SSC CGL 2020 Tier-I Official Paper (Held On : 13 Aug 2021 Shift 1) (Previous Year Paper)

The number of triangles in the figure is shown below:

Hence, ‘11’ is the correct answer.

The logic is:

Horses : Neigh → The sound made by Horse is Neigh.

Similarly,

Ducks : Quack → The sound made by Ducks is Quack.

Hence, ‘Ducks : Quack’ is the correct answer.

The paper when unfolded will appear as shown below:

Hence, 'option 3’ is the correct answer.

The description is as follows:

 

All options except 'Weight' are related to the skills of a person, while 'weight' is related to heaviness of a person.

Hence, ‘Weight’ is the odd one out.

The least possible Venn diagram is:          

  

 

Conclusions:

I. Some diagonals are polygons. → True (As All polygons are angles and All angles are diagonals → Some diagonals are polygons)

II. All diagonals are decagons. → False (As, there is no definite relation between diagonals and decagons. Hence, false)

III. No polygon is a cone. → True (As All polygons are angles, All angles are diagonals, All cones are cubes and No diagonal is a cube → No polygon is a cone)

IV. Some cubes are angles. → False (As All angles are diagonals and No diagonal is a cube → No cubes are angles)

Hence, both conclusions I and III follow.

Additional Information

1. ÷, –, +, ×, = 

Given equation: 42 * 7 * 64 * 11 * 6 * 4 

After replacing the * by the symbols in the given option and use BODMAS rule, we get:

42 ÷ 7 - 64 + 11 × 6 = 4

⇒ 6 – 64 + 66 = 4

⇒ 72 – 64 = 4

⇒ 8 ≠ 4

2. ×, +, –, ÷, = 

Given equation: 42 * 7 * 64 * 11 * 6 * 4 

After replacing the * by the symbols in the given option and use BODMAS rule, we get:

42 × 7 + 64 - 11 ÷ 6 = 4 

⇒ 294 + 64 – 1.84 = 4

⇒ 358 – 1.84 = 4

⇒ 356.16 ≠ 4

3. ÷, +, –, ×, =

Given equation: 42 * 7 * 64 * 11 * 6 * 4 

After replacing the * by the symbols in the given option and use BODMAS rule, we get:

42 ÷ 7 + 64 - 11 × 6 = 4 

⇒ 6 + 64 – 66 = 4

⇒ 70 – 66 = 4

⇒ 4 = 4

4. ×, –, +, ÷, =

Given equation: 42 * 7 * 64 * 11 * 6 * 4 

After replacing the * by the symbols in the given option and use BODMAS rule, we get:

42 × 7 - 64 + 11 ÷ 6 = 4 

⇒ 294 – 64 + 1.84 = 4

⇒ 295.84 – 64 = 4

⇒ 231.84 ≠ 4

Hence, ‘÷, +, –, ×, =’ is the correct answer.

The logic is:

First number : First number × (First number + 1) = Second number

1. 14 : 210 → 14 × (14 + 1) = 14 × 15 = 210 2. 18 : 342 → 18 × (18 + 1) = 18 × 19 = 342 

3. 17 : 307 → 17 × (17 + 1) = 17 × 18 = 306 ≠ 307

4. 12 : 156 → 12 × (12 + 1) = 12 × 13 = 156

Hence, '17 : 307' is the odd one out.

The logic is:

Hence, ‘139’ is the correct answer.

The pattern followed here is:

Here consonants of the words are coded according to their positional values in English alphabetical series.

• Consonants → same positional value

The Vowels are coded according to the opposite letter place value × 2.

• Vowels → Opposite letter positional value × 2.

For AROUND = 52182412144

For FIX = 63624

Similarly,

For PLASTIC = ?

Hence, ‘1612521920363’ is the correct answer.

The logic is:

Depression : Mood → Depression is a state when a person suffers with mood disorder.

Similarly,

Insomnia : ? → Insomnia is a sleep disorder. (Insomnia is a common sleep disorder that can make it hard to fall asleep, hard to stay asleep, or cause you to wake up too early and not be able to get back to sleep.)

Hence, ‘Sleep’ is the correct answer.

1. N, L, P, A, N, U, Z

Given series: L_UA_Z_N_AP_L_U_PZ

After filling the blanks with the letters in the option given above, we get:

LNUALZ / PNAAPN / LUUZPZ

No particular pattern is repeating itself when the sequence is divided into three equal parts.

2. P, N, L, Z, U, A, N

Given series: L_UA_Z_N_AP_L_U_PZ

After filling the blanks with the letters in the option given above, we get:

LPUANZ / LNZAPU / LAUNPZ

No particular pattern is repeating itself when the sequence is divided into three equal parts.

3. P, L, U, Z, N, A, N

Given series: L_UA_Z_N_AP_L_U_PZ

After filling the blanks with the letters in the option given above, we get:

LPUALZ / UNZAPN / LAUNPZ

No particular pattern is repeating itself when the sequence is divided into three equal parts.

4. N, P, L, U, Z, N, A

Given series: L_UA_Z_N_AP_L_U_PZ

After filling the blanks with the letters in the option given above, we get:

LNUAPZ / LNUAPZ / LNUAPZ

A particular pattern is repeating itself when the sequence is divided into three equal parts.

Hence, ‘N, P, L, U, Z, N, A’ is the correct answer.

Confusion PointsOption figure 1 & 2 both are the different.

The logic followed here is:

1. The images at the extreme left corner of the figure change their position by shifting 90º clockwise.

2. The image at the north-east corner shifts its position at the center in the next figure.

3. Figures at odd position have big triangle at south-east corner and figures at even position have small triangle at south-east corner.

The figure from among the given options that can replace the question mark (?) in the following series is shown below:

 

Hence, ‘option 1’ is the correct answer.

