SSC CGL 2024 Tier-I Official Paper (Held On: 24 Sept, 2024 Shift 3) (Previous Year Paper)

The logic followed here is: Number of letters in the word × 7 = Number code.

'BLEACHERS' is coded as '63'

'ABACAS' is coded as '42'

Similarly, 'FIZZY' will be coded as:

Hence, the correct answer is "35".

Counting the squares are there in the figure:

Hence, the correct answer is "13".

The logic followed here is:

Logic: Study of specific natural objects.

Here,

Botany : Plants → Botany is the study of Plants.

Similarly,

Conchology : Shells → Conchology is the study of Shells.

Hence, the correct answer is "Option 2".

The paper look when unfolded:

Hence, the correct answer is "Option 2".

The logic followed here is:

Hence, the correct answer is "VMX".

The logic followed here is: Cube of 2^nd number = 1^st number.

64 is related to 4 → 4³ = 64

216 is related to 6 → 63 = 216

Similarly, 729 is related to:→ 93 = 729

Hence, the correct answer is "9".

The logic followed here is:

Hence, the correct answer is "67, 71".

The logic followed here is:

'ZCJV' is coded as 'YBIU'

'GKMD' is coded as 'FJLC'

Similarly, 'EYQR' will be coded as:

Hence, the correct answer is "DXPQ".

Given word: 'SIGNATURE'

After arranged in alphabetical order:

Now,

there are only One alphabets "G" that will remain unchanged if each of the letters in the word 'SIGNATURE' is arranged in English alphabetical order.

Hence, the correct answer is "Option 2".

According to the given information:

The number of persons who can drive all the three vehicles = 6

Persons who can drive any two of these vehicles = 8 + 7 + 12 = 27

Thus, the difference between the number of persons who can drive all the three vehicles and that of persons who can drive any two of these vehicles = 27 - 6 = 21

Hence, the correct answer is "21".

The logic followed here is:

Logic: Positive Emotional States.

Here,

Love : Harmony → Love is associated with Harmony.

Similarly,

Peace : Bliss → Peace is associated with Bliss.

Hence, the correct answer is "Option 2".

The correct mirror image of the given figure is:

Hence, the correct answer is "Option 2".

The logic followed here is:

1)  HKI

2)  RUS

3)  MPN

4)  QUR

Hence, the correct answer is "QUR".

BODMAS Table: 

After decoding signs:

Given equation: 7 ÷ 5 − 30 × 6 + 4 = ?

After interchanging the signs then:

⇒ 7 × 5 + 30 ÷ 6 - 4 = ?

⇒ ? = 7 × 5 + 5 - 4 

⇒ ? = 35 + 5 - 4 

⇒ ? = 40 - 4 = 36

Hence, the correct answer is "36".

The logic followed here is:

Given Sequence: B_DE_GDGBK_IB_DK

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

Option 1) CODB → BCDE / OGDG / BKDI / BBDK

Option 2) CBDO → BCDE / BGDG / BKDI / BODK

Option 3) CBDP → BCDE / BGDG / BKDI / BPDK

Option 4) OCBD → BODE / CGDG / BKBI / BDDK

Hence, the correct answer is "CBDO".

Statements:

All grains are cereals.

Some cereals are fruits.

No fruit is a vegetable.

The least possible Venn diagram is as follows:

Conclusions:

I. All cereals can be vegetables. - False (because Some cereals are fruits and No fruit is a vegetable. so which part belongs to fruits cannot be vegetable)

II. All grains are fruits. - False (because there is no direct relation between grains and fruits. So it can be possible but not definite)

Hence, the correct answer is "Neither Conclusion I nor II follow".

The logic followed here is:

Logic: (1^st number)² - 9 = 2^nd number.

Now,

For, 7 : 40

⇒ 7² - 9 = 49 - 9 = 40.

For, 5 : 16

⇒ 5² - 9 = 25 - 9 = 16.

Similarly, for 9 ∶ ?

⇒ 9² - 9 = 81 - 9 = 72.

Hence, the correct answer is "Option 1".

BODMAS Table: 

After decoding signs:

Given equation: 1604 × 4 + 87 − 16 ÷ 16 = ?

After interchanging the signs then:

⇒ 1604 ÷ 4 - 87 + 16 × 16 = ?

⇒ ? = 401 - 87 + 16 × 16

⇒ ? = 401 - 87 + 256

⇒ ? = 657 - 87 = 570

Hence, the correct answer is "570".

The correct mirror image of the given figure is:

Hence, the correct answer is "Option 3".

The logic followed here is:

Bottom-right corner symbol shifts in top-left corner and flipped horizontally.

And another symbol is replaced by a new symbol.

Hence, the correct answer is "Option 2".

The logic followed here is:

Number is increased by 1 and shifting one step ahead in anticlockwise direction.

Vowels are increasing (A, E, I, O, U).

Hence, the correct answer is "Option 2".

Family Chart:

Given: Q is the mother of T. R is the husband of P. S is the father of Q. T is the brother of P. 

So, P is the "Daughter's daughter" of S.

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

For find the opposite surface of '5', 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.

3 is opposite to 4.

So, 1 will be opposite to 5.

Hence, the correct answer is "1".

The logic followed here is:

Logic: (1^st number × 4) + 10 = 2^nd number.

Now,

For, 26 : 114

⇒ (26 × 4) + 10 = 104 + 10 = 114.

For, 47 : 198

⇒ (47 × 4) + 10 = 188 + 10 = 198.

Similarly, for 15 ∶ 70

⇒ (15 × 4) + 10 = 60 + 10 = 70.

Hence, the correct answer is "Option 4".

The logic followed here is: 

Logic: Numbers are prime numbers except only one number.

1) 829: Prime number (divisible only by 1 and 829)

2) 821: Prime number (divisible only by 1 and 821)

3) 643: Prime number (divisible only by 1 and 643)

4) 237: Not prime number (divisible by 1, 3, 79 and 237)

Hence, the correct answer is "237".

The correct answer is Kirana gharana.

Key Points

• Gangubai Hangal was a renowned Indian classical vocalist of the Kirana gharana.
• The Kirana gharana is one of the prominent schools of Hindustani classical music.
• It is known for its melodious and improvisational style, focusing on the purity of notes and elaborate raga development.
• Gangubai Hangal was known for her deep and powerful voice which contributed significantly to the Kirana gharana's legacy.

Additional Information

• Kirana Gharana
• The Kirana gharana is named after the town of Kirana in Uttar Pradesh, India.
• This gharana emphasizes the importance of the 'sur' (note) and is known for its slow elaboration of ragas, known as 'alap'.
• Influential musicians from this gharana include Abdul Karim Khan, Bhimsen Joshi, and Roshan Ara Begum.
• Gharana
• The term 'gharana' means 'family' or 'lineage' in Hindi.
• In the context of Indian classical music, it refers to a system of social organization linking musicians or dancers by lineage or apprenticeship, and by adherence to a particular musical style.
• Each gharana has its own distinct style, techniques, and repertoire.
• Hindustani Classical Music
• It is the classical music of North India, distinct from the Carnatic music of South India.
• It has a rich history dating back to the Vedic times and has evolved over centuries, integrating various cultural influences.
• The music is characterized by its raga (melodic framework) and tala (rhythmic pattern).

The correct answer is Kolhapur.

Key Points

• Kolhapur, located in the state of Maharashtra, is known for its industrial development.
• The city is famous for its foundries and engineering industries, producing automotive parts, machinery, and casting components.
• Kolhapur is also renowned for the production of Kolhapuri chappals, a traditional handcrafted leather footwear.
• The industrial growth in Kolhapur has led to the establishment of several industrial estates and a thriving manufacturing sector.

Additional Information

• Industrial Districts in India
• Industrial districts in India are regions that have a high concentration of industries and manufacturing units.
• These regions often benefit from better infrastructure, connectivity, and access to raw materials and skilled labor.
• Examples include Pune, known for its automobile and IT industries, and Ludhiana, famous for its textile and bicycle manufacturing units.
• Economic Impact
• Industrial districts contribute significantly to the economy by generating employment, increasing exports, and contributing to the GDP.
• They also attract investments and foster innovation and technological advancements.
• Challenges
• Despite their importance, industrial districts often face challenges such as pollution, resource management, and the need for sustainable practices.
• Efforts are being made to address these issues through regulations, technological upgrades, and promoting green industries.

The correct answer is Behavioural condition.

Key Points

• Behavioural condition is not typically used as a classification criterion for drugs.
• Drugs are generally classified based on their chemical structure, molecular target, and pharmacological effect.
• Classification based on chemical structure involves grouping drugs with similar molecular frameworks.
• Molecular target classification groups drugs based on the biological molecules they interact with, while pharmacological effect classification is based on the therapeutic effects of the drugs.

Additional Information

• Chemical Structure:
• This classification method involves grouping drugs with similar chemical compositions or molecular frameworks.
• Examples include beta-lactams, benzodiazepines, and phenothiazines.
• Molecular Target:
• This method classifies drugs based on the specific biological molecules they interact with.
• For example, drugs that target enzymes, receptors, ion channels, or nucleic acids.
• Pharmacological Effect:
• Drugs are classified based on the therapeutic effects they produce.
• For example, antihypertensives, analgesics, antipyretics, and antidepressants.

The correct answer is Chauri-Chaura incident.

Key Points

• The Chauri-Chaura incident occurred on February 5, 1922, in the Chauri-Chaura town of the Gorakhpur district in the United Provinces (now Uttar Pradesh), British India.
• During this incident, a large group of protesters participating in the Non-Cooperation Movement clashed with police, who opened fire. In retaliation, the demonstrators attacked and set fire to a police station, killing all of its occupants.
• In response to the violence, Mahatma Gandhi, the leader of the Non-Cooperation Movement, called off the movement on February 12, 1922, emphasizing his principle of non-violence (Ahimsa).
• The Chauri-Chaura incident marked a significant turning point in the Indian independence movement, highlighting the challenges of maintaining non-violent discipline among the masses.

Additional Information

• Non-Cooperation Movement
• Launched by Mahatma Gandhi in 1920, the Non-Cooperation Movement aimed to resist British rule in India through non-violent means.
• It was a significant phase of the Indian independence movement, advocating for Swaraj or self-governance.
• Key strategies included the boycott of British goods, institutions, and services, and the promotion of Indian-made goods and institutions.
• The movement saw widespread participation but was suspended after the Chauri-Chaura incident to prevent further violence.
• Gandhian Principles
• Mahatma Gandhi's philosophy centered around non-violence (Ahimsa) and truth (Satya).
• Gandhi believed in the power of peaceful protest and civil disobedience as tools for social and political change.
• He emphasized the importance of self-reliance, community solidarity, and ethical living.
• Chauri-Chaura Town
• Chauri-Chaura is a town in the Gorakhpur district of Uttar Pradesh, India.
• The town gained historical significance due to the incident of February 1922, which had a profound impact on India's struggle for independence.
• Impact on Indian Independence Movement
• The suspension of the Non-Cooperation Movement led to a period of reflection and re-strategizing within the Indian National Congress.
• It underscored the need for disciplined, non-violent resistance and influenced future movements, such as the Civil Disobedience Movement and the Quit India Movement.
• The incident also highlighted the complexities of leading a mass movement and the potential for deviations from non-violent principles.

