SSC CGL 2024 Tier-I Official Paper (Held On: 19 Sept, 2024 Shift 2) (Previous Year Paper)

The logic followed here is:

Each vowel in the word ‘ISOLATED’ is changed to the following letter in the English alphabetical order,

Each consonant is changed to the preceding letter in the English alphabetical order. 

The letter which is the third from the left end = P

The second from the left end = R

Thus, One letter (P, Q, R) is there in the English alphabetical order between P and R.

Hence, the correct answer is "One".

The logic followed here is:

Given Sequence: _ B _ _ _ A L _ _ L _ _

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

Option 1) LALBBABA → L B A / L B A / L B A / L B A

Here, "L B A" pattern is followed.

Option 2) BBALLBAA → B B B / A L A / L L B / L A A

Option 3) LBAALLBA → L B B / A A A / L L L / L B A

Option 4) LABBAALL → L B A / B B A / L A A / L L L

Hence, the correct answer is "LALBBABA".

According to the given information:

Thus, the code for ‘raining’ = ko

Hence, the correct answer is "ko".

To find the opposite surface of 'R', we need to compare the common surface in the given dices.

So, From Dice 1 and Dice 2,

From both dices, X is common and on the same surface. So, all remaining surface will be opposite to each other.

S → R

M → T

X → O

T is opposite to M.

So, S will be opposite to R.

Hence, the correct answer is "S".

The logic followed here is:  √Second number  -  √First Number = 1

1)  784 : 961 → (28)2 : (31)2 = 31 - 28 = 3

2)  1764 : 1849 → (42)2 : (43)2 = 43 - 42 = 1

3)  1156 : 1225 → (34)2 : (35)2 = 35 - 34 = 1

4)  1521 : 1600 → (39)² : (40)² = 40 - 39 = 1

Hence, the correct answer is "784 : 961".

The logic followed here is:

Logic: Capital cities of countries.

Here,

New Zealand : Wellington → Wellington is the capital city of New Zealand.

Similarly,

Ethiopia : Addis Ababa → Addis Ababa is the capital city of Ethiopia.

Hence, the correct answer is "Option 4".

According to the given information:

Thus, the code for ‘gadgets’ is either 'ty' or 'dj'

Hence, the correct answer is "either 'ty' or 'dj'".

The logic followed here is:

Option 1) LKM

Option 2) XYW

Option 3) TSU

Option 4) HGI

Hence, the correct answer is "XYW".

BODMAS Table: 

After decoding signs:

Given equation: 16 − 2 × 42 + 13 ÷ 12 = ? 

After interchanging the signs then:

⇒ 16 ÷ 2 + 42 × 13 - 12 = ? 

⇒ ? = 8 + 42 × 13 - 12

⇒ ? = 8 + 546 - 12

⇒ ? = 554 - 12 = 542

Hence, the correct answer is "542".

The logic followed here is:

Hence, the correct answer is "262".

Statements:

All teachers are dancers.

All dancers are engineers.

Some engineers are beautiful.

The least possible Venn diagram is as follows:

Conclusions:

I. All teachers are beautiful. - False (because there is no direct relation given between teachers and beautiful, so it can be possible but not definite)

II. Some dancers are beautiful. - False (because there is no direct relation given between dancers and beautiful, so it can be possible but not definite)

Hence, the correct answer is "Neither conclusion I nor II follows".

Counting the semicircles are there in the given figure:

Hence, the correct answer is "7".

Family Chart:

Decoding the Symbols:

1. S & F * J ! I # M → S is the brother of F, F is the wife of J, J is the father of I and I is the daughter of M.

Here, contradiction between "J is the father of I and I is the daughter of M"

2. S * F & J ! I # M → S is the wife of F,  F is the brother of J, J is the father of I and I is the daughter of M.

Here, S is the M's husband's brother's wife.

3. S & F ! J * I # M → S is the brother of F, F is the father of J, J is the wife of I and I is the daughter of M.

Here, contradiction between "J is the wife of I and I is the daughter of M."

4. S & F # J ! I * M → S is the brother of F, F is the daughter of J, J is the father of I and I is the wife of M.

So, In Option 2, S is the husband's brother's wife of M.

Hence, the correct answer is "S * F & J ! I # M".

The logic followed here is:

Logic: (1st number × 5) - 10 = 2nd number.

Now,

For, 13 : 55

⇒ (13 × 5) - 10 = 65 - 10 = 55.

For, 23 : 105

⇒ (23 × 5) - 10 = 115 - 10 = 105.

Similarly, for 31 : 145

⇒ (31 × 5) - 10 = 155 - 10 = 145.

Hence, the correct answer is "Option 4".

The logic followed here is:

From figure 1 to 2 and figure 3 to 4:

Top-right, middle and bottom-middle symbols are interchanging their position in circular manner.

Middle-left and bottom-right symbols are interchanged and middle-right and bottom-left symbols are interchanged their position.

From figure 2 to 3 and figure 4 to 5:

Top-right,  bottom-right, bottom-middle and bottom-left symbols are interchanging their position in anticlockwise direction.

All 3 middle symbols are shifting their position in circular manner.

Hence, the correct answer is "Option 2".

The correct mirror image of the given figure is:

Hence, the correct answer is "Option 2".

The logic followed here is:

'MAIN' is written as 'OHBL'

'MAIL' is written as 'MHBL'

Similarly, 'MANY’ will be written as:

Hence, the correct answer is "ZMBL".

The logic followed here is:

From figure 1 to 2, and figure 3 to 4:

Top-left and bottom-right are interchanged and top-right and bottom-left are interchanged.

From figure 2 to 3, and figure 4 to 5:

Shapes are rotating their position in clockwise direction by 90° at the same position.

Hence, the correct answer is "Option 3".

The logic followed here is:

Logic: (1st number + 1) ÷ 2 = 2nd number.

Now,

For 97 : 49

⇒ (97 + 1) ÷ 2 = 98 ÷ 2 = 49.

For 115 : 58

⇒ (115 + 1) ÷ 2 = 116 ÷ 2 = 58.

Similarly, for 133 : ?

⇒ (133 + 1) ÷ 2 = 134 ÷ 2 = 67.

Hence, the correct answer is "Option 3".

The paper will look when unfolded:

Hence, the correct answer is "Option 1".

The logic followed here is:

'NQRP' is related to 'RUVT'

'DGLF' is related to 'HKPJ'

Similarly, 'KCUM' is related to:

Hence, the correct answer is "OGYQ".

Given equation: 8 _ 27 _ 9 _ 6 _ 5 = 0

Option 1) + - + +

Mathematical signs to sequentially place from left to right:

⇒ 8 + 27 - 9 + 6 + 5 = 0

⇒ 46 - 9 = 0

⇒ 37 = 0

Option 2) +, ÷, −, −

Mathematical signs to sequentially place from left to right:

⇒ 8 + 27 ÷ 9 - 6 - 5 = 0

⇒ 8 + 3 - 6 - 5 = 0

⇒ 11 - 6 - 5 = 0

⇒ 11 - 11 = 0

⇒ 0 = 0

Option 3) −, ÷, +, +

Mathematical signs to sequentially place from left to right:

⇒ 8 - 27 ÷ 9 + 6 + 5 = 0

⇒ 8 - 3 + 6 + 5 = 0

⇒ 19 - 3 = 0

⇒ 16 = 0

Option 4) −, +, ÷, +

Mathematical signs to sequentially place from left to right:

⇒ 8 - 27 + 9 ÷ 6 + 5 = 0

⇒ 8 - 27 + 1.5 + 5 = 0

⇒ 14.5 - 27 = 0

⇒ -12.5 = 0

Hence, the correct answer is "+, ÷, −, −".

The logic followed here is:

Logic: (1st number)² - 50 = 2nd number.

Now,

For, 13 : 119

⇒ 13² - 50 = 169 - 50 = 119.

For, 9 : 31

⇒ 9² - 50 = 81 - 50 = 31.

Similarly, for 8 ∶ ?

⇒ 8² - 50 = 64 - 50 = 14.

Hence, the correct answer is "Option 1".

The logic followed here is:

Hence, the correct answer is "GZTG".

To find the opposite surface of 'Ball', we need to compare the common surface in the given dices,

Alternate surfaces are opposite to each other in open dice.

Bat is opposite to Stick.

Net is opposite to Goal.

So, Field is opposite to Ball.

Hence, the correct answer is "Field".

The correct answer is Microfinance.

Key Points

• Microfinance refers to the provision of financial services to low-income individuals or those who do not have access to typical banking services.
• It includes a range of services such as microloans, savings, insurance, and money transfers.
• The primary aim of microfinance is to provide small loans to impoverished individuals to help them start or expand small businesses, thereby reducing poverty.
• Microfinance institutions (MFIs) play a critical role in providing these financial services, especially in developing countries.

Additional Information

• Microloan
• Microloans are small loans given to individuals who typically do not have access to traditional banking services.
• These loans are often used to start or grow small businesses and are a subset of microfinance services.
• Microloans can range from a few dollars to several thousand dollars, depending on the borrower's needs and the lender's policies.
• The concept of microloans was popularized by Nobel Laureate Muhammad Yunus and the Grameen Bank in Bangladesh.
• Homeloan
• A home loan, or mortgage, is a loan given by a bank or financial institution to individuals to purchase a home or property.
• Home loans are typically long-term loans with a repayment period ranging from 10 to 30 years.
• They require collateral, usually the property being purchased, to secure the loan.
• Interest rates on home loans can be fixed or variable, depending on the lender's policies and the borrower's agreement.
• Pradhan Mantri Awas Yojana (PMAY)
• PMAY is an initiative by the Government of India aimed at providing affordable housing to the urban poor.
• The scheme was launched on June 25, 2015, with the goal of constructing 20 million affordable houses by March 31, 2022.
• It has two components: PMAY-U (Urban) for the urban poor and PMAY-G (Gramin) for the rural poor.
• Beneficiaries under this scheme are identified through the Socio-Economic and Caste Census (SECC) of 2011.

The correct answer is diurnal.

Key Points

• Diurnal temperature variation refers to the fluctuation between the highest and lowest temperatures that occur within a single day.
• Such variations are common in many parts of the world and are influenced by factors such as altitude, latitude, and local weather patterns.
• Diurnal temperature changes are more pronounced in desert regions due to the lack of moisture which prevents the air from retaining heat.
• Understanding diurnal temperature variation is important for agriculture, meteorology, and various environmental studies.

Additional Information

• Annual Temperature Variation:
• This refers to the changes in temperature that occur over the course of a year.
• It is influenced by factors such as seasonal changes, earth's tilt, and orbit around the sun.
• Annual variations are more significant in temperate regions compared to tropical regions.
• Wet Bulb Temperature:
• It measures the lowest temperature that can be reached by evaporating water into the air.
• It is important in understanding the effects of heat and humidity on humans and is used in calculating heat index.
• Durian:
• Durian is a large, spiky fruit known for its strong odor.
• It is unrelated to temperature variations and is often called the "king of fruits" in Southeast Asia.

The correct answer is Cricket.

