SSC CGL 2024 Tier-I Official Paper (Held On: 13 Sept, 2024 Shift 3) (Previous Year Paper)

Given word: 'REDUCTION'

After arranged in alphabetical order:

 

After arranged in alphabetical order,

Third alphabet from the left end is: E,

Fourth alphabet from the left end is: I,

Now,

there are 3 letters between E and I.

 

Hence, the correct answer is "Option 3".

Decoding symbols:

So, by checking each option one by one:

i. Z & B ! T x J ? L → Z is the brother of B who is the wife of T who is the father of J who is the mother of L. 

Here, Z is L's mother's maternal uncle.

ii. Z ! B x T ? J & L → Z is the wife of B who is the father of T who is the mother of J who is the brother of L. 

Here, Z is L's maternal grandmother.

iii. Z ? B x T & J ! L → Z is the mother of B who is the father of T who is the brother of J who is the wife of L. 

Here, Z is L's wife's father's mother.

iv. Z ? B ! T & J x L → Z is the mother of B who is the wife of T who is the brother of J who is the father of L. 

Here, Z is L's father's brother's mother-in-law.

Hence, the correct answer is "Option 3".

The logic followed here is :-

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

As D is common on both the dices, it is clear that F, B, X, and N are adjacent to it, and the remaining i.e. A will be opposite to it.

Opposite pairs:

F → N

B → X

D → A

So, the opposite side of "D" will be side "A".

Hence, the correct answer is "Option 1".

The logic followed here is:

 

Joy - Peace, Problem - Concern, Worry - Fear all are Synonyms,

Whereas,

Misery - Happiness are Antonyms.

Hence, the correct answer is "Misery - Happiness".

The correct mirror image of the given figure is:

Hence, the correct answer is "Option 4".

The logic followed here is:

Hence, "Option 1" is the correct answer.

The logic followed here is:

1. For first box series →

The right diagonal elements interchange positions.

The second element of 3rd column is replaced by new element subsequently.

The rest of the elements shifts anticlockwise.

2) For second box series →

The elements at the corners shifts clockwise.

The rest of the elements shifts anticlockwise.

Thus, the final series is:

Hence, "Option 2" is the correct answer.

The logic followed here is:

Logic: 1st number × 3 = 2nd number; 2nd number + 4 = 3rd number; 3rd number + 5 = 4th number.

Now, for 9 – 27 – 31 – 36

⇒ 9 × 3 = 27;

⇒ 27 + 4 = 31;

⇒ 31 + 5 = 36.

for, 10 – 30 – 34 – 39

⇒ 10 × 3 = 30;

⇒ 30 + 4 = 34;

⇒ 34 + 5 = 39.

Similarly, for 5 – 15 – 19 – 24:

⇒ 5 × 3 = 15

⇒ 15 + 4 = 19

⇒ 19 + 5 = 24

Hence, the correct answer is "Option 4".

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

Thus, as marked there are a total of 11 squares.

Hence, "Option 1" is the correct answer.

The logic followed here is:

Logic: (1st number × 7) = 2nd number.

Now,

For, 63 : 441

⇒ (63 × 7) = 441.

For, 51 : 357

⇒ (51 × 7) = 357.

Similarly, for:

68 : ?

⇒ (68 × 7) = 476 (not equal to 340)

54 : ?

⇒ (54 × 7) = 378 (not equal to 324)

39 : ?

⇒ (39 × 7) = 273 (equal to 273)

89 : ?

⇒ (89 × 7) = 623 (not equal to 96)

Hence, the correct answer is "Option 3".

The logic followed here is:

Logic: Animal reserves and the animals they are famous for.

Here,

Kaziranga : Rhinoceros → Kaziranga is famous for its population of Rhinoceros.

Similarly,

Periyar : Elephants → Periyar is famous for its population of Elephants.

Hence, the correct answer is "Option 2".

The logic followed here is:

Logic: Profession and its related activity.

Here,

Magician : Magic  → Magic is an activity performed by a Magician.

Similarly,

Dancer : Dance  → Dance is an activity performed by a Dancer.

Hence, the correct answer is "Option 1".

Given series: _ O _ O _ _ O _ P _ O _

Option 1) PPOOPOP → P O P O O O O P P O O P → (It doesn't follow any pattern)

Option 2) OOPPPOO → O O O O P P O P P O O O → (It doesn't follow any pattern)

Option 3) OPOPOOP → O O P O O P O O P O O P → (It follows the pattern of: OOP - OOP - OOP - OOP)

Option 4) PPOOOPO → P O P O O O O O P P O O → (It doesn't follow any pattern)

Hence, "Option 3" is the correct answer.

The logic followed here is :-

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

As K is common on both the dices, it is clear that L, J, G, and I are adjacent to it, and the remaining i.e. H will be opposite to it.

Opposite pairs:

L → G

J → I

K → H

So, the opposite side of "G" will be side "L".

Hence, the correct answer is "Option 3".

According to the given Venn diagram:

The shaded region represents the number of women who are either smart or brave or both.

So, the number of women who are either smart or brave or both = 13 + 14 + 3 → 30

Hence, the correct answer is "Option 3".

The logic followed here is:

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

Thus, the final series is:

Hence, "Option 1" is the correct answer.

BODMAS Table: 

After decoding signs:

 

Given equation: 27 − 6 + 4 × 8 ÷ 6 = ?

After interchanging the signs then:

⇒ 27 ÷ 6 × 4 + 8 - 6 = ?

⇒ ? = 4.5 × 4 + 8 - 6

⇒ ? = 18 + 8 - 6

⇒ ? = 26 - 6

⇒ ? = 20

Hence, the correct answer is "20".

The logic followed here is:

Logic: (1st number × 9) = 2nd number.

Now,

For, 13 : 117

⇒ (13 × 9) = 117.

For, 23 : 207

⇒ (23 × 9) = 207.

Similarly, for 51 ∶ ?

⇒ (51 × 9) = 459.

Hence, the correct answer is "Option 4".

BODMAS Table: 

After decoding signs:

 

Given equation: 2864 × 4 - 168 ÷ 2 + 69 = ?

After interchanging the signs then:

⇒ 2864 ÷ 4 + 168 × 2 - 69 = ?

⇒ ? = 716 + 168 × 2 - 69

⇒ ? = 716 + 336 - 69

⇒ ? = 1052 - 69

⇒ ? = 983

Hence, the correct answer is "983".

The logic followed here is:

Hence, "Option 4" is the correct answer.

The logic followed here is:

'VAT' is coded as '43'

And,

'AWW' is coded as '47'

Similarly,

'ZHU' is coded as

Hence, "Option 3" is the correct answer.

The logic followed here is:

Option 1) 360 – 12 – 6

12² + 6³ = 360

144 + 216 = 360

360 = 360 (LHS = RHS)

Option 2) 82 – 8 – 2

82 + 23 = 82

64 + 8 = 82

72 ≠ 82 (LHS ≠ RHS)

Option 3) 63 – 6 – 3

62 + 33 = 63

36 + 27 = 63

63 = 63 (LHS = RHS)

Option 4)164 – 10 – 4

102 + 43 = 360

100 + 64 = 164

164 = 164 (LHS = RHS)

Thus, out of all the options '82 – 8 – 2' is the odd one out.

Hence, "Option 2" is the correct answer.

The figure which when put in place of '?' will logically complete the pattern is shown below:

Thus, option Figure 1 will logically complete the pattern given.

Hence, "Option 1" is the correct answer.

The logic followed here is:

'ESTATE' is written as '80'

And,

'FAMOUS' is written as '85'

Similarly,

'FOREST' is written as

Hence, "Option 1" is the correct answer.

According to the given Venn diagram:

The shaded region represents the number of people who play either only volleyball or only chess.

So, the number of people who play either only volleyball or only chess = 23 + 43 → 66

Hence, the correct answer is "Option 4".

The correct answer is Spines.

Key Points

• Spines are not a component of a flower; they are modified leaves, stems, or stipules that are typically hard, pointed, and meant for protection.
• Flowers are reproductive structures in angiosperms (flowering plants) and typically consist of four main parts: androecium, corolla, calyx, and gynoecium.
• The primary role of spines is to protect the plant from herbivores and reduce water loss, especially in arid environments.
• Unlike the other options listed (androecium, corolla, and calyx), spines do not play a role in the reproductive process of the flower.

Additional Information

• Androecium
• The androecium is the collective term for the stamens in a flower. Each stamen typically consists of a filament and an anther, where pollen is produced.
• It is the male reproductive part of the flower.
• Corolla
• The corolla is the collective term for the petals of a flower.
• Petals are often brightly colored to attract pollinators.
• Calyx
• The calyx is the collective term for the sepals of a flower.
• Sepals are typically green and provide protection to the flower bud before it opens.
• Gynoecium
• The gynoecium is the collective term for the female reproductive parts of a flower, consisting of one or more carpels.
• Each carpel typically includes an ovary, style, and stigma.

The correct answer is Benaras.

Key Points

• The Hindu College was established in the year 1791.
• It was founded in the city of Benaras, which is now known as Varanasi.
• The institution was set up to provide Western education to Indians.
• It played a significant role in the Indian Renaissance and the spread of modern education in India.

Additional Information

• Hindu College:
• The Hindu College in Benaras was one of the earliest institutions to provide a structured curriculum for Western-style education in India.
• It was founded by influential citizens of Benaras, including Raja Rammohan Roy.
• The college was established with the aim of imparting modern education in various fields such as science, mathematics, and the humanities.
• It later evolved into the Central Hindu School and eventually became a part of Banaras Hindu University (BHU).
• Banaras Hindu University (BHU):
• Established in 1916 by Pandit Madan Mohan Malaviya, BHU is one of the largest residential universities in Asia.
• BHU offers courses in various fields including arts, science, engineering, medicine, and agriculture.
• The university is renowned for its research and academic excellence.
• It has a sprawling campus spread over 1,300 acres in Varanasi.
• Raja Rammohan Roy:
• Raja Rammohan Roy was a prominent social and educational reformer in India during the early 19th century.
• He is often called the "Father of the Indian Renaissance" due to his efforts in promoting modern education and social reforms.
• He was an advocate for the abolition of practices such as Sati and child marriage.
• His contributions laid the foundation for the development of modern Indian society.

The correct answer is Between the south-eastern tip of India and the western coast of Sri Lanka.

Key Points

• The Gulf of Mannar is a large shallow bay forming part of the Laccadive Sea in the Indian Ocean.
• It lies between the south-eastern tip of India and the western coast of Sri Lanka, specifically between Tamil Nadu and the Northern Province of Sri Lanka.
• The Gulf of Mannar is renowned for its rich biodiversity and is home to the Gulf of Mannar Marine National Park, which is one of the richest coastal regions in Asia.
• This region is famous for its pearl banks and is often referred to as the Pearl of the Indian Ocean.