The correct option that indicates the arrangement of the given words in the order in which they appear in an English dictionary is shown below:

5. Fraud
6. Freedom
7. Freeze
8. Fringe
9. Frozen
10. Fryer

Hence, ‘5, 2, 1, 6, 4, 3’ is the correct answer.

Let total players = A

Players in Hotel X = 100/3% of A = A/3

Players in Hotel Y = A - A/3 = 2A/3

According to question:

A/3 + 20 = 50/100 × A

⇒ A/3 + 20 = A/2

⇒ A/2 - A/3 = 20

⇒ A/6 = 20

⇒ A = 20 × 6

⇒ A = 120

Now, Players in Hostel X = 120/3 = 40 

Players in Hostel Y = 120 - 40 = 80

Now, if 20 players from hotel X are shifted to hotel Y, then the number of players in hotel X becomes,

= 40 - 20 = 20

And the number of players in hotel X becomes what percent of the total number of players:

20/120 × 100

= 16.67%

Hence, ‘16.67%’ is the correct answer.

The pattern followed here is:

According to the alphabetical positions of the letters,

1. TPLH → T - 4 = P; P - 4 = L; L - 4 = H

2. NJFA → N - 4 = J; J - 4 = F; F - 5 = A

3. CYUQ → C - 4 = Y; Y - 4 = U; U - 4 = Q 4. FBXT → F - 4 = B; B - 4 = X; X - 4 = T

Hence, ‘NJFA’ is the correct answer.

The correct mirror image of the given combination when the mirror is placed at 'PQ' is shown below:

Hence, ‘option 1’ is the correct answer.

Additional Information

In a mirror image Left changes to right and the right changes to left vice-versa.

When placement of mirror not given, By default we will assume it in Vertical direction on the right side.

The Letter adjacent to the mirror after the mirror image will adjacent to it just with the left change to right and right change to left vice-versa.

The starting letter of the word will be at last after the mirror image drawn with left changes to right and right changes to left.

The pattern followed here:

Some women are researchers and introverts.

Some researchers are women and introverts.

Some introverts are researchers and women.

The Venn diagram that best illustrates the relationship among Women, Researchers, and Introverts is shown below:

Hence, ‘option 4’ is the correct answer.

Faces opposite to each other is shown below:

Clearly, '&' would be the symbol on the opposite side of the # symbol.

Hence, ‘&’ is the correct answer.

Reasoning  Puzzle  Question Find the missing number in the matrix.  Based on logic value  Fully explained answer and step-by-step solution.

 

The logic is:

Column wise: (2nd number - 1st number) × 4 = 3rd number.

Column 1: (15 - 4) × 4 = 11 × 4 = 44

Column 3: (21 - 6) × 4 = 15 × 4 = 60

Similarly,

Let the missing number be a

Column 2: (? - 7) × 4 = 68

⇒ ? - 7 = 68 ÷ 4

⇒ ? - 7 = 17

⇒ ? = 17 + 7

⇒ ? = 24

Hence, ‘24’ is the correct answer.

By using the symbols in the table given below, we can draw the following family tree:

Clearly, N is the brother of T.

Hence, ‘Brother’ is the correct answer.

The pattern followed here is:

According to the alphabetical positions of the letters,

Hence, ‘XVVU’ is the correct answer.

The pattern followed here is:

Letters at even position of the word are coded according to the equivalent opposite letter of a particular letter of the word and letters at odd position are coded according to the logic shown below:

Similarly,

Hence, ‘OMYIZRU’ is the correct answer.

The logic is:

12 : 60 → 12 × [(12 ÷ 2) - 1] = 12 × [6 - 1] = 12 × 5 = 60

Similarly,

16 : ? → 16 × [(16 ÷ 2) - 1] = 16 × [8 - 1] = 16 × 7 = 112

Hence, ‘112’ is the correct answer.

The logic is:

(25, 18, 225) → (25 × 18) ÷ 2 = 450 ÷ 2 = 225 

Similarly,

(24, 22, 264) → (24 × 22) ÷ 2 = 528 ÷ 2 = 264 

Hence, ‘(24, 22, 264)’ is the correct answer.

Important Points

• Do not get tensed by the complexity of the figures.
• Carefully understand the structure of the given question figure.
• The embedded figure may not be of exact size. It may be small or big.
• Sometimes the rotation or reflection of the figures may be confusing.
• So think wisely before selecting the answer.

The option figure that is embedded in the given figure when rotation is NOT allowed is shown below:

Hence, ‘option 2’ is the correct answer.

The correct answer is Umesh Sharma.

Key Points

• As of January 2021, Umesh Sharma was elected as the President of the Veterinary Council of India.
• Veterinary Council of India is a statutory body established under the Indian Veterinary Council Act 1984.
• It was established under the Ministry of Agriculture of the Government of India in 1984.
• Veterinary Council of India is based in New Delhi.
• The first elections to the council took place in 1999.
• Rajeev Arora is the additional chief secretary of Haryana.

He was the founder of Brahmo Samaj is NOT correct.

Key Points

• Brahmo Samaj was founded by Raja Ram Mohan Roy.
• Brahmo Samaj was a monotheistic sect of Hinduism.
• It was founded as Brahmo Sabha in 1828.
• Brahmo Sabha was later renamed as Brahmo Samaj.

Important Points

• Swami Dayananda Saraswati was a social reformer in the Indian subcontinent.
• His birthplace was Gujarat.
• The childhood name of Swami Dayananda Saraswathi is Mul Shankar.
• He is the founder of Arya Samaj.
• It was founded in 1875 as its headquarters in Mumbai.
• He is popularly called "Martin Luther of Hindustan".
• Satyartha Prakash is a famous work of Swami Dayananda Saraswati.
• 'Shuddhi movement' was started by Swami Dayananda Saraswathi.

