SSC CGL 2024 Tier-I Official Paper (Held On: 17 Sept, 2024 Shift 3) (Previous Year Paper)

The logic followed here is:

Given Sequence: _ P _ J _ P _ P _ _ P _

Now, putting the letters in the blank spaces given in the option one by one:

Option 1) JPPJPJP → J P P / J P P / J P P / J P P

Here, 'J P P' is the pattern.

Option 2) PPJJJJJ → P P P / J J P / J P J / J P J

Option 3) JPJPJPJ → J P P / J J P / P P J / P P J

Option 4) JJPPPPJ → J P J / J P P / P P P / P P J

Hence, the correct answer is "JPPJPJP".

The logic followed here is:

Top-right corner letter shifts in bottom-left corner and flipped horizontally.

And remaining letter is replaced by a new letter.

Hence, the correct answer is "Option 4".

According to the given information:

Thus farmers are there who have all four assets = 16

Hence, the correct answer is "16".

The logic followed here is:

Option 1) NPRS

Option 2) KMOP

Option 3) UVXW

Option 4) EGIJ

Hence, the correct answer is "UVXW".

Counting the triangles:

Thus, triangles are there in the given figure = 12

Hence, the correct answer is "12".

The correct mirror image of the given figure is:

Hence, the correct answer is "Option 2".

Family Chart:

Decoding the Symbols:

Given: J + K ÷ L − T × R

It means: J is the mother of K, K is the daughter of L, L is the brother of T and T is the sister of R.

S, J is the Wife of L.

Hence, the correct answer is "Wife".

BODMAS Table: 

After decoding signs:

Given equation: 6 ÷ 8 – 91 × 13 + 30 = ?

After interchanging the signs then:

⇒ 6 × 8 + 91 ÷ 13 - 30 = ?

⇒ ? = 6 × 8 + 7 - 30

⇒ ? = 48 + 7 - 30

⇒ ? = 55 - 30 = 25

Hence, the correct answer is "25".

The logic followed here is:

From figure 1 to 2, and figure 3 to 4:

Top-right and bottom-left symbols are interchanging their position.

Diagonal symbols are changing their position.

From figure 2 to 3, and figure 4 to 5:

All 4 corner symbols are shifting their position in clockwise direction.

Middle symbols are changing their position.

Hence, the correct answer is "Option 2".

The logic followed here is:

Hence, the correct answer is "PHN".

The logic followed here is:

Logic: Opposites.

Here,

Meek is the opposite of Bold.

Similarly,

Spend is the opposite of Hoard.

Hence, the correct answer is "Option 4".

BODMAS Table: 

After decoding signs:

| Signs | Decoded signs |
| --- | --- |
| + | - |
| - | + |
| × | ÷ |
| ÷ | × |

 

Given equation: 99 ÷ 41 + 208 × 2 – 35 = ?

After interchanging the signs then:

⇒ 99 × 41 - **208 ÷ 2** + 35 = ?

⇒ ? = **99 ×** **41** - 104 + 35

⇒ ? = **4059** - 104 **+ 35**

⇒ ? = **4094 - 104** = **3990**

Hence, the correct answer is "3990".

The correct mirror image of the given figure is:

Hence, the correct answer is "Option 4".

The logic followed here is:

Logic: (1^st number + 3^rd number) ÷ 2 = 2^nd number.

Now,

For, (54, 65, 76)

⇒ (54 + 76) ÷ 2 = 130 ÷ 2 = 65.

For, (48, 59, 70)

⇒ (48 + 70) ÷ 2 = 118 ÷ 2 = 59.

Similarly, For (38, 49, 60)

⇒ (38 + 60) ÷ 2 = 98 ÷ 2 = 49.

Hence, the correct answer is "Option 2".

The logic followed here is:

'OASIS' is written as 'RJTBN'

'OBESE' is written as 'DTFCN'

Similarly, 'OCCUR’ will be written as:

Hence, the correct answer is "QVDDN".

Given Statements:

Some circles are rectangles.

Some rectangles are squares.

All squares are triangles.

The least possible Venn diagram is as follows:

Conclusions:

I. No circle is a triangle. - False (because there is no direct relation given between circle and triangle, so it can be possible but not definite)

II. Some rectangles are triangles. - True (because Some rectangles are squares and All squares are triangles. So, Some rectangles are triangles it is definite)

III. All rectangles are triangles. - False (because Some rectangles are squares and All squares are triangles. So, All rectangles are triangles it is not definite)

Hence, the correct answer is "Only conclusion II follows".

Given word: 'MUSICAL'

After arranged in alphabetical order:

After arranged in alphabetical order,

Third alphabet from the left end is: I,

Third alphabet from the right end is: M,

Now,

there are 3 letters between I and M.

Hence, the correct answer is "Option 1".

The logic followed here is:

Hence, the correct answer is "UH 73".

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

1)  86 – 41 – 38 → (86 + 38) / 3 = 124 / 3 = 41.3

2)  118 – 93 – 161 → (118 + 161) / 3 = 279 / 3 = 93

3)  33 – 15 – 12 → (33 + 12) / 3 = 45 / 3 = 15

4)  52 – 36 – 56 → (52 + 56) / 3 = 108 / 3 = 36

Hence, the correct answer is "86 – 41 – 38".

The logic followed here is:

Logic: (1^st number × 3^rd number) × 2 = 2^nd number.

For example:

For (23, 230, 5):

⇒ (23 × 5) × 2

⇒ 115 × 2 = 230

For, (17, 272, 8)

⇒ (17 × 8) × 2

⇒ 136 × 2 = 272

Similarly, for (19, 304, 8):

⇒ (19 × 8) × 2

⇒ 152 × 2 = 304

Hence, the correct answer is "Option 4".

The logic followed here is:

Logic: (1^st number × 3) + 16 = 2^nd number.

Now,

For, 17 : 67

⇒ (17 × 3) + 16 = 51 + 16 = 67.

For, 34 : 118

⇒ (34 × 3) + 16 = 102 + 16 = 118.

Similarly, for 34 : 118

⇒ (34 × 3) + 16 = 102 + 16 = 118.

Hence, the correct answer is "Option 2".

The logic followed here is:

Hence, the correct answer is "1064".

The logic followed here is:

‘RUG’ is coded as ‘EPS’

‘SIT’ is coded as ‘RQG’

Similarly, ‘COP’ will be coded as:

Hence, the correct answer is "NAM".

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

So, From Dice 2 and Dice 3,

From both dices, Nose and Leg are common. So, remaining surface will be opposite to each other.

Opposites pairs:

Nose → Eyes

Leg → Ear

Mouth →  Hand

So, 'Mouth' will be the opposite face of 'Hand'.

Hence, the correct answer is "Mouth"

The paper will look when unfolded:

Hence, the correct answer is "Option 1".

The correct answer is 7.9 million.

Key Points

• According to the Census of India 2011, 'Other Religions and Persuasions (ORP)' comprise 7.9 million people.
• The ORP category includes those who do not identify with the six major religions listed in the census data: Hinduism, Islam, Christianity, Sikhism, Buddhism, and Jainism.
• These religions include smaller faiths and tribal religions practiced in various regions across India.
• This category helps in recognizing the diversity of beliefs and practices present in the country, reflecting the pluralistic nature of Indian society.

Additional Information

• Census of India
• The Census of India is a decennial exercise conducted by the Office of the Registrar General and Census Commissioner, India under the Ministry of Home Affairs.
• The first census in India was conducted in 1872, but the first complete and synchronous census was conducted in 1881.
• The census collects data on population size, distribution, and various socio-economic parameters such as literacy, employment, and housing.
• The latest census was conducted in 2011, and the next one was scheduled for 2021 but was postponed due to the COVID-19 pandemic.
• Religion Data in Census
• The religion data in the Indian Census includes information on various religious communities in the country.
• The major religions accounted for in the census are Hinduism, Islam, Christianity, Sikhism, Buddhism, and Jainism.
• In addition to these, the census also recognizes 'Other Religions and Persuasions (ORP)' and 'Religion not stated' categories.
• This data helps in understanding the religious composition and diversity of the Indian population.
• Importance of Census Data
• Census data is crucial for planning and policy-making by the government.
• It helps in the allocation of resources, development of infrastructure, and implementation of various welfare schemes.
• It also provides valuable insights for researchers and demographers studying population trends and socio-economic changes.
• Accurate census data is essential for ensuring effective governance and equitable distribution of services.

The correct answer is 6^th August 2021.

Key Points

• The Rajiv Gandhi Khel Ratna Award was renamed as the Major Dhyan Chand Khel Ratna Award on 6^th August 2021.
• This decision was taken by the Government of India to honor the legendary Indian hockey player Major Dhyan Chand.
• Major Dhyan Chand is widely regarded as one of the greatest field hockey players in history and led India to win gold medals in the Olympics.
• The renaming was announced by Prime Minister Narendra Modi, emphasizing the need to recognize the contributions of sportspersons who have brought glory to the nation.

Additional Information

• Rajiv Gandhi Khel Ratna Award:
• This award was the highest sporting honor of India, instituted in 1991-1992.
• It was named after Rajiv Gandhi, the former Prime Minister of India.
• The award is given annually by the Ministry of Youth Affairs and Sports, Government of India.
• The award comprises a medallion, a certificate, and a cash prize.
• It was given for outstanding performance in the field of sports over a period of four years.
• Major Dhyan Chand:
• Major Dhyan Chand was an Indian field hockey player, who is considered one of the greatest field hockey players of all time.
• He was part of the Indian team that won gold medals in the 1928, 1932, and 1936 Olympic Games.
• Known as "The Wizard" for his superb ball control, Dhyan Chand is a legendary figure in Indian and world hockey.
• His birthday, 29^th August, is celebrated as National Sports Day in India.
• Renaming of Awards:
• Renaming awards is a common practice to honor individuals who have made significant contributions in a particular field.
• It reflects the values and priorities of a nation at a given time.
• The renaming of the Rajiv Gandhi Khel Ratna Award to Major Dhyan Chand Khel Ratna Award highlights the importance of sports and the recognition of sports legends in India.

The correct answer is CP Radhakrishnan.

Key Points

• CP Radhakrishnan is an Indian politician and member of the Bharatiya Janata Party (BJP).
• He has served as a Member of Parliament from Coimbatore.
• He was appointed as the Governor of Jharkhand in February 2023.
• CP Radhakrishnan has held several important positions within the BJP and has been actively involved in various social and political activities.

Additional Information

• Governorship in India
• The Governor is the head of a state in India and is appointed by the President of India.
• The role of the Governor is largely ceremonial, but they have certain discretionary powers that can be significant in times of political instability.
• Governors are appointed for a term of five years but can remain in office at the pleasure of the President.
• The Governor's primary responsibility is to oversee the functioning of the state government and ensure that it operates within the framework of the Constitution.
• Bharatiya Janata Party (BJP)
• The BJP is one of the two major political parties in India and is known for its right-wing politics and nationalist ideologies.
• Founded in 1980, the BJP has grown to become the largest political party in terms of representation in the national parliament and state assemblies.
• The party's agenda includes economic liberalism, national security, and cultural nationalism.
• The BJP has produced several Prime Ministers, including the current Prime Minister, Narendra Modi.