Additional Information

• Gulf of Mannar Marine National Park
• It is a protected area in the Gulf of Mannar and covers an area of about 560 square kilometers.
• The park is known for its coral reefs, seagrasses, and mangroves, which provide habitat for numerous marine species.
• It is home to endangered species such as the dugong, sea turtles, and various species of dolphins and whales.
• The park is part of the larger Gulf of Mannar Biosphere Reserve, which is recognized by UNESCO.
• Geographical Significance
• The Gulf of Mannar acts as a natural boundary between India and Sri Lanka.
• This region is historically significant as it has been a major trading route for centuries.
• The Mannar Island, located in the gulf, is known for its historical importance and natural beauty.
• The shallow waters and coral reefs make it a popular destination for marine research and ecotourism.
• Environmental Concerns
• The region faces environmental challenges such as coral bleaching, overfishing, and pollution.
• Conservation efforts are in place to protect the unique marine biodiversity of the Gulf of Mannar.
• Local communities are involved in sustainable fishing practices to preserve the ecological balance.
• There are ongoing initiatives to promote awareness and education about marine conservation in the area.

The correct answer is Priya Malik.

Key Points

• Priya Malik won the Gold Medal in the World Cadet Wrestling Championship held in Budapest, Hungary in 2021.
• She competed in the 73 kg category.
• Her victory came shortly after India's Neeraj Chopra won a gold medal in javelin at the Tokyo Olympics, raising the country's spirits.
• Priya Malik's win is significant as it demonstrates the growing prowess of Indian wrestlers on the world stage.

Additional Information

• World Cadet Wrestling Championship
• This championship is an international wrestling competition for athletes aged 16-17 years.
• It is organized by the United World Wrestling (UWW), the international governing body for the sport of amateur wrestling.
• The event features various weight categories for both male and female wrestlers.
• The championship aims to provide a platform for young wrestlers to showcase their talent and gain international experience.
• United World Wrestling (UWW)
• UWW is the international governing body for the sport of wrestling, overseeing wrestling disciplines such as Freestyle, Greco-Roman, and Beach Wrestling.
• It was founded in 1912 and is headquartered in Corsier-sur-Vevey, Switzerland.
• UWW organizes various international competitions, including the World Wrestling Championships, Continental Championships, and the Olympic Games wrestling events.
• The organization is committed to promoting and developing the sport of wrestling globally, ensuring fair play and adherence to the rules.
• Indian Wrestlers on the Global Stage
• India has a rich history in wrestling, with the sport being one of the oldest and most popular in the country.
• Indian wrestlers have made significant strides on the international stage, winning medals in events such as the Olympics, Asian Games, Commonwealth Games, and World Championships.
• Notable Indian wrestlers include Sushil Kumar, Bajrang Punia, Vinesh Phogat, and Sakshi Malik.
• The success of wrestlers like Priya Malik highlights the continued growth and potential of Indian wrestling on the global stage.

The correct answer is Madras Music Academy.

Key Points

• The Sangeet Kala Acharya Award is given by the Madras Music Academy, which is one of the oldest and most prestigious music academies in India.
• This award has been conferred annually since 1993.
• The award is given to individuals who have made significant contributions to the field of music and dance.
• The Madras Music Academy is known for its annual music festival, which is one of the largest cultural events in South India.

Additional Information

• Madras Music Academy
• The Madras Music Academy was established in 1928 and is located in Chennai, Tamil Nadu.
• It was founded with the objective of promoting and preserving the classical music and dance traditions of India.
• The academy plays a pivotal role in the promotion of Carnatic music, which is a major genre of Indian classical music.
• It conducts the annual Madras Music Season, a six-week-long event featuring performances by renowned artists, lecture demonstrations, and workshops.
• Besides the Sangeet Kala Acharya Award, the academy also confers other prestigious awards such as the Sangita Kalanidhi, TTK Award, and the Sangita Kala Acharya Award.
• Calcutta School of Music
• Founded in 1915 by Phillipe Sandre, the Calcutta School of Music is one of the premier institutions in India for the study of Western classical music, jazz, and Indian music.
• The school offers training in a wide range of musical instruments, voice training, and music theory.
• It also organizes concerts, workshops, and masterclasses by renowned musicians from around the world.
• Shankar Mahadevan Academy
• Founded by the renowned Indian singer and composer Shankar Mahadevan, this academy offers online music courses in Indian classical and contemporary music.
• The academy aims to make quality music education accessible to learners worldwide through its digital platform.
• It offers courses for various skill levels, from beginners to advanced students, and covers vocal, instrumental, and theoretical aspects of music.
• Subramaniam Academy of Performing Arts (SaPa)
• Founded by violinist Dr. L. Subramaniam and singer Kavita Krishnamurti Subramaniam, SaPa is an institution dedicated to the teaching of Indian and global music.
• Located in Bangalore, the academy offers programs for students of all ages, focusing on both traditional and contemporary music education.
• SaPa also conducts workshops, festivals, and collaborations with international artists to promote cultural exchange and musical innovation.

The correct answer is Fees and fines.

Key Points

• Non-tax revenue refers to the recurring income earned by the government from sources other than taxes.
• Fees and fines are common examples of non-tax revenue. Fees are payments made to the government for specific services or activities (e.g., passport fees, court fees), while fines are penalties imposed for violating laws or regulations.
• Non-tax revenue sources help the government fund its operations without increasing the tax burden on citizens.
• Other examples of non-tax revenue include interest receipts on loans given by the government, dividends from public sector enterprises, and income from government property.

Additional Information

• Personal income tax:
• It is a direct tax levied on the income of individuals.
• The rate of personal income tax varies based on income slabs and is progressive in nature, meaning higher income earners pay a higher percentage in taxes.
• Corporate income tax:
• This is a tax imposed on the net income or profit of corporations.
• It is a significant source of revenue for the government and is also a direct tax.
• Value-added tax (VAT):
• VAT is an indirect tax levied on the value added to goods and services at each stage of production or distribution.
• Consumers ultimately bear the cost of VAT as it is included in the purchase price of goods and services.
• Non-tax revenue sources:
• These include all government revenue sources that do not fall under the category of taxes.
• Examples include fees for services, fines and penalties, dividends from public enterprises, interest receipts, and revenue from government properties.

The correct answer is Lavani.

Key Points

• Sulochana Chavan was a legendary singer known for her contributions to the traditional Marathi folk genre of Lavani music.
• Lavani is a genre of music popular in Maharashtra and is noted for its powerful rhythm and sensuous, yet socially significant, lyrics.
• Lavani has traditionally been performed by female artists and is recognized for its combination of song and dance.
• Chavan's contributions to Lavani music earned her widespread acclaim and numerous accolades, including the prestigious Sangeet Natak Akademi Award.

Additional Information

• Giddha
• Giddha is a popular folk dance of women in the Punjab region of India and Pakistan.
• It is performed by women during special occasions and festivals, characterized by energetic movements and traditional Punjabi attire.
• Jhumair
• Jhumair is a traditional folk dance and song of the Chotanagpur plateau region in Jharkhand, India.
• It is performed during harvest season and other community celebrations, featuring slow rhythmic movements and traditional instruments.
• Ghumar
• Ghumar is a traditional folk dance of Rajasthan, India, performed by women in colorful attire.
• The dance is noted for its graceful circular movements and is performed during festive occasions like weddings and other celebrations.
• Lavani
• Lavani is a traditional folk art form that combines song and dance, originating from Maharashtra, India.
• The word 'Lavani' is derived from the Marathi word 'Lavanya', which means beauty, signifying the grace and charm of the performance.
• Lavani songs are known for their fast tempo and are often performed to the beats of the dholki drum.
• The dance is characterized by its vibrant costumes, energetic footwork, and expressive gestures.

The correct answer is Tulsi.

Key Points

• Tulsi (Ocimum sanctum), also known as Holy Basil, is a revered medicinal plant in India.
• It is traditionally used in Ayurvedic medicine to treat cold, cough, and other respiratory ailments.
• Tulsi leaves are known to have antibacterial, antiviral, and antifungal properties, which help in relieving infections causing cold and cough.
• Consumption of Tulsi tea or Tulsi-infused water can boost immunity and provide relief from respiratory disorders.

Additional Information

• Babool (Acacia nilotica)
• Babool tree has various medicinal uses, including in dental care, skin care, and as a remedy for diarrhea and dysentery.
• Its bark and leaves are commonly used in traditional medicine.
• Jamun (Syzygium cumini)
• Jamun fruit and seeds are used in Ayurvedic medicine to manage diabetes.
• It is also beneficial for digestive health and in treating ulcers.
• Arjun (Terminalia arjuna)
• Arjun tree is well-known for its cardiovascular benefits.
• Its bark is used to treat heart diseases, hypertension, and cholesterol issues.

The correct answer is Mango showers.

Key Points

• Mango showers are the colloquial term used to describe pre-monsoon rains in India, particularly in Kerala, Karnataka, and Tamil Nadu.
• These showers usually occur between March and April and are known to aid the ripening of mangoes, which is why they are called "mango showers".
• They are also beneficial for other crops and help to mitigate the high temperatures experienced in the summer months.
• The showers are a result of thunderstorms caused by the confluence of moist air from the Indian Ocean and the Bay of Bengal.

Additional Information

• Kalbaisakhi
• Kalbaisakhi, also known as Nor'westers, are violent thunderstorms that occur in the northeastern and eastern parts of India, particularly in West Bengal and Odisha.
• These storms are common in the pre-monsoon season, especially in the months of April and May.
• They are characterized by high-speed winds, heavy rain, and sometimes hail, which can cause significant damage to crops and property.
• Nor Westers
• Nor'westers are a type of storm that occurs in the northeastern and eastern parts of India, particularly in West Bengal, Bihar, Jharkhand, and Odisha.
• They are caused by the interaction of hot, dry air from the northwest with moist air from the Bay of Bengal.
• These storms can bring heavy rainfall, strong winds, and hail, often leading to damage in the affected areas.
• Orange Showers
• There is no widely recognized meteorological phenomenon known as "Orange showers".
• This term does not correspond to any specific weather pattern or event in India.

The correct answer is Ashoka.

Key Points

• Piyadassi is a title which means "beloved of the gods" and it was used by Ashoka in his inscriptions.
• Ashoka was the third emperor of the Mauryan dynasty and is considered one of India's greatest rulers.
• Ashoka ruled from 268 to 232 BCE and his reign was marked by an extensive expansion of the Mauryan Empire.
• Ashoka embraced Buddhism after the Kalinga War and propagated the religion across Asia.

Additional Information

• Mauryan Empire
• The Mauryan Empire was founded by Chandragupta Maurya in 322 BCE after overthrowing the Nanda dynasty.
• It was one of the largest empires in the world at its peak, spanning from modern-day Afghanistan to Bengal and south into the Deccan Plateau.
• Chandragupta was succeeded by his son Bindusara, who expanded the empire further before passing it to Ashoka.
• The Mauryan administration was highly organized with a centralized government and a bureaucratic system.
• Kalinga War
• The Kalinga War was fought between the Mauryan Empire under Ashoka and the state of Kalinga around 261 BCE.
• This war is particularly significant because of the massive destruction and loss of life it caused.
• The aftermath of the war had a profound impact on Ashoka, leading him to embrace Buddhism and propagate peace and non-violence.
• Ashoka's transformation post-Kalinga War is well-documented in his rock and pillar edicts.
• Buddhism
• Buddhism is a religion and philosophy based on the teachings of Siddhartha Gautama (the Buddha) who lived in the 5th century BCE.
• Ashoka's patronage played a significant role in the spread of Buddhism across India and beyond.
• Ashoka convened the Third Buddhist Council at Pataliputra (modern-day Patna) to discuss the teachings and spread of Buddhism.
• His efforts in promoting Buddhism included sending emissaries and missionaries to various parts of Asia, including Sri Lanka, Southeast Asia, and Central Asia.

