SSC CGL 2024 Tier-I Official Paper (Held On: 19 Sept, 2024 Shift 3) (Previous Year Paper)

The logic followed here is:

Option 1) MOP

Option 2) NPR

Option 3) HJL

Option 4) TVX

Thus, out of all the options 'MOP' is the odd one out.

Hence, "Option 1" is the correct answer.

Given: 69 + 2485 × 5 – 912 ÷ 2 = ?480÷0.3×5+500190+9×27÷1.6=2480÷0.3×5+500190+9×27÷1.6=2

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

69 + 2485 × 5 – 912 ÷ 2 = ? ⇒ 69 - 2485 ÷ 5 + 912 × 2 = ?

⇒ 69 - 2485 ÷ 5 + 912 × 2 

⇒ 69 - 497 + 912 × 2 

⇒ 69 + 1824 - 497 

⇒ 1893 - 497 

⇒ 1396

Hence, "Option 1" is the correct answer.

The logic followed here is:

Logic: 1st number + 51 = 2nd number.

Now,

For, 11 : 62

⇒ 11 + 51 = 62.

For, 13 : 64

⇒ 13 + 51 = 64.

Similarly, for 16 ∶ 67

⇒ 16 + 51 = 67.

Hence, the correct answer is "Option 3".

Decoding symbols:

Given: H × I − J ÷ K + L

H × I → H is the mother of I.

I − J → I is the wife of J.

J ÷ K → J is the father of K.

K + L → K is the sister of L.

Thus, H is L's mother’s mother or maternal grandmother.

Hence, the correct answer is "Option 4".

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

Conclusions:

I. Some trousers are shirts → Follow (As some shirts are pants, all pants are jackets and all jackets are trousers. As whole pants comes in jackets and whole jacket comes in trousers and some part of pants is common with shirts, therefore it is true.)

II. All jackets are shirts → Does not follow (As some shirts are pants and all pants are jackets. So it is possible but not definite.)

∴ Here, Only conclusion I follows.​

Hence, the correct answer is "Option 4".

The logic followed here is:

Hence, "Option 2" is the correct answer.

The logic followed here is:​

Hence, "Option 3" is the correct answer.

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

Thus, as marked there are a total of 4 triangles.

Hence, "Option 3" is the correct answer.

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

Thus, option Figure 4 will look like the paper when unfolded.

Hence, "Option 4" is the correct answer.

The logic followed here is:

1. First box series → 1st Box and 3rd Box

The elements at the corners shift anticlockwise subsequently.

The rest of the elements shift anticlockwise subsequently.

2) Second box series → 2nd Box and 4th Box

The 1st row elements and first element of 3rd row shift clockwise subsequently.

The center element is replaced by the new element.

The rest of the elements shift clockwise subsequently.

Thus, the final series is:

Hence, "Option 1" is the correct answer.

Given series: I _ _ I _ I _ _ I _ J _

Option 1) I J I I J J I → I I J/ I I I/ I J I/J J I → (It doesn't follow any pattern)

Option 2) J I I I J J I → I J I/ I I I/ I J I/J J I → (It doesn't follow any pattern)

Option 3) J I J I J I I → I J I/ I J I/ I J I/ I J I → (It follows the pattern of: IJI - IJI - IJI - IJI)

Option 4) I I J J J I J → I I I/ I J I/ J J I/ I J J → (It doesn't follow any pattern)

Hence, "Option 3" is the correct answer.

The logic followed here is:

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

Now,

For, 4-5-41

⇒ (4)² + (5)²

⇒ 16 + 25 = 41.

For, 2-3-13

⇒ (2)² + (3)²

⇒ 4 + 9 = 13.

Similarly, for 5-6-61

⇒ (5)² + (6)²

⇒ 25 + 36 = 61.

Hence, the correct answer is "Option 3".

The logic followed here is:

1. All the elements shift clockwise and are flipped horizontally subsequently.

Thus, the final series is:

Hence, "Option 3" is the correct answer.

The logic followed here is:

Option 1) 56 – 31 – 205

(56 × 2) + (31 × 3) = 205

112 + 93 = 205

205 = 205 (LHS = RHS)

Option 2) 88 – 23 – 245

(88 × 2) + (23 × 3) = 245

176 + 69 = 245

245 = 245 (LHS = RHS)

Option 3) 22 – 13 – 85

(22 × 2) + (13 × 3) = 85

44 + 39 = 85

83 ≠ 85 (LHS ≠ RHS)

Option 4) 18 – 12 – 72

(18 × 2) + (12 × 3) = 72

36 + 36 = 72

72 = 72 (LHS = RHS)

Thus, out of all the options '22 – 13 – 85' is the odd one out.

Hence, "Option 3" is the correct answer.

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

Thus, option 1 will look like the paper when unfolded.

Hence, "Option 1" is the correct answer.

The logic followed here is:

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

Now,

For, (11, 1, 10)

⇒ (11 × 1) - 1

⇒ 11 - 1 = 10.

For, (13, 7, 90)

⇒ (13 × 7) - 1

⇒ 91 - 1 = 90.

Similarly, for (4, 5, 19)

⇒ (4 × 5) - 1

⇒ 20 - 1 = 19.

Hence, the correct answer is "Option 4".

According to the given Venn diagram:

The shaded region represents the number of students who like Hindi and Science both, but NOT Maths.

So, the number of students who like Hindi and Science both, but NOT Maths = 4

Hence, the correct answer is "Option 3".

The logic followed here is:

Logic: Developmental stages in a life cycle.

Here,

Larva : Insect → Larva is a developmental stage that eventually transforms into an Insect.

Similarly,

Fawn : Deer → Fawn is a developmental stage that eventually transforms into a Deer.

Hence, the correct answer is "Option 3".

The logic followed here is:

'PANTHER' is written as '0146283'

'ANOTHER' is written as '2864109'

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

Hence, "Option 4" is the correct answer.

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

∴ Here, given figure is embedded in the option figure 2.

Hence, the correct answer is "Option 2".

The logic followed here is:

‘CYCLE’ is coded as ‘242241522’

And,

‘FINAL’ is coded as ‘2118132615’

Similarly,

‘MUSIC’ is coded as

Hence, "Option 2" is the correct answer.

The logic followed here is:

‘MPQO’ is related to ‘KNSQ’

And,

‘DGHF’ is related to ‘BEJH’

Similarly,

‘EINW’ is related to

Hence, "Option 1" is the correct answer.

Given: 49 × 7 – 6 ÷ 8 +13 = ?480÷0.3×5+500190+9×27÷1.6=2480÷0.3×5+500190+9×27÷1.6=2480÷0.3×5+500190+9×27÷1.6=2480÷0.3×5+500190+9×27÷1.6=2

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

49 × 7 – 6 ÷ 8 +13 = ? ⇒ 49 ÷ 7 + 6 × 8 –13 = ?

⇒ 49 ÷ 7 + 6 × 8 –13 

⇒ 7 + 6 × 8 –13

⇒ 7 + 48 –13

⇒ 55 –13

⇒ 42 

Hence, "Option 3" is the correct answer.

The correct mirror image of the given figure is:

Hence, the correct answer is "Option 1".

Given word: 'DRAFTING'

After arranged in alphabetical order:

 

Now,

there is no alphabet that will remain unchanged if each of the letters in the word 'DRAFTING' is arranged in English alphabetical order.

Hence, the correct answer is "Option 4".

The correct answer is Madhya Pradesh.

Key Points

• The Madhya Pradesh government launched the 'Ankur' scheme in May 2021.
• This scheme aims to promote tree plantation among the public.
• Under this scheme, people are encouraged to plant saplings and take care of them.
• People will be awarded for successfully planting and nurturing the saplings.

Additional Information

• Environmental Importance of Tree Plantation
• Tree plantation plays a crucial role in maintaining ecological balance.
• Trees absorb carbon dioxide and release oxygen, helping to combat climate change.
• They provide habitat and food for wildlife, thus preserving biodiversity.
• Trees prevent soil erosion and help in maintaining the water cycle.
• Government Initiatives for Environmental Conservation
• Green India Mission: Launched by the Indian government to protect, restore and enhance India’s diminishing forest cover.
• National Afforestation Programme: Aimed at increasing forest and tree cover to restore ecological balance and improve the quality of life.
• Namami Gange Programme: An integrated conservation mission to clean and protect the Ganga River by controlling pollution and rejuvenating its ecosystem.
• Community Participation in Environmental Conservation
• Community participation is essential for the success of environmental conservation efforts.
• Public awareness campaigns can educate people about the importance of tree planting and environmental protection.
• Involving local communities in tree plantation drives can ensure better care and maintenance of the saplings.
• Incentive schemes like 'Ankur' can motivate more people to participate in such activities.

The correct answer is current.

Key Points

• The Right-Hand Thumb Rule is used to determine the direction of the magnetic field relative to the direction of the current.
• In this rule, the thumb points in the direction of the current, and the fingers curl around the conductor, indicating the direction of the magnetic field lines.
• The Right-Hand Thumb Rule helps visualize the 3-dimensional relationship between current and magnetic fields, crucial in electromagnetism.
• This rule is fundamental in understanding the behavior of electromagnets, electric motors, and generators.

Additional Information

• Current
• Current is the flow of electric charge in a conductor, typically measured in amperes (A).
• It is the movement of electrons through a conductive material, such as a wire.
• The direction of conventional current is taken as the direction in which positive charge carriers would move.
• In most practical scenarios, electrons (negative charge carriers) move in the opposite direction to the conventional current.
• Electric Field
• An electric field is a region around a charged particle or object within which a force would be exerted on other charged particles or objects.
• It is represented by field lines that point in the direction a positive test charge would move if placed in the field.
• The strength of the electric field is measured in volts per meter (V/m).
• Electric fields are fundamental to the behavior of electric charges and the operation of capacitors.
• Motion of the Conductor
• The motion of a conductor in a magnetic field induces a voltage across the conductor.
• This phenomenon is described by Faraday's Law of Electromagnetic Induction.
• The direction of the induced voltage is given by Lenz's Law, which states that the induced voltage will oppose the change in magnetic flux.
• This principle is applied in the operation of electric generators.
• Magnetic Field
• A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials.
• It is represented by field lines that indicate the direction and strength of the magnetic force.
• The strength of a magnetic field is measured in teslas (T) or gauss (G).
• Magnetic fields are fundamental to the operation of electromagnets, transformers, and magnetic storage devices.