Key Points

• The term "Agricultural shot" is used in Cricket to describe an unorthodox or poorly executed shot.
• In cricket, an agricultural shot typically refers to a wild swing of the bat, which is not played with the standard technique.
• This type of shot often results in the ball going in an unintended direction, usually not as per the plan of the batsman.
• It is generally considered to be a high-risk shot, often played in desperation or by less technically skilled players.

Additional Information

• Cricket Terminologies:
• Cover Drive: A classic cricket shot played by driving the ball through the covers with a vertical bat.
• Yorker: A delivery bowled right at the batsman's feet, typically aimed at the base of the stumps.
• Duckworth-Lewis Method: A complex mathematical formula used to calculate the target score for the team batting second in a limited-overs cricket match interrupted by weather or other circumstances.
• Leg Spin: A type of spin bowling where the ball spins from the leg side to the off side (for a right-handed batsman).
• All-rounder: A player who excels at both batting and bowling.
• Types of Shots in Cricket:
• Straight Drive: A shot played straight down the ground with a vertical bat.
• Sweep: A shot played to a ball pitched on or outside the leg stump, typically hit towards the leg side.
• Pull Shot: A cross-batted shot played to a short-pitched ball, usually hit towards the leg side.
• Cut Shot: A shot played to a ball pitched short and wide of off stump, hit towards the point or gully region.
• Hook Shot: A shot played to a bouncer, where the batsman hits the ball towards the leg side, typically into the stands.
• Cricket Fielding Positions:
• Slip: A fielding position behind the batsman on the off side, typically placed to catch edges from the bat.
• Gully: Positioned between the slips and point, intended to catch edges that slip through.
• Mid-wicket: A position on the leg side, in the region square of the wicket.
• Long-on: Positioned straight down the ground, typically near the boundary.
• Square Leg: A position on the leg side, square with the wicket.

The correct answer is Vakatakas.

Key Points

• The Vakatakas were an ancient Indian dynasty that ruled over parts of central and southern India between the 3rd and 5th centuries CE.
• They were contemporaries and successors of the Satavahanas and are known for their patronage of arts and culture, particularly the Ajanta Caves.
• The Vakataka kingdom included parts of modern-day Maharashtra and Madhya Pradesh, including regions like Vidarbha and northern Maharashtra.
• Pravarasena I and Prithivishena I were notable rulers of the Vakataka dynasty who expanded their territory and influence significantly.

Additional Information

• Vakataka Dynasty
• The Vakataka dynasty was founded by Vindhyashakti around the mid-3rd century CE.
• Their capital was located at Nandivardhana (near present-day Nagpur) and later at Pravarapura (modern-day Mansar).
• The Vakatakas were known to have matrimonial alliances with the Gupta Empire, which helped in the cultural and political exchange between the two dynasties.
• Their patronage was instrumental in the flourishing of the Ajanta Caves, which are renowned for their elaborate frescoes and sculptures.
• Maukharis
• The Maukharis were a dynasty that ruled in northern India, particularly in the region of Kanauj, during the 6th century CE.
• They were contemporaries of the later Guptas and played a significant role in the politics of northern India.
• Maitrakas
• The Maitrakas ruled over the region of Saurashtra in western India from the late 5th century to the 8th century CE.
• They established their capital at Vallabhi and were known for their contributions to education and culture.
• Chalukyas
• The Chalukyas were a powerful dynasty that ruled over large parts of southern and central India between the 6th and 12th centuries CE.
• They are primarily known for their architectural achievements, including the rock-cut temples at Badami and the temple complexes at Pattadakal and Aihole.

The correct answer is The ratio between energy flows at different points in a food chain.

Key Points

• Ecological efficiency refers to the efficiency with which energy is transferred from one trophic level to the next in a food chain.
• Typically, only about 10% of the energy at one trophic level is passed on to the next level; this is known as the 10% law.
• This concept helps in understanding the energy dynamics and the structure of ecosystems.
• Higher trophic levels receive less energy, which explains why there are typically fewer organisms at higher levels in a food chain.

Additional Information

• Trophic Levels
• In ecology, a trophic level is the position that an organism occupies in a food chain.
• Primary producers (plants) are at the first trophic level, primary consumers (herbivores) at the second, secondary consumers (carnivores that eat herbivores) at the third, and so on.
• Energy Flow
• Energy flow in an ecosystem is one-way: from the sun to producers, then to consumers, and finally to decomposers.
• Energy is lost at each trophic level, primarily as heat resulting from metabolism.
• 10% Law
• The 10% law states that only about 10% of the energy at one trophic level is transferred to the next trophic level.
• This rule was first introduced by Raymond Lindeman in 1942.
• Food Chain
• A food chain is a linear sequence of organisms through which nutrients and energy pass as one organism eats another.
• In a food chain, each organism occupies a different trophic level.

The correct answer is legislative relations between the Centre and the states.

Key Points

• Articles 245 to 255 in Part XI of the Indian Constitution deal with the legislative relations between the Centre and the states.
• These provisions outline the distribution of legislative powers between the Union and the State legislatures.
• Article 245 states the extent of laws made by Parliament and by the Legislatures of States.
• Article 246 deals with the subject-matter of laws made by Parliament and by the Legislatures of States, emphasizing the three lists: Union List, State List, and Concurrent List.

Additional Information

• Emergency Provisions (Articles 352 to 360)
• These articles cover the proclamation of emergency by the President of India.
• They include National Emergency (Article 352), President's Rule (Article 356), and Financial Emergency (Article 360).
• Financial Relations (Articles 268 to 293)
• These articles outline the financial relations between the Centre and the States.
• They cover topics such as distribution of revenues, grants-in-aid, and borrowing powers of the Centre and States.
• Tribunals (Articles 323A to 323B)
• These articles provide for the establishment of administrative tribunals for the adjudication or trial of disputes and offenses.
• Article 323A deals with administrative tribunals, while Article 323B deals with tribunals for other matters.

The correct answer is Rajasthan and Gujarat.

Key Points

• The Luni River is one of the major rivers in the western part of India, specifically in the states of Rajasthan and Gujarat.
• The river originates in the Pushkar Valley of the Aravalli Range near Ajmer, Rajasthan.
• The Luni River flows southwest and ultimately discharges into the Rann of Kutch in Gujarat.
• The river is known for its saline and brackish water, especially in its lower reaches.

Additional Information

• Geographical Origin
• The Luni River originates from the Naga hills of the Aravalli Range, near Ajmer in Rajasthan.
• The river is formed by the confluence of two streams, the Saraswati and the Sagarmati.
• Course and Tributaries
• The river flows in a south-westerly direction through the Thar Desert, covering a distance of about 495 kilometers.
• It receives several tributaries such as the Jojari, Guhiya, and Bandi rivers.
• Salinity and Endorheic Basin
• The Luni River is known for its high salinity, particularly in its lower reaches due to the high evaporation rate and low rainfall in the region.
• It is an endorheic river, meaning it does not drain into an ocean but ends in an inland basin, specifically the Rann of Kutch.
• Economic Importance
• The river's water is extensively used for irrigation in the arid and semi-arid regions of Rajasthan and Gujarat.
• Despite its saline nature, it supports a variety of agricultural activities and local ecosystems.

The correct answer is Chhattisgarh.

Key Points

• The 'Kaushalya Mattriva Yojana' was launched by the Chief Minister of Chhattisgarh, Bhupesh Baghel, in 2022.
• This scheme provides financial assistance of ₹5,000 to women giving birth to a second girl child.
• The initiative aims to promote the welfare and safe motherhood of women in the state.
• It is part of the state government's efforts to improve the gender ratio and support women’s health and empowerment.

Additional Information

• Kaushalya Mattriva Yojana:
• This scheme is specifically designed to provide financial support to mothers giving birth to a second girl child.
• The financial assistance aims to encourage families to value and support the birth of girl children, thereby addressing gender biases.
• It is part of a broader set of initiatives by the Chhattisgarh government to enhance maternal and child health.
• The scheme also aims to reduce the financial burden associated with childbirth and early child-rearing, particularly for families with limited resources.
• Other important initiatives by Chhattisgarh government:
• Godhan Nyay Yojana: This scheme aims to promote organic farming and provide financial support to farmers by purchasing cow dung from them.
• Rajiv Gandhi Kisan Nyay Yojana: This scheme aims to provide financial assistance to farmers, ensuring fair compensation for their crops and promoting agricultural productivity.
• Mor Zameen Mor Makaan: A housing scheme aimed at providing affordable housing to the economically weaker sections of society.
• Universal PDS: The state has implemented a universal Public Distribution System to ensure food security for all residents.

The correct answer is Double.

Key Points

• For many chemical reactions, the rate constant approximately doubles with a 10°C increase in temperature.
• This phenomenon is often referred to as the "rule of thumb" in chemical kinetics.
• The temperature dependence of the rate constant is described by the Arrhenius equation.
• According to the Arrhenius equation, the rate constant (k) is related to temperature (T) through the exponential factor involving activation energy (Ea) and the gas constant (R).

Additional Information

• Arrhenius Equation:
• The Arrhenius equation is k = A * exp(-Ea/(RT)), where k is the rate constant, A is the pre-exponential factor (frequency of collisions), Ea is the activation energy, R is the gas constant, and T is the temperature in Kelvin.
• This equation shows the relationship between the rate constant and temperature, indicating that as temperature increases, the rate constant increases exponentially.
• Activation Energy (Ea):
• Activation energy is the minimum energy that reacting molecules must possess to undergo a chemical reaction.
• Higher activation energy means that fewer molecules have sufficient energy to react at a given temperature.
• Gas Constant (R):
• The gas constant (R) is a physical constant that appears in many fundamental equations in physical sciences, such as the ideal gas law and the Arrhenius equation.
• The value of R is 8.314 J/(mol·K).
• Rule of Thumb in Chemical Kinetics:
• The "rule of thumb" in chemical kinetics suggests that for many reactions, the rate constant approximately doubles for every 10°C rise in temperature.
• This empirical rule helps chemists estimate the effect of temperature changes on reaction rates without detailed calculations.

The correct answer is 1911.

Key Points

• Vinayak Damodar Savarkar, also known as Veer Savarkar, was a prominent freedom fighter, social reformer, and writer.
• In 1910, Savarkar was arrested by the British for his involvement in revolutionary activities against the colonial regime.
• He was subsequently tried and sentenced to two life terms of imprisonment totaling fifty years and was deported to the Andaman Islands in 1911.
• The Cellular Jail in the Andaman and Nicobar Islands, also known as Kala Pani, was where he was imprisoned. This jail was notorious for its harsh conditions and was used by the British to exile political prisoners.

Additional Information

• Vinayak Damodar Savarkar:
• He was born on May 28, 1883, in Bhagur, a village near Nashik in Maharashtra, India.
• He was a staunch advocate of Hindutva and is credited with founding the ideology.
• Savarkar wrote several books, including "The First War of Indian Independence 1857," which presented the uprising of 1857 as a unified and national effort against British rule.
• After his release from prison, he continued his activities as a social reformer and political activist, although he was kept under strict surveillance by the British authorities.
• Cellular Jail (Kala Pani):
• The Cellular Jail is located in Port Blair, the capital of the Andaman and Nicobar Islands.
• It was constructed by the British between 1896 and 1906 to exile political prisoners.
• The jail is known for its solitary confinement cells, which were designed to isolate prisoners from each other and prevent them from communicating.
• Many prominent Indian freedom fighters, including Batukeshwar Dutt, Yogendra Shukla, and more, were imprisoned here.
• Today, the Cellular Jail is a national memorial monument.