The correct answer is Lord Jagannath.

Key Points

• The Rath Yatra of Puri is dedicated to Lord Jagannath, who is considered a form of Lord Vishnu.
• This annual festival involves a grand procession where the deities are taken out of the Jagannath Temple and placed in chariots, symbolizing their journey to their aunt's house.
• The festival is celebrated with much pomp and grandeur, attracting millions of devotees from across the world.
• The festival generally occurs in the month of June or July, based on the Hindu lunar calendar.

Additional Information

• Lord Jagannath:
• Lord Jagannath is a deity worshipped primarily in the Indian state of Odisha.
• He is considered a form of Lord Vishnu, and is also identified with Krishna and Buddha.
• The Jagannath Temple in Puri is one of the Char Dham pilgrimage sites, which are considered very sacred by Hindus.
• The temple is famous for its annual Rath Yatra, or chariot festival, in which the three main deities are pulled on huge and elaborately decorated temple cars.
• Rath Yatra:
• The Rath Yatra involves the transportation of deities on large chariots from the Jagannath Temple to the Gundicha Temple, which is about 3 kilometers away.
• The three deities—Lord Jagannath, his brother Balabhadra, and sister Subhadra—are taken out in three separate chariots.
• The chariots are pulled by thousands of devotees who consider it an auspicious act.
• The festival lasts for nine days, and the return journey of the deities is known as Bahuda Yatra.
• Significance:
• The Rath Yatra symbolizes the journey of Lord Jagannath along with his siblings to their maternal aunt's house.
• It represents the idea of the divine coming down to meet the devotees, breaking the barriers of the temple.
• The festival promotes the spirit of brotherhood and unity among people, irrespective of their caste or creed.

The correct answer is 12.

Key Points

• The President of India has the authority to nominate 12 members to the Rajya Sabha.
• These nominated members are chosen for their expertise in various fields such as literature, science, art, and social service.
• This provision is under Article 80 of the Indian Constitution.
• The purpose of these nominations is to ensure that the Rajya Sabha benefits from the experience and specialized knowledge of distinguished individuals.

Additional Information

• Rajya Sabha
• The Rajya Sabha is the upper house of the Parliament of India.
• It represents the states and union territories of the country.
• The Rajya Sabha has a maximum membership of 250 members.
• Out of these, 238 members are elected by the states and union territories, while 12 are nominated by the President.
• The Rajya Sabha is a permanent body and is not subject to dissolution, but one-third of its members retire every two years.
• Article 80 of the Indian Constitution
• This article deals with the composition of the Rajya Sabha.
• It specifies the maximum number of members, the method of election, and the nomination of members by the President.
• Under this article, the President nominates 12 members who have special knowledge or practical experience in various fields such as literature, science, art, and social service.
• Nominated Members
• The nominated members bring diverse expertise to the Rajya Sabha.
• They are not bound by party lines and can contribute independently to debates and legislation.
• Some notable nominated members include Lata Mangeshkar (singer), Sachin Tendulkar (cricketer), and Raghunath Mohapatra (sculptor).

The correct answer is Allowing the maximum foreign direct investment.

Key Points

• The Industrial Policy Statement of 1980 emphasized enhancing productivity, employment generation, and the optimum utilization of existing production capacity.
• It aimed at the promotion of export-oriented industries to improve foreign exchange earnings.
• The policy also focused on consumer protection against high prices and poor quality of goods.
• Allowing the maximum foreign direct investment (FDI) was not a primary objective of the Industrial Policy Statement of 1980.

Additional Information

• Industrial Policy Statement 1980
• The Industrial Policy Statement of 1980 was introduced to revitalize the Indian economy and make it more competitive.
• The policy emphasized modernization, technological upgradation, and the development of small and medium enterprises (SMEs).
• It aimed at achieving higher growth rates and industrial diversification to reduce regional disparities.
• The policy also sought to improve the efficiency of public sector enterprises and increase their contribution to the economy.
• Foreign Direct Investment (FDI)
• FDI involves investment from a foreign entity into the business interests of another country.
• It is a significant source of non-debt financial resources for the economic development of a country.
• FDI brings in capital, technology, and expertise, and can lead to job creation and infrastructure development.
• While FDI is important, it was not the main focus of the Industrial Policy of 1980, which was more oriented towards self-reliance and domestic industrial growth.

The correct answer is 2026.

Key Points

• India has been selected to host the BWF World Championships in 2026.
• This prestigious event is one of the most significant tournaments in the badminton world, drawing top players from across the globe.
• The decision to award India the hosting rights reflects the country's growing prominence in the sport of badminton.
• India has previously hosted the BWF World Championships in 2009, showcasing its capability to manage such large-scale international events.

Additional Information

• BWF World Championships
• The Badminton World Federation (BWF) World Championships is the most prestigious tournament in the sport of badminton.
• It was first held in 1977 and is now conducted annually, except in Olympic years.
• The tournament determines the world champions in five disciplines: men’s singles, women’s singles, men’s doubles, women’s doubles, and mixed doubles.
• Winning a title at the BWF World Championships is considered a significant achievement and highly coveted among badminton players.
• Badminton in India
• Badminton is one of the most popular sports in India, with a growing number of players achieving international success.
• India has produced several world-class badminton players such as Prakash Padukone, Pullela Gopichand, Saina Nehwal, and PV Sindhu.
• PV Sindhu made history by becoming the first Indian woman to win a silver medal in badminton at the 2016 Rio Olympics.
• The Badminton Association of India (BAI) is the governing body responsible for the development and promotion of badminton in the country.
• Significance of Hosting International Sports Events
• Hosting international sports events like the BWF World Championships can have significant benefits for the host country.It helps in boosting tourism and the local economy, generating global visibility, and enhancing the country’s reputation as a capable and reliable event organizer.
• Such events also inspire young athletes and contribute to the development of sports infrastructure.
• India's successful hosting of the 2010 Commonwealth Games and the 2017 FIFA U-17 World Cup are examples of the country’s ability to manage large-scale international sports events.

The correct answer is CH₃-Mg-Cl.

Key Points

• Grignard reagents are organomagnesium compounds which are commonly represented as R-Mg-X, where R is an organic group and X is a halogen.
• In the given options, CH₃-Mg-Cl (Methylmagnesium chloride) fits this representation.
• Grignard reagents are highly reactive and are used in various organic synthesis reactions, particularly for forming carbon-carbon bonds.
• They were discovered by the French chemist Victor Grignard, who was awarded the Nobel Prize in Chemistry in 1912 for this discovery.

Additional Information

• Grignard Reagent:
• Grignard reagents are typically prepared by reacting an alkyl or aryl halide with magnesium metal in an anhydrous ether solvent.
• They are strong nucleophiles and strong bases, making them very useful in organic chemistry for forming new carbon-carbon bonds.
• The general formula for a Grignard reagent is R-Mg-X, where R can be an alkyl, aryl, or vinyl group, and X is usually a halogen like chlorine, bromine, or iodine.
• Due to their high reactivity, Grignard reagents must be handled under an inert atmosphere (such as nitrogen or argon) to prevent them from reacting with moisture or oxygen in the air.
• Victor Grignard:
• François Auguste Victor Grignard was a French chemist who discovered the Grignard reagent, which revolutionized the field of organic chemistry.
• His discovery allowed for the formation of carbon-carbon bonds, which is fundamental in the synthesis of a wide variety of organic compounds.
• He was awarded the Nobel Prize in Chemistry in 1912 for this significant contribution.
• Applications of Grignard Reagents:
• Grignard reagents are used to form various types of carbon-carbon bonds, making them invaluable in the synthesis of alcohols, acids, and other organic compounds.
• They can react with carbonyl compounds (such as aldehydes and ketones) to form secondary and tertiary alcohols.
• They are also used in the synthesis of carboxylic acids by reacting with carbon dioxide.
• In addition to their use in academic research, Grignard reagents are also used in the pharmaceutical and chemical industries for the synthesis of complex organic molecules.

The correct answer is Rice.

Key Points

• Rice is typically grown during the Kharif season (June to November) and Rabi season (November to March) in India.
• The Zaid season falls between the Rabi and Kharif seasons, specifically from March to June.
• Crops like cucumber, watermelon, and muskmelon are commonly grown during the Zaid season due to their shorter growing period and summer adaptability.
• Rice requires a significant amount of water and is usually grown in areas with assured irrigation during the monsoon season (Kharif) or post-monsoon season (Rabi).

Additional Information

• Kharif Season
• The Kharif season is also known as the monsoon season, running from June to November.
• Crops grown during this season are typically sown with the beginning of the first rains in July, during the south-west monsoon season, and harvested in October.
• Major Kharif crops include rice, maize, sorghum, pearl millet, and cotton.
• Rabi Season
• The Rabi season is known as the winter season, running from November to March.
• Crops are sown in winter and harvested in the spring.
• Major Rabi crops include wheat, barley, peas, gram, and mustard.
• Zaid Season
• The Zaid season falls between the Kharif and Rabi seasons, specifically from March to June.
• Crops grown during this season have a shorter growing period and are typically summer crops.
• Examples of Zaid crops include cucumber, watermelon, muskmelon, and bitter gourd.

This detailed solution provides comprehensive information about the correct option and explains the different agricultural seasons in India.

The correct answer is Ashleigh Gardner.

Key Points

• Ashleigh Gardner is an Australian cricketer who plays as an all-rounder.
• She was awarded the Player of the Match in the semi-final of the T20 Women's World Cup 2023.
• In the semi-final match against India, Gardner made significant contributions with both bat and ball.
• Her performance was crucial in leading Australia to the final of the T20 Women's World Cup 2023.

Additional Information

• T20 Women's World Cup
• The T20 Women's World Cup is an international championship of Twenty20 cricket organized by the International Cricket Council (ICC).
• The tournament format consists of a round-robin group stage followed by knockout matches, including semi-finals and a final.
• It is one of the most prestigious events in women's cricket, featuring teams from around the world.
• The 2023 edition showcased excellent performances and competitive spirit among the participating teams.
• Semi-Final Match
• The semi-final match is a critical knockout stage in the tournament where the winning team proceeds to the final.
• In the 2023 edition, the semi-final between India and Australia was highly anticipated due to the strong track records of both teams.
• Key players' performances in semi-finals often play a pivotal role in determining the outcome of the match.
• Player of the Match
• The Player of the Match award is given to the standout performer in a particular match.
• This award recognizes the player who has made the most significant impact on the match's outcome through exceptional performance.
• In the context of the T20 Women's World Cup, this award is highly prestigious and highlights individual brilliance on a global stage.