The correct answer is Option 1: A-ii, B-i, C-iv, D-iii.

Key Points

• Devendra Jhajharia was awarded the Padma Bhushan 2022 (A-ii).
• Sumit Antil received the Padma Shri 2022 (B-i).
• Neeraj Chopra was honored with the Major Dhyan Chand Khel Ratna Award 2021 (C-iv).
• Shikhar Dhawan was conferred with the Arjun Award 2021 (D-iii).

Additional Information

• Devendra Jhajharia is an Indian Paralympic javelin thrower who has won multiple medals in the Paralympic Games. He was awarded the Padma Bhushan in 2022 for his contributions to sports.
• Sumit Antil is an Indian Paralympic javelin thrower who won a gold medal in the Tokyo 2020 Paralympics. He was awarded the Padma Shri in 2022 for his outstanding performance in sports.
• Neeraj Chopra is an Indian track and field athlete who competes in the javelin throw. He won a gold medal in the Tokyo 2020 Olympics and was honored with the Major Dhyan Chand Khel Ratna Award in 2021.
• Shikhar Dhawan is an Indian cricketer and an opening batsman for the Indian national team. He was awarded the Arjun Award in 2021 for his exceptional performance in cricket.

The correct answer is Articles 20 and 21.

Key Points

• Article 20 of the Indian Constitution deals with the protection of certain rights in the case of conviction for offenses. It prohibits ex-post-facto laws, double jeopardy, and self-incrimination.
• Article 21 guarantees the right to life and personal liberty. It states that no person shall be deprived of his life or personal liberty except according to the procedure established by law.
• Both Articles 20 and 21 are fundamental rights under Part III of the Indian Constitution and remain enforceable during times of emergency.
• Under Article 359 of the Constitution, the President can suspend the right to move any court for the enforcement of other fundamental rights during an emergency, but Articles 20 and 21 cannot be suspended.

Additional Information

• Article 15 and 17:
• Article 15 prohibits discrimination on grounds of religion, race, caste, sex, or place of birth.
• Article 17 abolishes untouchability and forbids its practice in any form.
• These articles can be suspended during an emergency.
• Article 14 and 16:
• Article 14 ensures equality before the law and equal protection of the laws within India.
• Article 16 guarantees equality of opportunity in matters of public employment.
• These articles can also be suspended during an emergency.
• Article 30 and 32:
• Article 30 provides rights of minorities to establish and administer educational institutions.
• Article 32 provides the right to constitutional remedies for the enforcement of Fundamental Rights.
• These articles can also be suspended during an emergency.

The correct answer is Majuli.

Key Points

• Majuli is the largest riverine island in the world.
• It is located on the Brahmaputra River in the Indian state of Assam.
• Majuli has a rich cultural heritage and is known for its Vaishnavite monasteries.
• The island was declared a district in 2016, making it the first island district of India.

Additional Information

• Peacock Island
• Peacock Island is a small riverine island in the Brahmaputra river, located in Guwahati, Assam.
• It is known for the Umananda Temple dedicated to Lord Shiva.
• Peacock Island is the smallest inhabited river island in the world.
• Bhavani Island
• Bhavani Island is located on the Krishna River in Andhra Pradesh, India.
• It is a popular tourist destination known for its lush greenery and recreational activities.
• Lakshadweep
• Lakshadweep is a group of islands located in the Arabian Sea off the southwestern coast of India.
• It is known for its beautiful beaches, coral reefs, and marine life.
• Unlike Majuli, Lakshadweep is not a riverine island but a group of coral atolls.

The correct answer is 1764.

Key Points

• The Battle of Buxar was fought on 22 October 1764.
• It was a significant battle between the British East India Company and the combined forces of the Mughal Emperor Shah Alam II, Nawab of Awadh Shuja-ud-Daula, and Nawab of Bengal Mir Qasim.
• The victory of the British East India Company in the Battle of Buxar solidified their position in India and paved the way for their eventual control over large parts of the Indian subcontinent.
• This battle is considered a pivotal point in Indian history as it marked the beginning of British colonial rule in India.

Additional Information

• British East India Company
• The British East India Company was an English company formed for the exploitation of trade with East and Southeast Asia and India, incorporated by royal charter on December 31, 1600.
• It eventually came to rule large areas of India with its own private armies, exercising military power and assuming administrative functions.
• Mughal Empire
• The Mughal Empire was an empire in the Indian subcontinent, founded in 1526.
• It was established and ruled by a Muslim dynasty with Turko-Mongol Chagatai roots from Central Asia and lasted until the mid-19th century.
• Nawab of Bengal
• The Nawab of Bengal was the hereditary ruler of Bengal and Bihar, with the capital at Murshidabad.
• The last independent Nawab of Bengal, Mir Qasim, was a key figure in the Battle of Buxar.
• Nawab of Awadh
• The Nawab of Awadh was the hereditary ruler of the state of Awadh (Oudh) in northern India.
• Shuja-ud-Daula, the Nawab of Awadh, was one of the key figures in the Battle of Buxar, fighting against the British East India Company.

The correct answer is Focusing on holistic development in agriculture.

Key Points

• The Second Green Revolution aimed at enhancing productivity sustainably through modern agricultural practices.
• It focused on diversification of crops, not just staple cereals, to ensure a balanced growth in agriculture.
• The initiative aimed to integrate technology and innovation in farming, including the use of biotechnology and precision farming.
• It also emphasized improving the income of farmers by promoting value addition, market linkages, and reducing the cost of cultivation.

Additional Information

• Green Revolution:
• The term "Green Revolution" refers to a series of research, development, and technology transfer initiatives that took place between the 1940s and the late 1960s.
• It involved the development of high-yielding variety (HYV) seeds, increased use of fertilizers and irrigation methods, which significantly boosted agricultural production in many developing countries, including India.
• Biotechnology in Agriculture:
• Biotechnology refers to the use of living organisms or their systems to develop or make products. In agriculture, it includes genetic engineering, molecular markers, molecular diagnostics, vaccines, and tissue culture.
• It helps in developing crops that are resistant to pests and diseases, reducing the need for chemical pesticides, and enhancing the nutritional value of crops.
• Precision Farming:
• Precision farming is an agricultural management concept based on observing, measuring, and responding to inter and intra-field variability in crops.
• It involves the use of technologies like GPS, sensors, and data analytics to optimize field-level management with regard to crop farming.
• Value Addition in Agriculture:
• Value addition refers to the process of changing or transforming a product from its original state to a more valuable state.
• In agriculture, it involves processes such as cleaning, sorting, grading, processing, packaging, and branding which enhance the marketability and profitability of agricultural produce.

The correct answer is Semmangudi Radhakrishna Srinivasa Iyer.

Key Points

• Semmangudi Radhakrishna Srinivasa Iyer was a renowned Carnatic vocalist, widely acknowledged for his significant contributions to classical music.
• He was awarded the prestigious Sangeetha Kalanidhi by the Music Academy in 1947 at the young age of 39, making him the youngest recipient at that time.
• His unique style and adherence to traditional values in music earned him immense respect and recognition in the Carnatic music community.
• Semmangudi's legacy continues to influence and inspire upcoming generations of Carnatic musicians.

Additional Information

• Sangeetha Kalanidhi
• The title Sangeetha Kalanidhi is considered one of the highest honors in the field of Carnatic music.
• It is awarded annually by the Music Academy, Chennai, to musicians who have made significant contributions to the field of Carnatic music.
• The award was instituted in 1929, and the recipients are chosen through a rigorous selection process by the academy.
• It serves as a recognition of excellence and lifetime achievement in Carnatic music.
• The Music Academy, Chennai
• Founded in 1928, the Music Academy is one of the foremost institutions dedicated to the promotion and preservation of Carnatic music.
• It is known for organizing the annual Madras Music Season, which is one of the largest and most prestigious festivals of Carnatic music.
• The academy plays a pivotal role in nurturing young talent and conducting research and education in the field of music.
• It has been instrumental in establishing high standards and fostering a rich cultural heritage in the realm of Carnatic music.
• Carnatic Music
• Carnatic music is a system of music commonly associated with the southern part of India, including the states of Tamil Nadu, Karnataka, Andhra Pradesh, and Kerala.
• It is characterized by its intricate compositions, complex rhythmic patterns, and emphasis on improvisation.
• The music is traditionally performed in a concert setting, featuring a vocalist accompanied by instruments such as the violin, mridangam, and tambura.
• Carnatic music has a rich history and has been passed down through generations, maintaining its classical roots while evolving with contemporary influences.

The correct answer is sodium.

Key Points

• Newlands' Octaves was an early attempt to classify the elements.
• John Newlands observed that when elements were arranged in order of increasing atomic masses, the properties of every eighth element were similar.
• In Newlands' Octaves, lithium and sodium showed similar properties.
• This observation was one of the early pieces of evidence that led to the development of the periodic table.

Additional Information

• Newlands' Octaves:
• Newlands' Octaves was proposed by John Newlands in 1864.
• It is based on the observation that elements exhibit similar properties at every eighth element when arranged by increasing atomic mass.
• This was analogous to the octaves in music, where every eighth note is similar to the first.
• Although this system had limitations and was not applicable to elements beyond calcium, it was an important step towards the development of the modern periodic table.
• Modern Periodic Table:
• The modern periodic table arranges elements by increasing atomic number, not atomic mass.
• It was developed by Dmitri Mendeleev and later refined by Henry Moseley.
• The periodic table is divided into periods (rows) and groups (columns).
• Elements in the same group have similar chemical properties due to having the same number of valence electrons.
• Properties of Alkali Metals (Group 1):
• Group 1 elements (alkali metals) include lithium, sodium, potassium, rubidium, cesium, and francium.
• These metals are highly reactive, especially with water, and form strong alkaline hydroxides.
• They have one electron in their outermost shell, which they readily lose to form +1 ions.
• They are soft metals with low melting points and are good conductors of electricity.

The correct answer is MV=PT.

Key Points

• Fisher's equation of exchange is represented as MV=PT.
• In this equation, M stands for the money supply in an economy.
• V represents the velocity of money, which is the rate at which money circulates in the economy.
• P is the price level, and T represents the total volume of transactions.