The correct answer is Bangalore.

Key Points

• The first women's international test cricket match hosted in India took place in Bangalore in 1976.
• This match was a significant milestone in the history of women's cricket in India.
• It marked the beginning of international women's cricket matches being played on Indian soil.
• Bangalore, known for its rich cricketing culture, provided a great platform for this historic event.

Additional Information

• Women's Test Cricket
• Women's Test cricket is the longest form of women's cricket and is considered the ultimate test of a player's endurance and skill.
• The first women's Test match was played between England and Australia in December 1934.
• Test matches are played over four days, unlike men's Test matches which are played over five days.
• Test cricket is governed by the International Cricket Council (ICC), which is the global governing body for the sport of cricket.
• History of Women's Cricket in India
• Women's cricket in India has a rich history, with the first recorded match being played in the early 1970s.
• The Women's Cricket Association of India (WCAI) was formed in 1973 to promote and develop women's cricket in the country.
• The Indian women's cricket team played its first Test match in 1976 against the West Indies.
• Over the years, women's cricket in India has grown significantly, with the team achieving considerable success in international tournaments.
• Bangalore
• Bangalore, now known as Bengaluru, is the capital city of the Indian state of Karnataka.
• The city is known for its vibrant culture, pleasant weather, and rich sporting history.
• Bangalore has been a prominent center for cricket in India, producing many renowned cricketers and hosting numerous international matches.
• The city is home to the M. Chinnaswamy Stadium, a premier cricket venue that has hosted several memorable matches over the years.

The correct answer is Raja Rammohun Roy.

Key Points

• Raja Rammohun Roy was a prominent social and educational reformer in India during the early 19^th century.
• He founded the Atmiya Sabha in 1814, which later became the Brahmo Samaj, to promote social and religious reforms.
• The Atmiya Sabha was established to fight against the social evils such as caste discrimination, and to promote monotheism and rational thinking.
• Rammohun Roy is often called the "Father of the Indian Renaissance" for his efforts to modernize Indian society.

Additional Information

• Raja Rammohun Roy:
• Born on May 22, 1772, in Radhanagar, Bengal, Rammohun Roy was a pioneer in the Indian social reform movement.
• He opposed practices such as sati (the burning of widows), child marriage, and caste discrimination, advocating for women's rights and education.
• Rammohun Roy was also instrumental in the establishment of the Hindu College in Calcutta in 1817, which later became Presidency College.
• His efforts in advocating for the abolition of sati led to the regulation banning the practice in 1829 by the British Governor-General, Lord William Bentinck.
• Atmiya Sabha:
• Founded in 1814, the Atmiya Sabha was an organization that aimed to discuss and propagate the ideas of monotheism and rationality.
• The Sabha's meetings involved discussions on the Upanishads, Vedas, and other scriptures to interpret them in a way that promoted a monotheistic view.
• It served as a platform for intellectuals to debate and discuss various social issues and the need for reforms in Hindu society.
• The Atmiya Sabha's activities laid the foundation for the later establishment of the Brahmo Samaj in 1828, which continued the reformist agenda.
• Brahmo Samaj:
• Founded by Raja Rammohun Roy in 1828, the Brahmo Samaj was a socio-religious reform movement that aimed to purify Hinduism and promote monotheism.
• The Brahmo Samaj sought to eliminate social evils such as caste discrimination, child marriage, and the dowry system.
• It emphasized the importance of education, especially for women, and supported widow remarriage.
• The movement had a significant impact on the Indian society and influenced many later reformers and movements.
• Social Reform in India:
• The early 19^th century saw the emergence of several social reform movements in India aimed at addressing various social issues and promoting modernization.
• Reformers like Raja Rammohun Roy, Ishwar Chandra Vidyasagar, and Swami Vivekananda played crucial roles in advocating for change and improving the status of women and marginalized communities.
• These movements laid the groundwork for future social and political developments in India, including the struggle for independence.
• The reforms also contributed to the development of a more progressive and inclusive society in India.

The correct answer is Mirzapur.

Key Points

• The Kajari Mahotsav is a famous festival primarily celebrated in Mirzapur district of Uttar Pradesh.
• This festival is named after the Kajari folk song and is celebrated in the month of August.
• The festival involves singing competitions, cultural programs, and various other traditional activities.
• Kajari songs are a unique form of traditional music that originated in the Mirzapur region.

Additional Information

• Kajari Songs:
• Kajari is a form of folk song originating from the Mirzapur region in Uttar Pradesh and is sung during the monsoon season.
• It is characterized by its melancholic tunes and themes that often revolve around the pain of separation and the longing for the beloved.
• Mirzapur District:
• Mirzapur is a city in the state of Uttar Pradesh, India, and is known for its scenic beauty, including waterfalls and temples.
• The district is culturally rich and known for various traditional festivals and celebrations.
• Other Festivals in Uttar Pradesh:
• Holi: The festival of colors celebrated with enthusiasm and vibrant colors across the state.
• Diwali: The festival of lights celebrated with great fervor, involving fireworks, decorations, and traditional rituals.
• Kumbh Mela: A major pilgrimage and festival held every twelve years at four river-bank pilgrimage sites in India, including Prayagraj in Uttar Pradesh.

The correct answer is Madhya Pradesh.

Key Points

• Matki dance is a traditional folk dance form from Madhya Pradesh, India.
• This dance is performed primarily by the womenfolk of the Malwa region of Madhya Pradesh.
• The dance is characterized by the intricate balancing of earthen pots (Matkis) on the head.
• It is generally performed on festive occasions and social gatherings.

Additional Information

• Folk Dances of Other States:
• Punjab: Punjab is famous for its energetic dance forms like Bhangra and Gidda. Bhangra is a vibrant dance performed by men, while Gidda is performed by women.
• Gujarat: Gujarat is renowned for its colorful dance forms such as Garba and Dandiya Raas. These dances are performed during the Navratri festival.
• Maharashtra: Lavani is a popular folk dance of Maharashtra known for its powerful rhythm and sensuous movements. It is performed to the beats of traditional music.
• Matki Dance:
• Matki dance involves women dressing in traditional attire and performing graceful movements while balancing earthen pots on their heads.
• The performance often includes multiple participants and may involve the stacking of pots to showcase skill and balance.
• The dance is a celebration of femininity, grace, and the cultural heritage of the Malwa region.
• It is an integral part of various cultural and social events, including weddings and festivals.

The correct answer is Teak.

Key Points

• Teak is a large deciduous tree that is native to tropical regions of South and Southeast Asia.
• It is known for its high-quality wood, which is durable, water-resistant, and resistant to pests and diseases.
• Teak wood is commonly used in the construction of furniture, boats, and buildings due to its strength and durability.
• The tree is also valued for its ability to regenerate from stumps and roots, making it a sustainable timber resource when managed properly.

Additional Information

• Tropical Deciduous Forests
• These forests, also known as monsoon forests, are found in regions that experience a distinct wet and dry season.They are characterized by trees that shed their leaves during the dry season to conserve water.
• Common species found in tropical deciduous forests include Teak, Sal, Sandalwood, and Bamboo.
• These forests support a diverse range of flora and fauna, providing habitat for numerous species of plants, animals, and insects.
• Other Tree Species in option: 
• Palms:
• Palms are more commonly associated with tropical rainforests and coastal regions.
• They are characterized by their large, compound leaves and unbranched trunks.
• Mahogany:
• Mahogany is a large tropical hardwood tree, valued for its reddish-brown timber.
• It is typically found in tropical rainforests rather than deciduous forests.
• Ebony:
• Ebony trees produce dense, dark wood that is highly valued for its beauty and hardness.
• These trees are mostly found in tropical rainforests of Africa, India, and Indonesia.

The correct answer is Sulochana Chavan.

Key Points

• Sulochana Chavan is a renowned Lavani singer from Maharashtra, India, known for her significant contribution to this traditional music genre.
• She was awarded the Padma Shri in 2022, one of India's highest civilian honors, recognizing her dedication to Indian arts and culture.
• Sulochana Chavan has been instrumental in popularizing the Lavani genre, which is a traditional folk music form of Maharashtra.
• Her work has earned her numerous accolades and widespread recognition, making her a celebrated figure in Indian music.

Additional Information

• Padma Shri
• The Padma Shri is the fourth highest civilian award in India, instituted in 1954.
• It is awarded to citizens of India in recognition of their distinguished contribution in various spheres such as arts, education, industry, literature, science, sports, medicine, social service, and public affairs.
• The award is conferred by the President of India at ceremonial functions held at Rashtrapati Bhavan.
• Recipients of the award are announced annually on Republic Day.
• Lavani
• Lavani is a genre of music popular in Maharashtra, India, known for its powerful rhythm and quick tempo.
• It is a combination of traditional song and dance, performed to the beats of Dholki, a percussion instrument.
• The word "Lavani" is derived from the word "Lavanya," which means beauty. It is traditionally performed by female performers wearing sarees, usually of nine yards.
• Lavani has played a significant role in the development of Marathi folk theatre.

The correct answer is HCOOH.

Key Points

• HCOOH, also known as formic acid, is the simplest carboxylic acid and is the most acidic among the given options.
• Formic acid has the highest acidity due to the absence of electron-donating alkyl groups which decrease the acidity by destabilizing the conjugate base.
• The other options, such as C2H5COOH (propanoic acid), C3H7COOH (butanoic acid), and CH3COOH (acetic acid), have alkyl groups that reduce their acidity.
• Formic acid is widely used in various industrial processes and has a higher acid dissociation constant (Ka) compared to other carboxylic acids listed.

Additional Information

• Carboxylic Acids
• Carboxylic acids are organic acids characterized by the presence of a carboxyl group (–COOH).
• The acidity of carboxylic acids is due to the resonance stabilization of the carboxylate anion (R-COO⁻) formed after deprotonation.
• Electron-withdrawing groups increase acidity by stabilizing the carboxylate anion, while electron-donating groups decrease acidity by destabilizing it.
• Acid Dissociation Constant (Ka)
• The acid dissociation constant (Ka) is a quantitative measure of the strength of an acid in solution.
• It is the equilibrium constant for the dissociation of an acid into its conjugate base and a proton (H⁺).
• Higher Ka values indicate stronger acids, as they dissociate more completely in solution.
• Formic acid has a higher Ka value compared to the other carboxylic acids listed, making it more acidic.
• Factors Influencing Acidity
• Inductive Effect: Electron-withdrawing groups increase acidity, while electron-donating groups decrease it.
• Resonance Effect: Resonance stabilization of the conjugate base increases acidity.
• Hybridization: Acidity increases with the s-character of the hybrid orbital holding the acidic hydrogen (sp > sp² > sp³).
• Solvation: Better solvation of the conjugate base in a solvent increases acidity.

The correct answer is Kannauj.

