SSC CGL 2024 Tier-I Official Paper (Held On: 09 Sept, 2024 Shift 1) (Previous Year Paper)

The least possible Venn diagram for the given statements is as shown below :

Conclusions:

I. All envelopes can never be parcels → Follow (As no card is a parcel and all cards are envelopes. As whole card comes in envelopes and cannot be parcel, therefore it is true.)

II. No card is a bag → Does not follow (As no card is a parcel, all cards are envelopes and some envelopes are bags. As no direct relation is given between card and bags, therefore it is possible but not definite.)

III. At least some bags are postcards → Does not follow (As no relation with postcards is given, therefore it is false).

∴ Here, Only conclusion I follows.

Hence, "Option 1" is the correct answer.

According to the given information,

Thus, according to the given codes the code 'aa' is for the word 'how'.

Hence, the correct answer is "Option 2".

The logic followed here is:

Hence, "Option 3" is the correct answer.

The mirror image of the given figure is as follow:

 

Hence, the correct answer is "Option 1".

The logic followed here is:

MEK: NVP

And,

GYT: TBG

Similarly,

JWH:?

Hence, "Option 1" is the correct answer.

Given series: L_P_QS_K_P_S_KP_Q_L_ _PQS

Option 1) KPLPQLPSKP→ LKPPQSLKPPQSLKPPQSLK PPQS → (It follows the pattern of: LKPPQS - LKPPQS - LKPPQS - LKPPQS)

Option 2) KPLPLQLSPK → LKPPQSLKPPLSQKPLQSLP KPQS → (It doesn't follow any pattern)

Option 3) KPLPQQLPSK → LKPPQSLKPPQSQKPLQPLS KPQS → (It doesn't follow any pattern)

Option 4) PKLPQPLSKP → LPPKQSLKPPQSPKPLQSLK PPQS → (It doesn't follow any pattern)

Hence, "Option 1" is the correct answer.

Given: 5 – 4 ÷ 9 + 80 x 10 = ?

According to the question, after ‘+’ and ‘–‘ are interchanged and ‘x’ and ‘÷’ are interchanged:

5 – 4 ÷ 9 + 80 x 10 = ? ⇒ 5 + 4 x 9 – 80 ÷ 10 = ?

⇒ 5 + 4 x 9 – 80 ÷ 10 = ?

⇒ 5 + 4 x 9 – 8 = ?

⇒ 5 + 36 – 8 = ?

⇒ 41 – 8 = ?

⇒ 33 = ?

Hence, "Option 1" is the correct answer.

The logic followed here is:

Given word: NIGHTMARES

Thus, Four letters will remain unchanged if each of the letters in the word 'NIGHTMARES' is re-arranged in the English alphabetical order from left to right.

Hence, the correct answer is "Four".

The logic followed here is:

1. All the elements shift anticlockwise including the center element.

Hence, "Option 1" is the correct answer.

The logic followed here is:

Hence, "Option 1" is the correct answer.

The least possible Venn diagram for the given statements is as shown below :

Conclusions:

I. All tears can never be rivers → Does not follow (As some tears are drops, some drops are streams and all streams are rivers. As no direct relation is given between tears and rivers, therefore it is false.)

II. Some drops are rivers → Follow (As some drops are streams and all streams are rivers. As whole streams comes in rivers and has some part common with drops, therefore it is true.)

III. All tears being streams is a possibility → Follow (As some tears are drops, some drops are streams. As no statements restricts tears from being streams, therefore it is possible.)

∴ Here, Both II and III conclusion follow.

Hence, "Option 4" is the correct answer.

The mirror image of the given figure is as follow:

Hence,  the correct answer is "option 3".

The logic followed here is:

Hence, "Option 2" is the correct answer.

The logic followed here is:

'NAME' is written as 'FNBO'

And,

'NANO' is written as 'POBO'

Similarly,

'NAIL' will be written as

Hence, "Option 2" is the correct answer.

The logic followed here is :-

From the given two different cubes the adjacent sides are as shown below :

As V is common on both the dices, it is clear that Q, Z, T, and L are adjacent to it, and the remaining i.e. A will be opposite to it.

Opposite pairs:

Z → L

Q → T

V → A

So, the opposite side of "Q" will be side "T".

Hence, the correct answer is "Option 1".

The logic followed here is:

Option 1) 335 - 119

335 - 119 = 216

216 = 216 (LHS = RHS)

Option 2) 358 - 152

358 - 152 = 206

206 ≠ 216 (LHS ≠ RHS)

Option 3) 317 - 101

317 - 101 = 216

216 = 216 (LHS = RHS)

Option 4) 308 - 92

308 - 92 = 216

216 = 216 (LHS = RHS)

Thus, out of all the options, '358 - 152' is the odd one out.

Hence, "Option 2" is the correct answer.

The logic followed here is: Multiply the number by 11.

19 is related to 209 → 19 × 11 = 209

27 is related to 297 → 27 × 11 = 297

Similarly, 61 is related: → 61 × 11 = 671

Hence, the correct answer is "671".

The logic followed here is:

1. All the elements shift clockwise including the center element.

Thus, the final series is:

Hence, "Option 2" is the correct answer.

Given: 4515 × 5 − 431 ÷ 3 + 821 = ?

According to the question, after ‘+’ and ‘–‘ are interchanged and ‘x’ and ‘÷’ are interchanged:

4515 × 5 − 431 ÷ 3 + 821 = ? ⇒ 4515 ÷ 5 + 431 × 3 – 821 = ?

⇒ 4515 ÷ 5 + 431 × 3 – 821 = ?

⇒ 903 + 431 × 3 – 821 = ?

⇒ 903 + 1293 – 821 = ?

⇒ 2196 – 821 = ?

⇒ 1375 = ?

Hence, "Option 3" is the correct answer.

The logic followed here is :-

From the given three different cubes the adjacent sides are as shown below :

As 25 is common on both the dices I and III, it is clear that 21, 23, 22, and 24 are adjacent to it, and the remaining i.e. 26 will be opposite to it.

Opposite pairs:

23 → 22

21 → 24

25 → 26

So, the opposite side of "24" will be side "21".

Hence, the correct answer is "Option 2".

The logic followed here is: The difference between the numbers in both pairs is 121.

First, let's find the difference between the numbers in each pair:

For the pair 31 and 152:

152 − 31 = 121

For the pair 47 and 168:

168 − 47 = 121

Now, we need to find the number that is related to 66 following the same logic. We do this by adding 121 to 66:

66 + 121 = 187

Thus, the number that is related to 66 is 187.

Hence, the correct answer is "187".

The logic followed here is: 2^nd number - 1^st number = 64.

(149, 213) → 213 - 149 = 64

(168, 232) → 232 - 168 = 64,

Let's check options one by one.

Option 1)  (162, 222) → 222 - 162 = 60 ≠ 64

Option 2)  (137, 211) → 211 - 137 = 74 ≠ 64

Option 3) (153, 217) → 217 - 153 = 64

Option 4)  (144, 198) → 198 - 144 = 54 ≠ 64

Hence, the correct answer is "(153, 217)".

The logic followed here is: Each of the other word pairs correctly matches a polygon with the number of its sides.

• Triangle has 3 sides
• Pentagon has 5 sides
• Square has 4 sides

Whenever, a Hexagon has 6 sides, not 8.

Therefore, "Hexagon - 8 sides" is the word pair that does not belong to the group.

Hence, the correct answer is "Hexagon - 8 sides".

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

1) Total triangles = [Number of parts × (1 + Number of parts)] ÷ 2

= [2 × (1 + 2)] ÷ 2

= [2 × 3] ÷ 2

= 6 ÷ 2

= 3

As there are 3 same types of triangles, therefore total number of triangles will be (3 × 3) = 9 

2. As marked there are total 12 triangles.

Thus, there are total (9 + 12) = 21 triangles.

Hence, "Option 1" is the correct answer.

Decoding symbols:

So,

i. A + K / J - P = D → A is mother of K who is the wife of J who is the father of P who is the brother of D.

Here, A is the mother of K.

ii. J % P = K + A = D → J is sister of P who is the brother of K who is the mother of A who is the brother of D.

Here, A is the son of k.

iii. K + A / P - D % J → K is mother of A who is the wife of P who is the father of D who is the sister of J.

Here, A is the daughter of K.

iv. D + K = A / P – J → D is mother of K who is the brother of A who is the wife of P who is the father of J.

Here, A is the sister of K.

Hence, the correct answer is "Option 3"$

The correct answer is Bharat Ratna.

Key Points

• Lata Mangeshkar is one of the most renowned playback singers in India, known for her melodious voice and a vast repertoire of songs in various Indian languages.
• She was awarded the Bharat Ratna, India's highest civilian award, in the year 2001 for her exceptional contribution to Indian music and arts.
• The Bharat Ratna is awarded in recognition of exceptional service/performance of the highest order in any field of human endeavour.
• Lata Mangeshkar is often referred to as the Swar Kokila for her contribution to Indian music over several decades.

Additional Information

• Filmfare Lifetime Achievement Award
• This award is presented by Filmfare as part of its annual Filmfare Awards for Hindi films.
• It is given to individuals for their outstanding contribution to the Indian film industry over their lifetime.
• Lata Mangeshkar received this award in 1993.
• Padma Vibhushan
• It is the second-highest civilian award in India, given for exceptional and distinguished service.
• Lata Mangeshkar was honored with the Padma Vibhushan in 1999.
• Dadasaheb Phalke Award
• This is India's highest award in cinema, presented annually at the National Film Awards ceremony by the Directorate of Film Festivals.
• Lata Mangeshkar received the Dadasaheb Phalke Award in 1989 for her outstanding contribution to the growth and development of Indian cinema.
• Bharat Ratna
• The Bharat Ratna is the highest civilian award of India, instituted in 1954.
• It is conferred for exceptional service/performance of the highest order in any field of human endeavour.
• The award does not carry any monetary grant or pension but is a recognition of the highest degree of service to the nation.
• Lata Mangeshkar received the Bharat Ratna in 2001.

The correct answer is Article 226.

Key Points

• Article 226 of the Indian Constitution empowers the High Courts to issue certain writs.
• This Article extends the writ jurisdiction to all cases where a fundamental right is violated, as well as other legal rights.
• Article 226 can be invoked not only for the enforcement of fundamental rights but also for any other purpose.
• The power conferred by Article 226 is wider in scope as compared to Article 32, which is restricted to the enforcement of Fundamental Rights only.

Additional Information

• Article 32
• Article 32 of the Indian Constitution provides the right to constitutional remedies, allowing individuals to move the Supreme Court directly in case of violation of fundamental rights.
• It is known as the "heart and soul" of the Constitution as described by Dr. B.R. Ambedkar.
• The Supreme Court can issue writs like habeas corpus, mandamus, prohibition, quo warranto, and certiorari for the enforcement of fundamental rights.
• This Article is a crucial provision for the protection of Fundamental Rights in India.
• Writs
• A writ is a formal written order issued by a court.
• Types of writs include habeas corpus (to produce the body), mandamus (to command), prohibition (to forbid), quo warranto (by what authority), and certiorari (to be informed).
• These writs are powerful tools in the hands of the judiciary to ensure the protection of rights and to enforce public duties.
• High Courts
• High Courts are the principal civil courts of original jurisdiction in each state and union territory.
• They also have appellate jurisdiction over the subordinate courts.
• High Courts have the power to issue writs under Article 226 for the enforcement of fundamental rights and for other purposes.
• Their jurisdiction is broader compared to the Supreme Court in terms of issuing writs.

The correct answer is Swami Sahajanand Saraswati.

