SSC CGL 2024 Tier-I Official Paper (Held On: 26 Sept, 2024 Shift 1) (Previous Year Paper)

Counting the triangles are there in the given figure:

Hence, the correct answer is "8".

The logic followed here is:

Hence, the correct answer is "AM45".

The option figure in which the given figure (X) is embedded:

Hence, the correct answer is "Option 2".

The logic followed here is:

FIGURE is coded as 'VSUGJW'

COPIED is coded as 'YMLSWX'

Similarly, the code for BLANKET:

Hence, the correct answer is "ZPANQWH".

The logic followed here is:

Logic: Simple and Basic are synonyms.

Here,

Simple : Basic → Simple is synonymous with Basic.

Similarly,

Difficult : Complex → Difficult is synonymous with Complex.

Hence, the correct answer is "Option 4".

Family Chart:

Decoding the Symbols:

Given: ‘M # N @ L × P + T

It means: M is the sister of N, N is the father of L, L is the wife of P and P is the son of T.

S, M is the 'Son’s wife’s father’s sister' of T.

Hence, the correct answer is "Son’s wife’s father’s sister".

BODMAS Table: 

After decoding signs:

Given equation: 229 ÷ 41 + 502 × 2 − 44 = ?

After interchanging the signs then:

⇒ 229 × 41 - 502 ÷ 2 + 44 = ?

⇒ ? = 229 × 41 - 251 + 44 = ?

⇒ ? = 9389 - 251 + 44

⇒ ? = 9433 - 251 = 9182

Hence, the correct answer is "9182".

The logic followed here is:

Small lines are decreasing by 3 from the previous figure.

Hence, the correct answer is "Option 3".

The logic followed here is:

Logic: (1^st number × 2^nd number) + 17 = 3^rd number.

Now,

For (3, 5, 32):

⇒ (3 × 5) + 17

= 15 + 17 = 32.

For (9, 1, 26)

⇒ (9 × 1) + 17

= 9 + 17 = 26.

Similarly, for Option 3: (6, 1, 23)

⇒ (6 × 1) + 17

= 6 + 17 = 23

Hence, the correct answer is Option (3).

BODMAS Table: 

Given equation: 4 – 5 ÷ 9 + 6 × 2 = ?

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

⇒ 4 + 5 × 9 – 6 ÷ 2 = ?

⇒ 4 + 45 – 3 = ?

⇒ 49 - 3 = ?

⇒ 46.

Hence, the correct answer is "Option 2".

The logic followed here is:

Hence, the correct answer is "MNQT".

The logic followed here is:

Given Sequence: Y _ Y _ _ Y _ T _ Y _ Y

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

Option 1) YTYTYT → Y Y Y / T Y Y / T T Y  / Y T Y

Option 2) TTYYTT → Y T Y / T Y Y / Y T T  / Y T Y

Option 3) TYTYYT → Y T Y / Y T Y / Y T Y / Y T Y

Here, 'Y T Y' is the pattern.

Option 4) TYTYTY → Y T Y / Y T Y / Y T T Y Y Y

Hence, the correct answer is "TYTYYT".

The correct mirror image of the given figure is:

Hence, the correct answer is "Option 2".

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

1)  82 – 46 – 10 → (82 + 10) / 2 = 92 / 2 = 46

2)  90 – 99 – 108 → (90 + 108) / 2 = 198 / 2 = 99

3)  67 – 45 – 23 → (67 + 23) / 2 = 90 / 2 = 45

4)  90 – 51 – 13 → (90 + 13) / 2 = 103 / 2 = 51.5 ≠ 51

Thus, "90 – 51 – 13" is the one that does NOT belong to that group.

Hence, the correct answer is "90 – 51 – 13".

The paper will look when unfolded:

Hence, the correct answer is "Option 3".

The logic followed here is:

1)  SUZY

2)  MPUT

3)  QSXW

4)  PRWV

Hence, the correct answer is "MPUT".

The logic followed here is:

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

Now,

For (108, 12, 9):

⇒ 12 × 9 = 108

For, (114, 19, 6)

⇒ 19 × 6 = 114

Similarly, for (136, 17, 8)

⇒ 17 × 8 = 136

Hence, the correct answer is "Option 1".

The logic followed here is: 1^st number + 2 = 2^nd number; 2^nd number - 3 = 3^rd number; 3^rd number + 4 = 4^th number.

17 – 19 – 16 – 20 → 17 + 2 = 19; 19 - 3 = 16; 16 + 4 = 20;

22 – 24 – 21 – 25 → 22 + 2 = 24; 24 - 3 = 21; 21 + 4 = 25;

Similarly,

Option 1) 15 – 17 – 14 – 18 → 15 + 2 = 17: 17 - 3 = 14; 14 + 4 = 18

Option 2) 16 – 18 – 12 – 16 → 16 + 2 = 18; 18 - 6 = 12; 12 + 4 = 16

Option 3) 11 – 13 – 10 – 15 → 11 + 2 = 13; 13 - 3 = 10; 10 + 5 = 15

Option 4) 18 – 20 – 16 – 14 → 18 + 2 = 20; 20 - 4 = 16; 16 - 2 = 14

Hence, the correct answer is "15 – 17 – 14 – 18".

Statements:

All pastries are chocolates.

Some chocolates are cookies.

All cookies are cakes.

The least possible Venn diagram is as follows:

Conclusions:

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

(II) Some cakes are chocolates. - True (because Some chocolates are cookies and All cookies are cakes. So, Some cakes are chocolates it is definite)

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

The logic followed here is:

All letters are flipped horizontally one-by-one in each next figure.

Hence, the correct answer is "Option 1".

The logic followed here is:

Thus, the code for 'T' = 5

Hence, the correct answer is "5".

Statements:

All red are green.

All green are pink.

No pink is blue.

The least possible Venn diagram is as follows:

Conclusions:

I. No green is blue. - True (because All green are pink and No pink is blue. So, No green is also blue)

II. Some blue are red. - False (because All red are green, All green are pink and No pink is blue. So, Some blue cannot red)

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

Given word: 'SPIDER'

After arranged in alphabetical order:

Now,

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

Hence, the correct answer is "Option 4".

The paper will look when unfolded:

Hence, the correct answer is "Option 1".

The logic followed here is:

Hence, the correct answer is "480".

The correct answer is Part V.

Key Points

• Part V of the Indian Constitution is titled "The Union".
• It encompasses provisions related to the President, the Vice-President, the Council of Ministers, and the Parliament.
• Articles 52 to 151 deal with the executive, legislative, and administrative functions of the Union Government.
• The election of the Vice President of India is specifically covered under Articles 63 to 71 in Part V of the Constitution.

Additional Information

• Part IV
• Part IV of the Indian Constitution is titled "Directive Principles of State Policy".
• It contains Articles 36 to 51.
• These articles guide the State in making laws and policies to establish a just society in the country.
• However, they are non-justiciable, meaning they are not enforceable by any court.
• Part III
• Part III of the Indian Constitution is titled "Fundamental Rights".
• It contains Articles 12 to 35.
• These articles guarantee essential rights such as equality, freedom, and protection against exploitation.
• These rights are justiciable, meaning they can be enforced by the courts.
• Part VI
• Part VI of the Indian Constitution is titled "The States".
• It contains Articles 152 to 237.
• This part deals with the executive, legislative, and administrative structure of the States.
• It outlines the roles and responsibilities of the Governor, State Legislature, and Council of Ministers.

The correct answer is Kabaddi.

Key Points

• The 'Baulk Line' is a critical feature in the game of Kabaddi.
• It is used to determine the playing area for raiders and defenders.
• A raider must cross the baulk line to make their raid valid.
• Crossing the baulk line puts pressure on the defenders and adds a strategic element to the game.

Additional Information

• Kabaddi
• Kabaddi is a contact team sport that originated in India.
• The objective of the game is for a single player, known as a "raider," to run into the opposing team's half of the court, tag out as many of their defenders as possible, and return to their own half of the court, all without being tackled by the defenders, and in a single breath.
• The baulk line plays a critical role during a raid. A raider must cross this line to make their raid valid, and it acts as a challenge for both raiders and defenders.
• Kabaddi has gained international recognition and is played in several countries, with professional leagues and tournaments being organized globally.
• Badminton
• Badminton is a racquet sport played using racquets to hit a shuttlecock across a net.
• It is usually played as a singles or doubles game.
• The boundaries of the playing area in badminton are defined by lines on the court, but terms like "baulk line" are not used in this sport.
• Tennis
• Tennis is a racquet sport that can be played individually against a single opponent (singles) or between two teams of two players each (doubles).
• The court is rectangular and bounded by lines, but does not include a "baulk line."
• Players use the tennis racquet to hit the ball over a net into the opponent's court.
• Kho-Kho
• Kho-Kho is a traditional Indian sport that involves two teams of twelve players each.
• The game is played on a rectangular field, and the main objective is to avoid being tagged by members of the opposing team.
• While the sport involves various lines and zones, such as the central line and the posts, the term "baulk line" is not applicable to Kho-Kho.

The correct answer is Light year is the unit of time..

Key Points

• A light year is not a unit of time; it is a unit of distance.
• A light year is the distance that light travels in one year in a vacuum.
• The concept of a light year is used in astronomy to measure vast distances between celestial objects.
• One light year is approximately 9.461 trillion kilometers (5.879 trillion miles).

Additional Information

• Light Year:
• A light year is defined as the distance that light travels in a vacuum in one Julian year (365.25 days).
• This distance is used in astronomy to express the scale of the universe.
• It allows astronomers to measure distances between stars, galaxies, and other celestial objects.
• Angstrom:
• An angstrom is a unit of length used primarily in the field of physics and chemistry.
• It is equal to 10^-10 meters or 0.1 nanometers.
• Angstroms are often used to measure wavelengths of light, atomic radii, and the size of molecules.
• Distance Measurement in Astronomy:
• Astronomers use various units to measure distances, including astronomical units (AU), light years, and parsecs.
• 1 AU is the average distance from the Earth to the Sun, approximately 149.6 million kilometers.
• 1 parsec is equal to about 3.26 light years or 31 trillion kilometers.
• These measurements help in understanding the vastness of space and the relative positions of celestial bodies.

The correct answer is It is an incentive to provide the benefits of advanced technology and decrease the cost of production.

Key Points

• Subsidies in the agriculture sector are primarily aimed at reducing the cost of production for farmers.
• They help in making advanced technologies and farming techniques more accessible to farmers, which can lead to increased productivity.
• By lowering production costs, subsidies can help in stabilizing food prices and ensuring food security.
• Such incentives also encourage farmers to adopt sustainable agricultural practices, which can lead to long-term benefits for the environment and the economy.

