SSC CGL 2024 Tier-I Official Paper (Held On: 25 Sept, 2024 Shift 3) (Previous Year Paper)

Given word: 'DOWNSIZE'

After arranged in alphabetical order:

 

Now,

there are Two alphabets "D" and "N" that will remain unchanged if each of the letters in the word 'DOWNSIZE' is arranged in English alphabetical order.

Hence, the correct answer is "Option 3".

The logic followed here is:

Logic: 1st number × 4 = 2nd number.

Now,

For, 19 is related to 76

⇒ 19 × 4 = 76.

For, 27 is related to 108

⇒ 27 × 4 = 108.

Similarly, for 49

⇒ 49 × 4 = 196.

So, 49 will be related to 196.

Hence, the correct answer is "Option 4".

Counting the triangles are there in the given figure:

Hence, the correct answer is "17".

The logic followed here is:

Given Sequence: C A R E T _ R C A _ E T C R C A R _ T C R _ A R E T _ R C A

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

Option 1) CRCCE → C A R E T C R / C A R E T C R / C A R C T C R / C A R E T E R / C A

Option 2) CRCEC → C A R E T C R / C A R E T C R / C A R C T C R / E A R E T C R / C A

Option 3) CERCC → C A R E T C R / C A E E T C R / C A R R T C R / C A R E T C R / C A

Option 4) CRECC → C A R E T C R / C A R E T C R / C A R E T C R / C A R E T C R / C A

Here, 'C A R E T C R' is the pattern followed.

Hence, the correct answer is "CRECC".

The logic followed here is:

'LISP' is related to 'ORHK'

'WAND' is related to 'DZMW'

Similarly, 'BOTE' is related to:

Hence, the correct answer is "YLGV".

BODMAS Table: 

After decoding signs:

 

Given equation: 61 − 42 × 6 ÷ 9 + 27 = ?

After interchanging the signs then:

⇒ 61 + 42 ÷ 6 × 9 - 27 = ?

⇒ ? = 61 + 7 × 9 - 27

⇒ ? = 61 + 63 - 27

⇒ ? = 124 - 27 = 97

Hence, the correct answer is "97".

BODMAS Table: 

After decoding signs:

 

Given equation: 2 × 16 ÷ 36 + 42 − 6 = ?

After interchanging the signs then:

⇒ 2 + 16 - 36 × 42 ÷ 6 = ?

⇒ ? = 2 + 16 - 36 × 7

⇒ ? = 2 + 16 - 252

⇒ ? = 18 - 252

⇒ ? = -234

Hence, the correct answer is "-234".

The logic followed here is:

Hence, the correct answer is "GKSN".

The logic followed here is:

Number of arc is increasing by 1 alternatively.

Hence, the correct answer is "Option 4".

The logic followed here is:

Option 1) UOW

Option 2) IAK

Option 3) PJR

Option 4) LFN

Hence, the correct answer is "IAK".

The correct mirror image of the given figure is:

Hence, the correct answer is "Option 3".

The paper will look when unfolded:

Hence, the correct answer is "Option 1".

The logic followed here is:

Hence, the correct answer is "58, 58".

The logic followed here is:

Logic: Opposites

Here,

‘Tame’ is related to ‘Wild’ → Tame and Wild are opposites.

Similarly,

‘Adjacent’ is related to ‘Distant’ → Adjacent and Distant are opposites.

Hence, the correct answer is "Option 2".

According to the given information:

The number of students who like to study all the three subjects = 4

The number of students who like to study biology only = 20

Thus, the difference between the number of students who like to study all the three subjects and that of students who like to study biology only = 20 - 4 = 16.

Hence, the correct answer is "16".

The correct mirror image of the given figure is:

Hence, the correct answer is "Option 1".

The logic followed here is:

Number of sides of figure is decreasing by 1.

Number of dots are decreasing by 1.

Number of sides - 1 = Number of dots.

Hence, the correct answer is "Option 2".

For find the opposite surface of '2', we need to compare the common surface in the given dices,

So, From Dice 1 and Dice 2,

From both dices, 2 is common and on the same surface. So, all remaining surface will be opposite to each other.

9 is opposite to 8.

5 is opposite to 6.

So, 3 will be opposite to 2.

Hence, the correct answer is "3".

According to the given information:

Thus, the code for 'A' = 2.

Hence, the correct answer is "2".

According to the given information:

So, people travel by bus and car both, but NOT by train = 9

Hence, the correct answer is "9".

The logic followed here is:

Logic: (1st number × 3) = 2nd number.

Now,

For, 14 : 42

⇒ (14 × 3) = 42.

For, 26 : 78

⇒ (26 × 3) = 78.

Similarly, for 47 : ?

⇒ (47 × 3) = 141.

Hence, the correct answer is "Option 4".

The logic followed here is: 1st number × 13 = 2nd number.

1) 15 : 195 → 15 × 13 = 195

2) 11 : 143 → 11 × 13 = 143

3) 13 : 169 → 13 × 13 = 169

4) 12 : 152 → 12 × 13 = 156 ≠ 152

Hence, the correct answer is "12 : 152".

The logic followed here is:

Logic: 1st number + 11 = 2nd number.

Now,

For, 45 : 56

⇒ 45 + 11 = 56.

For, 62 : 73

⇒ 62 + 11 = 73.

Similarly, for 19 ∶ ?

⇒ 19 + 11 = 30.

Hence, the correct answer is "Option 2".

The logic followed here is:

'CASE' is written as 'FTBD'

'CAKE' is written as 'FLBD'

Similarly, 'CALF' will be written as:

Hence, the correct answer is "GMBD".

Family Chart:

Decoding the Symbols:

 

Given: P + Q × R ÷ S ÷ T

It means: P is the husband of Q, R is the sister of Q, R is the daughter of S and S is the daughter of T.

So, Q is the "Daughter’s daughter" of T.

Hence, the correct answer is "Daughter’s daughter".

The correct answer is 2015.

Key Points

• Ujala Yojana was launched by the Government of India in 2015.
• The scheme is aimed at promoting energy efficiency by distributing LED bulbs to households at a subsidized rate.
• It was initially launched under the name "Domestic Efficient Lighting Programme (DELP)" and later renamed as Unnat Jyoti by Affordable LEDs for All (UJALA).
• The scheme is implemented by Energy Efficiency Services Limited (EESL), a joint venture of PSUs under the Ministry of Power.

Additional Information

• Energy Efficiency Services Limited (EESL)
• EESL is a public sector entity responsible for implementing energy efficiency projects in India.
• It was established under the Ministry of Power, Government of India.
• EESL works towards promoting energy conservation and implementing energy efficiency projects on a large scale.
• It plays a critical role in implementing the Ujala Yojana by procuring and distributing LED bulbs.
• LED Bulbs
• Light Emitting Diode (LED) bulbs are highly energy-efficient lighting solutions.
• They consume significantly less electricity compared to traditional incandescent and CFL bulbs.
• LED bulbs have a longer lifespan and provide better illumination.
• They are environmentally friendly and help reduce carbon emissions.
• Energy Efficiency
• Energy efficiency refers to the optimal use of energy to perform a task, reducing energy consumption without compromising on output.
• It involves adopting practices and technologies that consume less energy for the same level of service or output.
• Energy efficiency measures help in reducing energy bills, conserving natural resources, and mitigating climate change.
• It is a key component of sustainable development and energy policy.

The correct answer is Marginal Farmers Development Agency (MFDA).

Key Points

• The Marginal Farmers Development Agency (MFDA) was initiated by the Government of India to address the specific needs of marginal farmers.
• It aims to improve the socio-economic conditions of marginal farmers by providing them with necessary resources, training, and support.
• MFDA focuses on increasing agricultural productivity and income through various development programs and strategies tailored for the target group.
• The scheme includes components such as credit facilities, supply of inputs, irrigation, training, and market linkages to ensure comprehensive support for marginal farmers.

Additional Information

• Drought Prone Areas Programme (DPAP)
• DPAP is one of the earliest area development programs initiated by the Government of India to tackle the problems of drought in vulnerable areas.
• The primary objective is to minimize the adverse effects of drought on crop production, livestock, and productivity of land, water, and human resources.
• Hill Areas Development Programme (HADP)
• HADP was launched to ensure the socio-economic development of hill areas and their people.
• The program aims at addressing the unique problems faced by hill areas, including environmental degradation and lack of basic infrastructure.
• Command Area Development & Water Management Programme (CADP)
• CADP was initiated to enhance the utilization of irrigation potential created under major and medium irrigation projects.
• The program focuses on improving water use efficiency and ensuring equitable distribution of water.

The correct answer is 8.

Key Points

• India sent a team of 8 athletes to represent the country at the 1924 Paris Summer Olympics.
• This was the second time India participated in the Summer Olympics, the first being in 1900.
• The athletes competed in various track and field events.
• Although the Indian athletes did not win any medals at the 1924 Olympics, their participation marked an important step in India's Olympic history.

Additional Information

• 1924 Paris Summer Olympics
• The 1924 Summer Olympics, officially known as the Games of the VIII Olympiad, were held in Paris, France.
• This edition of the Olympic Games saw the participation of 3,089 athletes from 44 countries.
• These Games were notable for the introduction of the Olympic motto "Citius, Altius, Fortius" (Faster, Higher, Stronger).
• The 1924 Olympics also featured the first Olympic Village, where athletes could stay during the competitions.
• India's Participation in the Olympics
• India first participated in the Olympics in 1900, when Norman Pritchard represented the country and won two silver medals in athletics.
• After the 1924 Paris Olympics, India has consistently participated in every Summer Olympic Games.
• India's first gold medal was won by the men's field hockey team in the 1928 Amsterdam Olympics.
• Over the years, India has earned numerous medals in various sports, with significant achievements in field hockey, wrestling, shooting, and boxing.
• Track and Field Events
• Track and field is a sport comprising various competitive athletic contests based on running, jumping, and throwing.
• The sport is categorized under the umbrella of athletics, which also includes road running, cross country running, and racewalking.
• Track and field events are typically held on a running track and a grass field within a stadium.
• Some of the standard events in track and field include sprints, middle-distance and long-distance races, hurdles, relays, long jump, high jump, pole vault, shot put, discus throw, javelin throw, and hammer throw.

The correct answer is Baiga.

Key Points

• Arjun Singh Dhurve was honored with the Tulsi Samman for his exceptional contribution to the Baiga tribal music and dance form.
• The Baiga tribe is one of the prominent tribes in Madhya Pradesh, India, known for its vibrant culture and traditional practices.
• The Baiga dance is an integral part of their customs, often performed during festivals and important community events.
• The Tulsi Samman is a prestigious award given by the Madhya Pradesh government to individuals who have made significant contributions in the field of tribal art and culture.