The correct answer is Option 2.

Key Points

• Ishwar Chandra Vidyasagar was a prominent Indian educator, social reformer, and scholar in the 19^th century.
• He strongly protested against child marriage and polygamy and worked towards ending these practices.
• Due to his relentless efforts, the Hindu Widow Remarriage Act, 1856, was enacted, which legalized the remarriage of Hindu widows.
• Although he was a renowned scholar, he was primarily known for his scholarship in the Sanskrit language, not Persian.

Additional Information

• Ishwar Chandra Vidyasagar
• Born on 26 September 1820 in Birsingha, Bengal, he was one of the most prominent figures of the Bengal Renaissance.
• He was a key proponent of modern education and played a significant role in the establishment of several educational institutions in Bengal.
• Vidyasagar's contributions extended to the field of literature as well. He simplified and modernized the Bengali alphabet and prose, making it more accessible to the common people.
• He was also known for his philanthropic activities, dedicating his life to the upliftment of the underprivileged and the betterment of society.
• Hindu Widow Remarriage Act, 1856
• This act was a significant social reform in British India that legalized the remarriage of Hindu widows.
• The act was passed on 26 July 1856, due to the efforts of social reformers like Ishwar Chandra Vidyasagar.
• The act provided legal safeguards and rights to widows who chose to remarry, which was a revolutionary step towards gender equality in that era.
• Child Marriage
• Child marriage refers to any formal marriage or informal union where one or both parties are under 18 years of age.
• It is a deeply rooted tradition in some cultures but has significant negative consequences, including the health and education of the child.
• Social reformers like Ishwar Chandra Vidyasagar campaigned against child marriage to protect the rights and futures of young children.
• Polygamy
• Polygamy is the practice of marrying multiple spouses. In the context of Ishwar Chandra Vidyasagar's time, it often referred to men having multiple wives.
• Vidyasagar and other reformers opposed this practice as it often led to the exploitation and mistreatment of women.
• Efforts to outlaw polygamy were part of broader social reforms aimed at ensuring gender equality and women's rights.

The correct answer is The net force acting on the object is the difference between the two forces..

Key Points

• When two forces act in opposite directions on an object, the net force is the difference between the two forces.
• The direction of the net force is in the direction of the larger force.
• If the forces are equal in magnitude but opposite in direction, they cancel each other out, resulting in a net force of zero.
• The concept of net force is crucial in understanding how forces influence the motion of objects according to Newton's Laws of Motion.

Additional Information

• Force
• A force is a push or pull upon an object resulting from its interaction with another object. Forces can cause objects to accelerate, decelerate, remain in place, or change shape.
• Forces are vector quantities, meaning they have both magnitude and direction.
• Forces can be categorized into contact forces (e.g., friction, tension) and action-at-a-distance forces (e.g., gravitational, electromagnetic).
• Net Force
• The net force is the vector sum of all the forces acting on an object.
• It determines the overall effect of all the forces on the motion of the object.
• According to Newton's Second Law of Motion, the net force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma).
• Newton's Laws of Motion
• First Law (Law of Inertia): An object will remain at rest or in uniform motion in a straight line unless acted upon by a net external force.
• Second Law (Law of Acceleration): The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass (F = ma).
• Third Law (Action and Reaction): For every action, there is an equal and opposite reaction.
• Equilibrium
• An object is in equilibrium when the net force acting on it is zero.
• In equilibrium, an object at rest stays at rest, and an object in motion continues to move at a constant velocity.
• Equilibrium can be static (object at rest) or dynamic (object moving with constant velocity).

The correct answer is Industrial Policy 1991.

Key Points

• The Industrial Policy of 1991 marked a significant shift in India's industrial policy towards liberalisation, privatisation, and globalisation.
• This policy reduced the stringent regulations and licensing requirements imposed on private companies, thereby encouraging private sector participation.
• The policy aimed to increase competition in the market, attract foreign direct investment (FDI), and enhance industrial growth and efficiency.
• Key reforms included the abolition of industrial licensing for most industries, reduction of areas reserved for the public sector, and easing of foreign investment norms.

Additional Information

• Industrial Policy 1980
• The Industrial Policy of 1980 sought to promote indigenous technology and ensure a balanced regional growth of industries.
• It focused on the optimum utilisation of installed capacity, modernization of industries, and accelerated growth of the small-scale sector.
• Industrial Policy 1977
• This policy aimed at promoting the growth of small-scale and cottage industries to achieve employment generation and regional development.
• It emphasised decentralisation and self-reliance.
• Industrial Policy 1956
• The 1956 policy was a landmark in the history of Indian industrial policy, emphasizing the establishment of a socialist pattern of society.
• It classified industries into three schedules: Schedule A (exclusively in the public sector), Schedule B (progressively state-owned), and Schedule C (left to the private sector).

The correct answer is Antim Panghal.

Key Points

• Antim Panghal became the first Indian woman to win the Under-20 World Wrestling Championship in 2022.
• She clinched the gold medal in the 53 kg category at the championship held in Sofia, Bulgaria.
• Antim defeated Kazakhstan's Altyn Shagayeva in the final to achieve this historic feat.
• She is recognized for her remarkable wrestling skills and dedication to the sport, contributing to India's growing presence in international wrestling competitions.

Additional Information

• Under-20 World Wrestling Championship
• The Under-20 World Wrestling Championship is an annual event organized by the United World Wrestling (UWW).
• It features young wrestling talents from various countries competing in different weight categories.
• The championship serves as a platform for young wrestlers to showcase their skills on an international stage and gain recognition.
• Winning a medal at this championship is considered a significant achievement in a wrestler's career and often leads to further opportunities and support.
• United World Wrestling (UWW)
• UWW is the international governing body for the sport of wrestling, overseeing freestyle, Greco-Roman, and women's wrestling disciplines.
• It is responsible for organizing world championships, continental championships, and other international wrestling events.
• UWW aims to promote and develop the sport of wrestling globally, ensuring fair competition and adherence to the rules.
• The organization also works towards increasing the popularity and visibility of wrestling as a sport worldwide.
• Wrestling in India
• Wrestling has a rich history and cultural significance in India, with traditional forms like Kushti being practiced for centuries.
• Indian wrestlers have achieved notable success in international competitions, including the Olympics, Commonwealth Games, and Asian Games.
• The Wrestling Federation of India (WFI) is the national governing body for wrestling in India, responsible for organizing domestic competitions and selecting teams for international events.
• Initiatives and support from the government, sports authorities, and private organizations have contributed to the growth and development of wrestling in India.

The correct answer is Bering Strait.

Key Points

• The Bering Strait is a narrow passage of water that separates the Chukchi Peninsula of the Russian Far East from the Seward Peninsula of Alaska.
• It connects the Pacific Ocean to the Arctic Ocean, specifically linking the Bering Sea to the Chukchi Sea.
• The strait is named after Vitus Bering, a Danish explorer in the service of the Russian Empire, who explored the area in the early 18^th century.
• The Bering Strait is approximately 82 kilometers (51 miles) wide at its narrowest point, with a depth of about 30-50 meters (98-164 feet).

Additional Information

• Davis Strait
• The Davis Strait is a northern arm of the Labrador Sea, which lies between Greenland and Canada’s Baffin Island.
• It is known for its harsh weather conditions and is a major route for the migration of icebergs from the Arctic Ocean.
• Florida Strait
• The Florida Strait is a strait located south-southeast of the North American mainland, generally accepted to be the boundary between the Gulf of Mexico and the Atlantic Ocean.
• It lies between the Gulf of Mexico and the Atlantic Ocean, separating the U.S. state of Florida from Cuba.
• Messina Strait
• The Strait of Messina is a narrow passage between the eastern tip of Sicily and the western tip of Calabria in the southern part of Italy.
• It connects the Tyrrhenian Sea to the north with the Ionian Sea to the south.

The correct answer is Justice Verma Committee.

Key Points

• The Justice Verma Committee was formed to identify the existence of legal provisions for the implementation of some of the Fundamental Duties enshrined in the Constitution of India.
• The Committee was set up in 1999 under the chairmanship of Justice J.S. Verma, former Chief Justice of India.
• It reviewed the existing legal framework and suggested measures to make the Fundamental Duties more enforceable.
• The Committee's recommendations emphasized the role of education and awareness in promoting a sense of duty among citizens.

Additional Information

• Fundamental Duties
• The Fundamental Duties are a set of moral obligations for all citizens of India, enshrined in Part IV-A of the Constitution (Article 51A).
• These duties were added by the 42^nd Amendment Act, 1976, based on the recommendations of the Swaran Singh Committee.
• There are currently 11 Fundamental Duties, which include respect for the Constitution, national symbols, the promotion of harmony, and the protection of public property.
• The Fundamental Duties are non-justiciable, meaning they are not legally enforceable by the courts.
• Justice Verma Committee
• Justice J.S. Verma, a renowned jurist and former Chief Justice of India, chaired the committee.
• The Committee's focus was to enhance the implementation of Fundamental Duties through legal and educational reforms.
• It highlighted the need for incorporating the Fundamental Duties into the education system to instill a sense of responsibility in citizens from a young age.
• The Committee's report has been influential in shaping policies and programs aimed at promoting civic responsibility.
• 42^nd Amendment Act, 1976
• The 42^nd Amendment Act is also known as the "Mini-Constitution" due to its extensive changes to the Constitution of India.
• It introduced the Fundamental Duties, provided for the establishment of Administrative Tribunals, and made several changes to the Preamble and other parts of the Constitution.
• The Amendment was passed during the Emergency period under Prime Minister Indira Gandhi's government.
• It aimed to strengthen the central government's powers and bring about socio-economic reforms.

The correct answer is M.K. Stalin.

Key Points

• M.K. Stalin is the current Chief Minister of Tamil Nadu.
• He assumed office on 7 May 2021.
• M.K. Stalin is the son of the former Chief Minister of Tamil Nadu, M. Karunanidhi.
• He is a prominent leader of the Dravida Munnetra Kazhagam (DMK) party.

Additional Information

• M.K. Stalin:
• M.K. Stalin (Muthuvel Karunanidhi Stalin) was born on 1 March 1953 in Chennai, Tamil Nadu.
• He started his political career with the DMK party and held various positions within the party over the years.
• Before becoming the Chief Minister, he served as the Deputy Chief Minister of Tamil Nadu from 2009 to 2011.
• Stalin has also held the position of Mayor of Chennai from 1996 to 2002.
• Dravida Munnetra Kazhagam (DMK):
• The DMK is a political party in the state of Tamil Nadu and the Union Territory of Puducherry in India.
• It was founded by C.N. Annadurai in 1949 as a breakaway faction from the Dravidar Kazhagam.
• The party has a strong base among the Tamil population and has been a significant force in Tamil Nadu politics.
• The DMK has been involved in various state and national level political activities and has formed the government in Tamil Nadu multiple times.
• Autobiography:
• An autobiography is a self-written account of the life of a person.
• It provides insights into the personal and professional life of the author, including their experiences, challenges, and achievements.
• Autobiographies are valuable historical documents as they offer first-hand accounts of events and people from the perspective of the author.
• Chief Minister:
• The Chief Minister is the head of the government in a state in India.
• The Chief Minister is elected by the legislative assembly of the state and is appointed by the Governor.
• The role of the Chief Minister includes formulating policies, implementing laws, and overseeing the administration of the state government.
• The Chief Minister is also responsible for representing the state in various national and international forums.