Additional Information

• Subsidies
• Subsidies are financial aids provided by the government to support economic sectors, making goods and services more affordable.
• They can be in the form of direct cash transfers, tax breaks, or reduced prices.
• Agricultural Subsidies
• Agricultural subsidies are designed to support farmers by reducing the cost of inputs such as seeds, fertilizers, and machinery.
• These subsidies aim to enhance agricultural productivity and ensure food security.
• Food Security
• Food security means ensuring that all people have access to sufficient, safe, and nutritious food to maintain a healthy and active life.
• It is dependent on food availability, accessibility, and proper utilization.
• Sustainable Agriculture
• Sustainable agriculture involves farming practices that are environmentally friendly, economically viable, and socially responsible.
• It aims to meet the needs of the present without compromising the ability of future generations to meet their own needs.

The correct answer is Central Government.

Key Points

• The Environment (Protection) Act, 1986, is an Act of the Parliament of India, enacted in the wake of the Bhopal Gas Tragedy.
• The Act empowers the Central Government to make rules to regulate environmental pollution and to establish authorities for the purpose.
• It provides the Central Government with the authority to take all necessary measures to protect and improve the quality of the environment.
• It encompasses a wide range of regulations and standards to control pollution and manage hazardous substances.

Additional Information

• Environment (Protection) Act, 1986
• The Act was passed in response to the need for overarching legislation to protect and improve the environment.
• The Act gives the Central Government the power to take measures for environmental protection, such as setting standards for emissions or discharge of environmental pollutants from various sources.
• It provides a framework for coordinating the activities of various central and state authorities established under other laws to ensure environmental protection.
• The Act also includes provisions for penalties for contravention of the standards and restrictions set by the government.
• Key Provisions of the Act
• Section 3: Empowers the Central Government to take measures to protect and improve the environment and to prevent, control, and abate environmental pollution.
• Section 6: Empowers the Central Government to make rules to regulate environmental pollution, including standards for the quality of the environment, and the maximum allowable concentrations of various pollutants.
• Section 15: Provides for penalties for non-compliance with the provisions of the Act, including imprisonment and fines.
• Section 19: Allows any person to file a complaint in a court of law against anyone who fails to comply with the Act's provisions.
• Importance of the Act
• The Environment (Protection) Act, 1986, serves as a comprehensive framework for the protection and improvement of the environment in India.
• It addresses the need for a unified approach to environmental regulation and management.
• The Act empowers the government to take proactive measures to prevent environmental degradation and to respond effectively to environmental emergencies.
• It plays a crucial role in ensuring sustainable development and safeguarding public health and well-being.

The correct answer is Articles 168 to 212 in Part VI.

Key Points

• Articles 168 to 212 in Part VI of the Indian Constitution deal with the State Legislature.
• The State Legislature consists of the Governor and one or two Houses, as the case may be.
• In some states, the legislature is bicameral, consisting of the Legislative Assembly and the Legislative Council, while in others, it is unicameral, consisting only of the Legislative Assembly.
• These articles lay down the composition, duration, and functions of the State Legislatures.

Additional Information

• Article 168: This article specifies the composition of the State Legislature, which can be either unicameral (with only the Legislative Assembly) or bicameral (with both the Legislative Assembly and the Legislative Council).
• Article 170: This article deals with the composition of the Legislative Assemblies, including the number of seats and the delimitation of constituencies.
• Article 171: This article outlines the composition of the Legislative Councils, including the method of election and nomination of members.
• Article 174: This article mandates that the Governor shall summon the Legislature at least once every six months.
• Article 176: This article requires the Governor to address the Legislature at the commencement of the first session after each general election and at the commencement of the first session of each year.
• Article 178: This article provides for the election of the Speaker and Deputy Speaker of the Legislative Assembly.
• Article 194: This article grants certain privileges and immunities to the members of the State Legislature.
• Article 212: This article states that the validity of any proceedings in the Legislature of a state shall not be called in question on the ground of any alleged irregularity of procedure.

The correct answer is Lithosphere.

Key Points

• The Lithosphere is the rigid, outermost shell of the Earth, encompassing both the crust and the uppermost part of the mantle.
• It is divided into tectonic plates that float on the semi-fluid asthenosphere beneath it.
• The thickness of the Lithosphere varies, being thinnest at mid-ocean ridges and thickest beneath continental crust.
• The Lithosphere is crucial for plate tectonics, as its movements and interactions cause earthquakes, volcanic activity, and the creation of mountain ranges.

Additional Information

• Stratosphere
• The Stratosphere is the second major layer of Earth's atmosphere, lying above the Troposphere and below the Mesosphere.
• It extends from about 10 km to 50 km above Earth's surface.
• This layer contains the ozone layer, which absorbs and scatters ultraviolet solar radiation.
• Commercial jet aircraft typically fly in the lower stratosphere.
• Troposphere
• The Troposphere is the lowest layer of Earth's atmosphere, extending from Earth's surface up to about 8-15 km.
• It contains approximately 75% of the atmosphere's mass and 99% of its water vapor and aerosols.
• Weather events occur in the Troposphere.
• Mesosphere
• The Mesosphere is the layer of the atmosphere above the Stratosphere and below the Thermosphere.
• It extends from about 50 km to 85 km above the Earth's surface.
• This layer is where most meteorites burn up upon entering the Earth's atmosphere.
• The temperature decreases with altitude in the Mesosphere.

The correct answer is option 4.

Key Points

• Brahmo Samaj was founded by Raja Ram Mohan Roy in 1828.
• Ramakrishna Mission was founded by Swami Vivekanand in 1897.
• Prarthana Samaj was founded by Atmaram Pandurang in 1867.
• Arya Samaj was founded by Dayananda Saraswati in 1875.

Additional Information

• Brahmo Samaj:
• Founded by Raja Ram Mohan Roy in 1828.
• Played a significant role in the socio-religious reform movements in India.
• It aimed to purify Hinduism and to promote monotheism.
• It advocated for the abolition of practices like Sati and child marriage.
• Ramakrishna Mission:
• Founded by Swami Vivekanand in 1897.
• Named after Swami Vivekanand's guru, Ramakrishna Paramahamsa.
• It focuses on religious and educational activities, and also on social service.
• The mission aims to promote the ideas of Vedanta and Yoga.
• Prarthana Samaj:
• Founded by Atmaram Pandurang in 1867.
• It was a reform movement within Hinduism.
• The organization aimed at making social reforms like the abolition of caste system, widow remarriage, and women's education.
• The Samaj promoted monotheism and condemned idolatry.
• Arya Samaj:
• Founded by Dayananda Saraswati in 1875.
• It aimed at promoting the values of the Vedas.
• The movement sought to reform and rejuvenate Hinduism.
• It opposed the caste system and advocated for gender equality and education for all.

The correct answer is Three.

Key Points

• In West Bengal, rice is a major crop and is cultivated three times a year.
• The three main rice crops are known as Aus, Aman, and Boro.
• Aus rice is sown during the pre-monsoon season (March-April) and harvested in the summer (June-July).
• Aman rice is sown during the monsoon season (June-July) and harvested in the winter (November-December).
• Boro rice is sown during the winter season (December-January) and harvested in the summer (April-May).

Additional Information

• Aus Rice:
• Also known as autumn rice, it is one of the early varieties of rice cultivated in the pre-monsoon period.
• It is typically sown in March-April and harvested by June-July.
• This variety of rice requires less water compared to the other two, making it suitable for areas with less rainfall.
• Aman Rice:
• Also known as winter rice, it is a monsoon-season crop.
• It is sown in June-July and harvested by November-December.
• This variety of rice depends heavily on the monsoon rains for its cultivation.
• Boro Rice:
• Also known as summer rice, it is cultivated in the winter season.
• It is sown in December-January and harvested by April-May.
• This variety requires irrigation facilities as it is grown during the dry season.

The correct answer is Agriculture.

Key Points

• Agriculture is a sector that is highly dependent on seasons and climatic conditions, leading to seasonal unemployment.
• During the off-season, agricultural workers may not find work, leading to temporary unemployment.
• This type of unemployment typically affects rural areas where agriculture is the primary occupation.
• Seasonal unemployment in agriculture can be mitigated by providing alternate employment opportunities during the off-season.

Additional Information

• Seasonal Unemployment:
• Seasonal unemployment occurs when people are unemployed at certain times of the year because their work is seasonal in nature.
• Common sectors that face seasonal unemployment include agriculture, tourism, and construction.
• In agriculture, this type of unemployment happens between planting and harvesting seasons.
• To mitigate seasonal unemployment, workers often seek alternative employment during the off-season or may participate in government employment schemes.
• Cyclical Unemployment:
• Cyclical unemployment occurs due to the fluctuations in the business cycle—during recessions, demand for goods and services decreases, leading to job losses.
• This type of unemployment is typically temporary and can be addressed through economic policies that stimulate demand.
• Structural Unemployment:
• Structural unemployment is caused by mismatches between the skills of the unemployed and the skills needed for available jobs.
• This can be due to technological changes, shifts in the economy, or other long-term changes.
• Re-training and education programs can help reduce structural unemployment.
• Frictional Unemployment:
• Frictional unemployment occurs when people are temporarily unemployed while transitioning from one job to another.
• This is a natural form of unemployment and is often short-term.
• It is also considered a sign of a healthy economy as workers move to jobs that better match their skills and preferences.

The correct answer is Meerut.

Key Points

• The revolution of 1857, also known as the First War of Indian Independence, started from the cantonment of Meerut.
• The revolt began on May 10, 1857, when Indian soldiers in Meerut rebelled against the British officers.
• The immediate cause of the revolt was the introduction of new rifle cartridges rumored to be greased with cow and pig fat, offending both Hindu and Muslim soldiers.
• Meerut was one of the largest military cantonments in India at the time, making it a significant starting point for the uprising.

Additional Information

• Causes of the Revolt
• Political Causes: The annexation policies of the British, such as the Doctrine of Lapse, created widespread resentment among Indian rulers.
• Economic Causes: The heavy taxation and exploitation of Indian farmers, artisans, and the destruction of traditional industries led to economic distress.
• Social and Religious Causes: The British social reforms, which were seen as interference in traditional customs and religious practices, angered many Indians.
• Spread of the Revolt
• After Meerut, the revolt quickly spread to other parts of India, including Delhi, Kanpur, Lucknow, Jhansi, and Gwalior.
• Many Indian rulers and soldiers joined the rebellion, seeing it as an opportunity to regain their lost power and territories.
• Suppression of the Revolt
• The British used superior military force and strategic alliances with some Indian rulers to suppress the revolt.
• By 1858, the British had regained control over most of the rebellious areas, marking the end of the revolt.
• Consequences of the Revolt
• The revolt led to the end of the East India Company's rule in India, and the British Crown took direct control of the Indian administration.
• The British government made several administrative and military reforms to prevent any future rebellions.
• The revolt also marked the beginning of a stronger nationalistic movement in India, leading to the eventual struggle for independence.

The correct answer is Warm blooded.

Key Points

• Mammals and birds are both warm-blooded, meaning they can regulate their body temperature internally.
• This characteristic allows them to maintain a stable body temperature regardless of the external environment.
• Being warm-blooded helps these animals to remain active in a wide range of temperatures.
• Warm-blooded animals have higher metabolic rates compared to cold-blooded animals, which helps in supporting their active lifestyles.