Additional Information

• Money Supply (M)
• The total amount of monetary assets available in an economy at a specific time.
• It includes various components such as currency in circulation and demand deposits.
• Changes in the money supply can influence inflation, interest rates, and economic growth.
• Velocity of Money (V)
• The rate at which money is exchanged from one transaction to another.
• It indicates the efficiency of money in an economy; a higher velocity means money is changing hands more frequently.
• Velocity is calculated as the ratio of nominal GDP to the money supply.
• Price Level (P)
• The average of current prices across the entire spectrum of goods and services produced in the economy.
• It is often measured by price indices such as the Consumer Price Index (CPI) or the GDP deflator.
• Changes in the price level reflect inflation or deflation in the economy.
• Total Volume of Transactions (T)
• The total number of transactions or the total quantity of goods and services traded in the economy.
• This can include consumer spending, business investments, and government expenditures.
• The volume of transactions is an indicator of economic activity.

The correct answer is Tarang Sanchar.

Key Points

• Tarang Sanchar is a web portal launched by the Department of Telecommunications (DoT).
• The portal aims to provide information about mobile towers and their compliance with Electromagnetic Field (EMF) emission norms.
• It offers a user-friendly interface for the public to get detailed information on EMF emissions from mobile towers in their vicinity.
• The initiative is part of the government's efforts to address public concerns regarding the health hazards due to EMF emissions from mobile towers.

Additional Information

• Electromagnetic Field (EMF) Emissions
• EMF emissions are produced by mobile towers and other electronic devices.
• There are concerns about the potential health risks associated with long-term exposure to EMF emissions.
• The Department of Telecommunications (DoT) has set stringent norms for EMF emissions to ensure they are within safe limits.
• Department of Telecommunications (DoT)
• The DoT is a department of the Ministry of Communications of the executive branch of the Government of India.
• It is responsible for formulating developmental policies for the accelerated growth of the telecommunication services in the country.
• The DoT also enforces wireless regulatory measures, monitors the performance of licensees, and promotes private investment in the telecommunications sector.
• Mobile Towers
• Mobile towers are structures that support antennas and other communication equipment for cellular networks.
• They play a critical role in providing mobile network coverage and connectivity.
• Proper placement and regulation of mobile towers are essential to ensure efficient network performance and public safety.
• Public Health and Safety
• Public health and safety are major considerations in the regulation of EMF emissions from mobile towers.
• The government has implemented various measures to address public concerns and ensure that EMF emissions remain within safe limits.
• Initiatives like the Tarang Sanchar portal help increase transparency and public awareness regarding EMF emissions and mobile tower compliance.

The correct answer is Pitt’s India Act 1784.

Key Points

• Pitt's India Act 1784
• This Act, also known as the East India Company Act 1784, established dual control of British territories in India by the British government and the East India Company.
• The Act created a Board of Control to oversee the civil, military, and revenue affairs of the company, enhancing the British government's authority.
• It marked a significant step towards the establishment of British rule in India.
• The Pitt's India Act of 1784, also known as the East India Company Act of 1784, is sometimes referred to as the "half-loaf system". This act established a dual system of governance in India, separating the political and commercial functions of the East India Company.

Additional Information

• Regulating Act 1773
• The Regulating Act of 1773 was the first step taken by the British government to control and regulate the affairs of the East India Company in India.
• It was passed by the British Parliament to establish a system of governance for British territories in India.
• The Act created the position of Governor-General of Bengal and was the precursor to the later Governor-General of India.
• It marked the beginning of parliamentary control over the East India Company, transitioning from purely commercial to political governance.
• Government of India Act 1858
• This Act, passed after the Indian Rebellion of 1857, resulted in the British Crown assuming direct control over India, ending the rule of the East India Company.
• It established the office of the Secretary of State for India and the India Council to assist in governance.
• The Governor-General of India was now called the Viceroy, representing the British monarch.
• Charter Act 1813
• This Act renewed the East India Company's charter but ended its monopoly on trade with India, except for trade in tea and trade with China.
• It allowed other British merchants to trade with India, increasing competition and economic activity.
• The Act also provided for the promotion of education and the Christian religion in India.

The correct answer is Article 43.

Key Points

• Article 43 of the Indian Constitution is part of the Directive Principles of State Policy (DPSP).
• It states that 'The State shall endeavour to promote cottage industries on an individual or co-operative basis in rural areas'.
• The objective of this article is to promote economic growth and self-sufficiency in rural areas.
• It aims to provide economic opportunities and employment in rural areas through the promotion of small-scale industries.

Additional Information

• Directive Principles of State Policy (DPSP)
• The DPSP are guidelines or principles given to the federal institutes governing the state of India, to be kept in consideration while framing laws and policies.
• They are enshrined in Part IV (Articles 36-51) of the Indian Constitution.
• These principles are aimed at ensuring social and economic democracy through a welfare state.
• While they are non-justiciable, meaning they are not legally enforceable by the courts for their violation, they are considered fundamental in the governance of the country.
• Cottage Industries
• Cottage industries are small-scale, decentralized manufacturing businesses often operated out of a home rather than a purpose-built facility.
• They typically involve the production of handmade goods and are labor-intensive.
• These industries play a vital role in the economy by providing employment and promoting traditional crafts.
• In rural areas, cottage industries help in reducing unemployment and migration to urban areas by providing local employment opportunities.
• Rural Development
• Rural development involves improving the quality of life and economic well-being of people living in rural areas, often relatively isolated and sparsely populated areas.
• It focuses on the development of infrastructure, education, healthcare, and employment opportunities.
• Promoting cottage industries is one way to achieve rural development by utilizing local resources and skills.
• Government policies and programs often aim to support rural development through various schemes and initiatives.

The correct answer is Article 356.

Key Points

• Article 356 of the Indian Constitution deals with the imposition of President's Rule in states.
• This article is also known as the "President's Rule" or "State Emergency."
• It allows the President of India to take over the functions of the state government if the state government is unable to function according to the provisions of the Constitution.
• President's Rule can be imposed for a maximum period of six months at a time and can be extended with the approval of the Parliament, but not beyond three years.

Additional Information

• Article 352
• Article 352 of the Indian Constitution deals with the proclamation of National Emergency.
• It can be declared due to war, external aggression, or armed rebellion.
• During a National Emergency, the central government gains sweeping powers, and the state governments come under the direct control of the central government.
• This can be proclaimed by the President of India based on the written advice of the Cabinet.
• Article 358
• Article 358 provides for the suspension of the rights guaranteed by Article 19 during a National Emergency.
• It states that while a National Emergency is in operation, nothing in Article 19 shall restrict the power of the State to make any law or to take any executive action abridging or taking away the rights conferred by Article 19.
• Article 360
• Article 360 of the Indian Constitution deals with the proclamation of Financial Emergency.
• The President can declare a Financial Emergency if the financial stability or credit of India or any part of its territory is threatened.
• During a Financial Emergency, the executive authority of the Union extends to giving directions to any state to observe such canons of financial propriety as may be specified in the directions.
• However, no Financial Emergency has been declared in India so far.

The correct answer is Sher Shah Suri.

Key Points

• The Battle of Chausa was fought on June 26, 1539, between the Mughal Emperor Humayun and Sher Shah Suri, the founder of the Suri Empire.
• Sher Shah Suri defeated Humayun in this battle, leading to the establishment of the Suri Empire in North India.
• The defeat forced Humayun to flee from India, and Sher Shah Suri became the ruler of Delhi.
• Sher Shah Suri's victory at Chausa was a significant moment in Indian history, marking the rise of the Suri Empire and the temporary setback of the Mughal Empire.

Additional Information

• Humayun:
• Humayun was the second Mughal Emperor who ruled over the Indian subcontinent.
• He was the son of Babur, the founder of the Mughal Empire.
• Humayun's reign was interrupted by Sher Shah Suri, but he later regained the throne with the help of the Safavid Empire.
• Sher Shah Suri:
• Sher Shah Suri, originally named Farid Khan, was the founder of the Suri Empire in North India.
• He is known for his administrative reforms, including the introduction of the "Rupiya" and the improvement of the road network in India.
• Sher Shah's rule is considered a model of efficient governance and military strategy.
• Battle of Chausa:
• The Battle of Chausa took place near the town of Chausa, located in present-day Bihar, India.
• It was one of the significant battles that showcased the military prowess of Sher Shah Suri.
• This battle was crucial in the rise of Sher Shah Suri and the establishment of the Suri Dynasty.

The correct answer is 1951 and 1982.

Key Points

• India hosted the Asian Games for the first time in 1951.
• The 1951 Asian Games were the first edition of the event and were held in New Delhi.
• India again hosted the Asian Games in 1982.
• The 1982 Asian Games were also held in New Delhi, which is the capital of India.

Additional Information

• Asian Games
• The Asian Games, also known as Asiad, are a continental multi-sport event held every four years among athletes from all over Asia.
• The Games are regulated by the Olympic Council of Asia (OCA) under the supervision of the International Olympic Committee (IOC).
• The first Asian Games were held in New Delhi, India, in 1951.
• The Asian Games are the second largest multi-sport event after the Olympic Games.
• These games include disciplines such as athletics, swimming, archery, badminton, basketball, boxing, and many more.
• New Delhi
• New Delhi is the capital of India and serves as the seat of all three branches of the Government of India, including the Executive, the Legislature, and the Judiciary.
• It is also a prominent cultural and political hub in the country.
• New Delhi hosted the first and the ninth Asian Games, in 1951 and 1982 respectively.
• In preparation for the games, New Delhi underwent significant infrastructure development, including the construction of new sports venues and the improvement of transportation facilities.

The correct answer is smog.

Key Points

• Smog is a type of intense air pollution.
• It is composed of smoke and fog which is why it is called "smog".
• Smog is formed when sunlight reacts with pollutants like nitrogen oxides and at least one volatile organic compound (VOC) in the atmosphere.
• Smog is common in industrial areas and cities with a lot of vehicular traffic.

Additional Information

• Mist
• Mist consists of small droplets of water suspended in the air.
• It is typically created when warm, moist air meets sudden cooling.
• Mist is less dense than fog and visibility is generally clearer than in fog conditions.
• Blizzard
• A blizzard is a severe snowstorm characterized by strong sustained winds of at least 35 mph (56 km/h).
• It typically lasts for a prolonged period—typically three hours or more.
• Blizzards can cause whiteout conditions and significant snow accumulation.
• Frost
• Frost refers to a thin layer of ice that forms on solid surfaces when the temperature drops below freezing.
• It typically occurs in clear, calm conditions where there is a loss of radiative heat.
• Frost can damage crops and other plants.