The correct answer is Article 79 to 122.

Key Points

• Article 79 to 122 of the Indian Constitution deals with the Parliament of India and its proceedings.
• These articles cover various aspects such as the composition of Parliament, the powers, and functions of its members, and the procedures for conducting parliamentary business.
• Article 79 establishes a bicameral legislature at the federal level, consisting of the President and two Houses: the Council of States (Rajya Sabha) and the House of the People (Lok Sabha).
• Articles 85 to 122 provide detailed provisions regarding the functioning of Parliament, including sessions, prorogation, dissolution, the conduct of business, and the powers and privileges of its members.

Additional Information

• Article 51A
• Article 51A specifies the fundamental duties of every citizen of India.
• It was added by the 42^nd Amendment Act, 1976.
• These duties include respecting the Constitution, promoting harmony, and safeguarding public property, among others.
• Article 36 to 51
• These articles pertain to the Directive Principles of State Policy.
• They provide guidelines to the central and state governments for making policies aimed at establishing a just society.
• Article 12 to 35
• These articles cover the Fundamental Rights guaranteed to the citizens of India.
• They include rights like the right to equality, right to freedom, right against exploitation, right to freedom of religion, cultural and educational rights, and the right to constitutional remedies.

The correct answer is 1.64.

Key Points

• The annual exponential growth rate of India's population as per the 2011 Census is 1.64%.
• This growth rate is a measure of how quickly the population is increasing each year.
• It is calculated using the formula for exponential growth: P(t) = P₀ * e^rt, where r is the growth rate.
• The growth rate reflects both natural increase (births minus deaths) and net migration.

Additional Information

• Exponential Growth Rate
• Exponential growth occurs when the growth rate of a mathematical function's value is proportional to the function's current value.
• In the context of population, it means the population grows at a constant percentage rate per period.
• The formula for exponential growth is P(t) = P₀ * e^rt, where:
• P(t) is the future population.
• P₀ is the initial population.
• r is the growth rate.
• t is the time period.
• e is the base of the natural logarithm, approximately equal to 2.71828.
• Census
• A census is a systematic enumeration of a population, typically recording various details of individuals.
• The Census of India is a decennial (every 10 years) exercise conducted by the Office of the Registrar General and Census Commissioner, under the Ministry of Home Affairs.
• It provides vital data on population size, distribution, and various demographic characteristics.
• The 2011 Census was the 15^th National Census of the country since 1872.

The correct answer is Andhra Pradesh.

Key Points

• Kuchipudi is one of the eight major Indian classical dances.
• It originates from a village named Kuchelapuram in the Indian state of Andhra Pradesh.
• Kuchipudi has a unique blend of Natya (drama) and Nritta (pure dance).
• It involves fast rhythms and fluid movements, creating a perfect balance between control and abandon, strength and delicacy.

Additional Information

• Indian Classical Dances
• India has a rich cultural heritage, and classical dance forms are a significant part of it.
• There are eight recognized classical dance forms in India: Bharatanatyam, Kathak, Kathakali, Manipuri, Kuchipudi, Odissi, Mohiniyattam, and Sattriya.
• Each dance form originates from different regions and has its own unique style, costumes, and music.
• These dances are not only a form of art but also a way to express the deep cultural and traditional values of India.
• Natya Shastra
• It is an ancient Indian treatise on the performing arts, encompassing theatre, dance, and music.
• Attributed to Sage Bharata Muni, it is considered the foundational text for Indian classical dance forms.
• The Natya Shastra provides detailed guidelines on various aspects of dance, including gestures, postures, expressions, and stage design.
• It serves as a comprehensive guide for artists and scholars in understanding the traditional practices and techniques of Indian performing arts.
• Kuchipudi Dance Form
• Kuchipudi is characterized by its graceful movements, intricate footwork, and expressive gestures.
• Performers often depict stories from Hindu mythology, particularly the tales of Lord Krishna.
• The dance involves both male and female performers, with male dancers traditionally playing female roles as well.
• Musical accompaniment for Kuchipudi includes classical instruments like the mridangam, veena, and flute.

The correct answer is C3H8.

Key Points

• C3H8 is the chemical formula for propane, which is a saturated hydrocarbon.
• Saturated hydrocarbons contain only single bonds between carbon atoms.
• Propane is an alkane, which is a type of saturated hydrocarbon.
• Alkanes have the general formula CnH2n+2, and propane (C3H8) fits this formula.

Additional Information

• Saturated Hydrocarbons:
• Saturated hydrocarbons are hydrocarbons in which all the carbon atoms are connected by single bonds. They are also known as alkanes.
• Alkanes have the general formula CnH2n+2, where n is the number of carbon atoms.
• Examples include methane (CH4), ethane (C2H6), and propane (C3H8).
• Unsaturated Hydrocarbons:
• Unsaturated hydrocarbons contain at least one double or triple bond between carbon atoms.
• Alkenes are hydrocarbons with one or more double bonds (general formula CnH2n), such as ethene (C2H4) and propene (C3H6).
• Alkynes are hydrocarbons with one or more triple bonds (general formula CnH2n-2), such as ethyne (C2H2) and propyne (C3H4).
• Benzene (C6H6):
• Benzene is an aromatic hydrocarbon, characterized by a ring of six carbon atoms with alternating double bonds.
• Aromatic hydrocarbons have unique stability and properties due to their conjugated pi-electron systems.
• Hydrocarbon Classes:
• Hydrocarbons can be classified into alkanes (saturated), alkenes (unsaturated), alkynes (unsaturated), and aromatic hydrocarbons.
• Each class has distinct chemical properties and reactivities.

The correct answer is Ahmedabad.

Key Points

• The final match of IPL 2023 was hosted at the Narendra Modi Stadium in Ahmedabad.
• Ahmedabad is the largest city in the state of Gujarat, India.
• Narendra Modi Stadium, formerly known as Sardar Patel Stadium, is the largest cricket stadium in the world in terms of seating capacity, holding over 132,000 spectators.
• This stadium has also hosted several international matches and is a prominent venue for cricket in India.

Additional Information

• Indian Premier League (IPL)
• The IPL is a professional Twenty20 cricket league in India contested by ten teams based out of seven Indian cities and three Indian states.
• The league was founded by the Board of Control for Cricket in India (BCCI) in 2008.
• It is usually held between March and May of every year and has an exclusive window in the ICC Future Tours Programme.
• IPL is known for its high entertainment quotient and has a massive following both in India and globally.
• Narendra Modi Stadium
• Located in Ahmedabad, Gujarat, this stadium is the largest in the world with a seating capacity of over 132,000.
• It is named after the current Prime Minister of India, Narendra Modi, who was also the president of the Gujarat Cricket Association.
• The stadium was renovated in 2020 and has since hosted several key matches, including international fixtures.
• It is equipped with state-of-the-art facilities, including LED floodlights, a clubhouse, and an Olympic-sized swimming pool.
• IPL Final Matches
• The IPL final is the last match of the tournament that determines the champion of the season.
• It is one of the most-watched sporting events in India and attracts significant global viewership.
• Some of the most memorable matches in cricket history have been IPL finals, featuring thrilling finishes and standout performances.
• The venue for the final is usually selected to accommodate a large number of spectators and provide a grand stage for the culmination of the tournament.

The correct answer is 21^st.

Key Points

• The Rajiv Gandhi National Sadbhavana Award is given annually by the Indian National Congress party.
• The award aims to recognize outstanding contributions towards promoting peace, harmony, and national integration in India.
• Ustad Amjad Ali Khan, a renowned sarod maestro, was honored with the 21^st Rajiv Gandhi National Sadbhavana Award.
• The award was conferred on him in recognition of his efforts to promote communal harmony and peace through music.

Additional Information

• Rajiv Gandhi National Sadbhavana Award
• The award was instituted in 1992, in memory of Rajiv Gandhi, the former Prime Minister of India.
• It is awarded annually on his birth anniversary, 20^th August.
• Recipients of the award are given a citation and a cash prize.
• The award celebrates individuals and organizations who have made significant contributions towards promoting values of harmony and peace.
• Ustad Amjad Ali Khan
• He is an Indian classical musician who plays the sarod, a stringed instrument.
• He has been awarded several prestigious awards including the Padma Vibhushan, India's second-highest civilian honor.
• He comes from a long lineage of sarod players and has significantly contributed to the Indian classical music scene.
• His music is known for its spiritual depth and ability to transcend cultural barriers.

The correct answer is Gomat.

Key Points

• The term "Gomat" is derived from the Sanskrit word "Go" meaning cow, and it signifies a wealthy person who owned many cows in Vedic times.
• In Vedic society, wealth was often measured by the number of cows one possessed, making "Gomat" an indicator of prosperity and affluence.
• Cows were highly valued in Vedic culture for their milk, dung, and other resources, which were essential for agriculture and daily life.
• The ownership of cows symbolized economic stability and social status in ancient Vedic civilization.

Additional Information

• Gotra
• Gotra refers to a lineage or clan assigned to a Hindu at birth. It is a system of patrilineal classification and is often used to trace ancestry.
• It is used for identifying families and for purposes related to marriage, as marriages within the same Gotra are traditionally discouraged.
• Gobatscha
• Gobatscha is not a commonly recognized term in Vedic literature or Sanskrit.
• Gaveshana
• Gaveshana means search or investigation in Sanskrit. It is often used in contexts related to research or inquiry.

The correct answer is Uttar Pradesh.

Key Points

• Chandra Prabha Wildlife Sanctuary is located in the state of Uttar Pradesh, India.
• It was established in 1957 and is situated in the Chandauli district of Uttar Pradesh.
• The sanctuary is spread over an area of 78 square kilometers and is known for its scenic beauty and rich biodiversity.
• It is home to various species of flora and fauna, including leopards, blackbucks, and numerous birds.

Additional Information

• Wildlife Sanctuaries in India
• Wildlife sanctuaries are protected areas established by the government to conserve wildlife and their habitats.
• India has more than 500 wildlife sanctuaries, each hosting a diverse range of species and ecosystems.
• These sanctuaries play a crucial role in protecting endangered species and maintaining ecological balance.
• Examples of famous wildlife sanctuaries in India include Jim Corbett National Park in Uttarakhand, Kaziranga National Park in Assam, and Sundarbans National Park in West Bengal.
• Conservation Efforts
• Conservation efforts in wildlife sanctuaries include habitat management, anti-poaching measures, and community involvement.
• Various government and non-governmental organizations work together to ensure the protection and sustainability of these natural habitats.
• Conservation programs often involve scientific research, wildlife monitoring, and awareness campaigns to educate the public about the importance of wildlife protection.
• Programs like Project Tiger and Project Elephant have been launched in India to focus on the conservation of specific endangered species.
• Biodiversity in Uttar Pradesh
• Uttar Pradesh, being a large state in India, hosts a rich and varied biodiversity.
• The state has several wildlife sanctuaries and national parks, including Dudhwa National Park, Katarniaghat Wildlife Sanctuary, and Hastinapur Wildlife Sanctuary.
• These protected areas are home to species such as tigers, elephants, rhinoceroses, and numerous bird species.
• The government of Uttar Pradesh actively engages in conservation activities to protect its wildlife and natural resources.