The correct answer is Al-Biruni.

Key Points

• Al-Biruni was an Iranian scholar.
• He is considered the father of modern geodesy.
• Kitab-ul-Hind is an Arabic book authored by Al-Biruni in the 11th century.
• Kitab-ul-Hind is also known by the name of Tarikh-ul-Hind.
• The book deals with the “caste system” in India.
• He extensively compared the Indian culture with other cultures in this book.

Important Points

• Al-Biruni called Kerala as Malabar.
• Kitab al-Tafhim is the book authored by Al-Biruni in both Persian and Arabic languages.
• He traveled to the Indian subcontinent in 1017.
• Notable work of Al-Biruni are: 
• The Remaining Signs of Past Centuries.
• Gems.
• The Mas'udi Canon.
• Understanding Astrology.

Additional Information

• Al-Bukhari was a Persian Islamic scholar who was born in Bukhara.
• Al-Khwarizmi was a Muslim mathematician who introduced Hindu-Arabic numerals and the concepts of algebra into European mathematics.
• Al-Kindi is popularly called “the philosopher of the Arabs.”

The correct answer is Forensic science.

Key Points

• Forensic science is the use of scientific methods or expertise to investigate crimes or examine evidence that might be presented in a court of law.
• It is also known as criminalistics.
• It is the application of sciences such as physics, chemistry, biology, computer science, and engineering to matters of law and to the identification of various facts of the civilian investigation.
• Forensic science comprises a diverse array of disciplines, from fingerprint and DNA analysis to anthropology and wildlife forensics.
• Forensic scientists collect, preserve, and analyze scientific evidence during the course of an investigation.

Additional Information

• Psychology is the scientific study of the mind and behavior.
• Morphology is the study of the size, shape, and structure of animals, plants, and microorganisms and of the relationships of their constituent parts.
• Kalology is the study of beauty and the ways in which beauty influences society.

The correct answer is Four.

Key Points

• Union Budget of 2021-22 was the first-ever digital Union Budget.
• It was presented by Union Finance Minister Nirmala Sitharaman.
• As per the Union Budget of 2021-22, The central government has announced the setting up of four regional institutes of virology.
• Regional NIV will help in efficiently countering the threats of viral pandemics/epidemics in the future.
• Apart from the new virology institutes the budget proposals also include the setting up of a regional research platform for the WHO South-East Asia region.
• It also established the National Institute of One Health (NIO) at Nagpur for research in ‘One Health’.
• The Union Budget 2021 allocated Rs 2,663 crore for Health Research for the upcoming fiscal year 2021-22.

Additional Information

• The Budget proposals for 2021-22 rest on 6 pillars.

1. Health and Wellbeing
2. Physical & Financial Capital, and Infrastructure
3. Inclusive Development for Aspirational India
4. Reinvigorating Human Capital
5. Innovation and R&D
6. Minimum Government and Maximum Governance

The correct answer is Nubra.

Key Points

• In Leh, the first-ever ice climbing festival was celebrated in Nubra valley in January 2021.
• It was organized by Nubra Adventure Club.
• The ice climbing festival was organized to promote winter tourism in Nubra valley.
• The event was conducted on seven days.
• 18 participants across Nubra valley including 4 women have taken part in Ice climbing event.

Important Points

• Nubra Valley is the northernmost part of Jammu and Kashmir.
• Nubra is currently administered as a subdivision and a tehsil in the Leh district.
• It is known as the Orchard of Ladakh.
• It was originally called Ldumra(valley of flowers).
• Nubra was listed as one of the five natural and historical divisions of Ladakh by Alexander Cunningham.

Additional Information

• Ripchar Valley is located in Ladakh.
• Markha Valley is situated in the eastern part of Ladakh.
• Dras is a hill station located in the Kargil district of Ladakh.

The correct answer is Khetri.

Key Points

• Khetri is a town in the Jhunjhunu district of Rajasthan.
• Khetri is situated at the foothills of the Aravalli Range.
• Khetri is famous for its copper mines.
• Khetri mines are owned by Hindustan Copper Ltd.
• The Khetri region is highly contaminated by copper with higher values in the soil near overburden material and tailings.
• Regular mining in Khetri ceased in the year 1872.

Additional Information

• Keonjhar is a mineral-producing district of Odisha.
• It is famous for mining Iron ore, Manganese ore, Chromite, Quartzite, Bauxite, Gold, and Pyrophyllite.
• Satna is a city in Madhya Pradesh.
• Satna is in the limestone belts of India.
• Satna contributes around 8%–9% of India's total cement production.

the correct answer is Ram Prasad Bismil.

Important Points

• The Hindustan Republican Association was a revolutionary organization formed by Ram Prasad Bismil.
• Hindustan Republican Association was formed to fight against British colonial rule in India and achieves independence for the country through an armed rebellion if necessary.
• It was the first revolutionary movement at the national level against the British.
• The constitution for the Hindustan Republican Association was drafted in 1923 at Allahabad.
• Sachindra Nath Sanyal and Jogesh Chandra Chatterjee were other prominent members of the party.
• Kakori Train Action Day is associated with the Hindustan Republican Association.
• Hindustan Republican Association was later renamed as Hindustan Socialist Republican Association.

Additional Information

• Ram Prasad Bismil was an Indian revolutionary who participated in the Mainpuri Conspiracy and the Kakori Conspiracy.
• He was hanged by the British on 19 December 1927.
• Jatindranath Mukherjee was the principal leader of the Jugantar party(central association of revolutionary independence activists in Bengal).
• Surya Sen was a Bengali revolutionary who was best known for leading the 1930 Chittagong armoury raid.
• Lala Lajpat Rai was an Indian independence activist who was best known for his peaceful protest march against the all-British Simon Commission.
• He is popularly known as Punjab Kesari.