Additional Information

• Viviparity
• Viviparity is the mode of reproduction in which the embryo develops inside the body and is born live.
• Mammals generally exhibit viviparity, but birds do not; birds are oviparous, laying eggs outside their body.
• Pigmented Skin
• Pigmented skin contains pigments that provide color and protection against UV radiation.
• This characteristic can be found in many animals, including mammals and birds, but it is not a defining characteristic of both groups.
• Pneumatic Bones
• Pneumatic bones are lightweight bones that contain air spaces, making them less dense.
• This feature is primarily found in birds and helps them in flight; it is not a common characteristic of mammals.
• Warm Blooded
• Warm-blooded animals, also known as endotherms, can regulate their body temperature through internal mechanisms.
• This trait is a common characteristic of both mammals and birds, allowing them to thrive in various environmental conditions.

The correct answer is Diwan-i-Amir-i-kohi.

Key Points

• Diwan-i-Amir-i-kohi was established by Sultan Muhammad bin Tughlaq to promote agriculture.
• This department was responsible for the development and management of agricultural activities.
• The establishment of Diwan-i-Amir-i-kohi aimed at improving the agricultural productivity of the empire.
• This initiative was part of Muhammad Tughlaq's broader economic policies to strengthen the revenue system and support the agrarian sector.

Additional Information

• Diwan-i-Arz
• Diwan-i-Arz was a military department responsible for the organization and administration of the army.
• It managed recruitment, training, and maintenance of the army.
• Diwan-i-Insha
• Diwan-i-Insha was in charge of the royal correspondence.
• It dealt with the drafting and dispatching of royal edicts and letters.
• Diwan-i-Risalat
• Diwan-i-Risalat was the department of religious affairs and charity.
• It managed the administration of religious endowments (waqf) and provided support to scholars and clerics.

The correct answer is Chromosomes begin to condense.

Key Points

• During the leptotene stage, chromosomes start to condense and become visible under a microscope.
• This stage is the initial phase of prophase I in meiosis, where chromosomes appear as long, thread-like structures.
• The condensation of chromosomes helps in their subsequent alignment and pairing during meiosis.
• Chromosomal condensation is crucial for the accurate segregation of chromosomes into daughter cells.

Additional Information

• Leptotene Stage
• The leptotene stage is the first stage of prophase I in meiosis.
• During this stage, chromosomes start to condense and become visible as thin threads.
• It is followed by the zygotene stage, where homologous chromosomes begin to pair and synapse.
• Meiosis
• Meiosis is a type of cell division that reduces the chromosome number by half, resulting in four genetically distinct daughter cells.
• It consists of two consecutive divisions: meiosis I and meiosis II.
• Meiosis is essential for sexual reproduction and contributes to genetic diversity.
• Chromosomal Synapsis
• Synapsis is the pairing of homologous chromosomes during prophase I of meiosis.
• It occurs during the zygotene stage, following the leptotene stage.
• Synapsis is crucial for the exchange of genetic material between homologous chromosomes through crossing over.

The correct answer is Sitar.

Key Points

• Arvind Parikh is a renowned sitar player from India.
• He is a disciple of Ustad Vilayat Khan, one of the most influential sitar maestros.
• Parikh has performed extensively in India and abroad, contributing significantly to the global recognition of Indian classical music.
• He has been honored with the Padma Bhushan award, one of India's highest civilian awards, for his contributions to the field of music.

Additional Information

• Sitar
• The sitar is a plucked stringed instrument used in Hindustani classical music.
• It has a long neck and a gourd-shaped body, with 18-20 strings including both playing and sympathetic strings.
• Renowned sitar players include Ravi Shankar, Vilayat Khan, and Anoushka Shankar.
• The sitar is known for its deep, resonant sound and is often used to play ragas, which are melodic frameworks for improvisation in Indian classical music.
• Bansuri
• The bansuri is an Indian flute made from a single hollow shaft of bamboo with six or seven finger holes.
• It is traditionally used in Indian classical music and folk music.
• Notable bansuri players include Hariprasad Chaurasia and Pannalal Ghosh.
• Tabla
• The tabla is a pair of twin hand drums from the Indian subcontinent, used in traditional, classical, popular, and religious music.
• It consists of a small right-hand drum called the dayan and a larger left-hand drum called the bayan.
• Famous tabla players include Zakir Hussain and Alla Rakha.
• Sarangi
• The sarangi is a bowed, short-necked string instrument from India, Nepal, and Pakistan.
• It is used in Hindustani classical music and is known for its rich, resonant sound.
• Prominent sarangi players include Ram Narayan and Sabri Khan.

The correct answer is Asha Bhosle.

Key Points

• Asha Bhosle is a renowned playback singer in India, known for her versatile voice and extensive career in the Indian music industry.
• She has been honored with numerous awards and accolades, including the Nightingale of Asia Award in 1987.
• Asha Bhosle is the sister of the legendary playback singer Lata Mangeshkar, and together, they have made significant contributions to Indian music.
• Her career spans over seven decades, during which she has recorded songs for films, albums, and live performances, making her one of the most prolific singers in the world.

Additional Information

• Playback Singing
• Playback singing is a technique in Indian cinema where the singing voice is pre-recorded for use in movies, and the actors lip-sync to the songs on screen.
• This practice allows for professional singers to provide high-quality vocal performances, while actors focus on their acting and on-screen presence.
• Awards in the Music Industry
• Music awards are given to recognize and honor achievements in the music industry, including performances, compositions, and contributions to the field.
• Notable music awards include the Grammy Awards, the Academy Awards (Oscars) for Best Original Song, and national awards like the Filmfare Awards in India.
• Contribution of Asha Bhosle
• Asha Bhosle has sung in multiple languages, including Hindi, Marathi, Bengali, Gujarati, Punjabi, Tamil, Telugu, and many more.
• She is known for her versatility in singing various genres, including pop, classical, ghazals, and folk songs.
• Her collaborations with music directors like R.D. Burman, O.P. Nayyar, and A.R. Rahman have resulted in some of the most iconic songs in Indian cinema.
• The Nightingale of Asia Award
• The Nightingale of Asia Award is an honor given to exceptional singers who have made significant contributions to the music industry in Asia.
• It recognizes the impact and influence of the recipient's work on the music culture of the region.

The correct answer is 2 and 3, respectively.

Key Points

• The balanced chemical equation for the given reaction is: C2H5OH + 3 O2 → 2 CO2 + 3 H2O.
• In this balanced equation, 2 molecules of carbon dioxide (CO2) are produced.
• In this balanced equation, 3 molecules of water (H2O) are produced.
• Balancing a chemical equation involves making sure that the number of each type of atom on the reactant side is equal to the number on the product side.

Additional Information

• Balancing Chemical Equations:
• It is the process of ensuring that the number of atoms for each element is the same on both the reactant and product sides of the equation.
• Law of Conservation of Mass states that mass is neither created nor destroyed in a chemical reaction, so we must balance the equation accordingly.
• Steps to Balance a Chemical Equation:
• Write down the unbalanced equation.
• List the number of atoms of each element present in the reactants and products.
• Use coefficients to balance the elements one at a time.
• Make sure that all coefficients are in the lowest possible ratio.
• Double-check to ensure that the number of atoms for each element is balanced on both sides of the equation.
• Combustion Reactions:
• This type of reaction occurs when a substance combines with oxygen and releases energy in the form of light or heat.
• In the given equation, ethanol (C2H5OH) undergoes combustion in the presence of oxygen to produce carbon dioxide and water.

The correct answer is Nagaland.

Key Points

• Kuki dance is a traditional folk dance performed by the Kuki tribes primarily residing in Nagaland.
• This dance is a form of communal celebration, often performed during festivals and special occasions.
• The dance is characterized by vibrant costumes, energetic movements, and rhythmic drumming.
• Kuki dance reflects the rich cultural heritage and traditions of the Kuki community in Nagaland.

Additional Information

• Nagaland
• Nagaland is a state in the northeastern region of India, known for its diverse indigenous tribes, each with its own unique culture and traditions.
• The state is famous for its traditional festivals and dances, many of which are celebrated with great enthusiasm and community participation.
• The Hornbill Festival, held in December, is one of the most prominent festivals in Nagaland, showcasing the cultural heritage of the various tribes.
• Apart from Kuki dance, other traditional dances of Nagaland include the Naga Warrior Dance, Bamboo Dance, and the Zeliang Dance.
• Kuki Tribes
• The Kuki tribes are one of the many indigenous ethnic groups residing in northeastern India, particularly in the states of Nagaland, Manipur, and Assam.
• They have a rich cultural heritage that includes unique customs, traditional attires, folklore, and dances.
• Kuki tribes traditionally live in close-knit communities, and their social structure is based on clans and kinship.
• Their traditional dances, including the Kuki dance, are an integral part of their cultural identity and are performed during festivals, weddings, and community gatherings.

The correct answer is Muhammad Yunus.

Key Points

• Muhammad Yunus is a Bangladeshi social entrepreneur, banker, economist, and civil society leader.
• He is the founder of the Grameen Bank, which was established in 1983 to provide microloans to the impoverished without requiring collateral.
• Yunus and Grameen Bank were awarded the Nobel Peace Prize in 2006 for their efforts to create economic and social development from below.
• Muhammad Yunus is often referred to as the pioneer of microfinance and microcredit.

Additional Information

• Microfinance
• Microfinance is a category of financial services targeted at individuals and small businesses who lack access to conventional banking and related services.
• It includes microcredit, the provision of small loans to the impoverished to help them engage in productive activities or grow their small businesses.
• Microfinance institutions (MFIs) provide these services to low-income individuals or groups.
• The primary goal of microfinance is to reduce poverty and promote financial inclusion.
• Grameen Bank
• Grameen Bank is a Nobel Peace Prize-winning microfinance organization and community development bank founded in Bangladesh.
• It makes small loans to the impoverished without requiring collateral, based on the idea that the poor have skills that are underutilized.
• The bank is built on the principles of trust and solidarity, and it lends primarily to women.
• As of today, Grameen Bank has inspired similar microfinance institutions around the world.
• Nobel Peace Prize
• The Nobel Peace Prize is one of the five Nobel Prizes established by the will of Alfred Nobel, the inventor of dynamite, in 1895.
• It is awarded annually to individuals or organizations that have made significant contributions to peace.
• The award is presented in Oslo, Norway, and it includes a medal, a diploma, and a monetary grant.
• Muhammad Yunus and Grameen Bank received this honor in 2006 for their efforts in economic and social development through microcredit.
• Social Entrepreneurship
• Social entrepreneurship is the use of startup company techniques to develop, fund, and implement solutions to social, cultural, or environmental issues.
• Social entrepreneurs are individuals who pursue innovative ideas with the potential to solve community-based problems.
• Their goal is to create social value rather than personal or shareholder wealth.
• Muhammad Yunus is regarded as one of the leading figures in the field of social entrepreneurship.

The correct answer is tennis.