The correct answer is National Income is NNP at FC..

Key Points

• Net National Product (NNP) at Factor Cost (FC) is considered the national income.
• It is calculated by subtracting depreciation from Gross National Product (GNP).
• Factor Cost refers to the cost of factors of production (i.e., labor, capital, land) used to produce goods and services.
• National Income represents the total income earned by a country’s residents and businesses, including any income from abroad.

Additional Information

• Gross National Product (GNP)
• It is the total value of all goods and services produced by the residents of a country within a specific time period, usually a year.
• GNP includes the value of products made by a country’s citizens both domestically and abroad.
• Net National Product (NNP)
• It is calculated by subtracting the depreciation (wear and tear of capital goods) from the Gross National Product.
• NNP = GNP - Depreciation.
• Factor Cost
• It refers to the costs of the factors of production (labor, capital, land, and entrepreneurship) used in creating goods and services.
• Factor Cost does not include taxes or subsidies on products.
• Market Price (MP)
• It is the price at which goods and services are sold in the market.
• Market Price includes indirect taxes and excludes subsidies provided by the government.

The correct answer is PV Sindhu.

Key Points

• PV Sindhu is an Indian professional badminton player.
• She has won medals in multiple tournaments including the Olympics and World Championships.
• In 2022, she won the Swiss Open badminton tournament, adding to her numerous international titles.
• PV Sindhu is widely recognized for her contributions to the sport and has been honored with several prestigious awards in India.

Additional Information

• Swiss Open
• The Swiss Open is an annual badminton tournament held in Switzerland.
• It is part of the Badminton World Federation (BWF) World Tour.
• The tournament attracts top international players and is an important event in the badminton calendar.
• PV Sindhu's Achievements
• PV Sindhu won a silver medal at the 2016 Rio Olympics and a bronze medal at the 2020 Tokyo Olympics.
• She became the first Indian woman to win a gold medal at the BWF World Championships in 2019.
• She has been honored with major Indian awards including the Padma Bhushan, Padma Shri, and the Rajiv Gandhi Khel Ratna Award.
• Badminton World Federation (BWF)
• The BWF is the international governing body for the sport of badminton.
• It was founded in 1934 and is responsible for organizing major international tournaments such as the BWF World Championships and the BWF World Tour.
• The BWF also oversees the rules and regulations of the sport, ensuring fair play and standardization across all competitions.

The correct answer is Farmer's.

Key Points

• The Farmer's portal is a platform provided by the Government of India to facilitate easy access to information related to agriculture for farmers.
• It offers comprehensive details on various aspects of farming including crop management, pest control, weather forecasts, and government schemes.
• The portal is designed to empower farmers with knowledge and information that can help them make better decisions and improve their agricultural practices.
• This platform also acts as a single window for agricultural information from various governmental departments and agencies, making it easier for farmers to access the necessary data.

Additional Information

• Agricultural Portals and Their Importance
• eNAM - The National Agriculture Market (eNAM) is an online trading platform for agricultural commodities in India. It aims to create a unified national market for agricultural products by connecting existing APMC mandis.
• Kisan Call Center - This initiative provides farmers with information and advice on agriculture through toll-free calls. Farmers can get answers to their queries on agriculture-related issues from experts.
• M-Kisan - This platform uses mobile technology to provide farmers with critical information such as weather updates, pest control measures, and best practices for crop management.
• PM-KISAN - The Pradhan Mantri Kisan Samman Nidhi (PM-KISAN) scheme provides financial assistance to farmers. The scheme aims to supplement the financial needs of farmers in procuring various inputs to ensure proper crop health and appropriate yields.
• Government Schemes for Farmers
• Pradhan Mantri Fasal Bima Yojana (PMFBY) - This crop insurance scheme provides financial support to farmers in the event of crop failure due to natural calamities, pests, and diseases.
• Pradhan Mantri Krishi Sinchai Yojana (PMKSY) - It aims to enhance water use efficiency through various measures including the creation of protective irrigation facilities, the promotion of efficient water conveyance, and the adoption of water-saving technologies.
• Soil Health Card Scheme - This scheme aims to provide farmers with soil health cards that carry crop-wise recommendations for nutrients and fertilizers to help improve productivity.
• Rashtriya Krishi Vikas Yojana (RKVY) - This scheme aims to promote decentralized planning for agriculture and allied sectors and to ensure the holistic development of agriculture and allied sectors.

The correct answer is Online training portal for COVID-19 management.

Key Points

• iGOT stands for Integrated Government Online Training portal.
• It was launched by the Government of India in April 2020 to provide online training for COVID-19 management.
• The portal aims to enhance the capacity building of healthcare and frontline workers to fight against the COVID-19 pandemic.
• The platform offers various training modules on topics such as clinical management, infection prevention and control, quarantine and isolation, and more.

Additional Information

• COVID-19
• COVID-19 is an infectious disease caused by the novel coronavirus SARS-CoV-2.
• It was first identified in December 2019 in Wuhan, China, and has since led to a global pandemic.
• Symptoms of COVID-19 include fever, cough, and difficulty breathing, and it can lead to severe respiratory illness and death, particularly in older adults and those with underlying health conditions.
• Preventive measures include wearing masks, social distancing, hand hygiene, and vaccination.
• Capacity Building
• Capacity building refers to activities that improve an organization’s ability to achieve its mission and sustain itself over time.
• In the context of healthcare, it involves training and equipping healthcare workers with the necessary skills and knowledge to handle health crises effectively.
• It is essential for ensuring a prompt and effective response to health emergencies like the COVID-19 pandemic.
• Infection Prevention and Control (IPC)
• IPC is a scientific approach and practical solution designed to prevent harm caused by infection to patients and healthcare workers.
• It is a crucial part of healthcare delivery, especially during outbreaks of contagious diseases such as COVID-19.
• IPC measures include hand hygiene, use of personal protective equipment (PPE), and safe waste management.
• Quarantine and Isolation
• Quarantine is the separation and restriction of movement of people who may have been exposed to a contagious disease to see if they become sick.
• Isolation is the separation of sick individuals with a contagious disease from healthy individuals to prevent the spread of the disease.
• Both measures are critical in controlling the spread of infectious diseases like COVID-19.

The correct answer is Nagabhatta I.

Key Points

• Nagabhatta I was the founder of the Pratihara dynasty and reigned around the early 8^th century.
• He is credited with defending the western frontier of India against Arab invasions, particularly from the Umayyad Caliphate.
• Nagabhatta I established Bhinmal (modern-day Rajasthan) as his capital, marking the beginning of the Pratihara Empire's prominence.
• Under his leadership, the Pratihara dynasty began to rise as a significant power in northern India.

Additional Information

• Pratihara Dynasty:
• The Pratihara dynasty was an important power in northern India from the 6^th to the 11^th centuries.
• They are known for their role in protecting India from external invasions, particularly from Arab forces.
• The Pratiharas were known for their patronage of art and architecture, contributing significantly to the cultural heritage of India.
• Bhinmal:
• Bhinmal, also known as Srimal, was an ancient city located in the present-day Jalore district of Rajasthan.
• It served as the early capital of the Pratihara dynasty before they moved their capital to Kannauj.
• Bhinmal was a center of learning and culture, producing notable scholars like Brahmagupta.
• Umayyad Caliphate:
• The Umayyad Caliphate was the second of the four major caliphates established after the death of Muhammad.
• Under the Umayyads, the Muslim empire expanded rapidly, and their incursions reached into the Indian subcontinent.
• The Pratiharas, under Nagabhatta I, were able to resist these invasions and establish themselves as a formidable power.

The correct answer is Sarvepalli Radhakrishnan.

Key Points

• Teachers' Day in India is celebrated on September 5 annually to honor the birth anniversary of Dr. Sarvepalli Radhakrishnan.
• Dr. Radhakrishnan was a renowned philosopher, scholar, and educator who served as the first Vice President and the second President of India.
• He was an advocate for education and a prominent figure in the academic field, contributing significantly to the education sector in India.
• The tradition of celebrating Teachers' Day began in 1962 when Dr. Radhakrishnan became the President of India and suggested that his birthday be celebrated as Teachers' Day to recognize the contributions of teachers.

Additional Information

• Dr. Sarvepalli Radhakrishnan
• Born on September 5, 1888, in a small town of Tiruttani in Tamil Nadu, India.
• He was a distinguished scholar and academic who authored several books on philosophy and religion.
• Dr. Radhakrishnan was awarded several prestigious honors, including the Bharat Ratna, India's highest civilian award, in 1954.
• He played a key role in shaping India's educational policies and institutions during his tenure as an academician and a statesman.
• Teachers' Day
• Teachers' Day is a special day dedicated to honoring and appreciating teachers for their contribution to education and society.
• It is celebrated in various countries around the world, often on different dates.
• In India, Teachers' Day is marked by various events and activities organized in schools and educational institutions to acknowledge the hard work and dedication of teachers.
• Students often express their gratitude to their teachers through cards, gifts, and special performances on this day.
• Importance of Education
• Education is essential for the overall development of individuals and societies.
• It empowers people with knowledge, skills, and values necessary for personal growth and social progress.
• Teachers play a crucial role in imparting education and shaping the future of students by providing guidance, support, and inspiration.
• Recognizing and celebrating the contributions of teachers is important to highlight the value of education and encourage a culture of respect for educators.

Shortcut Trick

Let the total combined principal be P.

Effective rate of interest for 5 years = 8% × 5 = 40%.

Since total interest received is ₹1,600, 40% of P = 1600.

⇒ Total Principal P = (1600 × 100) ÷ 40 = ₹4,000

⇒ Sum lent to Ravi = Total Principal − Raju's Principal = 4000 − 3000 = ₹1,000

∴ The correct answer is ₹1,000.

Alternate Method

Given:

Principal lent to Raju (P₁) = ₹3,000

Principal lent to Ravi (P₂) = ₹x

Rate of Interest (R) = 8% per annum

Time (T) = 5 years

Total Simple Interest (SI) = ₹1,600

Formula Used:

Simple Interest = (P × R × T) ÷ 100

Calculations:

⇒ SI from Raju = (3000 × 8 × 5) ÷ 100

⇒ SI from Raju = 120000 ÷ 100 = ₹1,200

⇒ SI from Ravi = Total Interest − SI from Raju

⇒ SI from Ravi = 1600 − 1200 = ₹400

⇒ Now, SI from Ravi = (x × 8 × 5) ÷ 100

⇒ 400 = (x × 40) ÷ 100

⇒ 400 = 0.4x

⇒ x = 400 ÷ 0.4 = 1000

∴ The correct answer is ₹1,000.