The correct answer is ₹51,000 crore.

Key Points

• The disinvestment target for the year 2023-24 set by the Finance Ministry of India is ₹51,000 crore.
• This target is part of the Government of India's efforts to raise funds by selling its stakes in public sector enterprises.
• The aim of disinvestment is to reduce the fiscal burden and improve public finances by raising revenue from the sale of government assets.
• Disinvestment helps in achieving greater efficiency and competitiveness in public enterprises by involving private sector participation.

Additional Information

• Disinvestment:
• Disinvestment refers to the action of an organization or government selling or liquidating an asset or subsidiary.
• The primary objective of disinvestment is to reduce the financial burden on the government and to improve public sector efficiency.
• It also aims to increase the share of private sector investment in public sector enterprises and thereby improve their performance and competitiveness.
• In India, disinvestment is overseen by the Department of Investment and Public Asset Management (DIPAM).
• Fiscal Deficit:
• Fiscal deficit occurs when a government's total expenditures exceed the revenue that it generates, excluding money from borrowings.
• This deficit is an indication of the total borrowings needed by the government.
• Reducing the fiscal deficit is one of the reasons for government disinvestment, as it helps in managing public finances better.
• Public Sector Enterprises (PSEs):
• Public Sector Enterprises are government-owned corporations or state-owned enterprises created to undertake commercial activities on behalf of the government.
• They are often established to ensure the provision of public services and to promote economic development.
• Disinvestment in these enterprises is a way to involve private sector management and investment to improve efficiency and productivity.
• Department of Investment and Public Asset Management (DIPAM):
• DIPAM is a department under the Ministry of Finance in India responsible for the management of government investments in equity and the disinvestment of government equity in public sector undertakings.
• It formulates policies and strategies for disinvestment in public sector enterprises to achieve the overall economic goals of the government.

The correct answer is Padma Shri.

Key Points

• The Padma Shri is the fourth highest civilian award in the Republic of India, after the Bharat Ratna, Padma Vibhushan, and Padma Bhushan.
• The award is given for distinguished service in various fields including arts, education, industry, literature, science, sports, medicine, social service, and public affairs.
• Usha Barle, a renowned Pandwani singer, was conferred with the Padma Shri in 2023 for her contributions to the field of traditional folk music.
• Pandwani is a traditional folk singing style from Chhattisgarh, and Usha Barle's recognition highlights the importance of preserving and promoting regional art forms.

Additional Information

• Padma Awards
• The Padma Awards were instituted in 1954 and are announced every year on Republic Day.
• They are awarded to Indian citizens in recognition of their distinguished contributions in various disciplines and fields.
• The awards are categorized into three: Padma Vibhushan (for exceptional and distinguished service), Padma Bhushan (for distinguished service of high order), and Padma Shri (for distinguished service).
• These awards are conferred by the President of India at ceremonial functions held at Rashtrapati Bhawan usually around March/April every year.
• Pandwani
• Pandwani is a traditional Indian folk singing style performed mainly in Chhattisgarh and neighboring states.
• The story narrates the tales from the Indian epic Mahabharata with a unique blend of singing and storytelling.
• The performances typically involve a lead singer and supporting musicians, often using traditional instruments such as the tambura and harmonium.
• The Pandwani tradition is an important cultural heritage and is recognized for its rich narrative style and musical depth.

The correct answer is Bacteria.

Key Points

• Bacteria are single-celled organisms that lack a nucleus and other membrane-bound organelles, making them prokaryotic.
• Bacteria are among the most ancient and diverse forms of life on Earth, thriving in various environments.
• Bacteria can be classified into various shapes such as cocci (spherical), bacilli (rod-shaped), and spirilla (spiral-shaped).
• Bacteria play crucial roles in processes like nutrient cycling, fermentation, and as part of the microbiome in various organisms.

Additional Information

• Prokaryotic Cells
• Prokaryotic cells are primitive cells that lack a well-defined nucleus and membrane-bound organelles.
• They have a simple cell structure with genetic material (DNA) present in a nucleoid region.
• Examples include Bacteria and Archaea.
• Eukaryotic Cells
• Eukaryotic cells have a well-defined nucleus and various membrane-bound organelles such as mitochondria, endoplasmic reticulum, and Golgi apparatus.
• These cells are more complex and are found in organisms such as animals, plants, fungi, and protists.
• Examples include Plasmodium (a protist), Leishmania (a protist), and Yeast (a fungus).
• Plasmodium
• Plasmodium is a genus of parasitic protozoa that causes malaria in humans and other animals.
• It is a eukaryotic organism with a complex life cycle involving both an insect vector (mosquito) and a vertebrate host.
• Leishmania
• Leishmania is a genus of protozoan parasites, transmitted by the bite of infected sandflies.
• It causes diseases known as leishmaniasis in humans.
• Yeast
• Yeast is a type of fungus that is used in baking, brewing, and scientific research.
• It is a eukaryotic microorganism with a well-defined nucleus and other organelles.

The correct answer is 68.

Key Points

• Dmitri Mendeleev, a Russian chemist, predicted the existence of elements that were not yet discovered in his time.
• He named these elements using the prefix "eka-", which means "one beyond" in Sanskrit, based on their position in the periodic table.
• Eka-aluminium was predicted to be the element below aluminium in the periodic table.
• Mendeleev estimated the atomic mass of eka-aluminium to be around 68, which later turned out to be very close to the actual atomic mass of gallium (69.723).

Additional Information

• Mendeleev's Periodic Table
• In 1869, Dmitri Mendeleev created the first version of the periodic table, which arranged elements by increasing atomic mass.
• This arrangement revealed recurring (periodic) properties of the elements, which led to the periodic law.
• Mendeleev's table had gaps for elements that were not yet discovered, and he used the properties of neighboring elements to predict the properties of these unknown elements.
• Eka-Elements
• To predict the properties of undiscovered elements, Mendeleev used the term "eka" followed by the name of the element in the same group but in the previous period (e.g., eka-aluminium).
• Eka-aluminium was later discovered and named gallium.
• Other predicted elements included eka-boron (scandium) and eka-silicon (germanium).
• Gallium
• Gallium was discovered by French chemist Paul-Émile Lecoq de Boisbaudran in 1875.
• It has an atomic number of 31 and an atomic mass of 69.723.
• Gallium has a melting point of 29.76 °C (85.57 °F), which is low enough to melt in the hand.
• It is used in electronics, particularly in semiconductors and LEDs.

The correct answer is Ravi Shankar.

Key Points

• Ravi Shankar was a renowned Indian sitar virtuoso and composer who achieved international fame.
• He founded the National Orchestra of India, known as the Vadya Vrinda, in 1952.
• Ravi Shankar significantly contributed to popularizing Indian classical music in the Western world, collaborating with many famous artists including George Harrison of The Beatles.
• He was awarded India's highest civilian honor, the Bharat Ratna, in 1999 for his contributions to music.

Additional Information

• Sitar
• The sitar is a plucked stringed instrument used in Indian classical music.It is known for its rich and complex sound, created by the interaction of its strings and the resonating gourd.
• The sitar has been used in many forms of Indian music, including Hindustani classical music, film music, and popular music.
• National Orchestra of India (Vadya Vrinda)
• Founded by Ravi Shankar in 1952, the orchestra aimed to blend Indian classical instruments with Western orchestration techniques.
• It was one of the first attempts to create a symphonic orchestra that represented Indian traditional music.
• Contributions to Music
• Ravi Shankar collaborated with numerous Western musicians, introducing the sitar and Indian classical music to a global audience.
• He performed at major music festivals, including the Monterey Pop Festival and Woodstock.
• He also composed music for films, including the soundtrack for the movie "Gandhi" directed by Richard Attenborough.

The correct answer is Justice Verma Committee.

Key Points

• The Justice Verma Committee was established to recommend amendments to criminal laws to provide for quicker trial and enhanced punishment for criminals accused of committing sexual assault against women.
• One of the significant contributions of the Justice Verma Committee was identifying the existence of legal provisions for the implementation of some of the Fundamental Duties.
• The Committee underscored the importance of fundamental duties as a counterpart to fundamental rights, emphasizing their role in ensuring the proper functioning of democracy.
• The recommendations of the Justice Verma Committee have been instrumental in shaping reforms in India's legal framework, especially concerning women's safety and legal provisions related to fundamental duties.

Additional Information

• Fundamental Duties:
• Fundamental Duties are specified in Part IV-A of the Indian Constitution (Article 51A).
• They were added by the 42nd Amendment Act, 1976, upon the recommendations of the Swaran Singh Committee.
• Originally, ten duties were listed; the 86th Amendment Act, 2002, added an eleventh duty.
• These duties are moral obligations on all citizens to help promote a spirit of patriotism and to uphold the unity of India.
• Justice Verma Committee:
• The Justice Verma Committee was constituted in the wake of the 2012 Delhi gang-rape case.
• It was tasked with reviewing existing laws and suggesting changes to ensure the safety and dignity of women.
• Its report, submitted in January 2013, made several landmark recommendations for legal reforms.
• The Committee's work led to the enactment of the Criminal Law (Amendment) Act, 2013, which introduced stricter penalties for sexual crimes.
• Sarkaria Commission:
• Formed in 1983, the Sarkaria Commission was set up to review the working of the Indian federal system and the balance of power between the state and central governments.
• It submitted its report in 1988 with recommendations to improve center-state relations.
• Rajamannar Commission:
• The Rajamannar Committee, established by the Tamil Nadu government in 1969, was tasked with examining center-state relations.
• It recommended greater autonomy for states and the establishment of a permanent inter-state council.

The correct answer is Yugoslavia.

Key Points

• The first official summit of the Non-Aligned Movement (NAM) was held in Belgrade, Yugoslavia, in September 1961.
• The summit was a significant event as it marked the formal establishment of the Non-Aligned Movement.
• The NAM was founded during the Cold War as a group of states that were not formally aligned with or against any major power bloc.
• The movement aimed to create an independent path in world politics that would not be aligned with any major power bloc, particularly the United States or the Soviet Union.

Additional Information

• Non-Aligned Movement (NAM)
• The Non-Aligned Movement is a forum of 120 developing world states that are not formally aligned with or against any major power bloc.
• After the United Nations, it is the largest grouping of states worldwide.
• It originated in the Bandung Conference of 1955, and its first summit was held in Belgrade, Yugoslavia, in 1961.
• Key principles of NAM include mutual respect for territorial integrity and sovereignty, mutual non-aggression, mutual non-interference in domestic affairs, equality and mutual benefit, and peaceful coexistence.
• Bandung Conference
• The Bandung Conference, also known as the Asian-African Conference, was a meeting of Asian and African states, most of which were newly independent.
• It took place on 18-24 April 1955 in Bandung, Indonesia.
• The conference aimed to promote Afro-Asian economic and cultural cooperation and to oppose colonialism or neocolonialism by any nation.
• The conference was an important step in the establishment of the Non-Aligned Movement.
• Cold War
• The Cold War was a period of geopolitical tension between the Soviet Union and the United States and their respective allies, the Eastern Bloc and the Western Bloc, after World War II.
• The period is generally considered to span from 1947, the year the Truman Doctrine was announced, to 1991, the year the Soviet Union collapsed.
• It was marked by a rivalry between the two superpowers, which did not result in direct large-scale fighting between the two, but involved proxy wars, the arms race, and ideological competition.