Key Points

• Nirupama Sanjeev is one of the first Indian women to win a round at a Grand Slam tournament.
• She achieved this milestone at the 1998 Australian Open.
• Nirupama has represented India in several international tennis competitions including the Fed Cup.
• She has been a significant figure in Indian tennis, inspiring many young athletes to pursue the sport.

Additional Information

• Tennis:
• Tennis is a racket sport that can be played individually against a single opponent or between two teams of two players each.
• The modern game of tennis originated in Birmingham, England, in the late 19^th century as lawn tennis.
• The four Grand Slam tournaments are the most prestigious events in tennis: the Australian Open, the French Open, Wimbledon, and the US Open.
• India has produced several notable tennis players, including Leander Paes, Mahesh Bhupathi, Sania Mirza, and Rohan Bopanna.
• Badminton:
• Badminton is a racquet sport played using racquets to hit a shuttlecock across a net.
• It is commonly played as a casual outdoor activity in a yard or on a beach; formal games are played on a rectangular indoor court.
• India has a rich history in badminton, with players like Prakash Padukone, Pullela Gopichand, Saina Nehwal, and PV Sindhu achieving international success.
• Swimming:
• Swimming is an individual or team racing sport that requires the use of one's entire body to move through water.
• The sport takes place in pools or open water (e.g., in a sea or lake).
• India has been making strides in swimming, with swimmers like Virdhawal Khade and Sajan Prakash gaining recognition.
• Chess:
• Chess is a board game played between two players. It is sometimes called Western or international chess to distinguish it from related games such as xiangqi.
• The current form of the game emerged in Southern Europe during the second half of the 15^th century after evolving from a similar, much older game of Indian origin.
• India has a strong presence in the world of chess, with Viswanathan Anand being a former World Chess Champion and a grandmaster.

The correct answer is Narasimhavarman.

Key Points

• Narasimhavarman I, also known as Narasimhavarman Mahamalla, was one of the most famous Pallava kings, ruling from 630 to 668 AD.
• He is known for successfully defeating the Chalukya king Pulakeshin II and capturing their capital, Vatapi (present-day Badami).
• His victory over the Chalukyas earned him the title "Vatapikonda", meaning the conqueror of Vatapi.
• Narasimhavarman I was also a great patron of art and architecture, and he contributed significantly to the development of the Mahabalipuram monuments, which are now UNESCO World Heritage Sites.

Additional Information

• Pallava Dynasty
• The Pallavas were a powerful South Indian dynasty that ruled from the 3^rd century to the 9^th century AD, with their capital at Kanchipuram.
• They were known for their patronage of arts, architecture, and literature, and they played a significant role in the cultural development of South India.
• Their architectural achievements include the rock-cut temples and monolithic rathas (chariots) at Mahabalipuram.
• Chalukya Dynasty
• The Chalukyas were another influential South Indian dynasty that ruled over large parts of the Deccan region between the 6^th and 12^th centuries.
• Their capital was initially at Vatapi (Badami), and they are known for their contributions to temple architecture, including the rock-cut caves at Badami and the temples at Aihole and Pattadakal.
• The Chalukyas had a long-standing rivalry with the Pallavas, which led to numerous battles between the two dynasties.
• Vatapi (Badami)
• Vatapi, now known as Badami, was the capital of the early Chalukya dynasty in the 6^th century AD.
• It is renowned for its rock-cut cave temples and structural temples, which are fine examples of Chalukya architecture.
• After Narasimhavarman I's conquest, the city briefly fell under Pallava control before being retaken by the Chalukyas.
• Mahabalipuram (Mamallapuram)
• Mahabalipuram, also known as Mamallapuram, is a historic town on the Coromandel Coast, known for its group of sanctuaries carved out of rock in the 7^th and 8^th centuries.
• This site includes the famous Shore Temple, the Pancha Rathas, and the Descent of the Ganges relief.
• It is a UNESCO World Heritage Site and reflects the architectural prowess of the Pallavas during Narasimhavarman's reign.

The correct answer is PV Sindhu.

Key Points

• PV Sindhu is an Indian badminton player and one of the top players in the world.
• She won the Singapore Open 2022, adding another title to her illustrious career.
• Sindhu is known for her powerful smashes and excellent court coverage.
• She has also won medals in prestigious tournaments such as the Olympics and World Championships.

Additional Information

• Badminton Terminologies:
• Smash: A powerful overhead shot that is hit steeply downwards into the opponent's court.
• Drop Shot: A shot that is hit softly and lands just over the net in the opponent's court.
• Rally: A sequence of shots between the players, starting with a serve and ending when the shuttlecock lands on the ground.
• Net Play: Shots that are played close to the net, often involving quick reflexes and precise control.
• Singapore Open:
• The Singapore Open is an annual badminton tournament held in Singapore.
• It is part of the BWF World Tour and attracts top players from around the world.
• The tournament features events in men's singles, women's singles, men's doubles, women's doubles, and mixed doubles.
• Winning the Singapore Open is considered a significant achievement in the sport of badminton.
• 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 World Champion in 2019, winning gold at the BWF World Championships.
• Sindhu has been honored with several awards, including the Padma Bhushan and the Rajiv Gandhi Khel Ratna.
• She continues to be a leading figure in Indian sports and an inspiration to aspiring athletes.

The correct answer is Madhya Pradesh.

Key Points

• The 'Lokrangjan' festival is celebrated in the state of Madhya Pradesh, India.
• This festival is held annually and showcases the rich folk and tribal culture of the state.
• It features various cultural performances, including traditional music, dance, and art forms.
• The festival aims to preserve and promote the diverse cultural heritage of the tribal communities in Madhya Pradesh.

Additional Information

• Folk Culture:
• Folk culture refers to the traditional beliefs, customs, and practices of a community, often passed down orally from generation to generation.
• It includes various art forms, such as music, dance, storytelling, and crafts.
• Folk culture is often closely tied to the daily lives and rituals of the people within the community.
• Tribal Culture:
• Tribal culture encompasses the traditions and way of life of indigenous tribal communities.
• These cultures often include unique languages, rituals, art, and social structures.
• Tribal festivals and ceremonies play a significant role in preserving and celebrating their heritage.
• Madhya Pradesh:
• Madhya Pradesh is a state in central India, known for its rich cultural heritage and historical significance.
• The state is home to numerous tribes, each with its own distinct culture and traditions.
• Madhya Pradesh is also known for its historical monuments, wildlife sanctuaries, and natural beauty.
• Lokrang Festival:
• Lokrang is a cultural festival that celebrates the folk and tribal traditions of Madhya Pradesh.
• The festival features performances by artists from various tribal communities, showcasing their unique art forms.
• It is an important event for promoting cultural exchange and preserving the heritage of indigenous communities.

The correct answer is Directive Principles of State Policy.

Key Points

• The Directive Principles of State Policy are enumerated in Part IV of the Indian Constitution, from Articles 36 to 51.
• These principles are guidelines for the framing of laws by the government.
• They aim to create social and economic conditions under which citizens can lead a good life.
• Although these principles are non-justiciable, meaning they are not enforceable by any court, they are considered fundamental in the governance of the country.

Additional Information

• Fundamental Rights
• These are a set of rights that are guaranteed to all citizens of India by the Constitution.They are mentioned in Part III of the Indian Constitution from Articles 12 to 35.
• These rights are justiciable, meaning they are enforceable by the courts.
• Examples include the right to equality, right to freedom, right against exploitation, right to freedom of religion, cultural and educational rights, and right to constitutional remedies.
• Fundamental Duties
• These are moral obligations that are expected to be followed by all citizens of India.
• They are mentioned in Part IV-A of the Indian Constitution under Article 51A.
• There are 11 Fundamental Duties listed in the Constitution.
• They were added by the 42^nd Amendment Act, 1976, based on the recommendations of the Swaran Singh Committee.
• The Union
• This pertains to the structure and functioning of the Union Government in India.
• It is covered in Part V of the Indian Constitution from Articles 52 to 151.
• It includes provisions related to the President, Vice-President, Council of Ministers, Parliament, and other Union bodies.

The correct answer is Ganga river.

Key Points

• The Gandak river is a significant tributary of the Ganga river.
• It originates in the Himalayas in Nepal and flows through the Indian states of Bihar and Uttar Pradesh before joining the Ganga.
• The river is known as Narayani in its upper reaches in Nepal.
• The Gandak river basin is an important agricultural area in the region, providing water for irrigation and supporting a large population.

Additional Information

• Ganga River
• The Ganga is one of the major rivers of the Indian subcontinent, flowing through the northern plains of India and Bangladesh.
• It is considered sacred in Hinduism and is worshipped as the goddess Ganga.
• The river has a length of about 2,525 kilometers (1,569 miles).
• The Ganga river basin is the most populous basin in the world, supporting hundreds of millions of people who rely on it for their water needs.
• Yamuna River
• The Yamuna is the second-largest tributary river of the Ganga in northern India.
• It originates from the Yamunotri Glacier in the Himalayas and flows through several states including Uttarakhand, Haryana, Uttar Pradesh, and Delhi.
• The river meets the Ganga at the Triveni Sangam in Allahabad (Prayagraj).
• Kaveri River
• The Kaveri is one of the major rivers in southern India, flowing through the states of Karnataka and Tamil Nadu.
• It originates from the Brahmagiri hills in the Western Ghats and flows into the Bay of Bengal.
• The river is crucial for agriculture and irrigation in the regions it flows through.
• Brahmaputra River
• The Brahmaputra is one of the major rivers of the Indian subcontinent, flowing through Tibet, India, and Bangladesh.
• It originates from the Angsi Glacier in Tibet and flows through the Assam valley in India and the Ganges Delta in Bangladesh.
• The river is known for its powerful and unpredictable nature, causing frequent flooding in the regions it flows through.

Given:

Annual income of five schools:

Formula used:

Compare the semester fee (SF) with three times the donation (3D):

If SF < 3D, the condition is satisfied.

Calculations:

For School X:

SF = 43, D = 14, 3D = 14 × 3 = 42

⇒ 43 > 42 (Condition not satisfied)

For School Y:

SF = 56, D = 17, 3D = 17 × 3 = 51

⇒ 56 > 51 (Condition not satisfied)

For School Z:

SF = 39, D = 14, 3D = 14 × 3 = 42

⇒ 39 < 42 (Condition satisfied)

For School U:

SF = 89, D = 50, 3D = 50 ×  3 = 150

⇒ 89 < 150 (Condition satisfied)

For School V:

SF = 59, D = 19, 3D = 19 × 3 = 57

⇒ 59 > 57 (Condition not satisfied)

∴ The condition is satisfied for 2 schools: Z and U. The correct answer is option (1).

Given:

Total number of students:

2021 = 16,000

2022 = 18,000

Percentage of students in School 4:

2021 = 23%

2022 = 22%

Formula used:

Number of students = (Percentage × Total students) / 100

Difference = Students in 2022 - Students in 2021

Calculations:

Step 1: Calculate the number of students in School 4 for 2021:

Students in 2021 = (23 × 16,000) / 100

⇒ Students in 2021 = 3,680

Step 2: Calculate the number of students in School 4 for 2022:

Students in 2022 = (22 × 18,000) / 100

⇒ Students in 2022 = 3,960

Step 3: Calculate the difference in the number of students:

Difference = 3,960 - 3,680

⇒ Difference = 280

∴ The difference in the number of students in School 4 in 2022 with respect to 2021 is 280.