The correct answer is Uttar Pradesh.

Key Points

• The Sardhana Christian Fair is a significant event held in the state of Uttar Pradesh.
• The fair is held in the town of Sardhana, which is located in the Meerut district of Uttar Pradesh.
• This fair is an important religious event for the Christian community, particularly those following the Catholic faith.
• The fair is held at the Basilica of Our Lady of Graces, a historical church built by Begum Samru, a notable figure in the region.

Additional Information

• Uttar Pradesh
• Uttar Pradesh, often abbreviated as UP, is a state in northern India with Lucknow as its capital.
• It is the most populous state in India, with a rich cultural and historical heritage.
• Uttar Pradesh is known for its diverse religious and cultural events, including various fairs and festivals that draw people from across the country.
• The state is home to several important historical and religious sites, including the Taj Mahal, Varanasi, and Ayodhya.
• Basilica of Our Lady of Graces
• The Basilica of Our Lady of Graces, also known as Sardhana Church, is a prominent Catholic church located in Sardhana, Meerut.
• It was built by Begum Samru, a wealthy and influential figure of the 18th century, who converted to Christianity and constructed the church in 1822.
• The church is an architectural marvel, reflecting a blend of Indian and European styles, and is one of the most significant Christian pilgrimage sites in North India.
• The annual fair held at the church attracts thousands of devotees and visitors, making it a notable event in the region.

The correct answer is Maihar gharana.

Key Points

• Ali Akbar Khan was a renowned Indian classical musician, known for his mastery of the sarod.
• He was a prominent figure associated with the Maihar gharana, a school of music founded by his father, Allauddin Khan.
• The Maihar gharana is known for its emphasis on instrumental music, especially the sarod and sitar.
• Ali Akbar Khan contributed significantly to popularizing Indian classical music in the West and established the Ali Akbar College of Music in the USA.

Additional Information

• Gharanas in Indian Classical Music
• A gharana is a system of social organization linking musicians or dancers by lineage or apprenticeship, and by adherence to a particular musical style.
• Each gharana has its own unique style, repertoire, and teaching methodology, and is often named after the place it originated from.
• Maihar Gharana
• Founded by Ustad Allauddin Khan in the early 20th century, the Maihar gharana is known for its emphasis on instrumental music.
• It has produced several prominent musicians, including Ravi Shankar, Nikhil Banerjee, and Ali Akbar Khan.
• Other Gharanas
• Agra Gharana: Known for its khayal singing, it originated in Agra and places emphasis on the dhrupad style of singing.
• Gwalior Gharana: One of the oldest gharanas, it is known for its khayal singing and has a well-structured raga presentation.
• Kirana Gharana: Known for its melodious and emotional style, it emphasizes the purity of notes and is famed for its slow-tempo renditions.

The correct answer is Green revolution.

Key Points

• The Green Revolution refers to the set of research, development, and technology transfer initiatives occurring between the 1940s and the late 1960s that increased agricultural production worldwide, especially in developing countries.
• In India, the Green Revolution was launched in the 1960s with the introduction of high-yielding varieties (HYVs) of seeds, increased use of fertilizers, and improved irrigation methods to boost food production.
• The Green Revolution led to a significant increase in food grain production, particularly wheat and rice, transforming India from a food-deficient country to one of the world's leading agricultural nations.
• However, the benefits of the Green Revolution were not evenly distributed, creating regional disparities and widening the gap between large and small farmers, as well as between different regions of the country.

Additional Information

• Yellow Revolution:
• The Yellow Revolution refers to the increase in the production of oilseeds in India, especially mustard and sunflower, initiated in the late 1980s and early 1990s.
• It aimed at achieving self-sufficiency in edible oil production to reduce dependence on imports.
• Pink Revolution:
• The Pink Revolution refers to the technological advancement and growth in the production of meat and poultry in India.
• It focuses on improving the quality and quantity of meat production, especially in the poultry sector.
• Blue Revolution:
• The Blue Revolution refers to the rapid increase in fish production and aquaculture in India.
• It aims to enhance the productivity of fisheries and aquaculture through sustainable practices and technological advancements.

The correct answer is Irular tribe.

Key Points

• Irular tribe is one of the prominent tribes in Kerala, primarily residing in the Palakkad and Nilgiri Hills.
• The Irular community is known for its rich cultural heritage and traditional practices, including unique dance forms.
• Elakkakadi dance is a notable cultural expression of the Irular tribe, performed during various festivals and community gatherings.
• The dance is characterized by rhythmic movements and traditional music, reflecting the tribe's connection to nature and their spiritual beliefs.

Additional Information

• Irular Tribe
• The Irular tribe is primarily found in the Indian states of Tamil Nadu and Kerala.
• They are traditionally known for their skills in agriculture, hunting, and snake-catching.
• Irulars speak the Irula language, which is part of the Dravidian language family.
• They have a deep understanding of herbal medicine and natural remedies, which are integral to their traditional knowledge system.
• Kurumba Tribe
• The Kurumba tribe is another indigenous community in the Nilgiri Hills of Tamil Nadu and Kerala.
• They are known for their expertise in honey collection and basket weaving.
• The Kurumbas have a distinct cultural identity, with their own language and customs.
• Adiyan Tribe
• The Adiyan tribe is predominantly found in the Wayanad district of Kerala.
• They are traditionally agricultural laborers and have a rich oral tradition of folklore and stories.
• Mavilan Tribe
• The Mavilan tribe is found in the Kannur and Kasaragod districts of Kerala.
• They are known for their traditional agricultural practices and have a distinct cultural heritage.

The correct answer is 2) c.

Key Points

• Chloroplasts are organelles found in plant cells and eukaryotic algae that conduct photosynthesis.
• Chloroplasts contain chlorophyll, a green pigment that captures light energy.
• They are surrounded by a double membrane and contain their own DNA, similar to mitochondria.
• The main function of chloroplasts is to convert solar energy into chemical energy in the form of glucose through photosynthesis.

Additional Information

• Photosynthesis
• It is the process by which green plants and some other organisms use sunlight to synthesize foods with the help of chlorophyll.
• Photosynthesis involves the conversion of carbon dioxide and water into glucose and oxygen.
• The general equation for photosynthesis is: 6CO₂ + 6H₂O + light energy = C₆H₁₂O₆ + 6O₂.
• Chlorophyll
• Chlorophyll is the molecule that absorbs sunlight and uses its energy to synthesize carbohydrates from CO₂ and water.
• It is what gives plants their green color.
• Thylakoids
• Thylakoids are membrane-bound compartments inside chloroplasts and cyanobacteria.
• They are the site of the light-dependent reactions of photosynthesis.
• Thylakoids are arranged in stacks known as grana.
• Stroma
• Stroma is the fluid-filled space that surrounds the grana inside the chloroplast.
• It is the site of the light-independent reactions (Calvin cycle) of photosynthesis.

The correct answer is Kumara.

Key Points

• Under the Mauryan administration, the provincial administration was generally headed by a Kumara (Prince).
• The Kumara was usually a royal prince, often the king's own son, who was entrusted with governing a province.
• These provinces were known as 'Janapadas'.
• The appointment of Kumaras helped ensure loyalty and effective governance in the vast Mauryan Empire.

Additional Information

• Dandapala
• Dandapala was an official responsible for the maintenance of law and order in the provinces.
• He managed the police and ensured the enforcement of rules and regulations.
• Samaharta
• Samaharta was in charge of revenue collection in the Mauryan administration.
• He was responsible for assessing and collecting land revenue, trade taxes, and other dues.
• Prashasti
• Prashasti refers to an inscription or eulogy written in praise of a ruler or a noble person.
• These inscriptions were often composed by poets and scholars of the court.

The correct answer is chess.

Key Points

• Bhakti Pradeep Kulkarni is a renowned Indian chess player who has made significant contributions to the sport.
• She has represented India in numerous international chess tournaments, showcasing her skills and bringing laurels to the country.
• Bhakti has achieved several accolades in her chess career, including national and international titles.
• She continues to inspire young chess enthusiasts in India with her dedication and achievements in the game of chess.

Additional Information

• Chess:
• Chess is a two-player strategy board game played on an 8x8 grid known as a chessboard.
• Each player begins with 16 pieces: one king, one queen, two rooks, two knights, two bishops, and eight pawns.
• The objective of the game is to checkmate the opponent's king, which means the king is in a position to be captured ("in check") and there is no legal move to escape the threat.
• Chess is believed to have originated in India during the Gupta Empire, where its early form was known as Chaturanga.
• The game has a rich history and has been recognized as a sport by the International Olympic Committee.
• FIDE (Fédération Internationale des Échecs) is the international governing body for chess, which organizes world championships and other major tournaments.
• Notable Chess Tournaments:
• The Chess World Championship is the highest accolade in the sport, determining the world's best chess player.
• The Candidates Tournament is held to determine the challenger for the World Chess Championship.
• Other prestigious tournaments include the Chess Olympiad, the Tata Steel Chess Tournament, and the Grand Chess Tour.

The correct answer is Casein.

Key Points

• Casein is the most abundant protein in milk, making up about 80% of the total protein content in bovine milk.
• It is nutritionally valuable due to its high quality and rich composition of essential amino acids.
• Casein proteins are slow-digesting, providing a sustained release of amino acids into the bloodstream.
• It plays an important role in providing the necessary nutrients for the growth and development of young mammals.

Additional Information

• Actin
• Actin is a protein that forms microfilaments and is involved in various types of cell motility and structure.
• It is not a major component of milk but is essential in muscle contraction and cellular movements.
• Albumin
• Albumin is a type of protein found in blood plasma, egg white, and milk.
• While it is present in milk, it does not constitute the majority of milk proteins.
• It is important for maintaining the osmotic pressure necessary for proper distribution of body fluids between body tissues and the bloodstream.
• Pepsin
• Pepsin is a digestive enzyme that breaks down proteins into peptides.
• It is produced in the stomach and is not a component of milk.
• It functions in the digestive system to help in the digestion of proteins from food.
• Casein
• Casein is a family of related phosphoproteins that are commonly found in mammalian milk.
• It forms a gel or clot in the stomach, which makes it very efficient in nutrient supply.
• Casein is used in cheese-making and as a food additive for its binding properties.
• Its ability to provide a slow and sustained release of amino acids makes it highly beneficial for muscle repair and growth.