The correct answer is New Delhi.

Key Points

• Gandharva Mahavidyalaya is one of the oldest music and dance academies in India, which was founded by Pandit Vinaya Chandra Maudgalya.
• The institution is located in New Delhi, the capital city of India.
• It has been instrumental in promoting Indian classical music and dance since its inception.
• Gandharva Mahavidyalaya offers courses in various forms of Indian classical music and dance, and has produced many renowned artists.

Additional Information

• Gandharva Mahavidyalaya:
• Founded in 1939 by Pandit Vinaya Chandra Maudgalya, a disciple of Vinayakrao Patwardhan, who was a prominent figure in the field of Indian classical music.
• The institution has branches and affiliated centers across India, but its main branch is located in New Delhi.
• It offers training in various forms of Indian classical music, including vocal, instrumental, and dance forms such as Kathak.
• Gandharva Mahavidyalaya also conducts exams and offers diplomas and certificates in music and dance, which are recognized by many educational and cultural institutions.
• Indian Classical Music:
• Indian classical music is categorized into two major traditions: Hindustani (North Indian) and Carnatic (South Indian).
• It is known for its complex ragas (melodic frameworks) and talas (rhythmic cycles).
• Prominent instruments in Indian classical music include the sitar, tabla, veena, mridangam, and bansuri.
• Indian classical music has a rich history and has been passed down through generations via the guru-shishya parampara (teacher-student tradition).
• Indian Classical Dance:
• There are eight recognized classical dance forms in India: Bharatanatyam, Kathak, Kathakali, Odissi, Kuchipudi, Manipuri, Mohiniyattam, and Sattriya.
• These dance forms are known for their intricate footwork, expressive hand gestures, and elaborate costumes.
• Each dance form has its own unique style and storytelling techniques, often depicting stories from Indian mythology and history.
• Classical dance is an integral part of Indian culture and heritage, and institutions like Gandharva Mahavidyalaya play a significant role in preserving and promoting these art forms.

The correct answer is Industrial policy.

Key Points

• The Industrial Policy of India provides a framework of rules and regulations for setting up public and private industries.
• It aims to promote industrial development and economic growth across the country.
• The policy outlines incentives and support mechanisms for industries to foster entrepreneurship and innovation.
• It includes measures for liberalization, privatization, and globalization to enhance the competitiveness of Indian industries on a global scale.

Additional Information

• Industrial Policy
• The Industrial Policy is formulated by the government to regulate and promote industrial development in the country.
• It includes guidelines for the establishment and operation of industries, including licensing, foreign investment, and technology transfer.
• The policy is periodically updated to adapt to changing economic conditions and to address emerging challenges in the industrial sector.
• Key components of the Industrial Policy include the promotion of small-scale industries, the development of infrastructure, and support for research and development.
• Union Budget
• The Union Budget is an annual financial statement presented by the Government of India, detailing its estimated revenue and expenditure for the upcoming fiscal year.
• It includes proposals for taxation, allocation of funds to various sectors, and measures to promote economic growth and development.
• The budget is a key policy document that outlines the government's economic priorities and fiscal policies.
• Company's Financial Statement
• A company's financial statement is a formal record of its financial activities and performance over a specific period.
• It includes the balance sheet, income statement, and cash flow statement, providing insights into the company's financial health and profitability.
• Financial statements are used by stakeholders, including investors, creditors, and regulators, to assess the company's financial position and make informed decisions.
• Annual Report of the Trade Union
• The annual report of a trade union provides a summary of its activities, achievements, and financial status for the year.
• It includes information on membership, collective bargaining agreements, and initiatives undertaken to protect workers' rights and interests.
• The report serves as a communication tool to keep members informed and engaged with the union's efforts.

The correct answer is The Pawn.

Key Points

• A pawn can only move forward to the unoccupied square immediately in front of it on the same file.
• On its first move, a pawn has the option to move forward two squares.
• Pawns capture diagonally, one square forward and to the left or right.
• If a pawn reaches the opposite end of the board, it can be promoted to any other piece, typically a queen.

Additional Information

• Chess Pieces and Their Movements:
• The Bishop: The bishop moves diagonally any number of squares. Each bishop starts on one color (light or dark) and must always stay on that color.
• The Queen: The queen can move any number of squares along a rank, file, or diagonal. It is the most powerful piece in terms of mobility.
• The Rook: The rook can move any number of squares along a rank or file. Rooks also play a crucial role in the endgame due to their ability to control large portions of the board.
• The Knight: The knight moves in an "L" shape: two squares in one direction and then one square perpendicular. Knights are unique because they can "jump" over other pieces.
• The King: The king moves one square in any direction. The objective of the game is to checkmate the opponent's king.
• Special Pawn Moves:
• En Passant: This special pawn capture can occur if a pawn moves two squares forward from its starting position and lands beside an opponent's pawn. The opponent's pawn can capture it "en passant" (in passing), as if it had only moved one square.Pawn Promotion: When a pawn reaches the opposite end of the board, it can be promoted to any other piece, except a king, typically a queen.

The correct answer is Uttar Pradesh.

Key Points

• Mulayam Singh Yadav served as the Chief Minister of Uttar Pradesh three times. His first term was from 1989 to 1991, the second from 1993 to 1995, and the third from 2003 to 2007.
• He was a prominent political leader and the founder of the Samajwadi Party, a major political party in Uttar Pradesh.
• Mulayam Singh Yadav played a significant role in Uttar Pradesh's politics and was known for his influence among the backward classes and minorities.
• Under his leadership, various social and economic policies were implemented in Uttar Pradesh aimed at improving the welfare of the underprivileged sections of society.

Additional Information

• Chief Minister:
• The Chief Minister is the head of the government at the state level in India.
• He/she is responsible for the administration of the state and the implementation of state policies and laws.
• The Chief Minister is appointed by the Governor of the state, usually the leader of the majority party or coalition in the state legislative assembly.
• Samajwadi Party:
• The Samajwadi Party is a political party in India primarily based in the state of Uttar Pradesh.
• It was founded on October 4, 1992, by Mulayam Singh Yadav.
• The party's ideology is based on socialism and it aims to promote social justice and equality.
• Uttar Pradesh:
• Uttar Pradesh is the most populous state in India, located in the northern part of the country.
• The capital of Uttar Pradesh is Lucknow, and it is known for its rich cultural heritage and historical significance.
• Uttar Pradesh plays a crucial role in Indian politics due to its large number of parliamentary and legislative assembly seats.

The correct answer is Digisaksham.

Key Points

• Digisaksham is a joint initiative by the National Career Service (NCS) and Microsoft India.
• The program aims to enhance employability by providing digital skills training to the youth.
• It includes training in various digital skills, such as data analysis, coding, and software development.
• The initiative is targeted at job seekers registered on the NCS portal, helping them improve their job prospects.

Additional Information

• National Career Service (NCS)
• The National Career Service is an initiative by the Ministry of Labour and Employment, Government of India.
• It aims to bridge the gap between job seekers and employers by providing a unified platform for career-related services.
• NCS offers services like job matching, career counseling, vocational guidance, and information on skill development courses.
• The portal caters to all stakeholders in the employment ecosystem, including job seekers, employers, training providers, and placement organizations.
• Microsoft India
• Microsoft India is a subsidiary of Microsoft Corporation, a global technology company.
• It focuses on empowering every individual and organization in India to achieve more through digital transformation.
• Microsoft India offers a wide range of products and services, including software, cloud computing, and AI solutions.
• The company is committed to driving digital skills development and fostering innovation in the country.
• Digital Skills
• Digital skills refer to the ability to use digital devices, communication applications, and networks to access and manage information.
• These skills include basic computer literacy, internet navigation, data analysis, coding, and software development.
• Proficiency in digital skills is essential for individuals to participate effectively in the digital economy and the modern workforce.
• Continuous learning and upskilling in digital technologies are crucial for career advancement and employability in various industries.

This HTML code provides a detailed explanation of the question related to the joint digital skills initiative for employability by the National Career Service and Microsoft India. The "key points" highlight the main aspects of the Digisaksham initiative, while the "additional information" section provides a broader context about the National Career Service, Microsoft India, and the importance of digital skills.

The correct answer is Assam.

Key Points

• Me-dam-me-phi is a festival primarily celebrated in Assam.
• This festival is observed by the Tai-Ahom community of Assam.
• It is celebrated to commemorate the ancestors and seek their blessings.
• The festival involves offerings to the departed souls and rituals to honor the dead.

Additional Information

• Tai-Ahom Community
• The Tai-Ahom people are an ethnic group found mostly in Assam, India.
• They are descendants of the Tai people who migrated from Southeast Asia to Assam.
• The Ahom kingdom was established in the 13th century and played a significant role in the history of Assam.
• Their language and culture have influenced the region significantly.
• Festival Rituals
• The Me-dam-me-phi festival involves rituals like offering food, drink, and other items to the spirits of the ancestors.
• Priests perform the rituals, and the community members participate in the ceremony.
• It is believed that the blessings of the ancestors bring prosperity and protection to the community.
• Significance of Me-dam-me-phi
• The festival holds cultural and religious significance for the Tai-Ahom people.
• It reinforces the bond between the living and the deceased family members.
• The festival also serves as an occasion for the community to come together and celebrate their heritage.

The correct answer is Oil India Limited.

Key Points

• Oil India Limited is a joint sector industry in India, where both the government and private sector have stakes.
• It was formed under a partnership between the Government of India and Burmah Oil Company Limited, UK.
• The company is a Navratna company, indicating that it has been given financial and operational autonomy by the Government of India.
• Oil India Limited is primarily engaged in the exploration, development, and production of crude oil and natural gas, transportation of crude oil, and production of LPG.

Additional Information

• Joint Sector Industry:
• A joint sector industry is one where the government and the private sector collaborate in the ownership and management of the enterprise.
• The main objective of such an arrangement is to pool resources, expertise, and investments from both the public and private sectors to achieve economic development.
• These industries are characterized by shared ownership between the government (usually through public sector undertakings) and private companies or individuals.
• Examples of joint sector enterprises include Oil India Limited, which is a partnership between the Government of India and private stakeholders.
• Public Sector Enterprise:
• Public sector enterprises are organizations that are wholly or majority-owned by the government.
• Examples include Bharat Heavy Electricals Limited (BHEL) and Steel Authority of India Limited (SAIL).
• Private Sector Enterprise:
• Private sector enterprises are businesses owned by private individuals or corporations.
• An example is Tata Iron and Steel Company (TISCO), now known as Tata Steel.
• Navratna Companies:
• Navratna status is given to public sector enterprises in India that have comparative advantages, giving them greater autonomy in financial and operational decision-making.
• This status helps these companies to better compete in the global market.
• Oil India Limited is one such Navratna company.

The correct answer is Kashmir.