Key Points

• Swami Sahajanand Saraswati was a prominent peasant leader in India and was deeply involved in the peasant movement.
• He founded the Bihar Provincial Kisan Sabha (BPKS) in 1929 to address the issues of peasants and farmers in Bihar.
• The BPKS played a crucial role in mobilizing peasants against the zamindari system and for the rights of tenant farmers.
• Swami Sahajanand Saraswati later became the president of the All India Kisan Sabha, a nationwide peasant organization, further amplifying his influence in the peasant movement in India.

Additional Information

• Peasant Movement in India:
• The peasant movement in India was a series of political movements for the rights of the agricultural workers and against the feudal land system.
• These movements were significant in shaping the agrarian policies of the country and addressing issues like land reforms, tenancy rights, and fair wages.
• Zamindari System:
• The Zamindari system was a traditional land revenue system in India where landlords or zamindars collected rent from the peasants and paid a portion to the British colonial government.
• This system led to the exploitation of peasants, high rents, and forced labor, causing widespread agrarian distress.
• All India Kisan Sabha (AIKS):
• The AIKS was formed in 1936 as a national organization to address the issues faced by peasants across India.
• It worked towards the abolition of the zamindari system, fair land distribution, and better living conditions for agricultural workers.
• Swami Sahajanand Saraswati's Contributions:
• He was a key figure in the peasant movement and worked tirelessly for the upliftment of the agrarian community.
• His efforts led to significant reforms in the agrarian sector, and he is remembered as a champion of peasant rights in India.

The correct answer is Atmaram Pandurang.

Key Points

• Atmaram Pandurang was a social reformer and the founder of the Prarthana Samaj in Mumbai in 1867.
• The Prarthana Samaj was a movement for religious and social reform in Maharashtra based on earlier reform movements.
• The organization aimed to make people believe in one God and discard the caste system and superstitions.
• Atmaram Pandurang's efforts were directed towards promoting women's education and widow remarriage and opposing child marriage.

Additional Information

• Prarthana Samaj
• Prarthana Samaj was inspired by the Brahmo Samaj founded by Raja Ram Mohan Roy in Bengal.
• It was established by Atmaram Pandurang in 1867 to reform the social practices and religious beliefs of the Hindus in Maharashtra.
• The Samaj's principles were based on the teachings of Christianity and the ancient Hindu scriptures, advocating monotheism and discarding idol worship.
• Prominent members of the Prarthana Samaj included Justice Mahadev Govind Ranade, R.G. Bhandarkar, and N.G. Chandavarkar, who played significant roles in the social reform movements in Maharashtra.
• Brahmo Samaj
• Founded by Raja Ram Mohan Roy in 1828 in Kolkata, Brahmo Samaj was one of the earliest reform movements in India.
• The Samaj aimed at purifying Hinduism and promoting monotheism, rejecting the caste system, idol worship, and rituals.
• It played a crucial role in the abolition of the practice of Sati and child marriage, and the promotion of widow remarriage and women's education.
• Raja Ram Mohan Roy is often called the "Father of Modern India" due to his efforts to modernize Indian society.
• Social Reform Movements in India
• During the 19^th and early 20^th centuries, India witnessed several social reform movements aimed at addressing social issues like caste discrimination, gender inequality, and religious superstitions.
• Prominent social reformers included Raja Ram Mohan Roy, Ishwar Chandra Vidyasagar, Jyotirao Phule, and Swami Vivekananda.
• These movements played a significant role in shaping modern Indian society by advocating for education, women's rights, and social equality.
• The reforms laid the groundwork for India's struggle for independence and the subsequent formation of a democratic and secular nation.

The correct answer is Dinabandhu Mitra.

Key Points

• Dinabandhu Mitra was a notable Bengali dramatist of the 19^th century.
• He wrote the play 'Nil Darpan' in 1860-61, which portrays the plight of indigo farmers under British rule in Bengal.
• 'Nil Darpan' is considered a significant work in Bengali literature and played a crucial role in the Indigo revolt of 1859-60.
• The play was translated into English by Reverend James Long, which led to his imprisonment and fine due to its anti-British sentiments.

Additional Information

• Indigo Rebellion:
• The Indigo Rebellion (or Indigo Revolt) was a peasant movement and subsequent uprising against indigo planters in Bengal in 1859-60.
• Indigo farmers were forced to grow indigo under oppressive conditions and received meager payment.
• The rebellion highlighted the exploitation by British planters and led to reforms in the indigo cultivation practices.
• Bengali Literature:
• Bengali literature is one of the richest literary traditions in the Indian subcontinent, with works dating back to the 10^th century.
• It encompasses a wide range of genres, including poetry, drama, novels, short stories, and essays.
• Notable figures in Bengali literature include Rabindranath Tagore, Bankim Chandra Chattopadhyay, and Sarat Chandra Chattopadhyay.
• British Colonial Rule in India:
• British colonial rule in India lasted from 1858 to 1947, following the transfer of power from the East India Company to the British Crown after the Revolt of 1857.
• During this period, India was subjected to economic exploitation, cultural changes, and political reorganization.
• Significant events during British rule include the establishment of the Indian National Congress, the Partition of Bengal, and the struggle for Indian independence led by figures like Mahatma Gandhi, Jawaharlal Nehru, and Subhas Chandra Bose.

The correct answer is CsOH > KOH > NaOH > LiOH.

Key Points

• The basicity of alkali metal hydroxides increases as we move down the group in the periodic table.
• This is because the ionic nature and size of the cation increase as we move down the group.
• CsOH is the most basic among the given hydroxides because cesium is at the bottom of the alkali metal group.
• LiOH is the least basic because lithium is at the top of the alkali metal group.

Additional Information

• Basicity:
• Basicity refers to the ability of a substance to accept protons (H+ ions) or donate a pair of valence electrons to form a bond.
• In the context of alkali metal hydroxides, basicity is determined by the extent to which the hydroxide ion (OH-) is available to react with acids.
• Alkali Metal Hydroxides:
• Alkali metal hydroxides are compounds formed by alkali metals (Group 1 elements) and hydroxide ions (OH-).
• Examples include LiOH (Lithium Hydroxide), NaOH (Sodium Hydroxide), KOH (Potassium Hydroxide), and CsOH (Cesium Hydroxide).
• These hydroxides are strong bases and are highly soluble in water.
• Periodic Table Trends:
• As we move down Group 1 in the periodic table, the size of the alkali metal cations increases.
• This leads to a decrease in the lattice energy of the hydroxides, making them more soluble and hence more basic.
• Ionic Nature:
• The ionic nature of a compound refers to the extent to which it dissociates into ions in a solution.
• Alkali metal hydroxides dissociate completely in water, releasing hydroxide ions, which contribute to their basicity.

The correct answer is Pallava.

Key Points

• Mahendravarman I was a ruler of the Pallava dynasty in South India.
• He reigned from 600–630 CE and was known for his patronage of the arts and architecture.
• Mahendravarman I was succeeded by his son Narasimhavarman I, who continued the expansion of the Pallava dynasty.
• Under Mahendravarman I, the Pallavas saw significant cultural and architectural advancements, including the construction of rock-cut temples.

Additional Information

• Pallava Dynasty:
• The Pallava dynasty was a South Indian dynasty that existed from 275 CE to 897 CE, ruling regions of present-day Tamil Nadu, Andhra Pradesh, and Karnataka.
• The Pallavas were known for their contributions to Indian art and architecture, especially during the reigns of Mahendravarman I and Narasimhavarman I.
• They played a significant role in the transition from rock-cut architecture to stone temples.
• The Shore Temple at Mahabalipuram, built by Narasimhavarman II, is a UNESCO World Heritage site and an excellent example of Pallava architecture.
• Mahendravarman I:
• Mahendravarman I was a prominent ruler of the Pallava dynasty, known for his military and architectural achievements.
• He converted from Jainism to Shaivism and promoted the construction of Hindu temples.
• His reign marked the beginning of the Pallava expansion in South India, establishing their dominance in the region.
• He is credited with the construction of several rock-cut temples, including the famous rock-cut cave temples in Mahabalipuram.
• Chola Dynasty:
• The Chola dynasty was another prominent South Indian dynasty that ruled from the 9^th to the 13^th century CE.
• They are known for their naval prowess, extensive trade, and contributions to Tamil literature, art, and architecture.
• Raja Raja Chola I and his son Rajendra Chola I were notable rulers who expanded the Chola empire significantly.
• The Brihadeeswarar Temple in Thanjavur is one of the greatest architectural achievements of the Chola dynasty.
• Pandya Dynasty:
• The Pandya dynasty was an ancient Tamil dynasty that ruled parts of South India, primarily the region of present-day Tamil Nadu.
• They were known for their contributions to Tamil literature and their support of the Tamil Sangam, an assembly of poets and scholars.
• The Pandyas were also famous for their pearl fisheries along the southern coast.
• Madurai was the capital of the Pandya kingdom and a significant cultural and religious center.

The correct answer is Kolkata.

Key Points

• The first golf club in India, the Royal Calcutta Golf Club, was established in Kolkata in 1829.
• The Royal Calcutta Golf Club is also the oldest golf club outside the British Isles.
• It was initially started to serve the British community in India during the British colonial period.
• The club has a rich history and has hosted numerous prestigious golf tournaments over the years.

Additional Information

• Golf in India
• Golf was introduced to India during the British colonial period and has grown in popularity since then.
• India has produced several notable golfers, including Jeev Milkha Singh, Anirban Lahiri, and Aditi Ashok.
• Numerous golf courses have been established across the country, catering to both amateur and professional golfers.
• The Indian Golf Union (IGU) is the governing body for golf in India, responsible for promoting and developing the sport.
• The Royal Calcutta Golf Club
• Founded in 1829, the Royal Calcutta Golf Club is located in the Tollygunge area of Kolkata.
• The club's course is known for its challenging layout and beautiful surroundings.
• It has hosted many national and international golf tournaments, making it a significant venue in the golfing world.
• The club offers various facilities and amenities for its members, including a clubhouse, dining options, and recreational activities.

The correct answer is Inconsistent regulatory environment.

Key Points

• The regulatory environment in India is often perceived as inconsistent and subject to frequent changes, making it difficult for foreign investors to plan long-term investments.
• Issues such as complex tax laws, bureaucratic red tape, and sudden changes in policies can create uncertainty for foreign businesses.
• The lack of a unified regulatory framework across different states in India further complicates the investment landscape for foreign companies.
• Foreign investors may face challenges related to compliance with diverse and sometimes conflicting regulations at the central and state levels.

Additional Information

• Foreign Direct Investment (FDI)
• FDI refers to investments made by a company or individual in one country into business interests located in another country.
• This type of investment generally involves a significant degree of control and influence over the foreign business.
• FDI can provide a number of benefits to the host country, including capital inflow, technology transfer, and job creation.
• Regulatory Environment
• The regulatory environment encompasses the laws, regulations, and policies that govern business operations in a country.
• A stable and predictable regulatory environment is crucial for attracting and retaining foreign investment.
• Key regulatory aspects include taxation, labor laws, environmental regulations, and business registration processes.
• Bureaucratic Red Tape
• Bureaucratic red tape refers to the excessive and often unnecessary regulations and procedures that can hinder business operations.
• It can include lengthy approval processes, complex documentation requirements, and inefficient administrative procedures.
• Reducing bureaucratic red tape is important for creating a more business-friendly environment and encouraging investment.
• Policy Stability
• Policy stability refers to the consistency and predictability of government policies over time.
• Frequent changes in policies can create uncertainty and risk for investors, making it difficult to plan and execute long-term investments.
• Ensuring policy stability is important for building investor confidence and attracting foreign investment.

The correct answer is Grammy.

Key Points

• Pandit Vishwa Mohan Bhatt is an Indian classical musician who plays the Mohan Veena, a modified slide guitar.
• He won the Grammy Award in 1994 for his album "A Meeting by the River" in collaboration with Ry Cooder.
• The Grammy Awards are presented by the Recording Academy to recognize achievements in the music industry.
• Pandit Vishwa Mohan Bhatt's Grammy win brought global recognition to Indian classical music and the Mohan Veena.

