expression: (tan 55°) / (cot 35°)

Formula Used:

cot(90° − θ) = tan θ

Calculation:

⇒ Let us change the denominator into terms of tangent to simplify the fraction.

⇒ We know that 35° can be written as (90° − 55°).

⇒ So, cot 35° = cot(90° − 55°)

⇒ Applying the complementary angle formula cot(90° − θ) = tan θ, we get:

⇒ cot 35° = tan 55°

⇒ Now, substitute this equivalent value back into the original fraction:

⇒ (tan 55°) / (cot 35°) = (tan 55°) / (tan 55°)

⇒ Both the numerator and the denominator are exactly the same, so they cancel out.

⇒ = 1

∴ The correct answer is 1.

Additional Information

Complementary Angle Identities

Two angles are complementary if their sum is 90°. Key conversion identities include: sin(90° − θ) = cos θ, sec(90° − θ) = cosec θ.

Reciprocal Identities

Trigonometric functions can be inverted to find their reciprocals: sin θ = 1 / cosec θ, cos θ = 1 / sec θ, tan θ = 1 / cot θ.

Pythagorean Identities

Important squared relations: sin²θ + cos²θ = 1, sec²θ − tan²θ = 1, and cosec²θ − cot²θ = 1.

Shortcut Trick

Cost Price of Shirt = 1200 × (100 ÷ 120) = ₹1000

Cost Price of Trouser = 660 × (100 ÷ 110) = ₹600

Ratio of their Cost Prices = 1000 : 600 = 5 : 3

∴ The correct answer is 5 : 3.

Alternate Method

Given:

Selling Price of Shirt = ₹1,200 | Profit on Shirt = 20%

Selling Price of Trouser = ₹660 | Profit on Trouser = 10%

Formula Used:

Cost Price (CP) = [Selling Price × 100] ÷ [100 + Profit%]

Calculations:

⇒ CP of Shirt = (1200 × 100) ÷ (100 + 20)

⇒ CP of Shirt = 120000 ÷ 120 = ₹1000

⇒ CP of Trouser = (660 × 100) ÷ (100 + 10)

⇒ CP of Trouser = 66000 ÷ 110 = ₹600

⇒ Ratio of CPs = CP of Shirt : CP of Trouser

⇒ Ratio = 1000 : 600

⇒ Ratio = 5 : 3

∴ The correct answer is 5 : 3.

Additional Information

Loss Formula

When an item is sold at a loss, the Cost Price is calculated as: CP = [SP × 100] ÷ [100 − Loss%]

Marked Price and Discount

The Selling Price is determined by subtracting the discount percentage from the Marked Price (MP): SP = MP × [100 − Discount%] ÷ 100

Ratio Simplification

A ratio a : b essentially represents the fraction a ÷ b. It is always expressed in its simplest equivalent form by cancelling the Highest Common Factor among the values.

Shortcut Trick

For any continuous proportion, the square of the mean proportional is equal to the product of the first and third terms.

⇒ (1 / 5a)² = 4a² × x

⇒ 1 / 25a² = 4a² × x

⇒ x = 1 / 100a⁴

∴ The correct answer is 1 / (100a⁴).

Alternate Method

Given:

First term = 4a²

Mean proportional = 1 / (5a)

Third term = x

Formula Used:

Mean Proportional = √(First Term × Third Term)

Calculations:

⇒ 1 / (5a) = √(4a² × x)

⇒ Squaring both sides, we get:

⇒ [1 / (5a)]² = 4a² × x

⇒ 1 / (25a²) = 4a² × x

⇒ x = 1 / (25a² × 4a²)

⇒ x = 1 / (100a⁴)

∴ The correct answer is 1 / (100a⁴).

Additional Information

Mean Proportional

For two given numbers a and b, the mean proportional x is calculated as x = √(a × b). It serves as the geometric mean between the two values in a continued proportion.

Third Proportional

For two numbers a and b, the third proportional c satisfies the continued proportion a : b :: b : c. It is calculated directly using the formula c = b² / a.