Key Points

• Kannauj, located in western Uttar Pradesh, was an ancient city and a significant political center in northern India.
• The Maukharis were a dynasty that ruled over Kannauj before it became a prominent city under the later rulers such as the Gurjara-Pratiharas.
• Over time, Kannauj replaced Pataliputra (modern-day Patna) as the political and cultural center of northern India.
• Kannauj is known for its historical significance and its strategic location which made it a focal point of power struggles among various dynasties.

Additional Information

• Maukharis Dynasty:
• The Maukharis were a dynasty that ruled parts of northern India during the early medieval period, roughly between the 6^th and 7^th centuries.
• They initially served as vassals of the Gupta Empire but later established their independence.
• The most notable ruler of the Maukharis was Isanavarman, who expanded their territory and consolidated power.
• The Maukhari dynasty is known for its contribution to the political and cultural landscape of northern India during its time.
• Kannauj:
• Kannauj, also known as Kanyakubja, has a rich history dating back to ancient times.
• It was a major center of trade, culture, and politics and remained so through various dynasties, including the Maukharis, Harshavardhana, and the Gurjara-Pratiharas.
• Kannauj is famously known for its production of perfumes and essential oils, particularly the traditional "attar" (perfume).
• Throughout its history, Kannauj has been a coveted city, leading to numerous battles and power struggles among regional powers.
• Pataliputra:
• Pataliputra, modern-day Patna, was the capital city of several ancient Indian empires, including the Maurya and Gupta Empires.
• It was a major political, economic, and cultural hub during its peak.
• Over time, the political center shifted from Pataliputra to other cities like Kannauj as different dynasties rose to power.
• Pataliputra is historically significant for its role in the administration and governance of ancient Indian empires.

The correct answer is Nitric oxide.

Key Points

• Nitric oxide (NO) is not considered a greenhouse gas because it does not have a significant impact on the Earth's temperature or climate.
• Greenhouse gases are those that can absorb and emit infrared radiation, leading to the greenhouse effect. Nitric oxide does not significantly contribute to this process.
• Common greenhouse gases include carbon dioxide (CO2), methane (CH4), and nitrous oxide (N2O), all of which have a much higher potential to trap heat in the atmosphere.
• Nitric oxide is primarily known for its role in the formation of smog and its impact on the respiratory system, rather than its role as a greenhouse gas.

Additional Information

• Greenhouse Gases:
• Greenhouse gases are those gases in the Earth's atmosphere that trap heat. They allow sunlight to enter the atmosphere freely, but when the Earth's surface emits the energy back as infrared radiation (heat), greenhouse gases absorb and re-emit this radiation, warming the planet.
• The primary greenhouse gases in Earth's atmosphere are water vapor (H2O), carbon dioxide (CO2), methane (CH4), nitrous oxide (N2O), and ozone (O3).
• Human activities, such as burning fossil fuels, deforestation, and industrial processes, have significantly increased the concentrations of these greenhouse gases, leading to global warming and climate change.
• Carbon Dioxide (CO2):
• Carbon dioxide is a colorless, odorless gas that is essential for plant life through the process of photosynthesis.
• It is produced by the combustion of fossil fuels, deforestation, and various industrial processes.
• CO2 is the most significant long-lived greenhouse gas in Earth's atmosphere, contributing to global warming and climate change.
• Methane (CH4):
• Methane is a potent greenhouse gas with a global warming potential many times greater than that of CO2 over a 20-year period.
• It is produced naturally through the decomposition of organic matter and human activities such as livestock farming, rice cultivation, and landfills.
• Despite its shorter atmospheric lifetime compared to CO2, methane's impact on global warming is substantial.
• Nitrous Oxide (N2O):
• Nitrous oxide is a greenhouse gas that is more effective than CO2 at trapping heat in the atmosphere over a 100-year period.
• It is emitted from agricultural and industrial activities, as well as during the combustion of fossil fuels and solid waste.
• N2O also plays a role in the depletion of the ozone layer in the stratosphere.

The correct answer is Fullerene.

Key Points

• Fullerene is an allotrope of carbon that was discovered by Robert F. Curl, Harold W. Kroto, and Richard E. Smalley in 1985.
• It consists of carbon atoms arranged in a hollow sphere, ellipsoid, or tube.
• Fullerenes are also known as "Buckyballs" because the first discovered fullerene (C₆₀) resembles the shape of a soccer ball.
• Fullerenes have unique properties that make them useful in materials science, electronics, and nanotechnology.

Additional Information

• Graphene:
• Graphene is a single layer of carbon atoms arranged in a two-dimensional honeycomb lattice.
• It is known for its remarkable electrical, thermal, and mechanical properties.
• Graphene is used in various applications, including electronics, composites, and energy storage.
• Lonsdaleite:
• Lonsdaleite, also known as hexagonal diamond, is an allotrope of carbon with a hexagonal lattice structure.
• It is formed under high-pressure conditions, typically during meteorite impacts.
• Lonsdaleite is harder than diamond and has potential applications in cutting and drilling tools.
• Carbophene:
• Carbophene is a less commonly known term and is not widely recognized in the scientific community.
• It may be a misnomer or an incorrect term for other carbon allotropes such as graphene or fullerenes.

The correct answer is Surbahar.

Key Points

• Annapurna Devi was an Indian surbahar (bass sitar) player of the Maihar gharana.
• She was the daughter and disciple of Allauddin Khan, who was a prominent figure in Hindustani classical music.
• Annapurna Devi was considered one of the foremost exponents of the surbahar, a rare and difficult instrument to master.
• She was also the spouse of the renowned sitar player Ravi Shankar and significantly contributed to the popularity of Indian classical music.

Additional Information

• Surbahar:
• The surbahar, also known as the bass sitar, is a plucked string instrument used in the Hindustani classical music of North India.
• It is often referred to as the "bass sitar" because it is larger than the sitar and has a lower pitch.
• The surbahar is designed for the elaborate, slow-tempo alap section of a raga performance, with its deep, resonant tones.
• The instrument has a wide range and can produce a variety of tonal colors, making it suitable for both solo and accompanying performances.
• Maihar Gharana:
• The Maihar gharana is a school of Hindustani classical music named after the town of Maihar in Madhya Pradesh, India.
• It was founded by the legendary musician Allauddin Khan, who was a master of multiple instruments including the sarod, surbahar, and sitar.
• The gharana is known for its emphasis on the dhrupad style of alap, intricate raga improvisations, and the use of both vocal and instrumental techniques.
• Some of the prominent disciples of Allauddin Khan from the Maihar gharana include Ravi Shankar, Ali Akbar Khan, and Annapurna Devi.
• Hindustani Classical Music:
• Hindustani classical music is a tradition of Indian classical music that developed in North India around the 13^th and 14^th centuries.
• It is characterized by a variety of ragas (melodic frameworks) and talas (rhythmic cycles) used in compositions and improvisations.
• Instruments commonly used in Hindustani classical music include the sitar, tabla, sarod, surbahar, and bansuri, among others.
• This genre of music is known for its intricate ornamentation, improvisational freedom, and deep emotional expression.

The correct answer is Article 80.

Key Points

• Article 80 of the Constitution of India deals with the composition of the Council of States (Rajya Sabha).
• The Council of States is to be composed of representatives of the States and Union territories.
• The allocation of seats in the Council of States is specified in the Fourth Schedule of the Constitution.
• This article ensures that the representation of states and union territories in the Rajya Sabha is in accordance with the provisions laid out in the Fourth Schedule.

Additional Information

• Article 81
• Article 81 of the Constitution of India deals with the composition of the House of the People (Lok Sabha).
• It specifies the maximum number of seats in the Lok Sabha and the allocation of these seats among the States.
• Article 82
• Article 82 pertains to the readjustment of seats in the Lok Sabha and State Legislative Assemblies after each census.
• This article ensures periodic readjustments to reflect changes in the population.
• Article 83
• Article 83 specifies the duration of the Houses of Parliament.
• The Lok Sabha has a term of five years, unless dissolved sooner, while the Rajya Sabha is a permanent body with one-third of its members retiring every two years.
• Fourth Schedule
• The Fourth Schedule of the Constitution of India allocates seats in the Rajya Sabha to each State and Union territory.
• This schedule ensures balanced representation of the States and Union territories in the Council of States.

The correct answer is Gorter and Grendel.

Key Points

• In 1925, Dutch scientists Evert Gorter and François Grendel conducted a groundbreaking experiment involving erythrocyte (red blood cell) lipid extraction.
• They found that the lipids extracted from erythrocytes could form a monolayer, which was pivotal in understanding the molecular surface area versus lateral pressure.
• Their work was fundamental in the development of the fluid mosaic model of cell membranes.
• Gorter and Grendel's experiment was one of the first to suggest that cell membranes are composed of a lipid bilayer.

Additional Information

• Lipid Monolayers and Bilayers:
• A lipid monolayer is a single layer of lipid molecules arranged on the surface of the water, with their hydrophobic tails in the air and hydrophilic heads in the water.
• A lipid bilayer is a double layer of lipids that form the fundamental structure of cell membranes, with hydrophobic tails facing inward and hydrophilic heads facing outward.
• These structures are crucial for maintaining the integrity and functionality of cell membranes.
• Erythrocytes (Red Blood Cells):
• Erythrocytes are the most common type of blood cell and are responsible for transporting oxygen from the lungs to the body's tissues and carbon dioxide from the tissues back to the lungs.
• Their membrane composition and properties are essential for their deformability and functionality.
• Surface Area vs. Lateral Pressure:
• This concept involves measuring the surface area occupied by lipid molecules as a function of the lateral pressure applied to the monolayer.
• It is used to study the physical properties of lipid layers and understand cell membrane mechanics.
• Fluid Mosaic Model:
• Proposed by Singer and Nicolson in 1972, this model describes the cell membrane as a fluid combination of lipids and proteins.
• The lipids provide a flexible matrix, while proteins float in or on this lipid bilayer, contributing to various cellular functions.

The correct answer is I, II, III, IV.

Key Points

• Land revenue was one of the oldest forms of taxation implemented during ancient and medieval periods.
• Sales tax was introduced much later as a form of indirect tax on the sale of goods.
• MODVAT (Modified Value Added Tax) was introduced in 1986 to allow manufacturers to obtain credit for the excise duty paid on inputs.
• Service tax was introduced in 1994 as a tax levied on services provided.

Additional Information

• Land Revenue
• Land revenue is a tax or revenue imposed by the government on the ownership or use of land.
• It has been a major source of revenue for many governments historically, especially in agrarian societies.
• In India, during the British rule, land revenue was a significant component of colonial administration.
• Land revenue systems like the Zamindari, Ryotwari, and Mahalwari were implemented in different regions.
• Sales Tax
• Sales tax is a tax levied by the government on the sale of goods and services.
• It is generally a percentage of the sale price and is added to the total cost paid by the consumer.
• This form of tax is an important source of revenue for state governments.
• With the introduction of Goods and Services Tax (GST) in India in 2017, the sales tax regime was subsumed under GST.
• MODVAT (Modified Value Added Tax)
• MODVAT was introduced in India in 1986.
• It allowed manufacturers to claim credit for the excise duty paid on inputs used in the production of goods.
• This system aimed to avoid the cascading effect of taxes and to make the tax structure more efficient.
• MODVAT was later replaced by CENVAT (Central Value Added Tax) and subsequently subsumed under GST.
• Service Tax
• Service tax was introduced in India in 1994.
• It was levied on specified services provided by service providers.
• Service tax was a part of the indirect tax regime and contributed significantly to the government’s revenue.
• With the advent of GST, service tax was abolished and integrated into the new tax structure.