Key Points

• Queen Didda ruled Kashmir from 958 AD to 1003 AD, effectively becoming the ruler of the region.
• She is known for her strong administrative skills and for maintaining the sovereignty of Kashmir during her reign.
• Queen Didda's rule is noted for its stability and prosperity, despite the political challenges of the time.
• She was one of the few female rulers in Indian history to have independently governed a region for such an extended period.

Additional Information

• Kannauj
• Kannauj, during the 10th century, was a prominent city in northern India and was a part of the tripartite struggle among the Pratiharas, the Rashtrakutas, and the Palas.
• It was known for its cultural and political significance.
• Malwa
• Malwa is a historical region in west-central India, known for its rich cultural history and contributions to Indian architecture and art.
• During the 10th century, it was ruled by the Paramara dynasty.
• Delhi
• Delhi, during the 10th century, was not as prominent as it became later during the Sultanate period.
• It was under various regional rulers before becoming the capital of the Delhi Sultanate in the 12th century.
• Kashmir
• Kashmir has a history of complex political dynamics and was known for its rich cultural and intellectual heritage.
• Queen Didda's rule is an important part of Kashmir’s history, highlighting the role of women in governance during that period.

Shortcut Trick The maximum value of any trigonometric expression in the form a sinθ + b cosθ is directly given by the formula √(a2 + b2). Here, comparing the given expression with the standard form, we have a = 2 and b = 3. ⇒ Maximum value = √(22 + 32) ⇒ Maximum value = √(4 + 9) = √13 ∴ The correct answer is √13.

Alternate Method Given: Expression: 2sinθ + 3cosθ

Formula Used: sin(A + B) = sinA cosB + cosA sinB sin2α + cos2α = 1 Calculations: Let us express the given expression in the form of R sin(θ + α). Assume a = R cosα = 2 and b = R sinα = 3 ⇒ Squaring and adding both the assumption equations: ⇒ R2cos2α + R2sin2α = 22 + 32 ⇒ R2(cos2α + sin2α) = 4 + 9 ⇒ R2(1) = 13 ⇒ R = √13 Now, substituting the values back into the original expression: ⇒ 2sinθ + 3cosθ ⇒ (R cosα)sinθ + (R sinα)cosθ ⇒ R(sinθ cosα + cosθ sinα) ⇒ √13 sin(θ + α) The maximum value of the sine function, sin(θ + α), is always 1. ⇒ Maximum value = √13 × 1 = √13 ∴ The correct answer is √13.

Additional Information Extremum Values of a sinθ ± b cosθ The maximum value of the function is +√(a2 + b2) and the minimum value is −√(a2 + b2). Extremum Values of a sin2θ + b cos2θ If a > b, the maximum value is always a and the minimum value is b. Minimum Value of Reciprocal Squares The minimum value of a2tan2θ + b2cot2θ is always 2ab. (The maximum value tends to infinity).

Shortcut Trick

The radius, perpendicular distance, and half of the chord form a right-angled triangle.

Using the standard (6, 8, 10) Pythagorean triplet, the half-chord length corresponds to 8 cm.

The total length of the chord is simply twice its half-length.

Length of chord = 2 × 8 = 16 cm.

∴ The correct answer is 16 cm.

Alternate Method

Given:

Radius of the circle (OB) = 10 cm

Perpendicular distance from the centre to the chord (OM) = 6 cm

Formula Used:

By Pythagoras Theorem: (Hypotenuse)² = (Perpendicular)² + (Base)²

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

Calculations:

⇒ In right-angled triangle OMB, OB² = OM² + MB²

⇒ 10² = 6² + MB²

⇒ 100 = 36 + MB²

⇒ MB² = 100 − 36

⇒ MB² = 64

⇒ MB = √64 = 8 cm

⇒ Since the perpendicular OM bisects the chord AB, the total length of chord AB = 2 × MB

⇒ AB = 2 × 8 = 16 cm

∴ The correct answer is 16 cm.

Additional Information

Perpendicular to a Chord Theorem

A perpendicular drawn from the centre of a circle to a chord perfectly bisects the chord into two equal segments.

Common Pythagorean Triplets

Knowing core triplets like (3, 4, 5), (6, 8, 10), (5, 12, 13), and (8, 15, 17) helps in solving right-angled triangle geometry problems instantly without manual calculations.

Longest Chord

The longest chord of a circle passes through its centre, known as the diameter, where Length = 2 × radius.

Given:

Simplify the expression: 25 ÷ 5 + 3 of 2 – 4 + 6 of 3 ÷ 2 – 8 + 4

Formula Used:

Order of operations (BODMAS/BIDMAS): Brackets, Orders (i.e., powers and roots, etc.), Division and Multiplication (left-to-right), Addition and Subtraction (left-to-right)

Calculation:

Expression: 25 ÷ 5 + 3 of 2 – 4 + 6 of 3 ÷ 2 – 8 + 4

⇒ 5 + 6 – 4 + 18 ÷ 2 – 8 + 4

⇒ 11 – 4 + 9 – 8 + 4

⇒ 12

The correct answer is option 4.

Shortcut Trick If x + 1/x = k, then the positive value of x − 1/x = √(k2 − 4). Given k = 7, substitute the value directly into the formula: x − 1/x = √(72 − 4) x − 1/x = √(49 − 4) = √45 = 3√5 ∴ The correct answer is 3√5.

Alternate Method Given: x + 1/x = 7 (where x > 0) Formula Used: (a − b)2 = (a + b)2 − 4ab Calculations: Let a = x and b = 1/x. ⇒ (x − 1/x)2 = (x + 1/x)2 − 4 × (x) × (1/x) ⇒ (x − 1/x)2 = 72 − 4 × 1 ⇒ (x − 1/x)2 = 49 − 4 ⇒ (x − 1/x)2 = 45 Taking the square root on both sides: ⇒ x − 1/x = ±√45 ⇒ x − 1/x = ±3√5 Since the question specifically asks for the positive value: ⇒ x − 1/x = 3√5 ∴ The correct answer is 3√5.

Additional Information Squares of Inverse Sums If x + 1/x = k, then x2 + 1/x2 = k2 − 2. Cubes of Inverse Sums If x + 1/x = k, then x3 + 1/x3 = k3 − 3k. Negative Inverse Conversion If x − 1/x = k, then x + 1/x = ±√(k2 + 4). Cubes of Negative Inverse Sums If x − 1/x = k, then x3 − 1/x3 = k3 + 3k.

Shortcut Trick

Overall Multiplier = Price Multiplier × Weight Multiplier

Price Multiplier = SP ÷ CP = 44 ÷ 40 = 11 ÷ 10

Weight Multiplier = Claimed Weight ÷ Actual Weight = 1000 ÷ 800 = 5 ÷ 4

Overall Multiplier = (11 ÷ 10) × (5 ÷ 4) = 55 ÷ 40 = 11 ÷ 8

Gain % = [(11 − 8) ÷ 8] × 100 = (3 ÷ 8) × 100 = 37.5%

∴ The correct answer is 37.5%.

Alternate Method

Given:

⇒ Cost price (CP) for 1000g rice = ₹40

⇒ Selling price (SP) claimed for 1000g = ₹44

⇒ Weight removed per 1kg packet = 200g

⇒ Actual weight given to customer = 1000 − 200 = 800g

Formula Used:

⇒ Profit = SP − CP

⇒ Profit % = (Profit ÷ CP) × 100

Calculations:

⇒ Actual cost incurred by shopkeeper for 800g = (40 ÷ 1000) × 800 = ₹32

⇒ Selling price obtained for the 1kg packet = ₹44

⇒ Total Profit earned = 44 − 32 = ₹12

⇒ Overall Profit % = (12 ÷ 32) × 100

⇒ Overall Profit % = (3 ÷ 8) × 100 = 37.5%

∴ The correct answer is 37.5%.

Additional Information

Dishonest Dealer Concept

When a dealer sells goods at Cost Price but uses false weight, his profit % = [(True Weight − False Weight) ÷ False Weight] × 100.

Successive Profit Multiplier

If there is a markup in price as well as cheating in weight, Overall Multiplier = (SP ÷ CP) × (True Weight ÷ False Weight). Final Profit % = (Overall Multiplier − 1) × 100.

Mark-up and Discount

The selling price can be adjusted by adding a markup on the CP and then giving a discount. Effective SP = CP × (1 + Markup %) × (1 − Discount %).

Given:

Highest sales in 2022 = ₹584 crore

Lowest sales in 2018 = ₹205 crore

Formula used:

The mean of the highest and lowest sales is given by:

Mean = $\frac{\text{Highest sales} + \text{Lowest sales}}{2}$

Calculations:

Mean = $\frac{584 + 205}{2}$

⇒ Mean = $\frac{789}{2}$

⇒ Mean = 394.5 crore

The mean of the highest and lowest sales is ₹394.5 crore.

Shortcut Trick

For age-based questions, verifying the options with the given conditions is the fastest method.

Condition 1: 3 years ago, the person's age was 6 × his brother's age.

Checking Option 1 (15 years, 5 years): Ages 3 years ago were 12 and 2. Since 12 = 6 × 2, it is true.

Condition 2: 8 years later, person's age will be 2 × brother's age − 3.

Ages 8 years later will be 23 and 13. Since 23 = 2 × 13 − 3 = 26 − 3, it is also true.

∴ The correct answer is 15 years and 5 years.

Alternate Method

Let the present age of the person be P and his younger brother be B.

Step 1: Applying the condition for 3 years ago

⇒ P − 3 = 6 × (B − 3)

⇒ P − 3 = 6B − 18

⇒ P − 6B = −15     ... (Equation 1)

Step 2: Applying the condition for 8 years later

⇒ P + 8 = 2 × (B + 8) − 3

⇒ P + 8 = 2B + 16 − 3

⇒ P − 2B = 5     ... (Equation 2)

Step 3: Solving the linear equations

Subtracting Equation 1 from Equation 2:

⇒ (P − 2B) − (P − 6B) = 5 − (−15)

⇒ 4B = 20

⇒ B = 5

Substitute B = 5 in Equation 2:

⇒ P − 2(5) = 5

⇒ P − 10 = 5

⇒ P = 15

∴ The correct answer is 15 years and 5 years.

Additional Information

Present Age Variable Assumption

It is always best to assume the current age as the variable (like x). This way, age 't' years ago becomes (x − t) and age 't' years later easily becomes (x + t) without any confusion.

Constant Age Difference Concept

The difference between the ages of two individuals always remains constant at any point in time. If a person is 5 years older than their brother today, they will always remain 5 years older.

Cross-Multiplication Method

When solving age problems given in ratios over two different time periods, we can equate the cross-product differences of the ratios to the difference in the given time intervals directly.

Shortcut Trick

Direct ratio of Cost Price and Marked Price is: CP : MP = (100 − D%) : (100 + P%)

⇒ CP : MP = (100 − 5) : (100 + 4.5) = 95 : 104.5

⇒ CP : MP = 190 : 209 = 10 : 11

Given 11 units (MP) = ₹275 ⇒ 1 unit = ₹25

Thus, 10 units (CP) = 10 × 25 = ₹250.

∴ The correct answer is ₹250.