Additional Information

• Tribal Music and Dance Forms
• Mardana: This is a traditional form of music and dance practiced by the tribal communities in various regions of India. It involves the use of traditional instruments and vibrant costumes.
• Pawara: A tribal dance form from the Pawara community, known for its rhythmic movements and cultural significance. It is usually performed during festivals and special occasions.
• Baiga: The Baiga tribe's dance form, which Arjun Singh Dhurve is recognized for, is characterized by its unique steps, rhythmic beats, and cultural expressions. The dancers often wear traditional attire made from natural materials.
• Jhumta: A traditional dance form performed by various tribal communities, often during harvest seasons and festivals. It involves group formations and synchronized movements.
• Tulsi Samman
• The Tulsi Samman is an award given by the Madhya Pradesh government to recognize and honor individuals who have made outstanding contributions to tribal art and culture.
• It aims to preserve and promote the rich cultural heritage of the tribal communities in the region.
• The award is named after the renowned poet and saint, Tulsidas, and is a symbol of cultural pride and recognition.

The correct answer is Birsa Munda.

Key Points

• Birsa Munda was an Indian tribal freedom fighter, religious leader, and folk hero who belonged to the Munda tribe.
• He spearheaded a tribal religious movement and is known for leading the Munda Rebellion against British colonial rule.
• His birth anniversary is on November 15, which is now celebrated as 'Janjatiya Gaurav Divas'.
• Birsa Munda's leadership and efforts in the fight against British oppression made him a significant figure in Indian history.

Additional Information

• Birsa Munda
• Born on November 15, 1875, in Ulihatu, Jharkhand, he played a key role in the Indian independence movement.
• He is known for his role in the Munda Rebellion (Ulgulan) against the British Raj and the landlords.
• Birsa Munda aimed to establish the Munda Raj and remove the British government and missionaries from the region.
• Despite his early death at the age of 25 on June 9, 1900, his legacy as a symbol of tribal resistance continues to inspire many.
• Janjatiya Gaurav Divas
• Janjatiya Gaurav Divas is a day to honor the contributions of tribal communities to the history and culture of India.
• The government of India declared November 15 as Janjatiya Gaurav Divas in 2021 to commemorate Birsa Munda's birth anniversary.
• This day is celebrated with various events and programs to recognize the sacrifices and achievements of tribal leaders and communities.
• It aims to promote tribal culture, heritage, and rights, ensuring the preservation and promotion of their unique identity.
• Munda Rebellion (Ulgulan)
• The Munda Rebellion, also known as Ulgulan, was a significant uprising against British colonial rule in the late 19th century.
• Led by Birsa Munda, the rebellion was a reaction to the exploitative policies of the British and local landlords.
• It sought to establish self-rule and end the oppression faced by the tribal communities.
• The rebellion, though eventually suppressed, highlighted the resilience and determination of the tribal people in their fight for justice.

The correct answer is Sikhism.

Key Points

• Hola Mohalla is a significant festival celebrated by the Sikh community.
• The festival was started by the tenth Sikh Guru, Guru Gobind Singh, in 1701.
• Hola Mohalla is typically celebrated a day after the Hindu festival of Holi.
• It is a week-long festival that involves mock battles, exhibitions, martial arts, poetry, and music.

Additional Information

• Sikhism
• Sikhism is a monotheistic religion that originated in the Punjab region of the Indian subcontinent around the end of the 15th century.
• The fundamental beliefs of Sikhism, articulated in the sacred scripture Guru Granth Sahib, include faith and meditation on the name of the one creator, unity of all humankind, engaging in selfless service, striving for justice for the benefit and prosperity of all, and honest conduct and livelihood while living a householder's life.
• The Sikh community celebrates various festivals like Baisakhi, Guru Nanak Jayanti, and Hola Mohalla.
• Sikhism was founded by Guru Nanak and has ten gurus, with Guru Gobind Singh being the last human Guru. Guru Granth Sahib is the eternal Guru of the Sikhs.
• Guru Gobind Singh
• Guru Gobind Singh was the tenth Sikh Guru, a spiritual master, warrior, poet, and philosopher.
• He was born on 22 December 1666 in Patna, Bihar, India.
• He founded the Khalsa in 1699, a collective body of initiated Sikhs, and introduced the Five Ks, five articles of faith that Khalsa Sikhs wear at all times.
• Guru Gobind Singh also declared that the Guru Granth Sahib, the Sikh holy scripture, would be the eternal Guru after him.
• Holi
• Holi is a popular ancient Hindu festival, also known as the "festival of colors" or the "festival of love".
• The festival signifies the arrival of spring, the end of winter, the blossoming of love, and for many, it is a festive day to meet others, play and laugh, forget and forgive, and repair broken relationships.
• The festival is also celebrated as a thanksgiving for a good harvest.
• It lasts for a night and a day, starting on the evening of the Purnima (Full Moon day) falling in the Hindu calendar month of Phalguna, which falls around the middle of March in the Gregorian calendar.

The correct answer is 101st Constitutional Amendment Act.

Key Points

• The 101st Constitutional Amendment Act was passed by the Indian Parliament in August 2016.
• This amendment led to the introduction of the Goods and Services Tax (GST) in India, which came into effect on 1st July 2017.
• The GST Council was formed as per Article 279A of the Indian Constitution, inserted by the 101st Amendment.
• The GST Council is responsible for making recommendations on the rates of taxes, goods and services that may be subjected or exempted from GST, and other related matters.

Additional Information

• 102nd Constitutional Amendment Act
• This amendment came into effect in 2018.
• It granted constitutional status to the National Commission for Backward Classes (NCBC).
• Article 338B was inserted to provide for the NCBC.
• 100th Constitutional Amendment Act
• This amendment was passed in 2015.
• It dealt with the exchange of certain enclaves between India and Bangladesh and the demarcation of the boundary between the two countries.
• It facilitated the implementation of the Land Boundary Agreement (LBA) between India and Bangladesh.
• 103rd Constitutional Amendment Act
• This amendment came into effect in 2019.
• It introduced a 10% reservation for economically weaker sections (EWS) in government jobs and educational institutions.
• Articles 15(6) and 16(6) were added to the Constitution to enable this reservation.
• Goods and Services Tax (GST)
• GST is a comprehensive, multi-stage, destination-based tax that is levied on every value addition.
• It replaced multiple indirect taxes previously levied by the central and state governments.
• GST is categorized into Central GST (CGST), State GST (SGST), Integrated GST (IGST), and Union Territory GST (UTGST).
• The GST Council, chaired by the Union Finance Minister, is a federal body where both the center and the states participate in the decision-making process.

The correct answer is Acetic acid.

Key Points

• Acetic acid is the primary component of vinegar.
• It is commonly used as a souring agent in pickled vegetables and sauces.
• Acetic acid has a distinctive sour taste and pungent smell.
• In addition to its use in culinary applications, acetic acid is also used as a raw material in the production of various chemical compounds.

Additional Information

• Citric acid:
• Citric acid is a weak organic acid found naturally in citrus fruits like lemons and oranges.
• It is often used as a preservative and flavoring agent in foods and beverages.
• Citric acid is also used in cleaning products due to its ability to remove stains and limescale.
• Tartaric acid:
• Tartaric acid is a naturally occurring acid found in grapes and bananas.
• It is commonly used in baking powders and as an acidulant in foods and beverages.
• Tartaric acid is also used in the wine-making process to control acidity.
• Formic acid:
• Formic acid is a simple carboxylic acid found in the venom of ants and bees.
• It is used in agriculture for preserving livestock feed and as a miticide in beekeeping.
• Formic acid is also used in various industrial applications, including leather production and textile processing.

The correct answer is Lignite.

Key Points

• Lignite is a type of coal that is considered to be low-grade due to its relatively low energy content.
• It is brown in color and has a soft texture, making it easier to break apart compared to harder forms of coal.
• Lignite has a high moisture content, sometimes as much as 45-66%, which reduces its efficiency as a fuel source.
• Despite its lower quality, lignite is often used in electricity generation, particularly in power plants located near lignite mines.

Additional Information

• Metallurgical Coal
• Also known as coking coal, it is primarily used in steel production.
• It has a high carbon content and low impurities, which are essential for producing high-quality coke.
• Metallurgical coal undergoes a coking process where it is heated in the absence of air to remove volatile substances, leaving behind a solid carbon mass called coke.
• Bituminous Coal
• It is a middle-rank coal between sub-bituminous and anthracite.
• Bituminous coal is used extensively for electricity generation and as a fuel in industries.
• It has a higher carbon content than lignite and sub-bituminous coal, providing more energy when burned.
• Anthracite
• It is the highest rank of coal due to its high carbon content and low impurities.
• Anthracite is hard, shiny, and has the highest energy content of all coal types.
• It is used for heating residential and commercial buildings and in some industrial processes.

The correct answer is Argon.

Key Points

• Argon is an inert gas, which means it does not react with the materials being welded, making it ideal for welding processes.
• It is used as a shielding gas to protect the weld area from atmospheric gases such as oxygen, nitrogen, and water vapor that can cause defects in the weld.
• Argon provides a stable arc and excellent control over the welding process, which is crucial for welding materials like titanium, aluminium, stainless steel, and magnesium.
• It is often used in Gas Tungsten Arc Welding (GTAW) and Gas Metal Arc Welding (GMAW) processes.

Additional Information

• Gas Tungsten Arc Welding (GTAW)
• Also known as Tungsten Inert Gas (TIG) welding, GTAW uses a non-consumable tungsten electrode to produce the weld.
• Argon is the most commonly used shielding gas in GTAW due to its inert properties and ability to create a stable arc.
• GTAW is known for producing high-quality, precision welds and is often used for welding thin materials.
• Gas Metal Arc Welding (GMAW)
• Also known as Metal Inert Gas (MIG) welding, GMAW uses a continuous wire feed as an electrode and an inert or semi-inert gas mixture for shielding.
• Argon is often used in GMAW, sometimes mixed with other gases like carbon dioxide or helium to optimize the welding process for different materials.
• GMAW is widely used in industrial applications due to its high welding speed and ease of automation.
• Importance of Shielding Gas
• Shielding gases like argon protect the weld pool from contamination by atmospheric gases, which can cause porosity and weld defects.
• They help stabilize the arc, improve weld quality, and ensure a clean and strong weld.
• The choice of shielding gas depends on the materials being welded and the specific welding process used.

The correct answer is Tansen Samman.

Key Points

• The Tansen Samman is an esteemed award started by the Government of Madhya Pradesh in 1980.
• It is given to the exponents of Hindustani classical music in recognition of their contribution and excellence in the field.
• The award is named after the legendary musician Tansen, who was one of the nine gems in the court of Emperor Akbar.
• The Tansen Samman includes a cash prize and a citation, honoring the recipient's lifelong dedication to Indian classical music.