Fourth Proportional

For three distinct numbers a, b, and c, the fourth proportional d completes the standard proportion a : b :: c : d, yielding the formula d = (b × c) / a.

Compound Ratio

When two or more distinct ratios are multiplied together term by term (for example, a:b and c:d merge into ac:bd), the resulting combined ratio is known as a compound ratio.

Shortcut Trick

Overall ratio of Marked Price to Selling Price = 150 : 105 = 10 : 7

First discount (12.5% = 1/8) ratio = 8 : 7

Second discount ratio = Overall ratio ÷ First discount ratio = (10 / 8) : (7 / 7) = 5/4 : 1 = 5 : 4

Discount value = 5 − 4 = 1 part reduction on a base of 5 parts.

Second discount percentage = (1 / 5) × 100 = 20%

∴ The correct answer is 20%.

Alternate Method

Given:

Marked Price (MP) = ₹150

Final Selling Price (SP) = ₹105

First Discount (D₁) = 12.5%

Formula Used:

Selling Price = MP × (1 − D₁/100) × (1 − D₂/100)

Calculations:

⇒ Price after first discount = 150 − (12.5% of 150)

⇒ 12.5% in fraction = 1/8, so discount amount = 1/8 × 150 = ₹18.75

⇒ Price after first discount = 150 − 18.75 = ₹131.25

⇒ The second discount is given on ₹131.25 to make the final SP ₹105.

⇒ Value of second discount = 131.25 − 105 = ₹26.25

⇒ Second discount percentage = (26.25 / 131.25) × 100

⇒ D₂ = (1 / 5) × 100 = 20%

∴ The correct answer is 20%.

Additional Information

Equivalent Single Discount

For two successive discounts of a% and b%, the equivalent single discount is given by the formula: [a + b − (a × b) / 100]%.

Marked Price vs Cost Price

The direct relationship between Marked Price (MP) and Cost Price (CP) when profit and discount are given is: MP / CP = (100 + Profit%) / (100 − Discount%).

Buy X Get Y Free

When Y items are given free on the purchase of X items, the discount percentage is calculated on total items: Discount% = [Y / (X + Y)] × 100.

Shortcut Trick

Ratio of A and B (Present) = 3 : 4

Ratio of A and B (5 years ago) = 2 : 3

Notice that the difference between the present and past ratio parts is equal for both A (3 − 2 = 1) and B (4 − 3 = 1).

This 1 unit part corresponds to the age difference of 5 years.

Present age of A = 3 parts = 3 × 5 = 15 years.

∴ The correct answer is 15 years.

Alternate Method

Given:

Present ratio of ages of A and B = 3 : 4

Ratio of ages 5 years back = 2 : 3

Calculations:

⇒ Let the present ages of A and B be 3x and 4x respectively.

⇒ 5 years ago, age of A = 3x − 5

⇒ 5 years ago, age of B = 4x − 5

⇒ According to the question, the ratio 5 years ago was 2 : 3.

⇒ (3x − 5) / (4x − 5) = 2 / 3

⇒ By cross-multiplying, we get:

⇒ 3 × (3x − 5) = 2 × (4x − 5)

⇒ 9x − 15 = 8x − 10

⇒ 9x − 8x = 15 − 10

⇒ x = 5

⇒ Present age of A = 3x = 3 × 5 = 15 years.

∴ The correct answer is 15 years.

Additional Information

Constant Age Difference

The absolute difference between the ages of two persons remains constant throughout their lifetime. If A is 5 years older than B today, A will always be 5 years older than B, regardless of past or future scenarios.

Equalising Ratios for Age Problems

If the difference in the ratio parts of ages across two time periods is not equal for both persons, you can make them equal by multiplying the first ratio by the numerical difference of the second ratio, and vice versa.

Cross-Multiplication Method

For an initial ratio a:b, if both quantities change by exactly the same amount 'x' to form a new ratio c:d, then 1 unit part of the original ratio is given by x(c − d) ÷ (ad − bc).