The correct answer is President.

Key Points

• The ordinance making power is a significant legislative power vested in the President of India.
• As per Article 123 of the Indian Constitution, the President has the authority to promulgate ordinances when either of the two Houses of Parliament is not in session.
• The ordinance holds the same force and effect as an act of Parliament.
• This power is intended to address urgent situations requiring immediate legislative action.

Additional Information

• Ordinance:
• An ordinance is a temporary law enacted by the President of India on the advice of the Union Cabinet.
• It is promulgated when Parliament is not in session and immediate action is required.
• Ordinances must be approved by Parliament within six weeks of reassembling, or they cease to operate.
• Ordinances allow the government to take swift action on matters of urgency without waiting for Parliament to convene.
• Article 123 of the Indian Constitution:
• Article 123 grants the President the power to promulgate ordinances during the recess of Parliament.
• This provision ensures that the executive can address urgent issues without legislative delays.
• Ordinances must be laid before both Houses of Parliament and require their approval to become permanent laws.
• This power is subject to judicial review to prevent misuse and ensure adherence to constitutional principles.
• Separation of Powers:
• The Indian Constitution follows the principle of separation of powers among the executive, legislature, and judiciary.
• The President's ordinance-making power represents an overlap between the executive and legislative functions.
• This power is exercised with checks and balances to prevent abuse and ensure accountability.

The correct answer is 1931.

Key Points

• The Gandhi-Irwin Pact was signed on 5 March 1931 between Mahatma Gandhi and the then Viceroy of India, Lord Irwin.
• This pact marked the end of the civil disobedience movement led by the Indian National Congress against British rule in India.
• Under the terms of the pact, the British Government agreed to release all political prisoners who had been detained during the civil disobedience movement.
• Mahatma Gandhi, representing the Indian National Congress, agreed to discontinue the civil disobedience movement and participate in the Second Round Table Conference in London.

Additional Information

• Gandhi-Irwin Pact
• The Gandhi-Irwin Pact was a political agreement signed by Mahatma Gandhi and the Viceroy of India, Lord Irwin, in 1931.
• It was a significant event in the Indian freedom struggle, which led to the cessation of the civil disobedience movement that was initiated against the British Salt Tax.
• The pact included several concessions by the British Government, such as the withdrawal of all ordinances and prosecutions, release of prisoners arrested for participating in the salt march, and the return of confiscated properties of the satyagrahis.
• Gandhi agreed to attend the Second Round Table Conference in London as the sole representative of the Indian National Congress.
• Civil Disobedience Movement
• Launched by Mahatma Gandhi in 1930, the Civil Disobedience Movement aimed at disobeying certain laws, demands, and commands of the British government, particularly the Salt Act.
• The movement began with the Dandi March, where Gandhi and his followers marched to the Arabian Sea coast to produce salt from seawater in defiance of British laws.
• This movement attracted widespread participation from various sections of Indian society and was a crucial step towards India's independence.
• The movement saw significant government repression, including the arrest of thousands of protesters and leaders.
• Second Round Table Conference
• The Second Round Table Conference was held in London in September 1931 and was attended by Mahatma Gandhi as the representative of the Indian National Congress.
• The conference aimed to discuss constitutional reforms in India and the future governance of the country.
• Despite the discussions, the conference did not result in any significant agreements, leading to Gandhi's return to India and the resumption of the civil disobedience movement.
• The failure of the conference highlighted the deep divisions between the British authorities and Indian leaders regarding India's political future.

The correct answer is e-Shram Portal.

Key Points

• The e-Shram Portal was launched by the Ministry of Labour and Employment on 26 August 2021.
• This portal aims to create a National Database of Unorganized Workers (NDUW) to ensure their welfare and social security.
• It provides a unique identification number called Universal Account Number (UAN) to each worker.
• The registration on the portal is free of cost and is expected to benefit around 38 crore unorganized workers in India.

Additional Information

• PM Mentoring Yuva Scheme
• This scheme is aimed at mentoring young authors below the age of 30.
• The goal is to create a pool of authors who can write on a variety of subjects to promote the Indian culture and literature globally.
• PM Daksh Yojana
• Launched by the Ministry of Social Justice and Empowerment to provide skill training to marginalized groups.
• The aim is to enhance the competency levels of marginalized communities and improve their employability.
• Gram Ujala Scheme
• Launched by Convergence Energy Services Limited (CESL) to provide affordable LED bulbs in rural areas.
• The scheme aims to promote energy efficiency and reduce carbon emissions.

This HTML code contains detailed information about the correct answer "e-Shram Portal" including key points about the scheme and additional information about other related schemes.

The correct answer is Ampere.

Key Points

• The SI unit of electric current is the Ampere, symbolized as A.
• It is named after André-Marie Ampère, a French physicist and mathematician who is considered one of the founders of the science of classical electromagnetism.
• One Ampere is defined as the flow of one coulomb of charge per second.
• It is a fundamental unit in the International System of Units (SI), and it is used to quantify the amount of electric charge passing a point in an electric circuit per unit time.

Additional Information

• Ohm (Ω)
• The Ohm is the SI unit of electrical resistance.
• Named after German physicist Georg Simon Ohm, it quantifies the resistance in a circuit.
• One Ohm is defined as the resistance between two points of a conductor when a constant potential difference of 1 Volt, applied to these points, produces a current of 1 Ampere.
• Volt (V)
• The Volt is the SI unit of electric potential, electric potential difference, and electromotive force.
• Named after the Italian physicist Alessandro Volta, it measures the potential difference between two points.
• One Volt is defined as the potential difference that will drive one Ampere of current against one Ohm of resistance.
• Metre (m)
• The Metre is the SI unit of length.
• It is defined as the distance traveled by light in a vacuum in 1/299,792,458 seconds.
• It is a fundamental unit in the metric system and is used to measure distance.

The correct answer is Hyponatremia.

Key Points

• Hyponatremia is a condition where the sodium level in the blood is lower than normal.
• Sodium is an essential electrolyte that helps regulate water balance in and around cells.
• Common causes of hyponatremia include excessive fluid intake, certain medications, and conditions like heart failure or kidney disease.
• Symptoms of hyponatremia can range from mild (nausea, headache) to severe (confusion, seizures, coma).

Additional Information

• Hyperkalemia
• Hyperkalemia is a condition characterized by an abnormally high level of potassium in the blood.
• Potassium is crucial for normal cell function, especially in the heart, muscles, and nerves.
• Causes include kidney dysfunction, excessive potassium intake, and certain medications.
• Symptoms can include muscle fatigue, weakness, paralysis, and abnormal heart rhythms.
• Hypokalemia
• Hypokalemia is a condition where there is a lower than normal level of potassium in the blood.
• It can result from conditions like chronic kidney disease, excessive loss of potassium through urine, or inadequate potassium intake.
• Symptoms may include muscle cramps, weakness, fatigue, and cardiac arrhythmias.
• Treatment typically involves potassium supplements and addressing the underlying cause.
• Hypernatremia
• Hypernatremia is a condition where the sodium level in the blood is higher than normal.
• It often occurs due to dehydration, water loss, or excessive sodium intake.
• Symptoms can include thirst, confusion, muscle twitching, and, in severe cases, seizures or coma.
• Treatment focuses on slowly rehydrating the patient and addressing the cause of the sodium imbalance.

The correct answer is a novel feature of the constitution.

Key Points

• Dr. B.R. Ambedkar, the principal architect of the Indian Constitution, described the Directive Principles of State Policy (DPSP) as a novel feature of the Indian Constitution.
• The DPSPs are guidelines for the framing of laws by the government and are considered fundamental in the governance of the country.
• They are enshrined in Part IV of the Constitution, covering Articles 36 to 51.
• The DPSPs aim to create social and economic conditions under which the citizens can lead a good life and also establish social and economic democracy through a welfare state.

Additional Information

• Directive Principles of State Policy (DPSP)
• The DPSPs are non-justiciable, meaning they are not legally enforceable by the courts for their violation.However, they are considered fundamental in the governance of the country and aim to establish a just society.
• The principles have been borrowed from the Irish Constitution and are inspired by the Directive Principles of Social Policy.
• They cover social and economic rights, Gandhian principles, and international peace and security.
• Importance of DPSP
• DPSPs act as a yardstick for measuring the performance of the government in terms of achieving social and economic justice.
• They guide the government in policy-making and legislation.
• They aim to create a welfare state where all citizens have equal opportunities for growth and development.
• Despite being non-justiciable, they are fundamental in the governance of the country and aim to establish social and economic democracy.
• Relation with Fundamental Rights
• Fundamental Rights and DPSPs together form the conscience of the Indian Constitution.
• While Fundamental Rights are justiciable and enforceable by the courts, DPSPs are non-justiciable.
• Both aim to establish social, economic, and political justice in the country.
• In case of conflict, the courts have generally given precedence to Fundamental Rights over DPSPs.
• Amendments and DPSP
• The 42nd Amendment Act of 1976 added a few DPSPs, such as the protection of the environment, promotion of equal justice, and provision of free legal aid.
• The 44th Amendment Act of 1978 incorporated the principle that the state shall minimize inequalities in income, status, facilities, and opportunities.
• These amendments reflect the dynamic nature of the Constitution and its ability to adapt to changing needs and aspirations of society.

The correct answer is Rajasthan.

Key Points

• Tera Tali is a unique and traditional folk dance form predominantly performed in the state of Rajasthan.
• This dance is a significant part of the culture of the Kamar tribe of Rajasthan.
• It involves the use of manjiras (small brass cymbals) tied to different parts of the body, especially the hands and feet.
• The performance is often accompanied by singing and is a reflection of the vibrant and rich cultural heritage of Rajasthan.

Additional Information

• Rajasthan:
• Rajasthan is the largest state in India by area, located in the northwestern part of the country.
• It is known for its rich cultural heritage, historical forts, palaces, art, and traditional music and dance forms.
• Rajasthan is home to other famous folk dances like Ghoomar, Kalbelia, and Bhavai.
• The state's vibrant and colorful festivals like the Pushkar Camel Fair and the Jaipur Literature Festival attract tourists from all over the world.
• Assam:
• Assam is a state in northeastern India known for its tea plantations, wildlife, and the Brahmaputra River.
• Bihu is the most important and widely celebrated festival in Assam, involving traditional dance and music.
• The state is also known for its rich biodiversity and is home to the Kaziranga National Park, famous for the one-horned rhinoceros.
• Madhya Pradesh:
• Madhya Pradesh is located in central India and is often referred to as the "Heart of India" due to its geographical location.
• The state has a rich cultural history and is known for its historical sites like Khajuraho, Bhimbetka rock shelters, and Sanchi Stupa.
• Folk dances like Gaur, Phulpati, and Maanch are popular in Madhya Pradesh.
• Haryana:
• Haryana is a state in northern India, known for its agricultural prosperity and as the home of the Green Revolution.
• The state is known for traditional folk dances like Jhumar, Dhamal, and Phag.
• Haryana has a rich history and is known for its contributions to Indian sports, especially wrestling and kabaddi.