Additional Information

• Hindustani Classical Music
• Hindustani classical music is one of the two major traditions of Indian classical music, the other being Carnatic music.
• It originated in North India and has a rich history that dates back to the Vedic times.
• The music is characterized by its raga (melodic framework) and tala (rhythmic cycle).
• Some of the prominent exponents of Hindustani classical music include Ravi Shankar, Zakir Hussain, and Bhimsen Joshi.
• Tansen
• Tansen was a prominent figure in Indian classical music and is often regarded as one of the greatest musicians of all time.
• He was a musician in the court of Emperor Akbar in the 16th century.
• Tansen is known for his contributions to the development of Hindustani classical music, especially the dhrupad style of singing.
• His legacy continues to influence Indian music to this day.
• Other Notable Awards in Music
• Kumar Gandharva Samman: Named after the famous classical singer Kumar Gandharva, this award is given to young musicians for their contribution to Indian classical music.
• Kishore Kumar Samman: This award is given in the field of film music and is named after the legendary playback singer Kishore Kumar.
• Iqbal Samman: This award is given for outstanding contribution in the field of Urdu literature and music.

The correct answer is Mumbai.

Key Points

• The first successful textile mill in India was established in Mumbai in 1854.
• Mumbai was chosen due to its proximity to the cotton-growing regions, a well-connected port, and access to a large labor force.
• The establishment of this mill marked the beginning of the modern textile industry in India.
• Mumbai eventually became one of the leading centers for the textile industry, contributing significantly to India's economy.

Additional Information

• Textile Industry in India:
• The textile industry is one of the oldest and most significant industries in India, known for its vast variety of products including cotton, silk, and woolen textiles.
• India is one of the largest producers of textiles and garments in the world, and the industry plays a crucial role in the economy by providing employment to millions of people.
• The industry is divided into two broad segments: the organized mill sector and the unorganized decentralized sector.
• The organized mill sector comprises large-scale spinning and composite mills, while the decentralized sector includes handlooms, power looms, and knitting and garment units.
• Historical Context:
• The establishment of the first textile mill in Mumbai by Cowaszee Nanabhoy Davar in 1854 was a landmark event in the history of India's industrialization.
• This period saw the beginning of modern industry in India, with the textile sector leading the way.
• Mumbai's transformation into a major industrial hub can be attributed to the growth of the textile industry.
• Over time, the success of the textile mills in Mumbai attracted more entrepreneurs to invest in the industry, leading to the establishment of numerous mills in the region.
• Impact on Economy:
• The growth of the textile industry significantly contributed to India's economic development by generating employment, promoting trade, and boosting industrialization.
• The industry also played a vital role in the freedom struggle by supporting the Swadeshi movement, which encouraged the use of domestically produced goods over imported ones.
• Today, the textile industry continues to be a major contributor to India's GDP and exports, reflecting its enduring importance to the economy.

The correct answer is Rowlatt Act, 1919.

Key Points

• The Rowlatt Act was passed by the British government in India in March 1919.
• It gave the colonial government enormous powers to repress political activities in India.
• The act allowed for the detention of political prisoners without trial for two years.
• The act led to widespread outrage and protests across India, eventually culminating in the Jallianwala Bagh massacre.

Additional Information

• India Contract Act, 1872
• The Indian Contract Act, 1872, governs the law relating to contracts in India.
• It is based on the principles of English Common Law and deals with the formation and enforcement of contracts.
• Criminal Tribes Act, 1871
• The Criminal Tribes Act was enacted by the British government to categorize certain tribes as "criminals by birth".
• It subjected these communities to severe restrictions and surveillance, with the assumption that their criminal tendencies were hereditary.
• Indian Slavery Act, 1843
• The Indian Slavery Act, 1843, was an act passed by the British government to abolish slavery in India.
• It declared the practice of slavery illegal and provided for the emancipation of all slaves in India.

The correct answer is Ice Hockey.

Key Points

• The term "Blue Line" is a significant part of the game Ice Hockey.
• It refers to the lines that divide the hockey rink into three zones: the defending zone, neutral zone, and attacking zone.
• The blue lines play a crucial role in determining offside violations in the game of ice hockey.
• They are essential in regulating the flow of the game and ensuring that players adhere to the rules related to positioning on the rink.

Additional Information

• Offside Rule in Ice Hockey
• The offside rule is one of the fundamental rules in ice hockey, aimed at preventing players from gaining an unfair advantage.
• A player is considered offside if they enter the attacking zone before the puck crosses the blue line.
• If an offside violation occurs, the referee stops the play, and a faceoff is conducted in the neutral zone to resume the game.
• Understanding the offside rule is crucial for players and coaches as it impacts offensive strategies and game planning.
• Zones on the Rink
• The ice hockey rink is divided into three zones by two blue lines: the defending zone, the neutral zone, and the attacking zone.
• The defending zone is the area between the blue line closest to a team's goal and the end boards.
• The neutral zone lies between the two blue lines, where much of the game's transition play occurs.
• The attacking zone is the area between the blue line farthest from a team's goal and the end boards, where a team aims to score goals.
• Importance of the Blue Line
• The blue lines are critical for maintaining the structure and flow of the game, ensuring that players adhere to the rules of positioning.
• They are used to judge offside violations and play a role in determining where faceoffs occur on the rink.
• Blue lines also contribute to the strategic aspects of the game, influencing how teams execute their offensive and defensive plays.
• Properly understanding and leveraging the blue lines can provide a competitive edge to teams in ice hockey.

The correct answer is Shiva.

Key Points

• Pushyamitra Shunga was the founder and the first ruler of the Shunga Empire in North India.
• He was a Brahmin by birth and overthrew the Maurya dynasty to establish his rule.
• According to various historical inscriptions, Pushyamitra Shunga was a devotee of the Hindu god Shiva.
• His patronage of Brahmanism and the revival of Vedic practices is well-documented in ancient texts and inscriptions.

Additional Information

• Pushyamitra Shunga:
• Pushyamitra Shunga was the founder of the Shunga dynasty, which ruled over parts of the Indian subcontinent from around 185 to 75 BCE.
• He was originally a commander-in-chief (Senapati) of the Mauryan army and later declared himself king after assassinating the last Mauryan ruler, Brihadratha.
• His reign is noted for the revival of orthodox Brahmanical practices and the decline of Buddhism, which had been patronized by the Mauryan rulers.
• Shiva:
• Shiva is one of the principal deities of Hinduism, widely revered as the Supreme Being within Shaivism, one of the major traditions within contemporary Hinduism.
• Shiva is known as "The Destroyer" within the Trimurti, the Hindu trinity that includes Brahma and Vishnu.
• He is often depicted with a third eye on his forehead, a snake around his neck, and a trident in his hand, symbolizing his roles as creator, preserver, and destroyer.
• Shiva is worshipped in the form of Lingam and has many temples dedicated to him across India.
• Shunga Dynasty:
• The Shunga dynasty succeeded the Maurya dynasty and is known for its patronage of Brahmanism and the promotion of Sanskrit literature and arts.
• They are also credited with the construction of significant architectural monuments, including the famous Bharhut Stupa.
• The Shunga period saw a resurgence of Hindu culture and the establishment of several Brahmanical institutions.
• Inscriptions:
• Inscriptions are writings engraved on materials such as stone, metal, or pottery, often used in ancient times to record important events, edicts, and proclamations.
• These inscriptions provide valuable historical information and insights into the political, social, and religious life of the period.
• Inscriptions from the Shunga period have been found in various parts of India, offering evidence of the dynasty's rule and their cultural contributions.

The correct answer is Released a special stamp, 'Wheat Revolution'.

Key Points

• The special stamp 'Wheat Revolution' was released to commemorate the success and achievements of the Green Revolution in India.
• The Green Revolution was a significant agricultural reform that began in the 1960s, aiming to increase food grain production in India.
• This period marked the introduction of high-yielding variety (HYV) seeds, chemical fertilizers, and better irrigation techniques, which led to a substantial increase in crop production, particularly wheat and rice.
• The release of the 'Wheat Revolution' stamp symbolized the country's commitment to agricultural advancement and the farmers' pivotal role in achieving food security.

Additional Information

• Green Revolution
• The Green Revolution refers to a set of research, development, and technology transfer initiatives that occurred between the 1940s and the late 1960s.
• It led to the adoption of new agricultural techniques, including the use of high-yielding variety seeds, synthetic fertilizers, pesticides, and improved irrigation methods.
• The main goal was to increase food production and prevent famine in developing countries, particularly in Asia and Latin America.
• In India, the Green Revolution began in the 1960s and resulted in the country becoming self-sufficient in food grain production within a few decades.
• High-Yielding Variety (HYV) Seeds
• These seeds are genetically modified to produce higher yields compared to traditional varieties.
• They are more responsive to fertilizers and irrigation, leading to increased productivity per hectare.
• HYV seeds played a crucial role in the success of the Green Revolution, especially for crops like wheat and rice.
• Impact of the Green Revolution in India
• The Green Revolution transformed India from a food-deficient country to one of the world's leading agricultural producers.
• It increased the production of staple crops, reduced food imports, and helped achieve food security.
• However, it also led to environmental challenges, such as soil degradation, water depletion, and increased use of chemical fertilizers and pesticides.
• Despite these challenges, the Green Revolution is considered a major milestone in India's agricultural history.

The correct answer is Khan Abdul Ghaffar Khan.

Key Points

• Khan Abdul Ghaffar Khan was a prominent leader in the Indian independence movement, advocating for non-violent resistance against British rule.
• He founded the Khudai Khidmatgar ("Servants of God") movement in 1929, which was composed primarily of Pashtuns (Pathans) from the North-West Frontier Province.
• The movement emphasized social reform, education, and the upliftment of the Pashtun community, aligning closely with the principles of Mahatma Gandhi's non-violent resistance.
• Khan Abdul Ghaffar Khan is often referred to as the "Frontier Gandhi" due to his dedication to non-violence and his close association with Mahatma Gandhi.

Additional Information

• Khudai Khidmatgar Movement
• The Khudai Khidmatgar, also known as the "Red Shirts", was a non-violent movement founded by Khan Abdul Ghaffar Khan in 1929.
• The movement aimed to promote social and educational reform among the Pashtuns of the North-West Frontier Province (now Khyber Pakhtunkhwa in Pakistan).
• The Khudai Khidmatgar adopted a uniform of red shirts and practiced non-violent resistance, inspired by the teachings of Mahatma Gandhi.
• The movement played a significant role in the Indian independence struggle, particularly in mobilizing the Pashtun community against British colonial rule.
• Khan Abdul Ghaffar Khan
• Born in 1890 in what is now Pakistan, Khan Abdul Ghaffar Khan was a towering figure in the Indian independence movement.
• He was a close associate of Mahatma Gandhi and an ardent advocate of non-violence.
• He spent much of his life working for the social upliftment and political rights of the Pashtun community.
• Despite facing repression from both the British authorities and later the Pakistani government, Khan Abdul Ghaffar Khan remained committed to his principles of non-violence and social justice.

The correct answer is Article 39(b).