Shortcut Trick

The area of an equilateral triangle can be computed directly using the square of its side length.

Area = (√3 ÷ 4) × a²

Area = (√3 ÷ 4) × 18 × 18

Area = 81√3

∴ The correct answer is 81√3 cm².

Alternate Method

Given:

Side of the equilateral triangle (a) = 18 cm

Formula Used:

Area of an equilateral triangle = (√3 ÷ 4) × a²

Calculations:

⇒ Area = (√3 ÷ 4) × (18)²

⇒ Area = (√3 ÷ 4) × 324

⇒ Area = 81√3 cm²

∴ The correct answer is 81√3 cm².

Additional Information

Altitude of an Equilateral Triangle

The height or altitude (h) bisects the base and divides the triangle into two equal right-angled triangles. Formula: h = (√3 ÷ 2) × a.

Inradius of an Equilateral Triangle

The radius (r) of the inscribed circle touches all three sides from the inside. Formula: r = a ÷ (2√3).

Circumradius of an Equilateral Triangle

The radius (R) of the circumscribed circle touches all three outer vertices. Formula: R = a ÷ √3.

Shortcut Trick

Using the negative remainder concept:

When 19 is divided by 20, the remainder is −1.

Therefore, when 19²⁰⁰ is divided by 20, the remainder will be (−1)²⁰⁰.

Since 200 is an even power, (−1)²⁰⁰ = 1.

∴ The correct answer is 1.

Alternate Method

Given:

Expression: 19²⁰⁰ ÷ 20

Formula Used:

By the Binomial Remainder Theorem, (x − a)ⁿ ÷ x leaves a remainder of (−a)ⁿ.

Calculations:

⇒ The given expression is 19²⁰⁰ ÷ 20.

⇒ We can rewrite the base 19 as (20 − 1).

⇒ So, the expression becomes (20 − 1)²⁰⁰ ÷ 20.

⇒ When expanded using the Binomial Theorem, all terms containing a multiple of 20 will be perfectly divisible by 20.

⇒ The only term left to determine the remainder is the last term: (−1)²⁰⁰.

⇒ Remainder = (−1)²⁰⁰

⇒ Since the exponent 200 is an even number, (−1)²⁰⁰ = 1.

∴ The correct answer is 1.

Additional Information

Binomial Remainder Theorem

When (ax + 1)ⁿ is divided by a, the remainder is always 1. When (ax − 1)ⁿ is divided by a, the remainder is 1 if n is even, and (a − 1) if n is odd.

Negative Remainder Concept

A negative remainder simplifies calculations. For example, 19 ÷ 20 gives a normal remainder of 19, but it can also be taken as −1 (since 19 is 1 less than 20). The final remainder must always be converted back to positive if it ends up negative.

Fermat's Little Theorem

If p is a prime number and a is an integer not divisible by p, then a^p−1 divided by p always leaves a remainder of 1.

Shortcut Trick

By using the mixture and alligation rule directly on the interest rates:

Ratio of investments = (6% − 4.7%) : (4.7% − 4%) = 1.3 : 0.7 = 13 : 7

Total parts = 13 + 7 = 20 parts ≡ ₹2,00,000

Amount in 4% bank (13 parts) = (13 ÷ 20) × 2,00,000 = ₹1,30,000

Amount in 6% bank (7 parts) = (7 ÷ 20) × 2,00,000 = ₹70,000

∴ The correct answer is ₹1,30,000 and ₹70,000.

Alternate Method

Given:

Total principal = ₹2,00,000

Interest rate of Bank 1 = 4% p.a.

Interest rate of Bank 2 = 6% p.a.

Overall equivalent rate = 4.7% p.a. for 1 year

Formula Used:

Simple Interest = (P × R × T) ÷ 100

Calculations:

⇒ Total interest earned at 4.7% = (2,00,000 × 4.7 × 1) ÷ 100 = ₹9,400

⇒ Let the principal invested in Bank 1 be ₹x. Thus, the principal in Bank 2 will be ₹(2,00,000 − x).

