SSC CGL 2024 Tier-I Official Paper (Held On: 18 Sept, 2024 Shift 1) (Previous Year Paper)

The logic followed here is:

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

Now,

For, 3 — 20 — 7

⇒ (3 + 7) × 2

⇒ 10 × 2 = 20.

For, 5 — 26 — 8

⇒ (5 + 8) × 2

⇒ 13 × 2 = 26.

Similarly, for 8 — 28 — 6

⇒ (8 + 6) × 2

⇒ 14 × 2 = 28.

Hence, the correct answer is "Option 3".

According to the given information,

Thus, according to the given codes the code '7' is for the word 'she'.

Hence, the correct answer is "Option 1".

The logic followed here is:​

Hence, "Option 3" is the correct answer.

The correct mirror image of the given figure is:

Hence, the correct answer is "Option 4".

The logic followed here is :-

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

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

Opposite pairs:

A → E

B → F

C → D

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

Hence, the correct answer is "Option 3".

The logic followed here is:

Hence, "Option 3" is the correct answer.

The logic followed here is:

Logic: Taste association.

Here,

Sugar : Sweet → Sugar is associated with Sweet taste.

Similarly,

Habanero : Spicy → Habanero is associated with Spicy taste.

Hence, the correct answer is "Option 4".

The logic followed here is:

1. First box series →

The elements shift anticlockwise subsequently.

The top left corner element is replaced by new element.

2) Second box series →

The elements shift clockwise subsequently.

The top left corner element is replaced by new element.

Thus, the final series is:

Hence, "Option 1" is the correct answer.

The logic followed here is:

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

Now,

For (5, 80, 16):

⇒ 5 × 16 = 80.

(18, 108, 6)

⇒ 18 × 6 = 108.

Similarly, for (13, 91, 7)

⇒ 13 × 7 = 91.

Hence, the correct answer is "Option 2".

Decoding symbols:

Given: P × Q ÷ R − S × A

P × Q → P is the wife of Q.

Q ÷ R → Q is the father of R.

R − S → R is the brother of S.

S × A → S is the wife of A.

Thus, S is P's daughter.

Hence, the correct answer is "Option 2".

The logic followed here is:

1. First box series →

The left diagonal elements shift anticlockwise subsequently.

No change in the rest of the elements.

2) Second box series →

The right diagonal elements shift clockwise subsequently.

No change in the rest of the elements.

Thus, the final series is:

Hence, "Option 4" is the correct answer.

The image embedded in the given figure is as shown below :

∴ Here, given figure is embedded in the option figure 1.

Hence, the correct answer is "Option 1".

Given series: _ Y Z K X K X _ Z K X _ X Y _ K X K X Y Z _ X K X Y

Option 1) XYZKK → X Y Z K X K X Y Z K X Z X Y K K X K X Y Z K X K X Y → (It doesn't follow any pattern)

Option 2) XYKZK → X Y Z K X K X Y Z K X K X Y Z K X K X Y Z K X K X Y → (It follows the pattern of: XYZKXK - XYZKXK - XYZKXK - XYZKXK)

Option 3) KXYKZ → K Y Z K X K X X Z K X Y X Y K K X K X Y Z Z X K X Y → (It doesn't follow any pattern)

Option 4) KYXZK → K Y Z K X K X Y Z K X X X Y Z K X K X Y Z K X K X Y → (It doesn't follow any pattern)

Hence, "Option 2" is the correct answer.

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

Conclusions:

I. Some pineapples are plums → Follow (As some pineapples are oranges, all oranges are bananas and all bananas are plums. As whole oranges comes in bananas which whole comes in plums and some part of pineapples is common with oranges, therefore it is true.)

II. All oranges are plums → Follow (As all oranges are bananas and all bananas are plums. As whole oranges comes in bananas which whole comes in plums, therefore it is true.)

III. No pineapple is a banana → Does not follow (As some pineapples are oranges and all oranges are bananas. No direct relation is given between pineapples and bananas.

∴ Here, Only conclusions I and II follow.​

Hence, the correct answer is "Option 2".

Given: 1064 × 2 − 653 ÷ 2 + 89 = ?480÷0.3×5+500190+9×27÷1.6=2480÷0.3×5+500190+9×27÷1.6=2

According to the question, after '+' and '−' are interchanged and '×' and '÷' are interchanged:

1064 × 2 − 653 ÷ 2 + 89 = ? ⇒ 1064 ÷ 2 + 653 × 2 − 89 = ?

⇒ 1064 ÷ 2 + 653 × 2 − 89 = ?

⇒ 532 + 653 × 2 − 89 = ?

⇒ 532 + 1306 − 89 = ?

⇒ 1838 − 89 = ?

⇒ 1749 = ?

Hence, "Option 2" is the correct answer.

The logic followed here is:

So, checking each option one by one:

Option 1) 16 – 9 – 139

16 × 9 - 5 = 139

144 - 5 = 139

139 = 139 (LHS = RHS)

Option 2) 25 – 7 – 175

25 × 7 - 5 = 175

175 - 5 = 175

170 ≠ 175 (LHS ≠ RHS)

Option 3) 12 – 7 – 79

12 × 7 - 5 = 79

84 - 5 = 79

79 = 79 (LHS = RHS)

Option 4) 17 – 5 – 80

17 × 5 - 5 = 80

85 - 5 = 80

80 = 80 (LHS = RHS)

Thus, out of all the options '25 – 7 – 175' is the odd one out.

Hence, "Option 2" is the correct answer.

The logic followed here is:

Logic: (1^st number × 5) + 1 = 2^nd number.

Now,

For, 51 : 256

⇒ (51 × 5) + 1

⇒ 255 + 1 = 256.

For, 53 : 266

⇒ (53 × 5) + 1

⇒ 265 + 1 = 266.

Similarly, for 55 ∶ ?

⇒ (55 × 5) + 1

⇒ 275 + 1 = 276.

Hence, the correct answer is "Option 1".

The logic followed here is:

'TABU' is written as 'SBAV'

And,

'TANK' is written as 'SBML'

Similarly,

'TAIL' is written as

Hence, "Option 3" is the correct answer.

Given word: 'MENTOR'

After arranged in alphabetical order:

 

Now,

there is One alphabet "N" will remain unchanged if each of the letters in the word 'MENTOR' is arranged in English alphabetical order.

Hence, the correct answer is "Option 2".

Given: 37 ÷ 56 – 7 × 2 + 6 = ?480÷0.3×5+500190+9×27÷1.6=2480÷0.3×5+500190+9×27÷1.6=2

According to the question, after '÷' and '–' are interchanged and '×' and '+' are interchanged:

37 ÷ 56 – 7 × 2 + 6 = ? ⇒ 37 – 56 ÷ 7 + 2 × 6 = ?

⇒ 37 – 56 ÷ 7 + 2 × 6 = ?

⇒ 37 – 8 + 2 × 6 = ?

⇒ 37 + 12 – 8 = ?

⇒ 49 – 8 = ?

⇒ 41 = ?

Hence, "Option 1" is the correct answer.

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

Thus, as marked there are a total of 14 triangles.

Hence, "Option 2" is the correct answer.

The logic followed here is:

'DRIVE' is coded as '62749'

'RIDES' is coded as '76284'

Thus, the code of 'V' is '9'.

Hence, "Option 3" is the correct answer.

The logic followed here is:

Option 1) Tired - exhausted ⇒ Correct

Tired is a synonym of exhausted.

Option 2) Loud - noisy ⇒ Correct

Loud is a synonym of noisy.

Gynecologist specialize in uterus. 

Option 3) Close - open ⇒ Incorrect

Close is a antonym of open.

Option 4) Honest - truthful ⇒ Correct

Honest is a synonym of truthful.

Hence, "Option 3" is the correct answer

The figure which will look like the paper when unfolded is shown below:

Thus, option Figure 2 will look like the paper when unfolded.

Hence, "Option 2" is the correct answer.

The logic followed here is:

JAP: BSH

And,

NDZ: FVR

Similarly,

LCX:?

Hence, "Option 2" is the correct answer.

The correct answer is Due to the decomposition of silver chloride into silver and chlorine by light.

Key Points

• Silver chloride (AgCl) is a white crystalline solid.
• When exposed to sunlight, AgCl decomposes into silver (Ag) and chlorine gas (Cl₂).
• The silver formed is a grey metal, which imparts the grey color to the white AgCl.
• This reaction is a type of photodecomposition reaction, where light energy causes the chemical breakdown of a compound.

Additional Information

• Photodecomposition Reaction
• This is a chemical reaction in which a compound is broken down into simpler substances by the action of light.
• In the case of silver chloride, light energy breaks the bond between silver and chlorine atoms.
• Such reactions are common in photochemistry and are used in various applications including photography.
• Silver Chloride (AgCl)
• It is a white crystalline solid that is well known for its low solubility in water.
• AgCl is used in photographic films and papers, where it plays a crucial role in capturing images.
• Photographic Process
• Silver halides like AgCl are used in photographic films because they are sensitive to light.
• When exposed to light, they decompose to form silver, which creates the image on the film.
• This process is the basis for traditional film photography.

The correct answer is Ram Mohan Roy.

Key Points

• Raja Ram Mohan Roy is known as the father of the Indian Renaissance due to his significant contributions to social, religious, and educational reforms in India.
• He founded the Brahmo Samaj in 1828 in Calcutta with the aim of reforming and modernizing Indian society by promoting monotheism and rationalism.
• Raja Ram Mohan Roy campaigned vigorously against social evils like sati (the practice of widow immolation) and child marriage, advocating for the rights of women and other marginalized groups.
• He also played a crucial role in the abolition of sati in 1829, collaborating closely with the British government to ensure its eradication.

Additional Information

• Brahmo Samaj
• The Brahmo Samaj was one of the earliest socio-religious reform movements in India, aimed at eradicating social and religious practices that were considered regressive and superstitious.
• It advocated for the worship of one God without any rituals or priestly intermediaries, and emphasized the importance of rationality and human dignity.
• The Brahmo Samaj also worked towards promoting education, especially for women, and campaigned for social reforms such as the abolition of caste discrimination and the promotion of widow remarriage.
• It laid the foundation for many future reform movements in India and influenced prominent leaders and thinkers such as Keshab Chandra Sen and Rabindranath Tagore.
• Raja Ram Mohan Roy
• Born on May 22, 1772, in Radhanagar, Bengal, Raja Ram Mohan Roy was a polymath who made significant contributions to Indian society through his work as a social and educational reformer, and his advocacy for human rights.
• He was a proponent of modern scientific knowledge and believed in the importance of English education for the progress of India.
• In addition to founding the Brahmo Samaj, he also established the Atmiya Sabha (Society of Friends) in 1815 to debate and discuss theological and philosophical issues.
• Raja Ram Mohan Roy was also a prolific writer and published several works in Bengali, Hindi, Persian, and English, addressing various social, political, and religious issues.

The correct answer is Mahi and Sabarmati.