Additional Information

• Grammy Awards
• The Grammy Awards, or simply Grammys, are awards presented by the Recording Academy to recognize outstanding achievements in the music industry.
• The awards were first presented in 1959, and they are considered one of the most prestigious awards in the music industry.
• Categories include Best New Artist, Album of the Year, Record of the Year, and Song of the Year, among others.
• The Grammy Awards ceremony is held annually, and it features performances by prominent artists, as well as the presentation of those awards that have a more popular interest.
• Pandit Vishwa Mohan Bhatt
• Pandit Vishwa Mohan Bhatt is a renowned Indian classical musician known for his expertise in playing the Mohan Veena.
• The Mohan Veena is a modified version of the Hawaiian guitar, incorporating elements of the sitar and the veena to suit Indian classical music.
• Pandit Bhatt has received numerous awards and honors for his contribution to music, including the Padma Shri and the Padma Bhushan from the Government of India.
• His collaboration with American guitarist Ry Cooder on the album "A Meeting by the River" earned him the Grammy Award for Best World Music Album in 1994.
• Sangeet Natak Akademi Award
• The Sangeet Natak Akademi Award is the highest Indian recognition given to practicing artists in the field of performing arts, including music, dance, and drama.
• It is awarded annually by the Sangeet Natak Akademi, India's National Academy of Music, Dance & Drama.
• The award consists of a purse, a tamrapatra (a brass plaque), and a shawl.
• Many renowned artists in India, including musicians, dancers, and theater artists, have been recipients of this prestigious award.

The correct answer is Uttar Pradesh.

Key Points

• Lathmar Holi is a unique and traditional form of Holi celebration primarily observed in the towns of Barsana and Nandgaon in Uttar Pradesh.
• This festival is celebrated a few days before the main Holi festival and involves women playfully beating men with sticks, hence the name 'Lathmar' which translates to 'beating with sticks'.
• The celebration is rooted in the legend of Radha and Krishna, where Krishna visited Radha's village and teased her and her friends, leading to the women chasing him away with sticks.
• The festival attracts numerous tourists and devotees who come to witness and participate in the vibrant and joyful celebrations.

Additional Information

• Holi Festival
• Holi, also known as the 'Festival of Colors', is an ancient Hindu festival celebrated predominantly in India and Nepal.
• The festival signifies the victory of good over evil, the arrival of spring, and the end of winter. It also serves as a thanksgiving for a good harvest.
• The celebrations begin with a Holika bonfire on the night before Holi, where people gather, sing, dance, and perform rituals.
• The following day is celebrated with vibrant colors, water balloons, and traditional sweets. It is a day of festivity and joy, bringing people together to forgive and forget past grievances.
• Radha and Krishna
• Radha and Krishna are central figures in Hindu mythology and are considered the epitome of divine love.
• Krishna, an incarnation of Lord Vishnu, is known for his playful and mischievous nature, while Radha symbolizes the ultimate devotee and beloved of Krishna.
• Their love story is celebrated in various cultural and religious contexts, including dance, music, and festivals like Holi.
• The stories of their divine love have inspired countless artistic and literary works in Indian culture.
• Braj Region
• Braj, also known as Brij or Braj Bhoomi, is a region in Uttar Pradesh associated with the childhood and youthful pastimes of Lord Krishna.
• It includes important pilgrimage sites like Mathura, Vrindavan, Govardhan, Gokul, and Barsana.
• The region is known for its rich cultural heritage, including traditional music, dance, and festivals dedicated to Krishna.
• Braj attracts millions of devotees and tourists annually, making it a significant religious and cultural destination in India.

The correct answer is Physiological density.

Key Points

• Physiological density is a measure of population density that refers to the number of people per unit area of arable land.
• This measure provides insights into the pressure exerted by the population on the productive land area.
• It helps in understanding the carrying capacity of the land and the potential challenges in terms of food production and resource management.
• Physiological density is considered more meaningful than simple population density (people per total land area) because it focuses on the land that can actually support human life.

Additional Information

• Agricultural Density
• Agricultural density is a measure of the number of farmers per unit area of arable land.
• This metric helps in understanding the agricultural efficiency and the potential output of the farming sector.
• It provides insights into the level of development and technological advancement in agriculture within a region.
• Higher agricultural density may indicate less mechanization and more labor-intensive agriculture.
• Gross Density
• Gross density refers to the total population divided by the total land area.
• This measure does not differentiate between arable and non-arable land, giving an overall view of population distribution.
• It is useful for general demographic studies and urban planning purposes.
• However, it may not accurately reflect the actual human impact on the productive land resources.
• Net Density
• Net density is a less commonly used term and is not a standard measure in population geography.
• It might refer to a refined measure of population density considering specific criteria or exclusions.
• In most contexts, "net density" is not clearly defined or widely recognized.
• It is important to refer to more standardized measures like physiological or agricultural density for accurate analysis.

The correct answer is Narendra Modi.

Key Points

• Narendra Modi has been recognized in Time Magazine's list of '100 most influential people' multiple times, including in 2021, due to his significant impact on Indian and global politics.
• As the Prime Minister of India since 2014, Modi has implemented several major reforms and initiatives aimed at enhancing India's economic and social landscapes.
• His notable initiatives include the Digital India campaign, Make in India, Swachh Bharat Abhiyan, and Atmanirbhar Bharat, which have been influential in shaping India's development trajectory.
• Modi's leadership during the COVID-19 pandemic, including the rollout of the vaccination program and economic relief measures, has also been a key factor in his inclusion in the list.

Additional Information

• Time Magazine's 100 Most Influential People
• Time Magazine's list of the 100 most influential people is an annual ranking that recognizes the contributions and impact of various individuals across different fields.
• The list includes leaders, innovators, artists, and icons who have made significant strides in their respective domains and have had a profound influence on the world stage.
• Being featured in this list is considered a prestigious honor, highlighting the individual's influence and achievements on a global scale.
• The list is divided into various categories, including leaders, pioneers, titans, artists, and icons, reflecting the diverse areas of influence and impact.
• Narendra Modi's Major Initiatives
• Digital India: A campaign launched to ensure the government's services are made available to citizens electronically by improving online infrastructure and increasing Internet connectivity.
• Make in India: An initiative aimed at transforming India into a global manufacturing hub by encouraging both multinational and domestic companies to manufacture their products in India.
• Swachh Bharat Abhiyan: A national campaign initiated to clean the streets, roads, and infrastructure of India's cities and rural areas, promoting hygiene and sanitation.
• Atmanirbhar Bharat: A vision of making India a self-reliant nation by promoting local production and reducing dependence on imports, especially in the wake of the COVID-19 pandemic.

The correct answer is Jitendra Singh.

Key Points

• Jitendra Singh is the Union Minister of State (Independent Charge) for Science and Technology as of July 2023.
• He has been serving in this position since his appointment by the Prime Minister.
• Jitendra Singh is also known for his roles in various other ministries, including the Ministry of Earth Sciences and the Ministry of Personnel, Public Grievances, and Pensions.
• He has been actively involved in promoting scientific research and development in India.

Additional Information

• Union Minister of State (Independent Charge)
• In the Indian government, a Union Minister of State (Independent Charge) is a minister who is in charge of a ministry without an overseeing cabinet minister.
• These ministers have the same responsibilities as cabinet ministers in their respective ministries but report directly to the Prime Minister.
• Ministry of Science and Technology
• This ministry is responsible for the promotion of science and technology in India.
• It encompasses several departments, including the Department of Science and Technology (DST), the Department of Biotechnology (DBT), and the Department of Scientific and Industrial Research (DSIR).
• The ministry aims to boost scientific research, innovation, and development in various fields such as space, biotechnology, and renewable energy.
• Jitendra Singh
• Jitendra Singh is an Indian politician and a member of the Bharatiya Janata Party (BJP).
• He has held various significant positions in the government, contributing to policies and initiatives in multiple sectors.
• He is known for his work in improving governance and administration, as well as fostering scientific advancements in the country.

The correct answer is Deccan lava plateau.

Key Points

• The Deccan lava plateau is known for its black soil, also called regur soil, which is ideal for growing cotton and other crops.
• This region is one of the most fertile areas in India, making it a crucial agricultural zone.
• The black soil in the Deccan lava plateau retains moisture, which is beneficial for farming even during dry seasons.
• The soil is rich in essential nutrients like calcium carbonate, magnesium, potash, and lime, which support diverse agricultural activities.

Additional Information

• Black Soil
• Also known as regur soil, black soil is formed from volcanic rocks and lava-flow areas.
• This type of soil is highly moisture-retentive and is rich in iron, lime, calcium, potassium, aluminum, and magnesium.
• Black soil is predominantly found in the Deccan plateau, including parts of Maharashtra, Madhya Pradesh, Gujarat, Andhra Pradesh, and Tamil Nadu.
• It is especially suited for growing cotton, and thus it is also called "cotton soil."
• Deccan Plateau
• The Deccan Plateau is a large plateau in western and southern India, covering about 500,000 square kilometers.
• It is bounded by the Western Ghats on the west, the Eastern Ghats on the east, and the Satpura and Vindhya ranges to the north.
• The plateau is composed of ancient crystalline, igneous, and metamorphic rocks.
• The region experiences a semi-arid climate with distinct dry and wet seasons.
• Fertility of Soil
• Soil fertility refers to the ability of soil to sustain agricultural plant growth, i.e., to provide plant habitat and result in sustained and consistent yields of high quality.
• Fertile soil contains essential nutrients for plant growth, including nitrogen, phosphorus, and potassium.
• Other factors influencing soil fertility include soil structure, organic matter content, pH level, and moisture retention capacity.
• In agricultural practices, soil fertility is enhanced by adding organic and inorganic fertilizers, crop rotation, and other soil management practices.

The correct answer is He shall not be entitled, without payment of rent, to use his official residence..

Key Points

• The President of India is entitled to use his official residence, Rashtrapati Bhavan, without any payment of rent.
• According to Article 59 of the Indian Constitution, the President shall not be a member of either House of Parliament or a House of the Legislature of any State.
• During the term of office, the President's emoluments and allowances cannot be diminished as per Article 59(3).
• The President shall not hold any office of profit during his term as mentioned in Article 59(4).

Additional Information

• Article 59 of the Indian Constitution
• It outlines the conditions of the President's office in India.
• The President shall not be a member of either House of Parliament or a House of the Legislature of any State.
• The President shall not hold any other office of profit.
• The President's emoluments and allowances cannot be diminished during his term of office.
• Rashtrapati Bhavan
• It is the official residence of the President of India.
• Located in New Delhi, it was formerly known as the Viceroy's House.
• Rashtrapati Bhavan includes the main building, large gardens, staff quarters, stables, and other offices.
• Office of Profit
• An office of profit is a position that brings financial gain, advantage, or benefit.
• The President is not allowed to hold any office of profit to ensure independence and impartiality.

The correct answer is Madhya Pradesh.

Key Points

• Madhya Pradesh is the largest producer of pulses in India.
• The state contributes significantly to the total production of pulses in the country.
• The major pulses grown in Madhya Pradesh include chickpeas (gram), pigeon peas (arhar/tur), lentils, and moong beans.
• Madhya Pradesh has favorable climatic conditions and soil types that support the cultivation of a variety of pulses.