The correct answer is 1950.

Key Points

• The planning commission was an institution that formulated Five-Year Plans in India.
• The planning commission was set up in 1950.
• The planning commission is only an advisory body.

Additional Information 

• The headquarters of the Planning Commission is located in New Delhi.
• It was established on the recommendation of an advisory planning board under the chairmanship of K. C. Niyogi.
• The concept of planning was based on the Russian model introduced by Joseph Stalin.
• The Prime Minister is the chairman of the planning commission.
• Jawaharlal Nehru was the first chairman of the planning commission.
• The Deputy Chairman of the planning commission was appointed by the Union Cabinet.
• Gulzarilal Nanda was the first deputy Chairman of the Planning Commission.

The correct answer is Meghalaya.

Key Points

• Baghmara Pitcher Plant Sanctuary is a sanctuary located in the South Garo Hills district of Meghalaya.
• It is named after a rare insectivorous Pitcher Plant, Nepenthes khasiana.
• Nepenthes khasiana is endemic to the Meghalaya region only.
• The Pitcher plant is locally known as "Me'mang-Koksi" (Ghost Basket).
• Locals regard it as a sacred plant or the Ghost basket.
• Baghmara Pitcher Plant Sanctuary is the only Sanctuary in Meghalaya where in-situ conservation of Pitcher Plant is being carried out.
• The Sanctuary also protects soil erosion and provides water to a section of the surrounding villages.

Important Points

• Other national parks and sanctuaries in Meghalaya are:
• Balpakram
• Nokrek
• Nongkhyllem
• Narpuh
• Siju

Additional Information

• Important wildlife sanctuaries in Assam:
• Garampani Wildlife Sanctuary.
• Laokhowa Wildlife Sanctuary.
• Bura Chapori Wildlife Sanctuary.
• Bornadi Wildlife Sanctuary.
• Pobitora Wildlife Sanctuary.
• Important wildlife sanctuaries in Karnataka:
• Adichunchanagiri Hills Wildlife Sanctuary
• Arabithittu Wildlife Sanctuary
• Attiveri Bird Sanctuary
• Bhadra Wildlife Sanctuary
• Bhimgad Wildlife Sanctuary
• Important wildlife sanctuaries in Goa:
• Bhagwan Mahavir Sanctuary.
• Cotigao Wildlife Sanctuary.
• Bondla Wildlife Sanctuary.
• Salim Ali Bird Sanctuary.
• Madei Wildlife Sanctuary.

The correct answer is Nagaland.

Key Points

• In December 2020, the Ministry of Home Affairs declared the entire State of Nagaland as a ‘disturbed area’ for six more months under the Armed Forces (Special Powers) Act, 1958 (AFSPA).
• The decision has been taken as killings, loots and extortion cases have been going on in various parts of the state.
• AFSPA empowers security forces to conduct operations anywhere and arrest anyone without any prior warrant.
• The Armed Forces Special Powers Act has been in force in Nagaland for several decades.
• Nagaland acquired statehood in 1963.

Additional Information

• AFSPA was passed on 11th September 1958 for civil rights activists from Manipur.
• AFSPA was first enacted to deal with the Naga insurgency in the Assam region.
• The central government enacted the Armed Forces (Punjab and Chandigarh) Special Powers Act in 1983.

The correct answer is Raj Kamal Jha.

Important Points

• Raj Kamal Jha is a newspaper editor and novelist in India.
• He is currently the Editor-in-Chief of The Indian Express.
• He was honoured with the 3rd Rabindranath Tagore Literary Prize’ for his novel ‘The City and The Sea’ in December 2020.
• In September 2021, he was honoured with the Editor of The Year award by the India Chapter of the International Advertising Association.
• He was honoured with the Commonwealth Writers Prize for Best First Book for his work 'The Blue Bedspread' in 2000.
• He was awarded JCB Prize for Literature for his novel The City and The Sea in 2019.
• Notable works of Raj Kamal Jha are:
• The City And The Sea.
• She Will Build Him A City.
• Fireproof.
• Zwischen den Welten.
• If You Are Afraid of Heights.
• The Blue Bedspread.

Additional Information

• Siddhartha Sarma is an Indian English novelist from Assam best known for his novel "The Grasshopper's Run".
• Shekhar Gupta is an Indian journalist and the founder of the online newspaper 'ThePrint'.
• Rajdeep Sardesai is an Indian journalist and author who was honoured with the Padma Shri award in 2008.

The correct answer is 1931.

Key Points

• The Second Round Table Conference was held in London in 1931.
• It was held from 7 September 1931 to 1 December 1931.
• Gandhiji participated in the Second Round Table Conference.
• Gandhi-Irwin Pact is associated with the Second Round Table Conference.
• The Second Round Table Conference was considered a failure due to strong disagreements among the participants.
• Due to the failure of the second round table conference, Gandhiji resumed the civil disobedience movement in 1932.
• All three round table conferences were held in London.

The correct answer is 22.

Key Points

• RTGS stands for Real-Time Gross Settlements.
• RTGS is a system where there is a continuous and real-time settlement of fund transfers, individually on a transaction-by-transaction basis.
• Unique Transaction Reference (UTR) number is a 22 character code used to uniquely identify a transaction in the RTGS system.
• The unique Transaction Reference number is used to identify a specific bank transaction in India.
• All banks in India use UTR numbers for all types of money transfers.
• UTR numbers can recognize and keep an eye on financial transactions.
• NEFT transactions UTR numbers are 16 characters long.
• Example of a UTR number: XXXXRCYYYYMMDD########
• XXXX indicates IFSC (this is the first 4 characters) and is the bank code of the sender.
• R indicates the RTGS system.
• C indicates the transaction channel.
• YYYYMMDD indicates the date of the transaction in this order: year, month, and day.
• ######## indicates the sequence number.