Key Points

• The Mahi River is one of the major west-flowing rivers in India, flowing through the states of Madhya Pradesh, Rajasthan, and Gujarat before draining into the Arabian Sea.
• The Sabarmati River originates in the Aravalli Range of the Udaipur District of Rajasthan and flows into the Gulf of Khambhat in the Arabian Sea, making it a west-flowing river.
• Both rivers play a crucial role in the irrigation and water supply for the regions they flow through, supporting agriculture and local communities.
• West-flowing rivers like Mahi and Sabarmati are less common in India compared to the east-flowing rivers, which generally drain into the Bay of Bengal.

Additional Information

• West Flowing Rivers in India
• Most of the rivers in India flow eastwards towards the Bay of Bengal. However, there are a few significant rivers that flow westwards and drain into the Arabian Sea.
• Besides Mahi and Sabarmati, other notable west-flowing rivers include the Narmada and the Tapti.
• The Narmada River is one of the largest west-flowing rivers in India, originating from the Amarkantak Plateau and flowing through Madhya Pradesh and Gujarat before emptying into the Arabian Sea.
• The Tapti River originates from the Satpura Range of Madhya Pradesh and flows westward through Maharashtra and Gujarat, also draining into the Arabian Sea.
• Importance of River Systems
• Rivers are vital for the ecosystem as they provide water for drinking, irrigation, and industrial purposes.
• They also support a diverse range of flora and fauna, contributing to biodiversity.
• Rivers play a crucial role in shaping the landscape, influencing soil fertility and agricultural productivity.
• Many civilizations have historically settled along riverbanks, benefiting from the natural resources and transportation routes provided by rivers.
• Challenges Facing River Systems
• Pollution from industrial, agricultural, and domestic sources poses a significant threat to river ecosystems.
• Over-extraction of water for irrigation and other uses can lead to reduced water flow, affecting both human and ecological health.
• Climate change impacts, such as altered precipitation patterns and increased frequency of extreme weather events, can affect river flow and water availability.
• Efforts to manage and conserve river systems involve integrated water resource management, pollution control measures, and conservation initiatives to protect and restore river ecosystems.

The correct answer is 42^nd Constitutional Amendment Act.

Key Points

• The 42^nd Amendment Act, also known as the Constitution (Forty-second Amendment) Act, 1976, was enacted during the Emergency period in India.
• It is considered one of the most comprehensive amendments to the Indian Constitution and was brought in by the Indira Gandhi government.
• This amendment made significant changes to the Constitution, including the insertion of Part IV A, which deals with the Fundamental Duties of the citizens.
• The 42^nd Amendment also attempted to reduce the power of the judiciary and increase the power of the Parliament.

Additional Information

• Part IV A (Fundamental Duties)
• Part IV A was added to the Constitution of India by the 42^nd Amendment Act, 1976.
• It consists of Article 51A which enumerates the Fundamental Duties of the citizens of India.
• These duties are inspired by the Constitution of the former Soviet Union and aim to remind the citizens of their moral obligations towards the nation.
• There are currently 11 Fundamental Duties listed in Article 51A.
• 42^nd Amendment Act, 1976
• Known as a "mini-Constitution" due to its wide-ranging amendments.
• It made extensive changes to the Preamble, Fundamental Rights, Directive Principles, and the structure of the executive and judiciary.
• This amendment curtailed the power of the judiciary and strengthened the central government.
• Many of the provisions of this amendment were later undone by the 44^th Amendment Act, 1978.
• Emergency Period
• The Emergency in India was declared by President Fakhruddin Ali Ahmed upon the advice of Prime Minister Indira Gandhi from 1975 to 1977.
• This period is considered one of the most controversial times in Indian political history, marked by the suspension of elections and civil liberties.
• The 42^nd Amendment was passed during this period, and it significantly altered the balance of power between the executive, legislature, and judiciary.

The correct answer is Santhal Rebellion.

Key Points

• The Santhal Rebellion, also known as the Santhal Hool, was a tribal rebellion in present-day Jharkhand, India, against both British colonial authority and zamindari system.
• The rebellion was led by the four Murmu Brothers - Sidhu, Kanhu, Chand, and Bhairav.
• The Santhals were an indigenous tribal community that faced exploitation and oppression under the British rule and local landlords.
• The rebellion began on 30^th June 1855 and lasted until the suppression of the uprising by the British in 1856.

Additional Information

• Moplah Rebellion
• The Moplah Rebellion of 1921 was an armed revolt by the Mappila Muslims of Kerala against British authority and the Hindu landlords.
• The rebellion was part of the Khilafat Movement and was characterized by communal violence between Mappilas and Hindus.
• The British suppressed the rebellion with considerable violence, and it is considered one of the significant uprisings against colonial rule in India.
• Kol Rebellion
• The Kol Rebellion (1831-32) was a revolt by the Kols, a tribal community in the Chotanagpur region against British rule and local landlords.
• The rebellion was primarily against the introduction of new revenue policies and exploitation by moneylenders and landlords.
• The British eventually suppressed the rebellion, but it remains a significant event in the history of tribal resistance in India.
• Munda Rebellion
• The Munda Rebellion (1899-1900) was led by Birsa Munda, a tribal leader, against British colonial rule in the Chotanagpur Plateau area.
• The rebellion aimed to establish Munda Raj and remove British government and missionaries from the region.
• The British suppressed the rebellion, but Birsa Munda is remembered as a prominent tribal freedom fighter in Indian history.

The correct answer is the crust.

Key Points

• The crust is the outermost layer of the Earth, varying in thickness from about 5 km under the oceans to about 70 km under the continents.
• The crust is made up of a variety of rocks, primarily granite and basalt.
• It forms the solid surface of the Earth, where we live, and is divided into several tectonic plates.
• The Earth's crust is distinct from the underlying mantle and core, which are composed of different materials and have different physical properties.

Additional Information

• Volcano
• A volcano is an opening in the Earth's crust through which molten lava, ash, and gases are ejected.
• Volcanic activity is a result of the movement of tectonic plates and the presence of magma beneath the Earth's surface.
• Volcanoes can form on land or underwater, and they can create mountains and islands over time.
• The Core
• The core is the innermost part of the Earth, composed mainly of iron and nickel.
• It is divided into two parts: the solid inner core and the liquid outer core.
• The movement of the liquid outer core generates the Earth's magnetic field.
• The Mantle
• The mantle is the layer of the Earth located between the crust and the core.
• It is composed of silicate rocks that are rich in iron and magnesium.
• The mantle extends to a depth of about 2,900 kilometers (1,800 miles) and is involved in tectonic activity and convection currents.

The correct answer is Vidhan Parishad.

Key Points

• The Vidhan Parishad is the Upper House of the State Legislature in India.
• It is similar to the Rajya Sabha, which is the Upper House of the Parliament of India.
• Not all states in India have a Vidhan Parishad; it exists only in a few states like Uttar Pradesh, Bihar, Karnataka, Maharashtra, Andhra Pradesh, and Telangana.
• Members of the Vidhan Parishad are not directly elected by the public; they are elected by members of the Legislative Assembly, graduates, teachers, and local authority members.

Additional Information

• Rajya Sabha
• The Rajya Sabha is the Upper House of the Parliament of India.
• It represents the states and union territories of India.
• Members of the Rajya Sabha are elected by the elected members of the State Legislative Assemblies, and by the members of the Electoral college for Union Territories, using a single transferable vote system.
• The Rajya Sabha is a permanent body and is not subject to dissolution, but one-third of its members retire every two years.
• Lok Sabha
• The Lok Sabha is the Lower House of the Parliament of India.
• Members of the Lok Sabha are directly elected by the people of India through general elections held every five years.
• The Lok Sabha is also known as the House of the People.
• It plays a crucial role in the law-making process in India.
• Vidhan Sabha
• The Vidhan Sabha is the Lower House of the State Legislature in India.
• Members of the Vidhan Sabha are directly elected by the people of the respective states through general elections held every five years.
• The Vidhan Sabha is also known as the Legislative Assembly.
• It is responsible for making laws on matters listed in the State List and Concurrent List in the Seventh Schedule of the Indian Constitution.

The correct answer is Second.

Key Points

• India has the second-largest road network in the world, as per the Ministry of Road Transport and Highways' annual report (2020-2021).
• The total length of the road network in India is over 5.89 million kilometers.
• The Indian road network comprises National Highways, State Highways, and other roads.
• This extensive network facilitates efficient transportation and connectivity throughout the country.

Additional Information

• National Highways:
• National Highways are the primary roadways that connect major cities and regions across India.
• They are managed by the National Highways Authority of India (NHAI).
• National Highways constitute about 2% of the total road network but carry about 40% of the road traffic.
• State Highways:
• State Highways are roads that connect important cities within a state and link them with National Highways or highways of neighboring states.
• These highways are managed and maintained by the respective state governments.
• Other Roads:
• Other roads include district roads, rural roads, and urban roads that provide connectivity within districts, rural areas, and cities respectively.
• These roads are crucial for the last-mile connectivity and are maintained by local bodies such as Panchayats and Municipalities.
• Importance of Road Network:
• A well-developed road network is essential for economic growth, as it facilitates the movement of goods and people.
• It improves access to services such as healthcare and education, especially in rural areas.
• It also plays a critical role in national security by enabling the quick movement of defense forces and supplies.

The correct answer is Option 2: To increase the pH of the fresh milk from 6 to slightly alkaline.

Key Points

• Adding baking soda, which is a mild base, to fresh milk increases its pH, making it slightly alkaline.
• This process helps in delaying the souring of milk, which is caused by the production of lactic acid by bacteria.
• By making the milk slightly alkaline, the growth of acid-producing bacteria is slowed down, thereby preserving the milk for a longer duration.
• This practice is particularly useful in warmer climates where milk spoils faster due to higher temperatures.

Additional Information

• pH Scale
• The pH scale measures how acidic or basic a substance is, ranging from 0 to 14.
• A pH of 7 is considered neutral, below 7 is acidic, and above 7 is basic (alkaline).
• Milk typically has a pH of around 6, making it slightly acidic.
• Baking Soda (Sodium Bicarbonate)
• Baking soda, chemically known as sodium bicarbonate (NaHCO3), is a mild alkaline compound.
• It is commonly used in baking to cause dough to rise by producing carbon dioxide gas.
• In addition to its culinary uses, baking soda is also used for cleaning, deodorizing, and as an antacid.
• Lactic Acid and Milk Souring
• Lactic acid is produced by lactic acid bacteria during the fermentation of lactose (milk sugar).
• The accumulation of lactic acid lowers the pH of milk, leading to its souring.
• Souring is a natural process that occurs as milk ages and is influenced by temperature and bacterial activity.

The correct answer is atomic radius.

Key Points

• The atomic radius is measured in picometres (pm), where 1 pm = 10^-12 meters.
• The atomic radius is defined as the distance from the nucleus of an atom to the outer boundary of the electron cloud.
• The atomic radius varies across the periodic table, generally decreasing from left to right across a period and increasing down a group.
• Understanding atomic radius is crucial for explaining the physical and chemical properties of elements, including bonding and reactivity.