Shortcut Trick Evaluate each statement logically based on the geometric properties of a planar triangle. (a) The sum of interior angles is exactly 180°, which is a universally fundamental property. (True) (b) An equilateral triangle has three equal interior angles of 60° each. (False) (c) A triangle can have at most one obtuse angle (>90°), else the sum exceeds 180°. (True) (d) The triangle inequality theorem states the sum of two sides must be strictly greater than the third side. (False) ∴ The correct answer is (a) and (c).

Alternate Method Given: Four statements regarding the properties of a triangle.

B C A

c b a

Properties: ∠A + ∠B + ∠C = 180° a + b > c Max 1 obtuse ∠

Analysis: ⇒ Statement (a): By the Angle Sum Property of a triangle, the sum of all three interior angles is always equal to 180°. Hence, statement (a) is logically sound. ⇒ Statement (b): All interior angles of a triangle can indeed be identical. For example, in an equilateral triangle, all three angles measure exactly 60°. Hence, statement (b) is invalid. ⇒ Statement (c): If a triangle possessed two obtuse angles (both greater than 90°), their combined sum would be strictly greater than 180°, defying the core angle sum property. Thus, a triangle can possess at most one obtuse angle. Hence, statement (c) is logically sound. ⇒ Statement (d): The Triangle Inequality Theorem dictates that the sum of the lengths of any two sides of a triangle must always be strictly greater than the length of the third side. It can never be equal or less. Hence, statement (d) is invalid. ⇒ Conclusively, only statements (a) and (c) correctly describe the properties of a triangle. ∴ The correct answer is (a) and (c).

Additional Information Triangle Inequality Theorem For any triangle possessing sides a, b, and c, the theorem declares: a + b > c, a + c > b, and b + c > a. Furthermore, the difference between any two sides is always strictly less than the length of the third side. Exterior Angle Property The scalar measure of an exterior angle of a triangle is mathematically equivalent to the sum of the measures of its two opposing (remote) interior angles. Classification of Triangles (by Angles) Acute (all interior angles are strictly less than 90°), Right (possesses exactly one 90° angle), and Obtuse (possesses exactly one interior angle strictly greater than 90°).

Shortcut Trick

Let the efficiency of the slower pipe be 1 unit/min.

Efficiency of the faster pipe = 4 units/min (as it is 4 times as fast).

Combined efficiency of both pipes = 1 + 4 = 5 units/min.

Total capacity of the tank = Combined efficiency × Total time = 5 × 48 = 240 units.

Time taken by the slower pipe alone = Total capacity ÷ Efficiency = 240 ÷ 1 = 240 minutes.

∴ The correct answer is 240 minutes.

Alternate Method

Given:

One pipe can fill the tank 4 times as fast as another pipe.

Time taken by both pipes together to fill the tank = 48 minutes.

Formula Used:

Total Work = Time × Efficiency

Let the time taken by the slower pipe be x minutes. The time taken by the faster pipe will be x/4 minutes.

Calculations:

⇒ Total efficiency of both pipes working together = 1 + 4 = 5

⇒ Combined time taken = Total Work ÷ Total Efficiency

⇒ 48 = x ÷ 5

⇒ x = 48 × 5

⇒ x = 240

∴ The correct answer is 240 minutes.

Additional Information

Efficiency and Time Relationship

Efficiency is inversely proportional to the time taken. If pipe A is n times faster than pipe B, the ratio of time taken by A and B is 1 : n.

Pipes & Cisterns - Basic Combined Work Rule

If pipe A can fill a tank in x hours and pipe B can fill it in y hours, then together they can fill the tank in (x × y) ÷ (x + y) hours.

Filling vs Emptying Pipes

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

Given:

A number whose fifth part increased by 5 is equal to its third part decreased by 7.

Formula used:

Let the number be x.

According to the condition:

(x/5) + 5 = (x/3) - 7

Calculations:

Step 1: Write the equation:

(x/5) + 5 = (x/3) - 7

Step 2: Simplify:

⇒ x/5 - x/3 = -7 - 5

⇒ (3x - 5x) / 15 = -12

⇒ -2x / 15 = -12

Step 3: Solve for x:

⇒ x = (-12 × 15) / -2

⇒ x = 90

Step 4: Find half of the number:

Half of x = 90 / 2

⇒ Half of x = 45

∴ Half of the number is 45.

Shortcut Trick

The Total Surface Area (TSA) of a cone is directly the sum of its Curved Surface Area (CSA) and its Base Area.

Base Area = π × r² = (22/7) × 7 × 7 = 154 cm²

TSA = CSA + Base Area = 550 + 154 = 704 cm²

∴ The correct answer is 704.00 cm².

Alternate Method

Given:

Curved Surface Area (CSA) = 550 cm²

Radius of the base (r) = 7 cm

Formula Used:

CSA of a cone = πrl (where l is the slant height)

Total Surface Area (TSA) = πr(l + r)

Calculations:

⇒ π × r × l = 550

⇒ (22/7) × 7 × l = 550

⇒ 22 × l = 550

⇒ l = 550 ÷ 22 = 25 cm

⇒ TSA = π × r × (l + r)

⇒ TSA = (22/7) × 7 × (25 + 7)

⇒ TSA = 22 × 32

⇒ TSA = 704 cm²

∴ The correct answer is 704.00 cm².

Additional Information

Volume of a Cone

The volume of a cone is one-third the volume of a cylinder with the same base radius and height. Formula: V = (1/3)πr²h

Slant Height Relationship

The slant height (l), radius (r), and perpendicular height (h) form a right-angled triangle. By Pythagoras theorem: l² = r² + h²

Surface Area Comparison

Curved Surface Area is just the lateral conical covering (πrl). Total Surface Area includes both this lateral cover and the flat circular bottom base (πr²).

Shortcut Trick

Net equivalent % formula = x + y + (x × y) ÷ 100

Equivalent decrease = −25 − 25 + (−25 × −25) ÷ 100 = −43.75%

Equivalent increase = 25 + 25 + (25 × 25) ÷ 100 = 56.25%

Required % = [(56.25 − 43.75) ÷ 56.25] × 100

⇒ (12.5 ÷ 56.25) × 100 = (2 ÷ 9) × 100 = 22.22%

∴ The correct answer is 22.22%.

Alternate Method

Given:

Two successive percentage increases of 25%.

Two successive percentage decreases of 25%.

Formula Used:

Effective % change = A + B + (A × B) ÷ 100

Percentage less = [(Greater Value − Smaller Value) ÷ Greater Value] × 100

Calculation:

⇒ For two successive decreases of 25%:

⇒ Effective decrease = 25 + 25 − (25 × 25) ÷ 100

⇒ Effective decrease = 50 − 6.25 = 43.75%

⇒ For two successive increases of 25%:

⇒ Effective increase = 25 + 25 + (25 × 25) ÷ 100

⇒ Effective increase = 50 + 6.25 = 56.25%

⇒ Difference between the two values = 56.25% − 43.75% = 12.5%

⇒ Percentage by which decrease is less than increase = (Difference ÷ Effective increase) × 100

⇒ Required % = (12.5 ÷ 56.25) × 100

⇒ Required % = (1250 ÷ 5625) × 100

⇒ Required % = (2 ÷ 9) × 100 = 200 ÷ 9 %

⇒ Required % ≈ 22.22%

∴ The correct answer is 22.22%.

Additional Information

Successive Percentage Change

When a value is subjected to two successive changes of x% and y%, the net change is given by (x + y + xy ÷ 100)%. Use a positive sign for an increase and a negative sign for a decrease.

Percentage Less / More

To find by what percent A is less than B, always take the difference with respect to the base value B using the formula: [(B − A) ÷ B] × 100.

Successive Discounts

For two successive discounts of d₁% and d₂%, the single equivalent discount formula directly evaluates as (d₁ + d₂ − d₁ × d₂ ÷ 100)%.

Shortcut Trick

If sin A = cos B for acute angles A and B, then A + B = 90°.

Here, A = 48° + k and B = 13°.

⇒ 48° + k + 13° = 90°

⇒ 61° + k = 90°

⇒ k = 29°

∴ The correct answer is 29°.

Alternate Method

Given:

sin(48° + k) = cos13°

Formula Used:

cos θ = sin(90° − θ)

Calculation:

⇒ sin(48° + k) = cos13°

⇒ sin(48° + k) = sin(90° − 13°)

⇒ sin(48° + k) = sin77°

Comparing the angles on both sides (since both are acute angles):

⇒ 48° + k = 77°

⇒ k = 77° − 48°

⇒ k = 29°

∴ The correct answer is 29°.

Additional Information

Complementary Angles

Two angles are complementary if their sum is 90°. Trigonometric ratios of complementary angles follow the rule: sin(90° − θ) = cos θ, tan(90° − θ) = cot θ, and sec(90° − θ) = cosec θ.

Supplementary Angles

Two angles are supplementary if their sum is 180°. For supplementary angles: sin(180° − θ) = sin θ, but cos(180° − θ) = −cos θ.

Basic Trigonometric Identities

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

Shortcut Trick

The largest angle of a triangle corresponds to the largest part in the given ratio.

Total parts = 7 + 8 + 3 = 18 parts

Largest part = 8

Largest angle = (8 ÷ 18) × 180° = 80°

∴ The correct answer is 80°.

Alternate Method

Given:

The angles of the triangle are in the ratio 7 : 8 : 3.

Let the three angles of the triangle be 7x, 8x, and 3x.

Formula Used:

Sum of interior angles of a triangle = 180°

7x 8x 3x

A B C

Calculations:

⇒ 7x + 8x + 3x = 180°

⇒ 18x = 180°

⇒ x = 180° ÷ 18

⇒ x = 10°

⇒ The largest angle corresponds to the largest ratio part, which is 8x.

⇒ Largest angle = 8 × 10° = 80°

∴ The correct answer is 80°.

Additional Information

Angle Sum Property of a Triangle

The sum of all interior angles of any triangle is always exactly 180°. This fundamental property helps determine unknown angles when given in ratios or expressions.

Types of Triangles Based on Angles

An acute triangle has all angles less than 90°, a right triangle has exactly one 90° angle, and an obtuse triangle has one angle greater than 90°. The triangle in this problem is acute since its angles are 70°, 80°, and 30°.

Exterior Angle Theorem

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

Shortcut Trick

Put θ = 45° in the given expression.

We know cot 45° = 1, tan 45° = 1, and cos 45° = 1/√2.

⇒ Expression = 5 / (1 + 1²) + 3 / (1 + 1²) + 2 × (1/√2)²

⇒ Expression = 5/2 + 3/2 + 2 × (1/2)

⇒ Expression = 8/2 + 1 = 4 + 1 = 5

∴ The correct answer is 5.