The correct answer is Dharmapala.

Key Points

• Dharmapala was a prominent ruler of the Pala dynasty in ancient India.
• He ruled during the 8^th century and is known for his significant contributions to education and Buddhism.
• Under his patronage, the Vikramshila University was established to promote Buddhist learning and scholarship.
• Dharmapala's reign marked the height of the Pala Empire's influence and cultural achievements.

Additional Information

• Pala Dynasty
• The Pala Dynasty ruled parts of Bengal and Bihar in the Indian subcontinent from the 8^th to the 12^th century.
• The dynasty was founded by Gopala in the mid-8^th century.
• The Palas were known for their patronage of Buddhism and their support for educational institutions.
• They built several monasteries and universities, including Nalanda and Vikramshila.
• Vikramshila University
• Vikramshila University was one of the most important centers of Buddhist learning in ancient India.
• It was established by Dharmapala around the late 8^th or early 9^th century.
• The university attracted scholars from all over Asia, including Tibet, China, and Southeast Asia.
• It specialized in various fields of knowledge, including philosophy, logic, grammar, and metaphysics.
• Buddhist Learning
• During the Pala period, Buddhism flourished in India, and several monasteries and universities were established to promote its teachings.
• These institutions became centers of learning and attracted students from different parts of the world.
• They played a crucial role in the preservation and dissemination of Buddhist texts and teachings.

The correct answer is ₹1,000 per month.

Key Points

• The Tamil Nadu Pudhumi Penn Scheme was launched by the Government of Tamil Nadu in 2023.
• It aims to provide financial assistance specifically to girl students studying in government schools.
• Under this scheme, each girl student receives ₹1,000 per month as financial aid.
• The scheme is part of the government's initiative to promote education among girl children and reduce drop-out rates in schools.

Additional Information

• Educational Financial Assistance Schemes in India
• Pre-Matric Scholarship Scheme: This scheme provides financial assistance to Scheduled Caste (SC) students who are studying in Class IX and X to reduce the drop-out rate and promote secondary education.
• Post-Matric Scholarship Scheme: This scheme aims to provide financial support to students from economically weaker sections after they have completed their matriculation so that they can pursue higher education.
• National Means-cum-Merit Scholarship Scheme: It aims to provide scholarships to meritorious students of economically weaker sections to prevent their drop-out at Class VIII and encourage them to continue their education at the secondary stage.
• Maulana Azad National Scholarship Scheme: This scheme is specifically for meritorious girls belonging to minority communities to provide financial assistance for their higher education.
• Social Initiatives for Girl Child Education
• Beti Bachao Beti Padhao: Launched by the Government of India, this scheme aims to ensure the survival, protection, and education of the girl child.
• Sukanya Samriddhi Yojana: A savings scheme initiated for the girl child, it encourages parents to build a fund for the future education and marriage expenses of their girl child.
• Kasturba Gandhi Balika Vidyalaya: This scheme aims to provide residential schools at the upper primary level for girls belonging to SC, ST, OBC, minority communities, and BPL families.
• CBSE Udaan: It aims to address the low enrollment ratio of girl students in prestigious engineering institutions and provides free online resources for preparation for engineering entrance exams.

The correct answer is 0.02.

Key Points

• The Pradhan Mantri Fasal Bima Yojana (PMFBY) is a crop insurance scheme launched by the Government of India.
• The maximum premium payable by the farmers is 2% for all kharif food and oilseeds crops.
• The scheme aims to provide insurance coverage and financial support to farmers in the event of crop failure due to natural calamities, pests, and diseases.
• It helps stabilize the income of farmers to ensure their continuance in farming.

Additional Information

• Pradhan Mantri Fasal Bima Yojana (PMFBY)
• The scheme was launched on 18^th February 2016 by Prime Minister Narendra Modi.
• It aims to provide a comprehensive insurance cover against the failure of crops, thus helping in stabilizing the income of farmers.
• Farmers are required to pay a uniform premium of only 2% for all Kharif crops and 1.5% for all Rabi crops. For annual commercial and horticultural crops, the premium is 5%.
• The remaining premium is paid by the Government to provide full insured amount to the farmers against crop loss on account of natural calamities.
• Pradhan Mantri Krishi Sinchayee Yojana (PMKSY)
• PMKSY was launched to improve farm productivity and ensure better utilization of the resources in the country.
• The scheme’s motto is "Har Khet Ko Pani" (Water for every field).
• It focuses on efficient water conveyance and precision water application devices like drips, sprinklers, pivots, rain-guns, etc.
• The scheme aims at improving on-farm water use efficiency through a proper combination of surface and ground water.
• Soil Health Card Scheme
• The Soil Health Card Scheme was launched by the Government of India in February 2015.
• The scheme aims at promoting soil test-based and balanced use of fertilizers to enable farmers to realize higher yields at lower cost.
• Under the scheme, farmers are provided with Soil Health Cards, which will carry crop-wise recommendations of nutrients and fertilizers required for the individual farms.
• The Soil Health Cards will be issued once every three years for all land holdings in the country.

The correct answer is To facilitate bank loans for SC/ST and women entrepreneurs.

Key Points

• The 'Standup India' initiative was launched by the Government of India to encourage entrepreneurship among women and SC/ST communities.
• It aims to facilitate bank loans between ₹10 lakh and ₹1 crore to at least one SC/ST borrower and one woman borrower per bank branch for setting up greenfield enterprises.
• The scheme was launched on 5^th April 2016 by Prime Minister Narendra Modi.
• It seeks to leverage the institutional credit structure to reach out to the underserved sector of people such as SCs, STs, and women entrepreneurs.

Additional Information

• Start-up India
• It is an initiative by the Government of India to promote entrepreneurship and innovation by creating a conducive environment for startups.
• The scheme was launched on 16^th January 2016 by Prime Minister Narendra Modi.
• It includes benefits such as simplified processes, tax exemptions, and funding support to foster the growth of startups in the country.
• Start-up India aims to build a strong ecosystem for nurturing innovation and provide opportunities for budding entrepreneurs.
• Make in India
• Launched by the Government of India on 25^th September 2014, the Make in India initiative aims to transform India into a global manufacturing hub.
• The initiative encourages both multinational and domestic companies to manufacture their products in India.
• It aims to increase the manufacturing sector's growth rate to 12-14% per annum and create additional manufacturing jobs.
• Make in India focuses on 25 key sectors of the economy, including automobiles, textiles, and pharmaceuticals.
• Digital India
• Launched on 1^st July 2015, Digital India is an initiative by the Government of India to ensure that government services are made available to citizens electronically.
• The program aims to improve online infrastructure and increase internet connectivity across India.
• Digital India focuses on three core components: digital infrastructure, digital services, and digital literacy.
• The ultimate goal is to transform India into a digitally empowered society and knowledge economy.

The correct answer is To safeguard public property.

Key Points

• The fundamental duty of safeguarding public property is enshrined in Article 51A of the Indian Constitution.
• This duty emphasizes the importance of protecting and preserving public assets such as monuments, buildings, and other government properties.
• By writing their names on the walls of Humayun's tomb, the students are defacing a historical monument, which is considered public property.
• Citizens are expected to prevent such actions and also report any damage to public property to the authorities.

Additional Information

• Fundamental Duties
• Fundamental Duties are a set of moral obligations specified in the Indian Constitution that are expected to be followed by citizens.
• These duties were added by the 42^nd Amendment Act, 1976, on the recommendations of the Swaran Singh Committee.
• There are 11 Fundamental Duties listed under Article 51A of the Constitution.
• These duties serve as a constant reminder to citizens to uphold the spirit of patriotism and to protect the unity of India.
• Humayun's Tomb
• Humayun's Tomb is a UNESCO World Heritage Site located in Delhi, India.
• It was commissioned by Humayun's first wife, Empress Bega Begum, in 1565 AD.
• The tomb is a fine example of Mughal architecture and is often referred to as the precursor to the Taj Mahal.
• It is built of red sandstone and white marble and is surrounded by beautiful gardens.
• Significance of Protecting Public Property
• Public properties like historical monuments, parks, public buildings, and infrastructure are assets of national importance.
• They reflect the cultural heritage and history of the nation and are meant for the collective use and enjoyment of the public.
• Vandalism and damage to public property not only diminish their aesthetic and historical value but also result in financial loss.
• Protecting public property ensures that future generations can also learn from and appreciate these important assets.

The correct answer is Indian National Congress.

Key Points

• Mahatma Gandhi, a pivotal leader in the Indian independence movement, represented the Indian National Congress at the Second Round Table Conference.
• The Second Round Table Conference was held in London from September 7, 1931, to December 1, 1931.
• The Indian National Congress was a major political party in India that aimed to gain independence from British rule.
• Gandhi attended the conference to present the demands and aspirations of the Indian people and to negotiate for the country's independence.

Additional Information

• Round Table Conferences
• The Round Table Conferences were a series of three conferences organized by the British government to discuss constitutional reforms in India.
• The first conference was held in London from November 1930 to January 1931, the second from September to December 1931, and the third from November to December 1932.
• The conferences were attended by representatives of various Indian political parties, princely states, and British officials.
• The aim was to formulate plans for the future governance of India and address the demands of Indian leaders for greater self-rule.
• Indian National Congress
• The Indian National Congress (INC) was founded in 1885 and played a significant role in the Indian independence movement.
• It was the principal party that led the struggle against British colonial rule and advocated for the rights and self-governance of Indians.
• Key leaders included Mahatma Gandhi, Jawaharlal Nehru, Vallabhbhai Patel, and many others who contributed to India's fight for independence.
• The INC adopted various methods of protest, including non-violent civil disobedience, boycotts, and mass movements to challenge British policies.
• Mahatma Gandhi
• Mohandas Karamchand Gandhi, popularly known as Mahatma Gandhi, was a prominent leader in the Indian independence movement.
• He is best known for his philosophy of non-violence (Ahimsa) and civil disobedience (Satyagraha).
• Gandhi's leadership and methods inspired millions of Indians to participate in the freedom struggle through peaceful means.
• His efforts played a crucial role in India achieving independence from British rule on August 15, 1947.

The correct answer is Sea and air.

Key Points

• India's international trade is predominantly carried out through sea and air routes due to their efficiency in handling large volumes of goods and their ability to connect globally.
• Sea routes are especially important for transporting bulk commodities, such as crude oil, coal, and iron ore, as they can carry large quantities at a lower cost.
• Air routes, while more expensive, are crucial for the transportation of high-value and perishable goods, such as electronics, pharmaceuticals, and fresh produce, due to their speed and reliability.
• Major ports like Jawaharlal Nehru Port Trust (JNPT) and major airports like Indira Gandhi International Airport in Delhi play pivotal roles in facilitating India's international trade.