Additional Information

• Atomic Radius:
• The atomic radius is a measure of the size of an atom, usually the mean or typical distance from the nucleus to the boundary of the surrounding electron cloud.
• In a group, the atomic radius increases from top to bottom due to the addition of electron shells.
• In a period, the atomic radius decreases from left to right due to the increase in nuclear charge, which pulls electrons closer to the nucleus.
• Molar Mass:
• Molar mass is the mass of a given substance (chemical element or chemical compound) divided by the amount of substance, measured in grams per mole (g/mol).
• It is used to convert between the mass of a substance and the amount in moles.
• Atomic Density:
• Atomic density is the number of atoms per unit volume of a material.
• It is often measured in atoms per cubic centimeter (atoms/cm³).
• Spin Quantum Number:
• The spin quantum number (s) describes the intrinsic angular momentum (spin) of an electron within an atom.
• It can have values of +1/2 or -1/2, indicating the two possible spin states of an electron.

The correct answer is Amit Khare.

Key Points

• Amit Khare is a retired IAS officer of the 1985 batch from the Bihar cadre.
• He has served as the Secretary of the Ministry of Information and Broadcasting (I&B) and the Ministry of Human Resource Development (HRD), now known as the Ministry of Education.
• He is credited with the drafting of the National Education Policy (NEP) 2020, a significant reform in the Indian education system.
• In October 2021, Amit Khare was appointed as an advisor to the Prime Minister of India.

Additional Information

• Ministry of Information and Broadcasting (I&B)
• The Ministry of Information and Broadcasting is a branch of the Government of India responsible for formulating and administering the rules, regulations, and laws relating to information, broadcasting, the press, and films in India.
• The ministry is responsible for the administration of Prasar Bharati, the broadcasting corporation of India, and the regulation of content on television and radio.
• It plays a crucial role in disseminating information to the public and promoting cultural and artistic expression through various media platforms.
• National Education Policy (NEP) 2020
• NEP 2020 is a comprehensive framework to guide the development of education in the country, replacing the previous National Policy on Education, 1986.
• The policy aims to transform the Indian education system by 2040, with a focus on providing high-quality education, fostering critical thinking, and promoting innovation.
• Key highlights of NEP 2020 include the introduction of a new curricular structure (5+3+3+4), emphasis on early childhood care and education, and the integration of vocational education into mainstream education.
• The policy also emphasizes the use of technology in education, multilingualism, and the promotion of Indian culture and values.
• Advisor to the Prime Minister
• An advisor to the Prime Minister is a senior official appointed to provide expert advice and support on various matters of governance, policy formulation, and implementation.
• Advisors play a critical role in assisting the Prime Minister in making informed decisions and ensuring the effective execution of government initiatives.
• They bring specialized knowledge and experience to the administration, contributing to the overall efficiency and effectiveness of the government.

The correct answer is First.

Key Points

• The Harrod-Domar model was the foundation for the First Five Year Plan of India.
• This plan was implemented between 1951 and 1956.
• The model emphasizes the importance of savings and investments for economic growth.
• It was aimed at addressing the country's immediate need for development in agriculture, irrigation, and energy.

Additional Information

• Harrod-Domar Model:
• This model is an economic growth model that was developed independently by Sir Roy Harrod in 1939 and Evsey Domar in 1946.
• The model suggests that the economy's growth rate depends on the level of savings and the productivity of capital investment.
• The key equation of the Harrod-Domar model is: Growth Rate = Savings Ratio / Capital-Output Ratio.
• The model highlights the need for investment to spur economic growth and the importance of efficient use of capital.
• First Five Year Plan of India:
• Launched in 1951, the First Five Year Plan focused on the primary sector, especially agriculture.
• It aimed to increase food production, improve irrigation facilities, and develop transport and communication infrastructure.
• The plan was successful in achieving its targets and laid the foundation for future economic planning in India.
• It also emphasized the need for industrialization to create job opportunities and improve the standard of living.
• Importance of Savings and Investments:
• Savings provide the funds necessary for investment, which in turn leads to economic growth.
• Higher savings rates can lead to higher investment rates, which can enhance productive capacity and increase GDP.
• Efficient allocation and use of savings are crucial for maximizing the returns on investment.
• Government policies often aim to encourage savings and investments through various incentives and reforms.
• Development Goals of the First Plan:
• To increase agricultural production to ensure food security for the population.
• To develop irrigation facilities to support agriculture and manage water resources effectively.
• To build infrastructure such as roads, railways, and ports to facilitate trade and movement of goods.
• To address the immediate needs of the country in terms of employment and poverty alleviation.

The correct answer is 2008.

Key Points

• The Rashtriya Swasthya Bima Yojana (RSBY) was launched by the Government of India in 2008.
• RSBY aims to provide health insurance coverage for Below Poverty Line (BPL) families.
• Under this scheme, beneficiaries are entitled to hospitalization coverage up to Rs. 30,000 for most of the diseases that require hospitalization.
• The scheme was designed to protect poor families from financial liabilities arising out of health shocks that involve hospitalization.

Additional Information

• Rashtriya Swasthya Bima Yojana (RSBY)
• RSBY was introduced by the Ministry of Labour and Employment, Government of India.
• The scheme provides for cashless insurance for hospital services through an empaneled network of healthcare providers.
• Beneficiaries need to pay a registration fee of Rs. 30, and the government pays the premium to the insurer selected by the state government on the basis of a competitive bidding.
• The scheme covers all pre-existing conditions and there is no age limit for the beneficiaries.
• Transport expenses are also provided to the patient, within an overall limit of Rs. 1000.
• Health Insurance in India
• Health insurance in India is a type of insurance that covers medical expenses incurred by the insured due to health conditions.
• Health insurance policies can be purchased by individuals or families to cover a wide range of medical expenses, including hospitalization, surgery, and other medical treatments.
• Several government schemes like Ayushman Bharat, PM-JAY, and others aim to provide health insurance coverage to various segments of the population.
• The goal of these schemes is to reduce the financial burden of medical expenses and ensure access to quality healthcare services for all.

The correct answer is stock.

Key Points

• Money supply is considered a stock concept because it represents the total amount of money available in an economy at a specific point in time.
• Unlike a flow concept, which is measured over a period of time, a stock concept is measured at one particular time.
• The money supply includes various components such as currency in circulation, demand deposits, and other liquid assets.
• Central banks and government agencies typically measure the money supply to monitor and manage the economy.

Additional Information

• Stock vs. Flow Concepts
• A stock concept is a quantity measured at a particular point in time. Examples include the money supply, wealth, and the number of goods in inventory.
• A flow concept is a quantity measured over a period of time. Examples include income, expenditure, and production.
• Components of Money Supply
• M1: This is the narrow measure of money supply, which includes currency in circulation and demand deposits (checking accounts).
• M2: This is a broader measure, which includes M1 plus savings accounts, small time deposits, and non-institutional money market funds.
• M3: This includes M2 plus large time deposits, institutional money market funds, and other larger liquid assets.
• Importance of Money Supply
• The money supply is a critical factor in determining inflation levels, interest rates, and overall economic stability.
• Central banks, such as the Federal Reserve or the Reserve Bank of India, use measures of the money supply to implement monetary policy.
• Changes in the money supply can affect the economy's performance, influencing everything from consumer spending to business investment.

The correct answer is Governor.

Key Points

• The Governor is the constitutional head of a state in India, appointed by the President of India.
• On Republic Day, the Governor unfurls the national flag at the state capital as part of the official celebration.
• The Governor's role is to represent the central government in the state and act according to the advice of the state's Council of Ministers.
• The Governor has various powers and functions including legislative, executive, and discretionary powers as outlined in the Constitution of India.

Additional Information

• Republic Day
• Republic Day in India is celebrated on January 26^th every year to honor the date on which the Constitution of India came into effect in 1950.
• It marks the transition of India from a British Dominion to a republic.
• The day is celebrated with grand parades in the national capital, New Delhi, and state capitals, showcasing India's defense capability, cultural and social heritage.
• Role of the Governor
• The Governor is the nominal executive head of the state, functioning as a representative of the President of India.
• The Governor's powers and duties include summoning and proroguing the state legislature, granting pardons, and appointing the Chief Minister.
• Governors can also reserve bills passed by the state legislature for the President's consideration.
• Chief Minister
• The Chief Minister is the head of the government in a state and is responsible for running the state administration.
• They are appointed by the Governor and are usually the leader of the majority party in the state legislature.
• The Chief Minister advises the Governor on various matters related to the state's governance.

The correct answer is Keshav Prasad Maurya.

Key Points

• Keshav Prasad Maurya is one of the Deputy Chief Ministers of Uttar Pradesh as of July 2023.
• He has been serving as the Deputy Chief Minister since 19 March 2017, following the BJP's victory in the state assembly elections.
• Maurya has a significant political career and is known for his work in the Bharatiya Janata Party (BJP).
• He represents the Sirka constituency in the Uttar Pradesh Legislative Assembly.

Additional Information

• Deputy Chief Minister:
• A Deputy Chief Minister (DCM) is a member of the executive branch of the state government in India who typically acts as the second-in-command to the Chief Minister.
• The position is not a constitutional post but can be created by the ruling party to balance power within the party or coalition.
• DCMs may handle important portfolios and play a crucial role in the administration of the state.
• Uttar Pradesh:
• Uttar Pradesh is a state in northern India, known for its historical, cultural, and political significance.
• It is the most populous state in India and has a diverse socio-economic structure.
• The capital of Uttar Pradesh is Lucknow, and it has a major legislative and administrative setup.
• Bharatiya Janata Party (BJP):
• The Bharatiya Janata Party is one of the two major political parties in India, alongside the Indian National Congress.
• Founded in 1980, the BJP follows a right-wing political ideology and promotes policies that reflect Hindu nationalist positions.
• It is currently the ruling party in India and has a significant presence in various states, including Uttar Pradesh.
• Yogi Adityanath:
• Yogi Adityanath is the current Chief Minister of Uttar Pradesh, having assumed office on 19 March 2017.
• He is known for his strong leadership and focus on law and order, development, and cultural heritage of the state.
• Prior to becoming the Chief Minister, he served as a Member of Parliament from the Gorakhpur constituency for five consecutive terms.

The correct answer is Ravi Shankar.

Key Points

• Ravi Shankar was an Indian sitar virtuoso and a composer of Hindustani classical music.
• He was the best-known proponent of the sitar in the second half of the 20^th century and influenced many musicians worldwide.
• Ravi Shankar collaborated with many international artists, including George Harrison of The Beatles, which helped to popularize Indian classical music in the West.
• He received numerous awards and honors throughout his career, including Bharat Ratna in 1999, India's highest civilian award.