The correct answer is Sudarshan Sahoo.

Key Points

• Sudarshan Sahoo is an Indian sculpture artist from Puri in Odisha.
• He was awarded the Padma Vibhushan in 2021.
• Crafts Museum in Puri was developed by Sudarshan Sahoo.
• It was established in 1977.
• He was honoured with the National Award for stone carving in 1981.
• He was honoured with the Padma Shri in 1988.
• He received the Shilp Guru Award in the year 2003.

Additional Information

• Tarlochan Singh is an Indian politician who was honoured with the Padma Bhushan in 2021.
• Mouma Das is an Indian table tennis player who was honoured with the Padma Shri in 2021.
• Sumitra Mahajan is an Indian politician who was honoured with the Padma Bhushan in 2021.

Gopalpur is NOT correct.

Key Points

• Among the options, only Gopalpur is NOT a town/city on the west coast of India.
• Gopalpur is a city on the east coast of India.
• Gopalpur is a coastal town in Odisha.
• Gopalpur served as an important port for the seafarers of ancient Kalinga during ancient times.
• It was an important military port where soldiers used to embark on a journey to Burma during world war I.
• Gopalpur was the place where the first-ever modern hotel was built.
• Gopalpur port is a deep seaport developed on the banks of the Bay of Bengal.
• Gopalpur port is connected to the Kolkata-Chennai railway line and National Highways.

Additional Information

• Surat is a town/city on the west coast of India located in Gujarat.
• Karwar is a town/city on the west coast of India located in Karnataka.
• Mangalore is a town/city on the west coast of India located in Karnataka.

The correct answer is Convection.

Key Points

• Convection is a mode of heat transfer by the actual motion of matter.
• Convection is a process in which hot, less dense materials rise upward and are replaced by colder, more dense materials.
• Convection is possible only in fluids.
• Convection can be either be natural or forced.
• Gravity plays a major role in natural convection.
• Material is forced to move by a pump or by some other physical means in forced convection.
• Forced-air heating systems in the home, the human circulatory system are the common examples of forced convection.

Additional Information

• The three distinct modes of heat transfer are:

1. Conduction.
2. Convection.
3. Radiation.

• The process of heat transfer without any medium is called radiation.
• Conduction is the process of heat transfer between two adjacent parts of a body due to their temperature difference.
• Condensation is the process where water vapor becomes liquid.

The correct answer is Charan Singh.

Key Points

• Charan Singh was the 5th Prime Minister of India between 1979 and 1980.
• He was the 3rd Deputy Prime Minister of India.
• He became the Deputy Prime Minister of India in the Morarji Desai government in 1977.
• Charan Singh was the first prime minister in India to head a minority government in the center.
• He is one of the the shortest-serving prime minister in India whose tenre was of 170 days.
• Kisan ghat is the resting place of Charan Singh.

Additional Information

• Jagjivan Ram was the fourth deputy prime minister of India who served during the tenure of the Morarji Desai and Charan Singh governments.
• Devi Lal was the sixth Deputy Prime Minister of India who served during the tenure of V. P. Singh and Chandra Shekhar governments.
• Chandrasekhar was the eighth Prime Minister of India between 1990 and 1991.

The correct answer is Impulse.

Key Points

• Impulse is the effect of impulsive force.
• Impulse is the force that changes the momentum of a body,
• Impulse has the same dimension as that of linear momentum.
• Impulse(I) = Force(F) × Time(V).
• Dimensional formulae of force and time are [MLT^-2] and [T] respectively.
• Dimensional formulae of Impulse = MLT^-1 → (A)
• Linear momentum is defined as the product of the mass (m) of an object and the velocity (v) of the object.
• Linear momentum = Mass × Velocity
• Dimensional formulae of mass and velocity are [M] and [LT⁻¹] respectively.
• Dimensional formulae of Linear momentum = MLT^-1 → (B)
• ∴ (A) = (B)
• Dimensional formulae of Linear momentum= Dimensional formulae of Impulse.

The correct answer is Sutlej.

Key Points

• Gurdwara Patal Puri is situated at Rupnagar district in Punjab.
• It is built on the banks of the River Sutlej.
• Guru Hargobind in 1644 as well as Guru Har Rai in 1661 were cremated in Gurdwara Patal Puri.
• River Sutlej originates from Rakshastal Lake.
• Sutlej River is also known as Shatadru (Ancient name).
• River Sutlej is one of the major tributaries of the Indus.
• Sutlej River enters India through Shipki La pass.
• The Bhakra-Nangal Dam is constructed across the Sutlej river.
• The Indira Gandhi canal is situated on the Sutlej river.
• Major dams built across the Sutlej river are:
• Bhakra dam
• Nathpa Jhakri Dam
• Ropar barrage.

Additional Information

• Haridwar, Kanpur, Allahabad, Varanasi, Patna are the major cities situated on the banks of river Ganga.
• Noida, Mathura, Agra, Firozabad are the major cities situated on the banks of the river Yamuna.
• Mandi district in Himachal Pradesh is located on the banks of the river Beas.

The correct answer is Carolina Marin.

Key Points

• Carolina Marin is a professional badminton player from Spain.
• She became the World Champion in the women's singles three times**, 2014, 2015, and 2018.**
• She is the Second female badminton athlete to win three women's singles World Champion titles after Han Aiping of China.
• Carolina Marin won the Toyota Thailand Open Women’s Singles Title in Bangkok in January 2021.
• She defeated Taiwanese's Tai Tzu-Ying in the final.
• She consecutively won the European Championships title five times between 2014 and 2021 (2014, 2016, 2017, 2018, and 2021.)
• She is currently the brand ambassador of football major LaLiga for its promotion in other countries.
• She won the gold medal in 2016 Rio de Janeiro in the Women's singles.