Additional Information

• Seaborne Trade
• Seaborne trade refers to the transportation of goods over sea routes. It is the backbone of international trade due to its ability to move large quantities of goods efficiently and cost-effectively.
• Major Indian ports include Mumbai, Chennai, Kolkata, and Cochin, which handle significant portions of India's cargo.
• India's maritime industry is crucial for the economy, facilitating imports and exports worth billions of dollars annually.
• Seaborne trade enables India to connect with global markets, ensuring the availability of essential commodities and boosting export earnings.
• Airborne Trade
• Airborne trade involves the transportation of goods via air routes, leveraging the speed and reliability of air transport.
• It is particularly important for time-sensitive and high-value goods, including electronics, pharmaceuticals, and perishable items.
• Major Indian airports such as Indira Gandhi International Airport (Delhi), Chhatrapati Shivaji Maharaj International Airport (Mumbai), and Kempegowda International Airport (Bangalore) are key hubs for airborne trade.
• Airborne trade enhances India's connectivity with global markets, enabling businesses to meet international demands promptly.
• Railway and Road Transport
• While railway and road transport are vital for domestic trade, they play a relatively smaller role in international trade.
• Railways are used to transport goods to and from ports and airports, facilitating the movement of goods within the country.
• Road transport complements railways by providing last-mile connectivity and ensuring the efficient distribution of goods.
• Integration of rail and road transport with sea and air routes is essential for a seamless supply chain and logistics network.

The correct answer is About 0.1 µm.

Key Points

• PPLOs (Pleuro Pneumonia Like Organisms) are the smallest free-living organisms known.
• They are characterized by their extremely small size, about 0.1 µm in diameter.
• PPLOs lack a cell wall, which contributes to their small size and flexible shape.
• These organisms are responsible for causing respiratory diseases, particularly in livestock.

Additional Information

• PPLO (Pleuro Pneumonia Like Organisms):
• PPLOs are a group of bacteria that belong to the class Mollicutes.
• These organisms are known for their lack of a rigid cell wall, making them resistant to antibiotics that target cell wall synthesis.
• They are often referred to as mycoplasmas, with Mycoplasma pneumoniae being a well-known species causing atypical pneumonia in humans.
• PPLOs thrive in osmotically balanced environments due to their lack of a cell wall, which makes them susceptible to osmotic lysis in hypotonic conditions.
• Importance in Research:
• The small size and simple genetic structure of PPLOs make them important models for studying the minimal requirements for cellular life.
• Research on PPLOs has provided insights into the evolution of bacteria and the mechanisms of pathogenesis in respiratory infections.
• Clinical Relevance:
• PPLOs are significant pathogens in veterinary medicine, causing respiratory diseases such as contagious bovine pleuropneumonia.
• In humans, mycoplasmas can lead to a range of respiratory and urogenital infections, requiring specific diagnostic and treatment approaches.
• Due to their lack of a cell wall, infections caused by PPLOs are treated with antibiotics that inhibit protein synthesis, such as tetracyclines or macrolides.

Shortcut Trick

The length of the longest diagonal in a cuboid is calculated using the formula: √(l² + b² + h²)

⇒ Diagonal = √(1.2² + 1.3² + 1.5²)

⇒ Diagonal = √(1.44 + 1.69 + 2.25)

⇒ Diagonal = √5.38

⇒ Since 2.31² = 5.3361 and 2.32² = 5.3824, √5.38 is approximately 2.32.

∴ The correct answer is 2.32 cm.

Alternate Method

Given:

Length (l) = 1.5 cm

Breadth (b) = 1.3 cm

Height (h) = 1.2 cm

Formula Used:

Length of the longest diagonal of a cuboidal box = √(l² + b² + h²)

Calculations:

⇒ Diagonal = √(1.5² + 1.3² + 1.2²)

⇒ Diagonal = √(2.25 + 1.69 + 1.44)

⇒ Diagonal = √5.38

⇒ Upon checking squares of nearest numbers, 2.31² = 5.3361 and 2.32² = 5.3824.

⇒ The value 5.38 is closest to 5.3824.

⇒ Therefore, √5.38 ≈ 2.32 cm (correct to two decimal places).

∴ The correct answer is 2.32 cm.

Additional Information

Total Surface Area of a Cuboid

The total surface area encompasses all six rectangular faces of the cuboid. Formula: TSA = 2(lb + bh + hl)

Volume of a Cuboid

The volume represents the total three-dimensional space enclosed by the cuboid. Formula: V = l × b × h

Diagonal of a Cube

For a cube where all sides are strictly equal (l = b = h = a), the length of the longest diagonal simplifies directly to a√3.

Shortcut Trick

Given: x + 1/x = 65/8

Express 65/8 as a mixed fraction: 65/8 = 8 + 1/8

By comparing x + 1/x with 8 + 1/8, we can clearly see x = 8 or x = 1/8.

Since x is stated to be an integer, we take x = 8.

Therefore, x − 1/x = 8 − 1/8 = (64 − 1)/8 = 63/8.

∴ The correct answer is 63/8.

Alternate Method

Given:

x + 1/x = 65/8

x is an integer.

Calculation:

⇒ (x² + 1) / x = 65/8

By cross-multiplying, we get:

⇒ 8(x² + 1) = 65x

⇒ 8x² − 65x + 8 = 0

⇒ 8x² − 64x − x + 8 = 0

⇒ 8x(x − 8) − 1(x − 8) = 0

⇒ (8x − 1)(x − 8) = 0

⇒ x = 1/8 or x = 8

Since it is given that x is an integer, x = 8.

Now, we substitute x = 8 in the expression x − 1/x:

⇒ x − 1/x = 8 − 1/8

⇒ (64 − 1) / 8

⇒ 63/8

∴ The correct answer is 63/8.

Additional Information

Algebraic Identities for Reciprocals

The relationship between the sum and difference of a number and its reciprocal is given by: (x + 1/x)² − (x − 1/x)² = 4.

Nature of Roots in Quadratic Equations

For a quadratic equation ax² + bx + c = 0, the sum of roots is −b/a and the product of roots is c/a. This helps quickly verify roots.

Reciprocal Roots Rule

When the coefficient of x² equals the constant term (a = c), the product of the roots is 1, meaning the two roots are reciprocals of each other.

Shortcut Trick If two sides of a triangle are given as 'a' and 'b' (where a < b), the number of possible integer values for the third side is always 2a − 1. Here, the smaller side is a = 6 units. Number of integer values = 2 × 6 − 1 = 12 − 1 = 11. ∴ The correct answer is 11 values.

Alternate Method

A B C

6 12 x

|12 − 6| < x < 12 + 6

Given: Side a = 6 units Side b = 12 units Side c = x units Concept Used: The Triangle Inequality Theorem states that the length of the third side of a triangle must always lie between the difference and the sum of the other two sides. Calculations: ⇒ Difference of two sides < x < Sum of two sides ⇒ (12 − 6) < x < (12 + 6) ⇒ 6 < x < 18 ⇒ The possible integer values for x are greater than 6 and strictly less than 18. ⇒ The integer values are 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, and 17. ⇒ Total number of possible integer values = 18 − 6 − 1 = 11 ∴ The correct answer is 11 values.

Additional Information Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is strictly greater than the length of the remaining third side: a + b > c. Types of Triangles Based on Sides For sides a, b, and c (where c is the longest side): If c2 < a2 + b2 it is an acute triangle. If c2 = a2 + b2 it is a right triangle. If c2 > a2 + b2 it is an obtuse triangle. Perimeter of a Triangle The perimeter is the total length of the boundary, calculated as P = a + b + c. The semi-perimeter (s) used in Heron's formula is s = (a + b + c) ÷ 2.

Shortcut Trick

Time taken by Arvind to complete one round = 180 seconds = 3 minutes.

Time taken by Chetan to complete one round = 420 seconds = 7 minutes.

To find the time they meet for the first time at the starting point, we calculate the LCM of their individual lap times.

Meeting time = LCM(3, 7) = 21 minutes.

∴ The correct answer is 21 minutes.

Alternate Method

Given:

Length of circular track = 210 m

Time taken by Arvind (T₁) = 180 seconds

Time taken by Chetan (T₂) = 420 seconds

Formula Used:

Time to meet for the first time at the starting point = LCM of individual times taken to complete one round.

Calculations:

⇒ Prime factorization of 180 = 2² × 3² × 5

⇒ Prime factorization of 420 = 2² × 3 × 5 × 7

⇒ LCM(180, 420) = 2² × 3² × 5 × 7

⇒ LCM(180, 420) = 4 × 9 × 5 × 7 = 1260 seconds

⇒ Convert total seconds into minutes = 1260 ÷ 60

⇒ Time to meet = 21 minutes

∴ The correct answer is 21 minutes.

Additional Information

Meeting First Time at Starting Point

When multiple people start from the same point simultaneously, they meet again at the starting point exactly at the LCM of the times taken by each to complete one full round independently.

Meeting First Time Anywhere on Track

If running in the same direction, Time to meet = Track Length ÷ (S₁ − S₂). If running in opposite directions, Time to meet = Track Length ÷ (S₁ + S₂).

Number of Distinct Meeting Points

If the ratio of their speeds is a : b in its simplest form, they will meet at (a − b) distinct points if running in the same direction, and at (a + b) distinct points if running in opposite directions.

Calculations:

Step 1: Calculate the average of the low rates for company C:

Low rates: Hyderabad = 175, Mumbai = 180, Bengaluru = 185.

Average = (175 + 180 + 185) / 3 = 540 / 3 = 180.

Step 2: Compare the low rates with the average:

Hyderabad Low: 175 (less than 180).

Mumbai Low: 180 (equal to 180).

Bengaluru Low: 185 (greater than 180).

The low rates of shares for company C are less than the average in the Hyderabad stock exchange.

Shortcut Trick

Apply the divisibility rule of 11 directly:

Sum of alternate digits (odd places) = 6 + 5 + P = 11 + P

Sum of remaining digits (even places) = 9 + 4 = 13

Difference = (11 + P) − 13 = P − 2

For divisibility by 11, P − 2 = 0 ⇒ P = 2

∴ The correct answer is 2.

Alternate Method

Given:

A 5-digit number 6954P is exactly divisible by 11.

P is a single digit (0 to 9).

Method Used:

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 either 0 or a multiple of 11.

Calculations:

⇒ Sum of digits at odd places (1^st, 3^rd, 5^th from right) = P + 5 + 6 = 11 + P

⇒ Sum of digits at even places (2^nd, 4^th from right) = 4 + 9 = 13

⇒ Difference = (11 + P) − 13

⇒ Difference = P − 2

⇒ For the difference to be a multiple of 11, the only possible value for a single digit P (0 ≤ P ≤ 9) is when the difference is exactly 0.

⇒ P − 2 = 0

⇒ P = 2

∴ The correct answer is 2.