Additional Information

Simple Interest (SI)

Interest calculated strictly on the initial principal amount only. Formula: SI = (P × R × T) ÷ 100.

Effective Rate of Interest

In simple interest, the overall percentage growth over 'T' years is derived simply by taking R × T. Total Interest = (R × T)% of Principal.

Compound Interest (CI)

Interest calculated on both the initial principal and the previously accumulated interest. Formula for amount: A = P(1 + R/100)^T.

Shortcut Trick

Coinciding lines are two straight lines that lie perfectly on top of each other.

Since a single line is made up of a continuous, infinite series of points, two overlapping lines share every single point.

Thus, they intersect at every point along their entire length.

∴ The correct answer is infinitely many points of intersection.

Alternate Method

Given:

Two straight lines are coinciding with each other.

Concept Used:

If two linear equations L₁: a₁x + b₁y + c₁ = 0 and L₂: a₂x + b₂y + c₂ = 0 represent coinciding lines, their coefficients are strictly proportional: a₁/a₂ = b₁/b₂ = c₁/c₂.

Intersecting (1 point)

Parallel (0 points)

Coinciding (Infinite points)

Calculations:

⇒ When the equations are proportional, they mathematically represent the exact same straight line.

⇒ A straight line contains an infinitely large set of coordinate points (x, y).

⇒ Because the lines perfectly overlap, every point present on the first line also acts as a point on the second line.

⇒ Therefore, the number of shared (intersecting) points between them is strictly infinite.

∴ The correct answer is infinitely many points of intersection.

Additional Information

Intersecting Lines

When two lines have completely different slopes (a₁/a₂ ≠ b₁/b₂), they will intersect at exactly one unique point, resulting in a single solution.

Parallel Lines

When two lines have the same slope but different intercepts (a₁/a₂ = b₁/b₂ ≠ c₁/c₂), they never meet each other at any point, resulting in 0 solutions.

Coincident Lines

When two lines share both the same slope and same intercept (a₁/a₂ = b₁/b₂ = c₁/c₂), they overlap perfectly, resulting in infinitely many solutions.

Shortcut Trick

Put θ = 60° in the given expression.

Expression = (1 + sin 60° − cos 60°) ÷ (1 + sin 60° + cos 60°)

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

⇒ (1 + √3) ÷ (3 + √3) = (1 + √3) ÷ [√3(√3 + 1)] = 1/√3

Now check the options for θ = 60°: tan(60°/2) = tan 30° = 1/√3 (matches perfectly).

∴ The correct answer is tan(θ/2).

Alternate Method

Given:

(1 + sin θ − cos θ) ÷ (1 + sin θ + cos θ)

Formulas Used:

1 − cos θ = 2 sin²(θ/2)

1 + cos θ = 2 cos²(θ/2)

sin θ = 2 sin(θ/2) cos(θ/2)

Calculation:

⇒ Rearranging terms in the numerator and denominator:

⇒ [(1 − cos θ) + sin θ) ÷ [(1 + cos θ) + sin θ]

⇒ Substituting the half-angle formulas:

⇒ [2 sin²(θ/2) + 2 sin(θ/2) cos(θ/2)] ÷ [2 cos²(θ/2) + 2 sin(θ/2) cos(θ/2)]

⇒ Taking 2 sin(θ/2) common in numerator and 2 cos(θ/2) common in denominator:

⇒ [2 sin(θ/2) × (sin(θ/2) + cos(θ/2))] ÷ [2 cos(θ/2) × (cos(θ/2) + sin(θ/2))]

⇒ Cancelling the common factor (sin(θ/2) + cos(θ/2)) and 2:

⇒ sin(θ/2) ÷ cos(θ/2)

⇒ tan(θ/2)

∴ The correct answer is tan(θ/2).

Additional Information

Double Angle Formulas

sin 2θ = 2 sin θ cos θ and cos 2θ = cos² θ − sin² θ = 2 cos² θ − 1 = 1 − 2 sin² θ.

Half Angle Tangent Identity

tan(θ/2) can also be directly expressed as (1 − cos θ) ÷ sin θ or sin θ ÷ (1 + cos θ).

Value Substitution Technique

For trigonometric identities, substituting standard angles like 30°, 45°, or 60° often bypasses complex algebraic simplifications. Avoid angles that make the denominator zero.

Given:

Exports from Jewellery = 500 million US $

Total exports = 720 (Electronics) + 680 (Cotton) + 500 (Jewellery) + 450 (Steel) + 850 (Software) = 3,200 million US $

Formula used:

Percentage = (Part / Total) × 100

Calculations:

Percentage of Jewellery exports = (500 / 3200) × 100

⇒ Percentage = 15.63%

Jewellery exports were 15.63% of the total exports in 2020.

Shortcut Trick

Initial ratio (1 year ago) B : A = 2 : 1, Final ratio (after 9 years) B : A = 4 : 3.

Total time gap between the two ratios = 1 year + 9 years = 10 years.

Using cross-multiplication: Difference in cross parts = (2 × 3) − (4 × 1) = 6 − 4 = 2 parts.

Difference in values = (4 × 10) − (3 × 10) = 40 − 30 = 10 years.

2 parts = 10 years ⇒ 1 part = 5 years ⇒ 1 year ago A = 5 years, B = 10 years.

Sum of present ages = (5 + 1) + (10 + 1) = 17 years.

∴ The correct answer is 17 years.

Alternate Method

Given:

Ratio of ages of B to A one year ago = 2 : 1

Ratio of ages of B to A after 9 years = 4 : 3

Calculations:

⇒ Let the ages of A and B one year ago be x and 2x respectively.

⇒ Present age of A = x + 1

⇒ Present age of B = 2x + 1

⇒ Age of A after 9 years = x + 1 + 9 = x + 10

⇒ Age of B after 9 years = 2x + 1 + 9 = 2x + 10

⇒ The given ratio of B's age to A's age after 9 years is 4 : 3.

⇒ (2x + 10) ÷ (x + 10) = 4 ÷ 3

⇒ 3 × (2x + 10) = 4 × (x + 10)

⇒ 6x + 30 = 4x + 40

⇒ 6x − 4x = 40 − 30

⇒ 2x = 10

⇒ x = 5

⇒ Present age of A = 5 + 1 = 6 years.

⇒ Present age of B = 2 × 5 + 1 = 11 years.

⇒ Sum of present ages = 6 + 11 = 17 years.

∴ The correct answer is 17 years.

Additional Information

Constant Age Difference

The difference between the ages of two individuals always remains constant over time. If person A is 5 years older than person B today, A will always remain 5 years older than B.

Time Shift Concept

If the present age is x years, then the age 'n' years ago was (x − n) years, and the age 'n' years from now will be (x + n) years.

Cross-Multiplication Method

When an initial ratio a : b changes to a final ratio c : d after adding a common value 'v', then 1 part = [v × (c − d)] ÷ [(a × d) − (b × c)]. This shortcut is highly useful for age and mixture problems to avoid algebraic equations.

Shortcut Trick

The gain percentage can be found directly using ratios without using the actual selling price.

Given, Gain = 1/6 of Selling Price (SP).

Let SP = 6 units, then Gain = 1 unit.

Cost Price (CP) = SP − Gain = 6 − 1 = 5 units.

Gain % = (Gain ÷ CP) × 100 = (1 ÷ 5) × 100 = 20%.

∴ The correct answer is 20%.

Alternate Method

Given:

Selling Price (SP) = ₹25404

Gain = 1/6 of SP

Formula Used:

Cost Price (CP) = SP − Gain

Gain % = (Gain ÷ CP) × 100

Calculations:

⇒ Gain = (1 ÷ 6) × 25404 = ₹4234

⇒ CP = 25404 − 4234 = ��21170

⇒ Gain % = (4234 ÷ 21170) × 100

⇒ Gain % = (1 ÷ 5) × 100 = 20%

∴ The correct answer is 20%.

Additional Information

True Profit Percentage

Actual profit or loss is always calculated on the Cost Price (CP) unless explicitly stated otherwise. Therefore, Profit % = (Profit ÷ CP) × 100.

Profit Calculated on Selling Price

If the profit is given as a fraction of the Selling Price (SP), let the SP represent the denominator. The CP will then be (Denominator − Numerator). This ratio avoids complex calculations with actual values.

Marked Price and Discount

Discount is always calculated on the Marked Price (MP). The Selling Price is derived by subtracting the discount from the Marked Price (SP = MP − Discount).

Shortcut Trick Divide the given equation by cos2 θ: ⇒ 10 tan2 θ + 6 = 7 sec2 θ ⇒ 10 tan2 θ + 6 = 7(1 + tan2 θ) ⇒ 3 tan2 θ = 1 ⇒ tan θ = 1/√3 ∴ The correct answer is 1/√3. Alternate Method Given: 10 sin2 θ + 6 cos2 θ = 7, where 0 < θ < 90º Formula Used: sin2 θ + cos2 θ = 1 ⇒ cos2 θ = 1 − sin2 θ Calculation: ⇒ 10 sin2 θ + 6(1 − sin2 θ) = 7 ⇒ 10 sin2 θ + 6 − 6 sin2 θ = 7 ⇒ 4 sin2 θ = 1 ⇒ sin2 θ = 1/4 ⇒ sin θ = 1/2 (Since 0 < θ < 90º, sin θ is positive) ⇒ θ = 30º Now we need to find tan θ: ⇒ tan 30º = 1/√3 ∴ The correct answer is 1/√3. Additional Information Trigonometric Identities 1 + tan2 θ = sec2 θ and 1 + cot2 θ = cosec2 θ Standard Values sin 30º = 1/2, cos 30º = √3/2, tan 30º = 1/√3 Sign Convention In the first quadrant (0 to 90º), all trigonometric ratios are positive.

Shortcut Trick

Ratio of Length : Breadth : Height = 3 : 2 : 1

Given Breadth (2 units) = 12 cm ⇒ 1 unit = 6 cm

Length = 3 units = 18 cm, and Height = 1 unit = 6 cm

Volume = Length × Breadth × Height = 18 × 12 × 6 = 1296 cm³

∴ The correct answer is 1296 cm³.

Alternate Method

Given:

Ratio of Length, Breadth, and Height = 3 : 2 : 1

Breadth = 12 cm

Formula Used:

Volume of Cuboid = L × B × H

Calculations:

⇒ Let the Length, Breadth, and Height of the cuboid be 3x, 2x, and 1x respectively.

⇒ Breadth = 2x = 12 cm ⇒ x = 6 cm

⇒ Length = 3x = 3 × 6 = 18 cm

⇒ Height = x = 1 × 6 = 6 cm

⇒ Volume of the cuboid = 18 × 12 × 6

⇒ Volume = 216 × 6 = 1296 cm³

∴ The correct answer is 1296 cm³.

Additional Information

Total Surface Area of Cuboid

The total area of all 6 faces is given by TSA = 2 × (LB + BH + HL).