Additional Information

• Sitar
• The sitar is a plucked stringed instrument used mainly in Hindustani classical music.
• It derives its name from the Persian word 'sehtar', meaning "three strings," although modern sitars typically have seven strings.
• The sitar has a long neck with twenty metal frets and six or seven main strings.
• It is known for its distinctive sound and is used extensively in Indian classical music, as well as in fusion music and film scores.
• Hindustani Classical Music
• Hindustani classical music is one of the two main forms of Indian classical music, the other being Carnatic music.
• It originated in the Vedic ritual chants and has evolved over centuries to include a rich array of ragas and talas.
• This music form is primarily vocal-centric, but instrumental traditions such as the sitar, sarod, and tabla also play a significant role.
• Notable exponents of Hindustani classical music include Pandit Ravi Shankar, Ustad Zakir Hussain, and Pandit Bhimsen Joshi.
• George Harrison
• George Harrison was an English musician, singer-songwriter, and lead guitarist of the Beatles.
• He was deeply influenced by Indian classical music and incorporated elements of it into the Beatles' music.
• George Harrison was a student of Ravi Shankar and played the sitar on several Beatles tracks, including "Norwegian Wood" and "Within You Without You".
• His association with Ravi Shankar helped introduce Indian classical music to a global audience.
• Bharat Ratna
• Bharat Ratna is the highest civilian award of the Republic of India.
• It was instituted in 1954 and is awarded in recognition of exceptional service/performance of the highest order, without distinction of race, occupation, position, or sex.
• The award has been conferred on various distinguished individuals, including scientists, artists, and politicians.
• Ravi Shankar was awarded the Bharat Ratna in 1999 for his contributions to Indian classical music.

The correct answer is Kannada Literature.

Key Points

• Pampa, Ponna, and Ranna are renowned as the ‘Three Jewels’ (Ratnatraya) of Kannada literature.
• These poets made significant contributions during the period of Kannada classical literature which is also known as the Old Kannada period.
• Pampa is celebrated for his epic works like ‘Adi Purana’ and ‘Vikramarjuna Vijaya’.
• Ranna is known for his famous work ‘Sahasa Bhima Vijaya’ or ‘Gadayuddha’.

Additional Information

• Pampa
• Pampa, also known as Adikavi, is one of the earliest poets of Kannada literature.
• His significant works include ‘Adi Purana’, which narrates the story of the first Jain Tirthankara Rishabhadeva, and ‘Vikramarjuna Vijaya’, which is a Jain version of the Mahabharata.
• Ponna
• Ponna was another celebrated Kannada poet and a contemporary of Pampa.
• He is known for his work ‘Shanti Purana’ which is a Jain epic that narrates the life of Shantinatha, the 16^th Tirthankara.
• Ranna
• Ranna is considered one of the greatest Kannada poets.
• His notable work ‘Gadayuddha’ (The Battle of Clubs) is a retelling of the Mahabharata focusing on the duel between Bhima and Duryodhana.
• Kannada Literature
• Kannada literature has a rich heritage dating back over a thousand years.
• It includes a wide range of texts in poetry, prose, and drama, and is known for its classical and medieval works.
• The period of the ‘Three Jewels’ is particularly significant in the history of Kannada literature, often regarded as its golden age.

The correct answer is Padma Shri.

Key Points

• Annavarapu Rama Swamy is a renowned Indian classical violinist known for his significant contribution to Indian classical music.
• In recognition of his contributions, he was awarded the Padma Shri in 2021.
• The Padma Shri is the fourth highest civilian award in India.
• This award is given for distinguished service in any field including the arts, education, industry, literature, science, sports, medicine, social service, and public affairs.

Additional Information

• Padma Awards
• The Padma Awards are one of the highest civilian honors of India announced annually on the eve of Republic Day.
• They are given in three categories: Padma Vibhushan (for exceptional and distinguished service), Padma Bhushan (for distinguished service of a high order), and Padma Shri (for distinguished service in any field).
• The awards were instituted in 1954 and are awarded by the Government of India.
• The Padma Awards recognize achievements in all fields of activities or disciplines where an element of public service is involved.
• Sangeet Natak Akademi
• It is the national academy for music, dance, and drama in India.
• It was created by a resolution of the Ministry of Education, Government of India in 1952 and became functional the following year.
• The Akademi's fellowship and awards are considered the highest national recognition given to practicing artists.
• It is an autonomous body under the Ministry of Culture, Government of India.
• Padma Bhushan
• The Padma Bhushan is the third-highest civilian award in the Republic of India, preceded by the Bharat Ratna and the Padma Vibhushan and followed by the Padma Shri.
• It is awarded for "distinguished service of a high order... without distinction of race, occupation, position, or sex."
• The award was established on 2 January 1954 by the President of India.
• Padma Vibhushan
• The Padma Vibhushan is the second-highest civilian award of the Republic of India, after the Bharat Ratna.
• It is awarded for "exceptional and distinguished service," without distinction of race, occupation, position, or sex.
• The award was instituted by the Government of India on 2 January 1954.

The correct answer is Pitt's India Act of 1784.

Key Points

• The Pitt's India Act of 1784 was introduced to address the shortcomings of the Regulating Act of 1773.
• This act established the Board of Control in England to control and supervise the administration of British India.
• The Board of Control consisted of six members, including two members of the British cabinet and was headed by the British Secretary of State.
• This act also created a dual system of governance, with the East India Company retaining its commercial functions, while the British government assumed control over political affairs.

Additional Information

• Charter Act of 1813
• This act ended the trade monopoly of the East India Company in India, except for trade in tea and trade with China.
• It allowed Christian missionaries to come to India and spread their religion.
• The act also provided for the promotion of education in India and allocated funds for the same.
• Regulating Act of 1773
• This was the first step taken by the British government to control and regulate the affairs of the East India Company in India.
• It established a system of dual governance, where the British government and the East India Company both had a say in the administration of India.
• The act established a Governor-General of Bengal with an executive council of four members to assist him.
• Charter Act of 1793
• This act extended the charter of the East India Company for another 20 years.
• It allowed the company to retain its monopoly over trade in India.
• The act also increased the salaries of the company's officials to reduce corruption.
• Pitt's India Act of 1784
• This act was introduced by the British Prime Minister William Pitt the Younger.
• It established the Board of Control to oversee the affairs of the East India Company in India.
• The act created a dual system of governance, with the company retaining its commercial functions and the British government assuming control over political affairs.
• The Board of Control consisted of six members, including two members of the British cabinet, and was headed by the British Secretary of State.

The correct answer is Odisha.

Key Points

• Odisha emerged as the overall champion in both men and women categories in the first Janjatiya Khel Mahotsav.
• The event saw participation from various states, but Odisha dominated in terms of overall performance.
• Odisha's athletes showcased exceptional talent and skills, leading to their top position.
• The state has been investing in sports infrastructure and athlete development, which is reflected in their performance.

Additional Information

• Janjatiya Khel Mahotsav
• The Janjatiya Khel Mahotsav is a national sports event organized to promote and encourage the participation of tribal communities in sports.
• This event aims to recognize and nurture the sporting talent in tribal areas, providing a platform for athletes to showcase their skills.
• The initiative is part of the government's efforts to promote sports among marginalized communities and ensure their inclusion in the national sports ecosystem.
• The event includes various sports competitions, workshops, and cultural programs to celebrate the heritage and talent of tribal communities.
• Sports Infrastructure in Odisha
• Odisha has been making significant strides in developing sports infrastructure, including the construction of world-class facilities and training centers.
• The state has hosted several national and international sports events, showcasing its capability to support large-scale sports competitions.
• Odisha's government has been proactive in providing financial support and incentives to athletes, ensuring they have the resources needed to excel.
• The focus on sports development has helped Odisha produce top athletes who compete at national and international levels.

The correct answer is Samiti.

Key Points

• The Varna system is a social stratification system found in the Vedic period in ancient India.
• The four main Varnas are Brahmanas, Kshatriyas, Vaishyas, and Shudras.
• Brahmanas were priests and teachers, Kshatriyas were warriors and rulers, Vaishyas were traders, and Shudras were laborers.
• Samiti, however, was not a Varna; it was an assembly of people in the Vedic period used for administrative and deliberative purposes.

Additional Information

• Varna System
• The Varna system is an ancient social hierarchy found in Vedic India, dividing society into four main classes: Brahmanas, Kshatriyas, Vaishyas, and Shudras.
• Brahmanas were considered the highest Varna, responsible for religious rituals, teaching, and maintaining sacred knowledge.
• Kshatriyas were the warrior class and rulers, responsible for protecting and governing the land.
• Vaishyas were the merchant and artisan class, responsible for trade, agriculture, and commerce.
• Shudras were the laborers and service providers, meant to serve the other three Varnas.
• Samiti
• Samiti was not a Varna but an important assembly in the Vedic period.
• It was a gathering of people, often involved in decision-making, administrative matters, and the governance of the community.
• The Samiti worked alongside the Sabha, another assembly, to discuss and deliberate on important issues.
• The role of the Samiti was significant in the participatory governance system of Vedic society.

The correct answer is Geeta Phogat.

Key Points

• Geeta Phogat is the first Indian woman wrestler to qualify for the Olympic Games.
• She represented India at the 2012 London Olympics in the women's freestyle 55 kg category.
• Geeta Phogat is also a gold medalist at the Commonwealth Games, winning in 2010.
• She hails from a family of wrestlers, with her sisters Babita and Vinesh Phogat also being notable wrestlers.

Additional Information

• Olympic Games
• The Olympic Games are a major international multi-sport event held every four years.
• They are divided into Summer and Winter Olympic Games, with each occurring every four years but two years apart.
• The modern Olympic Games were inspired by the ancient Olympic Games held in Olympia, Greece, from the 8^th century BC to the 4^th century AD.
• The International Olympic Committee (IOC) is the governing body of the Olympic Movement, with the Olympic Charter defining its structure and authority.
• Commonwealth Games
• The Commonwealth Games is an international multi-sport event involving athletes from the Commonwealth of Nations.
• The event was first held in 1930 and, with the exception of 1942 and 1946 (cancelled due to World War II), has taken place every four years since then.
• The games are overseen by the Commonwealth Games Federation (CGF), which ensures the event is held in a manner consistent with its principles of equality and inclusivity.
• The Commonwealth Games include a range of sports and are known for their spirit of friendly competition and camaraderie among the member countries.
• Freestyle Wrestling
• Freestyle wrestling is a style of amateur wrestling that is practiced worldwide.
• In freestyle wrestling, athletes can use their legs in both offense and defense and are allowed to hold their opponents above or below the waist.
• It is one of the two styles of wrestling contested in the Olympic Games, the other being Greco-Roman wrestling.
• Freestyle wrestling emphasizes both the physical and technical aspects of the sport, making it a dynamic and challenging discipline.
• Phogat Family
• The Phogat family from Haryana, India, is renowned for their achievements in wrestling.
• Mahavir Singh Phogat, the patriarch of the family, is a former wrestler and coach who trained his daughters in the sport.
• Geeta, Babita, Ritu, and Vinesh Phogat have all achieved significant success in wrestling, making the family a symbol of perseverance and excellence.
• Their story was popularized in the 2016 Bollywood film "Dangal," which portrayed Mahavir Phogat's efforts to train his daughters to become world-class wrestlers.

The correct answer is Fucoxanthin.

Key Points

• Fucoxanthin is a marine carotenoid found in brown algae such as wakame and hijiki.
• It is responsible for giving brown seaweed its characteristic color.
• Fucoxanthin has been studied for its potential health benefits, including its effects on metabolism and weight management.
• It also possesses antioxidant properties, which help in protecting cells from damage.