Alternate Method

Given:

Marked Price (MP) = ₹275

Discount (D) = 5%

Profit (P) = 4.5%

Formula Used:

Selling Price (SP) = MP × (100 − D%) / 100

Selling Price (SP) = CP × (100 + P%) / 100

Equating both SP formulations gives the direct relation: CP / MP = (100 − D%) / (100 + P%)

Calculations:

⇒ CP / 275 = (100 − 5) / (100 + 4.5)

⇒ CP / 275 = 95 / 104.5

⇒ CP = 275 × (950 / 1045)

⇒ CP = 275 × (190 / 209)

⇒ CP = 275 × (10 / 11)

⇒ CP = 25 × 10

⇒ CP = 250

∴ The correct answer is ₹250.

Additional Information

Direct Relation Between CP and MP

When profit percentage (P%) and discount percentage (D%) are simultaneously given, the ratio of Cost Price to Marked Price is directly derived as CP / MP = (100 − D) / (100 + P).

Successive Discounts

When two successive discounts of a% and b% are offered, the effective discount percentage is evaluated by the formula: Equivalent Discount = (a + b − ab / 100)%.

Markup Percentage

The percentage by which the marked price is intentionally raised above the true cost price is the markup percentage. Formula used is Markup % = [(MP − CP) / CP] × 100.

Shortcut Trick

For two intersecting circles, the length of the common tangent PQ can be directly calculated using the relationship between the extended common chord XY and the inner chord AB.

The extended line bisects the common tangents and the external segments are equal (XA = YB).

Direct formula: PQ = √(XY² − AB²)

⇒ PQ = √(5² − 3²)

⇒ PQ = √(25 − 9) = √16 = 4 cm.

∴ The correct answer is 4 cm.

Alternate Method

Given:

Length of common chord AB = 3 cm

Extended transversal segment XY = 5 cm

Formula Used:

Tangent-Secant Theorem: PT² = PA × PB

            P Q R S X Y A B PQ = 4 cm

Calculations:

⇒ For the first circle, XP is a tangent and XAB is a secant → XP² = XA × XB

⇒ For the second circle, XQ is a tangent and XAB is a secant → XQ² = XA × XB

⇒ Therefore, XP = XQ, meaning X is the midpoint of PQ.

⇒ Similarly, Y is the midpoint of RS, so YR = YS.

⇒ Direct common tangents are equal in length, so PQ = RS → XP = YR.

⇒ XP² = YR² ⇒ XA × XB = YB × YA

⇒ Let XA = x and YB = y. We know XB = x + 3 and YA = y + 3.

⇒ x(x + 3) = y(y + 3) ⇒ x² + 3x = y² + 3y

⇒ (x − y)(x + y + 3) = 0 ⇒ x = y (since lengths must be positive).

⇒ Thus, XA = YB.

⇒ Given XY = XA + AB + YB = 5 cm.

⇒ 2 × XA + 3 = 5 ⇒ 2 × XA = 2 ⇒ XA = 1 cm.

⇒ Now, XB = XA + AB = 1 + 3 = 4 cm.

⇒ XP² = 1 × 4 = 4 ⇒ XP = 2 cm.

⇒ Length of PQ = 2 × XP = 2 × 2 = 4 cm.

∴ The correct answer is 4 cm.

Additional Information

Tangent-Secant Theorem

If a tangent and a secant intersect in the exterior of a circle, the square of the tangent equals the product of the secant segment and its external part (PT² = PA × PB).

Common Chord Property

The common chord of two intersecting circles completely bisects their direct common tangents.

Direct Common Tangent (DCT)

For two circles with radii R and r, and distance between centers d, the length of DCT = √[d² − (R − r)²].

Shortcut Trick

Since the total capacity of the tank is constant, the number of buckets is inversely proportional to the bucket's capacity.

New bucket capacity = 2/5 of the original capacity.

⇒ Required number of buckets = 5/2 × original number of buckets.

⇒ Required number of buckets = 5/2 × 50 = 125.

∴ The correct answer is 125 buckets.

Alternate Method

Given:

Number of buckets required initially = 50

New capacity of the bucket = 2/5 of the present capacity

Formula Used:

Total Tank Capacity = Number of buckets × Capacity of one bucket

Calculations:

⇒ Let the original capacity of one bucket be C.

⇒ Total capacity of the tank = 50 × C = 50C

⇒ New capacity of one bucket = 2/5 × C = 0.4C

⇒ Let the number of new buckets required be N.

⇒ Total capacity with new buckets = N × 0.4C

⇒ Since the total tank capacity remains exactly the same, we equate both expressions:

⇒ N × 0.4C = 50C

⇒ N × (2/5) = 50

⇒ N = 50 × (5/2)

⇒ N = 25 × 5 = 125

∴ The correct answer is 125 buckets.

Additional Information

Inverse Proportion

When the product of two variables is constant (e.g., Number of buckets × Capacity), they are inversely proportional. If capacity is multiplied by a factor, the number of buckets is divided by the same factor.

Direct Proportion

Two quantities are directly proportional if their ratio is constant. If one quantity increases, the other increases at the same rate (y = k × x).

Unitary Method

A foundational mathematical technique for solving problems by first finding the value of a single unit, then multiplying to find the required equivalent value.

Work Equivalence (M₁D₁ = M₂D₂)

This inverse concept applies similarly to time and work problems, where total work is constant. The product of workers and days remains equal across scenarios.

Given:

Total sales of Shop S2 for both years = 70 + 90 = 160 thousand pens

Total sales of Shop S5 for both years = 80 + 60 = 140 thousand pens

Formula used:

The ratio of the total sales of Shop S2 to the total sales of Shop S5 is given by:

Ratio = $\frac{\text{Total sales of Shop S2}}{\text{Total sales of Shop S5}}$

Calculations:

Ratio = $\frac{160}{140}$

⇒ Ratio = $\frac{16}{14}$

⇒ Ratio = 8:7

The ratio of the total sales of Shop S2 to the total sales of Shop S5 is 8:7.

Shortcut Trick

Total Work = LCM(12, 16) = 48 units.

Efficiency of A = 48 ÷ 12 = 4 units/day.

Efficiency of B = 48 ÷ 16 = 3 units/day.

Work done by A and B in 6 days = (4 + 3) × 6 = 42 units.

Fraction of work left = (48 − 42) ÷ 48 = 6 / 48 = 1 / 8.

∴ The correct answer is 1/8.

Alternate Method

Given:

Time taken by A = 12 days

Time taken by B = 16 days

Duration they worked together = 6 days

Formula Used:

Total Work = Efficiency × Time

Calculations:

⇒ Combined efficiency of A and B = 4 + 3 = 7 units/day

⇒ Total work completed in 6 days = 7 × 6 = 42 units

⇒ Remaining work = Total Work − Work Completed

⇒ Remaining work = 48 − 42 = 6 units

⇒ Fraction of work left = Remaining Work ÷ Total Work

⇒ Fraction of work left = 6 ÷ 48 = 1/8

∴ The correct answer is 1/8.

Additional Information

Pipes and Cisterns

It follows the exact same LCM concept as Time and Work. A pipe filling a tank has positive efficiency, while a leak or emptying pipe acts with negative efficiency.

Work and Wages

Total wages earned are always distributed in the exact ratio of the total work done by each person. If they work for the same number of days, wages are directly proportional to their efficiencies.

Alternate Days Work

When people work on alternate days, calculate the combined work done in one complete cycle (e.g., 2 days) and scale it up using multiples to reach closest to the total work.

Shortcut Trick

In a cyclic quadrilateral, opposite angles sum to 180°.

⇒ ∠PQR = 180° − ∠PSR = 180° − 120° = 60°

The angle subtended by a diameter at the circumference is always 90°.

⇒ ∠PRQ = 90°

In right-angled ΔPQR, ∠QPR = 180° − (90° + 60°) = 30°

∴ The correct answer is 30°.

Alternate Method

Given:

PQ is the diameter of the circle with centre O.

PQRS is a cyclic quadrilateral inscribed in the circle.

∠PSR = 120°

O P Q R S

120°

60°

90°

? ∴ ∠QPR = 30°

Formula Used:

1. The sum of opposite angles in a cyclic quadrilateral is 180°.
2. The angle subtended by a diameter at the circumference of a circle is 90°.
3. The sum of interior angles of a triangle is 180°.

Calculation:

Since PQRS is a cyclic quadrilateral, the sum of its opposite angles is 180°.

⇒ ∠PSR + ∠PQR = 180°

⇒ 120° + ∠PQR = 180°

⇒ ∠PQR = 180° − 120° = 60°

Since PQ is the diameter, the angle formed in the semicircle is a right angle.

⇒ ∠PRQ = 90°

Now, consider ΔPQR. The sum of all angles in a triangle is 180°.

⇒ ∠QPR + ∠PQR + ∠PRQ = 180°

⇒ ∠QPR + 60° + 90° = 180°

⇒ ∠QPR + 150° = 180°

⇒ ∠QPR = 180° − 150° = 30°

∴ The correct answer is 30°.

Additional Information

Cyclic Quadrilateral Property

A quadrilateral whose all four vertices lie on a circle is called a cyclic quadrilateral. The sum of either pair of opposite angles is exactly 180° (∠A + ∠C = 180° and ∠B + ∠D = 180°).

Angle in a Semicircle

According to Thales's Theorem, any angle subtended by a diameter of a circle at any point on the circumference is always a right angle (90°).

Angle Subtended by an Arc

The angle subtended by an arc at the centre is double the angle subtended by it at any remaining part of the circle.

Shortcut Trick

The Volume and Curved Surface Area (CSA) of a cylinder are directly related.

Since Volume = πr²h and CSA = 2πrh, we can substitute CSA into the volume formula.

Volume = (CSA ÷ 2) × r

Volume = (792 ÷ 2) × 10.5

Volume = 396 × 10.5 = 4158 cm³

∴ The correct answer is 4158 cm³.

Alternate Method

Given:

Radius of base (r) = 10.5 cm

Curved Surface Area (CSA) = 792 cm²

Formula Used:

CSA of cylinder = 2πrh

Volume of cylinder = πr²h

Calculations:

⇒ 2 × (22 ÷ 7) × 10.5 × h = 792

⇒ 2 × 22 × 1.5 × h = 792

⇒ 66 × h = 792

⇒ h = 792 ÷ 66 = 12 cm

Now, substitute the value of height (h) to calculate the volume:

⇒ Volume = πr²h

⇒ Volume = (22 ÷ 7) × 10.5 × 10.5 × 12

⇒ Volume = 22 × 1.5 × 10.5 × 12

⇒ Volume = 33 × 126

⇒ Volume = 4158 cm³

∴ The correct answer is 4158 cm³.

Additional Information

Total Surface Area of a Cylinder

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

Volume of a Cone vs Cylinder

A cone with the same base radius and height has exactly one-third the volume of the corresponding cylinder: V = (1 ÷ 3)πr²h.

Hollow Cylinder

For a hollow cylinder with outer radius R and inner radius r, the volume of the material used is given by V = π(R² − r²)h.

Shortcut Trick

Put t = 30° to quickly evaluate the expression and eliminate options.