Key Points

• Article 39(b) is a part of the Directive Principles of State Policy (DPSP) in the Indian Constitution.
• It emphasizes that the ownership and control of the material resources of the community should be distributed in a manner that best serves the common good.
• This principle aims to reduce economic inequalities and ensure equitable distribution of resources.
• Article 39(b) reflects the socialist philosophy embedded in the Indian Constitution, focusing on social justice and economic welfare.

Additional Information

• Directive Principles of State Policy (DPSP)
• The DPSPs are guidelines for the framing of laws by the government. These principles are considered fundamental in the governance of the country.
• They are not enforceable by any court, but the principles laid down are considered in the governance of the state.
• The aim of the DPSPs is to create social and economic conditions under which citizens can lead a good life.
• They are mentioned in Part IV of the Indian Constitution, from Articles 36 to 51.
• Article 39(a)
• It directs the state to ensure that the citizens, men, and women equally, have the right to an adequate means of livelihood.
• Article 39(c)
• It aims to ensure that the operation of the economic system does not result in the concentration of wealth and means of production to the common detriment.
• Article 39(d)
• It mandates that there is equal pay for equal work for both men and women.

The correct answer is Khasi Hills.

Key Points

• Mawsynram is located in the southern ranges of the Khasi Hills in the Indian state of Meghalaya.
• It is renowned for receiving the highest average annual rainfall in the world, with an average of about 467 inches (11,871 mm).
• The Khasi Hills are part of the larger Patkai range, which extends across India and into Myanmar.
• Mawsynram's record rainfall is due to its location on the windward side of the hills, where the monsoon winds from the Bay of Bengal are funneled and cause heavy precipitation.

Additional Information

• Shevaroy hills
• The Shevaroy Hills are a range of hills located in the Eastern Ghats in Tamil Nadu, India.
• They are known for their coffee plantations and the hill station of Yercaud.
• Mizo hills
• The Mizo Hills, also known as Lushai Hills, are part of the Patkai range in the state of Mizoram, India.
• These hills are characterized by dense forests and a significant tribal population.
• Naga hills
• The Naga Hills are part of the Purvanchal Range in Nagaland, India.
• These hills are named after the Naga tribes who inhabit the region.
• Khasi hills
• The Khasi Hills are part of the Garo-Khasi range in Meghalaya, India.
• They are known for their unique living root bridges and the highest rainfall in Mawsynram.

The correct answer is 4th.

Key Points

• India finished in the 4th position in the overall medals tally at the Commonwealth Games 2022.
• India secured a total of 61 medals in the competition.
• Out of these, India won 22 gold medals, 16 silver medals, and 23 bronze medals.
• The performance was remarkable considering the tough competition from other participating nations.

Additional Information

• Commonwealth Games
• The Commonwealth Games is an international multi-sport event involving athletes from the Commonwealth of Nations.
• The event was first held in 1930 and has been conducted every four years since then, except for 1942 and 1946 due to World War II.
• The games are overseen by the Commonwealth Games Federation (CGF), which also controls the sporting program and selects the host cities.
• The Commonwealth Games feature sports that are popular in the Commonwealth countries, including athletics, swimming, rugby sevens, and netball.
• India's Performance in Commonwealth Games
• India has been participating in the Commonwealth Games since 1934.
• India's best performance came in the 2010 Commonwealth Games held in Delhi, where it finished second in the overall medals tally.
• India has consistently performed well in sports like shooting, wrestling, weightlifting, and boxing in the Commonwealth Games.
• The 2022 Commonwealth Games saw India achieving significant success in various disciplines, showcasing the country's sporting talent and determination.
• Medals Tally
• The medals tally is a table that lists the number of gold, silver, and bronze medals won by each country participating in a multi-sport event.
• The ranking in the medals tally is usually determined by the number of gold medals won, followed by the number of silver and bronze medals.
• In the event of a tie, the total number of medals won is considered.
• The medals tally provides an overall picture of the performance of countries in the event and highlights the leading sports nations.

The correct answer is Pratihara, Rashtrakuta, and Pala.

Key Points

• The Tripartite Struggle was a prolonged conflict between the Pratihara, Rashtrakuta, and Pala dynasties.
• This struggle took place during the 8th to 10th centuries over the control of the city of Kannauj, which was a symbol of prestige and power.
• Kannauj was strategically significant as it was located in the fertile Ganges plain and was a prominent center of commerce and culture.
• The struggle weakened all three dynasties over time, paving the way for other powers to rise in the Indian subcontinent.

Additional Information

• Pratihara Dynasty
• The Pratiharas, also known as the Gurjara-Pratiharas, were a powerful dynasty in northern India.
• They played a crucial role in resisting Arab invasions and preserving the cultural heritage of India.
• Their rule is marked by significant developments in art, architecture, and the spread of Hinduism.
• Rashtrakuta Dynasty
• The Rashtrakutas were a prominent dynasty in southern and central India.
• They were known for their military prowess and their patronage of arts and literature.
• The famous rock-cut temples of Ellora, including the Kailasa temple, were built during their reign.
• Pala Dynasty
• The Palas were a major dynasty in eastern India, particularly in the regions of Bengal and Bihar.
• They were great patrons of Buddhism and played a significant role in the spread of the religion to Tibet and Southeast Asia.
• The Nalanda University flourished under their patronage, becoming a renowned center of learning.

The correct answer is Camillo Golgi.

Key Points

• Camillo Golgi was an Italian physician, pathologist, and scientist.
• He is renowned for discovering the Golgi apparatus, a complex of vesicles and folded membranes within the cytoplasm of eukaryotic cells.
• The Golgi apparatus plays a critical role in sorting and packaging proteins and lipids for secretion or delivery to other organelles.
• Golgi's discovery was a pivotal moment in cell biology, and he was awarded the Nobel Prize in Physiology or Medicine in 1906 for his contributions.

Additional Information

• Golgi Apparatus
• The Golgi apparatus, also known as the Golgi complex, is an organelle found in most eukaryotic cells.
• It is involved in the modification, sorting, and packaging of proteins and lipids for secretion or use within the cell.
• The Golgi apparatus is composed of a series of flattened, stacked pouches called cisternae.
• The cis face of the Golgi apparatus is the receiving side, while the trans face is the shipping side.
• Camillo Golgi's Contributions
• Camillo Golgi developed a staining technique called the "black reaction" or "Golgi stain," which allowed him to visualize the Golgi apparatus and other cellular structures.
• This staining method involves using silver nitrate to selectively stain the nervous system tissue, enabling the detailed study of neurons and their connections.
• Golgi's work laid the foundation for our understanding of the organization and function of the nervous system.
• His discoveries have had a lasting impact on the field of cell biology and neuroscience.
• Other Notable Researchers
• William Harvey: An English physician who made significant contributions to the understanding of the circulatory system, particularly the function of the heart and blood vessels.
• Francis Crick: A British molecular biologist, biophysicist, and neuroscientist who co-discovered the structure of DNA with James Watson.
• Louis Pasteur: A French biologist, microbiologist, and chemist renowned for his discoveries of the principles of vaccination, microbial fermentation, and pasteurization.

The correct answer is Savings as well as government expenditure are injections to the circular flow..

Key Points

• Savings are considered a leakage from the circular flow of income as they represent income not spent on consumption.
• Government expenditure is an injection into the circular flow as it represents spending that adds to the overall demand in the economy.
• In a closed economy with no government intervention, savings would be equal to investment.
• The correct representation would be that investment and government spending are injections, while savings, taxes, and imports are leakages.

Additional Information

• Circular Flow of Income
• The circular flow of income is a model that depicts how money flows through the economy between households and firms.
• In this model, households provide factors of production (labor, land, capital) to firms, and in return, they receive income (wages, rent, interest, profit).
• Households then use this income to purchase goods and services from firms, completing the cycle.
• Leakages and Injections
• Leakages are non-consumption uses of income, including savings, taxes, and imports. They represent money that leaves the circular flow.
• Injections are additions to the income of firms, including investment, government spending, and exports. They represent money entering the circular flow.
• In a balanced circular flow, the total value of leakages is equal to the total value of injections.
• Real Flow and Monetary Flow
• The real flow refers to the flow of physical goods and services and the factors of production.
• The monetary flow refers to the flow of money in the form of income and expenditure.
• In the circular flow model, the real flow and monetary flow move in opposite directions to each other.

The correct answer is zero order reaction.

Key Points

• In a zero-order reaction, the rate of reaction is independent of the concentration of the reactants.
• These reactions are commonly seen in catalytic processes where the catalyst surface is saturated by the reactants.
• In such reactions, the rate of reaction remains constant until the reactant concentration is significantly depleted.
• Zero-order kinetics is often observed in heterogeneous catalysis and other situations where a limiting step is saturated.

Additional Information

• First-order Reaction
• A first-order reaction is one where the rate is directly proportional to the concentration of one reactant.
• The rate law for a first-order reaction can be expressed as: Rate = k[A], where k is the rate constant and [A] is the concentration of the reactant.
• These reactions often involve unimolecular processes.
• Examples include radioactive decay and certain substitution reactions in organic chemistry.
• Second-order Reaction
• A second-order reaction's rate is proportional to the square of the concentration of one reactant or the product of the concentrations of two reactants.
• The rate law can be written as: Rate = k[A]^2 or Rate = k[A][B], depending on the reaction mechanism.
• These reactions often involve bimolecular processes.
• Examples include the reaction between hydrogen and iodine to form hydrogen iodide.
• Third-order Reaction
• A third-order reaction's rate depends on the concentration of three reactants or the concentration of one reactant raised to the third power, or other combinations summing to three.
• The rate law can be expressed as: Rate = k[A]^3 or other combinations.
• These reactions are less common compared to first and second-order reactions.
• An example is the reaction between nitric oxide and oxygen to form nitrogen dioxide.

The correct answer is New Delhi.

Key Points

• The Sangeet Natak Akademi is India’s national academy for music, dance, and drama.
• It was established by the Indian education ministry in 1952 and is presently an autonomous body of the Ministry of Culture.
• The registered office of the Sangeet Natak Akademi is located in New Delhi, India.
• It aims to promote and preserve the vast intangible cultural heritage of India through various initiatives and awards.

Additional Information

• Sangeet Natak Akademi Awards
• These are national honors conferred by the Sangeet Natak Akademi on performing artists and practitioners of traditional arts.
• The awards are given annually in the fields of music, dance, and drama.
• Awardees receive a cash prize, a shawl, and a citation.
• Fellowship of the Akademi (Akademi Ratna)
• It is the highest honor conferred by the Sangeet Natak Akademi.
• The fellowship is awarded to individuals who have made an exceptional contribution to the field of performing arts.
• Fellows receive a cash prize, a shawl, and a citation.
• Training and Scholarships
• The Akademi provides scholarships to young artists for advanced training in performing arts.
• It also organizes workshops and seminars to promote traditional art forms.
• The aim is to preserve and propagate the rich cultural heritage of India.
• Documentation and Publication
• The Akademi undertakes documentation and publication of various art forms.
• This includes audio and video recordings, books, and journals.
• The aim is to create a repository of the vast intangible cultural heritage of India.