The correct answer is Bairam Khan.

Key Points

• Bairam Khan was a prominent noble and general of the Mughal Empire during the reign of Emperor Akbar.
• He served as the regent and guardian of young Akbar, managing the affairs of the state and the military.
• Bairam Khan was conferred with the title of "Khan-i-Khanan," which means "Khan of Khans" or "Chief of Chiefs."
• His loyalty and strategic expertise were instrumental in consolidating Mughal rule in India.

Additional Information

• Vakil: The term "Vakil" in the Mughal administration refers to a representative or a high-ranking official who acted on behalf of the emperor. The Vakil was responsible for supervising the administration and ensuring that the emperor's orders were implemented.
• Khan-i-Khanan: This title was one of the highest honors in the Mughal hierarchy, bestowed upon individuals who demonstrated exceptional loyalty and service to the emperor. It signified supreme authority and respect.
• Akbar's Reign: Akbar, also known as Akbar the Great, was the third Mughal emperor who ruled from 1556 to 1605. His reign is noted for significant administrative, cultural, and military achievements, including the establishment of a centralized government and promoting a policy of religious tolerance.
• Birbal and Raja Todar Mal: While Birbal was one of Akbar's trusted advisors and known for his wit and wisdom, Raja Todar Mal served as Akbar's finance minister and played a crucial role in the implementation of revenue reforms. Neither of them held the title of "Khan-i-Khanan."

Shortcut Trick

Downstream speed (D) : Upstream speed (U) = Distance Ratio (in same time) = 10 : 4 = 5 : 2.

Let D = 5k and U = 2k. Total time = (30 ÷ 5k) + (30 ÷ 2k) = 14 ⇒ 6/k + 15/k = 14 ⇒ 21/k = 14 ⇒ k = 3/2.

Speed of stream = (D − U) ÷ 2 = (5k − 2k) ÷ 2 = 3k/2.

Substitute k = 3/2 ⇒ Speed of stream = (3 × 3/2) ÷ 2 = 9/4 km/h.

∴ The correct answer is 9/4 km/h.

Alternate Method

Given:

Total distance for a one-way trip = 30 km.

Total time for the round trip = 14 hours.

Distance covered downstream in a given time = 10 km.

Distance covered upstream in the same time = 4 km.

Formula Used:

Time = Distance ÷ Speed

Speed of stream = (Downstream Speed − Upstream Speed) ÷ 2

Calculations:

Let the speed of the boat in still water be u km/h and the speed of the stream be v km/h.

⇒ Downstream speed (D) = u + v

⇒ Upstream speed (U) = u − v

Since the time taken is the same for 10 km downstream and 4 km upstream:

⇒ 10 ÷ (u + v) = 4 ÷ (u − v)

⇒ (u + v) ÷ (u − v) = 10/4 = 5/2

Let Downstream speed D = 5k and Upstream speed U = 2k.

Total time taken for the round trip is 14 hours:

⇒ 30 ÷ D + 30 ÷ U = 14

⇒ 30 ÷ 5k + 30 ÷ 2k = 14

⇒ 6/k + 15/k = 14

⇒ 21/k = 14

⇒ k = 21 ÷ 14 = 3/2

Now, calculate the absolute speeds D and U:

⇒ Downstream Speed (D) = 5 × 3/2 = 15/2 = 7.5 km/h

⇒ Upstream Speed (U) = 2 × 3/2 = 3 km/h

Speed of the stream (v) = (D − U) ÷ 2

⇒ v = (7.5 − 3) ÷ 2 = 4.5 ÷ 2 = 9/4 km/h

∴ The correct answer is 9/4 km/h.

Additional Information

Downstream and Upstream Speeds

When a boat moves with the stream, its effective speed increases (D = u + v). When it moves against the stream, its effective speed decreases (U = u − v).

Boat in Still Water

The speed of a boat in still water (u) can be calculated directly if downstream and upstream speeds are known: u = (D + U) ÷ 2.

Speed of the Stream

The speed of the stream or current (v) is half the difference between the downstream and upstream speeds: v = (D − U) ÷ 2.

Constant Time Principle

If the time taken to travel two different distances is the same, the ratio of the distances is directly proportional to the ratio of their respective speeds (D₁ ÷ D₂ = S₁ ÷ S₂).

Shortcut Trick

For two intersecting circles with a common tangent AB, and intersection points C and D:

The sum of the angles subtended by the common tangent at the intersection points is always 180°.

⇒ ∠ACB + ∠ADB = 180°

⇒ ∠ACB + 68° = 180°

⇒ ∠ACB = 112°

∴ The correct answer is 112°.

Alternate Method

Given:

∠ADB = 68°

Formula Used:

Alternate Segment Theorem: The angle between a tangent and a chord is equal to the angle subtended by the chord in the alternate segment.

Sum of interior angles of a triangle is 180°.

68°

112°

P Q

A B C D

Calculations:

⇒ Let AB be the common tangent touching the circles at A and B.

⇒ By the Alternate Segment Theorem for the first circle, the angle between tangent AB and chord AC is equal to the angle in the alternate segment.

⇒ ∠BAC = ∠ADC

⇒ Similarly, for the second circle, the angle between tangent AB and chord BC is equal to the angle in the alternate segment.

⇒ ∠ABC = ∠BDC

⇒ In ΔABC, the sum of all interior angles is 180°.

⇒ ∠ACB + ∠BAC + ∠ABC = 180°

⇒ Substituting the corresponding alternate angles into the equation:

⇒ ∠ACB + ∠ADC + ∠BDC = 180°

⇒ We know from the figure that ∠ADC + ∠BDC = ∠ADB.

⇒ ∠ACB + ∠ADB = 180°

⇒ ∠ACB + 68° = 180°

⇒ ∠ACB = 180° − 68°

⇒ ∠ACB = 112°

∴ The correct answer is 112°.

Additional Information

Alternate Segment Theorem

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

Direct Common Tangent (DCT)

Length of DCT = √(d² − (r₁ − r₂)²), where d is the distance between the centres and r₁, r₂ are the radii.

Transverse Common Tangent (TCT)

Length of TCT = √(d² − (r₁ + r₂)²), applicable when circles are strictly disjoint.

Common Chord of Intersecting Circles

The line joining the centres of two intersecting circles acts as the perpendicular bisector of their common chord.

Shortcut Trick

17 ÷ 18 gives a remainder of −1, so 17¹³ ÷ 18 gives a remainder of (−1)¹³ = −1.

21 ÷ 18 gives a remainder of 3.

Combined remainder = −1 − 3 = −4.

Since the remainder is negative, add the divisor: 18 − 4 = 14.

∴ The correct answer is 14.

Alternate Method

Given:

Expression: (17¹³ − 21) ÷ 18

Concept Used:

Remainder Theorem: When dividing a polynomial by (x − a), the remainder is found by evaluating the expression at x = a.

Calculations:

⇒ We can express the base 17 as (18 − 1).

⇒ The given expression becomes: ((18 − 1)¹³ − 21) ÷ 18

⇒ When expanding (18 − 1)¹³ using the binomial theorem, all terms containing 18 are perfectly divisible by 18.

⇒ The only remaining term from the expansion is the constant term: (−1)¹³ = −1.

⇒ Now, divide the constant 21 by 18, which gives a remainder of 3.

⇒ Total remainder expression = −1 − 3 = −4.

⇒ A remainder of −4 means the value is 4 less than the divisor.

⇒ Actual positive remainder = 18 − 4 = 14.

∴ The correct answer is 14.

Additional Information

Remainder of aⁿ ÷ (a+1)

When aⁿ is divided by (a+1), the remainder is 1 if n is an even number, and the remainder is 'a' (or −1) if n is an odd number.

Remainder of aⁿ ÷ (a−1)

When aⁿ is divided by (a−1), the remainder is always 1, regardless of whether the power n is odd or even.

Negative Remainder Concept

A negative remainder mathematically indicates how much less the number is from the next multiple of the divisor. The actual positive remainder is obtained by adding the divisor: Actual Remainder = Divisor − Negative Remainder.

Given:

Expression: 16 ÷ 2 - 7 × 15 ÷ 3 + 5 × 5 + 4 × 3 - 10

Formula Used:

Apply the BODMAS/BIDMAS rule (Brackets, Orders (i.e. powers and square roots, etc.), Division and Multiplication, Addition and Subtraction)

Calculation:

Start with division and multiplication from left to right:

16 ÷ 2 = 8

7 × 15 = 105

105 ÷ 3 = 35

5 × 5 = 25

4 × 3 = 12

Now substitute these values back into the expression:

8 - 35 + 25 + 12 - 10

Next, perform addition and subtraction from left to right:

8 - 35 = -27

-27 + 25 = -2

-2 + 12 = 10

10 - 10 = 0

The simplified result is 0.

The correct answer is option 3.

Given:

District A: Male = 40, Female = 30

District B: Male = 25, Female = 35

District C: Male = 20, Female = 30

District D: Male = 45, Female = 25

Formula used:

To find the total number of male and female volunteers in all districts:

Total Male Volunteers = Male in A + Male in B + Male in C + Male in D

Total Female Volunteers = Female in A + Female in B + Female in C + Female in D

Calculations:

Total Male Volunteers = 40 + 25 + 20 + 45 = 130

Total Female Volunteers = 30 + 35 + 30 + 25 = 120

Total number of male and female volunteers = 130 + 120 = 250

Total number of male and female volunteers is 250.

Shortcut Trick

Curved Surface Area (CSA) of cylinder = 2πrh

Given 2πrh = 880

r × h = 880 ÷ (2 × 22 ÷ 7)

r × h = 880 × (7 ÷ 44) = 140

∴ The correct answer is 140 cm².

Alternate Method

Given:

Curved surface area of the cylinder = 880 cm²

Formula Used:

Curved surface area of a cylinder = 2πrh

Where r = radius and h = height

Calculations:

⇒ 2 × (22 ÷ 7) × r × h = 880

⇒ (44 ÷ 7) × r × h = 880

⇒ r × h = 880 × (7 ÷ 44)

⇒ r × h = 20 × 7

⇒ r × h = 140 cm²

∴ The correct answer is 140 cm².

Additional Information

Total Surface Area of Cylinder

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

Volume of Cylinder

The volume represents the total space occupied by the cylinder: V = πr²h.

Surface Area of Cone

For a cone with base radius r and slant height l, the Curved Surface Area = πrl and Total Surface Area = πr(r + l).

Shortcut Trick

The minute hand covers 2π radians (a full circle) in 60 minutes.

Angle turned in 1 minute = 2π ÷ 60 = π/30 radians.

Angle turned in 25 minutes = 25 × π/30 = 5π/6 radians.

∴ The correct answer is 5π/6 radians.