⇒ Total Interest = Interest from Bank 1 + Interest from Bank 2

⇒ [(x × 4 × 1) ÷ 100] + [((2,00,000 − x) × 6 × 1) ÷ 100] = 9400

⇒ 4x + 12,00,000 − 6x = 9,40,000

⇒ 2x = 12,00,000 − 9,40,000

⇒ 2x = 2,60,000

⇒ x = ₹1,30,000

⇒ Investment in Bank 2 = 2,00,000 − 1,30,000 = ₹70,000

∴ The correct answer is ₹1,30,000 and ₹70,000.

Additional Information

Mixtures and Alligations

It is an analytical method used to find the ratio in which two quantities must be mixed to obtain a desired average value. Rule: (Quantity of Cheaper) ÷ (Quantity of Dearer) = (Price of Dearer − Average Price) ÷ (Average Price − Price of Cheaper).

Simple Interest (SI)

It is the interest calculated only on the original principal amount. The principal remains constant throughout the tenure. Formula: SI = (P × R × T) ÷ 100.

Overall / Effective Interest Rate

When multiple investments are made at different interest rates, the overall interest rate is the weighted average of the individual rates based on the amount of capital invested in each scheme.

Shortcut Trick

Put A = 30° in the given expression to quickly find the answer.

⇒ sin 30° / (1 + cos 30°) + cot 30°

⇒ (1 / 2) / (1 + √3 / 2) + √3

⇒ 1 / (2 + √3) + √3 = (2 − √3) + √3 = 2

Checking the given options for A = 30°, we get cosec 30° = 2.

∴ The correct answer is cosec A.

Alternate Method

Given:

Expression = sin A / (1 + cos A) + cot A

Formula Used:

⇒ cot A = cos A / sin A

⇒ sin² A + cos² A = 1

Calculations:

⇒ Expression = sin A / (1 + cos A) + (cos A / sin A)

⇒ Taking the LCM of the denominators:

⇒ Expression = [sin² A + cos A × (1 + cos A)] / [sin A × (1 + cos A)]

⇒ Expression = [sin² A + cos A + cos² A] / [sin A × (1 + cos A)]

⇒ Grouping the squared terms together:

⇒ Expression = [(sin² A + cos² A) + cos A] / [sin A × (1 + cos A)]

⇒ Expression = (1 + cos A) / [sin A × (1 + cos A)]

⇒ Expression = 1 / sin A

⇒ Expression = cosec A

∴ The correct answer is cosec A.

Additional Information

Basic Trigonometric Identities

• sin² θ + cos² θ = 1
• 1 + tan² θ = sec² θ
• 1 + cot² θ = cosec² θ

Reciprocal Relations

• sin θ × cosec θ = 1
• cos θ × sec θ = 1
• tan θ × cot θ = 1

Half Angle Concept

The term sin A / (1 + cos A) directly simplifies to tan(A / 2). Understanding these relations can help save time during exam simplification problems.

Shortcut Trick

Rationalize the given expression inside the square root:

⇒ √[(1 − sinθ) / (1 + sinθ)] = √[(1 − sinθ)² / (1 − sin²θ)]

⇒ (1 − sinθ) / cosθ = secθ − tanθ

Given tanθ = 8/15, from right-angled triplet (8, 15, 17), secθ = Hypotenuse / Base = 17/15.

⇒ secθ − tanθ = 17/15 − 8/15 = 9/15 = 3/5

∴ The correct answer is 3/5.

Alternate Method

Given:

tanθ = 8 / 15

We know that tanθ = Perpendicular (P) / Base (B)

Let P = 8 and B = 15.

By Pythagoras Theorem:

⇒ Hypotenuse (H) = √(P² + B²)

⇒ H = √(8² + 15²) = √(64 + 225) = √289 = 17

⇒ sinθ = P / H = 8 / 17

Evaluating the Expression:

⇒ √(1 − sinθ) / √(1 + sinθ)

Substitute the value of sinθ:

⇒ √(1 − 8/17) / √(1 + 8/17)

⇒ √((17 − 8)/17) / √((17 + 8)/17)

⇒ √(9/17) / √(25/17)

⇒ √(9 / 25) = 3 / 5

∴ The correct answer is 3/5.