Shortcut Trick

Given the ratio of Circumference to Diameter = 22 : 7

Here, 22 units of Circumference → 33 cm

1 unit → 33 ÷ 22 = 1.5 cm

Diameter (7 units) = 7 × 1.5 = 10.5 cm

Radius = Diameter ÷ 2 = 10.5 ÷ 2 = 5.25 cm

∴ The correct answer is 5.25 cm.

Alternate Method

Given:

Circumference of the circle (C) = 33 cm

Ratio of Circumference (C) to Diameter (D) = 22 : 7

Formula Used:

Diameter (D) = Circumference × (7 ÷ 22)

Radius (r) = Diameter ÷ 2

Calculations:

⇒ C ÷ D = 22 ÷ 7

⇒ 33 ÷ D = 22 ÷ 7

⇒ D = (33 × 7) ÷ 22

⇒ D = (3 × 7) ÷ 2

⇒ D = 21 ÷ 2 = 10.5 cm

⇒ Radius (r) = 10.5 ÷ 2 = 5.25 cm

∴ The correct answer is 5.25 cm.

Additional Information

Circumference of a Circle

The total boundary length is given by C = 2πr or C = πD, where π ≈ 22 ÷ 7.

Area of a Circle

The total space enclosed within the boundary is evaluated as Area = πr².

Semicircle Properties

The perimeter of a semicircle includes both the curved path and the straight diameter: P = πr + 2r. Its Area is πr² ÷ 2.

Shortcut Trick

Ratio of Initial Income : Expenditure : Savings = 18600 : 12400 : 6200 = 3 : 2 : 1.

We know, Income = Expenditure + Savings ⇒ 3 = 2 + 1.

Applying the net percentage change equation (Weighted Average):

⇒ 3 × 15% = (2 × 8%) + (1 × x%)

⇒ 45% = 16% + x% ⇒ x = 29%

∴ The correct answer is 29%.

Alternate Method

Given:

Initial Income = ₹18,600

Initial Expenditure = ₹12,400

Formula Used:

Savings = Income − Expenditure

Percentage Increase = (Increase ÷ Original Value) × 100

Calculations:

⇒ Initial Savings = ₹18,600 − ₹12,400 = ₹6,200

⇒ New Income = ₹18,600 × (100 + 15)% = ₹18,600 × 1.15 = ₹21,390

⇒ New Expenditure = ₹12,400 × (100 + 8)% = ₹12,400 × 1.08 = ₹13,392

⇒ New Savings = ₹21,390 − ₹13,392 = ₹7,998

⇒ Increase in Savings = ₹7,998 − ₹6,200 = ₹1,798

⇒ Percentage increase in Savings = (1,798 ÷ 6,200) × 100 = 29%

∴ The correct answer is 29%.

Additional Information

Income, Expenditure & Savings Relation

Income = Expenditure + Savings. Any percentage change in income is strictly distributed between expenditure and savings based on their proportional weightage.

Weighted Average Method

When a quantity is divided into two parts in the ratio a : b, and the parts change by x% and y%, the net percentage change is given by (ax + by) ÷ (a + b).

Percentage Change Formula

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

Shortcut Trick

Let the base average be 60 (average of 12 students).

The other 18 students have an average of 80, which is +20 more than the base.

Total excess score brought by these 18 students = 18 × 20 = +360.

Distribute this excess equally among all 30 students = 360 ÷ 30 = +12.

Overall Average = Base Average + Distributed Excess = 60 + 12 = 72.

∴ The correct answer is 72 marks.

Alternate Method

Given:

Total students in the class = 30

Average score of 12 students = 60

Number of remaining students = 30 − 12 = 18

Average score of the 18 students = 80

Formula Used:

Combined Average = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂)

Avg: 60

12 students

Avg: 80

18 students

Combined Avg = 72

Diff = 12 Diff = 8

Calculations:

⇒ Total score of the first 12 students = 12 × 60 = 720

⇒ Total score of the remaining 18 students = 18 × 80 = 1440

⇒ Total score of all 30 students = 720 + 1440 = 2160

⇒ Average score of the class = 2160 ÷ 30

⇒ Average = 72

∴ The correct answer is 72 marks.

Additional Information

Weighted Average

When two groups have different sizes and averages, their combined average is found using the formula: A = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Deviation Method

Select a base average (usually the smaller average). Find the net deviation of all other values from this base. The combined average is simply Base Average + (Total Net Deviation ÷ Total Number of Items).

Alligation Method

The ratio of the number of items in two groups is inversely proportional to the difference between their individual averages and the combined average. Ratio n₁ : n₂ = (A₂ − A) : (A − A₁).

Shortcut Trick

Find (a+b)² = a² + b² + 2ab = 148 + 2 × 54 = 256 ⇒ a+b = 16

Find (a-b)² = a² + b² − 2ab = 148 − 2 × 54 = 40 ⇒ a-b = √40 = 2√10

Divide the sum by the difference: (a+b) / (a-b) = 16 / 2√10 = 8/√10

∴ The correct answer is 8/√10.

Alternate Method

Given:

a² + b² = 148

ab = 54

Formula Used:

(a+b)² = a² + b² + 2ab

(a-b)² = a² + b² − 2ab

Calculations:

⇒ (a+b)² = 148 + 2 × 54

⇒ (a+b)² = 148 + 108 = 256

⇒ a + b = √256 = 16

⇒ (a-b)² = 148 − 2 × 54

⇒ (a-b)² = 148 − 108 = 40

⇒ a − b = √40 = 2√10

⇒ Required value = (a+b) / (a-b)

⇒ Required value = 16 / 2√10

⇒ Required value = 8 / √10

∴ The correct answer is 8/√10.

Additional Information

Important Algebraic Identities

(a+b)² + (a-b)² = 2(a² + b²)

Difference of Squares Formula

a² − b² = (a+b)(a-b) — useful to expand or factorise equations quickly.

Relating Sum and Difference

(a+b)² − (a-b)² = 4ab is another crucial formula to relate sum, difference, and product directly.

Given:

The table shows the number of pages printed by printers P1 and P2 over three days:

Formula used:

Total pages = Pages printed by P1 + Pages printed by P2

Difference = Total pages by P1 - Total pages by P2

Calculations:

Total pages printed by P1:

⇒ 200 + 150 + 225 = 575

Total pages printed by P2:

⇒ 250 + 300 + 210 = 760

Difference between total pages printed by P1 and P2:

⇒ 760 - 575 = 185

∴ The difference is 185 pages.

Given:

9 × N of 4 ÷ 4 - 15 × 4 = 12

Formula:

BODMAS

Calculation:

9 × N of 4 ÷ 4 - 15 × 4 = 12

9 × N4 ÷ 4 - 15 × 4 = 12

9 × N - 15 × 4 = 12

9 × N - 60 = 12

9 × N = 72

N = 8

The correct answer is N = 8.

Shortcut Trick

By the Midpoint Theorem, the line segment joining the midpoints of any two sides of a triangle is parallel to the third side and is equal to half of its length.

Since S and T are the midpoints of sides PQ and PR respectively, ST is parallel to QR.

Length of ST = QR ÷ 2

Length of ST = 12 ÷ 2 = 6 cm

∴ The correct answer is 6 cm.

Alternate Method

Given:

In triangle PQR, S and T are the midpoints of the sides PQ and PR respectively.

The length of the side QR = 12 cm.

ST is parallel to QR.

Formula Used:

If two triangles are similar, the ratio of their corresponding sides is equal.

Midpoint property: PS = PQ ÷ 2 and PT = PR ÷ 2.

P Q R S T ST = ? QR = 12 cm

Calculations:

⇒ In triangle PST and triangle PQR, angle P is common.

⇒ Since ST is parallel to QR, corresponding angles are equal: angle PST = angle PQR.

⇒ Therefore, triangle PST is similar to triangle PQR (by AA similarity criterion).

⇒ The ratio of corresponding sides will be equal: ST / QR = PS / PQ

⇒ Since S is the midpoint of PQ, the ratio of PS to PQ is 1 / 2.

⇒ ST / 12 = 1 / 2

⇒ ST = 12 × (1 / 2)

⇒ ST = 6 cm

∴ The correct answer is 6 cm.

Additional Information

Midpoint Theorem

The line segment joining the midpoints of two sides of a triangle is always parallel to the third side and is exactly half of its length.

Basic Proportionality Theorem (Thales Theorem)

If a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides proportionally in the same ratio.

Properties of Similar Triangles

When two triangles are similar, the ratio of their corresponding sides is equal, and the ratio of their areas is equal to the square of the ratio of their corresponding sides.

Shortcut Trick

The given expression follows the multiple-angle trigonometric identity directly.

Identity: sin(3θ) = 3sin(θ) − 4sin³(θ)

Here, the angle is θ = 15^°.

Expression = 3sin(15^°) − 4sin³(15^°) = sin(3 × 15^°)

⇒ sin(45^°) = 1/√2

∴ The correct answer is 1/√2.

Alternate Method

Given:

Expression: 3sin 15^° − 4sin³ 15^°

Formula Used:

Standard exact value: sin 15^° = (√3 − 1) / 2√2

Algebraic identity: (a − b)³ = a³ − 3a²b + 3ab² − b³

Calculations:

⇒ 3sin 15^° = 3 × [(√3 − 1) / 2√2] = (3√3 − 3) / 2√2

⇒ sin³ 15^° = [(√3 − 1) / 2√2]³

⇒ Expand Numerator: (√3 − 1)³ = (√3)³ − 3(√3)²(1) + 3(√3)(1)² − (1)³

⇒ (√3 − 1)³ = 3√3 − 9 + 3√3 − 1 = 6√3 − 10

⇒ Cube Denominator: (2√2)³ = 16√2

⇒ 4sin³ 15^° = 4 × [(6√3 − 10) / 16√2] = (6√3 − 10) / 4√2

⇒ Now substitute the expanded parts into the original expression:

⇒ Expression = [(3√3 − 3) / 2√2] − [(6√3 − 10) / 4√2]

⇒ Make denominators equal by multiplying the first term by 2/2:

⇒ Expression = [(6√3 − 6) / 4√2] − [(6√3 − 10) / 4√2]

⇒ Combine terms: [6√3 − 6 − 6√3 + 10] / 4√2

⇒ Expression = 4 / 4√2 = 1/√2

∴ The correct answer is 1/√2.

Additional Information

Multiple Angle Identity for Cosine

Similar to the sine function, the cosine triple angle formula is given by: cos(3θ) = 4cos³(θ) − 3cos(θ).

Double Angle Identities

Important double angle formulas include: sin(2θ) = 2sin(θ)cos(θ) and cos(2θ) = cos²(θ) − sin²(θ) = 1 − 2sin²(θ).

Standard Trigonometric Values for 15^°

For competitive exams, it is highly useful to memorize these common exact values: sin(15^°) = (√3 − 1) / 2√2 and cos(15^°) = (√3 + 1) / 2√2.