Additional Information

• Pulses
• Pulses are an important group of crops that belong to the legume family.
• They are a rich source of protein, dietary fiber, vitamins, and minerals, making them an essential part of a balanced diet.
• Pulses are also known for their ability to fix nitrogen in the soil, which improves soil fertility and reduces the need for chemical fertilizers.
• Common types of pulses include chickpeas, lentils, beans, peas, and pigeon peas.
• Agriculture in Madhya Pradesh
• Madhya Pradesh is known as the "Heart of India" due to its geographical location.
• The state has a diverse agro-climatic condition that supports the cultivation of a wide range of crops.
• Apart from pulses, Madhya Pradesh is also a leading producer of wheat, soybean, and oilseeds in India.
• The state government has implemented various agricultural schemes and policies to support farmers and enhance agricultural productivity.
• Importance of Pulses in India
• India is the largest producer and consumer of pulses in the world.
• Pulses play a crucial role in ensuring food security and nutrition in the country.
• They are a key component of the Indian diet, especially for vegetarians who rely on pulses as a primary source of protein.
• The cultivation of pulses also contributes to sustainable agricultural practices and environmental conservation.

The correct answer is Microfinance Institutions Network.

Key Points

• The Microfinance Institutions Network (MFIN) is an industry association that represents the microfinance industry in India.
• It is a self-regulatory organization (SRO) recognized by the Reserve Bank of India (RBI).
• The primary objective of MFIN is to work towards the robust development of the microfinance sector by promoting responsible lending practices and ensuring client protection.
• MFIN also works on policy advocacy, capacity building, and facilitating knowledge sharing among its members.

Additional Information

• Self-help Group Association
• Self-help groups (SHGs) are small voluntary associations of people, typically women, who come together to form a group for the purpose of savings and credit.
• SHGs promote savings among members and use the pooled resources to provide small loans to members.
• They play a crucial role in financial inclusion and empowerment of marginalized communities, especially women.
• Microfinance and Investments Regulatory Authority
• There is no known regulatory authority by this name specifically for microfinance and investments in India.
• Microfinance institutions in India are primarily regulated by the Reserve Bank of India (RBI).
• NABARD (National Bank for Agriculture and Rural Development)
• NABARD is an apex development bank in India, established to provide and regulate credit and other facilities for the promotion and development of agriculture, small-scale industries, cottage and village industries, handicrafts, and other rural crafts.
• It plays a pivotal role in the development of the rural sector in India.
• While NABARD supports microfinance activities, it is not an industry association or SRO specifically for the microfinance sector.

The correct answer is Tissue-organ grade organisation - Euplectella

Key Points

• Cellular grade organisation refers to a level of organization where cells are functionally differentiated but do not form tissues. Sycon is an example of this grade.
• Protoplasmic grade organisation is the simplest level of organisation where all life functions are confined within a single cell. Paramecium is an example of this grade.
• Cell-tissue grade organisation involves groups of similar cells performing a common function. Jellyfish represent this grade.
• Tissue-organ grade organisation refers to tissues grouped into organs, each performing different functions. 
• ​Euplectella (Venus's flower basket) is a type of glass sponge and does not exhibit tissue-organ grade organization. Instead, it is an example of cellular grade organization, characteristic of Porifera (sponges), where the organization is primarily at the cellular level and not organized into true tissues and organs.

Additional Information

• Cellular Grade Organisation
• This refers to organisms whose body is made up of cells that are functionally differentiated but do not form true tissues.
• Sponges (Porifera) like Sycon exhibit this type of organisation.
• These organisms have a simple body structure with cells specialized for different functions such as feeding, reproduction, and protection.
• Protoplasmic Grade Organisation
• It is the simplest level of organization where all life functions occur within a single cell.
• Protozoans like Paramecium demonstrate this type of organization.
• These organisms are unicellular and perform all necessary life processes within one cell.
• Cell-Tissue Grade Organisation
• Organisms with this level of organisation have cells grouped into tissues.
• Coelenterates, such as jellyfish, exhibit this type of organisation.
• These organisms have more complex structures with different tissues performing various functions.
• Tissue-Organ Grade Organisation
• This level of organisation involves tissues grouped into organs, each performing specific functions.
• Euplectella, a genus of glass sponges, is an example of cellular grade organisation grade.

The correct answer is A flower with both androecium and gynoecium.

Key Points

• Monoecious plants have both male (androecium) and female (gynoecium) reproductive organs in the same plant.
• This characteristic allows monoecious plants to self-pollinate, although cross-pollination can also occur.
• Examples of monoecious plants include maize, cucumbers, and squash.
• Monoecious plants are different from dioecious plants, which have male and female reproductive organs on separate plants.

Additional Information

• Androecium
• The androecium is the male reproductive part of a flower.
• It consists of stamens, each of which has a filament topped by an anther where pollen is produced.
• Gynoecium
• The gynoecium is the female reproductive part of a flower.
• It consists of one or more carpels, each containing an ovary, style, and stigma.
• The ovary contains ovules, which develop into seeds upon fertilization.
• Dioecious
• Dioecious plants have male and female reproductive organs on separate plants.
• This means that a dioecious plant is either male or female, not both.
• Examples of dioecious plants include holly, willow, and asparagus.
• Pollination
• Pollination is the process by which pollen is transferred from the male part of a plant to the female part, allowing fertilization to occur.
• Pollination can be carried out by wind, water, insects, birds, and other animals.
• Self-pollination occurs within the same flower or plant, while cross-pollination occurs between different plants.

The correct answer is 1973.

Key Points

• Project Tiger was launched by the Government of India on 1^st April 1973.
• The aim of this initiative was to ensure a viable population of Bengal tigers in their natural habitats and to protect them from extinction.
• This project is administered by the National Tiger Conservation Authority (NTCA).
• Initially, nine tiger reserves were covered under this project, which has now expanded to cover more than 50 tiger reserves across India.

Additional Information

• National Tiger Conservation Authority (NTCA)
• The NTCA was constituted under the provisions of the Wildlife (Protection) Act, 1972, as amended in 2006, for strengthening tiger conservation.
• It is a statutory body under the Ministry of Environment, Forest and Climate Change.
• The NTCA is responsible for the implementation of Project Tiger to ensure the protection of tigers.
• It regularly assesses tiger populations, habitat status, and the success of conservation efforts.
• Wildlife (Protection) Act, 1972
• This Act was enacted by the Indian Parliament to ensure the protection of wild animals, birds, and plants.
• It aims to safeguard the country's ecological and environmental security.
• The Act provides for the creation of protected areas such as national parks and wildlife sanctuaries.
• It also lays down the legal framework for the protection and management of wildlife habitats.
• Conservation of Bengal Tigers
• Bengal tigers (Panthera tigris tigris) are native to the Indian subcontinent and are classified as Endangered on the IUCN Red List.
• Efforts to conserve Bengal tigers include anti-poaching measures, habitat restoration, and community involvement.
• India is home to more than 70% of the world's wild tiger population.
• Effective conservation strategies have led to an increase in the tiger population in recent years.

The correct answer is Decomposition of calcium carbonate.

Key Points

• The decomposition of calcium carbonate (CaCO₃) results in the formation of calcium oxide (CaO) and carbon dioxide (CO₂).
• This reaction is represented by the equation: CaCO₃ → CaO + CO₂.
• The oxidation states of the elements do not change during this decomposition reaction, indicating it is not a redox reaction.
• In contrast, decomposition reactions that are redox reactions involve changes in the oxidation states of the participating elements.

Additional Information

• Decomposition Reactions
• A decomposition reaction is a type of chemical reaction in which a single compound breaks down into two or more simpler compounds or elements.
• These reactions often require an input of energy in the form of heat, light, or electricity.
• Examples include the decomposition of water into hydrogen and oxygen (2H₂O → 2H₂ + O₂) and the decomposition of potassium chlorate into potassium chloride and oxygen (2KClO₃ → 2KCl + 3O₂).
• Redox Reactions
• Redox reactions, or oxidation-reduction reactions, involve the transfer of electrons between two substances.
• Oxidation refers to the loss of electrons, while reduction refers to the gain of electrons.
• In a redox reaction, one substance is oxidized (loses electrons) and another is reduced (gains electrons).
• An example is the decomposition of sodium hydride (NaH), where NaH decomposes into sodium metal (Na) and hydrogen gas (H₂), with sodium being reduced and hydrogen being oxidized.
• Oxidation States
• The oxidation state (or number) of an element in a compound is a measure of the degree of oxidation of that element.
• It is defined as the charge an atom would have if all bonds were completely ionic.
• Changes in oxidation states during a reaction indicate a redox process.

The correct answer is Mumbai.

Key Points

• The Board of Control for Cricket in India (BCCI) is the national governing body for cricket in India.
• The BCCI headquarters is located at the Wankhede Stadium in Mumbai, Maharashtra.
• Mumbai is known as the cricket capital of India, having produced many legendary cricketers.
• The BCCI was formed in December 1928 and is affiliated with the International Cricket Council (ICC).

Additional Information

• BCCI (Board of Control for Cricket in India)
• The BCCI is the wealthiest cricket board in the world, managing the Indian cricket team and organizing domestic and international cricket matches in India.
• It oversees cricketing activities in India and has the authority to select players, umpires, and officials to participate in international cricket events.
• The BCCI also organizes the Indian Premier League (IPL), one of the world's most popular and lucrative T20 cricket leagues.
• It is responsible for developing cricket infrastructure and promoting the sport at all levels in the country.
• Wankhede Stadium
• Wankhede Stadium, located in Mumbai, is one of the most iconic cricket stadiums in India.
• The stadium has hosted numerous memorable cricket matches, including the 2011 ICC Cricket World Cup final, where India emerged victorious.
• It has a seating capacity of around 33,000 spectators and serves as the home ground for the Mumbai cricket team and the Mumbai Indians (IPL team).
• Named after Sheshrao Krishnarao Wankhede, a politician and the then President of the Mumbai Cricket Association, the stadium was established in 1974.
• International Cricket Council (ICC)
• The ICC is the global governing body for cricket, responsible for overseeing and regulating the international game.
• Headquartered in Dubai, United Arab Emirates, the ICC organizes major international cricket tournaments like the ICC Cricket World Cup, ICC T20 World Cup, and ICC Champions Trophy.
• The ICC ensures that the rules and regulations of cricket are uniformly followed by its member countries.
• It also appoints match officials for international matches and works towards the development and promotion of cricket worldwide.

The correct answer is MK Stalin.

Key Points

• M.K. Stalin is the current Chief Minister of Tamil Nadu, having assumed office on May 7, 2021.
• He is the son of the former Chief Minister of Tamil Nadu, M. Karunanidhi.
• Stalin is a prominent leader of the Dravida Munnetra Kazhagam (DMK) party.
• Under his leadership, the DMK won the 2021 Tamil Nadu Legislative Assembly election, securing a majority of the seats.

Additional Information

• Dravida Munnetra Kazhagam (DMK)
• DMK is a major political party in the state of Tamil Nadu and the union territory of Puducherry.
• It was founded by C.N. Annadurai in 1949 as a breakaway faction from the Dravidar Kazhagam (DK) led by Periyar E.V. Ramasamy.
• The party's ideologies are centered on social justice, self-respect, and Tamil nationalism.
• DMK has been a dominant force in Tamil Nadu politics, having ruled the state multiple times since its inception.
• M.K. Stalin's Political Career
• M.K. Stalin began his political career at an early age and has held various positions within the DMK party.
• He has served as the Mayor of Chennai and as the Deputy Chief Minister of Tamil Nadu.
• Stalin was appointed as the Working President of DMK in 2017, following the declining health of his father, M. Karunanidhi.
• He played a pivotal role in leading the party to victory in the 2021 Tamil Nadu Legislative Assembly election.
• Tamil Nadu Legislative Assembly
• The Tamil Nadu Legislative Assembly is the unicameral legislature of the state of Tamil Nadu in India.
• It consists of 234 members who are elected by the people of Tamil Nadu through direct elections.
• The assembly is responsible for making laws on state subjects and has the power to pass budgets and financial bills.
• It plays a crucial role in the governance and administration of the state.
• 2021 Tamil Nadu Legislative Assembly Election
• The 2021 Tamil Nadu Legislative Assembly election was held on April 6, 2021.
• The DMK-led alliance won a decisive victory, securing 159 out of 234 seats.
• The AIADMK, which was the ruling party at the time, managed to win only 75 seats.
• This election marked the return of DMK to power after a decade in opposition.