Additional Information

Divisibility Rule of 3 and 9

A number is divisible by 3 (or 9) if the sum of its digits is exactly divisible by 3 (or 9).

Divisibility Rule of 4 and 8

A number is divisible by 4 if its last two digits form a number divisible by 4. It is divisible by 8 if its last three digits form a number divisible by 8.

Combined Divisibility Rule of 7, 11, and 13

Form alternating sums of blocks of three digits from right to left. If the resultant difference is divisible by 7, 11, or 13, then the original number is also divisible by 7, 11, or 13 respectively.

Shortcut Trick

Bank's earning rate from the stock market = (30,000 ÷ 3,00,000) × 100 = 10%

Interest rate paid to depositors = 6%

Net profit margin for the bank = 10% − 6% = 4%

Final earnings of the bank = 4% of 60,00,000

Final earnings = (4 ÷ 100) × 60,00,000 = ₹2,40,000

∴ The correct answer is ₹2,40,000.

Alternate Method

Given:

Total amount deposited in the bank = ₹60,00,000

Interest given on deposited amount = 6%

Earnings from stock market = ₹30,000 for every ₹3,00,000

Calculations:

⇒ Total interest paid to depositors = 6% of ₹60,00,000

⇒ Total interest paid = (6 ÷ 100) × 60,00,000 = ₹3,60,000

⇒ Total money invested in the stock market by the bank = ₹60,00,000

⇒ Bank's total earnings from the market = (60,00,000 ÷ 3,00,000) × 30,000

⇒ Bank's total earnings = 20 × 30,000 = ₹6,00,000

⇒ Final earnings of the bank = Earnings from stock market − Interest paid to depositors

⇒ Final earnings = 6,00,000 − 3,60,000 = ₹2,40,000

∴ The correct answer is ₹2,40,000.

Additional Information

Return on Investment (ROI)

ROI is a performance measure used to evaluate the efficiency of an investment. Formula: ROI = (Net Profit ÷ Cost of Investment) × 100.

Net Yield / Margin

It represents the core difference between the rate an institution earns on investments and the rate it pays on its obligations. Net Margin = Earning Rate − Borrowing Rate.

Simple Percentage Computation

When base values are uniform throughout the problem, percentages can be subtracted directly to calculate final absolute differences (e.g., 10% − 6% = 4% on the common base).

Formula Used:

Average expenditure = (Total expenditure) / (Number of years).

Calculations:

Step 1: Calculate the total expenditure on transport:

Total transport expenditure = 38 + 45 + 57 + 63

Total = 203 ₹ lakh.

Step 2: Calculate the average:

Average expenditure = Total expenditure / Number of years

Average = 203 / 4

Average = 50.75 ₹ lakh.

The average expenditure on transport per year during the given period is 50.75 ₹ lakh.

Shortcut Trick

Given cosec x = 5/4 ⇒ Hypotenuse = 5, Perpendicular = 4, and Base = 3.

From this, sin x = 4/5, cos x = 3/5, tan x = 4/3, and cot x = 3/4.

⇒ Numerator = 5 × (4/5)² + 8 × (3/4)² = 16/5 + 9/2 = 77/10

⇒ Denominator = 27 × (4/3)² − 125 × (3/5)² = 48 − 45 = 3

⇒ Final Value = (77/10) ÷ 3 = 77/30

∴ The correct answer is 77/30.

Alternate Method

Given:

⇒ cosec x = 5/4

We know that cosec x = Hypotenuse / Perpendicular

Calculations:

⇒ Let Hypotenuse (H) = 5 and Perpendicular (P) = 4

Using Pythagoras theorem:

⇒ Base (B) = √(H² − P²)

⇒ Base (B) = √(5² − 4²) = √(25 − 16) = √9 = 3

Now, finding the required trigonometric ratios:

⇒ sin x = P / H = 4/5

⇒ cos x = B / H = 3/5

⇒ tan x = P / B = 4/3

⇒ cot x = B / P = 3/4

Solving the numerator of the expression: 5sin²x + 8cot²x

⇒ Numerator = 5 × (4/5)² + 8 × (3/4)²

⇒ Numerator = 5 × (16/25) + 8 × (9/16)

⇒ Numerator = 16/5 + 9/2 = (32 + 45) / 10 = 77/10

Solving the denominator of the expression: 27tan²x − 125cos²x

⇒ Denominator = 27 × (4/3)² − 125 × (3/5)²

⇒ Denominator = 27 × (16/9) − 125 × (9/25)

⇒ Denominator = (3 × 16) − (5 × 9) = 48 − 45 = 3

Dividing numerator by denominator:

⇒ Value = (77/10) ÷ 3 = 77/30

∴ The correct answer is 77/30.

Additional Information

Trigonometric Ratios

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

Pythagoras Theorem

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

Trigonometric Identities

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

Shortcut Trick

Ratio of milk quantities = 42 : 28 = 3 : 2

Weighted Avg = (Ratio₁ × P₁ + Ratio₂ × P₂) ÷ (Ratio₁ + Ratio₂)

⇒ Average = (3 × 25 + 2 × 40) ÷ (3 + 2)

⇒ Average = (75 + 80) ÷ 5 = 155 ÷ 5 = ₹31

∴ The correct answer is ₹31.

Alternate Method

Given:

Quantity of first milk = 42 litres, Price = ₹25/litre

Quantity of second milk = 28 litres, Price = ₹40/litre

Formula Used:

Average Price = Total Cost ÷ Total Quantity

₹25

₹40

₹ x Average

40 − x

x − 25 42 L → 3 parts 28 L → 2 parts

Calculations:

⇒ Total Cost = (42 × 25) + (28 × 40)

⇒ Total Cost = 1050 + 1120 = 2170

⇒ Total Quantity = 42 + 28 = 70

⇒ Average Price = 2170 ÷ 70

⇒ Average Price = 31

Alternatively, using the Alligation method from the diagram:

⇒ (40 − x) ÷ (x − 25) = 42 ÷ 28

⇒ (40 − x) ÷ (x − 25) = 3 ÷ 2

⇒ 2 × (40 − x) = 3 × (x − 25)

⇒ 80 − 2x = 3x − 75

⇒ 5x = 155 ⇒ x = 31

∴ The correct answer is ₹31.

Additional Information

Rule of Alligation

When two ingredients of different prices are mixed, the ratio of their quantities is given by: (Cheaper Quantity) ÷ (Dearer Quantity) = (Price of Dearer − Mean Price) ÷ (Mean Price − Price of Cheaper).

Weighted Average

It calculates the average of values with different weights or quantities. Formula: A_w = (w₁x₁ + w₂x₂) ÷ (w₁ + w₂).

Shortcut Trick

Since D is the midpoint of BC, the hypotenuses BD and CD are equal.

The perpendiculars from D to sides AB and AC are also equal, so DL = DM.

In right-angled ΔBLD and ΔCMD, both the hypotenuses and one side are equal.

By RHS congruence, ΔBLD ≅ ΔCMD, which implies the base angles are equal (∠B = ∠C).

If two angles of a triangle are equal, the opposite sides are also equal (AB = AC).

∴ The correct answer is isosceles triangle.

Alternate Method

               A B C D L M

Given:

In ΔABC, D is the mid-point of BC ⇒ BD = DC

DL ⊥ AB and DM ⊥ AC ⇒ ∠BLD = ∠CMD = 90°

Lengths of the perpendiculars are equal ⇒ DL = DM

Proof:

In right-angled ΔBLD and right-angled ΔCMD:

⇒ ∠BLD = ∠CMD = 90° (Given)

⇒ Hypotenuse BD = Hypotenuse DC (D is the mid-point of BC)

⇒ Side DL = Side DM (Given)

⇒ By RHS (Right angle - Hypotenuse - Side) congruence criterion, ΔBLD ≅ ΔCMD.

Therefore, the corresponding parts are equal:

⇒ ∠LBD = ∠MCD (CPCT)

⇒ ∠B = ∠C

In any triangle, the sides opposite to equal angles are always equal in length.

⇒ AB = AC

⇒ Since two sides of ΔABC are equal, it must be an isosceles triangle.

∴ The correct answer is isosceles triangle.

Additional Information

Congruence Criteria (RHS)

Two right-angled triangles are congruent if the hypotenuse and one side of one triangle are respectively equal to the hypotenuse and one side of the second triangle.

Properties of Isosceles Triangles

An isosceles triangle has at least two equal sides and two equal base angles. The median drawn to the unequal side always acts as the altitude and the angle bisector.

Equidistant Property of an Angle Bisector

Any point lying on the bisector of an angle is strictly equidistant from the arms of that angle. Here, D being equidistant from AB and AC implies AD bisects ∠A.

Shortcut Trick

Effective Multiplier = (100 − D₁)% × (100 − D₂)%

SP = 18500 × 80% × 95%

SP = 18500 × (4/5) × (19/20)

SP = 18500 × (76/100) = 14060

∴ The correct answer is ₹14,060.

Alternate Method

Given:

Marked Price (MP) of TV set = ₹18,500

First successive discount (D₁) = 20%

Second successive discount (D₂) = 5%

Formula Used:

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

Calculations:

⇒ Price after 1^st discount = 18500 × (100 − 20)/100

⇒ Price after 1^st discount = 18500 × 80/100 = 18500 × 4/5 = ₹14,800

⇒ Price after 2^nd discount = 14800 × (100 − 5)/100

⇒ Final Selling Price = 14800 × 95/100 = 148 × 95

⇒ Final Selling Price = ₹14,060

∴ The correct answer is ₹14,060.

Additional Information

Successive Discount Formula

For two successive discounts of x% and y%, the equivalent single discount can also be found directly using the formula: [x + y − (x × y) ÷ 100]%.

Marked Price (MP)

The listed, printed, or tagged price of an article. Unless specified otherwise, all discount percentages are always calculated on the Marked Price.

Selling Price (SP)

The final amount paid by a customer. It connects directly to Marked Price via SP = MP − Discount, and to Cost Price via SP = CP + Profit.

Shortcut Trick

Since Expenditure is 66(2/3)% = 2/3 of Income, let Income = 3 units, Expenditure = 2 units, and Savings = 1 unit.

Using weighted average: (Exp Ratio × Exp % Change) + (Sav Ratio × Sav % Change) = (Inc Ratio × Inc % Change)

⇒ 2 × 20% + 1 × x = 3 × 25%

⇒ 40% + x = 75%

⇒ x = 35%

∴ The correct answer is 35%.

Alternate Method

Given:

⇒ Expenditure = 66(2/3)% of Income = 2/3 × Income

Let initial Income = 300

Calculations:

⇒ Initial Expenditure = 2/3 × 300 = 200

⇒ Initial Savings = Income − Expenditure = 300 − 200 = 100

⇒ New Income = 300 + (25/100 × 300) = 300 + 75 = 375

⇒ New Expenditure = 200 + (20/100 × 200) = 200 + 40 = 240

⇒ New Savings = New Income − New Expenditure = 375 − 240 = 135

⇒ Increase in Savings = 135 − 100 = 35

⇒ Percentage effect on Savings = (35 ÷ 100) × 100% = 35%

∴ The correct answer is 35%.