Shortcut Trick

Given, Gain = 1/5 of Selling Price (SP).

Let SP = 5 units. Then, Gain = 1 unit.

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

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

Note: The actual SP value (₹40620) is not needed for calculating the percentage.

∴ The correct answer is 25%.

Alternate Method

Given:

Selling Price (SP) = ₹40620

Gain = 1/5 of SP

Formula Used:

CP = SP − Gain

Gain % = (Gain ÷ CP) × 100

Calculations:

⇒ Gain = (1 ÷ 5) × 40620 = ₹8124

⇒ CP = 40620 − 8124 = ₹32496

⇒ Gain % = (8124 ÷ 32496) × 100

⇒ Gain % = (1 ÷ 4) × 100 = 25%

∴ The correct answer is 25%.

Additional Information

Base of Calculation

Profit and loss percentages are always calculated on the true Cost Price (CP) unless specifically stated otherwise in the question.

Standard Profit Formula

Profit % = (Profit ÷ CP) × 100, where Profit = SP − CP.

Profit on Selling Price

When profit is calculated based on SP instead of CP, the actual Cost Price is smaller than SP, making the actual real profit percentage (on CP) higher than the given profit percentage on SP.

Shortcut Trick

Using the effective discount formula: Equivalent Discount = a + b − (a×b)/100

Equivalent discount of 15% and 20% = 15 + 20 − (15×20)/100 = 35 − 3 = 32%

Equivalent discount of 32% and 5% = 32 + 5 − (32×5)/100 = 37 − 1.6 = 35.4%

∴ The correct answer is 35.4%.

Alternate Method

Given:

Successive discount series = 15%, 20%, 5%

Formula Used:

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

Calculations:

⇒ Let the Marked Price (MP) be 100.

⇒ Price after 1st discount of 15% = 100 × (100 − 15)/100 = 85

⇒ Price after 2nd discount of 20% = 85 × (100 − 20)/100 = 85 × 80/100 = 68

⇒ Final Selling Price after 3rd discount of 5% = 68 × (100 − 5)/100 = 68 × 95/100 = 64.6

⇒ Single Equivalent Discount = MP − Final SP

⇒ Equivalent Discount = 100 − 64.6 = 35.4%

∴ The correct answer is 35.4%.

Additional Information

Successive Discount Formula

For two consecutive discounts a% and b%, the single equivalent discount is [a + b − (a×b)/100]%.

Marked Price (MP) & Cost Price (CP) Relation

The direct relationship is given by: CP/MP = (100 − Discount%) ÷ (100 + Profit%).

Successive Percentage Multiplier

For a series of discounts D₁, D₂, D₃, the final value = Initial Base × (1 − D₁/100) × (1 − D₂/100) × (1 − D₃/100).

Given:

Formula used:

Total sales ratio = Total sales of B1 ÷ Total sales of B2

Calculations:

Total sales of B1 = 195 + 230 + 250

⇒ Total sales of B1 = 675

Total sales of B2 = 284 + 300 + 416

⇒ Total sales of B2 = 1000

Ratio = Total sales of B1 : Total sales of B2

⇒ Ratio = 675 : 1000

⇒ Ratio = 27 : 40

∴ The ratio of the total sales of bicycles B1 to B2 is 27:40.

Shortcut Trick

If a pipe fills a tank in x mins and another empties it in y mins (where y > x), the net time taken to fill the tank is (x × y) ÷ (y − x).

Here, x = 10 mins and y = 12 mins.

Time = (10 × 12) ÷ (12 − 10) = 120 ÷ 2 = 60 minutes.

Converting into hours: 60 minutes = 1 hour.

∴ The correct answer is 1 hour.

Alternate Method

Given:

Time taken by Tap 1 to fill = 10 minutes

Time taken by Tap 2 to empty = 12 minutes

Calculations:

⇒ Net efficiency of both taps working together = 6 − 5 = 1 unit/min

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

⇒ Total time = 60 ÷ 1 = 60 minutes

⇒ Time in hours = 60 ÷ 60 = 1 hour

∴ The correct answer is 1 hour.

Additional Information

Pipes and Cisterns Sign Convention

An inlet pipe fills the tank, so its work and efficiency are taken as positive (+). An outlet pipe empties the tank, so its work and efficiency are taken as negative (−).

Total Work using LCM

In Time & Work and Pipes & Cisterns, assume the total capacity of the tank as the LCM of all given times to avoid fractions in calculation.

Efficiency

Efficiency is inversely proportional to time. When combined, simply add the positive and negative efficiencies to find the net work rate per unit time.

Given:

Formula used:

Average production = (Sum of production for selected years) ÷ (Number of years)

Difference in averages = Average production of A - Average production of C

Calculations:

Sum of production for A (2016-2018) = 38,000 + 41,000 + 53,000

⇒ 38,000 + 41,000 + 53,000 = 132,000

Average production of A = 132,000 ÷ 3 = 44,000

Sum of production for C (2016-2018) = 51,000 + 57,000 + 63,000

⇒ 51,000 + 57,000 + 63,000 = 171,000

Average production of C = 171,000 ÷ 3 = 57,000

Difference in averages = 44,000 - 57,000

⇒ Difference = -13,000

∴ The difference in average production of A and C during 2016-2018 is 13,000 lakhs.

Shortcut Trick

Given the cost of 15 apples and 12 oranges is ₹447, we divide by 3 to get the cost of 5 apples and 4 oranges as ₹149.

An inconsistent statement will have the same ratio of apples to oranges (5:4) but an equivalent cost that does not match ₹149.

In the fourth statement, 35 apples and 28 oranges are in a 5:4 ratio.

However, dividing its cost ₹1,029 by 7 gives ₹147 for 5 apples and 4 oranges.

Since ₹147 ≠ ₹149, it contradicts the given data, leading to no possible solutions.

∴ The correct answer is Purchased 35 apples and 28 oranges and paid a sum of ₹1,029.

Alternate Method

Given:

Cost of 15 apples and 12 oranges = ₹447

Formula Used:

For a system of linear equations a₁x + b₁y = c₁ and a₂x + b₂y = c₂:

If a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the system is inconsistent and yields no possible solution.

Calculation:

⇒ Let the price of 1 apple be a and 1 orange be o.

⇒ 15a + 12o = 447

⇒ Checking Statement 1: 25a + 20o = 745

⇒ 15/25 = 12/20 = 447/745 (ratio simplifies to 3/5 for all). Consistent (Infinite solutions).

⇒ Checking Statement 2: 10a + 13o = 353

⇒ 15/10 ≠ 12/13. Consistent (Unique solution).

⇒ Checking Statement 3: 12a + 8o = 340

⇒ 15/12 ≠ 12/8. Consistent (Unique solution).

⇒ Checking Statement 4: 35a + 28o = 1029

⇒ a₁/a₂ = 15/35 = 3/7

⇒ b₁/b₂ = 12/28 = 3/7

⇒ c₁/c₂ = 447/1029 = 149/343

⇒ Here, a₁/a₂ = b₁/b₂ ≠ c₁/c₂

⇒ Because the ratios of coefficients are equal but differ from the constant ratio, the system is inconsistent.

∴ The correct answer is Purchased 35 apples and 28 oranges and paid a sum of ₹1,029.

Additional Information

Unique Solution

A system of equations has a unique solution if the lines intersect at exactly one point. Condition: a₁/a₂ ≠ b₁/b₂.

No Solution (Inconsistent System)

A system has no solution if the lines are perfectly parallel and never intersect. Condition: a₁/a₂ = b₁/b₂ ≠ c₁/c₂.

Infinite Solutions (Dependent System)

A system has infinitely many solutions if the equations represent the exact same line. Condition: a₁/a₂ = b₁/b₂ = c₁/c₂.

Shortcut Trick

The exterior angle of a triangle is equal to the sum of the two opposite interior angles. For ΔTQP, the exterior angle at Q is ∠PQR.

∠PQR = ∠PTQ + ∠TPQ = 27° + 55° = 82°.

By the Alternate Segment Theorem, the angle between the tangent TP and chord PQ equals the angle in the alternate segment: ∠PRQ = ∠TPQ = 55°.

In ΔPQR, ∠QPR = 180° − (∠PQR + ∠PRQ) = 180° − (82° + 55°) = 43°.

The central angle ∠ROQ is twice the angle subtended at the circumference by the same arc: ∠ROQ = 2 × ∠QPR = 2 × 43° = 86°.

∴ The correct answer is 86°.

Alternate Method

Given:

T is a point outside a circle with center O.

TP is a tangent to the circle at point P.

TQR is a secant intersecting the circle at points Q and R (with Q nearer to T).

∠PTQ = 27° and ∠TPQ = 55°.

T P Q R O 27° 55° ?

Formula Used:

1. The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
2. Alternate Segment Theorem: The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.
3. Angle at the center is double the angle subtended by the same arc at any remaining part of the circle.

Calculations:

In ΔTQP, the side TQ is extended to R forming the exterior angle ∠PQR.

⇒ ∠PQR = ∠PTQ + ∠TPQ

⇒ ∠PQR = 27° + 55° = 82°

Applying the Alternate Segment Theorem for the tangent TP and the chord PQ:

⇒ ∠PRQ = ∠TPQ

⇒ ∠PRQ = 55°

Now, applying the angle sum property in the inscribed triangle ΔPQR:

⇒ ∠QPR = 180° − (∠PQR + ∠PRQ)

⇒ ∠QPR = 180° − (82° + 55°)

⇒ ∠QPR = 180° − 137° = 43°

The arc QR subtends ∠ROQ at the center and ∠QPR at the circumference. Therefore:

⇒ ∠ROQ = 2 × ∠QPR

⇒ ∠ROQ = 2 × 43° = 86°

∴ The correct answer is 86°.

Additional Information

Alternate Segment Theorem

The angle between a tangent to a circle and a chord through the point of contact is strictly equal to the angle subtended by that chord in the alternate segment of the circle.

Central Angle Theorem

The angle subtended by an arc at the center of a circle is double the angle subtended by the same arc at any point on the remaining part of the circle (∠Center = 2 × ∠Circumference).

Exterior Angle of a Triangle

The measure of an exterior angle of a triangle is exactly equal to the sum of the measures of its two remote interior angles.

Shortcut Trick

If x² − x + 1 = 0, it is a standard algebraic result that x³ = −1.

We need to evaluate x¹² + x⁶ + 1.

Rewrite the expression in terms of x³: (x³)⁴ + (x³)² + 1.

Substitute x³ = −1: (−1)⁴ + (−1)² + 1.

⇒ 1 + 1 + 1 = 3.

∴ The correct answer is 3.