Additional Information

• Carotenoids:
• Carotenoids are pigments found in plants, algae, and photosynthetic bacteria. They play a critical role in plant health by absorbing light energy for use in photosynthesis.
• They are divided into two classes: xanthophylls (which contain oxygen) and carotenes (which are purely hydrocarbons and contain no oxygen).
• Carotenoids are important for human health as they have antioxidant properties and can be converted into vitamin A in the body.
• Astaxanthin:
• Astaxanthin is another marine carotenoid, primarily found in microalgae, shrimp, and salmon. It is known for its strong antioxidant properties.
• It is often used as a dietary supplement for its potential health benefits, including eye and skin health, and anti-inflammatory effects.
• Neoxanthin:
• Neoxanthin is a type of xanthophyll found in green leafy vegetables and plays a role in the light-harvesting complex of photosynthesis.
• It has been studied for its potential anti-cancer properties.
• β-cryptoxanthin:
• β-cryptoxanthin is a carotenoid found in fruits such as oranges, papayas, and tangerines.
• It is a provitamin A carotenoid, meaning it can be converted into vitamin A in the body.
• It has been associated with various health benefits, including bone health and reducing the risk of certain chronic diseases.

The correct answer is To prohibit strikes, lockouts and layoffs in units engaged in essential defence services.

Key Points

• The Essential Defence Services Bill, 2021 was introduced to ensure the uninterrupted supply of essential services concerning the nation's defence and security.
• The Bill aims to prohibit strikes, lockouts, and layoffs in units engaged in essential defence services to maintain operational efficiency and readiness.
• This legislation empowers the government to declare any service in the defence sector as an essential defence service if its stoppage would affect the nation’s security, defence preparedness, or the functioning of the armed forces.
• Violations of the provisions of this Bill can lead to penalties, including imprisonment and fines, to ensure strict compliance and maintain discipline within these critical sectors.

Additional Information

• Foreign Direct Investment (FDI) in Defence Services:
• The Indian government allows up to 74% FDI in the defence sector through automatic route and beyond 74% through the government route wherever it is likely to result in access to modern technology or for other reasons to be recorded.
• FDI in the defence sector is aimed at boosting domestic manufacturing and reducing dependency on imports.
• Women in Defence Services:
• India has been progressively increasing the induction of women in various branches of the defence forces.
• Women are now allowed in combat roles, and the government has also taken steps to ensure permanent commission for women officers.
• One Rank One Pension (OROP):
• OROP is a long-standing demand of the Indian armed forces for uniform pension to be paid to military personnel retiring in the same rank with the same length of service, irrespective of their date of retirement.
• The scheme aims to bridge the gap between the pensions of current and past pensioners at periodic intervals.

Shortcut Trick

Total additional interest percentage = Time × Extra Rate = 6 × 7% = 42%

This 42% additional interest on the principal corresponds to ₹1512.

Since Principal is always exactly 100%, we calculate:

Principal (100%) = (1512 ÷ 42) × 100

Principal = 36 × 100 = ₹3600

∴ The correct answer is ₹3600.

Alternate Method

Given:

Time (T) = 6 years

Increase in Rate of Interest = 7%

Increase in Simple Interest = ₹1512

Formula Used:

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

Difference in SI = (P × Change in Rate × T) ÷ 100

Calculations:

⇒ Let the initial principal sum be P.

⇒ The additional interest earned is strictly due to the extra 7% rate applied over the 6 years duration.

⇒ Total extra interest percentage = 7% × 6 = 42%

⇒ According to the question, this 42% of P equals ₹1512.

⇒ (42 ÷ 100) × P = 1512

⇒ P = (1512 × 100) ÷ 42

⇒ P = 36 × 100

⇒ P = 3600

∴ The correct answer is ₹3600.

Additional Information

Direct Proportionality in Simple Interest

Simple Interest is directly proportional to the Principal, Rate, and Time. If only the Rate increases by an amount, say r%, the extra interest is calculated directly as Extra SI = (P × r × T) ÷ 100.

Amount in Simple Interest

The final amount obtained after time T is the sum of the Principal and the accumulated Simple Interest. Formula: A = P + SI = P × [1 + (R × T) ÷ 100].

Simple Interest vs Compound Interest

In Simple Interest, the principal remains the same every year and interest does not compound. In Compound Interest, the interest earned in previous years is added back to the principal, yielding interest on interest.

Shortcut Trick

The perpendicular from the centre bisects the chord, so half the chord is 32 ÷ 2 = 16 cm.

The radius, distance from the centre, and half-chord form a right-angled triangle.

Recognize the common Pythagorean triplet (12, 16, 20), which is a multiple of the (3, 4, 5) base triplet.

Hence, the radius of the circle is 20 cm.

∴ The correct answer is 20 cm.

Alternate Method

Given:

Length of the chord (AB) = 32 cm

Distance of the chord from the centre (OM) = 12 cm

Formula Used:

Pythagoras Theorem: Hypotenuse² = Perpendicular² + Base²

Calculations:

⇒ The perpendicular drawn from the centre of a circle to a chord bisects the chord.

⇒ Therefore, AM = MB = 32 ÷ 2 = 16 cm.

⇒ In right-angled triangle ΔOMB,

⇒ OB² = OM² + MB²

⇒ Radius (r)² = 12² + 16²

⇒ r² = 144 + 256

⇒ r² = 400

⇒ r = √400 = 20 cm

∴ The correct answer is 20 cm.

Additional Information

Perpendicular from Centre to Chord

A perpendicular drawn from the centre of a circle to a chord bisects the chord into two equal halves. Conversely, a line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.

Equal Chords and Distance

Equal chords of a circle are equidistant from the centre. Similarly, chords that are equidistant from the centre of a circle are always equal in length.

Intersecting Chords Theorem

If two chords AB and CD intersect internally at a point P, then the product of their respective line segments is always equal: AP × PB = CP × PD.

Shortcut Trick Effective discount formula = x + y − (x × y) ÷ 100 Discount I = 15 + 20 − (15 × 20) ÷ 100 = 32% Discount II = 25 + 10 − (25 × 10) ÷ 100 = 32.5% Discount III = 33% Selling price is minimum when discount is maximum, which is 33%. ∴ The correct answer is Scheme III.

Alternate Method Given: Three discount schemes: I. Successive discounts of 15% and 20% II. Successive discount of 25% and 10% III. A discount of 33% Formula Used: Effective Discount = x + y − (x × y) ÷ 100 Selling Price (SP) = Marked Price × (100 − Discount%) ÷ 100

Calculations: ⇒ Scheme I Discount = 15 + 20 − (15 × 20) ÷ 100 = 35 − 3 = 32% ⇒ Scheme I Selling Price = 100% − 32% = 68% of Marked Price ⇒ Scheme II Discount = 25 + 10 − (25 × 10) ÷ 100 = 35 − 2.5 = 32.5% ⇒ Scheme II Selling Price = 100% − 32.5% = 67.5% of Marked Price ⇒ Scheme III Discount = 33% ⇒ Scheme III Selling Price = 100% − 33% = 67% of Marked Price ⇒ Comparing the Selling Prices: 67% < 67.5% < 68%. ⇒ The Selling Price is clearly minimum under Scheme III. ∴ The correct answer is Scheme III.

Additional Information Successive Discounts When two discounts x% and y% are given consecutively on an item, the equivalent single discount is given by the formula [x + y − (x × y) ÷ 100]%. Marked Price vs Selling Price The selling price is calculated after applying the discount on the marked price. It can be found directly using the formula: SP = MP × (100 − Discount%) ÷ 100. Maximising Discount for Consumer For a consumer, if the sum of successive discounts given in different schemes is the same, the sequence with the largest numerical difference between the individual discounts always yields a higher overall discount.

Shortcut Trick

Sum of Length (L) and Breadth (B) = Perimeter ÷ 2 = 48 ÷ 2 = 24 m

Given the difference L − B = 6 m

L = (24 + 6) ÷ 2 = 15 m

B = (24 − 6) ÷ 2 = 9 m

Area = L × B = 15 × 9 = 135 m²

∴ The correct answer is 135 m².

Alternate Method

Given:

Perimeter of rectangular garden = 48 m

Length (L) = Breadth (B) + 6 m

Formula Used:

Perimeter of rectangle = 2(L + B)

Area of rectangle = L × B

Calculation:

⇒ 2(B + 6 + B) = 48

⇒ 2(2B + 6) = 48

⇒ 2B + 6 = 24

⇒ 2B = 18

⇒ B = 9 m

⇒ L = 9 + 6 = 15 m

⇒ Area = L × B

⇒ Area = 15 × 9 = 135 m²

∴ The correct answer is 135 m².

Additional Information

Properties of a Rectangle

Opposite sides are equal and parallel. All interior angles are exactly 90°.

Diagonal of a Rectangle

Length of diagonal = √(L² + B²). Diagonals are equal in length and bisect each other.

Square as a Rectangle

A square is a special type of rectangle where Length = Breadth. For a square, Area = Side² and Perimeter = 4 × Side.

Shortcut Trick

Side of each cube (a) = ³√512 = 8 cm.

When 'n' cubes are joined end-to-end, the total surface area = (4n + 2)a².

For 3 cubes, Surface area = (4 × 3 + 2)a² = 14a².

⇒ 14 × (8)² = 14 × 64 = 896 cm².

∴ The correct answer is 896 cm².

Alternate Method

Given:

Volume of each cube = 512 cm³

Number of cubes joined = 3

Formula Used:

Volume of cube = a³ (where a is the side length)

Surface area of cuboid = 2 × (lb + bh + hl)

Calculations:

⇒ a³ = 512

⇒ a = ³√512 = 8 cm

When 3 cubes are joined end to end, a resulting cuboid is formed with dimensions:

⇒ Length (l) = 8 + 8 + 8 = 24 cm

⇒ Breadth (b) = 8 cm

⇒ Height (h) = 8 cm

⇒ Surface Area = 2 × (lb + bh + hl)

⇒ Surface Area = 2 × (24×8 + 8×8 + 8×24)

⇒ Surface Area = 2 × (192 + 64 + 192)

⇒ Surface Area = 2 × 448 = 896 cm²

∴ The correct answer is 896 cm².

Additional Information

Total Surface Area of Cubes Joined

When 'n' cubes of side 'a' are joined in a row, the exposed surface area becomes (4n + 2)a² because (2n − 2) connecting faces become hidden.

Volume of the New Cuboid

The volume of the resulting cuboid is simply the sum of the individual volumes, calculated as n × a³.

Diagonal of a Cuboid

The length of the longest internal diagonal inside a cuboid is determined using the formula √(l² + b² + h²).

Shortcut Trick

By using the remainder directly, substitute r = 3 into the given polynomial expression.

⇒ r² + 6r + 7

⇒ (3)² + 6 × 3 + 7

⇒ 9 + 18 + 7 = 34

Now, find the remainder when 34 is divided by 8.

⇒ 34 ÷ 8 gives a remainder of 2.