Lateral Surface Area of Cuboid

The area of the 4 walls (excluding the top and bottom faces) is given by LSA = 2 × H × (L + B).

Diagonal of a Cuboid

The length of the longest possible line inside a cuboid is given by D = √(L² + B² + H²).

Volume of a Cube

For a cube where all edges are equal to length a, the volume simplifies to V = a³.

Shortcut Trick

Total ratio parts = 1 + 2 + 3 = 6 parts.

Difference in ratio parts between the largest and smallest angle = 3 − 1 = 2 parts.

Sum of interior angles of a triangle is always 180°.

Required difference = (2 ÷ 6) × 180° = 60°.

∴ The correct answer is 60°.

Alternate Method

Given:

The ratio of the three angles of a triangle is 1 : 2 : 3.

Formula Used:

Sum of all internal angles in a triangle = 180°

1x (Smallest)

3x (Largest)

2x

1x + 2x + 3x = 180°

Calculations:

Let the three angles of the triangle be x, 2x, and 3x.

⇒ x + 2x + 3x = 180°

⇒ 6x = 180°

⇒ x = 30°

The smallest angle = x = 30°

The largest angle = 3x = 3 × 30° = 90°

⇒ Difference between the largest and smallest angle = 90° − 30° = 60°

∴ The correct answer is 60°.

Additional Information

Sum of Angles Property

The sum of all three interior angles of a planar triangle is always exactly 180° regardless of the triangle's shape or dimensions.

Right-Angled Triangle

Since the largest angle in this specific triangle is exactly 90°, it forms a right-angled triangle where the sum of the two remaining acute angles is also 90°.

Polygon Exterior Angles

For any convex polygon, including a triangle, the sum of its exterior angles (taking one at each vertex) is always exactly 360°.

Shortcut Trick

Put A = 15° directly in the given expression.

3A = 45°, 2A = 30°, and 4A = 60°.

Expression = [11√2 × cos 45° + 10 × sin 30°] ÷ [7√3 × sin 60°]

Expression = [11√2 × (1/√2) + 10 × (1/2)] ÷ [7√3 × (√3/2)]

Expression = (11 + 5) ÷ (21/2) = 16 × (2/21) = 32/21

∴ The correct answer is 32/21.

Alternate Method

Given:

A = 15°

Expression: (11√2 cos 3A + 10 sin 2A) ÷ (7√3 sin 4A)

Formula Used:

sin 30° = 1/2

cos 45° = 1/√2

sin 60° = √3/2

Calculations:

⇒ 3A = 3 × 15° = 45°

⇒ 2A = 2 × 15° = 30°

⇒ 4A = 4 × 15° = 60°

Substitute these values into the given expression:

⇒ Numerator = 11√2 × cos 45° + 10 × sin 30°

⇒ Numerator = 11√2 × (1/√2) + 10 × (1/2)

⇒ Numerator = 11 + 5 = 16

⇒ Denominator = 7√3 × sin 60°

⇒ Denominator = 7√3 × (√3/2)

⇒ Denominator = (7 × 3) ÷ 2 = 21/2

⇒ Value = Numerator ÷ Denominator

⇒ Value = 16 ÷ (21/2) = 16 × (2/21) = 32/21

∴ The correct answer is 32/21.

Additional Information

Standard Trigonometric Values

sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2

cos 30° = √3/2, cos 45° = 1/√2, cos 60° = 1/2

Double Angle Formulas

sin 2A = 2 sin A cos A

cos 2A = cos² A − sin² A

Triple Angle Formulas

sin 3A = 3 sin A − 4 sin³ A

cos 3A = 4 cos³ A − 3 cos A

Shortcut Trick

Difference in ratio parts = 5 − 2 = 3 parts.

Value of 3 parts = 210

Value of 1 part = 210 ÷ 3 = 70

Smaller number = 2 parts = 2 × 70 = 140

∴ The correct answer is 140.

Alternate Method

Given:

Ratio of two numbers = 2 : 5

Difference between the numbers = 210

Concept Used:

When two numbers are in a given ratio, they can be represented as multiples of a common variable x.

Calculations:

⇒ Let the smaller number be 2x and the larger number be 5x.

⇒ Difference = 5x − 2x = 3x

⇒ Given difference = 210

⇒ 3x = 210

⇒ x = 210 ÷ 3 = 70

⇒ Smaller number = 2x = 2 × 70 = 140

∴ The correct answer is 140.

Additional Information

Properties of Ratio

A ratio compares two quantities of the same kind. If the ratio is a : b, the absolute quantities can be expressed as ax and bx, where x is a common multiplier.

Equivalent Ratios

Multiplying or dividing both terms of a ratio by the same non-zero number produces an equivalent ratio (e.g., 2 : 5 = 4 : 10 = 6 : 15).

Compound Ratio

When two or more ratios are multiplied term by term, the resulting ratio is called a compound ratio. For a : b and c : d, the compound ratio is ac : bd.

Shortcut Trick

Use the successive percentage formula to find the net profit or loss directly:

Net Profit % = x + y + (x × y) ÷ 100

Here, the markup is positive (x = +25) and the discount is negative (y = −18).

Net Profit % = 25 − 18 − (25 × 18) ÷ 100

Net Profit % = 7 − 4.5 = 2.5%

∴ The correct answer is 2.5%.

Alternate Method

Given:

Markup percentage = 25% above cost price

Discount percentage = 18% on marked price

Formula Used:

MP = CP × (100 + Markup%) ÷ 100

SP = MP × (100 − Discount%) ÷ 100

Gain % = [(SP − CP) ÷ CP] × 100

Calculations:

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

⇒ Market Price (MP) = 100 + (25% of 100) = 125

⇒ Selling Price (SP) = MP − (18% of MP)

⇒ SP = 125 − (18 ÷ 100 × 125)

⇒ SP = 125 − 22.5 = 102.5

⇒ Gain = SP − CP = 102.5 − 100 = 2.5

⇒ Gain % = (2.5 ÷ 100) × 100 = 2.5%

∴ The correct answer is 2.5%.

Additional Information

Successive Percentage Rule

When two continuous percentage changes x and y occur, the net percentage change is given by (x + y + xy÷100)%. Use (+) for profit/markup and (−) for discount/loss.

Marked Price (MP) and Selling Price (SP)

The price on the label of an article is the Marked Price. Selling Price is calculated after deducting the discount from the Marked Price. Formula: SP = MP × (1 − Discount÷100).

Direct Relationship Formula

A very useful relation between Cost Price and Marked Price is: CP ÷ MP = (100 − Discount%) ÷ (100 + Profit%).

Given:

Total donors = 240

Angle for blood group A = 90°

Angle for blood group B = 120°

Angle for blood group O = 80°

Angle for blood group AB = 70°

Formula Used:

Number of donors for a blood group = (Angle for that blood group / 360°) × Total donors

Calculations:

Number of donors for blood group A = (90° / 360°) × 240 = 60

Number of donors for blood group O = (80° / 360°) × 240 = 160/3

Number of donors for blood group AB = (70° / 360°) × 240 = 140/3

Average donors of blood groups A, O, and AB = (60 + 160/3 + 140/3) / 3 = (480/3) / 3 = 160/3

Number of donors for blood group B = (120° / 360°) × 240 = 80

Ratio of average donors of blood groups A, O, and AB to the donors of blood group B = 160/3 : 80 = 2 : 3.

The ratio of the average donors of blood groups 'A', 'O', and 'AB' to the donors of blood group 'B' is 2 : 3.

Shortcut Trick

Total capacity of the cistern = LCM of 4, 5, and 3 = 60 units.

Efficiency of Pipe A = 60 ÷ 4 = 15 units/hr; Pipe B = 60 ÷ 5 = 12 units/hr.

Efficiency of Pipe C (emptying) = −(60 ÷ 3) = −20 units/hr.

Combined efficiency = 15 + 12 − 20 = 7 units/hr.

Time to fill = 60 / 7 hours = (60 / 7) × 60 = 3600 / 7 ≈ 514 minutes.

∴ The correct answer is 514 minutes.

Alternate Method

Given:

Time taken by Pipe A to fill the cistern = 4 hours

Time taken by Pipe B to fill the cistern = 5 hours

Time taken by Pipe C to empty the cistern = 3 hours

Formula Used:

Total Work = LCM of times taken

Efficiency = Total Work ÷ Time

Total Time = Total Work ÷ Combined Efficiency

Calculations:

⇒ Combined efficiency of all pipes working together = 15 + 12 − 20

⇒ Net combined efficiency = 7 units/hr

⇒ Time taken to fill the cistern = Total Work ÷ Net efficiency = 60 / 7 hours

⇒ Convert hours into minutes = (60 / 7) × 60 minutes

⇒ Total time in minutes = 3600 / 7 = 514.285... minutes

⇒ Rounding to the nearest minute, we get 514 minutes.

∴ The correct answer is 514 minutes.

Additional Information

Pipes and Cisterns Basics

If a pipe can fill a tank in x hours, its part filled in 1 hour is 1/x. If a pipe empties the tank in y hours, its part emptied in 1 hour is 1/y (always taken as negative).

Combined Work of Filling and Emptying

When both inlet (filling) and outlet (emptying) pipes are open simultaneously, the net part filled in 1 hour is (1/x) − (1/y). If the total sum is positive, the tank is filling; if negative, the tank is emptying.

LCM Method Trick

To avoid tricky fractions, always assume the total capacity of the tank as the LCM of the given individual times. Then compute efficiency (work done per hour) for each pipe, treating emptying pipes as having negative efficiency.

Given:

Marks for each question and the number of students who answered correctly are given in table:

Formula used:

Average marks =  (Total marks obtained) / (total number of students)

Number of students in the class = Answered correctly + Answered incorrectly + Not Attempted

Calculations:

Total marks obtained for Q1 = 20 × 1 = 20

Total marks obtained for Q2 = 15 × 2 = 30

Total marks obtained for Q3 = 5 × 20 = 100

Total marks obtained for Q4 = 10 × 12 = 120

Total marks obtained = 20 + 30 + 100 + 120 = 270

Total number of students in the class = 40

Average marks = $\frac{270}{40} = 6.75$

The average marks obtained by the class in the examination is 6.75.

Confusion Points 

Number of students in the class = Students who attempted the question correctly + students who did not attempt question correctly + students who didn't attempt the question.

So,

Question 1 = 20 + 10 + 10 = 40

Question 2 = 15 + 15 + 10 = 40 

Question 3 = 20 + 20 + 0 = 40 

Question 4 = 12 + 15 + 13 = 40 

Thus, the total number of students = 40

Because there are only three choices for each question, if we add the number of students for each choice then we will get the total number of students.

Shortcut Trick

Let a = 1 and b = 1. Since a + b = c, we get c = 2.

Substitute these values into the expression: a³ + b³ − c³ + 3abc

⇒ 1³ + 1³ − 2³ + 3 × 1 × 1 × 2

⇒ 1 + 1 − 8 + 6 = 0

∴ The correct answer is 0.