The correct answer is Rudradaman I.

Key Points

• Rudradaman I was a ruler of the Western Kshatrapas, a dynasty of Saka rulers who controlled the region of western India.
• The Sudarshana lake was a significant water reservoir in the region, and its maintenance and improvements were important for agriculture and the local economy.
• Rudradaman I is known for his achievements in infrastructure, including the repair and maintenance of the Sudarshana lake, which is recorded in the Junagarh inscription.
• He was one of the most prominent rulers of the Western Kshatrapas and is known for his military prowess and administrative skills.

Additional Information

• Western Kshatrapas
• The Western Kshatrapas were a dynasty of Saka rulers who ruled over the western and central parts of India, including Gujarat, Maharashtra, Rajasthan, and Madhya Pradesh.
• The term "Kshatrapa" refers to "satraps" or provincial governors of the ancient Persian Achaemenid Empire, who later became independent rulers in the region.
• The Western Kshatrapas are known for their contributions to trade, commerce, and infrastructure development in their territories.
• They issued a large number of coins, which provide valuable information about their reigns and the socio-economic conditions of their times.
• Sudarshana Lake
• The Sudarshana lake was originally built during the reign of the Mauryan emperor Chandragupta Maurya and was later maintained and improved by various rulers, including Rudradaman I.
• This lake was a significant source of irrigation and played a crucial role in supporting agriculture in the region.
• The lake's maintenance and expansion are detailed in the Junagarh inscription, which highlights the importance of water management and infrastructure in ancient India.
• The lake's name, "Sudarshana," means "beautiful" or "pleasant to look at," reflecting its significance and appeal.
• Junagarh Inscription
• The Junagarh inscription is a rock edict located in the Girnar hills near Junagarh in the state of Gujarat, India.
• It records the achievements of Rudradaman I, particularly his efforts in repairing and maintaining the Sudarshana lake after it was damaged by floods.
• The inscription is written in Sanskrit and is one of the earliest known examples of classical Sanskrit literature in Indian epigraphy.
• It provides valuable historical information about the region, the rulers, and their administrative capabilities.

Shortcut Trick

The perimeter of a triangle is the sum of its three sides.

If two triangles each have two equal corresponding sides (say a and b) and their total perimeters (P) are equal, the third side can be found directly.

Third side of 1^st Triangle = P − (a + b)

Third side of 2^nd Triangle = P − (a + b)

Since the third sides are also equal, all three corresponding sides of both triangles match.

Thus, the triangles are strictly congruent by the SSS (Side-Side-Side) rule.

∴ The correct answer is These have two equal sides and the same perimeter.

Alternate Method

a b c Perimeter = a + b + c

a b d Perimeter = a + b + d

Given:

Let the two equal corresponding sides of both triangles be a and b.

Let the unknown third side of the first triangle be c and the second triangle be d.

The perimeters of both triangles are given to be identical.

Calculations:

⇒ Perimeter of 1^st Triangle = a + b + c

⇒ Perimeter of 2^nd Triangle = a + b + d

Equating the two perimeters as given:

⇒ a + b + c = a + b + d

Subtracting (a + b) from both sides:

⇒ c = d

⇒ All three sides of the first triangle are precisely equal to the corresponding three sides of the second triangle (a = a, b = b, c = d).

⇒ Therefore, by the SSS (Side-Side-Side) congruence criterion, the two triangles are congruent.

Analyzing other options logically:

⇒ Same area and same base (or height): Area = 1/2 × base × height. Triangles with the same base and height will have the same area, but they can be of completely different shapes (e.g., one right-angled and one obtuse-angled). Hence, they are not necessarily congruent.

⇒ One side and one angle equal: This provides insufficient data. At least three specific matching components (like SAS, ASA, or AAS) are required to confirm congruence.

∴ The correct answer is These have two equal sides and the same perimeter.

Additional Information

Congruence Criteria for Triangles

Two triangles are strictly congruent if they satisfy any of these five standard rules: SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle), AAS (Angle-Angle-Side), or RHS (Right angle-Hypotenuse-Side).

Similarity vs Congruence

Congruent triangles have identical shape and size (ratio of corresponding sides is exactly 1:1). Similar triangles share the same shape but can differ in size (ratio of sides is equal but not necessarily 1). All congruent triangles are similar, but similar triangles are rarely congruent.

Base and Area Property

If two distinct triangles are situated on the same base and bounded between the same parallel lines, their areas will be exactly equal. However, their perimeter, side lengths, and interior angles can differ substantially.

Shortcut Trick

By using the rule of Alligation:

Ratio of Boys to Girls = (Average of Girls − Overall Average) : (Overall Average − Average of Boys)

Ratio = (80 − 78) : (78 − 75)

Ratio = 2 : 3

∴ The correct answer is 2 : 3.

Alternate Method

Given:

Average marks of boys = 75

Average marks of girls = 80

Overall average marks = 78

Alligation Method

Boys

75

Girls

80

Overall

78

2

3

Ratio = 2 : 3

(80 − 78) (78 − 75)

Formula Used:

Overall Average = (Total marks of boys + Total marks of girls) ÷ Total number of students

Calculation:

Let the number of boys be x and the number of girls be y.

⇒ Total marks of boys = 75 × x

⇒ Total marks of girls = 80 × y

⇒ Total marks of all students = 78 × (x + y)

⇒ 75x + 80y = 78(x + y)

⇒ 75x + 80y = 78x + 78y

⇒ 80y − 78y = 78x − 75x

⇒ 2y = 3x

⇒ x/y = 2/3

∴ The correct answer is 2 : 3.

Additional Information

Alligation Rule

It is a method used to easily find the ratio in which two or more items or groups at given averages/prices must be mixed to produce a desired overall average.

Weighted Average

When combining two groups with different individual averages, the combined overall average is weighted by the respective number of quantities in each group: Average = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Ratio & Proportion

A ratio establishes a mathematical relationship between two quantities of the same kind, represented as a fraction. If a : b = x : y, then a/b = x/y.

Shortcut Trick

In an equilateral triangle, all three sides are equal.

Sum of two sides = 16 cm

One side = 16 ÷ 2 = 8 cm

Hence, the third side is also 8 cm.

∴ The correct answer is 8 cm.

Alternate Method

A B C a a a a + a = 16 cm

Given:

The sum of two sides of an equilateral triangle = 16 cm

Property Used:

An equilateral triangle has all three sides equal in length. Let the length of each side be a.

Calculation:

⇒ a + a = 16

⇒ 2a = 16

⇒ a = 16 ÷ 2

⇒ a = 8 cm

Since all sides are equal, the third side is also equal to a.

⇒ Third side = 8 cm

∴ The correct answer is 8 cm.

Additional Information

Area of Equilateral Triangle

The area of an equilateral triangle with side a is given by Area = (√3 ÷ 4) × a².

Height of Equilateral Triangle

The altitude or height is calculated as Height = (√3 ÷ 2) × a.

Perimeter of Equilateral Triangle

The perimeter is simply the sum of all sides, Perimeter = 3 × a.

Inradius and Circumradius

Inradius (r) = a ÷ (2√3) and Circumradius (R) = a ÷ √3.

Given:

Data on candidates for centres P, Q, and R across years 2016 to 2020, with total and present candidates provided.

Formula used:

Absentees = Total - Present

Calculation:

For each year, calculate total absentees for all centres and identify the second highest.

2016:

⇒ P = 45 - 42 = 3

⇒ Q = 56 - 45 = 11

⇒ R = 55 - 50 = 5

Total Absentees = 3 + 11 + 5 = 19

2017:

⇒ P = 50 - 45 = 5

⇒ Q = 75 - 55 = 20

⇒ R = 45 - 40 = 5

Total Absentees = 5 + 20 + 5 = 30

2018:

⇒ P = 64 - 60 = 4

⇒ Q = 65 - 45 = 20

⇒ R = 65 - 55 = 10

Total Absentees = 4 + 20 + 10 = 34

2019:

⇒ P = 80 - 69 = 11

⇒ Q = 84 - 72 = 12

⇒ R = 70 - 62 = 8

Total Absentees = 11 + 12 + 8 = 31

2020:

⇒ P = 55 - 44 = 11

⇒ Q = 80 - 66 = 14

⇒ R = 90 - 85 = 5

Total Absentees = 11 + 14 + 5 = 30

The absentee totals by year are: 19 (2016), 30 (2017), 34 (2018), 31 (2019), 30 (2020).

∴ The second highest absentee total is 31 in 2019.

Shortcut Trick

For a number to be divisible by 6, it must be divisible by both 2 and 3.

Sum of the digits of 217924k = 2 + 1 + 7 + 9 + 2 + 4 + k = 25 + k.

To be divisible by 3, the sum (25 + k) must be a multiple of 3.

Checking the options (4, 6, 2, 0), only 25 + 2 = 27 is exactly divisible by 3.

Since the unit digit 2 is even, it is also divisible by 2.

∴ The correct answer is k = 2.

Alternate Method

Given:

Number = 217924k

The number is strictly divisible by 6.

Concept Used:

Divisibility rule of 6: A number must be completely divisible by both 2 and 3.

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

Divisibility rule of 3: The sum of all individual digits of the number must be divisible by 3.

Calculations:

⇒ Sum of all the digits = 2 + 1 + 7 + 9 + 2 + 4 + k

⇒ Sum of the digits = 25 + k

⇒ For the number to be divisible by 3, the value of (25 + k) must be exactly divisible by 3.

⇒ Let us substitute and check the given options for k:

⇒ Option 1: k = 4 ⇒ Sum = 25 + 4 = 29 (Not divisible by 3)

⇒ Option 2: k = 6 ⇒ Sum = 25 + 6 = 31 (Not divisible by 3)

⇒ Option 3: k = 2 ⇒ Sum = 25 + 2 = 27 (Divisible by 3)

⇒ Option 4: k = 0 ⇒ Sum = 25 + 0 = 25 (Not divisible by 3)

⇒ The only value from the options that satisfies the divisibility condition for 3 is k = 2.

⇒ Furthermore, because 2 is an even number, it inherently satisfies the divisibility rule of 2.

∴ The correct answer is k = 2.

Additional Information

Divisibility Rule of 4

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

Divisibility Rule of 8

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

Divisibility Rule of 9

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

Divisibility Rule of 11

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

Identify Highest Scores in Each Subject:

From the table:

Maths: Highest score is 100% of 150 marks

Chemistry: Highest score is 80% of 130 marks

Physics: Highest score is 90% of 120 marks

Geography: Highest score is 95% of 100 marks

History: Highest score is 90% of 60 marks

Computer Science: Highest score is 90% of 40 marks

Calculate the Marks for Each Subject Based on Percentage:

Maths: 150 × 100% = 150 marks

Chemistry: 130 × 80% = 104 marks

Physics: 120 × 90% = 108 marks

Geography: 100 × 95% = 95 marks

History: 60 × 90% = 54 marks

Computer Science: 40 × 90% = 36 marks

Calculate the total obtained marks: 150 + 104 + 108 + 95 + 54 + 36 = 547

Calculate the total maximum possible marks: 150 + 130 + 120 + 100 + 60 + 40 = 600

Calculate the overall percentage:

Percentage = (547 / 600) × 100 = $91 \frac{1}{6}\%$

The exact overall percentage score obtained by that student would be $91 \frac{1}{6}\%$

Shortcut Trick

Use the complementary angle identity: sin² θ + sin² (90° − θ) = 1.