Additional Information

• Ashwini Ponnappa is a professional badminton player from India.
• PV Sindhu is an Indian professional badminton player who won a bronze medal for India in 2020 Tokyo Olympics.
• Tai Tzu-Ying is a Taiwanese professional badminton player who won the gold medal in the 2018 Asian Games.

The correct answer is PM-WANI.

Key Points

• PM WANI stands for Prime Minister Wi-Fi Access Network Interface.
• PM-WANI is a scheme aimed at helping accelerate the uptake of broadband internet services.
• The initiative aims to elevate wireless internet connectivity in the country.
• The PM-WANI scheme was announced by the Department of Telecommunications.
• The government aims to provide internet connectivity through Public Data Offices (PDOs) as a part of the WANI scheme.
• Under this scheme, no license fee will be charged for providing broadband internet services.
• The WANI framework contains the following components:
• Public Data Office (PDO).
• Public Data Office Aggregator (PDOA).
• App Provider.
• Central Registry.

The correct answer is Nephron.

Key Points

• Kidneys are bean-shaped organs in the human body.
• Excretion is the major function of the kidney.
• Kidney act as the filter of the body.
• Nephron is the structural and functional unit of the kidney.
• The cup-shaped part of the nephron is called the bowman's capsule.
• Nephron composed of a renal corpuscle and a renal tubule
• There are around 1,000,000 nephrons in each human kidney.
• The study of kidneys is called nephrology.

Additional Information

• The cortex is the outermost layer of the brain.
• The ureter is a tube that carries urine from the kidney to the urinary bladder.
• Medulla is the terminal part of the brainstem.

The correct answer is energy.

Key Points

• The electron volt is the unit of energy commonly used in atomic and nuclear physics.
• The amount of energy gained by the charge of a single electron moved across an electric potential difference of 1 volt is called an electron volt.
• It is commonly used in atomic and nuclear physics applications.
• The symbol of electron volt is eV.
• 1eV = 1.602×10-19 J
• 1eV = 1.602 × 10−12 erg

Important Points

• Other units of energy are:
• Joule is the SI unit of Energy.
• The unit Joule is named after the English physicist James Prescott Joule.
• Kg⋅m²⋅s^-2 is the SI base unit of Joule.
• The calorie is a unit of energy commonly used in nutrition.

Additional Information

• Watt is the SI unit of Power.
• The unit watt is named after scientist James Watt.
• Volt is the SI unit of potential difference.
• Ampere is the SI unit of current.

The correct answer is Royal Challengers Bangalore.

Key Points

• AB de Villiers is a former professional cricketer from South Africa.
• He holds the record for the fastest ODI century in just 31 balls.
• He played for Royal Challengers Bangalore in IPL 2020.
• Delhi Daredevils(now Delhi Capitals) was is his first franchise in the Indian Premier League.
• He moved to Royal Challengers Bangalore (RCB) in the league's fourth season.

Important Points

• The Royal Challengers Bangalore is a franchise cricket team based in Karnataka.
• It was founded in 2008.
• Royal Challengers Bangalore have won their maiden IPL trophy in 2025.
• Rajat Patidar is the current captain of the Royal Challengers Bangalore.

Additional Information

• Teams that played IPL games 2020 are:

1. Delhi Capitals.
2. Royal Challengers Bangalore.
3. Kolkata Knight Riders.
4. Mumbai Indians.
5. Chennai Super Kings.
6. Punjab Kings.
7. Rajasthan Royals.
8. Sunrisers Hyderabad.

Shortcut TrickWe can use the formula 

(x3 + y3) = (x + y) [(x + y)² - 3xy]

Given x + y = 4 and $\rm \frac{1}{x} + \frac{1}{y} = \frac{16}{15}$ 

After solving we get xy = 15/4

Now, From formula, we get 

x3 + y3 = (4) [(4)2 - 3 (15/4)]

x3 + y3 = 4 [16 - 45/4]

x3 + y3 = 4 [19/4] = 19

∴  (x3 + y3) = 19

Alternate Method

Given:

x + y = 4

$\rm \frac{1}{x} + \frac{1}{y} = \frac{16}{15}$

Formula used:

(x + y)² = x² + y² + 2xy

(x³ + y³) = (x + y)(x² - xy + y²)

Calculation:

$\rm \frac{1}{x} + \frac{1}{y} = \frac{16}{15}$

⇒ (y + x)/xy = 16/15 

⇒ 4/xy = 16/15

⇒ xy = 15/4            (1) 

x + y = 4

By squaring both sides.

x2 + y2 + 2xy = 16 

⇒ x2 + y2 = 16 - 2xy 

⇒  x2 + y2 = 16 - 2 × 15/4  

⇒ x2 + y2 = 16 - 15/2 = 17/2       (2)

∵ (x3 + y3) = (x + y)(x2 - xy + y2)

⇒ (x3 + y3) = 4 × (17/2 - 15/4)

∴ (x3 + y3) = 19

Given:

676xy is divisible by 3, 7 & 11

Concept:

When 676xy is divisible by 3, 7 &11, it will also be divisible by the LCM of 3, 7 &11. 

Dividend = Divisor × Quotient + Remainder

Calculation:

LCM (3, 7, 11) = 231

By taking the largest 5-digit number 67699 and divide it by 231.

∵ 67699 = 231 × 293 + 16

⇒ 67699 = 67683 + 16 

⇒ 67699 - 16 = 67683 (completely divisible by 231)

∴ 67683 = 676xy (where x = 8, y = 3)

(3x - 5y) = 3 × 8 - 5 × 3

⇒ 24 - 15 = 9 

∴ The required result = 9

Given:

ΔABC ~ ΔPQR

Area of ΔABC = 64 cm²

Area of ΔPQR = 81 cm²

PT = 10.8 cm 

AD is the median of ΔABC.