Alternate Method

Given:

Expression: 5 / (1 + cot²θ) + 3 / (1 + tan²θ) + 2cos²θ

Formula Used:

1 + cot²θ = cosec²θ

1 + tan²θ = sec²θ

1 / cosecθ = sinθ

1 / secθ = cosθ

sin²θ + cos²θ = 1

Calculations:

⇒ 5 / cosec²θ + 3 / sec²θ + 2cos²θ

⇒ 5sin²θ + 3cos²θ + 2cos²θ

⇒ 5sin²θ + 5cos²θ

⇒ 5 × (sin²θ + cos²θ)

⇒ 5 × 1 = 5

∴ The correct answer is 5.

Additional Information

Basic Trigonometric Identities

sin²θ + cos²θ = 1, sec²θ − tan²θ = 1, and cosec²θ − cot²θ = 1.

Reciprocal Relations

sinθ = 1 / cosecθ, cosθ = 1 / secθ, and tanθ = 1 / cotθ.

Value Substitution Method

For expressions independent of θ, substituting standard angles like 0°, 30°, 45°, 60°, or 90° directly yields the answer, provided no term becomes undefined.

Given:

The table shows the exports (in units) of three companies over the years 2017 to 2021.

Formula used:

Total exports for a year = Exports by Company P + Exports by Company Q + Exports by Company R.

Compare total exports for given pairs of years.

Calculations:

Step 1: Calculate total exports for each year:

2017: 2000 + 1500 + 1400 = 4900

2018: 4000 + 3000 + 2000 = 9000

2019: 2000 + 1500 + 2000 = 5500

2020: 3000 + 3000 + 3000 = 9000

2021: 4000 + 3000 + 3000 = 10000

Step 2: Compare total exports for the given pairs of years:

2020 and 2021: 9000 ≠ 10000

2018 and 2019: 9000 ≠ 5500

2017 and 2018: 4900 ≠ 9000

2018 and 2020: 9000 = 9000

∴ The total exports are equal for the years 2018 and 2020. The correct answer is option (4).

Calculations:

Step 1: Total production for Delhi (2019 to 2022):

Delhi (2019): 600

Delhi (2020): 450

Delhi (2021): 350

Delhi (2022): 800

Total = 600 + 450 + 350 + 800 = 2200

Average production for Delhi = 2200 / 4 = 550

Step 2: Total production for Kolkata (2019 to 2022):

Kolkata (2019): 500

Kolkata (2020): 400

Kolkata (2021): 250

Kolkata (2022): 650

Total = 500 + 400 + 250 + 650 = 1800

Average production for Kolkata = 1800 / 4 = 450

Step 3: Calculate the ratio:

Ratio = Average production of Delhi : Average production of Kolkata

⇒ Ratio = 550 : 450

⇒ Ratio = 11 : 9

∴ The ratio between the average production in Delhi and Kolkata during 2019 to 2022 is  11 : 9

Shortcut Trick

Let the number of chairs be C.

If 1 person sits per chair, 1 person is left over → Total People = C + 1.

If 2 persons sit per chair, 1 chair is vacant → Total People = 2 × (C − 1).

Equating both: C + 1 = 2C − 2 ⇒ C = 3.

∴ The correct answer is 3 chairs.

Alternate Method

Given:

⇒ Case 1: If 1 person sits on each chair, exactly 1 person has no chair.

⇒ Case 2: If 2 persons sit on each chair, exactly 1 chair is vacant.

Formula Used:

⇒ Total People = (Occupied Chairs × People per chair) + People standing

Case 1: 1 Person / Chair Case 2: 2 Persons / Chair

1 person without chair

Vacant

Result: 3 Chairs, 4 People

Calculations:

⇒ Let the total number of chairs be C.

⇒ Let the total number of persons be P.

⇒ In Case 1, 1 person sits on each chair, leaving 1 person without a chair.

⇒ Total People (P) = C + 1     ... (Equation 1)

⇒ In Case 2, 2 persons sit on each chair, leaving 1 chair vacant.

⇒ This implies the total people are sitting on exactly (C − 1) chairs.

⇒ Total People (P) = 2 × (C − 1)

⇒ P = 2C − 2     ... (Equation 2)

⇒ Equating Equation 1 and Equation 2, we get:

⇒ C + 1 = 2C − 2

⇒ 2C − C = 1 + 2

⇒ C = 3

⇒ The total number of chairs in the room is 3.

∴ The correct answer is 3 chairs.

Additional Information

Linear Equations in Two Variables

Word problems with two unknown quantities are typically solved by forming two independent equations (like variables C and P in this case) and equating them to find the required value.

Allocation & Distribution Problems

When distributing items into specific groups, the total number of items always equals the (capacity per group × number of groups used) plus any remaining items.

Shortcut Trick

Apply the successive percentage change formula: Net Change = a + b + (a × b) ÷ 100

Here, a = +8 (increase) and b = −18 (discount).

Net Change = 8 − 18 + (8 × −18) ÷ 100

Net Change = −10 − 1.44 = −11.44%

The negative sign indicates an actual discount of 11.44% on the original price.

∴ The correct answer is 11.44%.

Alternate Method

Given:

Initial price markup = +8%

Discount announced = 18%

Calculations:

Let the original price of the commodity be 100.

⇒ Marked Price after 8% increase = 100 + 8% of 100

⇒ Marked Price = 100 + 8 = 108

⇒ Final Selling Price after 18% discount = Marked Price − 18% of Marked Price

⇒ Final Selling Price = 108 − (18 ÷ 100) × 108

⇒ Final Selling Price = 108 − 19.44 = 88.56

⇒ Total discount amount = Original Price − Final Selling Price

⇒ Total discount amount = 100 − 88.56 = 11.44

⇒ Since the assumed original price is 100, the overall discount percentage is strictly equal to the discount amount.

∴ The correct answer is 11.44%.

Additional Information

Successive Percentage Change

When a value is sequentially changed by a% and then by b%, the effective overall percentage change is a + b + (a × b) ÷ 100. Always use a positive sign for an increase and a negative sign for a decrease or discount.

Discount vs Markup

Markup is always calculated on the Cost Price to arrive at the Marked Price (MP). A discount is invariably provided on the Marked Price to arrive at the Final Selling Price.

Effective Discount of Two Successive Discounts

For two sequential discounts of x% and y%, the net single equivalent discount magnitude is given directly by x + y − (x × y) ÷ 100.

Shortcut Trick

When running in the same direction on a circular track, the relative speed is the difference of their speeds.

Relative Speed = 22 − 11 = 11 m/s

Time to meet = Track Length ÷ Relative Speed

Time = 220 ÷ 11 = 20 seconds

∴ The correct answer is 20 seconds.

Alternate Method

Given:

Length of the circular track = 220 m

Speed of Ravi = 22 m/s

Speed of Kapil = 11 m/s

Formula Used:

Time = Distance ÷ Relative Speed

Relative speed (same direction) = Speed 1 − Speed 2

Calculations:

⇒ Relative speed = 22 − 11 = 11 m/s

⇒ Time taken to meet = Total distance ÷ Relative speed

⇒ Time taken = 220 ÷ 11

⇒ Time taken = 20 seconds

∴ The correct answer is 20 seconds.

Additional Information

Meeting at the Starting Point

The time taken to meet exactly at the starting point for the first time is the LCM of the individual times taken by each runner to complete one full round.

Relative Speed in Opposite Direction

When two bodies move in opposite directions around a track, their relative speed is the sum of their individual speeds (S₁ + S₂).

Number of Distinct Meeting Points

When running in the same direction, the total number of distinct meeting points on the circular track is the difference of their speeds when expressed in their simplest ratio.

Shortcut Trick

To obtain the greatest possible profit, the books must be bought at the lowest possible price and sold at the highest possible price.

Maximum Profit per book = Maximum Selling Price − Minimum Cost Price = 300 − 200 = ₹100.

Greatest profit for 25 books = 25 × 100 = ₹2500.

∴ The correct answer is ₹2500.

Alternate Method

Given:

Cost price of books ranges from ₹200 to ₹260.

Selling price of books ranges from ₹250 to ₹300.

Total number of books to be sold = 25.

Formula Used:

Profit = Selling Price (SP) − Cost Price (CP)

Calculation:

⇒ For maximum profit, the cost price must be the minimum possible and the selling price must be the maximum possible.

⇒ Minimum Cost Price (CP) of one book = ₹200

⇒ Maximum Selling Price (SP) of one book = ₹300

⇒ Maximum profit per book = Maximum SP − Minimum CP

⇒ Maximum profit per book = 300 − 200 = ₹100

⇒ Total greatest possible profit for 25 books = 25 × 100 = ₹2500

∴ The correct answer is ₹2500.

Additional Information

Maximum vs Minimum Profit

Maximum profit is obtained when the Cost Price is minimum and Selling Price is maximum. Conversely, minimum profit (or maximum loss) occurs when the Cost Price is maximum and Selling Price is minimum.

Profit and Loss Percentage

Profit % = (Profit ÷ CP) × 100. Loss % = (Loss ÷ CP) × 100. Percentage is always calculated on the original Cost Price unless specified otherwise in the question.

Selling Price Formula

When Profit % is given: SP = CP × (100 + Profit %) ÷ 100. When Loss % is given: SP = CP × (100 − Loss %) ÷ 100.

Shortcut Trick

C's share fraction = 5 ÷ (2 + 3 + 5) = 1/2 of the distributable profit.

Profit remaining after charity = 100% − 9% = 91% of total profit.

C's share = 1/2 × 91% of 2,50,000

C's share = (45.5 ÷ 100) × 2,50,000 = 1,13,750

∴ The correct answer is ₹1,13,750.

Alternate Method

Given:

Ratio of investments of A, B and C = 2 : 3 : 5

Total profit = ₹2,50,000

Percentage of profit given to charity = 9%

Formula Used:

Distributable Profit = Total Profit × ((100 − Charity%) ÷ 100)

Profit Share = (Individual Ratio ÷ Total Ratio) × Distributable Profit

Calculations:

⇒ Distributable profit = 2,50,000 × ((100 − 9) ÷ 100)

⇒ Distributable profit = 2,50,000 × (91 ÷ 100) = ₹2,27,500

⇒ Total ratio parts = 2 + 3 + 5 = 10 parts

⇒ C's profit share ratio = 5 parts

⇒ Share of C = (5 ÷ 10) × 2,27,500

⇒ Share of C = 1/2 × 2,27,500 = 1,13,750

∴ The correct answer is ₹1,13,750.

Additional Information

Simple Partnership

When all partners invest their capitals for the same time period, the profit is distributed exactly in the ratio of their invested capitals.

Compound Partnership

When partners invest for different time periods, the profit ratio is calculated dynamically as (Capital₁ × Time₁) : (Capital₂ × Time₂).

Active vs Sleeping Partner

An active partner gets a separate fixed share or salary for managing the business operations, and the remaining profit is then divided among all partners based on their capital investments.

Shortcut Trick

Ratio of Boys to Girls = (150 − 60) : 60 = 90 : 60 = 3 : 2.