Shortcut Trick

In a kite, if a diagonal splits unequal adjacent angles (∠ABD ≠ ∠ADB), it must be the axis of symmetry.

This means the diagonal BD divides the kite into two identical triangles, making ∠C = ∠A.

In ΔABD, the sum of all angles is 180°.

⇒ ∠A = 180° − (54° + 50°) = 76°

⇒ ∠C = 76°

∴ The correct answer is 76°.

Alternate Method

Given:

ABCD is a kite.

∠ABD = 54°

∠ADB = 50°

Method Used:

A kite has two pairs of equal adjacent sides, and its main diagonal divides it into two congruent triangles.

The sum of all interior angles in any triangle is always 180°.

Calculation:

⇒ For the given kite ABCD, if AB were equal to AD, ΔABD would be an isosceles triangle, making ∠ABD = ∠ADB.

⇒ However, given ∠ABD (54°) ≠ ∠ADB (50°), we conclude that AB ≠ AD.

⇒ Therefore, the adjacent equal sides must be AB = BC and AD = CD.

⇒ In this configuration, the diagonal BD acts as the axis of symmetry.

⇒ Because BD is the axis of symmetry, it divides the kite into two congruent triangles: ΔABD ≅ ΔCBD.

⇒ In ΔABD, the sum of internal angles is 180°.

⇒ ∠DAB + ∠ABD + ∠ADB = 180°

⇒ ∠DAB + 54° + 50° = 180°

⇒ ∠DAB + 104° = 180°

⇒ ∠DAB = 180° − 104° = 76°

⇒ Due to the exact congruence of ΔABD and ΔCBD, the corresponding angles are perfectly equal, meaning ∠DCB = ∠DAB.

⇒ ∠DCB = 76°

∴ The correct answer is 76°.

Additional Information

Properties of a Kite

A kite is a quadrilateral with two distinct pairs of equal-length sides that are adjacent to each other. The opposite angles where the unequal sides meet are always equal.

Diagonals of a Kite

The diagonals of a kite are always perpendicular to each other, intersecting at exactly 90°. The main diagonal (axis of symmetry) acts as the perpendicular bisector of the other diagonal.

Axis of Symmetry

The diagonal connecting the common vertices of the equal sides acts as the axis of symmetry. It bisects the angles at its endpoints and cleanly divides the kite into two identical congruent triangles.

Shortcut Trick

Let Y = 100 units ⇒ X = 130 units and Z = 125 units.

Difference (X − Z) = 130 − 125 = 5 units → 5000 votes.

Total cast votes = X + Y + Z = 130 + 100 + 125 = 355 units.

Total cast votes = 355 × (5000 ÷ 5) = 3,55,000.

Total voters list = (3,55,000 ÷ 71) × 100 = 5,00,000.

∴ The correct option is A.

Alternate Method

Given:

X = 130% of Y, Z = 125% of Y, X − Z = 5000, Cast votes = 71% of Total list.

Formula Used:

Number of votes = (Percentage ÷ 100) × Base value; Total Voters = (Cast Votes ÷ Cast Percentage) × 100.

Y (100u)

Z (125u)

X (130u)

Diff = 5u 5u = 5000

Total Votes Cast = X + Y + Z = 355 units

⇒ Let Y = 100u

⇒ X = 100u + 30% of 100u = 130u

⇒ Z = 100u + 25% of 100u = 125u

⇒ X − Z = 130u − 125u = 5u

⇒ 5u = 5000 ⇒ 1u = 1000

⇒ Total votes cast = 130u + 100u + 125u = 355u

⇒ Total votes cast = 355 × 1000 = 3,55,000

⇒ 71% of Total voters = 3,55,000

⇒ Total voters in list = (3,55,000 ÷ 71) × 100

⇒ Total voters in list = 5000 × 100 = 5,00,000

∴ The correct option is A.

Additional Information

Percentage Comparison

If A is x% more than B, then A = B × (1 + x/100). If A is x% less than B, then A = B × (1 − x/100).

Election Terminology

Voter List Total = Votes Polled + Non-voters. Votes Polled = Valid Votes + Invalid Votes.

Common Base Method

When multiple variables are compared to one common variable, assume the common variable as 100 to simplify calculations.

Given:

The number of candidates appearing for the posts is as follows:

Formula used:

Average number of candidates for posts C, D, E, and F = (Candidates for C + D + E + F) / 4

Calculations:

Average number = (8600 + 9900 + 5300 + 5100) / 4

⇒ Average number = 28900 / 4

⇒ Average number = 7225

Answer:

The average number of candidates appearing for posts C, D, E, and F is 7225.

Given:

The pie chart shows the expenditures on various sports in a country. The expenditure on cricket is ₹72 lakh, and the angles corresponding to various sports are given as:

 Tennis = 85°

 Golf = 48°

 Swimming = 45°

 Basketball = 52°

 Cricket = 60°

Step 1: Calculate the total expenditure

The expenditure on cricket corresponds to 60° of the pie chart, and the total expenditure is ₹72 lakh for cricket. We calculate the total expenditure using the relation between the angle and the expenditure:

Total Expenditure = (Expenditure on Cricket) × (360° / Angle corresponding to Cricket)

Total Expenditure = 72 × (360 / 60) = 72 × 6 = ₹432 lakh

Step 2: Calculate the expenditure on Tennis, Golf, Swimming, and Basketball

Now, we calculate the expenditure on tennis, golf, swimming, and basketball using their respective angles:

 Tennis: Expenditure on Tennis = (85 / 360) × 432 = ₹102 lakh

 Golf: Expenditure on Golf = (48 / 360) × 432 = ₹57.6 lakh

 Swimming: Expenditure on Swimming = (45 / 360) × 432 = ₹54 lakh

 Basketball: Expenditure on Basketball = (52 / 360) × 432 = ₹62.4 lakh

Step 3: Calculate the total expenditure on Tennis, Golf, Swimming, and Basketball

Total Expenditure on Tennis, Golf, Swimming, and Basketball = ₹102 + ₹57.6 + ₹54 + ₹62.4 = ₹276 lakh

Step 4: Calculate the average expenditure on these sports

The average expenditure is given by:

Average Expenditure = (Total Expenditure on Tennis, Golf, Swimming, and Basketball) / 4

Average Expenditure = ₹276 lakh / 4 = ₹69 lakh

Final Answer:

The average expenditure on Tennis, Golf, Swimming, and Basketball is ₹69 lakh.

Shortcut Trick

Given the sum of the sides a + b = 9 and their product ab = 18.

By finding two numbers that add up to 9 and multiply to 18, we can easily identify the sides as a = 6 and b = 3.

We need to find the value of (a² − ab + b²).

Substitute the values directly: 6² − (6 × 3) + 3²

⇒ 36 − 18 + 9 = 27

∴ The correct answer is 27.

Alternate Method

Given:

Length of the rectangular field = a

Breadth of the rectangular field = b

Sum of length and breadth (a + b) = 9 cm

Area of the rectangular field (ab) = 18 cm²

Formula Used:

(a + b)² = a² + b² + 2ab

Calculations:

From the standard algebraic identity, we can modify the required expression as follows:

⇒ a² − ab + b² = (a² + b² + 2ab) − 3ab

⇒ a² − ab + b² = (a + b)² − 3ab

Now, substitute the given values (a + b = 9 and ab = 18) into the modified equation:

⇒ a² − ab + b² = (9)² − 3 × 18

⇒ a² − ab + b² = 81 − 54

⇒ a² − ab + b² = 27

∴ The correct answer is 27.

Additional Information

Perimeter of a Rectangle

The total boundary length of a rectangle, calculated using the formula P = 2(l + b).

Diagonal of a Rectangle

The longest straight line distance between two opposite corners, given by the Pythagorean relationship d = √(l² + b²).

Algebraic Cube Identities

The expression calculated above frequently appears in the sum of cubes formula: a³ + b³ = (a + b)(a² − ab + b²).

Shortcut Trick

An even number can never completely divide an odd number.

By observing the given options, we can see that 7497 is an odd number.

Therefore, it is impossible for 7497 to be a multiple of 8, which is an even number.

∴ The correct answer is 7497.

Alternate Method

Given:

Numbers: 7344, 5544, 7497, 4608

Concept Used:

Divisibility Rule of 8: A number is divisible by 8 if the number formed by its last three digits is perfectly divisible by 8.

Calculations:

Checking each option for divisibility by 8:

⇒ For 7344: The last 3 digits are 344.

⇒ 344 ÷ 8 = 43 (Divisible by 8)

⇒ For 5544: The last 3 digits are 544.

⇒ 544 ÷ 8 = 68 (Divisible by 8)

⇒ For 7497: The last 3 digits are 497.

⇒ 497 is an odd number, hence it is not divisible by 8.

⇒ For 4608: The last 3 digits are 608.

⇒ 608 ÷ 8 = 76 (Divisible by 8)

∴ The correct answer is 7497.

Additional Information

Divisibility Rule for 2

A number is divisible by 2 if its last digit is an even number (0, 2, 4, 6, 8).

Divisibility Rule for 4

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

Divisibility Rule for 16

A number is divisible by 16 if the number formed by its last four digits is perfectly divisible by 16.

Divisibility Rule for 3 & 9

A number is divisible by 3 or 9 if the sum of all its digits is completely divisible by 3 or 9, respectively.

Given:

The formula to calculate the income is:

Income = Expenditure + (Expenditure × Profit%) / 100

Step 1: Plug in the given values for Company C in the year 2003:

 Expenditure = ₹50 crores

 Profit % = 55%

Step 2: Calculate the income using the formula:

Income = 50 + (50 × 55) / 100

Income = 50 + 27.5

Income = ₹77.5 crores

Final Answer:

The income of Company C in the year 2003 is ₹77.5 crores.

Shortcut Trick

A central angle of 120° means the sector is exactly 1/3 of a full circle.

Area of full circle = Area of sector × 3 = 462 × 3 = 1386 cm².

⇒ πr² = 1386

⇒ (22 ÷ 7) × r² = 1386

⇒ r² = 1386 × (7 ÷ 22) = 63 × 7 = 441

∴ The correct answer is 21 cm.