Group the complementary angle terms: −(sin² 40° + sin² 50°) − (sin² 35° + sin² 55°) = −1 − 1 = −2.

Sum the standard angle values: sin² 30° + sin² 45° + sin² 45° + sin² 60°.

⇒ (1/2)² + (1/√2)² + (1/√2)² + (√3/2)² = 1/4 + 1/2 + 1/2 + 3/4 = 2.

Total value = 2 − 2 = 0.

∴ The correct answer is 0.

Alternate Method

Given:

sin² 30° − sin² 40° + sin² 45° − sin² 55° − sin² 35° + sin² 45° − sin² 50° + sin² 60°

Formula Used:

sin (90° − θ) = cos θ

sin² θ + cos² θ = 1

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

Calculations:

⇒ Rearranging terms to group standard and complementary angles:

⇒ (sin² 30° + sin² 45° + sin² 45° + sin² 60°) − (sin² 40° + sin² 50°) − (sin² 35° + sin² 55°)

⇒ Substitute the standard trigonometric values:

⇒ Sum of standard values = (1/2)² + (1/√2)² + (1/√2)² + (√3/2)²

⇒ Sum of standard values = 1/4 + 1/2 + 1/2 + 3/4 = 2

⇒ For the complementary angles, convert sine to cosine:

⇒ sin 50° = sin (90° − 40°) = cos 40°

⇒ sin 55° = sin (90° − 35°) = cos 35°

⇒ Applying the Pythagorean identity sin² θ + cos² θ = 1:

⇒ − (sin² 40° + cos² 40°) = −1

⇒ − (sin² 35° + cos² 35°) = −1

⇒ Combining all the calculated parts together:

⇒ Total Expression Value = 2 − 1 − 1 = 0

∴ The correct answer is 0.

Additional Information

Complementary Angle Identities

Trigonometric functions of complementary angles satisfy: sin(90° − θ) = cos θ, tan(90° − θ) = cot θ, and sec(90° − θ) = cosec θ.

Pythagorean Trigonometric Identities

The foundational identities are sin² θ + cos² θ = 1, 1 + tan² θ = sec² θ, and 1 + cot² θ = cosec² θ.

Standard Trigonometric Values

Frequently used standard values include sin 30° = cos 60° = 1/2, sin 45° = cos 45° = 1/√2, and sin 60° = cos 30° = √3/2.

Shortcut Trick

If A + B = 90°, then cos²A + cos²B = 1.

Here, A = 29° and B = 61°.

Since 29° + 61° = 90°, the condition is satisfied.

⇒ cos²29° + cos²61° = 1

∴ The correct answer is 1.

Alternate Method

Given:

Expression: cos²29° + cos²61°

Formula Used:

cos(90° − θ) = sinθ

sin²θ + cos²θ = 1

Calculations:

⇒ cos61° = cos(90° − 29°)

⇒ cos61° = sin29°

Substitute this back into the given expression:

⇒ cos²29° + (sin29°)²

⇒ cos²29° + sin²29°

⇒ 1

∴ The correct answer is 1.

Additional Information

Complementary Angles

Two angles are complementary if their sum is 90°. The standard conversions are: sin(90° − θ) = cosθ, tan(90° − θ) = cotθ, and sec(90° − θ) = cosecθ.

Pythagorean Identities

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

Sum to 1 Properties

For any two angles A and B where A + B = 90°, we have the direct rules: sin²A + sin²B = 1, cos²A + cos²B = 1, tanA × tanB = 1, and cotA × cotB = 1.

Shortcut Trick

Use the direct ratio formula between Cost Price (CP) and Marked Price (MP):

CP : MP = (100 − Discount%) : (100 + Profit%)

CP : MP = (100 − 40) : (100 + 10) = 60 : 110 = 6 : 11

Given MP = 11 units = ₹550 → 1 unit = ₹50

CP = 6 units = 6 × 50 = ₹300

∴ The correct answer is ₹300.

Alternate Method

Given: Marked Price (MP) = ₹550, Discount = 40%, Profit = 10%

Formula: Selling Price (SP) = MP × (100 − Discount%)/100; Cost Price (CP) = SP × 100/(100 + Profit%)

CP

SP

MP

+10%

−40% MP = ₹550 SP = ₹330 CP = ?

⇒ Selling Price (SP) = 550 − (40% of 550)

⇒ SP = 550 − (0.40 × 550)

⇒ SP = 550 − 220 = ₹330

⇒ SP is 10% more than CP → SP = 1.10 × CP

⇒ 330 = 1.10 × CP

⇒ CP = 330 ÷ 1.1 = ₹300

∴ The correct answer is ₹300.

Additional Information

Marked Price and Selling Price

Marked Price (MP) is the tag price, and the discount is always calculated on this value to reach the Selling Price (SP = MP − Discount).

Effective Percentage Method

If two successive discounts d₁% and d₂% are given, the single equivalent discount is (d₁ + d₂ − (d₁×d₂)/100)%.

Mark-up Percentage

Mark-up is the increase over the CP to set the MP. Mark-up % = [(MP − CP) ÷ CP] × 100.

Given:

Minimum marks required to pass in History = 105

Maximum marks in History = 175

Calculation for each student:

Ram:

History percentage = 54%

Marks in History = (54 / 100) × 175 = 94.5 (rounded to 95)

Result: Ram does not pass, as 95 < 105.

Mohan:

History percentage = 51%

Marks in History = (51 / 100) × 175 = 89.25 (rounded to 89)

Result: Mohan does not pass, as 89 < 105.

Shyam:

History percentage = 75%

Marks in History = (75 / 100) × 175 = 131.25 (rounded to 131)

Result: Shyam passes, as 131 > 105.

Conclusion:

Only Shyam passes in History, so the number of students who pass the exam in History is 1.

Shortcut Trick

Since fill pipe P is 21 times faster than Q, the efficiency ratio of P to Q is 21 : 1.

Total capacity of the cistern = Efficiency of Q × Time taken by Q = 1 × 110 = 110 units.

Combined efficiency of P and Q working together = 21 + 1 = 22 units/min.

Time taken when both pipes are open = Total capacity ÷ Combined efficiency = 110 ÷ 22 = 5 minutes.

∴ The correct answer is 5 minutes.

Alternate Method

Given:

Time taken by Q to fill the cistern = 110 minutes.

Efficiency of P is 21 times the efficiency of Q.

Method Used:

Let efficiency of Q = 1 unit/min.

Then, efficiency of P = 21 units/min.

Time taken by P to fill the cistern alone = 110 ÷ 21 minutes.

⇒ Combined efficiency of P and Q = 21 + 1 = 22 units/min.

⇒ Time taken by both pipes together = Total Work ÷ Combined Efficiency

⇒ Time taken = 110 ÷ 22 = 5 minutes.

∴ The correct answer is 5 minutes.

Additional Information

Inlet and Outlet Pipes

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

Combined Work Formula

If Pipe A fills a tank in x hours and Pipe B fills it in y hours, the time taken to fill the tank when both are opened together is given by (x × y) ÷ (x + y).

Work and Efficiency Relation

Time taken to complete a certain amount of work is inversely proportional to the efficiency of the worker or pipe. Total Work = Time × Efficiency.

Shortcut Trick

We know that for a cone, V ∝ r² × h

⇒ Ratio of heights (h₁/h₂) = (V₁/V₂) × (r₂/r₁)²

⇒ h₁/h₂ = (3/5) × (5/2)²

⇒ h₁/h₂ = (3/5) × (25/4) = 75/20 = 15/4

∴ The correct answer is 15 : 4.

Alternate Method

Given:

Ratio of radii of two cones (r₁ : r₂) = 2 : 5

Ratio of volumes of two cones (V₁ : V₂) = 3 : 5

Formula Used:

Volume of a cone = 1/3 × π × r² × h

Calculations:

⇒ Let the radii of the two cones be 2r and 5r respectively.

⇒ Let their heights be h₁ and h₂ respectively.

⇒ Volume of the first cone (V₁) = 1/3 × π × (2r)² × h₁ = 1/3 × π × 4r² × h₁

⇒ Volume of the second cone (V₂) = 1/3 × π × (5r)² × h₂ = 1/3 × π × 25r² × h₂

⇒ Ratio of their volumes = V₁ / V₂ = 3 / 5

⇒ (1/3 × π × 4r² × h₁) / (1/3 × π × 25r² × h₂) = 3 / 5

⇒ (4 × h₁) / (25 × h₂) = 3 / 5

⇒ h₁ / h₂ = (3 × 25) / (5 × 4)

⇒ h₁ / h₂ = 75 / 20 = 15 / 4

⇒ The ratio of their heights is 15 : 4.

∴ The correct answer is 15 : 4.

Additional Information

Volume of a Cone

The amount of space occupied by a cone. Formula: V = 1/3 × π × r² × h.

Curved Surface Area (CSA)

The area of the curved surface excluding the base. Formula: CSA = π × r × l, where l = √(r² + h²).

Total Surface Area (TSA)

The sum of the curved surface area and the base area. Formula: TSA = π × r × (r + l).

Frustum of a Cone

The lower part of a cone cut by a parallel plane. Volume = 1/3 × π × h × (R² + r² + R × r).

Shortcut Trick

Recognize the numerator x³ + 15x² + 75x + 125 as the perfect cube (x + 5)³.

Recognize the denominator x² − 25 as the difference of squares (x − 5)(x + 5).

Expression becomes: [ (x + 5)³ ÷ ((x − 5)(x + 5)) ] × (x − 5).

Canceling common terms (x − 5) and (x + 5) yields (x + 5)².

Expanding (x + 5)² gives x² + 10x + 25.

∴ The correct answer is x² + 10x + 25.

Alternate Method

Given:

Expression: (x³ + 15x² + 75x + 125) / (x² − 25) × (x − 5)

Formula Used:

1. (a + b)³ = a³ + 3a²b + 3ab² + b³
2. a² − b² = (a − b)(a + b)
3. (a + b)² = a² + 2ab + b²

Calculations:

⇒ Let us factorise the numerator: x³ + 15x² + 75x + 125

⇒ Numerator = (x)³ + 3(x)²(5) + 3(x)(5)² + (5)³ = (x + 5)³

⇒ Factorise the denominator: x² − 25 = (x)² − (5)² = (x − 5)(x + 5)

⇒ Substitute these into the given expression:

⇒ [ (x + 5)³ / ( (x − 5)(x + 5) ) ] × (x − 5)

⇒ Cancel out the common factor (x − 5) from the numerator multiplier and denominator.

⇒ The expression simplifies to: (x + 5)³ / (x + 5)

⇒ Cancel out one (x + 5) term to get: (x + 5)²

⇒ Expand using the formula (a + b)²:

⇒ (x + 5)² = x² + 2(x)(5) + 5² = x² + 10x + 25

∴ The correct answer is x² + 10x + 25.

Additional Information

Cubic Identities

Formulas for the sum and difference of cubes: a³ + b³ = (a + b)(a² − ab + b²) and a³ − b³ = (a − b)(a² + ab + b²).

Perfect Cube Expansions

Expanded forms: (a + b)³ = a³ + b³ + 3ab(a + b) and (a − b)³ = a³ − b³ − 3ab(a − b).

Simplification of Rational Expressions

Always factorise both the numerator and the denominator into their simplest forms before attempting to cancel common terms.

Shortcut Trick Solve the given expression directly using the BODMAS rule: ⇒ (11k − 9) ÷ (36 − 35 + k2) = 1 ⇒ (11k − 9) = (1 + k2) ⇒ k2 − 11k + 10 = 0 ⇒ (k − 10)(k − 1) = 0 ⇒ k = 1, 10 ∴ The correct answer is 1, 10.