∴ The correct answer is 2.

Alternate Method

Given:

A number r leaves a remainder of 3 when divided by 8.

Formula Used:

Dividend = Divisor × Quotient + Remainder

Calculations:

Let the quotient be k. We can write the number as:

⇒ r = 8k + 3

Substitute the value of r in the expression (r² + 6r + 7):

⇒ (8k + 3)² + 6 × (8k + 3) + 7

⇒ (64k² + 48k + 9) + (48k + 18) + 7

⇒ 64k² + 96k + 34

Now, break 34 into 32 + 2 to factor out 8:

⇒ 64k² + 96k + 32 + 2

⇒ 8 × (8k² + 12k + 4) + 2

The first term is completely divisible by 8 as it is a multiple of 8.

⇒ The remaining part is 2.

∴ The correct answer is 2.

Additional Information

Remainder Theorem Application

When solving questions related to remainders of algebraic expressions, if a variable x leaves remainder a when divided by n, the expression P(x) will leave the same remainder as P(a) divided by n.

Modular Arithmetic Properties

Addition Rule: (A + B) mod n = [(A mod n) + (B mod n)] mod n. This allows simplifying each term individually before taking the final remainder.

Divisibility Rule of 8

A number is exactly divisible by 8 if the number formed by its last three digits is a multiple of 8.

Shortcut Trick

Using the trigonometric identity: 1 + tan²(t) = sec²(t)

⇒ tan²(t) = (x/y)² − 1 = (x² − y²) / y²

⇒ tan(t) = √(x² − y²) / y

⇒ cot(t) = 1 / tan(t) = y / √(x² − y²)

∴ The correct answer is y / √(x² − y²).

Alternate Method

Given:

sec(t) = x / y

Formula Used:

sec(θ) = Hypotenuse / Base

cot(θ) = Base / Perpendicular

Calculations:

⇒ Let the right-angled triangle be ΔABC with a right angle at B and angle t at vertex C.

⇒ From sec(t) = x / y, we assign Hypotenuse (AC) = x and Base (BC) = y.

⇒ By Pythagoras Theorem: Hypotenuse² = Perpendicular² + Base²

⇒ x² = Perpendicular² + y²

⇒ Perpendicular (AB) = √(x² − y²)

⇒ The value of cot(t) represents the ratio of the Base to the Perpendicular.

⇒ cot(t) = y / √(x² − y²)

∴ The correct answer is y / √(x² − y²).

Additional Information

Trigonometric Identities

The fundamental identities linking basic trigonometric functions are: sin²(θ) + cos²(θ) = 1, sec²(θ) − tan²(θ) = 1, and cosec²(θ) − cot²(θ) = 1.

Pythagoras Theorem

For any right-angled triangle, the square of the longest side (hypotenuse) equals the sum of the squares of the base and perpendicular: H² = P² + B².

Trigonometric Ratios

Basic ratios include sin(θ) = P / H, cos(θ) = B / H, and tan(θ) = P / B. Their corresponding reciprocals are cosec(θ), sec(θ), and cot(θ).

Given:

$\left[ \left( 5\frac{1}{2} - 3\frac{1}{3} \right) \text{ of } 1\frac{5}{13} + 1 - 2 \times 3\frac{5}{2} \div 2\frac{3}{4} \right]$

Formula used:

1. Mixed fraction: $(a\frac{b}{c} = \frac{ac + b}{c})$
2. "of" means multiplication
3. Division: $(a \div b = a \times \frac{1}{b})$

Calculations:

Convert all mixed fractions into improper fractions:

$5(\frac{1}{2} = \frac{11}{2}), 3(\frac{1}{3} = \frac{10}{3}), 1(\frac{5}{13} = \frac{18}{13}), 3(\frac{5}{2} = \frac{11}{2}), 2(\frac{3}{4} = \frac{11}{4})$

Step 1: Subtraction inside brackets

$(\frac{11}{2} - \frac{10}{3} = \frac{(11 \times 3) - (10 \times 2)}{6})$

⇒ $(\frac{33 - 20}{6} = \frac{13}{6})$

Step 2: Multiply with $(1\frac{5}{13} = \frac{18}{13})$

$(\frac{13}{6} \times \frac{18}{13} = \frac{13 \times 18}{6 \times 13})$

⇒ $(\frac{18}{6} = 3)$

Step 3: Simplify $(2 \times 3\frac{5}{2} \div 2\frac{3}{4})$

$(3\frac{5}{2} = \frac{11}{2}, 2\frac{3}{4} = \frac{11}{4})$

Division:

$(\frac{11}{2} \div \frac{11}{4} = \frac{11}{2} \times \frac{4}{11} = \frac{44}{22} = 2)$

Multiply by 2: $(2 \times 2 = 4)$

Step 4: Add and subtract remaining terms  (3 + 1 - 4)

⇒ 3 + 1 = 4

⇒ 4 - 4 = 0

∴ The simplified value is 0.

Shortcut Trick

The mean proportional between any two numbers a and b is given directly by the square root of their product.

⇒ Mean proportional = √(a × b)

⇒ Mean proportional = √(36 × 100)

⇒ Mean proportional = 6 × 10 = 60

∴ The correct answer is 60.

Alternate Method

Given:

The two given numbers are 36 and 100.

Formula Used:

If x is the mean proportional between a and b, then a : x :: x : b, which implies the product of extremes equals the product of means.

Calculations:

⇒ Let the mean proportional be x.

⇒ 36 : x = x : 100

⇒ x × x = 36 × 100

⇒ x² = 3600

⇒ x = √3600

⇒ x = 60

∴ The correct answer is 60.

Additional Information

Mean Proportional

For any two numbers a and b, the mean proportional is their geometric mean, calculated as √(a × b).

Third Proportional

For two numbers a and b, the third proportional c satisfies a : b :: b : c, which results in the formula c = b² ÷ a.

Fourth Proportional

For three numbers a, b, and c, the fourth proportional d satisfies a : b :: c : d, yielding the formula d = (b × c) ÷ a.

Compound Ratio

When two or more ratios are multiplied together, the resultant ratio is called a compound ratio. For ratios a:b and c:d, it is (a × c) : (b × d).

Shortcut Trick

Assume total voters = 100, then Males = 45 and Females = 55.

Literate males = 40% of 45 = 18.

Illiterate females = 40% of 55 = 22.

Required % = [(22 − 18) ÷ 18] × 100 = (4 ÷ 18) × 100 = 22.22%.

∴ The correct answer is 22.22%.

Alternate Method

Given:

Total males = 45% of total voters

Illiterate males = 60% of total males

Literate females = 60% of total females

Formula Used:

Percentage More = (Difference ÷ Base Value) × 100

Calculations:

⇒ Let the total number of voters in the constituency be 100.

⇒ Number of male voters = 45% of 100 = 45

⇒ Number of female voters = 100 − 45 = 55

⇒ Since 60% of males are illiterate, literate males = (100 − 60)% of 45

⇒ Literate males = 40% of 45 = 18

⇒ Since 60% of females are literate, illiterate females = (100 − 60)% of 55

⇒ Illiterate females = 40% of 55 = 22

⇒ Difference between illiterate females and literate males = 22 − 18 = 4

⇒ Required percentage = (4 ÷ 18) × 100

⇒ Required percentage = (2 ÷ 9) × 100 = 22.22%

∴ The correct answer is 22.22%.

Additional Information

Base Value in Percentages

The term immediately following 'than' or 'of' in a question generally determines the base value (denominator) for percentage calculations.

Percentage Change

Formula: [(Final Value − Initial Value) ÷ Initial Value] × 100. A positive result indicates a percentage increase, and a negative result indicates a decrease.

Successive Percentage

When multiple percentage changes occur successively, the effective percentage change is calculated using: a + b + (a × b) ÷ 100.

Shortcut Trick

For an odd number of consecutive terms, the average is always the exact middle number.

Average (Middle number) = Total Sum ÷ Number of terms

Middle number = 80 ÷ 5 = 16

Since the middle (3^rd) number is 16, the consecutive series is: 14, 15, 16, 17, 18.

∴ The correct answer is 18.

Alternate Method

Given:

The sum of five consecutive numbers is 80.

Formula Used:

If the first number is x, then 5 consecutive numbers are x, x + 1, x + 2, x + 3, and x + 4.

14 +

15 +

16 + Average

17 +

18 Largest

= 80

Sum of 5 Consecutive Numbers = 80

Calculations:

⇒ Total Sum = x + (x + 1) + (x + 2) + (x + 3) + (x + 4)

⇒ 5x + 10 = 80

⇒ 5x = 80 − 10

⇒ 5x = 70

⇒ x = 70 ÷ 5 = 14

⇒ The smallest number is 14.

⇒ The largest number = x + 4 = 14 + 4 = 18.

∴ The correct answer is 18.

Additional Information

Consecutive Numbers

These are numbers that follow each other continuously in order from smallest to largest without any gaps. Example: n, n+1, n+2.

Average of an Arithmetic Progression

The average of any sequence of equally spaced numbers (like consecutive numbers) is always calculated as (First Term + Last Term) ÷ 2, which is identical to the exact middle term.

Consecutive Even or Odd Numbers

If the first even or odd number in a sequence is x, the next consecutive even or odd numbers will progress by a difference of 2, i.e., x+2, x+4, x+6, etc.

Shortcut Trick

Net Gain % = Sum of (Part fraction × Gain/Loss % of that part)

Gain on ⁵/₈ part = 20%

Gain on remaining ³/₈ part = 0% (since sold at Cost Price)

Net Gain % = (⁵/₈ × 20%) + (³/₈ × 0%)

Net Gain % = ¹⁰⁰/₈% = 12.5% = 12 ¹/₂%

∴ The correct answer is 12 ¹/₂%.

Alternate Method

Given:

Fraction of articles sold at 20% gain = ⁵/₈

Remaining fraction sold at cost price (0% gain) = 1 − ⁵/₈ = ³/₈

Formula Used:

Overall Gain % = (Total Profit ÷ Total Cost Price) × 100

Calculations:

Let the total number of articles be 8 and the CP of each article be 100.

⇒ Total CP = 8 × 100 = 800

⇒ SP of 5 articles (sold at 20% gain) = 5 × 100 × 1.20 = 600

⇒ SP of 3 articles (sold at CP) = 3 × 100 = 300

⇒ Total SP = 600 + 300 = 900

⇒ Total Profit = Total SP − Total CP = 900 − 800 = 100

⇒ Overall Gain % = (100 ÷ 800) × 100

⇒ Overall Gain % = ¹⁰⁰/₈ = 12.5% = 12 ¹/₂%

∴ The correct answer is 12 ¹/₂%.

Additional Information

Net Profit Percentage Rule

When goods are sold in multiple parts, the net profit or loss percentage is the weighted average of the profit/loss percentages of each part: Net % = (Part₁ × %₁) + (Part₂ × %₂) + ...

Selling at Cost Price

If any part of the total quantity is sold exactly at the cost price, the profit or loss on that particular fraction is 0%, effectively making its contribution to the net percentage equation zero.