Alternate Method

Given:

Area of sector = 462 cm²

Central angle (θ) = 120°

Formula Used:

Area of a sector = (θ ÷ 360°) × πr²

Calculations:

⇒ 462 = (120° ÷ 360°) × (22 ÷ 7) × r²

⇒ 462 = (1 ÷ 3) × (22 ÷ 7) × r²

⇒ r² = (462 × 3 × 7) ÷ 22

⇒ r² = 21 × 3 × 7

⇒ r² = 441

⇒ r = √441

⇒ r = 21

∴ The correct answer is 21 cm.

Additional Information

Area of a Sector

The area of a sector of a circle with radius r and central angle θ (in degrees) represents a proportional slice of the entire circle's area. It is calculated using the formula: A = (θ ÷ 360°) × πr².

Length of an Arc

The arc length of a sector is a fraction of the circle's total circumference. It is calculated by multiplying the full circumference by the fraction of the angle: L = (θ ÷ 360°) × 2πr.

Perimeter of a Sector

The complete boundary or perimeter of a sector includes the curved arc length along with the two straight radii enclosing it. The formula is: Perimeter = L + 2r.

Given:

The expression to solve is: [15 ÷ 3 × 7 – 8 of 2 + 6].

Formula used:

Follow the order of operations (BIDMAS/PEMDAS), which stands for:

 B = Brackets

 I = Indices (Powers)

 D = Division

 M = Multiplication

 A = Addition

 S = Subtraction

Calculations:

The given expression is: 15 ÷ 3 × 7 – 8 × 2 + 6

Step 1: Perform division and multiplication from left to right:

15 ÷ 3 = 5

5 × 7 = 35

8 × 2 = 16

Now the expression becomes: 35 – 16 + 6

Step 2: Perform subtraction and addition from left to right:

35 – 16 = 19

19 + 6 = 25

Answer:

The value of the expression is 25.

Shortcut Trick

Total Capacity = LCM(20, 30) = 60 units.

Efficiency: P1 = 3 units/hr, P2 = 2 units/hr → Total Efficiency = 5 units/hr.

If P1 worked till the end, it would fill an extra 5 × 3 = 15 units.

New Total Work = 60 + 15 = 75 units (assuming both worked continuously from start to end).

Total time = 75 ÷ 5 = 15 hours.

∴ The correct answer is 15 hours.

Alternate Method

Given:

Time taken by P1 = 20 hours

Time taken by P2 = 30 hours

Concept Used:

Total Work = LCM of times taken by individual pipes.

Efficiency = Total Work ÷ Time taken.

Calculation:

⇒ Total efficiency of P1 and P2 = 3 + 2 = 5 units/hr.

⇒ P1 was closed 5 hours before completion, so P2 worked alone for the last 5 hours.

⇒ Work done by P2 in the last 5 hours = 5 × 2 = 10 units.

⇒ Remaining work to be done by both initially = 60 − 10 = 50 units.

⇒ Time taken by P1 and P2 together = 50 ÷ 5 = 10 hours.

⇒ Total time taken to fill the cistern = 10 + 5 = 15 hours.

∴ The correct answer is 15 hours.

Additional Information

Inlet vs Outlet Pipes

Inlet pipes add water (positive efficiency), while outlet pipes drain water (negative efficiency). Net efficiency is the algebraic sum of their individual efficiencies.

Leaving Before Completion Shortcut

If a pipe is closed 'x' hours before the work is completed, simply assume it worked until the end and add its extra work (x × efficiency) to the original Total Work. Then, divide by the combined efficiency of all pipes.

Alternating Pipes

When pipes operate alternately for an hour each, calculate the total work done in a 2-hour cycle and then multiply to find the maximum complete cycles that fit into the total capacity.

Shortcut Trick

The given expression is [(1 + cos θ) × (1 − cos θ)] ÷ [(1 + sin θ) × (1 − sin θ)]

Using the algebraic identity (a + b) × (a − b) = a² − b², the expression becomes (1 − cos² θ) ÷ (1 − sin² θ)

Using the fundamental trigonometric identities, we know that 1 − cos² θ = sin² θ and 1 − sin² θ = cos² θ

Substituting these, Expression = sin² θ ÷ cos² θ = tan² θ

Since tan θ = 5/8, we get (5/8)² = 25/64

∴ The correct answer is 25/64.

Alternate Method

Given: tan θ = 5/8

We know that tan θ = Perpendicular (P) ÷ Base (B)

⇒ Let Perpendicular P = 5 and Base B = 8

Formula Used: Pythagoras Theorem: H² = P² + B²

⇒ Hypotenuse (H) = √(5² + 8²)

⇒ H = √(25 + 64) = √89

⇒ sin θ = P ÷ H = 5/√89

⇒ cos θ = B ÷ H = 8/√89

Now, evaluate the given expression:

⇒ [(1 + cos θ) × (1 − cos θ)] ÷ [(1 + sin θ) × (1 − sin θ)]

⇒ [(1 + 8/√89) × (1 − 8/√89)] ÷ [(1 + 5/√89) × (1 − 5/√89)]

⇒ [1 − (8/√89)²] ÷ [1 − (5/√89)²]

⇒ [1 − 64/89] ÷ [1 − 25/89]

⇒ [(89 − 64) ÷ 89] ÷ [(89 − 25) ÷ 89]

⇒ (25/89) ÷ (64/89)

⇒ 25/64

∴ The correct answer is 25/64.

Additional Information

Trigonometric Identities

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

Algebraic Identities

The difference of squares formula (a + b) × (a − b) = a² − b² is highly useful for simplifying conjugate trigonometric expressions.

Trigonometric Ratios

In a right-angled triangle, sin θ = Perpendicular ÷ Hypotenuse, cos θ = Base ÷ Hypotenuse, and tan θ = Perpendicular ÷ Base.

Shortcut Trick

When profit is given as a fraction of Selling Price (SP), the actual SP value (₹23358) is not needed.

Given Gain = 1/6 of SP.

Let SP = 6 units, then Gain = 1 unit.

Cost Price (CP) = SP − Gain = 6 − 1 = 5 units.

Gain % = (Gain ÷ CP) × 100 = (1 ÷ 5) × 100 = 20%.

∴ The correct answer is 20%.

Alternate Method

Given:

Selling Price (SP) = ₹23358

Gain = 1/6 of SP

Formula Used:

Gain = SP − CP

Gain % = (Gain ÷ CP) × 100

Calculations:

⇒ Gain = (1 ÷ 6) × 23358 = ₹3893

⇒ CP = SP − Gain

⇒ CP = 23358 − 3893 = ₹19465

⇒ Gain % = (3893 ÷ 19465) × 100

⇒ Gain % = (1 ÷ 5) × 100 = 20%

∴ The correct answer is 20%.

Additional Information

Profit/Loss on Selling Price

Sometimes profit or loss is stated as a percentage of the Selling Price, but actual profit/loss percentage must always be calculated on the Cost Price (CP). True Profit % = Profit ÷ (SP − Profit) × 100.

Marked Price & Discount

Discount is always calculated on the Marked Price (MP). Equivalent discount of two successive discounts d₁% and d₂% is given by (d₁ + d₂ − d₁×d₂÷100)%.

Dishonest Dealer

If a trader sells goods at cost price but uses a false weight, the Profit % = (True Weight − False Weight) ÷ False Weight × 100.

Shortcut Trick

Since the time taken by both to meet is the same, the ratio of distances covered is equal to the ratio of their speeds.

Ratio of speeds = Policeman : Thief = 27 : 22

Difference in ratio = 27 − 22 = 5 units

This 5 units represents the initial gap of 195 m ⇒ 1 unit = 195 ÷ 5 = 39 m

Distance covered by the thief = 22 units = 22 × 39 = 858 m

∴ The correct answer is 858 m.

Alternate Method

Given:

Initial distance between Policeman and Thief = 195 m

Speed of Policeman (S_P) = 27 km/h

Speed of Thief (S_T) = 22 km/h

Formula Used:

Relative Speed (same direction) = S₁ − S₂

Time = Distance ÷ Relative Speed

Distance = Speed × Time

Calculations:

⇒ Relative Speed = 27 − 22 = 5 km/h

⇒ Convert Relative Speed into m/s = 5 × (5 ÷ 18) = 25 ÷ 18 m/s

⇒ Time taken by Policeman to overtake = 195 ÷ (25 ÷ 18)

⇒ Time = (195 × 18) ÷ 25 = 140.4 seconds

⇒ Speed of Thief in m/s = 22 × (5 ÷ 18) = 55 ÷ 9 m/s

⇒ Distance covered by the Thief = Speed × Time = (55 ÷ 9) × 140.4

⇒ Distance = 55 × 15.6 = 858 m

∴ The correct answer is 858 m.

Additional Information

Relative Speed Concept

When two bodies move in the same direction, relative speed is the difference of their individual speeds (S₁ − S₂). When moving in opposite directions, relative speed is their sum (S₁ + S₂).

Conversion of Units

To convert speed from km/h to m/s, multiply by (5 ÷ 18). Conversely, to convert from m/s to km/h, multiply by (18 ÷ 5).

Proportionality of Distance and Speed

If the time taken is constant, the distance traveled by an object is directly proportional to its speed (D₁ ÷ D₂ = S₁ ÷ S₂). This allows solving chase problems instantly using ratio methods.

Shortcut Trick

Test the options directly by substituting the values into the given equations.

Check the values: a = −12, x = 48, and y = 4.

Equation 2: 4x + ay = 4(48) + (−12)(4) = 192 − 48 = 144. Also a² = (−12)² = 144. (Matches)

Equation 1: ax + (a² + 1)y = (−12)(48) + (144 + 1)(4) = −576 + 580 = 4. (Matches)

∴ The correct answer is If a = −12, then x = 48, y = 4 are the solutions.

Alternate Method

Given:

Equation (i): ax + (a² + 1)y = 4

Equation (ii): 4x + ay = a²

Calculations:

⇒ From equation (ii), 4x = a² − ay ⇒ x = (a² − ay) ÷ 4

⇒ Substitute x in equation (i):

⇒ a[(a² − ay) ÷ 4] + (a² + 1)y = 4

⇒ a³ − a²y + 4y(a² + 1) = 16

⇒ a³ − a²y + 4a²y + 4y = 16

⇒ y(3a² + 4) = 16 − a³

⇒ y = (16 − a³) ÷ (3a² + 4)

⇒ Now check for a = −12:

⇒ y = [16 − (−12)³] ÷ [3(−12)² + 4]

⇒ y = (16 + 1728) ÷ (3 × 144 + 4)

⇒ y = 1744 ÷ 436 = 4

⇒ Substitute y = 4 and a = −12 in equation (ii):

⇒ 4x + (−12)(4) = (−12)²

⇒ 4x − 48 = 144

⇒ 4x = 192

⇒ x = 48

∴ The correct answer is If a = −12, then x = 48, y = 4 are the solutions.