Alternate Method Given: (88 ÷ 8 × k − 3 × 3) / (62 − 7 × 5 + k2) = 1 Method Used: According to the BODMAS rule, perform division first, then multiplication, addition, and subtraction sequentially. Calculations: ⇒ Step 1: Simplify the Numerator ⇒ Numerator = 88 ÷ 8 × k − 3 × 3 ⇒ Numerator = 11 × k − 9 = 11k − 9 ⇒ Step 2: Simplify the Denominator ⇒ Denominator = 62 − 7 × 5 + k2 ⇒ Denominator = 36 − 35 + k2 = 1 + k2 ⇒ Step 3: Substitute both simplified values back into the equation ⇒ (11k − 9) / (1 + k2) = 1 ⇒ 11k − 9 = 1 + k2 ⇒ k2 − 11k + 10 = 0 ⇒ k2 − 10k − k + 10 = 0 ⇒ k(k − 10) − 1(k − 10) = 0 ⇒ (k − 1)(k − 10) = 0 ⇒ k = 1 or k = 10 ∴ The correct answer is 1, 10.

Additional Information BODMAS Rule It dictates the correct order of mathematical operations: Brackets, Orders (powers/roots), Division, Multiplication, Addition, and Subtraction. Quadratic Equation Roots For the standard quadratic equation form ax2 + bx + c = 0, the Sum of the roots = −b/a and the Product of the roots = c/a. Nature of Roots The nature of roots is determined by the Discriminant (D = b2 − 4ac). If D > 0, roots are real and distinct. If D = 0, roots are real and equal. If D < 0, roots are imaginary.

Shortcut Trick

The half-lengths of the parallel chords are 10 ÷ 2 = 5 cm and 24 ÷ 2 = 12 cm.

Notice the common Pythagorean triplet (5, 12, 13) where 13 represents the hypotenuse (radius).

The distances from the centre for these chords would be 12 cm and 5 cm.

Difference in perpendicular distances = 12 − 5 = 7 cm, which perfectly matches the given PQ.

Thus, the radius is 13 cm, meaning the diameter is 2 × 13 = 26 cm.

∴ The correct answer is 26 cm.

Alternate Method

Given:

AB = 10 cm, CD = 24 cm, PQ = 7 cm

OP ⊥ AB and OQ ⊥ CD

O Q P C D A B 12 cm 5 cm 7 cm x r r

Formula Used:

A perpendicular from the centre to a chord bisects the chord into two equal halves.

Pythagoras Theorem: Hypotenuse² = Base² + Perpendicular²

Calculations:

⇒ AP = AB ÷ 2 = 10 ÷ 2 = 5 cm

⇒ CQ = CD ÷ 2 = 24 ÷ 2 = 12 cm

⇒ Let the radius of the circle be r and the distance OQ = x.

⇒ Since chords are on the same side of the centre, OP = OQ + PQ = x + 7

⇒ In right-angled ΔOQC: r² = OQ² + CQ²

⇒ r² = x² + 12² = x² + 144  --- (i)

⇒ In right-angled ΔOPA: r² = OP² + AP²

⇒ r² = (x + 7)² + 5² = x² + 14x + 49 + 25 = x² + 14x + 74  --- (ii)

⇒ Equating (i) and (ii): x² + 144 = x² + 14x + 74

⇒ 14x = 144 − 74 = 70

⇒ x = 5 cm

⇒ Substitute the value of x in (i):

⇒ r² = 5² + 144 = 25 + 144 = 169

⇒ r = √169 = 13 cm

⇒ Diameter of the circle = 2 × 13 = 26 cm

∴ The correct answer is 26 cm.

Additional Information

Perpendicular from Centre to Chord

A perpendicular drawn from the centre of a circle to any chord bisects the chord into two equal halves. This forms a right-angled triangle where the radius is the hypotenuse.

Distance Between Parallel Chords

If two parallel chords lie on the same side of the centre, the distance between them is the difference of their perpendicular distances from the centre (d = |d₁ − d₂|). If they lie on opposite sides, the distance is the sum (d = d₁ + d₂).

Pythagorean Triplets in Circle Geometry

Commonly repeating triplets like (3, 4, 5) or (5, 12, 13) or (8, 15, 17) very frequently satisfy the values of the radius, half-chord, and perpendicular distance. Spotting them saves calculation time.

Shortcut Trick

Given expression: (sec²θ − 1) × cot²θ × (1 + cot²θ)

Using standard trigonometric identities: sec²θ − 1 = tan²θ and 1 + cot²θ = cosec²θ

⇒ Expression = tan²θ × cot²θ × cosec²θ

Since tanθ × cotθ = 1, the expression simplifies to: 1 × cosec²θ = cosec²θ

⇒ (5/3)² = 25/9

∴ The correct answer is 25/9.

Alternate Method

Given: cosecθ = 5/3

Since cosecθ = Hypotenuse ÷ Perpendicular, let Hypotenuse = 5 and Perpendicular = 3.

By Pythagoras theorem: Base = √(5² − 3²) = √16 = 4.

From the right-angled triangle, secθ = Hypotenuse ÷ Base = 5/4

And cotθ = Base ÷ Perpendicular = 4/3

Substitute these values into the given expression: (sec²θ − 1) × cot²θ × (1 + cot²θ)

⇒ ((5/4)² − 1) × (4/3)² × (1 + (4/3)²)

⇒ (25/16 − 1) × (16/9) × (1 + 16/9)

⇒ (9/16) × (16/9) × (25/9)

⇒ 1 × (25/9) = 25/9

∴ The correct answer is 25/9.

Additional Information

Pythagorean Identities

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

Reciprocal Identities

Trigonometric functions are related by their reciprocals: sinθ × cosecθ = 1, cosθ × secθ = 1, and tanθ × cotθ = 1.

Quotient Identities

Tangent and cotangent ratios can be expressed as: tanθ = sinθ ÷ cosθ and cotθ = cosθ ÷ sinθ.

Shortcut Trick Length of Direct Common Tangent (DCT) = √[d2 − (R − r)2] Given: Distance (d) = 37 cm, Radii R = 22 cm, r = 10 cm ⇒ DCT = √[372 − (22 − 10)2] ⇒ DCT = √[372 − 122] = √(1369 − 144) ⇒ DCT = √1225 = 35 cm ∴ The correct answer is 35 cm.

Alternate Method

M Q O1 O2 P

22 cm 10 cm 37 cm 12 cm MQ = ?

Given: ⇒ Radius of the larger circle (R) = 22 cm ⇒ Radius of the smaller circle (r) = 10 cm ⇒ Distance between centres (d) = 37 cm

Formula Used: ⇒ Length of Direct Common Tangent (MQ) = √[d2 − (R − r)2]

Calculation: ⇒ MQ = √[372 − (22 − 10)2] ⇒ MQ = √[372 − 122] Using the identity a2 − b2 = (a − b)(a + b): ⇒ MQ = √[(37 − 12) × (37 + 12)] ⇒ MQ = √(25 × 49) ⇒ MQ = 5 × 7 = 35 cm ∴ The correct answer is 35 cm.

Additional Information Transverse Common Tangent (TCT) The length of the transverse common tangent is given by the formula: TCT = √[d2 − (R + r)2], where the tangent line crosses the line joining the centres. Number of Common Tangents If d > R + r, there are exactly 4 common tangents. If d = R + r, there are 3 common tangents. If R − r < d < R + r, there are exactly 2 common tangents. If d = R − r, there is only 1 common tangent.

Shortcut Trick

When two persons run in the same direction, their relative speed is the difference of their speeds.

Relative speed = 12 − 6 = 6 m/s.

Time to meet next = Track length ÷ Relative speed.

Time = 840 ÷ 6 = 140 seconds.

∴ The correct answer is 140 s.

Alternate Method

Given:

⇒ Total length of the circular track = 840 m

⇒ Speed of runner A = 6 m/s

⇒ Speed of runner B = 12 m/s

Formula Used:

⇒ Time taken to meet = Distance ÷ Relative Speed

Calculation:

⇒ Since both A and B are running in the same direction, their relative speed will be the difference between their speeds.

⇒ Relative Speed = Speed of B − Speed of A

⇒ Relative Speed = 12 − 6 = 6 m/s

⇒ Time taken to meet next = 840 ÷ 6

⇒ Time = 140 s

∴ The correct answer is 140 s.

Additional Information

Meeting Anywhere on Track (First Time)

Time = Track Length ÷ Relative Speed. If moving in the same direction, RS = |S₁ − S₂|. If opposite, RS = S₁ + S₂.

Meeting at the Starting Point (First Time)

Time = LCM of the individual times taken to complete one full round = LCM(L÷S₁, L÷S₂).

Number of Distinct Meeting Points

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

Shortcut Trick

Assume all 40 coins are of the lowest denomination (₹1), so total value = 40 × 1 = ₹40.

Remaining value to be adjusted = 100 − 40 = ₹60.

A ₹5 coin contributes ₹4 extra and a ₹2 coin contributes ₹1 extra over the ₹1 base.

Thus, 4 × (Number of ₹5 coins) + 1 × (Number of ₹2 coins) = 60.

From the options, only 13 gives a valid value (4 × 13 = 52, which is ≤ 60). Other options exceed 60.

∴ The correct answer is 13 coins.

Alternate Method

Given:

Total amount paid = ₹100

Total number of coins = 40

Available denominations = ₹5, ₹2, and ₹1

Calculations:

Let the number of ₹5, ₹2, and ₹1 coins be x, y, and z respectively.

⇒ Total coins: x + y + z = 40 .... (Equation 1)

⇒ Total value: 5x + 2y + z = 100 .... (Equation 2)

⇒ Subtract Equation 1 from Equation 2:

⇒ (5x + 2y + z) − (x + y + z) = 100 − 40

⇒ 4x + y = 60

⇒ y = 60 − 4x

Since the number of coins cannot be negative, y ≥ 0.

⇒ 60 − 4x ≥ 0

⇒ 4x ≤ 60

⇒ x ≤ 15

Now, checking the given options for the possible value of x (₹5 coins):

⇒ Checking 16: Not possible, as x ≤ 15

⇒ Checking 17: Not possible

⇒ Checking 18: Not possible

⇒ Checking 13: Possible

Verification for x = 13:

⇒ y = 60 − 4 × 13 = 60 − 52 = 8

⇒ z = 40 − (13 + 8) = 40 − 21 = 19

⇒ Total value = (5 × 13) + (2 × 8) + (1 × 19) = 65 + 16 + 19 = ₹100 (Satisfies condition)

∴ The correct answer is 13 coins.

Additional Information

Linear Equations in Two Variables

When checking for integer solutions in linear equations (Diophantine equations), using boundary conditions (such as making sure the number of items is ≥ 0) helps eliminate incorrect options quickly.

Denomination Problems

In coin-based questions, always form two distinct equations: one representing the physical count of coins and the other representing their total financial value.

Deviation Method for Values

By assigning the minimum denomination value to all the items, the remaining residual value can be split among the higher denominations based on their difference from the base minimum value.

Shortcut Trick

Winner gets 59.375% (19/32) and loser gets 40.625% (13/32) of valid votes.

Difference = 19/32 − 13/32 = 3/16 of valid votes = 2484.

Total valid votes = 2484 × 16/3 = 13248.

Total valid votes = Total Voters × (100 − y)% × 90%.

16000 × ((100 − y)/100) × (90/100) = 13248 ⇒ 144 × (100 − y) = 13248 ⇒ 100 − y = 92 ⇒ y = 8.

∴ The correct answer is y = 8.