Alternate Method

Given:

x² + 1 = x

Formula Used:

a³ + b³ = (a + b)(a² − ab + b²)

Calculation:

⇒ x² − x + 1 = 0

Multiplying both sides by (x + 1):

⇒ (x + 1)(x² − x + 1) = 0 × (x + 1)

Using the identity for a³ + b³:

⇒ x³ + 1³ = 0

⇒ x³ = −1

Now, substitute x³ = −1 into the given expression: x¹² + x⁶ + 1

⇒ (x³)⁴ + (x³)² + 1

⇒ (−1)⁴ + (−1)² + 1

⇒ 1 + 1 + 1

⇒ 3

∴ The correct answer is 3.

Additional Information

Sum of Cubes Identity

The identity a³ + b³ = (a + b)(a² − ab + b²) is very useful for powers of complex roots. If x² − x + 1 = 0, then multiplying by (x + 1) gives x³ = −1.

Difference of Cubes Identity

The identity a³ − b³ = (a − b)(a² + ab + b²) is used similarly. If x² + x + 1 = 0, then multiplying by (x − 1) gives x³ = 1.

Complex Roots of Unity

The roots of x² + x + 1 = 0 are known as the non-real cube roots of unity (denoted as ω and ω²). They satisfy ω³ = 1 and 1 + ω + ω² = 0.

Shortcut Trick

The well is dug in the shape of a cylinder.

Volume of earth dug out = Volume of the cylinder = πr²h

⇒ Volume = (22/7) × 14 × 14 × 23

⇒ Volume = 22 × 2 × 14 × 23 = 14,168

∴ The correct answer is 14,168 cm³.

Alternate Method

Given:

Radius of the well (r) = 14 cm

Depth of the well (h) = 23 cm

Formula Used:

Volume of a cylinder = π × r² × h

Calculation:

⇒ Volume of earth taken out = (22/7) × (14)² × 23

⇒ Volume = (22/7) × 196 × 23

⇒ Volume = 22 × 28 × 23

⇒ Volume = 616 × 23

⇒ Volume = 14,168 cm³

∴ The correct answer is 14,168 cm³.

Additional Information

Curved Surface Area (CSA) of a Cylinder

The area of the outer curved surface is calculated using the formula: CSA = 2πrh.

Total Surface Area (TSA) of a Cylinder

The total surface area includes the curved surface plus the top and bottom circular bases: TSA = 2πr(r + h).

Volume of a Hollow Cylinder

If a cylinder is hollow with outer radius R and inner radius r, its material volume is given by: Volume = π(R² − r²)h.

Relation with a Cone

A cone having the exact same base radius and height as a cylinder will always have one-third of the cylinder's volume: V = (1/3)πr²h.

Shortcut Trick

Total amount if no repayment is made = 335180 + (26% of 335180) = 335180 + 87146.8 = ₹422326.8

Since ₹23060 is paid towards the principal at the end of Year 1, interest is saved for Year 2 on this amount.

Total effective value of this repayment = Principal paid + Interest saved for 1 year = 23060 + (13% of 23060) = 23060 + 2997.8 = ₹26057.8

Remaining amount to pay at the end of 2nd year = Total Amount − Total effective repayment

⇒ Amount to pay = 422326.8 − 26057.8 = ₹396269

∴ The correct answer is ₹396269.

Alternate Method

Given:

Principal (P) = ₹335180

Rate (R) = 13% per annum

Repayment at the end of Year 1 = ₹23060 (towards principal)

Calculations:

⇒ Interest accrued during the 1st year (I₁) = (335180 × 13) ÷ 100 = ₹43573.4

⇒ Pranav repays ₹23060 towards the principal at the end of the 1st year.

⇒ Remaining principal for the 2nd year = 335180 − 23060 = ₹312120

⇒ Interest accrued during the 2nd year (I₂) = (312120 × 13) ÷ 100 = ₹40575.6

⇒ Total pending dues at the end of the 2nd year = Remaining Principal + 1st Year Interest + 2nd Year Interest

⇒ Total Amount = 312120 + 43573.4 + 40575.6 = ₹396269

∴ The correct answer is ₹396269.

Additional Information

Simple Interest (SI)

Calculated strictly on the principal amount over a period. Formula: SI = (P × R × T) ÷ 100.

Partial Repayment Rule

When a borrower makes a partial payment, it typically covers the accrued interest first. However, if explicitly mentioned (as in this case), it is deducted directly from the principal, which sequentially lowers the interest for future periods.

SI vs CI Difference

In Simple Interest, unpaid accrued interest does not earn further interest. In Compound Interest, accrued interest is added to the principal (A = P(1 + R/100)^T), creating an "interest on interest" effect.

Shortcut Trick

The angle subtended by an arc at the center is twice the angle subtended at any point on the remaining part of the circle.

⇒ m∠QPS = m∠QOS ÷ 2 = 20x° ÷ 2 = 10x°

In a cyclic quadrilateral, opposite angles are supplementary (sum to 180°).

⇒ m∠QPS + m∠QRS = 180°

⇒ 10x° + 26x° = 180° ⇒ 36x = 180 ⇒ x = 5

∴ The correct answer is 5.

Alternate Method

Given:

m∠QOS = 20x°

m∠QRS = 26x°

Formula Used:

1. Angle at the center = 2 × Angle at the circumference
2. Sum of opposite angles in a cyclic quadrilateral = 180°

Calculations:

By the angle at the center theorem, the angle subtended by the arc QS at the remaining part of the circle (at vertex P) is half the angle subtended at the center.

⇒ m∠QPS = m∠QOS ÷ 2

⇒ m∠QPS = 20x° ÷ 2 = 10x°

Since all the vertices P, Q, R, and S lie exactly on the circumference of the circle, PQRS is a cyclic quadrilateral.

In a cyclic quadrilateral, the sum of any pair of opposite angles is equal to 180°.

⇒ m∠QPS + m∠QRS = 180°

Substitute the respective angle values:

⇒ 10x° + 26x° = 180°

⇒ 36x = 180

⇒ x = 180 ÷ 36

⇒ x = 5

∴ The correct answer is 5.

Additional Information

Cyclic Quadrilateral

A quadrilateral whose all four vertices lie on the circumference of a circle. The sum of either pair of opposite angles is always exactly 180°.

Angle at Center Theorem

The angle subtended by an arc at the center of the circle is double the angle subtended by the same arc at any point on the remaining part of the circle.

Angles in the Same Segment

Angles subtended by the same arc at any point on the circumference of the circle are always equal to each other.

Shortcut Trick

Relative speed in the same direction = 8 − 3 = 5 rounds per hour.

This means Priya gains 5 complete rounds over Seema in 60 minutes.

They will cross each other when Priya gains exactly 1 complete round.

Time to gain 1 round = 60 ÷ 5 = 12 minutes.

Crossing time = 11:54 p.m. + 12 minutes = 12:06 a.m.

∴ The correct answer is 12:06 a.m.

Alternate Method

Given:

⇒ Speed of Seema = 3 rounds/hr

⇒ Speed of Priya = 8 rounds/hr

⇒ Starting time = 11:54 p.m.

Formula Used:

⇒ Relative Speed (same direction) = Speed₁ − Speed₂

⇒ Time to cross for the first time = 1 round distance ÷ Relative Speed

Calculations:

⇒ Since they are moving in the same direction, relative speed = 8 − 3 = 5 rounds/hr

⇒ They cross each other when the faster person (Priya) covers exactly 1 more round than the slower person (Seema).

⇒ Time taken to gain 1 round = 1 ÷ 5 hours

⇒ Time taken in minutes = (1/5) × 60 = 12 minutes

⇒ Crossing time = 11:54 p.m. + 12 minutes

⇒ 11:54 p.m. + 6 minutes = 12:00 a.m. (Midnight)

⇒ 12:00 a.m. + 6 minutes = 12:06 a.m.

∴ The correct answer is 12:06 a.m.

Additional Information

Circular Motion (Same Direction)

When two bodies move in the same direction on a circular track, the time taken to meet for the first time anywhere on the track is calculated as Track Length ÷ (Speed₁ − Speed₂).

Circular Motion (Opposite Direction)

When two bodies move in opposite directions, their relative speed increases. The time taken to meet for the first time is Track Length ÷ (Speed₁ + Speed₂).

Meeting at the Starting Point

The time taken for both runners to meet exactly at the starting point is the LCM (Least Common Multiple) of the individual times taken by each to complete one full round.

Number of Distinct Meeting Points

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

Given:

We need to calculate what percentage of the average sales of branches B1, B2, and B3 in 2001 is the average sales of branches B1, B3, and B6 in 2000.

Calculations:

Step 1: Calculate the average sales of branches B1, B2, and B3 in 2001:

Sales in 2001:

B1 = 105, B2 = 65, B3 = 110

Total sales = 105 + 65 + 110 = 280

Average sales in 2001 = 280 ÷ 3 = 93.33

Step 2: Calculate the average sales of branches B1, B3, and B6 in 2000:

Sales in 2000:

B1 = 80, B3 = 95, B6 = 70

Total sales = 80 + 95 + 70 = 245

Average sales in 2000 = 245 ÷ 3 = 81.67

Step 3: Calculate the percentage:

Percentage = (Average sales in 2000 ÷ Average sales in 2001) × 100

⇒ Percentage = (81.67 ÷ 93.33)  × 100

⇒ Percentage = 87.5%

∴ The average sales in 2001 is approximately 87.5% of the average sales in 2000.

Shortcut Trick

Given x ∝ y³ / z², we can directly compare the two states using ratio method:

x₁ / x₂ = (y₁ / y₂)³ × (z₂ / z₁)²

(1/36) / x₂ = (2 / 3)³ × (5 / 3)² = (8 / 27) × (25 / 9) = 200 / 243

x₂ = (1/36) × (243 / 200) = 27 / 800

800x₂ = 800 × (27 / 800) = 27

∴ The correct answer is 27.

Alternate Method

Given:

x varies directly with the cube of y and inversely with the square of z.

⇒ x = k × (y³ / z²), where k is a constant.

Calculations:

Substitute the given values (x = 1/36, y = 2, z = 3) to find k:

⇒ 1/36 = k × (2³ / 3²)

⇒ 1/36 = k × (8 / 9)

⇒ k = (1/36) × (9 / 8) = 1 / 32

Now, substitute y = 3 and z = 5 to find the new value of x:

⇒ x = (1/32) × (3³ / 5²)

⇒ x = (1/32) × (27 / 25)

⇒ x = 27 / 800

Find the required value of 800x:

⇒ 800x = 800 × (27 / 800) = 27

∴ The correct answer is 27.

Additional Information

Direct Variation

If two quantities x and y vary directly, their ratio is constant: x / y = k or x = ky. As one increases, the other increases proportionally.

Inverse Variation

If x and y vary inversely, their product is constant: x × y = k or x = k / y. As one increases, the other decreases proportionally.

Joint Variation

When a quantity varies directly with one variable and inversely with another, the equations combine as shown in this problem: x = k × (y / z).

Shortcut Trick

Directly substitute the standard value of cosec 60°.