Alternate Method

Given:

a + b = c ⇒ a + b − c = 0

Formula Used:

If x + y + z = 0, then x³ + y³ + z³ − 3xyz = 0

Calculations:

We need to evaluate a³ + b³ − c³ + 3abc.

Let x = a, y = b, and z = −c.

⇒ x + y + z = a + b − c = 0

Applying the algebraic formula since the sum of variables is zero:

⇒ a³ + b³ + (−c)³ − 3 × a × b × (−c) = 0

⇒ a³ + b³ − c³ + 3abc = 0

∴ The correct answer is 0.

Additional Information

Sum of Cubes Identity

The general identity is x³ + y³ + z³ − 3xyz = (x + y + z)(x² + y² + z² − xy − yz − zx).

Special Case for Zero Sum

If the sum of variables x + y + z = 0, then the entire right side of the equation becomes 0, leading to x³ + y³ + z³ = 3xyz.

Sum and Difference of Two Cubes

a³ + b³ = (a + b)(a² − ab + b²) and a³ − b³ = (a − b)(a² + ab + b²).

Square of a Trinomial

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca).

Shortcut Trick

Ratio of Initial Income : Expenditure : Savings = 16500 : 12000 : 4500 ⇒ 11 : 8 : 3.

Using the weighted average method: (Total Ratio × Income Change) = (E_ratio × Expenditure Change) + (S_ratio × Savings Change)

11 × 24 = 8 × (−10) + 3 × x

264 = −80 + 3x ⇒ 3x = 344 ⇒ x = 114.66%

Rounding off to the nearest integer, we get 115%.

∴ The correct answer is 115%.

Alternate Method

Given:

Initial Income = ₹16,500

Initial Expenditure = ₹12,000

⇒ Initial Savings = Income − Expenditure = 16,500 − 12,000 = ₹4,500

Calculations:

⇒ New Income = 16,500 + (24% of 16,500) = 16,500 × 1.24 = ₹20,460

⇒ New Expenditure = 12,000 − (10% of 12,000) = 12,000 × 0.90 = ₹10,800

⇒ New Savings = New Income − New Expenditure = 20,460 − 10,800 = ₹9,660

⇒ Increase in Savings = 9,660 − 4,500 = ₹5,160

⇒ Percentage Increase = (Increase ÷ Initial Savings) × 100

⇒ Percentage Increase = (5,160 ÷ 4,500) × 100 = 114.66%

⇒ Percentage Increase ≈ 115% (rounded off to the nearest integer)

∴ The correct answer is 115%.

Additional Information

Income, Expenditure, and Savings Relationship

Income = Expenditure + Savings. Any absolute change in income is exactly equal to the sum of absolute changes in expenditure and savings.

Percentage Change Formula

Percentage Change = [(Final Value − Initial Value) ÷ Initial Value] × 100. A positive value represents a growth/increase while a negative value signifies a reduction.

Weighted Average Method for Change

If an entity is divided into two parts, the overall percentage change is the weighted average of the changes of its parts: Total % Change = (R₁ × C₁ + R₂ × C₂) ÷ (R₁ + R₂), where R represents the ratio and C represents the percentage change.

Shortcut Trick

According to the Midpoint Theorem, joining the midpoints of all three sides of a triangle divides it into four identical (congruent) smaller triangles.

The four resulting triangles are △ADF, △BDE, △ECF, and △DEF.

Therefore, △DEF is exactly congruent in shape and size to △ADF, △BDE, and △ECF.

However, the parent triangle △ABC is four times larger in area compared to △DEF, meaning they are similar, but NOT congruent.

∴ The correct answer is △ABC.

Alternate Method

Given:

D, E, and F are the mid-points of sides AB, BC, and AC respectively in △ABC.

A B C D E F

Formula Used:

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

Calculations:

⇒ By Midpoint Theorem, DE ∥ AC and DE = ¹÷₂ × AC = AF = FC

⇒ EF ∥ AB and EF = ¹÷₂ × AB = AD = DB

⇒ DF ∥ BC and DF = ¹÷₂ × BC = BE = EC

⇒ Now, let's compare △DEF and △BDE:

⇒ DE = DE (Common side)

⇒ EF = BD (Since EF is half of AB)

⇒ DF = BE (Since DF is half of BC)

⇒ By SSS (Side-Side-Side) congruence criterion, △DEF ≅ △BDE.

⇒ Similarly, by comparing the sides, we can establish that △DEF ≅ △ECF and △DEF ≅ △ADF.

⇒ Since △ABC is the main enclosing triangle, its side lengths are precisely double the sides of △DEF, which makes them similar but significantly different in overall size.

⇒ Hence, △DEF is NOT congruent to △ABC.

∴ The correct answer is △ABC.

Additional Information

Midpoint Theorem

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

Congruent Triangles

Two triangles are completely congruent if all three corresponding sides and corresponding angles are identical. The ratio of their areas is always 1:1.

Similar Triangles

If two triangles have equal corresponding angles and strictly proportional side lengths, they are similar. The ratio of their respective areas is equivalent to the square of the ratio of their corresponding side lengths.

Shortcut Trick

Sum of digits = 4 + 3 + 6 + P + 5 = 18 + P.

For the number to be divisible by 3, the sum of its digits must be a multiple of 3.

Since 18 is perfectly divisible by 3, P must also be a multiple of 3.

Possible single-digit values for P are 0, 3, 6, and 9.

The largest possible value among them is 9.

∴ The correct answer is 9.

Alternate Method

Given:

The number 436P5 is divisible by 3.

Concept Used:

Divisibility rule of 3: A number is exactly divisible by 3 if the sum of all its digits is divisible by 3.

Calculations:

⇒ Sum of the digits = 4 + 3 + 6 + P + 5

⇒ Sum of the digits = 18 + P

⇒ For (18 + P) to be a multiple of 3, it can take values such as 18, 21, 24, 27, 30, etc.

⇒ Therefore, the possible values for P are (18 − 18) = 0, (21 − 18) = 3, (24 − 18) = 6, and (27 − 18) = 9.

⇒ The next value would be (30 − 18) = 12, which is not a valid single digit.

⇒ Thus, the possible single-digit values for P are 0, 3, 6, and 9.

⇒ The question requires the largest value, which is 9.

∴ The correct answer is 9.

Additional Information

Divisibility by 9

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

Divisibility by 4

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

Divisibility by 11

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

Shortcut Trick

By the Angle Bisector Theorem, the ratio of the base segments equals the ratio of the adjacent sides.

⇒ BD : DC = AB : AC

⇒ BD : DC = 6 : 12 = 1 : 2

⇒ DC corresponds to 2 parts out of the 3 total parts of BC.

⇒ DC = (2 ÷ 3) × 9 = 6 cm.

∴ The correct answer is 6 cm.

Alternate Method

A B C D

6 cm 12 cm

9 cm

1 part 2 parts

Given:

⇒ In triangle ABC, AB = 6 cm, AC = 12 cm, and BC = 9 cm.

⇒ AD is the angle bisector of ∠BAC.

Formula Used:

⇒ The Internal Angle Bisector Theorem states that an angle bisector of a triangle divides the opposite side into two segments proportional to the other two sides.

⇒ AB ÷ AC = BD ÷ DC

Calculation:

⇒ 6 ÷ 12 = BD ÷ DC

⇒ BD ÷ DC = 1 ÷ 2

⇒ The segments BD and DC are in the ratio 1 : 2.

⇒ We know that BC = BD + DC = 9 cm.

⇒ Total ratio parts of BC = 1 + 2 = 3 parts.

⇒ 3 parts = 9 cm

⇒ 1 part = 3 cm

⇒ Since DC corresponds to 2 parts:

⇒ DC = 2 × 3 cm = 6 cm.

∴ The correct answer is 6 cm.

Additional Information

External Angle Bisector Theorem

If the external angle bisector of an angle intersects the opposite side extended, it divides the opposite side externally in the ratio of the other two sides. Formula: AB ÷ AC = BD ÷ CD.

Incenter of a Triangle

The point of intersection of all three internal angle bisectors is called the incenter. It is equidistant from all three sides, and the perpendicular distance to the sides is the inradius (r = Area ÷ Semi-perimeter).

Length of Angle Bisector

The length of the internal angle bisector AD can be found using the sides of the triangle and the base segments. Formula: AD² = (AB × AC) − (BD × DC).

Shortcut Trick

Difference in squares of radii = (Half of larger chord)² − (Half of smaller chord)²

⇒ R² − r² = (18 ÷ 2)² − (6 ÷ 2)²

⇒ R² − r² = 9² − 3²

⇒ R² − r² = 81 − 9 = 72

∴ The correct answer is 72 cm².

Alternate Method

O M A B C D

r R d

3 cm

9 cm

Given:

Length of inner chord (CD) = 6 cm

Length of outer chord (AB) = 18 cm

Formula Used:

A perpendicular drawn from the center of a circle to a chord bisects the chord.

Pythagoras Theorem: Hypotenuse² = Base² + Perpendicular²

Calculations:

⇒ Let O be the center of the concentric circles and OM (d) be the perpendicular drawn to the chords.

⇒ Half of inner chord (MD) = 6 ÷ 2 = 3 cm

⇒ Half of outer chord (MB) = 18 ÷ 2 = 9 cm

⇒ Let the inner radius be r and outer radius be R.

⇒ In right-angled ΔOMD:

⇒ r² = d² + 3²

⇒ d² = r² − 9     ... (Equation 1)

⇒ In right-angled ΔOMB:

⇒ R² = d² + 9²

⇒ d² = R² − 81     ... (Equation 2)

⇒ Equating Equation 1 and Equation 2:

⇒ r² − 9 = R² − 81

⇒ R² − r² = 81 − 9

⇒ R² − r² = 72

∴ The correct answer is 72 cm².

Additional Information

Perpendicular to a Chord

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

Area of Annulus (Ring)

The region bounded between two concentric circles is called an annulus. Its area is calculated by subtracting the inner circle area from the outer circle area: Area = π(R² − r²).

Concentric Circles

Circles that share the same center point but have different radii are known as concentric circles. Any line intersecting both circles forms segments that are symmetrical around the perpendicular from the center.

Shortcut Trick

Ratio of students in the two groups = 12 : 18 = 2 : 3

Let the assumed average be 62 (the smaller average).

Deviation of the second group from the assumed average = 78 − 62 = +16

Total deviation across the 3 parts = +16 × 3 = +48

Net increase in overall average = +48 ÷ (2 + 3) parts = +48 ÷ 5 = +9.6

∴ The correct answer is 71.6 marks.

Alternate Method

Given:

Total number of students in the class = 30

Average score of 12 students = 62

Average score of remaining (30 − 12) = 18 students = 78

Formula Used:

Average = Sum of Observations ÷ Total number of Observations

Weighted Average Concept

12 Students (Group 1) Average = 62

18 Students (Group 2) Average = 78

Overall Class Average = 71.6

Calculations:

⇒ Total sum of scores of the first 12 students = 12 × 62 = 744

⇒ Total sum of scores of the remaining 18 students = 18 × 78 = 1404

⇒ Total sum of scores of all 30 students = 744 + 1404 = 2148

⇒ Average score of the class = 2148 ÷ 30

⇒ Average score of the class = 214.8 ÷ 3 = 71.6

∴ The correct answer is 71.6 marks.