Let the assumed average weight be 48 kg.

Deviation of boys' average from 48 kg = 52 − 48 = +4 kg.

Total deviation = 3 × 4 = +12 kg.

Net change in average = +12 ÷ (3 + 2) = +2.4 kg.

∴ The correct answer is 50.4 kg.

Alternate Method

Given:

Total students = 150

Number of girls = 60

Number of boys = 150 − 60 = 90

Average weight of boys = 52 kg

Average weight of girls = 48 kg

Boys: 52

Girls: 48

Avg: x

Ratio: 3

Ratio: 2

x − 48 52 − x

Formula Used:

Using Alligation Rule, Ratio of Boys to Girls = (Avg of Girls − Combined Avg) : (Combined Avg − Avg of Boys)

Calculation:

⇒ Ratio of Boys to Girls = 90 : 60 = 3 : 2

Let the combined average weight of the class be x kg.

⇒ (x − 48) ÷ (52 − x) = 3 ÷ 2

⇒ 2 × (x − 48) = 3 × (52 − x)

⇒ 2x − 96 = 156 − 3x

⇒ 5x = 252

⇒ x = 50.4

∴ The correct answer is 50.4 kg.

Additional Information

Weighted Average

When two groups of sizes N₁ and N₂ have averages A₁ and A₂, the combined average is given by (N₁ × A₁ + N₂ × A₂) ÷ (N₁ + N₂).

Alligation Rule

A quick method to find the ratio in which two groups are mixed. The ratio of quantity of group 1 to group 2 is (Mean − A₂) : (A₁ − Mean), assuming A₁ > Mean > A₂.

Deviation Method

Reduces calculation by assuming a base average. Net average = Assumed Average + (Total Deviation ÷ Total Items).

Shortcut Trick

The expression follows the identity: a³ − b³ = (a − b)(a² + ab + b²).

Substitute a = 16 and b = 6, yielding a − b = 10 and ab = 16 × 6 = 96.

The numerator becomes 3 × 10 × (16² + 6² + 96) = 30(16² + 6² + 96).

Given the RHS is 30, the denominator (16² + 6² + Q) must be equal to (16² + 6² + 96).

By direct comparison, Q = 96.

∴ The correct answer is 96.

Alternate Method

Given:

[3(16³ − 6³)] ÷ (16² + 6² + Q) = 30

Formula Used:

a³ − b³ = (a − b)(a² + b² + ab)

Calculations:

⇒ 16³ − 6³ = (16 − 6)(16² + 6² + 16 × 6)

⇒ 16³ − 6³ = 10(16² + 6² + 96)

⇒ [3 × 10(16² + 6² + 96)] ÷ (16² + 6² + Q) = 30

⇒ 30(16² + 6² + 96) ÷ (16² + 6² + Q) = 30

⇒ 16² + 6² + 96 = 16² + 6² + Q

⇒ Q = 96

∴ The correct answer is 96.

Additional Information

Sum of Cubes Identity

a³ + b³ = (a + b)(a² − ab + b²). The middle term of the trinomial factor is always negative.

Difference of Squares

a² − b² = (a − b)(a + b). This is highly useful for simplifying numerators and denominators.

Algebraic Expansions

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

Shortcut Trick

For a right-angled triangle, the circumcenter always lies at the midpoint of the hypotenuse.

This means the hypotenuse acts as the diameter of the circumcircle.

Using the standard Pythagorean triplet (7, 24, 25), we multiply by 4 to get sides (28, 96, 100), making hypotenuse AC = 100 cm.

Circumradius (R) = 100 ÷ 2 = 50 cm.

Area of circumcircle = π × R² = 3.14 × 2500 = 7850 cm².

∴ The correct answer is 7850 cm².

Alternate Method

Given:

m(AB) = 28 cm

m(BC) = 96 cm

∠ABC = 90°

π = 3.14

Formula Used:

Hypotenuse = √(Perpendicular² + Base²)

Circumradius (R) = Hypotenuse ÷ 2

Area of Circle = π × R²

Calculations:

⇒ In right-angled ΔABC, apply Pythagorean theorem to find hypotenuse AC:

⇒ AC = √(AB² + BC²)

⇒ AC = √(28² + 96²)

⇒ AC = √(784 + 9216)

⇒ AC = √10000 = 100 cm

⇒ The hypotenuse of a right-angled triangle forms the diameter of its circumcircle.

⇒ Circumradius (R) = AC ÷ 2

⇒ Circumradius (R) = 100 ÷ 2 = 50 cm

⇒ Area of Circumcircle = 3.14 × (50)²

⇒ Area = 3.14 × 2500

⇒ Area = 7850 cm²

∴ The correct answer is 7850 cm².

Additional Information

Circumcenter of a Right-Angled Triangle

The circumcenter lies exactly at the midpoint of the hypotenuse, making the circumradius equal to half the hypotenuse (R = h ÷ 2).

Inradius of a Right-Angled Triangle

The radius of the inscribed circle can be calculated directly using the formula r = (Perpendicular + Base − Hypotenuse) ÷ 2.

Pythagorean Triplets

A set of three positive integers (a, b, c) that satisfy a² + b² = c². Common primitive triplets include (3, 4, 5), (5, 12, 13), and (7, 24, 25).

Shortcut Trick

Interest for 4 years = 2625 − 1875 = ₹750

Rate % = (750 × 100) ÷ (1875 × 4) = 10%

Interest on ₹24000 for 9 years at 10% = (24000 × 10 × 9) ÷ 100 = ₹21600

Total Amount = Principal + Interest = 24000 + 21600 = ₹45600

∴ The correct answer is ₹45600.

Alternate Method

Given:

Principal P₁ = ₹1875, Amount A₁ = ₹2625, Time T₁ = 4 years

Principal P₂ = ₹24000, Time T₂ = 9 years

Formula Used:

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

Amount = P + SI

Calculations:

⇒ SI₁ = A₁ − P₁ = 2625 − 1875 = ₹750

⇒ R = (SI₁ × 100) ÷ (P₁ × T₁)

⇒ R = (750 × 100) ÷ (1875 × 4) = 75000 ÷ 7500 = 10%

⇒ SI₂ = (P₂ × R × T₂) ÷ 100

⇒ SI₂ = (24000 × 10 × 9) ÷ 100 = ₹21600

⇒ A₂ = P₂ + SI₂

⇒ A₂ = 24000 + 21600 = ₹45600

∴ The correct answer is ₹45600.

Additional Information

Simple Interest vs Compound Interest

In Simple Interest, the principal remains constant throughout the entire period, whereas in Compound Interest, the principal periodically increases as the generated interest is added to it.

Direct Proportion in Simple Interest

For a fixed principal amount and rate of interest, the total Simple Interest earned is directly proportional to the elapsed time period.

Shortcut Trick

Given tanA + cotA = 2. Since cotA = 1 ÷ tanA, the equation is in the form x + 1/x = 2, which always means x = 1.

Therefore, tanA = 1 and cotA = 1, which implies the angle A = 45°.

For A = 45°, secA = √2.

Substitute these directly into 7tan⁸A − 6cot⁸A + 8sec²A:

⇒ 7(1)⁸ − 6(1)⁸ + 8(√2)² = 7 − 6 + 16 = 17

∴ The correct answer is 17.

Alternate Method

Given:

tanA + cotA = 2 (where A is an acute angle)

Formula Used:

cotA = 1 ÷ tanA

sec²A = 1 + tan²A

Calculations:

⇒ tanA + 1/tanA = 2

⇒ (tan²A + 1) ÷ tanA = 2

⇒ tan²A − 2tanA + 1 = 0

⇒ (tanA − 1)² = 0

⇒ tanA = 1

⇒ Since A is an acute angle, A = 45°

⇒ cotA = 1 and secA = √2

Now, evaluate the given expression: 7tan⁸A − 6cot⁸A + 8sec²A

⇒ 7(1)⁸ − 6(1)⁸ + 8(√2)²

⇒ 7(1) − 6(1) + 8(2)

⇒ 7 − 6 + 16

⇒ 1 + 16 = 17

∴ The correct answer is 17.

Additional Information

Trigonometric Identities

Fundamental reciprocal and Pythagorean identities include cotθ = 1 ÷ tanθ and sec²θ − tan²θ = 1.

Algebraic Special Case

If any positive real number and its reciprocal sum to exactly 2 (x + 1/x = 2), then the number itself must be strictly equal to 1 (x = 1).

Trigonometric Values at 45°

For an acute angle of 45° (or π ÷ 4 radians), the values are tan(45°) = 1, cot(45°) = 1, and sec(45°) = √2.

Shortcut Trick

For a system of linear equations to have no solution, the lines must be parallel.

Condition: a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Compare the given equation 5x + 7y − 8 = 0 with Option 1: 5x + 7y − 16 = 0.

We get 5/5 = 7/7 ≠ −8/−16 ⇒ 1 = 1 ≠ 1/2. This perfectly satisfies the no solution condition.

∴ The correct answer is 5x + 7y - 16 = 0.

Alternate Method

Given:

Equation 1: 5x + 7y − 8 = 0

Here, a₁ = 5, b₁ = 7, c₁ = −8

Formula Used:

For equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 to have no solution (parallel lines):

Condition: a₁/a₂ = b₁/b₂ ≠ c₁/c₂

5x + 7y - 8 = 0

5x + 7y - 16 = 0 Lines are parallel Distance is constant ∴ No Solution

Calculations:

Let's check the given options systematically:

Option 1: 5x + 7y − 16 = 0

⇒ a₂ = 5, b₂ = 7, c₂ = −16

⇒ a₁/a₂ = 5/5 = 1

⇒ b₁/b₂ = 7/7 = 1

⇒ c₁/c₂ = −8/−16 = 1/2

⇒ Since 1 = 1 ≠ 1/2, it has no solution.

Option 2: 7x + 5y − 8 = 0

⇒ a₁/a₂ = 5/7, b₁/b₂ = 7/5

⇒ Since 5/7 ≠ 7/5, the lines intersect (unique solution).

Option 3: 5x − 7y − 8 = 0

⇒ a₁/a₂ = 5/5 = 1, b₁/b₂ = 7/−7 = −1

⇒ Since 1 ≠ −1, the lines intersect (unique solution).

Option 4: 10x + 14y − 16 = 0

⇒ a₁/a₂ = 5/10 = 1/2, b₁/b₂ = 7/14 = 1/2, c₁/c₂ = −8/−16 = 1/2

⇒ Since 1/2 = 1/2 = 1/2, the lines are exactly coincident (infinite solutions).

∴ The correct answer is 5x + 7y - 16 = 0.

Additional Information

Unique Solution (Intersecting Lines)

The system has exactly one solution if the lines intersect at a single point. Condition: a₁/a₂ ≠ b₁/b₂.

Infinite Solutions (Coincident Lines)

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

No Solution (Parallel Lines)

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

Shortcut Trick

By the remainder theorem, when 9 is divided by 4, the remainder is 1.

Therefore, the remainder of 9²⁰ divided by 4 is strictly 1²⁰ = 1.