Expression = [(1 − sin(60°)) / (1 + sin(60°))] × [(cos(30°) + sin(30°)) / (cos(30°) − sin(30°))]

Expression = [(1 − √3/2) / (1 + √3/2)] × [(√3/2 + 1/2) / (√3/2 − 1/2)]

Expression = (7 − 4√3) × (2 + √3) = 2 − √3.

Checking given options for t = 30°, Option 2 gives: (1 − tan(30°)) / (1 + tan(30°)) = (1 − 1/√3) / (1 + 1/√3) = (√3 − 1) / (√3 + 1) = 2 − √3.

∴ The correct answer is (1 − tan(t)) / (1 + tan(t)).

Alternate Method

Given:

Expression = [(1 − sin(2t)) / (1 + sin(2t))] × [(cos(t) + sin(t)) / (cos(t) − sin(t))]

Formula Used:

1 = sin²(t) + cos²(t)

sin(2t) = 2sin(t)cos(t)

(a − b)² = a² + b² − 2ab

Calculations:

⇒ 1 − sin(2t) = cos²(t) + sin²(t) − 2sin(t)cos(t) = (cos(t) − sin(t))²

⇒ 1 + sin(2t) = cos²(t) + sin²(t) + 2sin(t)cos(t) = (cos(t) + sin(t))²

Substitute these values into the given expression:

⇒ Expression = [(cos(t) − sin(t))² / (cos(t) + sin(t))²] × [(cos(t) + sin(t)) / (cos(t) − sin(t))]

Cancel out the common terms from numerator and denominator:

⇒ Expression = (cos(t) − sin(t)) / (cos(t) + sin(t))

Divide numerator and denominator by cos(t):

⇒ Expression = [1 − (sin(t) / cos(t))] / [1 + (sin(t) / cos(t))]

⇒ Expression = (1 − tan(t)) / (1 + tan(t))

∴ The correct answer is (1 − tan(t)) / (1 + tan(t)).

Additional Information

Double Angle Formulas

sin(2θ) = 2sin(θ)cos(θ) and cos(2θ) = cos²(θ) − sin²(θ).

Tangent Addition Formula

tan(A − B) = (tan(A) − tan(B)) / (1 + tan(A)tan(B)). Applying this for A = 45° gives tan(45° − θ) = (1 − tan(θ)) / (1 + tan(θ)).

Pythagorean Identities

sin²(θ) + cos²(θ) = 1, which is commonly used to convert 1 ± sin(2θ) into perfect square expressions like (cos(θ) ± sin(θ))².

Shortcut Trick

Average age of the 6 children = Total sum ÷ Number of children = 90 ÷ 6 = 15 years.

Since the ages are in an arithmetic progression with a difference of 2, the average lies exactly in the middle of the series.

For 6 children (even number), the average of 15 years lies directly between the 3rd and 4th child.

Therefore, the age of the 3rd child is 14 years and the 4th child is 16 years.

Adding the 2-year interval consecutively: 5th child = 18 years, 6th child (eldest) = 20 years.

∴ The correct answer is 20 years.

Alternate Method

Given:

Number of children = 6

Interval between births = 2 years

Sum of all their ages = 90 years

Calculations:

Let the age of the youngest child be x years.

Since the age interval is 2 years, the ages of the 6 children can be written as:

⇒ x, (x + 2), (x + 4), (x + 6), (x + 8), and (x + 10)

According to the given condition, the total sum of their ages is 90 years:

⇒ x + (x + 2) + (x + 4) + (x + 6) + (x + 8) + (x + 10) = 90

⇒ 6x + (2 + 4 + 6 + 8 + 10) = 90

⇒ 6x + 30 = 90

⇒ 6x = 90 − 30

⇒ 6x = 60

⇒ x = 10

The age of the youngest child is 10 years.

The age of the eldest child is represented by (x + 10):

⇒ Age of the eldest child = 10 + 10 = 20 years

∴ The correct answer is 20 years.

Additional Information

Arithmetic Progression (AP)

A mathematical sequence where the difference between consecutive terms remains strictly constant. Here, the children's ages form an AP with a common difference of 2.

Average of an Arithmetic Progression

The average (or mean) of an AP directly correlates to its middle value. If the sequence has an even number of terms, the average lands exactly halfway between the two central terms. It can also be determined by the formula: Average = (First Term + Last Term) ÷ 2.

Sum of an AP

The sum of any arithmetic sequence can be rapidly computed if the average is known: Sum = n × Average, where n represents the total number of terms in the sequence.

Shortcut Trick

Simple Interest for 10 years = ₹2,366 − ₹1,456 = ₹910.

Interest for 1 year = ₹910 ÷ 10 = ₹91.

Rate % = (Interest for 1 year ÷ Principal) × 100

Rate % = (91 ÷ 1456) × 100 = 6.25%.

∴ The correct answer is 6.25%.

Alternate Method

Given:

Principal (P) = ₹1,456

Amount (A) = ₹2,366

Time (T) = 10 years

Formula Used:

Simple Interest (I) = Amount (A) − Principal (P)

Rate (R) = (I × 100) ÷ (P × T)

Calculations:

⇒ SI = ₹2,366 − ₹1,456

⇒ SI = ₹910

⇒ R = (910 × 100) ÷ (1456 × 10)

⇒ R = 91000 ÷ 14560

⇒ R = 9100 ÷ 1456

⇒ R = 25 ÷ 4

⇒ R = 6.25%

∴ The correct answer is 6.25%.

Additional Information

Simple Interest (SI)

It is the interest calculated only on the initial principal amount. The principal remains constant throughout the tenure. Formula: SI = (P × R × T) ÷ 100.

Amount (A)

It is the total sum of money payable at the end of the time period, which includes the original principal and the interest accumulated. Formula: A = P + SI.

Compound Interest (CI)

Interest calculated on both the initial principal and the accumulated interest from previous periods. Formula: A = P(1 + R ÷ 100)^T.

Given:

Total profit in 1997 =  135 + 115 + 90 + 70 = 410 lakh

Total profit in 2000 = 150 + 180 + 210 + 230 = 770 lakh

Formula used:

The formula to calculate the percentage increase is:

$\frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100$

Calculations:

Percentage increase = $\frac{770 - 410}{410} \times 100$

⇒ Percentage increase = $\frac{360}{410} \times 100$

⇒ Percentage increase = 0.878 × 100

⇒ Percentage increase = 87.8%

The percentage increase in the annual profit of the departmental store from 1997 to 2000 is 87.8%.

Shortcut Trick

Use the direct identity: 1 / (1 + tan² θ) = cos² θ.

So, the fraction 1 / (1 + tan² 58°) becomes cos² 58°.

Since 58° and 32° are complementary angles (58° + 32° = 90°), we know cos 58° = sin 32°.

Substitute this back into the expression: 1 − sin² 32° + sin² 32°.

The terms − sin² 32° and + sin² 32° cancel each other out.

∴ The correct answer is 1.

Alternate Method

Given:

1 − sin² 32° + 1 / (1 + tan² 58°)

Formula Used:

1 + tan² θ = sec² θ

1 / sec² θ = cos² θ

cos(90° − θ) = sin θ

Calculations:

⇒ 1 − sin² 32° + 1 / (1 + tan² 58°)

⇒ 1 − sin² 32° + 1 / sec² 58°

⇒ 1 − sin² 32° + cos² 58°

⇒ 1 − sin² 32° + cos²(90° − 32°)

⇒ 1 − sin² 32° + sin² 32°

⇒ 1 − (sin² 32° − sin² 32°)

⇒ 1

∴ The correct answer is 1.

Additional Information

Trigonometric Identities

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

Complementary Angles

For any angle θ, the ratios of complementary angles relate as: sin(90° − θ) = cos θ, tan(90° − θ) = cot θ, and sec(90° − θ) = cosec θ.

Shortcut Trick

Ratio of number of matches = 20 : 25 = 4 : 5.

Let assumed average be 15, then the deviation of the second group = 24 − 15 = +9.

Net average deviation = (+9 × 5 parts) ÷ (4 + 5) total parts = +45 ÷ 9 = +5.

Overall Average = 15 + 5 = 20.

∴ The correct answer is 20 runs.

Alternate Method

Given:

Average score in the first 20 matches = 15

Average score in the next 25 matches = 24

Total number of matches = 20 + 25 = 45

Formula Used:

Combined Average = (Sum of all observations) ÷ (Total number of observations)

20 matches Avg = 15

25 matches Avg = 24

Total 45 matches, Combined Avg = ?

Calculations:

⇒ Total score in the first 20 matches = 20 × 15 = 300

⇒ Total score in the next 25 matches = 25 × 24 = 600

⇒ Total score in all 45 matches = 300 + 600 = 900

⇒ Average score in all 45 matches = 900 ÷ 45 = 20

∴ The correct answer is 20 runs.

Additional Information

Weighted Average

When combining two sets of data with sizes n₁ and n₂ and averages A₁ and A₂ respectively, the combined average is calculated as (n₁ × A₁ + n₂ × A₂) ÷ (n₁ + n₂).

Net Deviation Method

This method simplifies average calculations by adjusting a baseline. True Average = Assumed Average + (Sum of Net Deviations ÷ Total Count). It is highly useful when numbers are large or group ratios are simpler to compute.

Given:

Percentage of students in Class 8 = 15%

Percentage of students in Class 10 = 25%

Percentage of students in Class 12 = 18%

Number of students admitted to Class 8 = 600

Formula used:

The total number of students can be calculated using the formula:

$\frac{\text{Number of students in Class 8} \times 100}{\text{Percentage of students in Class 8}}$

Calculations:

Calculate the total number of students admitted to the school:

$\frac{600 \times 100}{15} = 4000$

Step 2: Find the number of students in Class 10 and Class 12:

Class 10 : $\frac{25}{100} \times 4000 = 1000$

Class 12 : $\frac{18}{100} \times 4000 = 720$

Calculate the difference in the number of students between Class 10 and Class 12:

Difference: 1000 - 720 = 280 

The number of students in Class 10 is 280 more than in Class 12.

Shortcut Trick

Ratio of speeds of Policeman to Thief = 10 : 9

Since the time taken is constant, ratio of distances covered = 10 : 9

Difference in distance ratio = 1 unit = 200 m = 0.2 km

Distance covered by thief = 9 units = 9 × 0.2 = 1.8 km

∴ The correct answer is 1.8 km.

Alternate Method

Given:

Speed of Policeman = 10 km/h

Speed of Thief = 9 km/h

Initial distance between them = 200 m = 0.2 km

Formula Used:

Relative Speed = Speed of A − Speed of B (in same direction)

Time = Distance ÷ Relative Speed

Distance = Speed × Time

Calculations:

⇒ Relative speed = 10 − 9 = 1 km/h

⇒ Time taken to catch the thief = 0.2 km ÷ 1 km/h = 0.2 h

⇒ Distance covered by thief before being caught = Speed of Thief × Time

⇒ Distance = 9 km/h × 0.2 h = 1.8 km

∴ The correct answer is 1.8 km.

Additional Information

Relative Speed Concept

When two bodies move in the same direction, their relative speed is the difference of their speeds (V₁ − V₂). When moving in opposite directions, it is the sum of their speeds (V₁ + V₂).

Unit Conversion

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

Average Speed

When equal distances are covered at different speeds x and y, the average speed for the entire journey is given by (2 × x × y) ÷ (x + y).