PT is the median of ΔPQR

Concept: 

The ratio of the areas of two similar triangles is equal to the ratio of the squares of the corresponding sides & medians. 

$\text{Area of (ΔABC)}\over{\text{Area of (ΔPQR)}}$ = $({AD\over PT})^2$

Calculation:

$\text{Area of (ΔABC)}\over{\text{Area of (ΔPQR)}}$ = $({AD\over PT})^2$

⇒ $64\over81$ = $({AD\over PT})^2$ 

⇒​ $AD\over PT$ = $\sqrt{64\over81}$ = $8\over9$

⇒​ $AD\over 10.8$ = $8\over9$

⇒​ AD = ${8\times10.8}\over9$ = 9.6 cm

∴ AD = 9.6 cm

Shortcut Trick

Fruits bought at 15 for Rs. 140

Equal quantity of bought at 10 for Rs. 120

Fruits sold at Rs. 132/dozen

Let, the total quantity of fruits = 30

 

Profit percent = (330 - 320)/320 × 100  = $3 \frac{1}{8}$%

∴ The required profit percent is  $3 \frac{1}{8}$%.

Alternate Method

Given:

Fruits at 15 for Rs. 140 = Fruits at 10 for Rs. 120

Fruits sold at Rs. 132/dozen

Formula used:

Profit > Loss 

Profit = SP - CP 

Profit percent = Profit/CP × 100 

Calculation:

Let, Total fruit brought

⇒ LCM (10 and 15) = 30

So, CP of 30 fruits at the rate of 15 for Rs. 140

⇒ 140/15 × 30 = Rs. 280

Similarly, CP of 30 fruits at 10 for Rs. 120,

⇒ 120/10 × 30 = Rs. 360

So, Total CP of 60 fruits = 280 + 360 = Rs. 640

Now,

⇒ SP of 12 fruits = Rs. 132

⇒ SP of 1 fruit = Rs. 11

⇒ SP of 60 fruits = Rs. 11 × 60 = Rs. 660

So, Profit = SP - CP = Rs.660 - Rs. 640 

⇒ Rs. 20 

Profit percent = 20/640 × 100 = $3 \frac{1}{8}$

∴ The required profit percent is  $3 \frac{1}{8}$%.

Calculation:

Total no of cars sold by dealer B in January, April and June = 600 + 580 + 620 = 1800

Total no of cars sold by dealers A and D in March = 628 + 722 = 1350

∴ The required ratio = 1800 : 1350 

⇒ 4 : 3

Given:

$\rm x+ \frac{1}{x} = 4$

Formulas used:

(x + y)2 = x2 + y2 + 2xy

⇒  x2 + y2 = (x + y)2 -  2xy

(x + y)³ = x³ + y³ + 3xy(x + y)

Calculation:

$\rm x+ \frac{1}{x} = 4$

By squaring and cubing both sides 

⇒ x² + 1/x² = 16 - 2 × 1 = 14       (1)

⇒ x³ + 1/x³ = 64 - 3 × 4 = 52       (2) 

(1) × (2) 

(x2 + 1/x2) × (x3 + 1/x3) = 14 × 52  = 728

⇒ x⁵ + 1/x + x + 1/x⁵ = 728 

⇒ x5 + 1/x5 = 728 - 4                   ∵ [$\rm x+ \frac{1}{x} = 4$]

∴ x5 + 1/x5 = 724

Given:

Average of 28 numbers = 77

Average of first 14 numbers = 74

Average of last 15 numbers = 84 

Formula used:

Average = Sum of observations ÷ No of observations

Calculation:

Value of 14th number = (Sum of first 14 numbers +  Sum of last 15 numbers) - Sum of 28 numbers  

⇒ 14th Number = (14 × 74 + 15 × 84 - 28 × 77)

⇒ 1036 + 1260 - 2156 = 140 

Average of remaining 27 numbers = (Sum of 28 numbers - 14th number) ÷ 27 

⇒ (2156 - 140) ÷ 27 = 2016 ÷ 27 

⇒ 74.66 or 74.7

∴ The required result = 74.7 

Alternate Method

Average of 28 numbers = 77

Average of first 14 numbers = 74

Average of last 15 numbers = 84

Deviation on first 14 numbers = 74 - 77 = - 3 × 14 = - 42 

Deviation on last 15 numbers = 84 - 77 = 7 × 15 = 105

14th number = 77 - 42 + 105 = 140

∴ Average of remaining 27 numbers = (28 × 77 - 140) ÷ 27 = 74.7

Follow BODMAS rule to solve this question, as per the order given below, 

Given:

20 ÷ 5 of 8 × [9 ÷ 6 × (6 - 3)] - (10 ÷ 2 of 20) 

⇒ 20 ÷ 40 × [9 ÷ 6 × 3] - (10 ÷ 40 )

⇒ 20 ÷ 40 × [1.5 × 3] - (10 ÷ 40 )

⇒ 0.5 × 4.5 - 0.25 

⇒ 2.25 - 0.25

⇒ 2 

∴ 20 ÷ 5 of 8 × [9 ÷ 6 × (6 - 3)] - (10 ÷ 2 of 20) = 2

Given:

A can complete the work in 24 days

A and B work on alternate days with B beginning the work, can complete the work in $11 \frac{1}{3}$ days

Formula Used:

Total Work = Efficiency × Time

Calculation:

Let the total work be 24 units

⇒ Efficiency of A = 24/24 = 1 unit

According to the question,

A works on 2nd, 4th, 6th, 8th, 10th and (1/3) of 12th day

⇒ A works for 5(1/3) = 16/3 days

⇒ B works for = $11 \frac{1}{3}$ - $\frac{16}{3}$ = $\frac{34-16}{3}$ = 6 days

⇒ Work done by A in 16/3 days = 16/3 units

Remaining work = 24 - (16/3) = 56/3 units

⇒ 56/3 units are completed by B in 6 days

(7/9)th part of 24 = (24 × 7)/9 = 56/3 units

∴ B alone will complete (7/9)th of the original work in 6 days.