Additional Information

Linear Equations in Two Variables

A system of two linear equations a₁x + b₁y = c₁ and a₂x + b₂y = c₂ can have a unique solution, infinite solutions, or no solution depending on the coefficients.

Unique Solution Condition

The system has a unique solution if and only if a₁ ÷ a₂ ≠ b₁ ÷ b₂. In this case, the intersecting lines meet at exactly one point (x, y).

Substitution Method

To solve by substitution, express one variable in terms of the other using one equation and substitute this expression into the second equation to find the exact value.

Shortcut Trick

Convert the discount percentages into fractional multiplying factors:

6% discount ⇒ 1 − 3/50 = 47/50

4% discount ⇒ 1 − 1/25 = 24/25

Net SP = MP × Successive Multipliers

Net SP = 12500 × 47/50 × 47/50 × 24/25 = 12500 × (53016 / 62500)

Net SP = 53016 / 5 = 10603.20

∴ The correct answer is ₹10,603.20.

Alternate Method

Given:

Marked Price (MP) = ₹12,500

First discount (D₁) = 6%

Second discount (D₂) = 6%

Third discount (D₃) = 4%

Formula Used:

Selling Price (SP) = MP × (1 − D₁/100) × (1 − D₂/100) × (1 − D₃/100)

Calculations:

⇒ Net SP = 12500 × (1 − 6/100) × (1 − 6/100) × (1 − 4/100)

⇒ Net SP = 12500 × (94/100) × (94/100) × (96/100)

⇒ Net SP = 12500 × (47/50) × (47/50) × (24/25)

⇒ Net SP = (12500 / 62500) × 47 × 47 × 24

⇒ Net SP = (1/5) × 2209 × 24

⇒ Net SP = 53016 / 5

⇒ Net SP = 10603.20

∴ The correct answer is ₹10,603.20.

Additional Information

Marked Price and Successive Discounts

Discount is generally calculated on the Marked Price of the article. For successive discounts, the single final selling price is computed continuously on the updated price using the formula SP = MP × (100 − D₁)/100 × (100 − D₂)/100...

Equivalent Discount Formula

If two successive discounts of a% and b% are given, the single equivalent discount rate is calculated as (a + b − ab/100)%.

Fraction Method for Quicker Calculation

Converting standard percentages to lowest fractions simplifies continuous multiplication. For example, a discount of x% leads to a multiplier of (1 − x/100).

Shortcut Trick

Final payment = Total amount of initial loan − Equivalent amount of partial repayment

Total Due = P × (1 + 2R) − Repayment × (1 + R)

⇒ Total Due = 598397 × (1 + 0.40) − 24284 × (1 + 0.20)

⇒ Total Due = 598397 × 1.4 − 24284 × 1.2

⇒ Total Due = 837755.8 − 29140.8 = 808615

∴ The correct answer is ₹808615.

Alternate Method

Given:

Principal (P) = ₹598397, Rate (R) = 20% per annum

Repayment of Principal at end of 1st year = ₹24284

Calculations:

⇒ Outstanding principal for 2nd year = 598397 − 24284 = ₹574113

⇒ Interest for 1st year = 20% of 598397 = ₹119679.4

⇒ Interest for 2nd year = 20% of 574113 = ₹114822.6

⇒ Total Interest = 119679.4 + 114822.6 = ₹234502

⇒ Total Amount to pay = Pending Principal + Total Interest

⇒ Total Amount = 574113 + 234502 = ₹808615

∴ The correct answer is ₹808615.

Additional Information

Simple Interest vs Compound Interest

In Simple Interest, interest is calculated only on the remaining principal, meaning accumulated past interest does not earn further interest.

Simple Interest Formula

The interest accumulated over time is given directly by the formula I = (P × R × T) ÷ 100.

Equivalent Amount Method

For simple interest partial payments, the total outstanding balance can be easily determined by evaluating the final amount of the initial principal and subtracting the respective equivalent amounts of each partial repayment up to the end term.

Shortcut Trick

According to Apollonius's Theorem, the sum of the squares of any two sides of a triangle equals twice the sum of the square of the median to the third side and the square of half the third side.

In ΔPQR, PT is the median to QR, which means T is the midpoint of QR.

⇒ PQ² + PR² = 2 × (PT² + QT²)

This is a standard direct theorem identity and requires no further calculation.

∴ The correct answer is PQ² + PR² = 2(PT² + QT²).

Alternate Method

P Q R T S

Given:

⇒ ΔPQR with median PT, meaning QT = TR.

Derivation:

⇒ Draw an altitude PS perpendicular to QR as shown in the figure.

⇒ In right-angled ΔPSQ: PQ² = PS² + QS²

⇒ In right-angled ΔPSR: PR² = PS² + SR²

⇒ Adding both equations: PQ² + PR² = 2PS² + QS² + SR²

⇒ From the figure, we can observe that QS = QT − ST and SR = ST + TR.

⇒ Since T is the midpoint, TR = QT. Therefore, SR = QT + ST.

⇒ Substituting these values: PQ² + PR² = 2PS² + (QT − ST)² + (QT + ST)²

⇒ Using the algebraic identity (a − b)² + (a + b)² = 2a² + 2b²:

⇒ PQ² + PR² = 2PS² + 2QT² + 2ST²

⇒ Rearranging terms: PQ² + PR² = 2(PS² + ST²) + 2QT²

⇒ In right-angled ΔPST, by Pythagoras theorem: PS² + ST² = PT²

⇒ Therefore: PQ² + PR² = 2PT² + 2QT² = 2(PT² + QT²)

∴ The correct answer is PQ² + PR² = 2(PT² + QT²).

Additional Information

Apollonius's Theorem

It is a fundamental geometric theorem relating the length of a median of a triangle to the lengths of its sides, stating that AB² + AC² = 2(AD² + BD²) for median AD.

Stewart's Theorem

A broader generalisation for any cevian AD dividing the side BC into segments m and n: AB² × n + AC² × m = BC(AD² + mn).

Properties of a Median

A median naturally divides the triangle into two smaller triangles of exactly equal areas. All three medians intersect at the centroid, dividing each median strictly in a 2:1 ratio.

Formula used:

1. Qualified candidates = ({Number of appeared candidates}) × ({Percentage of qualified candidates}) / 100

Calculations:

Step 1: Calculate qualified candidates in 2008:

Qualified candidates in 2008 = A × 70 / 100 = 0.7A .

Step 2: Calculate qualified candidates in 2009:

Qualified candidates in 2009 = 2A × 80 / 100 = 1.6A .

Step 3: Total qualified candidates:

0.7A + 1.6A = 552 .

⇒2.3A = 552 .

⇒ A = 552 / 2.3 = 240 .

Step 4: Calculate appeared candidates in 2009:

Number of appeared candidates in 2009 = 2A = 2 × 240 = 480 .

∴ The number of appeared candidates from State Z in 2009 is 480.

Shortcut Trick

Total Surface Area of a closed cube = 6a²

6a² = 1152

a² = 192

a = √192 = 8√3 cm

∴ The correct answer is 8√3 cm.

Alternate Method

Given:

Total surface area of the closed cube = 1152 cm²

Formula Used:

Total Surface Area of a cube = 6a²

where 'a' is the length of the side of the cube.

Calculations:

⇒ 6a² = 1152

⇒ a² = 1152 ÷ 6

⇒ a² = 192

⇒ a = √192

⇒ a = √(64 × 3)

⇒ a = 8√3 cm

∴ The correct answer is 8√3 cm.

Additional Information

Lateral Surface Area

The lateral surface area includes only the 4 side faces of the cube, excluding the top and bottom faces. Formula: LSA = 4a².

Volume of a Cube

The volume represents the total three-dimensional space enclosed by the cube. Formula: Volume = a³.

Main Diagonal of a Cube

The longest straight line segment that can be drawn inside a cube connecting two opposite corners. Formula: Main Diagonal = a√3.

Face Diagonal

The diagonal line extending across any single flat, square face of the cube. Formula: Face Diagonal = a√2.

Shortcut Trick

By using the value substitution method from the standard angle values:

Let θ = 45°, then tan 45° = 1 and cot 45° = 1.

Substitute the values in the equation: tan θ + cot θ = 1 + 1 = 2.

This strictly satisfies the given condition, hence θ = 45°.

∴ The correct answer is 45°.

Alternate Method

Given:

tan θ + cot θ = 2

Formula Used:

cot θ = 1 / tan θ

Calculation:

⇒ tan θ + 1 / tan θ = 2

⇒ (tan² θ + 1) / tan θ = 2

⇒ tan² θ + 1 = 2 tan θ

⇒ tan² θ − 2 tan θ + 1 = 0

⇒ (tan θ − 1)² = 0

⇒ tan θ − 1 = 0

⇒ tan θ = 1

⇒ tan θ = tan 45°

⇒ θ = 45°

∴ The correct answer is 45°.

Additional Information

Reciprocal Trigonometric Identities

tan θ = 1 / cot θ, sin θ = 1 / csc θ, and cos θ = 1 / sec θ.

Pythagorean Identities

sin² θ + cos² θ = 1, sec² θ − tan² θ = 1, and csc² θ − cot² θ = 1.

Standard Angle Values

tan 30° = 1 / √3, tan 45° = 1, and tan 60° = √3.

Shortcut Trick

Using the Tangent-Secant Theorem: Tangent² = External part × Total secant

⇒ 21² = 14 × (14 + BC)

⇒ 441 = 14 × (14 + BC)

⇒ 14 + BC = 31.5 ⇒ BC = 17.5

∴ The correct answer is 17.5 units.

Alternate Method

Given:

Length of the tangent from point A (AT) = 21 units

Length of the external secant segment (AB) = 14 units

Let the length of the internal secant segment (BC) be x units.