Additional Information

Pythagorean Triplets

Common right-angled triangle side ratios (P, B, H) include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25). Memorizing these saves calculation time in exams.

Basic Trigonometric Ratios

For a right-angled triangle: sinθ = Perpendicular / Hypotenuse, cosθ = Base / Hypotenuse, and tanθ = Perpendicular / Base.

Important Trigonometric Identities

sin²θ + cos²θ = 1 | sec²θ − tan²θ = 1 | cosec²θ − cot²θ = 1

Shortcut Trick

The Curved Surface Area of a cone is given by πrl.

Since π contains 22 and r = 9, the answer must be a multiple of both 11 and 9.

Checking the options for divisibility by 9 (sum of digits must be a multiple of 9):

Only 1386 has a digit sum of 18 (1 + 3 + 8 + 6 = 18), which is perfectly divisible by 9.

∴ The correct answer is 1386 cm².

Alternate Method

Given:

Radius of the base (r) = 9 cm

Slant height (l) = 49 cm

Formula Used:

Curved Surface Area (CSA) of a right circular cone = πrl

Calculations:

⇒ CSA = (22/7) × 9 × 49

⇒ CSA = 22 × 9 × 7

⇒ CSA = 198 × 7

⇒ CSA = 1386 cm²

∴ The correct answer is 1386 cm².

Additional Information

Total Surface Area of a Cone

The total surface area includes the curved surface area and the area of the circular base. Formula: TSA = πr(l + r).

Volume of a Cone

The volume represents the total capacity of the cone. Formula: V = (1/3)πr²h, where h is the perpendicular height of the cone.

Slant Height of a Cone

The slant height (l), radius (r), and height (h) form a right-angled triangle. Formula: l = √(r² + h²).

Shortcut Trick

According to the fundamental properties of circles, congruent arcs always subtend equal chords.

Since the arc PAQ is congruent to the arc RBS, their corresponding chords PQ and RS must be exactly equal in length.

⇒ PQ = RS

Therefore, the ratio of the lengths of chord PQ to chord RS is 1 : 1.

∴ The correct answer is 1 : 1.

Alternate Method

Given:

Arcs PAQ and RBS of a circle are congruent.

Formula Used:

Congruent arcs of a circle subtend equal angles at the center of the circle.

Two triangles are congruent by the SAS criterion if two sides and the included angle are equal.

O

P Q A

R S B

Ratio of Chords PQ : RS = 1 : 1

Calculations:

⇒ Let the circle have center O and radius r.

⇒ Since arc PAQ ≅ arc RBS, the angles they subtend at the center are strictly equal.

⇒ ∠POQ = ∠ROS

⇒ In ΔPOQ and ΔROS:

⇒ OP = OR (Radii of the same circle)

⇒ OQ = OS (Radii of the same circle)

⇒ ∠POQ = ∠ROS (As proved above)

⇒ By SAS congruence criterion, ΔPOQ ≅ ΔROS

⇒ Because corresponding parts of congruent triangles are equal (CPCTC), PQ = RS.

⇒ Ratio of PQ : RS = 1 : 1

∴ The correct answer is 1 : 1.

Additional Information

Equal Chords Property

Equal chords of a circle subtend equal angles at the center. Conversely, if angles subtended by chords are equal, the chords are equal.

Perpendicular from Center

A perpendicular line drawn from the center of a circle to a chord bisects the chord into two equal halves.

Equidistant Chords

Chords of equal lengths in a circle or in congruent circles are at an equal perpendicular distance from the center.

Congruent Arcs

If two arcs of a circle are equal in measure, their corresponding minor or major chords are identically equal in length.

Shortcut Trick

Winner's vote share = 88%

Loser's vote share = 100% − 88% = 12%

Majority (Difference) = 88% − 12% = 76%

76% of total votes = 684

Total votes = (684 ÷ 76) × 100 = 900

∴ The correct answer is 900 votes.