Alternate Method

Given:

Time duration for which the minute hand turned = 25 minutes

Formula Used:

Angle (in radians) = (Given minutes ÷ Total minutes in 1 hour) × 2π

Calculations:

⇒ Total minutes in a complete clock revolution = 60 minutes

⇒ Fraction of the clock covered by the minute hand = 25 ÷ 60 = 5/12

⇒ The total angle for one full revolution = 360° = 2π radians

⇒ Angle turned in 25 minutes = (5/12) × 2π

⇒ Angle turned = 10π ÷ 12 = 5π/6 radians

∴ The correct answer is 5π/6 radians.

Additional Information

Speed of the Minute Hand

The minute hand moves at a constant speed of 6° per minute (which is equivalent to π/30 radians per minute).

Speed of the Hour Hand

The hour hand moves much slower at a speed of 0.5° per minute (which is equivalent to π/360 radians per minute).

Angle Between Clock Hands

The exact internal angle θ between the hour and minute hand at H hours and M minutes can be calculated using the formula: θ = |30H − 11/2 M| degrees.

Degrees to Radians Conversion

To convert an angle from degrees into radians, simply multiply the value by π/180. For example, an angle of 150° corresponds to 150° × (π/180) = 5π/6 radians.

Shortcut Trick

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

Let the assumed average of the class be 62 (the average of the first group).

Deviation of the second group from the assumed average = 72 − 62 = +10.

Net average deviation = (3 × 10) ÷ (2 + 3) = 30 ÷ 5 = +6.

Overall average = Assumed Average + Net deviation = 62 + 6 = 68.

∴ The correct answer is 68.

Alternate Method

Given:

Total number of students = 30

Average score of 12 students = 62

Average score of the remaining students = 72

Formula Used:

Overall Average = (Total sum of all scores) ÷ (Total number of students)

Avg = 62 12 students (Ratio: 2)

Avg = 72 18 students (Ratio: 3)

Overall Avg = 68

Diff = 6

Diff = 4

Calculations:

⇒ Number of remaining students = 30 − 12 = 18

⇒ Sum of scores of 12 students = 12 × 62 = 744

⇒ Sum of scores of 18 students = 18 × 72 = 1296

⇒ Total sum of scores of 30 students = 744 + 1296 = 2040

⇒ Overall Average = 2040 ÷ 30 = 68

∴ The correct answer is 68.

Additional Information

Weighted Average

When two groups with sizes N₁ and N₂ have averages A₁ and A₂ respectively, the combined average is calculated as (N₁A₁ + N₂A₂) ÷ (N₁ + N₂).

Deviation Method

To simplify calculations, select an assumed average. The overall average equals the assumed average plus the net deviation divided by the total number of observations.

Alligation Method

The ratio in which two quantities are mixed is inversely proportional to the differences of their individual averages from the overall average.

Shortcut Trick

Put A = 45° ⇒ m = a sec 45° = a√2 and y = b tan 45° = b

Substitute in b²m² − a²y² + (a²y²) / (b²m²) + cos²A

⇒ b²(a√2)² − a²(b)² + (a²b²) / (b²(a√2)²) + cos²45°

⇒ 2a²b² − a²b² + (a²b²) / (2a²b²) + (1/√2)²

⇒ a²b² + 1/2 + 1/2 = a²b² + 1

∴ The correct answer is a²b² + 1.

Alternate Method

Given:

m = a secA ⇒ secA = m/a

y = b tanA ⇒ tanA = y/b

Formula Used:

sec²A − tan²A = 1

sin²A + cos²A = 1

Calculations:

⇒ (m/a)² − (y/b)² = 1

⇒ (b²m² − a²y²) / (a²b²) = 1

⇒ b²m² − a²y² = a²b²

Evaluating the fractional term:

⇒ (a²y²) / (b²m²) = (a²b² tan²A) / (a²b² sec²A)

⇒ tan²A / sec²A = (sin²A / cos²A) × cos²A = sin²A

Substitute these calculated values into the main expression:

⇒ b²m² − a²y² + (a²y²) / (b²m²) + cos²A

⇒ a²b² + sin²A + cos²A

⇒ a²b² + 1

∴ The correct answer is a²b² + 1.

Additional Information

Trigonometric Identities

sin²θ + cos²θ = 1

sec²θ − tan²θ = 1

cosec²θ − cot²θ = 1

Trigonometric Conversions

tanθ = sinθ / cosθ ⇒ tanθ × cosθ = sinθ

secθ = 1 / cosθ ⇒ secθ × cosθ = 1

Shortcut Trick

When two pipes can fill a tank in x and y minutes individually, the time taken to fill the tank together is (x × y) ÷ (x + y).

Time taken = (36 × 45) ÷ (36 + 45)

⇒ Time taken = 1620 ÷ 81

⇒ Time taken = 20 minutes

∴ The correct answer is 20 minutes.

Alternate Method

Given:

Time taken by Pipe P to fill the tank = 36 minutes

Time taken by Pipe Q to fill the tank = 45 minutes

Calculations:

Total capacity of the tank = LCM of 36 and 45 = 180 units

⇒ Combined efficiency of P and Q = 5 + 4 = 9 units/minute

⇒ Time taken to fill the tank together = Total Work ÷ Combined Efficiency

⇒ Time taken = 180 ÷ 9

⇒ Time taken = 20 minutes

∴ The correct answer is 20 minutes.

Additional Information

Inlet and Outlet Pipes

An inlet pipe fills the tank and its work (efficiency) is always taken as positive. An outlet pipe empties the tank and its work (efficiency) is always taken as negative.

Unitary Method for Time & Work

If a pipe can fill a tank in x hours, the part of the tank filled in 1 hour is 1/x. If two pipes operate simultaneously, their 1 hour work is given by (1/x + 1/y).

Three Pipes Working Together

If three pipes A, B and C have efficiencies e_a, e_b, and e_c respectively, the total time required to fill a tank is calculated as Total Capacity ÷ (e_a + e_b + e_c).

Given:

State A

Literate percentage = 55%

Literate male to female ratio = 5 : 6

Illiterate male to female ratio = 4 : 5

Literate male population = 5,00,000

Formula used:

Total population = Literate population + Illiterate population

For the male population, we use the ratio to calculate the total number of males, then calculate the total population using the given literate male count.

Calculations:

Let the total population of State A be P.

Literate population = 55% of P = (55/100) × P

Literate male population = 5/11 of literate population (since male to female ratio is 5 : 6)

⇒ 5/11 × (55/100) × P = 5,00,000

⇒ (5 × 55 × P)/(11 × 100) = 5,00,000

⇒ (275P)/1100 = 5,00,000

⇒ 275P = 5,00,000 × 1100

⇒ 275P = 5,50,00,000

⇒ P = 5,50,00,000 / 275

⇒ P = 20,00,000

Total population of State A = 20,00,000

Given:

Data table of car production (in lakhs) by six brands between 2018 and 2023:

Formula used:

Identify the maximum number of cars produced in any year by scanning the data across each row for every year.

Calculations:

For each year, compare the production data across all brands:

2018: A = 36, B = 23, C = 45, D = 35, E = 57, F = 43 → Maximum = E (57)

2019: A = 40, B = 37, C = 47, D = 37, E = 56, F = 45 → Maximum = E (56)

2020: A = 42, B = 28, C = 51, D = 42, E = 52, F = 56 → Maximum = F (56)

2021: A = 45, B = 32, C = 54, D = 32, E = 38, F = 41 → Maximum = C (54)

2022: A = 47, B = 34, C = 56, D = 36, E = 48, F = 52 → Maximum = C (56)

2023: A = 42, B = 26, C = 48, D = 28, E = 42, F = 40 → Maximum = C (48)

Answer:

The highest production was by Brand E in 2018 with 57 lakhs.

Shortcut Trick

Total list price of 6 smartphones = 6 × 25000 = Rs. 1,50,000

Trade discount = 30% of Total list price

Trade discount = (30 ÷ 100) × 1,50,000 = Rs. 45,000

∴ The correct answer is Rs. 45,000.

Alternate Method

Given:

Number of smartphones purchased = 6

Price of each smartphone = Rs. 25,000

Trade discount percentage = 30%

Formula Used:

Total List Price = Number of items × Price per item

Discount Amount = Total List Price × (Discount % ÷ 100)

Calculations:

⇒ Total List Price = 6 × 25,000

⇒ Total List Price = Rs. 1,50,000

⇒ Discount Amount = 1,50,000 × (30 ÷ 100)

⇒ Discount Amount = 1,50,000 × 0.30

⇒ Discount Amount = Rs. 45,000

∴ The correct answer is Rs. 45,000.

Additional Information

Trade Discount

It is a reduction in the list price of goods allowed by a manufacturer or wholesaler to retailers, typically for bulk purchases. It is not recorded individually in accounting books.

Cash Discount

A cash discount is allowed to encourage prompt payment. It is always calculated on the net invoice price (i.e., List Price − Trade Discount).

Successive Discounts

When two discounts a% and b% are given successively, the effective discount percentage is given by the formula: Effective Discount = (a + b − (a × b) ÷ 100)%.

Shortcut Trick

The mean proportional between two numbers a and b is given by √(a × b).

Required Mean = √(12.96 × 0.16)

We know that √12.96 = 3.6 and √0.16 = 0.4

⇒ Mean = 3.6 × 0.4 = 1.44

∴ The correct answer is 1.44.

Alternate Method

Given:

First number (a) = 12.96

Second number (b) = 0.16

a = 12.96 First Value

b = 0.16 Second Value

Mean Proportional x = √(a × b)

Formula Used:

Mean proportional (x) = √(a × b)

Calculations:

⇒ x = √(12.96 × 0.16)

⇒ x = √( (1296 ÷ 100) × (16 ÷ 100) )

⇒ x = √(20736 ÷ 10000)

⇒ x = 144 ÷ 100

⇒ x = 1.44

∴ The correct answer is 1.44.

Additional Information

Third Proportional

If a : b :: b : c, then c is called the third proportional to a and b. Formula: c = b² ÷ a.

Fourth Proportional

If a : b :: c : d, then d is called the fourth proportional to a, b, and c. Formula: d = (b × c) ÷ a.

First Proportional

If a : b :: c : d, then a is called the first proportional to b, c, and d. Formula: a = (b × c) ÷ d.

Shortcut Trick

For similar triangles, the ratio of corresponding sides is equal to the ratio of their perimeters.

⇒ RS ÷ AB = Perimeter of ΔRST ÷ Perimeter of ΔABC

⇒ RS ÷ 24 = 26 ÷ 39

⇒ RS ÷ 24 = 2 ÷ 3

⇒ RS = 16 cm

∴ The correct answer is 16 cm.

Alternate Method

Given:

Perimeter of ΔRST = 26 cm

Perimeter of ΔABC = 39 cm

Side AB = 24 cm

R S T RS = ? P = 26 cm

A B C AB = 24 cm P = 39 cm

ΔRST ∼ ΔABC

Formula Used:

For two similar triangles, the ratio of their perimeters is strictly equal to the ratio of their corresponding sides.