Radius of the circle = 14 units (Not required for this calculation)

Formula Used:

Tangent-Secant Theorem: AT² = AB × AC

A T B C O 21 14 x

Calculations:

The total length of the secant segment from point A is AC.

⇒ AC = AB + BC

⇒ AC = 14 + x

Applying the Tangent-Secant theorem:

⇒ 21² = 14 × (14 + x)

⇒ 441 = 14 × (14 + x)

⇒ 14 + x = 441 ÷ 14

⇒ 14 + x = 31.5

⇒ x = 31.5 − 14

⇒ x = 17.5 units

∴ The correct answer is 17.5 units.

Additional Information

Tangent-Secant Theorem

If a tangent and a secant are drawn to a circle from an exterior point, the square of the tangent is equal to the product of the external secant segment and the total secant length: AT² = AB × AC.

Intersecting Secants Theorem

If two secants are drawn to a circle from one exterior point, the product of the external segment and the total length of each secant are equal: AB × AC = AD × AE.

Intersecting Chords Theorem

When two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord: AE × EC = BE × ED.

Shortcut Trick

Put a standard value for the acute angle, let A = 60°.

Then, cos 60° = 1/2.

Given expression = √[(1 − 1/2) / (1 + 1/2)] + √[(1 + 1/2) / (1 − 1/2)]

⇒ √(1/3) + √3 = 1/√3 + √3 = 4/√3

Checking Option 1: 2 cosec 60° = 2 × (2/√3) = 4/√3. This matches our result perfectly.

∴ The correct answer is 2 cosec A.

Alternate Method

Given:

Expression = √[(1 − cos A) / (1 + cos A)] + √[(1 + cos A) / (1 − cos A)]

Formula Used:

1 − cos² A = sin² A

1 / sin A = cosec A

Calculations:

⇒ Let the given expression be E.

⇒ Multiplying numerator and denominator by their respective conjugates inside the root:

⇒ E = √[(1 − cos A)² / (1 − cos² A)] + √[(1 + cos A)² / (1 − cos² A)]

⇒ Using the identity 1 − cos² A = sin² A:

⇒ E = √[(1 − cos A)² / sin² A] + √[(1 + cos A)² / sin² A]

⇒ Since A is an acute angle, sin A, (1 − cos A), and (1 + cos A) are all positive.

⇒ E = [(1 − cos A) / sin A] + [(1 + cos A) / sin A]

⇒ E = (1 − cos A + 1 + cos A) / sin A

⇒ E = 2 / sin A

⇒ E = 2 cosec A

∴ The correct answer is 2 cosec A.

Additional Information

Trigonometric Identities

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

Reciprocal Relations

The reciprocal trigonometric functions are given by: cosec θ = 1 / sin θ, sec θ = 1 / cos θ, and cot θ = 1 / tan θ.

Half-Angle Formulas

Important half-angle formulas derived from double angles: 1 − cos θ = 2 sin²(θ/2) and 1 + cos θ = 2 cos²(θ/2).

Shortcut Trick

Let x = 25, substituting in the given expression.

The numerator becomes x⁴ + 2x² + 1 − x² = x⁴ + x² + 1.

The denominator is x² + x + 1.

Using the identity x⁴ + x² + 1 = (x² + x + 1)(x² − x + 1), the expression simplifies to x² − x + 1.

Value = 25² − 25 + 1 = 625 − 24 = 601.

∴ The correct answer is 601.

Alternate Method

Given:

[25⁴ + 1250 + 1 − 25²] ÷ [25² + 25 + 1]

Formula Used:

(a² + 1)² = a⁴ + 2a² + 1

x² − y² = (x + y)(x − y)

Calculation:

⇒ Numerator = 25⁴ + 2 × 625 + 1 − 25²

⇒ Numerator = 25⁴ + 2 × 25² + 1 − 25²

⇒ Numerator = (25² + 1)² − 25²

Applying the algebraic identity for the difference of squares:

⇒ Numerator = (25² + 1 + 25)(25² + 1 − 25)

⇒ Numerator = (25² + 25 + 1)(25² − 25 + 1)

Now, divide the simplified numerator by the given denominator:

⇒ Expression = [(25² + 25 + 1)(25² − 25 + 1)] ÷ [25² + 25 + 1]

⇒ Expression = 25² − 25 + 1

⇒ Expression = 625 − 25 + 1 = 601

∴ The correct answer is 601.

Additional Information

Standard Algebraic Identities

An important factored identity often used in simplification is a⁴ + a²b² + b⁴ = (a² + ab + b²)(a² − ab + b²).

Difference of Squares

a² − b² = (a − b)(a + b) is commonly used to simplify complex numeric expressions by converting them into smaller factors.

Shortcut Trick

Initial Ratio (Boys : Girls) = 5 : 6 | Final Ratio = 8 : 11 | Change = −40 : +50

Equate the difference of the ratio cross-products to the value cross-products:

⇒ |(5 × 11) − (6 × 8)| units = |(8 × 50) − (11 × (−40))|

⇒ 7 units = (400 + 440) = 840 ⇒ 1 unit = 120

⇒ Initial Girls = 6 units = 6 × 120 = 720

∴ The correct answer is 720 girls.

Alternate Method

Given:

Initial ratio of boys to girls = 5 : 6

Boys who left the college = 40

Girls who joined the college = 50

Final ratio of boys to girls = 8 : 11

Calculations:

Let the initial number of boys be 5x and the initial number of girls be 6x.

After changes, the new number of boys = 5x − 40

And the new number of girls = 6x + 50

According to the given condition, we equate the new quantities to the final ratio:

⇒ (5x − 40) ÷ (6x + 50) = 8 ÷ 11

⇒ 11 × (5x − 40) = 8 × (6x + 50)

⇒ 55x − 440 = 48x + 400

⇒ 55x − 48x = 400 + 440

⇒ 7x = 840

⇒ x = 120

The initial number of girls = 6x

⇒ 6 × 120 = 720

∴ The correct answer is 720 girls.

Additional Information

Cross-Multiplication Method

Used rapidly when two quantities in a given ratio change by specific amounts to form a new ratio. Equate the difference of the ratio cross-products directly to the difference of the value cross-products.

Compound Ratio

When two or more ratios are multiplied together, it forms a compound ratio. For ratios a:b and c:d, the compound ratio is ac:bd.

Proportion Rule

An equation stating that two ratios are equal is called a proportion. If a ÷ b = c ÷ d, then a × d = b × c (product of extremes equals product of means).

Shortcut Trick

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

Let the assumed base average of the class be 62.

Deviation of the second group = 68 − 62 = +6

Average increase for the whole class = (3 × 6) ÷ (2 + 3) = 18 ÷ 5 = +3.6

Overall Average = 62 + 3.6 = 65.6

∴ The correct answer is 65.6 marks.

Alternate Method

Given:

Total number of students = 30

Average score of 12 students (A₁) = 62

Average score of remaining 18 students (A₂) = 68

Group 1 (12 st) Group 2 (18 st)

62

68

Avg

68 − Avg

Avg − 62

Ratio = 12 : 18

Formula Used:

Combined Average = (n₁ × A₁ + n₂ × A₂) ÷ (n₁ + n₂)

Calculations:

⇒ Combined Average = (12 × 62 + 18 × 68) ÷ 30

⇒ Combined Average = (744 + 1224) ÷ 30

⇒ Combined Average = 1968 ÷ 30

⇒ Combined Average = 65.6

∴ The correct answer is 65.6 marks.

Additional Information

Weighted Average

Used when combining multiple groups of different sizes. The combined average is calculated as A = (n₁A₁ + n₂A₂ + ...) ÷ (n₁ + n₂ + ...).

Alligation Method

A visual cross-multiplication tool used to quickly find the quantity ratio. If two groups mix to form an average A, their quantity ratio is given as (A₂ − A) : (A − A₁).

Deviation Method

Assume a base average, calculate the net deviation of all values from it, and adjust the assumed base by dividing the net deviation by the total number of items.

The correct answer is: Chaos.

Key Points

• "Chaos" means a state of utter confusion and disorder. (अव्यवस्था)
• Example: The sudden blackout during the event caused chaos among the audience.
• It is used to describe situations where no order or organization is present, often leading to panic or disturbance. (अराजकता)
• Example: The marketplace was in chaos after the unexpected announcement of a curfew.

Therefore, the correct answer is: Chaos.

Additional Information

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

• Pathos (भावुकता): A quality in art or literature that evokes pity or sadness.
• Example: The movie was filled with pathos, leaving the audience in tears.
• Engross (डुबाना): To occupy the attention or focus of someone completely.
• Example: She was so engrossed in her book that she didn’t notice the time.
• Innuendos (इशारे): Indirect or subtle references, often implying something derogatory.
• Example: The politician’s speech was filled with innuendos about his opponent.

The correct answer is Option 1.

 

Key Points

• The phrase "bite the bullet" means to force oneself to perform an unpleasant action in a difficult situation.
• The other options do not fit the context of the sentence.
• "Dodge the bullet" means to avoid a difficult or dangerous situation, which is not the case here.
• "Shoot the bullet" and "ignore the bullet" are not commonly used idioms and do not convey the intended meaning.

Therefore, the correct answer is Option 1.

 

Correct Sentence: "After reflecting on the matter regarding his boss's selfish attitude, Raju decided to bite the bullet to resolve the matter."

The correct answer is option 3.

Key Points

The misspelled word is 'occurence'.

• The correct spelling would be 'occurrence'.
• The word 'occurrence' is a noun which means 'an incident or event'.
• Therefore, the correct answer would be 'occurrence'.

The correct answer is Option 2.

 

Key Points

• Passive voice structure: The object of the active voice sentence becomes the subject of the passive voice sentence.
• In the active voice, the sentence is: "Meenakshi could have sent a reply."
• To convert to passive voice: Place the object "a reply" at the beginning.
• Change the active verb form "could have sent" to "could have been sent" (perfect modal passive).
• Add the preposition "by" followed by the original subject "Meenakshi."

Therefore, the correct answer is Option 2.

 

Correct Sentence: "A reply could have been sent by Meenakshi."

 

Additional Information

• Option 1: "A reply could have be sent by Meenakshi." – Incorrect verb form (should be "been" instead of "be").
• Option 3: "A reply could have sent by Meenakshi." – Incorrect structure (missing "been").
• Option 4: "A reply could have been send by Meenakshi." – Incorrect verb form (should be "sent" instead of "send").