Alternate Method

Given:

Winning candidate's vote share = 88%

Majority (difference in votes) = 684

Formula Used:

Total Votes = 100%

Losing Candidate % = 100% − Winning Candidate %

Majority = Winning % − Losing %

Calculations:

⇒ Let the total number of polled votes be x.

⇒ Votes secured by the winning candidate = 0.88x

⇒ Votes secured by the losing candidate = x − 0.88x = 0.12x

⇒ Difference in their votes (Majority) = 0.88x − 0.12x = 0.76x

⇒ According to the problem, the majority is 684 votes.

⇒ 0.76x = 684

⇒ x = 684 ÷ 0.76

⇒ x = (684 × 100) ÷ 76

⇒ x = 9 × 100

⇒ x = 900

∴ The correct answer is 900 votes.

Additional Information

Valid and Invalid Votes

If some votes are declared invalid, then Valid Votes = Total Votes − Invalid Votes. The percentages of the winner and loser are typically calculated based on valid votes, unless specified otherwise.

Multiple Candidates Election

When there are more than two candidates, the majority is calculated as the difference between the votes of the winner and the runner-up (the candidate with the second-highest votes).

Percentage Base Method

Finding the total quantity when a part is given is a direct application of base percentage. Total Value = Given Value ÷ Percentage Fraction.

Given:

We are asked to find the largest fraction among: $\frac{2}{3}, \frac{6}{53}, \frac{79}{90}, \frac{33}{44}$

Approach:

To compare these fractions, we will either:

Convert them to decimal form by dividing the numerator by the denominator, or

Find a common denominator to compare them directly.

Calculations:

1. For $\frac{2}{3}$

$\frac{2}{3} \approx 0.6667$

2. For $\frac{6}{53}$

$\frac{6}{53} \approx 0.1132$

3. For $\frac{79}{90}$

$\frac{79}{90} \approx 0.8778$

4. For $\frac{33}{44}$

$\frac{33}{44} = \frac{3}{4} = 0.75$

Conclusion:

The largest fraction is $\frac{79}{90}$ as it has the highest value of approximately 0.8778.

Shortcut Trick

In an isosceles triangle, the altitude drawn to the unequal side bisects the base.

This splits the triangle into two identical right-angled triangles.

Using the standard Pythagorean triplet (8, 15, 17), if the hypotenuse = 17 and perpendicular = 15, then the base (QT) = 8.

Since the altitude bisects the base, the total length of QR is twice the length of QT.

⇒ QR = 2 × 8 = 16 cm.

∴ The correct answer is 16 cm.

Alternate Method

Given:

⇒ In ΔPQR, PQ = PR = 17 cm.

⇒ PT ⊥ QR and PT = 15 cm.

P Q R T 17 cm 17 cm 15 cm x x

Concept Used:

⇒ The altitude drawn from the vertex of an isosceles triangle to its base exactly bisects the base.

⇒ ∴ QT = TR = QR ÷ 2.

Calculation:

⇒ In right-angled ΔPQT, apply Pythagoras theorem:

⇒ PQ² = PT² + QT²

⇒ 17² = 15² + QT²

⇒ 289 = 225 + QT²

⇒ QT² = 289 − 225 = 64

⇒ QT = √64 = 8 cm.

⇒ Since QT is exactly half of the base QR:

⇒ QR = 2 × QT

⇒ QR = 2 × 8 = 16 cm.

∴ The correct answer is 16 cm.

Additional Information

Properties of Isosceles Triangles

In an isosceles triangle, the two angles opposite to the equal sides are also equal. Furthermore, the altitude drawn to the unequal side functions as an angle bisector and median simultaneously.

Pythagoras Theorem

Applicable only to right-angled triangles, it dictates that the square of the longest side (hypotenuse) equals the sum of the squares of the other two sides: a² + b² = c².

Common Pythagorean Triplets

Memorising frequent integer combinations dramatically reduces exam calculation time. Core examples include (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), and (9, 40, 41).