Assuming Favourable Values

To avoid complex fractional calculations, it is always best to assume the total quantity to be equal to the LCM of the denominators of the given fractions (e.g., assuming total 8 articles when dealing with ⁵/₈).

Shortcut Trick

Part of the tank filled by X in 20 hours = 20 ÷ 60 = 1/3

Part of the tank filled by Y in 20 hours = 20 ÷ 72 = 5/18

Total part filled by both in 20 hours = 1/3 + 5/18 = 6/18 + 5/18 = 11/18

Part of the tank left empty = 1 − 11/18 = 7/18

∴ The correct answer is 7/18.

Alternate Method

Given:

Time taken by Pipe X to fill the tank = 60 hours

Time taken by Pipe Y to fill the tank = 72 hours

Time for which both pipes are opened = 20 hours

Concept Used:

Total capacity of the tank = LCM of the individual times

Efficiency = Total capacity ÷ Time taken

Calculations:

⇒ Total efficiency of X and Y together = 6 + 5 = 11 units/hour

⇒ Work done by both pipes in 20 hours = 11 × 20 = 220 units

⇒ Remaining work (empty part of the tank) = Total capacity − Work done

⇒ Remaining work = 360 − 220 = 140 units

⇒ Fraction of the tank left empty = Remaining work ÷ Total capacity

⇒ Fraction left = 140 ÷ 360 = 14/36 = 7/18

∴ The correct answer is 7/18.

Additional Information

Inlet and Outlet Pipes

An inlet pipe fills a tank and its work is considered positive, whereas an outlet pipe empties the tank and its work is considered negative.

Combined Time Formula

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

Total Capacity (LCM Method)

The most efficient way to solve pipe & cistern problems is to assume the total capacity of the tank as the LCM of all given times.

Shortcut Trick

If x = a − √b and a² − b = 1, then 1/x = a + √b.

Here, 7² − (4√3)² = 49 − 48 = 1, so 1/x = 7 + 4√3.

⇒ x + 1/x = (7 − 4√3) + (7 + 4√3) = 14

⇒ x² + 1/x² = k² − 2 = 14² − 2 = 196 − 2 = 194

∴ The correct answer is 194.

Alternate Method

Given:

x = 7 − 4√3

Formula Used:

(a − b)(a + b) = a² − b²

(a + b)² = a² + b² + 2ab

Calculations:

⇒ 1/x = 1 / (7 − 4√3)

⇒ Rationalizing the denominator by multiplying the numerator and denominator by (7 + 4√3):

⇒ 1/x = (7 + 4√3) / [(7 − 4√3)(7 + 4√3)]

⇒ 1/x = (7 + 4√3) / (7² − (4√3)²)

⇒ 1/x = (7 + 4√3) / (49 − 48)

⇒ 1/x = 7 + 4√3

⇒ Now, x + 1/x = (7 − 4√3) + (7 + 4√3)

⇒ x + 1/x = 14

⇒ Squaring both sides:

⇒ (x + 1/x)² = 14²

⇒ x² + 1/x² + 2(x)(1/x) = 196

⇒ x² + 1/x² + 2 = 196

⇒ x² + 1/x² = 196 − 2 = 194

∴ The correct answer is 194.

Additional Information

Rationalization Concept

If the difference between the squares of two terms in a binomial surd is 1, then its reciprocal is simply its conjugate. For example, if a = √p + √q and p − q = 1, then 1/a = √p − √q.

Important Algebraic Identities

If x + 1/x = k, then x² + 1/x² = k² − 2.

Higher Power Identities

If x + 1/x = k, then x³ + 1/x³ = k³ − 3k. Also, x⁴ + 1/x⁴ = (k² − 2)² − 2.

Given:

Fuel and Transport expenditures for 2015 to 2019:

2015 = 98, 2016 = 112, 2017 = 101, 2018 = 133, 2019 = 142

Formula used:

Average = (Sum of all expenditures) ÷ (Number of years)

Calculations:

Step 1: Add all expenditures

98 + 112 + 101 + 133 + 142 = 586

Step 2: Divide the sum by 5 (number of years)

Average = 586 ÷ 5

⇒ Average = 117.2

∴ The average amount paid by the company on fuel and transport per year is ₹1,17,200.

Shortcut Trick

Solve for x, y, z in terms of a using basic elimination: we get x = 3a, y = 6 − 4a, and z = 6 − 5a.

The difference between y and z is y − z = (6 − 4a) − (6 − 5a) = a.

For x, y, z to be three consecutive integers, the difference between any two must be an integer and cannot exceed 2.

From the given options, substituting a = 1 gives x = 3, y = 2, z = 1, which perfectly forms a set of three consecutive integers (1, 2, 3).

∴ The correct answer is 1.

Alternate Method

Given:

x − y + z = 2a ... (i)

x + 4y − 2z = 12 − 3a ... (ii)

2x − 3y + 4z = 6 − 2a ... (iii)

Calculations:

⇒ Multiply equation (i) by 2 and add to equation (ii):

⇒ 2(x − y + z) + (x + 4y − 2z) = 4a + 12 − 3a

⇒ 2x − 2y + 2z + x + 4y − 2z = a + 12

⇒ 3x + 2y = a + 12 ... (iv)

⇒ Multiply equation (i) by 4 and subtract equation (iii):

⇒ 4(x − y + z) − (2x − 3y + 4z) = 8a − (6 − 2a)

⇒ 4x − 4y + 4z − 2x + 3y − 4z = 8a − 6 + 2a

⇒ 2x − y = 10a − 6 ... (v)

⇒ Multiply equation (v) by 2 and add to equation (iv):

⇒ 2(2x − y) + (3x + 2y) = 2(10a − 6) + (a + 12)

⇒ 4x − 2y + 3x + 2y = 20a − 12 + a + 12

⇒ 7x = 21a

⇒ x = 3a

⇒ Substitute x = 3a into equation (v):

⇒ 2(3a) − y = 10a − 6

⇒ 6a − y = 10a − 6

⇒ y = 6 − 4a

⇒ Substitute x = 3a and y = 6 − 4a into equation (i):

⇒ 3a − (6 − 4a) + z = 2a

⇒ 7a − 6 + z = 2a

⇒ z = 6 − 5a

⇒ Since x, y, and z are consecutive integers, let us check the difference between them.

⇒ y − z = (6 − 4a) − (6 − 5a) = a

⇒ For the variables to be consecutive integers, the absolute difference between any two cannot exceed 2.

⇒ Substitute a = 1 from the given options:

⇒ x = 3(1) = 3

⇒ y = 6 − 4(1) = 2

⇒ z = 6 − 5(1) = 1

⇒ The values 1, 2, and 3 correctly form a set of three consecutive integers.

∴ The correct answer is 1.

Additional Information

System of Linear Equations

A system of three linear equations can be solved by progressively eliminating variables until a single variable remains. Substitution and coefficient matching (addition/subtraction) are the most common algebraic methods.

Consecutive Integers

A sequence of integers where each number is exactly 1 more than the previous number. The maximum difference between the largest and the smallest number in a set of three consecutive integers is exactly 2.

Given:

Expenditure on advertisement: ₹2.1 crore

Percentage of total expenditure on advertisement: 15%

Percentage of total expenditure on transport: 12.5%

Percentage of total expenditure on taxes: 10%

Formula used:

Total Expenditure = Expenditure on Advertisement ÷ (Percentage on Advertisement ÷ 100)

Expenditure on Transport = Total Expenditure × (Percentage on Transport ÷ 100)

Expenditure on Taxes = Total Expenditure × (Percentage on Taxes ÷ 100)

Difference = Expenditure on Transport - Expenditure on Taxes

Calculations:

Step 1: Calculate Total Expenditure

Total Expenditure = ₹2.1 crore ÷ (15 ÷ 100)

⇒ Total Expenditure = ₹2.1 crore ÷ 0.15

⇒ Total Expenditure = ₹14 crore

Step 2: Calculate Expenditure on Transport

Expenditure on Transport = ₹14 crore × (12.5 ÷ 100)

⇒ Expenditure on Transport = ₹14 crore × 0.125

⇒ Expenditure on Transport = ₹1.75 crore

Step 3: Calculate Expenditure on Taxes

Expenditure on Taxes = ₹14 crore × (10 ÷ 100)

⇒ Expenditure on Taxes = ₹14 crore × 0.10

⇒ Expenditure on Taxes = ₹1.4 crore

Step 4: Calculate the Difference

Difference = Expenditure on Transport - Expenditure on Taxes

⇒ Difference = ₹1.75 crore - ₹1.4 crore

⇒ Difference = ₹0.35 crore

∴ The difference between the expenditure on transport and taxes is ₹35 Lakh.

Shortcut Trick

Given that sin(θ) = cos(θ).

Divide both sides by cos(θ) to get the ratio.

⇒ sin(θ) ÷ cos(θ) = 1

⇒ tan(θ) = 1

From the standard trigonometric table, we know that tan(45°) = 1.

∴ The correct answer is 45°.

Alternate Method

Given:

An equation where sin(θ) and cos(θ) are equal.

Concept Used:

In a right-angled triangle, if the Perpendicular and Base are equal in length, the corresponding acute angle θ is 45°.

Calculations:

⇒ Let Perpendicular = 1 unit and Base = 1 unit.

⇒ Hypotenuse = √(1² + 1²) = √2 units.

⇒ sin(45°) = Perpendicular ÷ Hypotenuse = 1 ÷ √2

⇒ cos(45°) = Base ÷ Hypotenuse = 1 ÷ √2

⇒ Therefore, at θ = 45°, both trigonometric functions yield the exact same value of 1 ÷ √2.

∴ The correct answer is 45°.

Additional Information

Complementary Angles Rule

Since sin(90° − θ) = cos(θ), equating sin(θ) = cos(θ) implies sin(θ) = sin(90° − θ). Solving this gives θ = 90° − θ, which means 2θ = 90° ⇒ θ = 45°.

Key Trigonometric Values at 45°

At 45°, both sine and cosine are equal (1 ÷ √2), both tangent and cotangent are equal (1), and both secant and cosecant are equal (√2).

Quadrants and Equality

Sine and Cosine are positive and equal at 45° (1^st quadrant). They are both negative and equal at 225° (3^rd quadrant) where both are −(1 ÷ √2).

Given:

Number of televisions sold:

Formula used:

Difference = (Sales of A + B + C in 2017) - (Sales of B + C + D in 2019)

Calculations:

Sales of A, B, and C in 2017:

190 + 204 + 212 = 606

Sales of B, C, and D in 2019:

192 + 205 + 203 = 600

Difference:

606 - 600 = 6

∴ The difference between the sales is 6 televisions.

Shortcut Trick

The central angle subtended by an arc is directly proportional to its arc length.

Given the ratio of Arc AB to Arc BC is 1 : 5, the ratio of their central angles ∠AOB : ∠BOC is also 1 : 5.

Since AOC is a diameter, the total angle on the straight line is 180°.

∠BOC = [5 ÷ (1 + 5)] × 180°

∠BOC = (5 ÷ 6) × 180° = 150°.