Shortcut Trick

By the divisibility rule of 11, the absolute difference of alternate digits' sum must be 0 or a multiple of 11.

A → |(7 + 2 + 1) − (1 + 7 + 2)| = 0 (Divisible)

B → |(1 + 7 + 1) − (7 + 2 + 0)| = 0 (Divisible)

C → |(6 + 6 + 4) − (4 + 1)| = 11 (Divisible)

D → |(7 + 6 + 4) − (5 + 1 + 8)| = 3 (Not divisible)

∴ The correct answer is D.

Alternate Method

Given:

Numbers: A = 712712, B = 177210, C = 64614, D = 756148

Formula Used:

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

Calculations:

Checking each given number step-by-step:

⇒ For A = 712712: |(7 + 2 + 1) − (1 + 7 + 2)| = |10 − 10| = 0

Since the difference is 0, 712712 is divisible by 11.

⇒ For B = 177210: |(1 + 7 + 1) − (7 + 2 + 0)| = |9 − 9| = 0

Since the difference is 0, 177210 is divisible by 11.

⇒ For C = 64614: |(6 + 6 + 4) − (4 + 1)| = |16 − 5| = 11

Since the difference is 11, 64614 is divisible by 11.

⇒ For D = 756148: |(7 + 6 + 4) − (5 + 1 + 8)| = |17 − 14| = 3

Since the difference is 3, which is neither 0 nor a multiple of 11, 756148 is NOT divisible by 11.

∴ The correct answer is D.

Additional Information

Divisibility Rule of 3

A number is divisible by 3 if the sum of all its digits is divisible by 3.

Divisibility Rule of 4

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

Divisibility Rule of 8

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

Divisibility Rule of 9

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

Shortcut Trick

Total percentage increase = ((2295 − 1000) ÷ 1000) × 100 = 129.5%

By checking the given options, if the second increment is 70% (Option 3), then the first increment is 35%.

Applying the successive percentage formula: 35% + 70% + (35 × 70) ÷ 100 = 105% + 24.5% = 129.5%

This matches our calculated total increase perfectly, confirming that the second increase is indeed 70%.

∴ The correct answer is 70%.

Alternate Method

Given:

Initial Price = ₹1000

Final Price = ₹2295

Formula Used:

Effective Percentage Change = a + b + (a × b) ÷ 100

Calculations:

⇒ Total percentage increase = ((2295 − 1000) ÷ 1000) × 100 = 129.5%

⇒ Let the first percentage increase be x% and the second percentage increase be 2x%.

⇒ Using the successive percentage change formula: x + 2x + (x × 2x) ÷ 100 = 129.5

⇒ 3x + x² ÷ 50 = 129.5

⇒ Multiplying the entire equation by 50 to eliminate the fraction:

⇒ 150x + x² = 6475

⇒ x² + 150x − 6475 = 0

⇒ x² + 185x − 35x − 6475 = 0

⇒ x(x + 185) − 35(x + 185) = 0

⇒ (x − 35)(x + 185) = 0

⇒ Since a percentage increment must be positive, x = 35%

⇒ Therefore, the second increment is 2x = 2 × 35% = 70%

∴ The correct answer is 70%.

Additional Information

Successive Percentage Change

When a value undergoes two successive percentage changes of a% and b%, the net effective change is given by the formula: Effective % = a + b + (a × b) ÷ 100. This is useful for saving steps instead of calculating intermediate values.

Multiplier Method

For successive increases, the final value can be directly calculated by multiplying the base value with respective multiplying factors: Final Value = Initial Value × (1 + a ÷ 100) × (1 + b ÷ 100).

Absolute Percentage Change

No matter how many intermediate changes occur, the total net percentage change over a period can always be found using the starting and ending amounts: % Change = ((Final Value − Initial Value) ÷ Initial Value) × 100.

Shortcut Trick

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

⇒ ∠C = 180° − (95° + 52°) = 180° − 147° = 33°

Since ΔABC ≅ ΔFDE, their corresponding parts must be equal (CPCT).

⇒ The side corresponding to AB is FD, so FD = 17 cm.

⇒ The angle corresponding to ∠C is ∠E, so ∠E = 33°.

∴ The correct answer is DF=17 cm, ∠E=33°.

Alternate Method

Given:

ΔABC ≅ ΔFDE

AB = 17 cm, ∠B = 52°, ∠A = 95°

A B C

95° 52° 33° 17 cm

F D E

95° 52° 33° 17 cm

Formula Used:

Sum of interior angles of a triangle = 180°

Corresponding Parts of Congruent Triangles (CPCT)

Calculation:

⇒ In ΔABC, sum of all angles is 180°

⇒ ∠A + ∠B + ∠C = 180°

⇒ 95° + 52° + ∠C = 180°

⇒ 147° + ∠C = 180°

⇒ ∠C = 180° − 147° = 33°

Since it is given that ΔABC ≅ ΔFDE, the vertices map to each other in the exact order.

⇒ The mapping of vertices is: A → F, B → D, C → E

⇒ The side corresponding to AB is FD. Therefore, FD = AB = 17 cm.

⇒ The angle corresponding to ∠C is ∠E. Therefore, ∠E = ∠C = 33°.

∴ The correct answer is DF=17 cm, ∠E=33°.

Additional Information

Congruent Triangles (CPCT)

When two triangles are congruent, all their corresponding sides and corresponding angles are perfectly equal. This foundational property is known as Corresponding Parts of Congruent Triangles (CPCT).

Conditions for Triangle Congruence

Two triangles can be proven to be congruent strictly using these five specific criteria: SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle), AAS (Angle-Angle-Side), and RHS (Right angle-Hypotenuse-Side).

Similarity vs Congruence

Congruent triangles are completely identical in both shape and size, meaning the ratio of their corresponding sides is exactly 1:1. On the other hand, similar triangles share the same shape but differ in size, meaning their sides are only proportionally related.

The correct answer is Option 1.

Key Points

• The error lies in the phrase "is further to".
• The correct word should be 'Farther' instead of 'Further' as the sentence is talking about the distance.
• The word "farther" is not used with the preposition "to" when indicating distance.
• The correct preposition in this context is "from" to indicate comparative distance.
• Hence, the correct expression should be "Mr. Ali’s shop is farther from the hospital."

Therefore, the correct answer is Option 1.

Correct Sentence: "Mr. Ali’s shop is farther from the hospital."

The correct answer is: Option 1 i.e. Innovative.

Key Points

• The word "creative" means having the quality or power of creating, resulting from originality of thought or expression. (रचनात्मक)
• Example: She has a creative mind that always comes up with unique ideas.
• "Innovative" refers to introducing new ideas, methods, or products; it is synonymous with being inventive and original. (नवीन)
• Example: The company's innovative technology has revolutionized the industry.

Therefore, the correct answer is: Option 1 i.e. Innovative.

Additional Information

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

• Likeable (पसंद करने योग्य): Pleasant, friendly, and easy to like.
• Example: He is a very likeable person who makes friends easily.
• Pedantic (पांडित्यपूर्ण): Overly concerned with minute details or formalisms, especially in teaching.
• Example: His lectures were so pedantic that students found them boring.
• Good (अच्छा): To be desired or approved of.
• Example: She did a good job on the project.

The correct answer is Option (1) i.e. CBEAD

Key Points

• (C) "I have missed" is the starting part because "I" is the subject and "have missed" is the verb phrase. It tells us what action is happening, which is the missing of something.
• (B) "my daily routine" comes after (C) because it tells us what the speaker has missed. Without this part, the sentence would be incomplete.
• (E) "at the workplace" specifies the location where the routine was carried out. This fits logically right after "my daily routine" to clarify the context.
• (A) "a lot" tells us how much the speaker misses their routine. This phrase is best placed after "I have missed" and before the time frame. It emphasizes the extent of what is being missed.
• (D) "since my retirement" gives us the time frame or the starting point of when the missing began.
  
• Final Sentence: "I have missed my daily routine at the workplace a lot since my retirement."

Therefore, the correct answer is CBEAD.

Additional Information

• Option 2 (BACED): The phrase "a lot" is placed too early, disrupting the logical flow and making the sentence sound unnatural.
• Option 3 (BEADC): The phrase "since my retirement" is misplaced, disrupting the natural flow of the sentence.
• Option 4 (BADEC):  "a lot" is incorrectly placed before "my daily routine" and "since my retirement" is out of place.

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

Key Points

• The word "stunning" means extremely impressive or attractive; something that is breathtakingly beautiful. (मनमोहक)
• Example: The stunning sunset left everyone in awe.
• "Mundane" refers to something that is lacking interest or excitement; dull or ordinary. (निरस)
• Example: She longed to escape her mundane routine and travel the world.
• Hence, we can infer that the opposite of 'stunning' is 'mundane'.

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

Additional Information

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

• Toxic (विषैला): Containing or being poisonous material especially when capable of causing death or serious debilitation.
• Example: The factory was shut down due to the emission of toxic fumes.
• Superfluous (अतिरिक्त): Unnecessary, especially through being more than enough.
• Example: She cleared out all the superfluous items from her closet.
• Smashing (शानदार): Excellent; wonderful.
• Example: They threw a smashing party to celebrate their victory.

The correct answer is Option 2.

Key Points

• In active voice, the subject performs the action expressed by the verb. The original sentence in passive voice is: "The concert tickets were bought by my friend." Here, the subject (concert tickets) receives the action.
• To convert this into active voice, the object "concert tickets" becomes the direct object in the active sentence.
• In active voice, the correct sentence would be: "My friend bought the concert tickets."
• The correct verb form to use here is the past tense "bought," which reflects the action completed in the past.
• Therefore, the correct transformation of the sentence into active voice is "My friend bought the concert tickets."

Therefore, the correct answer is Option 2.

Correct Sentence: "My friend bought the concert tickets."

Additional Information

• Option 1: "My friend has bought the concert tickets." – This is in the present perfect tense, which is incorrect for the context, as the original sentence refers to a simple past action.
• Option 3: "My friend buys the concert tickets." – This is in the simple present tense, which implies a habitual action, not a specific past event.
• Option 4: "My friend is buying the concert tickets." – This is in the present continuous tense, which refers to an action happening right now, not in the past.

The correct answer is: Option 2 i.e. Gloomy.

Key Points

• The word "morbid" means an unhealthy interest in disturbing and unpleasant subjects, especially death and disease. (रुग्ण)
• Example: He had a morbid fascination with horror movies.
• "Gloomy" refers to a feeling of deep sadness, depression, or a lack of hope. (उदास)
• Example: The weather was gloomy, reflecting his mood after hearing the bad news.
• Hence, we can infer that the word similar in meaning to 'morbid' is 'gloomy'.

Therefore, the correct answer is: Option 2 i.e. Gloomy.

Additional Information

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

• Blithe (आनंदित): Showing a casual and cheerful indifference considered to be callous or improper.
• Example: She spoke with blithe confidence about her future plans.
• Grand (भव्य): Magnificent and imposing in appearance, size, or style.
• Example: The grand palace was a sight to behold.
• Bet (शर्त): Risk something, usually a sum of money, against someone else's on the basis of the outcome of an unpredictable event.
• Example: He placed a bet on his favorite team to win the championship.