Alternate Method

⇒ Let total eligible voters = 16,000

⇒ Total votes cast = 16,000 × ((100 − y)/100)

⇒ Total valid votes = 16,000 × ((100 − y)/100) × (90/100) = 144 × (100 − y)

⇒ Winner's share of valid votes = 59.375% = 59375/100000 = 19/32

⇒ Loser's share of valid votes = 100% − 59.375% = 40.625% = 13/32

⇒ Majority difference = 19/32 − 13/32 = 6/32 = 3/16

⇒ Majority votes = (3/16) × Valid votes = 2484

⇒ Valid votes = 2484 × (16/3) = 828 × 16 = 13,248

⇒ Equating the valid votes: 144 × (100 − y) = 13,248

⇒ 100 − y = 13,248 ÷ 144

⇒ 100 − y = 92

⇒ y = 8

∴ The correct answer is y = 8.

Additional Information

Valid vs Total Votes

In election problems, always calculate winning margins based on the correct base specified in the question. Total valid votes = Total votes cast − Invalid votes.

Successive Percentage Change

When consecutive filtering steps occur (e.g., people not voting, then invalid votes), chain the multiplying factors: Final = Initial × (1 − R₁/100) × (1 − R₂/100).

Fraction to Percentage Conversions

Memorizing eighths and sixteenths speeds up calculations: 1/16 = 6.25%, 3/16 = 18.75%, and 1/32 = 3.125%. Thus, 59.375% = 50% + 9.375% = 1/2 + 3/32 = 19/32.

Shortcut Trick

The final amount to be paid can be found by subtracting the future value of the part payment from the total future value of the original loan.

Total amount for 2 years (if no payment was made) = 422092 × (1 + 40%) = 422092 × 1.4 = ₹590928.8

Value of the 1^st year repayment at the end of the 2^nd year = 21679 × (1 + 20%) = 21679 × 1.2 = ₹26014.8

Remaining amount to be paid = 590928.8 − 26014.8 = ₹564914

∴ The correct answer is ₹564914.

Alternate Method

Given:

Principal (P) = ₹4,22,092

Rate (R) = 20% per annum simple interest

Principal repaid at the end of Year 1 = ₹21,679

Year 0 P = ₹4,22,092

Year 1 Paid: ₹21,679 Int 1: ₹84,418.4

Year 2 Final Dues Int 2: ₹80,082.6

20% on ₹4,22,092

20% on ₹4,00,413

New Principal = 4,22,092 − 21,679 = ₹4,00,413 Total Dues = New P + Int 1 + Int 2 = ₹5,64,914

Calculations:

⇒ Interest accrued in the 1^st year = (P × R × T) ÷ 100

⇒ Interest for 1^st year = (422092 × 20 × 1) ÷ 100 = ₹84,418.4

⇒ At the end of Year 1, Mohan repays ₹21,679 towards the principal. The remaining principal becomes:

⇒ New Principal for 2^nd year = 422092 − 21679 = ₹4,00,413

⇒ Interest accrued in the 2^nd year on the new principal = (400413 × 20 × 1) ÷ 100 = ₹80,082.6

⇒ The total amount to be paid at the end of the 2^nd year consists of the new principal and the accumulated unpaid interest for both years.

⇒ Total pending interest = 84418.4 + 80082.6 = ₹1,64,501

⇒ Total amount paid at the end of the 2^nd year = New Principal + Total Pending Interest

⇒ Total amount = 400413 + 164501 = ₹5,64,914

∴ The correct answer is ₹564914.

Additional Information

Simple Interest Formula

Simple interest is calculated exclusively on the principal amount. The formula is SI = (P × R × T) ÷ 100, where P is principal, R is rate, and T is time.

Merchant's Rule for Partial Payments

When partial payments are made, both the initial debt and the payments earn simple interest until the date of final settlement. Final Balance = Future Value of Debt − Future Value of Payments.

Difference Between Simple & Compound Interest

In Simple Interest, accumulated interest does not earn further interest. In Compound Interest, accrued interest is added to the principal to earn interest in subsequent periods.

Shortcut Trick

The given expression is in the form of (a² − b²) / (a + b).

We know the algebraic identity a² − b² = (a − b)(a + b).

Substituting x for a and 3 for b, the numerator breaks down to (x − 3)(x + 3).

Dividing by the denominator (x + 3), the common term cancels out entirely.

This leaves exactly (x − 3).

∴ The correct answer is x − 3.

Alternate Method

Given:

Expression: (x² − 9) / (x + 3)

Formula Used:

Difference of squares: a² − b² = (a − b)(a + b)

Calculations:

⇒ (x² − 9) / (x + 3)

⇒ (x² − 3²) / (x + 3)

⇒ Expand the numerator using the formula:

⇒ [(x − 3)(x + 3)] / (x + 3)

⇒ Cancel the common factor (x + 3) from both the numerator and the denominator:

⇒ x − 3

∴ The correct answer is x − 3.

Additional Information

Difference of Squares

This is one of the most frequently used identities in algebraic simplification: a² − b² = (a − b)(a + b).

Perfect Square Trinomials

For standard expansions: (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b².

Sum and Difference of Cubes

Higher power factorization formulas: a³ + b³ = (a + b)(a² − ab + b²) and a³ − b³ = (a − b)(a² + ab + b²).

Simplification Rules

Always factorize algebraic numerators and denominators to their simplest forms before attempting to cancel or cross-multiply terms.

Shortcut Trick

In the fraction 17/27, the denominator represents the total work and the numerator represents the completed work.

17 units of work take = 34 days

1 unit of work takes = 34 ÷ 17 = 2 days

Total work (27 units) takes = 27 × 2 = 54 days

∴ The correct answer is 54 days.

Alternate Method

Given:

Fraction of the order completed = 17/27

Time taken to complete this fraction of the order = 34 days

Formula Used:

Total Time = Time taken for partial work ÷ Fraction of work completed

Calculations:

⇒ Let the number of days required to complete the entire order be x days.

⇒ Complete order is considered as 1 whole unit of work.

⇒ As per the given data, 17/27 of the work is completed in 34 days.

⇒ (17/27) × x = 34

⇒ x = 34 × (27/17)

⇒ x = 2 × 27

⇒ x = 54 days

∴ The correct answer is 54 days.

Additional Information

Unitary Method

It involves finding the value of a single unit and then multiplying that unit value by the required number of units to find the total value.

Work and Time Relationship

Work done is directly proportional to the time taken (assuming constant efficiency). If a fraction of work W is completed in T days, the entire work is completed in T ÷ W days.

Fraction to Whole Conversion

The denominator of a fraction generally represents the total parts, and the numerator represents the considered parts. Equating these parts directly to the given values leads to much faster calculations.

Shortcut Trick

In dishonest dealer problems where goods are sold at cost price, the profit comes entirely from the false weight.

Profit % = [(True Weight − False Weight) ÷ False Weight] × 100

Here, True Weight = 1000 g and False Weight = 870 g.

Profit % = [(1000 − 870) ÷ 870] × 100

Profit % = (130 ÷ 870) × 100 = (1300 ÷ 87)% ≈ 14.94%

∴ The correct answer is 14.94%.

Alternate Method

Given:

Professed Selling Price = Cost Price

True weight for 1 kg = 1000 g

False weight used by the grocer = 870 g

Formula Used:

Profit % = [(Error) ÷ (True Value − Error)] × 100

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

Calculations:

⇒ Let the cost price of 1 g of rice be ₹1.

⇒ Selling Price charged to the customer (for 1000 g) = ₹1000.

⇒ Actual Cost Price incurred by the grocer (for providing 870 g) = ₹870.

⇒ Profit = Selling Price − Cost Price

⇒ Profit = 1000 − 870 = ₹130

⇒ Profit Percentage = (Profit ÷ Cost Price) × 100

⇒ Profit Percentage = (130 ÷ 870) × 100

⇒ Profit Percentage = (1300 ÷ 87)% ≈ 14.942%

⇒ Rounding off to two decimal places, we get 14.94%.

∴ The correct answer is 14.94%.

Additional Information

Dishonest Dealer (Selling at CP)

When a dealer professes to sell items at Cost Price but uses a false weight, his profit percentage is given strictly by the error in weight over the false weight given: Profit % = [Error ÷ (True Weight − Error)] × 100.

Dishonest Dealer (Selling at Profit/Loss)

If the dealer marks up his goods by x% and also uses a false weight, the effective profit is determined using a multiplying factor: Final SP/CP ratio = [(100 + x) / 100] × [True Weight / False Weight].

Fraud During Purchasing & Selling

If a dealer cheats by x% while buying (takes more) and y% while selling (gives less), his overall profit percentage is calculated effectively by taking the ratio of the quantities exchanged: [(100 + x) ÷ (100 − y) − 1] × 100.

Given:

Maths = 65

Physics = 80

Chemistry = 75

English = 85

Physical Education = 90

Formula used:

Total Marks = Maths + Physics + Chemistry + English + Physical Education

Percentage = $\frac{\text{Total Marks}}{\text{Maximum Marks}} \times 100$

Calculations:

Total Marks = 65 + 80 + 75 + 85 + 90

⇒ Total Marks = 395

Percentage = $\frac{395}{500} \times 100$

⇒ Percentage = 79%

∴ Tarun obtained 79% marks in all the subjects taken together.

The correct answer is Option 3.

 

Key Points

• In passive voice, the object of the active voice sentence becomes the subject.
• The verb "saw" in active voice is changed to "was seen" in passive voice.
• The subject of the active voice sentence "Ishika" becomes the agent in the passive voice sentence, introduced by the preposition "by".
• The location "in the forest" remains unchanged and is placed at the end of the sentence.

Therefore, the correct answer is Option 3.

 

Correct Sentence: "The tiger was seen by Ishika in the forest."

 

Additional Information

• Option 1: "The tiger saw by Ishika in the forest." – Incorrect structure; the verb form "saw" does not fit in passive voice.
• Option 2: "The tiger was seen by the forest in Ishika." – Incorrect order and context; "in Ishika" is incorrect placement.
• Option 4: "The tiger sees Ishika in the forest." – This is in active voice, not passive.

The correct answer is: Option 3.

Key Points

• The word "vouch" means to assert or confirm as a result of one's own experience that something is true or accurately so described. (गवाही देना)
• Example: I can vouch for his honesty; he has always been truthful with me.
• "Invalidate" refers to something that makes an argument, statement, or theory unsound or erroneous; to disprove or nullify. (अमान्य करना)
• Example: The discovery of new evidence might invalidate the previous theory.

Hence, we can infer that the opposite of 'vouch' is 'invalidate'.

Additional Information

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

• Maintain (बनाए रखना): To keep in an existing state; to preserve or retain.
• Example: It's essential to maintain a healthy lifestyle for overall well-being.
• Certify (प्रमाणित करना): To confirm or attest, typically in writing; to officially recognize as meeting certain standards.
• Example: The document was certified by a notary public.
• Witness (गवाह): Someone who sees an event and can report or give evidence about it.
• Example: She was the only witness to the accident.

The correct answer is Option 1.

 

Key Points

• The primary issue in the sentence is the use of the term "uninterrupted art."
• The correct term should be "sequential art" which accurately describes the format of a graphic novel or traditional comics.
• Sequential art refers to the use of a series of images to convey a story, which is a defining characteristic of graphic novels.
• Thus, the error is located in the use of "uninterrupted art" instead of "sequential art."

Therefore, the correct answer is Option 1.

 

Correct Sentence: "'Spy Family' is a graphic novel that is a narrative work in which the story is conveyed to the reader using sequential art in a traditional comics format."

 

Additional Information

• Option 2: "existential art in a traditional" – This term is not relevant to the context of graphic novels or traditional comics format.
• Option 3: "sedimental art in a traditional" – This term is not relevant to the context of graphic novels or traditional comics format.
• Option 4: "longitudinal art in a traditional" – This term is not relevant to the context of graphic novels or traditional comics format.