Expression = 5/√3 − 2/√3

⇒ (5 − 2) / √3

⇒ 3 / √3 = √3

∴ The correct answer is √3.

Alternate Method

Given:

The expression is (5/√3 − cosec 60°)

Formula Used:

cosec θ = 1 / sin θ

sin 60° = √3 / 2 ⇒ cosec 60° = 2 / √3

Calculation:

⇒ 5/√3 − cosec 60°

Substitute the value of cosec 60°:

⇒ 5/√3 − 2/√3

Since the denominators are identical, subtract the numerators directly:

⇒ (5 − 2) / √3

⇒ 3 / √3

Rationalize the denominator by multiplying the numerator and denominator by √3:

⇒ (3 × √3) / (√3 × √3)

⇒ 3√3 / 3

⇒ √3

∴ The correct answer is √3.

Additional Information

Trigonometric Values for 60°

sin 60° = √3/2, cos 60° = 1/2, and tan 60° = √3.

Reciprocal Trigonometric Identities

cosec θ = 1 / sin θ, sec θ = 1 / cos θ, and cot θ = 1 / tan θ.

Rationalization

To remove a square root from the denominator, multiply both numerator and denominator by the square root. Example: a / √b = a√b / b.

Shortcut Trick

For a number to be divisible by 6, it must be even (divisible by 2) and its digit sum must be divisible by 3.

Eliminate odd options immediately: 12581 and 62233 are rejected.

Check 128530: Sum = 1 + 2 + 8 + 5 + 3 + 0 = 19 (Not divisible by 3).

Check 438750: Sum = 4 + 3 + 8 + 7 + 5 + 0 = 27 (Divisible by 3). It satisfies both conditions.

∴ The correct answer is 438750.

Alternate Method

Given:

Numbers to check: 128530, 12581, 438750, and 62233

Concept Used:

Divisibility by 6: A number is divisible by 6 if and only if it is divisible by both 2 and 3.

Divisibility by 2: The unit digit must be an even number (0, 2, 4, 6, 8).

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

Calculations:

⇒ Let us evaluate the given options one by one:

⇒ Option 1: 128530

⇒ Unit digit is 0, which is even (divisible by 2).

⇒ Sum of digits = 1 + 2 + 8 + 5 + 3 + 0 = 19

⇒ Since 19 is not divisible by 3, 128530 is not divisible by 6.

⇒ Option 2: 12581

⇒ Unit digit is 1, which is odd (not divisible by 2). So, it is not divisible by 6.

⇒ Option 3: 438750

⇒ Unit digit is 0, which is even (divisible by 2).

⇒ Sum of digits = 4 + 3 + 8 + 7 + 5 + 0 = 27

⇒ Since 27 is divisible by 3, and the number is even, 438750 is divisible by 6.

⇒ Option 4: 62233

⇒ Unit digit is 3, which is odd (not divisible by 2). So, it is not divisible by 6.

∴ The correct answer is 438750.

Additional Information

Divisibility Rule of 4

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

Divisibility Rule of 8

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

Divisibility Rule of 9

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

Divisibility Rule of 11

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

Shortcut Trick

Put θ = 30° in the given equation.

⇒ x = sec 30° + tan 30° = 2/√3 + 1/√3 = 3/√3 = √3

We need to find sin 30°, which is exactly 1/2.

Now, substitute x = √3 in Option 3:

⇒ (x² − 1) / (1 + x²) = (3 − 1) / (1 + 3) = 2 / 4 = 1/2

∴ The correct answer is (x² − 1)/(1 + x²).

Alternate Method

Given:

secθ + tanθ = x   --- (1)

Formula Used:

sec²θ − tan²θ = 1

Calculations:

⇒ (secθ + tanθ) × (secθ − tanθ) = 1

⇒ x × (secθ − tanθ) = 1

⇒ secθ − tanθ = 1/x   --- (2)

Adding equation (1) and equation (2):

⇒ 2 × secθ = x + 1/x ⇒ secθ = (x² + 1) / 2x

Subtracting equation (2) from equation (1):

⇒ 2 × tanθ = x − 1/x ⇒ tanθ = (x² − 1) / 2x

From trigonometric ratios, we know secθ = Hypotenuse / Base and tanθ = Perpendicular / Base.

Comparing the values, we get Base = 2x, Perpendicular = x² − 1, and Hypotenuse = x² + 1.

⇒ sinθ = Perpendicular / Hypotenuse

⇒ sinθ = (x² − 1) / (x² + 1)

∴ The correct answer is (x² − 1)/(1 + x²).

Additional Information

Pythagorean Triplets

In a right-angled triangle, hypotenuse² = perpendicular² + base². A highly useful variable triplet for trigonometry limits is (2x, x² − 1, x² + 1).

Trigonometric Identities

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

Relation between Trigonometric Ratios

Basic conversions include tanθ = sinθ ÷ cosθ and secθ = 1 ÷ cosθ. Dividing these gives the direct relation sinθ = tanθ ÷ secθ.

The correct answer is 'flour'.

 

Key Points

• The word "flour" refers to a powder obtained by grinding grain, typically wheat, and used to make bread, cakes, and pastry. (आटा)
• Kneading is a process typically associated with making dough, which is made from flour.
• The other options: "flora," "flavour," and "floor" do not fit the context of kneading.

​Therefore, the correct answer is 'Option 4'.

Complete Sentence: Ruby gave her mother a hug and offered to knead the flour.

Additional Information

• Flora (फूल पौधे): The plants of a particular region, habitat, or geological period.
• Flavour (स्वाद): The distinctive taste of a food or drink.
• Floor (मंजिल): The lower surface of a room, on which one may walk.

The correct answer is: Option 1 i.e. alliteration.

Key Points

• The word "alliteration" means the repetition of usually initial consonant sounds in two or more neighboring words or syllables. (अनुप्रास)
• Example: Sally sells sea shells by the sea shore.
• "Epithet" refers to a descriptive term or phrase expressing a quality or characteristic of the person or thing mentioned. (विशेषण)
• Example: Alexander the Great.
• Hence, we can infer that the use of initial consonant sounds repetitively is called 'alliteration'.

Therefore, the correct answer is: Option 1 i.e. alliteration.

Additional Information

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

• Epithet (विशेषण): A descriptive term or phrase expressing a quality or characteristic of the person or thing mentioned.
• Example: He earned the epithet "Alexander the Great" for his conquests.
• Simile (उपमा): A figure of speech involving the comparison of one thing with another thing of a different kind, used to make a description more emphatic or vivid.
• Example: As brave as a lion.
• Allegory (रूपक): A narrative in which characters and events represent abstract ideas or concepts.
• Example: George Orwell's "Animal Farm" is an allegory of the Russian Revolution.

The correct answer is 'bear'.

 

Key Points

• The word "bear" means to endure or tolerate something difficult or unpleasant. (सहन करना)
• It is the correct word used in the context of enduring an insult.
• The other options: "bier," "bare," and "beer" do not fit the context.

​Therefore, the correct answer is 'Option 4'.

Complete Sentence: The mother couldn’t bear such an insult from her son in front of the others.

Additional Information

• Bier (मय्यत): A movable frame on which a coffin or corpse is placed before burial or cremation.
• Bare (नंगा): Without covering or clothing; naked.
• Beer (बीयर): An alcoholic drink made from yeast-fermented malt flavored with hops.

The correct answer is Option 2.

Key Points

• The sentence begins in the past tense ("While I was walking"), so the rest of the sentence should also be consistent in tense.
• The phrase "my friend calls me" should be in the past tense to match the first part of the sentence. It should be "my friend called me."
• Additionally, the verb "ask" should be in the past tense. It should be "asked."
• Thus, the correct phrase is "my friend called me and asked."

Therefore, the correct answer is 2.

 

Correct Sentence: "While I was walking to the store, my friend called me and asked for a favour."

Additional Information

• 1: "While I was walking" – This part is grammatically correct and does not contain any errors.
• 3: "to the store" – This part is grammatically correct and does not contain any errors.
• 4: "for a favour" – This part is grammatically correct and does not contain any errors.
• 2: "my friend calls me and ask" – The error here is the incorrect use of present tense verbs "calls" and "ask" instead of the past tense "called" and "asked."

The correct answer is: Option 2.

Key Points

• The phrase "Turned up their nose at her" means to treat someone with disdain or contempt. (उससे घृणा या तिरस्कार से पेश आना)
• Example: The wealthy neighbors turned up their noses at her old, worn-out clothes.
• "Impoverished" means being in a state of poverty or lacking basic resources. (गरीब)
• Example: Due to her impoverished state, she often skipped meals to save money.

Therefore, the correct answer is: Option 2.

Additional Information

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

• Came in handy (काम आया): To be useful or convenient.
• Example: The extra blanket came in handy during the cold night.
• Turned the corner (मोड़ मुड़ना): To pass a critical point and start to improve.
• Example: After months of hard work, the company finally turned the corner and began making a profit.
• Gave a wide space to her (उससे दूरी बनाकर रखी): To avoid someone or something.
• Example: The children gave a wide space to the barking dog on the street.

The correct answer is: early.

Key Points

• The word "late" means after the expected or usual time; it can refer to something that happens or is done after the scheduled time. (देर से)
• Example: She arrived late to the meeting because of the traffic.
• "Early" refers to something that happens before the expected or usual time; it is the opposite of late. (जल्दी)
• Example: She arrived early to the meeting to prepare her presentation.
• Hence, we can infer that the opposite of 'late' is 'early'.

Therefore, the correct answer is: early.

Additional Information

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

• Exit (निकास): A way out of a building, room, or vehicle.
• Example: The emergency exit is located at the back of the room.
• Small (छोटा): Of a size that is less than normal or usual.
• Example: She has a small apartment in the city.
• Suffer (सहन करना): To experience pain, distress, or hardship.
• Example: He continues to suffer from chronic back pain.

The correct answer is: Option 2.

Key Points

• The word "baron" refers to a member of the lowest order of the British nobility or a person of great power in a particular area of business. (बैरन)
• Example: The oil baron owned several refineries and was one of the wealthiest men in the country.
• "Bayonet" refers to a blade that may be fixed to the muzzle of a rifle and used to stab an opponent in hand-to-hand fighting. (बायोनेट)
• Example: The soldier fixed his bayonet to the end of his rifle before charging at the enemy.
• Hence, the correct word "bayonet" means a blade fixed to the end of a gun. (बायोनेट)
• Example: The soldier struck the enemy with the bayonet.

Therefore, the correct answer is: Option 2.

Additional Information

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

• Baron (बैरन): A member of the lowest order of the British nobility or a powerful person in a particular business.
• Example: The media baron owns several television networks.
• Bigot (कट्टरपंथी): A person who is intolerant towards those holding different opinions.
• Example: She refused to listen to his ideas because she is a bigot.
• Baton (लाठी): A short stick or staff used by police officers, conductors, or in relay races.
• Example: The conductor used his baton to lead the orchestra.