∴ The correct answer is 150°.

Alternate Method

Given:

AOC is the diameter of the circle with centre O.

Arc AB = (1 ÷ 5) × Arc BC

Concept Used:

The angle subtended by an arc at the centre is directly proportional to the length of the arc.

The angle of a straight line (diameter) at the centre is 180°.

Calculation:

⇒ Let the length of Arc AB be x.

⇒ Then, the length of Arc BC = 5x.

⇒ Ratio of ∠AOB to ∠BOC = Arc AB : Arc BC = x : 5x = 1 : 5.

⇒ Let ∠AOB = k and ∠BOC = 5k.

⇒ Since AOC is a straight line, the sum of angles on one side is 180°.

⇒ ∠AOB + ∠BOC = 180°

⇒ k + 5k = 180°

⇒ 6k = 180°

⇒ k = 30°

⇒ Therefore, ∠BOC = 5k = 5 × 30° = 150°

∴ The correct answer is 150°.

Additional Information

Arc and Angle Proportion Rule

In any given circle, the central angle subtended by an arc is completely proportional to its physical arc length. (∠ ∝ Arc length).

Semicircle Property

The arc of a semicircle subtends exactly 180° at the centre of the circle and 90° at any point on the remaining circumference.

Inscribed Angle Theorem

The angle subtended by an arc at the centre is exactly double the angle subtended by the same arc at any remaining part of the circle's boundary.

Given:

Total number of fruits sold in a day: 1,200

Percentage of bananas sold: 40%

Formula used:

Number of bananas sold = Total fruits sold × (Percentage of bananas ÷ 100)

Calculations:

Number of bananas sold = 1,200 × (40 ÷ 100)

⇒ Number of bananas sold = 1,200 × 0.40

⇒ Number of bananas sold = 480

∴ The number of bananas sold is 480.

Shortcut Trick

The perimeter of a circular sector is the sum of its arc length and twice its radius.

Perimeter (P) = 2r + (θ ÷ 360°) × 2πr

⇒ P = (2 × 24) + (60° ÷ 360°) × 2 × 3.14 × 24

⇒ P = 48 + (1 ÷ 6) × 150.72

⇒ P = 48 + 25.12 = 73.12 cm

∴ The correct answer is 73.12 cm.

Alternate Method

Given:

Radius (r) = 24 cm

Central angle (θ) = 60°

π = 3.14

Formula Used:

Length of Arc (l) = (θ ÷ 360°) × 2πr

Perimeter of Sector = Length of Arc + 2 × Radius

Calculations:

⇒ Length of Arc (l) = (60° ÷ 360°) × 2 × 3.14 × 24

⇒ l = (1 ÷ 6) × 150.72

⇒ l = 25.12 cm

⇒ Perimeter = 25.12 + 24 + 24

⇒ Perimeter = 73.12 cm

∴ The correct answer is 73.12 cm.

Additional Information

Area of a Sector

The area enclosed by the sector is calculated using the formula: Area = (θ ÷ 360°) × πr² or Area = (1 ÷ 2) × l × r.

Length of an Arc

The curve joining the two radii is called the arc. Its length is a fraction of the circle's full circumference: l = (θ ÷ 360°) × 2πr.

Perimeter of a Semicircle

A semicircle is a special sector with a central angle of 180°. Its perimeter formula simplifies to: P = πr + 2r = r(π + 2).

Shortcut Trick

If an equation is of the form a sin A + b cos A = c, and satisfies a² + b² = c².

Then we can directly write the result: tan A = a / b.

Here, a = 3, b = 4, and c = 5.

Since 3² + 4² = 5² (9 + 16 = 25), the condition is perfectly satisfied.

⇒ tan A = 3 / 4

∴ The correct answer is 3/4.

Alternate Method

Given:

⇒ 3 sin A + 4 cos A = 5

Formula Used:

⇒ sin² A + cos² A = 1

⇒ tan A = sin A ÷ cos A

Calculations:

⇒ 3 sin A + 4 cos A = 5

⇒ Dividing the entire equation by 5, we get:

⇒ (3/5) sin A + (4/5) cos A = 1

⇒ We know the fundamental trigonometric identity: sin A × sin A + cos A × cos A = 1

⇒ By comparing the two equations directly, we can conclude:

⇒ sin A = 3/5

⇒ cos A = 4/5

⇒ Now, substitute these values to find tan A:

⇒ tan A = sin A ÷ cos A

⇒ tan A = (3/5) ÷ (4/5)

⇒ tan A = 3/4

∴ The correct answer is 3/4.

Additional Information

General Trigonometric Rule

For any given equation of the form a sin θ + b cos θ = c, if the condition a² + b² = c² holds true, then sin θ = a/c, cos θ = b/c, and tan θ = a/b.

Pythagorean Triplets

A set of three positive integers (a, b, c) that satisfy the relation a² + b² = c². Common examples include (3, 4, 5), (5, 12, 13), (7, 24, 25), and (8, 15, 17).

Fundamental Trigonometric Identities

• sin² θ + cos² θ = 1

• 1 + tan² θ = sec² θ

• 1 + cot² θ = cosec² θ

Shortcut Trick

Let downstream speed = D and upstream speed = U.

We have: 24/U + 40/D = 17  and  12/U + 32/D = 10.

Double the 2^nd equation and subtract the 1^st: 24/D = 3 ⇒ D = 8 km/h.

Substitute D = 8 in 1^st equation: 24/U + 5 = 17 ⇒ U = 2 km/h.

Speed of stream = (D − U) ÷ 2 = (8 − 2) ÷ 2 = 3 km/h.

∴ The correct answer is 3 km/h.

Alternate Method

Given:

Distance upstream (D_u1) = 24 km, Distance downstream (D_d1) = 40 km, Time (T₁) = 17 hours

Distance upstream (D_u2) = 12 km, Distance downstream (D_d2) = 32 km, Time (T₂) = 10 hours

Formula Used:

Time = Distance ÷ Speed

Speed of Stream = (D − U) ÷ 2

Calculation:

Let the upstream speed be U and downstream speed be D.

⇒ 24/U + 40/D = 17   --- (Equation 1)

⇒ 12/U + 32/D = 10   --- (Equation 2)

Multiply Equation 2 by 2:

⇒ 24/U + 64/D = 20   --- (Equation 3)

Subtract Equation 1 from Equation 3:

⇒ (64 − 40)/D = 20 − 17

⇒ 24/D = 3

⇒ D = 8 km/h

Substitute D = 8 in Equation 1:

⇒ 24/U + 40/8 = 17

⇒ 24/U + 5 = 17

⇒ 24/U = 12

⇒ U = 2 km/h

Now calculate the speed of the stream:

⇒ Speed of stream = (8 − 2) ÷ 2

⇒ Speed of stream = 6 ÷ 2 = 3 km/h

∴ The correct answer is 3 km/h.

Additional Information

Upstream Speed

The speed of the boat traveling against the direction of the flow of the stream. U = Speed of Boat − Speed of Stream.

Downstream Speed

The speed of the boat traveling in the direction of the flow of the stream. D = Speed of Boat + Speed of Stream.

Speed of Boat in Still Water

The average of downstream and upstream speeds. V_b = (D + U) ÷ 2.

Speed of Stream

Half of the difference between downstream and upstream speeds. V_s = (D − U) ÷ 2.

Shortcut Trick

For congruent triangles ΔABC ≅ ΔQPR, the vertices map to each other in the exact order written.

Mappings: A → Q, B → P, C → R.

Side PQ maps directly to AB, so PQ = AB = 7 cm.

Angle R maps directly to angle C, so ∠R = ∠C = 75°.

This matches the given condition perfectly without calculating any other unknown sides or angles.

∴ The correct answer is PQ = 7 cm and ∠R = 75°.

Alternate Method

A B C 7 cm 8 cm 65° 75°

Q P R 7 cm 8 cm 65° 75° ∆ABC ≅ ∆QPR

Given:

⇒ ΔABC ≅ ΔQPR

⇒ AB = 7 cm, AC = 8 cm

⇒ ∠B = 65°, ∠C = 75°

Concept Used:

⇒ CPCT (Corresponding Parts of Congruent Triangles): If two triangles are congruent, their corresponding angles and sides are precisely equal.

Calculations:

⇒ Since ΔABC ≅ ΔQPR, the exact correspondence is given by A ↔ Q, B ↔ P, and C ↔ R.

⇒ Corresponding sides match: AB = QP, BC = PR, and AC = QR

⇒ Corresponding angles match: ∠A = ∠Q, ∠B = ∠P, and ∠C = ∠R

⇒ Applying the given dimensions directly to the mapped parts:

⇒ PQ = AB = 7 cm

⇒ QR = AC = 8 cm

⇒ ∠P = ∠B = 65°

⇒ ∠R = ∠C = 75°

⇒ Also, by the angle sum property, ∠A = 180° − (65° + 75°) = 40°, meaning ∠Q = 40°.

⇒ Comparing our derived values with the given options, we find that PQ = 7 cm and ∠R = 75° is the only correct statement.

∴ The correct answer is PQ = 7 cm and ∠R = 75°.

Additional Information

Congruence Criteria

Two triangles are strictly congruent if they satisfy SSS, SAS, ASA, AAS, or RHS congruency conditions.

CPCT Theorem

CPCT denotes 'Corresponding Parts of Congruent Triangles'. It establishes that if two triangles are congruent, all their equivalent angles and sides are identical.

Angle Sum Property

The total sum of interior angles in any triangle is exactly 180°. This helps determine the third angle if two are known (∠A = 180° − ∠B − ∠C).

Similar vs Congruent

Congruent triangles have an identical shape and exact size, whereas similar triangles possess the identical shape but may differ proportionally in size.

The correct answer is option 2.

Key Points

The misspelled word is 'Mischievious'.

• The correct spelling would be 'Mischievous'.
• The word 'Mischievous' is an adjective which means 'showing a fondness for causing trouble in a playful way'. (शरारती)
• For Example -
• The child had a mischievous glint in his eye as he planned his next prank.
• Therefore, the correct answer would be 'Mischievous'.

 

Additional Information

Let's understand the meanings of the other marked words - 

• Conscientious - wishing to do one's work or duty well and thoroughly (कर्तव्यनिष्ठ)
• Ostentatious - characterized by vulgar or pretentious display; designed to impress or attract notice (आडंबरपूर्ण)
• Accommodate - (of physical space, especially a building) provide lodging or sufficient space for (आवास देना)

The correct answer is 'diffuses'.

 

Key Points

• The word "diffuses" means to spread or cause to spread over a wide area or among a large number of people.
• Helium is used with oxygen for asthma treatment because it diffuses very easily.
• The other options: "diffusez," "difuses," and "diffuzes" are not correctly spelled.

​Therefore, the correct answer is 'Option 3'.

Complete Sentence: Helium is used with oxygen for asthma treatment because it diffuses very easily.

Additional Information

• Diffusez: Incorrect spelling, does not convey the intended meaning.
• Difuses: Incorrect spelling, should be "diffuses".
• Diffuzes: Incorrect spelling, does not convey the intended meaning.