Additional Information

Income, Expenditure & Savings Relationship

Income = Expenditure + Savings. The overall percentage change in income acts as the weighted average of the percentage changes in expenditure and savings.

Mixture & Alligation Approach

This problem type can also be effectively solved using the Alligation cross method on percentage changes, where Ratio of Expenditure to Savings = (Savings % Change − Income % Change) : (Income % Change − Expenditure % Change).

Percentage Change Formula

Percentage Change = (Final Value − Initial Value) ÷ Initial Value × 100%. A positive resulting value indicates an increase, while a negative value signifies a decrease.

Shortcut Trick

Let the function be a sum of direct and inverse variations: f = ax + b/y.

For x = 0, y = 1: f = 15 ⇒ a(0) + b/1 = 15 ⇒ b = 15.

For x = 1, y = 15: f = 2 ⇒ a(1) + 15/15 = 2 ⇒ a = 1.

The determined function is f = x + 15/y.

For x = 5 and y = 3: f = 5 + 15/3 = 5 + 5 = 10.

∴ The correct answer is 10.

Alternate Method

Given:

A function f(x, y) consists of two parts:

One part varies directly with x.

One part varies inversely with y.

f = 15 when x = 0 and y = 1.

f = 2 when x = 1 and y = 15.

Formula Used:

f = a × x + b ÷ y

Calculations:

⇒ Step 1: Substitute the first condition (x = 0, y = 1, f = 15) into the formula.

⇒ 15 = a × 0 + b ÷ 1

⇒ b = 15

⇒ Step 2: Substitute the second condition (x = 1, y = 15, f = 2) into the formula.

⇒ 2 = a × 1 + 15 ÷ 15

⇒ 2 = a + 1

⇒ a = 1

⇒ The complete function equation is f = x + 15/y.

⇒ Step 3: Find f when x = 5 and y = 3.

⇒ f = 5 + 15 ÷ 3

⇒ f = 5 + 5

⇒ f = 10

∴ The correct answer is 10.

Additional Information

Direct Variation

If two quantities A and B are in direct variation, they increase or decrease together proportionally. The equation takes the form A = k × B, where k is a constant.

Inverse Variation

If two quantities A and B are in inverse variation, an increase in one causes a proportional decrease in the other. The equation takes the form A = k ÷ B.

Partial Variation

When a value depends on the sum of two or more independent variables or conditions, it is expressed as a combined sum of multiple variations. Example: f = a × x + b ÷ y.

Shortcut Trick

Total Simple Interest = ₹8,928 − ₹7,200 = ₹1,728

Interest for 1 year = 8% of ₹7,200 = ₹576

Time = Total Interest ÷ 1 Year Interest = 1728 ÷ 576 = 3 years

∴ The correct answer is 3 years.

Alternate Method

Given:

Principal (P) = ₹7,200

Amount (A) = ₹8,928

Rate (R) = 8% per annum

Formula Used:

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

Calculations:

⇒ SI = Amount − Principal

⇒ SI = 8928 − 7200 = ₹1728

⇒ Using formula: SI = (P × R × T) ÷ 100

⇒ 1728 = (7200 × 8 × T) ÷ 100

⇒ 1728 = 72 × 8 × T

⇒ 1728 = 576 × T

⇒ T = 1728 ÷ 576

⇒ T = 3

∴ The correct answer is 3 years.

Additional Information

Simple Interest Formula

Simple Interest is calculated on the principal amount for the entire duration. SI = (P × R × T) ÷ 100.

Amount in Simple Interest

The total amount (A) at the end of the time period is the sum of Principal and Simple Interest. A = P + SI = P[1 + (R × T) ÷ 100].

Difference between SI and CI

For the first year, SI and CI are exactly the same if compounded annually. For subsequent years, CI is calculated on the accumulated amount, whereas SI remains constant each year.

Shortcut Trick

Given tan(A) = 3/5 = Perpendicular / Base.

Hypotenuse = √(3² + 5²) = √(9 + 25) = √34.

cos(A) = Base / Hypotenuse = 5 / √34.

∴ The correct answer is 5/√34.

Alternate Method

Given:

tan(A) = 3/5

Formula Used:

tan(A) = Perpendicular / Base

cos(A) = Base / Hypotenuse

Hypotenuse² = Perpendicular² + Base²

Calculations:

⇒ Let Perpendicular (P) = 3x and Base (B) = 5x

⇒ Hypotenuse (H) = √(P² + B²)

⇒ H = √((3x)² + (5x)²)

⇒ H = √(9x² + 25x²)

⇒ H = √(34x²) = x√34

⇒ cos(A) = B / H

⇒ cos(A) = 5x / (x√34)

⇒ cos(A) = 5 / √34

∴ The correct answer is 5/√34.

Additional Information

Trigonometric Identities

Three fundamental identities are: sin²(A) + cos²(A) = 1, 1 + tan²(A) = sec²(A), and 1 + cot²(A) = cosec²(A).

Reciprocal Relations

The reciprocal of sine is cosecant, cosine is secant, and tangent is cotangent: cosec(A) = 1/sin(A), sec(A) = 1/cos(A), cot(A) = 1/tan(A).

Pythagoras Theorem

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

Calculations:

Step 1: Calculate the total sales of manufacturer Q:

Total sales of Q = 280 + 290 + 310 + 310 + 330

Total sales of Q = 1,520 thousand.

Step 2: Calculate the total sales of manufacturer S:

Total sales of S = 400 + 420 + 430 + 440 + 450

Total sales of S = 2,140 thousand.

Step 3: Find the ratio:

Ratio = Total sales of Q : Total sales of S

Ratio = 1,520 : 2,140

Divide both terms by 20:

Ratio = 76 : 107.

The ratio of the total sales of manufacturer Q to manufacturer S is 76:107.

Shortcut Trick

Divide the given equation by cos θ:

tan θ + 1 = √3 ⇒ tan θ = √3 − 1

Since cot θ = 1 / tan θ, we get cot θ = 1 / (√3 − 1)

Multiply numerator and denominator by (√3 + 1):

cot θ = (√3 + 1) / ((√3)² − 1²) = (√3 + 1) / 2

∴ The correct answer is (√3 + 1)/2.

Alternate Method

Given:

sin θ + cos θ = √3 cos θ

Calculations:

⇒ sin θ = √3 cos θ − cos θ

Take cos θ common on the right side:

⇒ sin θ = (√3 − 1) cos θ

We know that cot θ = cos θ / sin θ

⇒ cos θ / sin θ = 1 / (√3 − 1)

⇒ cot θ = 1 / (√3 − 1)

Rationalising the denominator by multiplying and dividing by (√3 + 1):

⇒ cot θ = [1 × (√3 + 1)] / [(√3 − 1) × (√3 + 1)]

Using the algebraic identity (a − b)(a + b) = a² − b²:

⇒ cot θ = (√3 + 1) / (3 − 1)

⇒ cot θ = (√3 + 1) / 2

∴ The correct answer is (√3 + 1)/2.

Additional Information

Trigonometric Identities

sin² θ + cos² θ = 1, 1 + tan² θ = sec² θ, and 1 + cot² θ = cosec² θ.

Trigonometric Ratios

tan θ = sin θ / cos θ and cot θ = cos θ / sin θ. The reciprocal relations are sin θ = 1 / cosec θ, cos θ = 1 / sec θ, and tan θ = 1 / cot θ.

Rationalisation

When the denominator contains a surd, we multiply the numerator and denominator by its conjugate to rationalize it, applying the rule (a + b)(a − b) = a² − b².

Calculation:

$-\frac{7}{6} + \frac{4}{9} \div \frac{5}{3} \times \frac{1}{16}$

⇒ $-\frac{7}{6} + \frac{4}{9} \times \frac{3}{5} \times \frac{1}{16}$

⇒ $-\frac{7}{6} + \frac{1}{3} \times \frac{1}{5} \times \frac{1}{4}$

⇒ $-\frac{7}{6} + \frac{1}{60}$

⇒ $\frac{(-70+1)}{60}$ = $\frac{-69}{60}$ = $-\frac{23}{20}$

Hence, the correct answer is option (1).

Shortcut Trick

The perimeter of a sector is the sum of its arc length (l) and twice its radius (2r).

Arc Length (l) = (θ ÷ 360°) × 2πr

l = (15° ÷ 360°) × 2 × 3.14 × 14

l = (1 ÷ 24) × 87.92 ≈ 3.66 cm

Perimeter = l + 2r = 3.66 + 2(14) = 3.66 + 28 = 31.66 cm

∴ The correct answer is 31.66 cm.

Alternate Method

Given:

Angle of the circular arc (θ) = 15°

Radius of the circular arc (r) = 14 cm

Value of π = 3.14

Formula Used:

Length of Arc (l) = (θ ÷ 360°) × 2πr

Perimeter of a Sector = Length of Arc + 2 × Radius

Calculations:

⇒ l = (15 ÷ 360) × 2 × 3.14 × 14

⇒ l = (1 ÷ 24) × 87.92

⇒ l = 87.92 ÷ 24

⇒ l ≈ 3.6633 cm

⇒ Perimeter of the Sector = l + 2r

⇒ Perimeter = 3.6633 + (2 × 14)

⇒ Perimeter = 3.6633 + 28

⇒ Perimeter = 31.6633 cm

⇒ Rounding up to two decimal places, Perimeter = 31.66 cm

∴ The correct answer is 31.66 cm.

Additional Information

Area of a Sector

The area of a sector with central angle θ is given by A = (θ ÷ 360°) × πr². Alternatively, if the arc length l is known, the area can be directly calculated as A = (l × r) ÷ 2.

Length of a Circular Arc

The arc length represents the distance along the curved edge of the circle and is calculated as l = (θ ÷ 360°) × 2πr.

Perimeter vs Arc Length

Students often confuse the perimeter of a sector with the arc length. The arc length is merely the curved boundary, while the full perimeter of the sector must include the two straight radii boundary lines, establishing the formula: Perimeter = l + 2r.

Shortcut Trick

By the Remainder Theorem, the remainder when f(m) is divided by (m − 2) is simply f(2).

Substitute m = 2 into the given polynomial directly:

f(2) = 2⁵ + 5(2)⁴ − 3(2) + 7

f(2) = 32 + 80 − 6 + 7 = 113

∴ The correct answer is 113.

Alternate Method

Given:

Polynomial f(m) = m⁵ + 5m⁴ − 3m + 7

Divisor = (m − 2)

Formula Used:

Remainder Theorem: When a polynomial P(x) is divided by (x − a), the remainder is P(a).

Calculation:

Equate the divisor to zero to find the value of m:

⇒ m − 2 = 0

⇒ m = 2

Now, substitute m = 2 in the given polynomial f(m) to find the remainder:

⇒ Remainder = f(2) = (2)⁵ + 5(2)⁴ − 3(2) + 7

⇒ f(2) = 32 + 5 × 16 − 6 + 7

⇒ f(2) = 32 + 80 − 6 + 7

⇒ f(2) = 112 − 6 + 7

⇒ f(2) = 106 + 7

⇒ f(2) = 113

∴ The correct answer is 113.