⇒ If ΔRST ∼ ΔABC, then Perimeter(ΔRST) ÷ Perimeter(ΔABC) = RS ÷ AB

Calculations:

⇒ 26 ÷ 39 = RS ÷ 24

⇒ 2 ÷ 3 = RS ÷ 24

⇒ RS = (2 × 24) ÷ 3

⇒ RS = 48 ÷ 3

⇒ RS = 16 cm

∴ The correct answer is 16 cm.

Additional Information

Ratio of Areas of Similar Triangles

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Area(Δ1) ÷ Area(Δ2) = (Side1 ÷ Side2)².

Ratio of Perimeters of Similar Triangles

The ratio of perimeters is exactly equal to the ratio of corresponding sides, altitudes, or medians. P1 ÷ P2 = S1 ÷ S2.

Congruence vs Similarity

Congruent triangles are identical in both shape and size (ratio 1:1), whereas similar triangles have the same shape but are proportionally scaled relative to each other.

Shortcut Trick According to the angle sum property of triangles, the sum of all three interior angles must equal exactly 180°. We can quickly sum the values in each option to find the valid set. Sum of {30°, 60°, 90°} = 30° + 60° + 90° = 180°, which perfectly forms a valid triangle. ∴ The correct answer is {30°, 60°, 90°}.

Alternate Method Given: Four sets of possible angles for a triangle:

1. {30°, 60°, 90°}
2. {80°, 50°, 90°}
3. {30°, 50°, 110°}
4. {30°, 50°, 90°} Formula Used: The sum of the interior angles of a valid triangle is exactly 180°.

60°

30° 90° A B C

Calculations: We check the sum of the angles for each given option: ⇒ Option 1: 30° + 60° + 90° = 180° (Valid Triangle) ⇒ Option 2: 80° + 50° + 90° = 220° (Invalid) ⇒ Option 3: 30° + 50° + 110° = 190° (Invalid) ⇒ Option 4: 30° + 50° + 90° = 170° (Invalid) ⇒ Only the first set of angles gives a sum of exactly 180°. ∴ The correct answer is {30°, 60°, 90°}.

Additional Information Types of Triangles by Angles Acute Triangle (all angles < 90°), Right Triangle (one angle = 90°), and Obtuse Triangle (one angle > 90°). Exterior Angle Property The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles. Triangle Inequality Theorem For any valid triangle, the sum of the lengths of any two sides must always be strictly greater than the length of the remaining third side.

Shortcut Trick

We know the standard algebraic identity: (a + b)² − (a − b)² = 4ab.

By reversing the terms, we get: (a − b)² − (a + b)² = −4ab.

Substitute a = g and b = h directly into the given expression.

⇒ (g − h)² − (g + h)² + 4gh = −4gh + 4gh = 0.

∴ The correct answer is 0.

Alternate Method

Given:

Expression: (g − h)² − (g + h)² + 4gh

Formula Used:

(a − b)² = a² + b² − 2ab

(a + b)² = a² + b² + 2ab

Calculations:

Expand the first term (g − h)²:

⇒ g² + h² − 2gh

Expand the second term (g + h)²:

⇒ g² + h² + 2gh

Substitute both expansions back into the original expression:

⇒ (g² + h² − 2gh) − (g² + h² + 2gh) + 4gh

Distribute the negative sign over the second term:

⇒ g² + h² − 2gh − g² − h² − 2gh + 4gh

Group the like terms together:

⇒ (g² − g²) + (h² − h²) − 2gh − 2gh + 4gh

⇒ 0 + 0 − 4gh + 4gh

⇒ 0

∴ The correct answer is 0.

Additional Information

Sum of Squares Identity

When adding the squares of sum and difference, the middle terms cancel out: (a + b)² + (a − b)² = 2(a² + b²).

Difference of Squares

The difference between the squares of two terms can be factored into a product: a² − b² = (a − b)(a + b).

Cube Formulas

The sum and difference formulas for perfect cubes are frequently used in simplification: (a ± b)³ = a³ ± b³ ± 3ab(a ± b).

Given:

State Q: Percentage of population below poverty line = 19%

State S: Percentage of population below poverty line = 15%

Male to Female ratio below poverty line for:

State Q: 2:5

State S: 1:6

Population of State Q = 14,700

Population of State S = 14,700

Formula used:

Number of males below poverty line = Total population × Percentage below poverty line × (Male ratio / Total ratio)

Calculations:

For State Q:

Population below poverty line = 14,700 × (19 / 100) = 2,793

Male ratio = 2, Female ratio = 5

Total ratio = 2 + 5 = 7

Number of males below poverty line = 2,793 × (2 / 7) = 798 males

For State S:

Population below poverty line = 14,700 × (15 / 100) = 2,205

Male ratio = 1, Female ratio = 6

Total ratio = 1 + 6 = 7

Number of males below poverty line = 2,205 × (1 / 7) = 315 males

Total number of males below poverty line in States Q and S:

Total males below poverty line = 798 (from Q) + 315 (from S) = 1,113 males

Shortcut Trick

Total Simple Interest (SI) percentage for 2 years at 15% p.a. = 15 × 2 = 30%.

Amount after the first 2 years = 10000 × 1.30 = ₹13,000.

The lender includes the interest in the principal, so the new principal for the next 2 years is ₹13,000.

Amount after the next 2 years = 13000 × 1.30 = ₹16,900.

∴ The correct answer is ₹16,900.

Alternate Method

Given:

Initial Principal (P₁) = ₹10,000

Rate of Interest (R) = 15% p.a.

Time period (T₁) = 2 years

Formula Used:

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

Amount (A) = P + SI

Year 0 Year 2 Year 4

₹10,000 ₹13,000 ₹16,900

(New P)

SI = ₹3,000

SI = ₹3,900

Total Amount Paid = ₹16,900

Calculations:

⇒ SI for the first 2 years = (10000 × 15 × 2) ÷ 100 = ₹3,000

⇒ Amount at the end of 2 years = 10000 + 3000 = ₹13,000

⇒ The new principal for the next 2 years (P₂) = ₹13,000

⇒ SI for the next 2 years = (13000 × 15 × 2) ÷ 100 = ₹3,900

⇒ Total amount paid = P₂ + SI for the next 2 years = 13000 + 3900 = ₹16,900

∴ The correct answer is ₹16,900.

Additional Information

Simple Interest (SI)

Simple interest is calculated uniformly on the initial principal throughout the loan period. The generic formula is SI = (P × R × T) ÷ 100.

Amount Formula

The total amount paid at the end of the term is the sum of the principal and the interest earned. Amount = P + SI.

Compounding Effect in SI

If simple interest is added to the principal midway during an extended period, the subsequent interest is calculated on the new increased principal. This effectively functions like compound interest for the new term block.

Shortcut Trick

When the arc length and radius of a sector are given, its area can be found directly using the formula:

Area = ½ × l × r

Where l = length of the arc, and r = radius of the circle.

⇒ Area = ½ × 7 × 12

⇒ Area = 7 × 6 = 42 cm²

∴ The correct answer is 42 cm².

Alternate Method

Given:

Radius of the sector (r) = 12 cm

Length of the arc (l) = 7 cm

Formula Used:

Angle subtended at the centre, θ = Arc length ÷ Radius (in radians)

Area of a sector = ½ × r² × θ

Calculations:

⇒ θ = l ÷ r = 7 ÷ 12 radians

⇒ Area of the sector = ½ × (12)² × (7 ÷ 12)

⇒ Area = ½ × 144 × (7 ÷ 12)

⇒ Area = ½ × 12 × 7

⇒ Area = 6 × 7 = 42 cm²

∴ The correct answer is 42 cm².

Additional Information

Area of Sector (Degree Measure)

If the central angle is given in degrees (θ), the area of the sector is calculated as Area = (θ ÷ 360°) × πr².

Length of an Arc (Degree Measure)

The length of an arc that subtends an angle θ at the centre is given by l = (θ ÷ 360°) × 2πr.

Perimeter of a Sector

The total perimeter of a sector consists of the arc length along with the two enclosing radii. Thus, Perimeter = l + 2r.

Shortcut Trick

Ratio of Expenditure to Savings = 80% : 20% = 4 : 1.

We apply the weighted average method: Income = Expenditure + Savings.

Change in Income = (Exp Ratio × Exp Change + Sav Ratio × Sav Change) ÷ Total Ratio

⇒ 15% = (4 × 5% + 1 × x%) ÷ 5

⇒ 75% = 20% + x% ⇒ x = 55%

∴ The correct answer is 55%.

Alternate Method

Given:

Kalyan spends 80% of his income.

Increase in income = 15%.

Increase in expenditure = 5%.

Formula Used:

Savings = Income − Expenditure

Percentage Increase = (Increase Value ÷ Initial Value) × 100

Calculation:

⇒ Let initial income be 100 units.

⇒ Initial expenditure = 80% of 100 = 80 units

⇒ Initial savings = 100 − 80 = 20 units

⇒ After a 15% increase, new income = 100 + 15 = 115 units

⇒ After a 5% increase, new expenditure = 80 + (5% of 80) = 80 + 4 = 84 units

⇒ New savings = New income − New expenditure = 115 − 84 = 31 units

⇒ Increase in savings = 31 − 20 = 11 units

⇒ Percentage increase in savings = (11 ÷ 20) × 100 = 55%

∴ The correct answer is 55%.

Additional Information

Income, Expenditure & Savings

The basic relationship is Income = Expenditure + Savings. Any percentage change in income impacts both expenditure and savings proportional to their initial distributions.

Percentage Change Formula

The percentage change is always calculated on the base original value: % Change = [(New Value − Old Value) ÷ Old Value] × 100.

Weighted Average (Alligation) Method

When a whole is divided into two parts in the ratio n₁ : n₂, the net percentage change can be calculated directly as: Net % = (n₁ × x₁ + n₂ × x₂) ÷ (n₁ + n₂).

Shortcut Trick

Ratio 1 (Profit on Price): 100 → 120 = 5 : 6

Ratio 2 (Profit on Weight): 950 → 1000 = 19 : 20

Net Ratio (CP : SP) = (5 × 19) : (6 × 20) = 95 : 120 = 19 : 24

Profit % = [(24 − 19) / 19] × 100 = 5/19 × 100 = 500/19 %

∴ The correct option is (B).

Alternate Method

Given:

Declared Profit % = 20%

False Weight = 950 gm instead of 1 kg (1000 gm)

Formula Used:

Overall Profit % = [(Total SP − Effective CP) / Effective CP] × 100

0 g 950 g (Real CP) 1200 g (Final SP)

Profit = 250 units (Seller spends for 950g) (Seller earns for 1200g)

Calculations:

⇒ Let the Cost Price (CP) of 1 gm = Re. 1

⇒ CP of 1000 gm = Rs. 1000

⇒ Seller sells 1 kg at 20% profit, so SP = 1000 + 20% of 1000 = Rs. 1200

⇒ Seller uses 950 gm instead of 1000 gm, so Effective CP = Rs. 950

⇒ Total Profit = SP − Effective CP = 1200 − 950 = Rs. 250

⇒ Profit % = (Profit / Effective CP) × 100

⇒ Profit % = (250 / 950) × 100

⇒ Profit % = 25/95 × 100 = 5/19 × 100 = 500/19 %

∴ The correct option is (B).