Shortcut Trick

For any algebraic equation of the form x + 1/x = 2, the value of x is always equal to 1.

Substitute x = 1 directly into the target expression.

⇒ x²⁷²³ + 1 / x³⁸³⁶

⇒ (1)²⁷²³ + 1 / (1)³⁸³⁶

⇒ 1 + 1 = 2

∴ The correct answer is 2.

Alternate Method

Given:

x + 1/x = 2

Calculation:

⇒ (x² + 1) / x = 2

⇒ x² + 1 = 2x

⇒ x² − 2x + 1 = 0

⇒ (x − 1)² = 0

⇒ x − 1 = 0

⇒ x = 1

Now, substitute x = 1 in the given expression:

⇒ x²⁷²³ + 1 / x³⁸³⁶

⇒ (1)²⁷²³ + 1 / (1)³⁸³⁶

⇒ 1 + 1/1

⇒ 1 + 1

⇒ 2

∴ The correct answer is 2.

Additional Information

Algebraic Identity Case 1

If x + 1/x = −2, then the value of x is always −1. Here, x raised to any even power is 1, and x raised to any odd power is −1.

Algebraic Identity Case 2

If x + 1/x = 1, then x³ = −1. This is derived from multiplying the equation by x and using the sum of cubes formula.

Algebraic Identity Case 3

If x + 1/x = −1, then x³ = 1. This rule is extremely useful for reducing high powers of x by substituting x³ = 1.

Algebraic Identity Case 4

If x + 1/x = √3 or x² + 1/x² = 1, then x⁶ = −1.

Shortcut Trick

The given expression is directly based on the standard algebraic identity for the difference of cubes.

Form: (a³ − b³) ÷ (a² + ab + b²) = a − b

Here, a = 7.3 and b = 4.7.

Result = 7.3 − 4.7 = 2.6

∴ The correct answer is 2.6.

Alternate Method

Given:

Expression: [ (7.3)³ − (4.7)³ ] ÷ [ (7.3)² + 7.3 × 4.7 + (4.7)² ]

Formula Used:

a³ − b³ = (a − b)(a² + ab + b²)

Calculation:

⇒ Let a = 7.3 and b = 4.7

⇒ The given expression takes the form: (a³ − b³) ÷ (a² + ab + b²)

⇒ Substitute the formula for a³ − b³ into the numerator:

⇒ Expression = [ (a − b)(a² + ab + b²) ] ÷ (a² + ab + b²)

⇒ The common term (a² + ab + b²) in the numerator and denominator cancels out.

⇒ Expression = a − b

⇒ Expression = 7.3 − 4.7

⇒ Expression = 2.6

∴ The correct answer is 2.6.

Additional Information

Sum of Cubes Identity

a³ + b³ = (a + b)(a² − ab + b²). If the numerator has a '+' sign, the simplified result becomes (a + b).

Difference of Squares

a² − b² = (a − b)(a + b). This is extensively used to simplify fractional arithmetic expressions.

Perfect Square Identities

(a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b².

Given:

Marked price of the book (MP) = ₹270

The marked price is 20% more than the cost price (CP).

Percentage cost of paper = 25% of the total cost price.

Formula used:

We are given that the marked price is 20% more than the cost price. This can be represented as:

$\text{MP} = \text{CP} \times (1 + \frac{20}{100}) = \text{CP} \times 1.20$

We can calculate the cost price (CP) from the given marked price (MP):

$\text{CP} = \frac{\text{MP}}{1.20}$

Calculations:

Using the above formula, we find the cost price:

$\text{CP} = \frac{270}{1.20} = 225$

Next, we calculate the cost of paper, which is 25% of the cost price:

$\text{Cost of paper} = \text{CP} \times \frac{25}{100} = 225 \times 0.25 = 56.25$

Conclusion:

The cost of paper to publish a single copy of the book is ₹56.25.