Adding the constant 2, the overall remainder becomes 1 + 2 = 3.

∴ The correct answer is 3.

Alternate Method

Given:

Expression: (9²⁰ + 2) ÷ 4

Formula Used:

The remainder of (ax + r)ⁿ ÷ a is equal to rⁿ

Calculation:

⇒ 9 can be expressed in terms of the divisor 4 as (4 × 2) + 1

⇒ Substitute this into the given expression: ((4 × 2 + 1)²⁰ + 2) ÷ 4

⇒ Applying the binomial remainder formula, the remainder of (4 × 2 + 1)²⁰ ÷ 4 reduces to 1²⁰

⇒ 1²⁰ = 1

⇒ Total remainder = Remainder of 9²⁰ + Remainder of 2

⇒ Total remainder = 1 + 2

⇒ Total remainder = 3

Since 3 is less than the divisor 4, the final remainder remains 3.

∴ The correct answer is 3.

Additional Information

Modular Arithmetic Addition

The remainder of a sum is the sum of the individual remainders: (A + B) mod N = [(A mod N) + (B mod N)] mod N.

Binomial Remainder Theorem

For any expression (ax + r)ⁿ divided by 'a', every term in the expansion except the last contains 'a' and is fully divisible. The remainder depends solely on rⁿ.

Negative Remainder Strategy

If solving an expression like 3²⁰ ÷ 4, writing 3 as (4 − 1) gives a remainder of (−1)²⁰ = 1. This significantly speeds up calculations for bases close to the divisor.

Shortcut Trick

In a circle, length of a tangent = √(distance from centre² − radius²).

Length of tangent AB = √(17² − 8²) = √(289 − 64) = √225 = 15 cm.

The quadrilateral ABCD consists of two congruent right-angled triangles (ΔABC and ΔADC).

Total Area = 2 × Area of ΔABC = 2 × (1/2 × Base × Height).

Area of Quadrilateral ABCD = 15 × 8 = 120 cm².

∴ The correct answer is 120 cm².

Alternate Method

      A B C D 17 cm 8 cm 8 cm 15 cm 15 cm

Given:

Radius of the circle (r) = BC = CD = 8 cm

Distance of external point A from centre C (AC) = 17 cm

Formula Used:

Area of a right-angled triangle = (1/2) × base × height

Calculations:

⇒ In a circle, a tangent is always perpendicular to the radius at the point of contact.

⇒ Therefore, ∠ABC = 90° and ∠ADC = 90°

⇒ Using Pythagoras theorem in right-angled ΔABC:

⇒ AB² + BC² = AC²

⇒ AB² + 8² = 17²

⇒ AB² = 289 − 64 = 225

⇒ AB = 15 cm

⇒ Area of right-angled ΔABC = (1/2) × 15 × 8 = 60 cm²

⇒ Due to symmetry, the tangents drawn from an external point are equal in length, making ΔABC ≅ ΔADC.

⇒ Area of ΔADC = 60 cm²

⇒ Total Area of Quadrilateral ABCD = Area of ΔABC + Area of ΔADC

⇒ Total Area = 60 + 60 = 120 cm²

∴ The correct answer is 120 cm².

Additional Information

Properties of Tangents

Tangents drawn from an external point to a circle are equal in length. Hence, the tangent segments AB and AD are completely identical.

Tangent-Radius Theorem

A tangent to a circle is fundamentally perpendicular to the radius drawn to the exact point of tangency, creating a perfect 90° angle.

Pythagorean Triplets

Sets of integers satisfying the condition a² + b² = c² speed up calculations immensely. Common triplets include (3, 4, 5), (5, 12, 13), and naturally (8, 15, 17).

The correct answer is: Option 3.

Key Points

• The word "enthralled" means to capture the fascinated attention of someone; to hold spellbound. (मंत्रमुग्ध करना)
• Example: The magician's performance enthralled the audience.
• "Bored" refers to feeling weary and impatient because one is unoccupied or lacks interest in one's current activity. (उबाऊ)
• Example: She was bored by the lecture on ancient history.
• Hence, we can infer that the opposite of 'enthralled' is 'bored'.

Therefore, the correct answer is: Option 3.

Additional Information

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

• Interested (रुचि): Showing curiosity or concern about something or someone.
• Example: He was very interested in learning about different cultures.
• Agreement (समझौता): Harmony or accordance in opinion or feeling.
• Example: They reached an agreement after hours of negotiation.
• Intrigued (जिज्ञासु): Arouse the curiosity or interest of; fascinate.
• Example: The detective was intrigued by the mysterious case.

The correct answer is: Low.

Key Points

• The word "high" means something that is of great vertical extent or elevated. (ऊँचा)
• Example: The high mountain peaks were covered with snow.
• "Low" refers to something that is situated not far above the ground or that is not very tall. (निचा)
• Example: The low fence around the garden barely reached my knees.
• Hence, we can infer that the opposite of 'high' is 'low'.

Therefore, the correct answer is: Low.

Additional Information

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

• Sturdy (मजबूत): Strongly and solidly built.
• Example: The sturdy table can hold a lot of weight.
• Hale (स्वस्थ): In good health, especially in old age.
• Example: Despite his age, he remains hale and active.
• Elevated (उन्नत): Situated or placed higher than the surrounding area.
• Example: The elevated platform gave us a better view of the performance.

The correct answer is Option 4.

Key Points

• The statement discusses the historical context of the Soviet army's actions after the Second World War.
• Part S is the first part as it sets the time frame, "after the Second World War":
• "after the Second World War"
• Part Q follows next to introduce the subject, "the east European countries":
• "the east European countries"
• Part O is the third part as it describes the action taken by the Soviet army, "that the Soviet army had liberated":
• "that the Soviet army had liberated"
• Part P comes next detailing what happened to these countries, "from the fascist forces came under":
• "from the fascist forces came under"
• Part R concludes by explaining the outcome, "the control of the USSR":
• "the control of the USSR"
• The correct sequence is SQOPR.

Coherent sentence after rearrangement:

"After the Second World War, the east European countries that the Soviet army had liberated from the fascist forces came under the control of the USSR."

The correct answer is: Option 3.

Key Points

• The word "beeline" means to go straight towards something quickly and directly. (सीधे और जल्दी जाना)
• Example: As soon as the gates opened, the fans made a beeline for the stage.
• "going straight and quick" accurately describes the action of making a beeline. (सीधे और जल्दी जाना)
• Example: When the lunch bell rang, the students made a beeline for the cafeteria.
• Hence, the correct substitution for "making a beeline" is "going straight and quick".

Therefore, the correct answer is: Option 3.

Additional Information

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

• Behaving abnormally (असामान्य रूप से व्यवहार करना): Acting in a way that is not typical or usual.
• Example: He was behaving abnormally after hearing the shocking news.
• Selecting inconsiderately (अविवेकपूर्ण चयन करना): Choosing something without careful thought or consideration.
• Example: She was selecting inconsiderately when she picked her team without knowing their skills.
• Immediately responding to things (तुरंत प्रतिक्रिया देना): Reacting to something at once.
• Example: He has a habit of immediately responding to things without thinking them through.

The correct answer is **Option 3**.

 

**Key Points**

- Passive voice structure: The object of the active voice sentence becomes the subject of the passive voice sentence.
- In the active voice, the sentence is: "Rohit did not slap Kunal."
- To convert to passive voice: Place the object "Kunal" at the beginning.
- Change the active verb form "did not slap" to "was not slapped" (simple past passive).
- Add the preposition "by" followed by the original subject "Rohit."

Therefore, the correct answer is **Option 3**.

 

**Correct Sentence:** *"Kunal was not slapped by Rohit."*

 

**Additional Information**

- **Option 1:** "Kunal was not being slapped by Rohit." – Incorrect tense (past continuous passive).
- **Option 2:** "Kunal had not been slapped by Rohit." – Incorrect tense (past perfect passive).
- **Option 4:** "Kunal has not been slapped by Rohit." – Incorrect tense (present perfect passive).

The correct answer is: Option 4.

Key Points

• The idiom "the tip of the iceberg" means:
• A small, noticeable part of a problem, the total size of which is really much greater. (समस्या का एक छोटा, ध्यान देने योग्य भाग, जिसका कुल आकार वास्तव में बहुत बड़ा है।)
• Example: The corruption scandal involving the mayor was just the tip of the iceberg; there were many more officials involved in illegal activities.

Therefore, the correct answer is: Option 4.

Additional Information

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

• Option 1: A small, noticeable part of a problem, the total size of which is non-existent. (समस्या का एक छोटा, ध्यान देने योग्य भाग, जिसका कुल आकार अस्तित्वहीन है।)
• Example: He thought the minor complaint was the entire issue, not realizing it was a small, noticeable part of a problem that didn't exist.
• Option 2: A small, noticeable part of a problem, the total size of which is solvable. (समस्या का एक छोटा, ध्यान देने योग्य भाग, जिसका कुल आकार हल करने योग्य है।)
• Example: The broken window was just a small, noticeable part of a problem that was easily solvable.
• Option 3: A large part of a problem which is really noticeable. (एक समस्या का बड़ा हिस्सा जो वास्तव में ध्यान देने योग्य है।)
• Example: The visible pollution is a large part of the problem which is really noticeable to everyone.

The correct answer is Option 1.

Key Points

• The sequence starts with a general introduction.
• Part A is the first because it introduces the radio telescope, "The world’s biggest radio telescope is in Puerto Rico.":
• "The world’s biggest radio telescope is in Puerto Rico."
• Part B follows next to provide its name, "It is called Arecibo dish.":
• "It is called Arecibo dish."
• Part D is the third to give historical context, "It is also 300 metres across." :
• "It is also 300 metres across"
• Part C comes last to add specific detail, "It was built 40 years ago.":
• "​It was built 40 years ago."
• The correct sequence is ABDC.

Coherent sentence after rearrangement:

"The world’s biggest radio telescope is in Puerto Rico. It is called Arecibo dish. It is also 300 metres across. It was built 40 years ago."

The correct answer is: Option 2.

Key Points

• The word "Optimist" means someone who tends to look on the more favorable side of events or conditions and expects the most favorable outcome. (आशावादी)
• Example: Even during the toughest times, she remained an optimist, always believing that things would get better.
• "Optimist" refers to a person who habitually expects good things to happen and sees the positive side of life. (आशावादी)
• Example: John is an optimist who always encourages his friends to see the silver lining in every situation.

Therefore, the correct answer is: Option 2.

Additional Information

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

• Saint (संत): A person acknowledged as holy or virtuous and regarded in Christian faith as being in heaven after death.
• Example: Mother Teresa is regarded as a saint for her selfless service to humanity.
• Pessimist (निराशावादी): A person who tends to see the worst aspect of things or believes that the worst will happen.
• Example: Unlike his optimistic friend, Mark is a pessimist who always expects things to go wrong.
• Scientist (वैज्ञानिक): A person who conducts scientific research to advance knowledge in an area of interest.
• Example: Marie Curie was a pioneering scientist known for her work on radioactivity.