Additional Information

Dishonest Dealer Weight Profit

Profit % due to false weight = [ (True Weight − False Weight) / False Weight ] × 100

Combined Profit Rule

If a dealer sells at x% profit and uses y% less weight, the overall profit multiplier is [(100 + x) / (100 − y)].

Successive Percentage for Profit

Total Gain % = [ (100 + g₁)/100 × (100 + g₂)/100 − 1 ] × 100, where g₁ and g₂ are individual gains.

Shortcut Trick

The expression follows the algebraic pattern: (a² + b² + c² + 2ab + 2bc + 2ca) ÷ (a + b + c).

By using the standard identity, the numerator is equivalent to (a + b + c)².

Thus, the given expression simplifies directly to (a + b + c).

Here, substituting a = 1.5, b = 2.5, and c = 3.5.

Result = 1.5 + 2.5 + 3.5 = 7.5.

∴ The correct answer is 7.5.

Alternate Method

Given:

Expression: (1.5 × 1.5 + 2.5 × 2.5 + 3.5 × 3.5 + 2 × 1.5 × 2.5 + 2 × 2.5 × 3.5 + 2 × 1.5 × 3.5) ÷ (1.5 + 2.5 + 3.5)

Formula Used:

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

Calculation:

Let a = 1.5, b = 2.5, and c = 3.5

⇒ Numerator = a² + b² + c² + 2ab + 2bc + 2ca

⇒ Numerator = (a + b + c)²

⇒ Denominator = a + b + c

⇒ Expression = (a + b + c)² ÷ (a + b + c)

⇒ Expression = a + b + c

⇒ Expression = 1.5 + 2.5 + 3.5

⇒ Expression = 7.5

∴ The correct answer is 7.5.

Additional Information

Square of a Binomial

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

Difference of Squares

a² − b² = (a − b)(a + b)

Sum of Cubes Identity

a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca)

Conditional Zero Sum

If a + b + c = 0, then a³ + b³ + c³ = 3abc

Shortcut Trick

Substitute A = 60° directly (cosec 60° = 2/√3, sec 60° = 2).

First term = (2/√3) ÷ (2/√3 − 1) + (2/√3) ÷ (2/√3 + 1) = 2/(2 − √3) + 2/(2 + √3).

First term = 2(2 + √3) + 2(2 − √3) = 8.

Second term = 2 ÷ (2 − 1) + 2 ÷ (2 + 1) = 2/1 + 2/3 = 8/3.

Given expression = 8 × (8/3)⁻¹ = 8 × 3/8 = 3.

∴ The correct answer is 3.

Alternate Method

Given:

Expression: [cosecA / (cosecA − 1) + cosecA / (cosecA + 1)] × [secA / (secA − 1) + secA / (secA + 1)]⁻¹

Calculations:

⇒ Let's simplify the first part:

⇒ cosecA / (cosecA − 1) + cosecA / (cosecA + 1)

⇒ = cosecA × [ (cosecA + 1 + cosecA − 1) ÷ (cosec²A − 1) ]

⇒ = cosecA × [ 2cosecA ÷ cot²A ] = 2cosec²A ÷ cot²A

⇒ = 2(1/sin²A) ÷ (cos²A/sin²A) = 2/cos²A = 2sec²A

⇒ Now, simplify the second part:

⇒ secA / (secA − 1) + secA / (secA + 1)

⇒ = secA × [ (secA + 1 + secA − 1) ÷ (sec²A − 1) ]

⇒ = secA × [ 2secA ÷ tan²A ] = 2sec²A ÷ tan²A

⇒ = 2(1/cos²A) ÷ (sin²A/cos²A) = 2/sin²A = 2cosec²A

⇒ Substitute both simplified parts into the original expression:

⇒ Result = 2sec²A × (2cosec²A)⁻¹ = 2sec²A ÷ 2cosec²A

⇒ Result = (1/cos²A) ÷ (1/sin²A) = sin²A ÷ cos²A = tan²A

⇒ Given A = 60°, substitute the value:

⇒ tan²60° = (√3)² = 3

∴ The correct answer is 3.

Additional Information

Trigonometric Identities

Important squared identities: sin²θ + cos²θ = 1 | sec²θ − tan²θ = 1 | cosec²θ − cot²θ = 1.

Reciprocal Relations

sinθ = 1 ÷ cosecθ | cosθ = 1 ÷ secθ | tanθ = 1 ÷ cotθ.

Standard Values at 60°

sin 60° = √3/2 | cos 60° = 1/2 | tan 60° = √3 | sec 60° = 2 | cosec 60° = 2/√3.

Shortcut Trick

Let b, c, and f be the unit prices of biscuit, chocolate, and coffee.

Given: 2b + c = 69; 2c + 3f = 127; 3b + 4c + 2f = 229.

From the first, c = 69 − 2b. Substitute this in the other two equations:

4b − 3f = 11 and 5b − 2f = 47. Solving these gives b = 17, f = 19.

Hence, c = 69 − 2(17) = 35. Total cost of 1 set = 17 + 35 + 19 = 71.

∴ The correct answer is ₹355.

Alternate Method

Let the cost of 1 biscuit, 1 chocolate, and 1 coffee be b, c, and f respectively.

Given:

⇒ 2b + c = 69   --- (1)

⇒ 2c + 3f = 127   --- (2)

⇒ 3b + 4c + 2f = 229   --- (3)

From (1), we get c = 69 − 2b.

Substitute c in (2):

⇒ 2(69 − 2b) + 3f = 127

⇒ 138 − 4b + 3f = 127

⇒ 4b − 3f = 11   --- (4)

Substitute c in (3):

⇒ 3b + 4(69 − 2b) + 2f = 229

⇒ 3b + 276 − 8b + 2f = 229

⇒ 5b − 2f = 47   --- (5)

Multiply (4) by 2 and (5) by 3:

⇒ 8b − 6f = 22

⇒ 15b − 6f = 141

Subtracting the first equation from the second:

⇒ 7b = 119 ⇒ b = 17

Putting b = 17 in (1):

⇒ 2(17) + c = 69 ⇒ c = 69 − 34 = 35

Putting c = 35 in (2):

⇒ 2(35) + 3f = 127 ⇒ 3f = 127 − 70 = 57 ⇒ f = 19

Total cost of 5 of each item = 5 × (b + c + f)

⇒ 5 × (17 + 35 + 19) = 5 × 71 = 355

∴ The correct answer is ₹355.

Additional Information

System of Linear Equations

A collection of one or more linear equations involving the same variables. They can be solved systematically using substitution, elimination, or matrix methods to find the unique solution values.

Substitution Method

A method for solving systems of equations by expressing one variable in terms of the others and substituting it into the remaining equations to reduce the number of variables.

Elimination Method

A process of multiplying equations by specific constants so that adding or subtracting them eliminates one of the variables entirely, allowing for easier resolution.

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

Key Points

• The word "prosperity" means a state of being successful, usually by making a lot of money. (समृद्धि )
• Example: The country enjoyed a period of peace and prosperity after the war.
• "Failure" refers to the lack of success or the inability to meet an expectation. (असफलता)
• Example: The project ended in failure due to poor planning.
• Hence, we can infer that the opposite of 'prosperity' is 'failure'.

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

Additional Information

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

• Security (सुरक्षा): The state of being free from danger or threat.
• Example: The government has increased security at all airports.
• Secure (सुरक्षित): Fixed or fastened so as not to give way, become loose, or be lost.
• Example: Please make sure the doors are secure before you leave.
• Affluence (धनाढ्यता): The state of having a great deal of money; wealth.
• Example: His affluence is evident from the luxurious lifestyle he leads.

The correct answer is 'Option 2'.

Key Points

• The word "Ambassador" (Option 2) fits perfectly as it refers to an important person who represents his/her country in a foreign country.

Hence, the correct answer is 'Option 2'.

Explanation: An ambassador is an official representative of a country's government who is appointed to represent their country in a foreign nation.

Additional Information

• Option 1: "Amateur" refers to a non-professional and does not describe the role of representing a country in a foreign land.
• Option 3: "Altruist" refers to a person with selfless concern for others, but does not pertain to the official role of an ambassador.
• Option 4: "Anonymous" means unknown, which is not suitable for the description of an ambassador.

The correct answer is Option 3.

 

Key Points

• Passive voice structure: The object of the active voice sentence becomes the subject of the passive voice sentence.
• In the active voice, the sentence is: "Martha made pancakes yesterday."
• To convert to passive voice: Place the object "pancakes" at the beginning.
• Change the active verb form "made" to "were made" (past simple passive).
• Add the preposition "by" followed by the original subject "Martha."
• Ensure the time phrase "yesterday" remains at the end of the sentence for clarity.

Therefore, the correct answer is Option 3.

 

Correct Sentence: "Pancakes were made by Martha yesterday."

 

Additional Information

• Option 1: "Pancakes had been made by Martha yesterday." – Incorrect tense (past perfect passive).
• Option 2: "Pancakes were being made by Martha yesterday." – Incorrect tense (past continuous passive).
• Option 4: "Pancakes was made by Martha yesterday." – Subject-verb agreement error.

The correct answer is Option 3.

 

Key Points

• The sentence construction begins by stating the existence of multiple reasons, establishing the context:
• "There are several reasons why even parents would also benefit from knowing more about how and why children do the things that they do."
• Sentence P is the first sentence as it introduces the existence of several reasons:
• "There are several reasons"
• Sentence Q follows as it begins to explain who benefits from these reasons:
• "why even parents would"
• Sentence O is the next part as it expands on the benefits that parents would have:
• "also benefit from knowing"
• Sentence R follows as it begins to detail what parents would know more about:
• "more about how and why"
• Sentence S is the concluding part as it completes the thought with the specific actions of children:
• "children do the things that they do"
• The correct sequence is PQORS.

 

Coherent sentence after rearrangement:

"There are several reasons why even parents would also benefit from knowing more about how and why children do the things that they do."

The correct answer is: Option 3.

Key Points

• The idiom "through thick and thin" means supporting someone in all circumstances, good or bad, no matter how difficult. (अच्छे और बुरे समय में)
• Example: True friends will stand by you through thick and thin, no matter what happens.

Therefore, the correct answer is: Option 3.

Additional Information

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

• Option 1: Under easy situations wherein she could ensure Monica's safety and security (सिर्फ आसान परिस्थितियों में समर्थन देना): Supporting only in simple and safe conditions.
• Example: She helps her friends only when things are going smoothly.
• Option 2: In all the happy moments, personally spending most of her time (सिर्फ खुशहाल क्षणों में साथ रहना): Being with someone only during joyful times.
• Example: She enjoys celebrating with friends in their happy times.
• Option 4: In the moments when Monica couldn't complete some simple tasks (सिर्फ तब मदद करना जब वे सरल कार्यों में असफल हों): Helping someone only when they struggle with simple tasks.
• Example: She only steps in to help when her friend faces minor difficulties.