Given:

The degrees representing each sport in the pie chart are:

Soccer = 35°

Badminton = 45°

Tennis = 65°

Rugby = 25°

Cricket = 100°

Swimming = 60°

Hockey = 30°

₹42,000 were spent on Soccer, which corresponds to 35°.

We need to find the total expenditure and the percentage comparison of money spent on Badminton, Tennis, and Rugby versus Cricket, Swimming, and Hockey.

Formula used:

The total expenditure can be found by using the ratio of expenditure on Soccer (35°) to the total degrees (360°):

$\frac{42,000}{35} = \frac{T}{360}$

Calculations:

From the equation, we can find the total expenditure (T):

$T = \frac{42,000 \times 360}{35} = 432,000$

So, the total expenditure is ₹432,000.

Now, let's calculate the money spent on Badminton, Tennis, and Rugby:

The degrees for Badminton, Tennis, and Rugby = 45° + 65° + 25° = 135°.

So, the money spent on these sports is:

$\frac{135}{360} \times 432,000 = 162,000$

Thus, ₹162,000 is spent on Badminton, Tennis, and Rugby.

Next, we calculate the money spent on Cricket, Swimming, and Hockey:

The degrees for Cricket, Swimming, and Hockey = 100° + 60° + 30° = 190°.

So, the money spent on these sports is:

$\frac{190}{360} \times 432,000 = 228,000$

Thus, ₹228,000 is spent on Cricket, Swimming, and Hockey.

Now, we need to find the percentage difference between the money spent on Badminton, Tennis, and Rugby and the money spent on Cricket, Swimming, and Hockey:

The percentage difference is given by:

$\frac{162,000 - 228,000}{228,000} \times 100 = \frac{-66,000}{228,000} \times 100 = -28.95\%$

The money spent on Badminton, Tennis, and Rugby is 28.95% less than the money spent on Cricket, Swimming, and Hockey.

The correct answer is Option 4.

Key Points

• The structure "lest + subject + should" is used to express a fear or a precaution, and "should" is the correct form in this context.
• The phrase "lest he should be weak" correctly follows this construction, conveying that Rohan took actions to prevent weakness.
• The other options (such as "lest he would be weak" or "lest he would not be weak") are incorrect because they do not align with the grammatical structure of "lest".

Therefore, the correct answer is Option 4.

Correct Sentence: "Rohan exercised and ate a lot, lest he should be weak."

Additional Information

• Option 1: "lest he should not weak" – The phrase "should not weak" is grammatically incorrect because "weak" is an adjective, not a verb.
• Option 2: "lest he would be weak" – The use of "would" is incorrect after "lest"; "should" is the appropriate modal verb in this context.
• Option 3: "lest he would not be weak" – "would" is unnecessary and incorrect here as "should" is used after "lest".

The correct answer is: Option 4 i.e. Serenity.

Key Points

• The word "annoyance" means a feeling of irritation or displeasure. (चिड़चिड़ापन)
• Example: His constant interruptions were a source of annoyance for everyone.
• "Serenity" refers to a state of being calm, peaceful, and untroubled. (शांति)
• Example: The serenity of the mountain landscape brought a sense of peace to her mind.
• Hence, we can infer that the opposite of 'annoyance' is 'serenity'.

Therefore, the correct answer is: Option 4 i.e. Serenity.

Additional Information

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

• Susceptibility (संवेदनशीलता): The state of being likely to be influenced or harmed by something.
• Example: His susceptibility to colds increased during the winter months.
• Absurdity (मूर्खता): The quality or state of being ridiculous or wildly unreasonable.
• Example: The absurdity of his argument was evident to everyone in the room.
• Gnawer (कुतरने वाला): An animal that gnaws, typically a rodent.
• Example: The gnawer had chewed through the wooden beams in the attic.

The correct answer is 'perennial'.

Key Points

• The word "perennial" means lasting or existing for a long or apparently infinite time; enduring or continually recurring. (अनवरत)
• A river that flows continuously throughout the year regardless of the season is considered perennial.
• The other options: "biennial," "biannual," and "annual" do not fit the context as appropriately.