Additional Information

Weighted Average

When two or more distinct groups are combined into a single group, their overall average is calculated using the formula: A_w = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Average of Arithmetic Progression

If the data points are in an Arithmetic Progression (AP) with a constant difference, the average is simply the middle term or (First Term + Last Term) ÷ 2.

Property of Constant Change

If a constant 'k' is added to, subtracted from, multiplied by, or divided into every single observation in a given dataset, the new average also undergoes the exact same mathematical operation by 'k'.

Shortcut Trick

Since both the policeman and the thief are moving in the same direction, relative speed = 82 − 64 = 18 km/h.

Convert relative speed to m/s = 18 × (5/18) = 5 m/s.

Time required to cover 1300 m gap = Distance ÷ Relative speed = 1300 ÷ 5 = 260 sec.

∴ The correct answer is 260 sec.

Alternate Method

Given:

Speed of Policeman (S₁) = 82 km/h

Speed of Thief (S₂) = 64 km/h

Initial distance between them (D) = 1300 m

Formula Used:

Relative Speed (when moving in the same direction) = S₁ − S₂

Time = Distance ÷ Relative Speed

1 km/h = 5/18 m/s

Calculations:

⇒ Relative Speed = 82 − 64 = 18 km/h

⇒ Convert speed into m/s: 18 × (5/18) = 5 m/s

⇒ Time taken to catch the thief = Distance ÷ Relative Speed

⇒ Time taken = 1300 ÷ 5 = 260 seconds

∴ The correct answer is 260 sec.

Additional Information

Relative Speed

When two bodies are moving in the same direction, their relative speed is the difference of their individual speeds: S_R = S₁ − S₂. When moving in opposite directions, it is the sum of their individual speeds: S_R = S₁ + S₂.

Speed Unit Conversion

To convert speed from km/h to m/s, multiply the value by 5/18. To convert speed from m/s to km/h, multiply the value by 18/5.

Average Speed

Average speed is the Total Distance Covered divided by Total Time Taken. If a person covers two equal distances at speeds x and y, Average Speed = (2 × x × y) ÷ (x + y).

Given:

Total expenditure = ₹10,70,000

Expenditure on drawing = 19%

Formula used:

Central angle = (Expenditure on drawing / Total expenditure) × 360°

Calculations:

Central angle of drawing = (19 / 100) × 360°

⇒ Central angle of drawing = 68.4°

∴ The correct answer is option (3).

Shortcut Trick

Sum of the first 12 natural numbers = 12 × 13 ÷ 2 = 78.

The factors of 78 are 1, 2, 3, 6, 13, 26, 39, and 78.

Among the given options (6, 12, 4, 8), only 6 perfectly divides 78.

Since 6 has more than two factors (1, 2, 3, 6), it is a composite number.

∴ The correct answer is 6.

Alternate Method

Given:

The series of the first 12 natural numbers: 1, 2, 3, ... 12.

Formula Used:

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

Calculations:

⇒ Sum = 12 × (12 + 1) ÷ 2

⇒ Sum = 12 × 13 ÷ 2

⇒ Sum = 6 × 13 = 78

⇒ Now, check the divisibility of 78 by the given composite numbers in the options:

⇒ 78 ÷ 6 = 13 (perfectly divisible)

⇒ 78 ÷ 12 = 6.5 (not perfectly divisible)

⇒ 78 ÷ 4 = 19.5 (not perfectly divisible)

⇒ 78 ÷ 8 = 9.75 (not perfectly divisible)

⇒ Thus, 6 is the required composite number that divides the sum completely.

∴ The correct answer is 6.

Additional Information

Sum of First n Natural Numbers

The sum of the first n consecutive natural numbers is given by the formula S = n(n + 1) ÷ 2.

Composite Numbers

A composite number is a positive integer that has at least one divisor other than 1 and itself. The smallest composite number is 4.

Sum of Squares of First n Natural Numbers

The sum of squares is given by the formula S = n(n + 1)(2n + 1) ÷ 6.

Sum of Cubes of First n Natural Numbers

The sum of cubes is given by the formula S = [n(n + 1) ÷ 2]².

The correct answer is Option 3.

Key Points

• The primary issue in the sentence is the omission of the auxiliary verb "was" before the term "driving."
• In the past continuous tense, which is used to describe an action that was happening at a specific time in the past, the verb "was" is required as an auxiliary for the subject "Jerry."
• The correct phrase should be "Jerry was driving," ensuring grammatical correctness and clarity.
• Thus, the error is located in part 3 of the sentence.

Therefore, the correct answer is Option 3.

Correct Sentence: "Jerry was driving home late when the incident took place."

Additional Information

• 1: "took place." – This part of the sentence is grammatically correct and does not require any changes.
• 2: "when the incident – This part of the sentence is grammatically correct and does not require any changes.
• 4: "home late" – This part of the sentence is grammatically correct and does not require any changes.

The correct answer is: Option 2.

Key Points

• The word "Unforseen" is incorrectly spelt. The correct spelling is "Unforeseen". (अप्रत्याशित)
• Example: The unforeseen event disrupted the entire schedule.
• "Unforeseen" refers to something that was not predicted or expected. (अप्रत्याशित)
• Example: The unforeseen expenses threw their budget into disarray.

Therefore, the correct answer is: Option 2.

Additional Information

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

• Possession (मालिकाना हक): The state of having, owning, or controlling something.
• Example: The ring was his most treasured possession.
• Referred (संदर्भित): Mentioned or directed to for information.
• Example: She referred to the manual for instructions.
• Weather (मौसम): The state of the atmosphere at a place and time regarding heat, dryness, sunshine, wind, rain, etc.
• Example: The weather forecast predicts rain for tomorrow.

The correct answer is: Option 1.

Key Points

• The word "evoke" means to bring or recall a feeling, memory, or image to mind. (याद दिलाना)
• Example: The old photographs evoked a sense of nostalgia.
• "Induce" means to succeed in persuading or leading someone to do something; to bring about or give rise to. (प्रेरित करना)
• Example: The advertisement induced many people to buy the new product.

Therefore, the correct answer is: Option 1.

Additional Information

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

• Abandon (त्याग देना): To leave someone or something completely; to desert.
• Example: They had to abandon the house due to flooding.
• Divert (मोड़ देना): To cause someone or something to change course or turn from one direction to another.
• Example: The police diverted traffic to avoid the accident site.
• Reduce (कम करना): To make smaller or less in amount, degree, or size.
• Example: The company aims to reduce its carbon emissions by 50% by 2030.

The correct answer is: Option 3.

Key Points

• The word "convalescing" means to recover one's health and strength over a period of time after an illness or medical treatment. (स्वास्थ्य लाभ करना)
• Example: After the surgery, he spent several weeks convalescing at home.
• "Recovering" means to return to a normal state of health, mind, or strength. (स्वस्थ होना)
• Example: She is slowly recovering from her injuries.

Therefore, the correct answer is: Option 3.

Additional Information

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

• Striking (हड़ताली): Attracting attention by reason of being unusual, extreme, or prominent.
• Example: The striking colors of the sunset were breathtaking.
• Traveling (यात्रा करना): Make a journey, typically of some length or abroad.
• Example: She enjoys traveling to new places and exploring different cultures.
• Ailing (बीमार): In poor health.
• Example: The ailing patient was taken to the hospital for better treatment.

The correct answer is: Option 1.

Key Points

• The word "restaurant" means a place where people pay to sit and eat meals that are cooked and served on the premises. (रेस्टोरेंट)
• Example: They decided to celebrate their anniversary at a fancy restaurant.
• "Delicious" refers to food that has a very pleasant taste or smell. (स्वादिष्ट)
• Example: The cake she baked was absolutely delicious.

Therefore, the correct answer is: Option 1.

Additional Information

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

• Resturant (रेस्टोरेंट): Incorrect spelling of the word "restaurant".
• Example: N/A (Incorrect spelling)
• Restourant (रेस्टोरेंट): Incorrect spelling of the word "restaurant".
• Example: N/A (Incorrect spelling)
• Reasturant (रेस्टोरेंट): Incorrect spelling of the word "restaurant".
• Example: N/A (Incorrect spelling)

The correct answer is: Option 3.

Key Points

• The word "Intelligible" means something that is clear and easy to understand. (सुबोध)
• Example: His speech was intelligible to everyone present at the meeting.
• "Incoherent" refers to something that is unclear, confusing, or not logically connected. (असंबद्ध)
• Example: Her argument was incoherent and difficult to follow.
• Hence, we can infer that the opposite of 'Intelligible' is 'Incoherent'.

Therefore, the correct answer is: Option 3.

Additional Information

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

• Sociable (मिलनसार): Willing to talk and engage in activities with other people; friendly.
• Example: He is a sociable person who enjoys meeting new people.
• Embezzle (गबन करना): To steal or misappropriate money placed in one's trust or belonging to the organization for which one works.
• Example: The accountant was found guilty of embezzling company funds.
• Deliberate (सोच समझकर किया हुआ): Done consciously and intentionally.
• Example: It was a deliberate attempt to discredit the politician.

The correct answer is Option 2.

Key Points

• The primary issue in the sentence is the usage of the adverb "violently."
• The sentence should use the adjective "violent" to correctly describe the subject "lions."
• The correct phrase should be "became violent," ensuring grammatical correctness and clarity.
• Thus, the error is located in part 2 of the sentence.

Therefore, the correct answer is Option 2.

Correct Sentence: "The lions became violent when the visitors started throwing stones at them."

Additional Information

• 1: "stones at them." – This part of the sentence is grammatically correct and does not require any changes.
• 3: "started throwing" – This part of the sentence is grammatically correct and does not require any changes.
• 4: "when the visitors" – This part of the sentence is grammatically correct and does not require any changes.

The correct answer is: Option 1.

Key Points

• The word "Gist" means the essence or main point of any passage, lecture, or book. (सार)
• Example: The gist of her argument was that education should be accessible to all.
• "Gist" refers to the most important pieces of information about something, or the central idea. (सारांश)
• Example: After reading the entire book, he could easily summarize the gist of the story.
• Hence, we can infer that the word 'Gist' is the correct word for the essence or main point of any passage, lecture, or book.

Therefore, the correct answer is: Option 1.

Additional Information

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

• Significance (महत्व): The quality of being worthy of attention; importance.
• Example: The significance of the discovery cannot be underestimated.
• Meaning (अर्थ): What is meant by a word, text, concept, or action.
• Example: He was searching for the meaning of the poem.
• Heart (हृदय): The central or most important part.
• Example: The heart of the matter is the lack of communication.

The correct answer is: Option 3.

Key Points

• The word "Grotesque" means something that is comically or repulsively ugly or distorted. (विकृत)
• Example: The painting depicted a grotesque figure with exaggerated features.
• "Ugly" refers to something that is unpleasant or repulsive, especially in appearance. (बदसूरत)
• Example: The building was so ugly that it was often referred to as an eyesore.

Therefore, the correct answer is: Option 3.

Additional Information

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