Additional Information

Remainder Theorem

If a polynomial p(x) of degree n ≥ 1 is divided by a linear polynomial (x − a), then the remainder is exactly p(a). This entirely eliminates the need to perform long division to find the remainder.

Factor Theorem

A special case of the Remainder Theorem which states that a polynomial p(x) has a factor (x − a) if and only if p(a) = 0. In simpler terms, if the remainder is zero, the divisor perfectly divides the polynomial.

Division Algorithm for Polynomials

For any polynomials p(x) and g(x) with g(x) ≠ 0, there exist unique polynomials q(x) and r(x) such that p(x) = g(x) × q(x) + r(x), where either r(x) = 0 or the degree of r(x) is strictly less than the degree of g(x).

Shortcut Trick

From the first statement, 18% of y + 24% of b = 30 ⇒ 3y + 4b = 500.

From the second statement, 3y − 2b = 200.

Subtracting the second equation from the first gives 6b = 300 ⇒ b = 50.

Substituting b = 50 into 3y − 2b = 200 gives 3y = 300 ⇒ y = 100.

The ratio of y to b is 100 : 50 = 2 : 1, so y² : b² = 2² : 1² = 4 : 1.

∴ The correct answer is 4 : 1.

Alternate Method

Given:

18% of yellow marbles (y) + 24% of blue marbles (b) = 30

Three times yellow marbles − Two times blue marbles = 200

y = 100 1 unit 1 unit

b = 50 1 unit

Ratio y² : b² = 2² : 1² = 4 : 1

Calculations:

⇒ (18/100)y + (24/100)b = 30

⇒ 18y + 24b = 3000

Dividing the entire equation by 6, we get:

⇒ 3y + 4b = 500   ... (Equation 1)

From the second condition given in the question:

⇒ 3y − 2b = 200   ... (Equation 2)

Subtracting Equation 2 from Equation 1:

⇒ (3y + 4b) − (3y − 2b) = 500 − 200

⇒ 6b = 300

⇒ b = 50

Substitute b = 50 into Equation 2:

⇒ 3y − 2(50) = 200

⇒ 3y − 100 = 200

⇒ 3y = 300

⇒ y = 100

Now, finding the required ratio of y² to b²:

⇒ y/b = 100/50 = 2/1

⇒ y² : b² = 2² : 1² = 4 : 1

∴ The correct answer is 4 : 1.

Additional Information

Duplicate Ratio

The ratio of the squares of two numbers is known as their duplicate ratio. For a base ratio of a : b, the duplicate ratio is calculated as a² : b².

Sub-duplicate Ratio

The ratio of the square roots of two values is known as the sub-duplicate ratio. For a base ratio of a : b, it is written as √a : √b.

Linear Equations Elimination

When solving two linear equations, making the coefficient of one variable identical and subtracting the equations is a highly efficient way to eliminate that variable and find the other.

Shortcut Trick By the triangle inequality theorem, any side of a triangle must be strictly less than half of its perimeter. Maximum possible length of a single side < Perimeter ÷ 2 Maximum possible length < 52 ÷ 2 Maximum possible length < 26 units ∴ The correct answer is < 26 units.

Alternate Method Given: Perimeter of the triangle = 52 units. Formula Used: According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

a b c Perimeter = a + b + c = 52 Triangle Inequality: a + b > c

Calculations: ⇒ Let the sides of the triangle be a, b, and c. ⇒ The perimeter is the sum of all sides: a + b + c = 52 ⇒ From the triangle inequality theorem: a + b > c ⇒ Add 'c' to both sides of the inequality: ⇒ a + b + c > c + c ⇒ Substitute the perimeter value (a + b + c = 52): ⇒ 52 > 2c ⇒ Divide by 2: ⇒ 26 > c, which means c < 26. ⇒ By applying the exact same logic to the other sides, we get a < 26 and b < 26. ⇒ Thus, all sides must be strictly less than 26 units in length. ∴ The correct answer is < 26 units.

Additional Information Triangle Inequality Theorem (Difference) The difference between the lengths of any two sides of a triangle is always strictly less than the length of the third side (|a − b| < c). Perimeter & Semi-perimeter The perimeter is the total boundary length (P = a + b + c). The semi-perimeter (s) is half of the perimeter (s = P ÷ 2), heavily used in Heron's formula for the area: Area = √[s(s − a)(s − b)(s − c)]. Types of Triangles Based on Sides Scalene (all three sides are of different lengths), Isosceles (exactly two sides are equal), and Equilateral (all three sides are equal, each measuring exactly one-third of the perimeter).

Given:

The number of students who passed the GATE exam in:

2018 = 140

2019 = 180

We need to calculate the percentage increase in the number of students in 2019 over 2018.

Formula Used:

Percentage Increase = ((Increase in Value) / (Original Value)) × 100

Calculations:

Step 1: Calculate the increase in the number of students:

Increase = 2019 value - 2018 value

Increase = 180 - 140 = 40.

Step 2: Apply the formula:

Percentage Increase = (40 / 140) × 100

Percentage Increase = 4000 / 140 = 28.57 (rounded off to two decimal places).

The percentage increase in the number of students in 2019 over 2018 is 28.57%.

Shortcut Trick

Total Capacity (Total Work) = LCM(12, 18, 8) = 72 units.

Combined Efficiency = (72 ÷ 12) + (72 ÷ 18) − (72 ÷ 8)

⇒ 6 + 4 − 9 = 1 unit/hour.

Time required to fill the tank = Total Capacity ÷ Combined Efficiency = 72 ÷ 1 = 72 hours.

∴ The correct answer is 72 hours.

Alternate Method

Given:

Time taken by Pipe 1 to fill the tank = 12 hours

Time taken by Pipe 2 to fill the tank = 18 hours

Time taken by Pipe 3 to empty the tank = 8 hours

Method Used: LCM Method

Calculations:

⇒ The total combined efficiency when all three pipes are opened simultaneously = 6 + 4 − 9 = 1 unit/hour.

⇒ Time taken to fill the empty tank = Total Work ÷ Net Efficiency

⇒ Time taken = 72 ÷ 1 = 72 hours.

∴ The correct answer is 72 hours.

Additional Information

Inlet and Outlet Pipes

An inlet pipe fills the tank (positive efficiency), while an outlet pipe empties it (negative efficiency). The net efficiency of the system is the algebraic sum of their individual efficiencies.

Time and Work Concept

Pipes and cisterns problems are identical to regular time and work problems. The total volume of the tank acts as the Total Work, and the filling/emptying rates act as the working Efficiency.

Fractional Method

If inlet pipes fill a tank in x and y hours, and an outlet pipe empties it in z hours, the net part of the tank filled in 1 hour is given by (1/x) + (1/y) − (1/z).

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

Key Points

• The word "mesmerised" means to hold the attention of someone to the exclusion of all else or so as to transfix them. (मंत्रमुग्ध)
• Example: The magician's performance left the audience mesmerised.
• "Fascinated" refers to being intensely interested in or attracted by someone or something. (मोहित)
• Example: She was fascinated by the intricate details of the painting.

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

Additional Information

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

• Irritated (चिढ़): Feeling annoyed or slightly angry.
• Example: She was irritated by the constant noise outside her window.
• Annoyed (परेशान): Slightly angry; irritated.
• Example: He was annoyed by the delay in the delivery of his package.
• Bored (उबाऊ): Feeling weary and impatient because one is unoccupied or lacks interest in one's current activity.
• Example: The lecture was so dull that the students appeared bored.

The correct answer is: Option 3.

Key Points

• The idiom "Easy does it" means to do something slowly and carefully. (धीरे और सावधानी से कुछ करना)
• Example: When lifting heavy objects, remember, easy does it to avoid injury.
• "Easy does it" advises caution and a measured approach. (सावधानी और मापी हुई पद्धति की सलाह देना)
• Example: While painting the delicate artwork, easy does it to ensure precision.

Therefore, the correct answer is: Option 3.

Additional Information

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

• Option 1: Used to tell someone to do something rarely (किसी को कुछ कम बार करने के लिए कहना)
• Example: You should eat junk food rarely to maintain good health.
• Option 2: Used to tell someone to do something violently (किसी को कुछ हिंसक रूप से करने के लिए कहना)
• Example: He was told to act violently in the action scene of the movie.
• Option 4: Used to tell someone to do something horridly (किसी को कुछ भयानक तरीके से करने के लिए कहना)
• Example: The villain in the story was asked to act horridly to frighten the audience.

The correct answer is: Dishonest.

Key Points

• The word "candid" means being straightforward, honest, and open in speech or expression. (स्पष्टवादी)
• Example: She gave a candid interview about her experiences in the industry.
• "Dishonest" refers to someone who is not honest, who lies or cheats. (बेईमान)
• Example: His dishonest behavior led to a loss of trust among his colleagues.
• Hence, we can infer that the opposite of 'candid' is 'dishonest'.

Therefore, the correct answer is: Dishonest.

Additional Information

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

• Tender (कोमल): Showing gentleness and concern or sympathy.
• Example: She touched the injured bird with tender care.
• Guileless (निर्दोष): Innocent and without deception.
• Example: His guileless smile won everyone's heart.
• Irrelevant (अप्रासंगिक): Not connected with or relevant to something.
• Example: His comments were completely irrelevant to the discussion.

The correct answer is option 1.

Key Points

The misspelled word is 'necesity'.

• The correct spelling would be 'necessity'.
• The word 'necessity' is a noun which means 'the fact of being required or indispensable'. (आवश्यकता)
• For Example -
• Food and water are basic necessities for human survival.
• Therefore, the correct answer would be 'necesity'.

Additional Information

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

• Educational - relating to the provision of education (शैक्षिक)
• Excellent - (उत्कृष्ट)
• Technology - the application of scientific knowledge for practical purposes (प्रौद्योगिकी)

The correct answer is: Option 2 i.e. Do something in an inexpensive and easy way.

Key Points

• The idiom "Cut corners" means to do something in the easiest, cheapest, or fastest way. (किसी काम को आसान, सस्ता, या सबसे तेज़ तरीके से करना)
• Example: The contractor cut corners by using low-quality materials, which caused the building to collapse.
• "Cut corners" refers to avoiding steps that are necessary or skipping important details to save time or money. (समय या पैसे बचाने के लिए आवश्यक कामों को छोड़ना)
• Example: When you cut corners on safety, you put everyone at risk.

Therefore, the correct answer is: Option 2 i.e. Do something in an inexpensive and easy way.

Additional Information

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

• Complete a work (काम पूरा करना): To finish a task or project.
• Example: She managed to complete the work before the deadline.
• Find a suitable place (उचित स्थान खोजें): To locate an appropriate location for something.
• Example: They need to find a suitable place for the new office.
• Clear a space (जगह साफ़ करना): To remove obstacles or clutter from an area.
• Example: We need to clear a space in the garage for the new car.