Alternate Method

Given:

Time taken by A to finish a task alone = 24 days 

Calculation:

Let the total work be = 1 

A alone can finish the task in 24 days 

⇒ A's one-day work = 1/24 

A and B complete the whole task in = $11 \frac{1}{3}$ days 

A and B work on alternate days, with B beginning so, we can say B will work only 6 days 

⇒ A will work only $11 \frac{1}{3}$ - 6 = $5 \frac{1}{3}$ days 

If A's one day work = 1/24 of work A completes in 1 day

⇒ A's $5 \frac{1}{3}$ days work = 1/24 × $5 \frac{1}{3}$ = 1/24 × 16/3 

⇒ 2/9 

Remaining work = 1 - 2/9 = 7/9 

∴ B does the 7/9th part of the work in 6 days. 

Note-

B, A, B, A, B, A, B, A, B, A, B, A/3

B completely 6-day work

That's why we have taken the 6 days work by B alone.

Given:

8(x + y)3 - 27(x - y)3 = (5y - x)(Ax2 + By2 + Cxy)

Formulas used:

(a + b)³ = a³ + b³ + 3ab(a + b)

(a - b)3 = a3 - b3 - 3ab(a - b)

Calculation: 

8(x + y)3 - 27(x - y)3 = (5y - x)(Ax2 + By2 + Cxy)

⇒ 8[x³ + y³ + 3xy(x + y)] - 27[x3 - y3 - 3xy(x - y)] = 5Ax²y + 5By³ + 5Cxy² - Ax³ - Bxy² - Cx²y

⇒ - 19x3 + 35y3 + 105x2y - 57xy2 = - Ax3 + 5By3 + 5Ax2y - Cx2y + 5Cxy2 - Bxy2 

By comparing the values of A, B & C in LHS & RHS

⇒ - 19x3 = - Ax3 

∴ A = 19 

⇒ + 35y3 = 5By3

⇒ 35 = 5B 

∴ B = 7

⇒ 105x2y = 5Ax2y - Cx2y

⇒ 105x2y = x2y(5A - C)

⇒ 105 = 5A - C 

⇒ 105 = 5 × 19 - C   [A = 19] 

∴ C = - 10 

Now, (A + B - C) = 19 + 7 - (- 10) 

⇒ 26 + 10 = 36 

∴ The required result = 36

 Alternate Method 

Given 

8(x + y)3 - 27(x - y)3 = (5y - x)(Ax2 + By2 + Cxy)

Concept Used 

a3 - b3 = (a - b)(a2 + b2 + ab) 

Calculation 

8(x + y)3 - 27(x - y)3 = (5y - x)(Ax2 + By2 + Cxy)

⇒ [2(x + y)]3 - [3(x - y)]3 = (5y - x)(Ax2 + By2 + Cxy)

⇒ We compare above equation with formula 

So, Ax2 = [2(x + y)]2, By2 = [3(x - y)]2, Cxy = 2(x + y) × 3(x - y)

Now, Let the value of x and y be 1 and -1 respectively 

Then , A = 0 , B = 36 and C =  0

A + B - C = 0 + 36 - 0 = 36

∴ The required answer is 36.

Given:

Side (a) of rhombus = 13 cm 

Length of one diagonal (p) = 24 cm 

Formulas used:

a = (p² + q² )^1/2÷ 2

Area of rhombus = 1/2 × p × q

Calculation:

Let the other diagonal of rhombus = q 

⇒ (24² + q²)^1/2 ÷ 2 = 13

⇒ (576 + q²)^1/2 = 26

⇒ 576 + q² = (26)²

⇒ 676 - 576 = q²

⇒ q = √100 = 10 

Area = 1/2 × 24 × 10 = 120 cm²

∴ The correct answer is 120 cm².

Shortcut Trick

Given: cos² θ / (cot² θ + sin² θ − 1) = 3

Using identity sin² θ − 1 = −cos² θ, denominator becomes cot² θ − cos² θ.

Substituting θ = 60°: cos² 60° = 1/4; cot² 60° = 1/3; sin² 60° = 3/4.

⇒ (1/4) / (1/3 + 3/4 − 1) = (1/4) / (1/12) = 3. Condition satisfied.

Value = tan 60° + cosec 60° = √3 + 2/√3 = (3 + 2)/√3 = 5/√3 = 5√3/3.

∴ The correct answer is 5√3/3.

Alternate Method

Given: cos² θ / (cot² θ − cos² θ) = 3

Formula Used: cot² θ = cos² θ / sin² θ

⇒ cos² θ / [cos² θ (1 / sin² θ − 1)] = 3

⇒ 1 / [(1 − sin² θ) / sin² θ] = 3

⇒ sin² θ / cos² θ = 3

⇒ tan² θ = 3 ⇒ tan θ = √3 ⇒ θ = 60°

⇒ tan 60° + cosec 60° = √3 + 2/√3

⇒ (3 + 2) / √3 = 5 / √3 = 5√3 / 3

∴ The correct answer is 5√3/3.

Additional Information

Trigonometric Identities

Fundamental identities include sin² θ + cos² θ = 1 and 1 + tan² θ = sec² θ.

Specific Values

For θ = 60°, sin = √3/2, cos = 1/2, tan = √3, and cosec = 2/√3.

Rationalization

To remove a radical from the denominator, multiply the numerator and denominator by the radical: a/√b = (a√b)/b.

Given:

ΔABC ~ ΔPQR and $\rm \frac{ar(\Delta ABC)}{ar (\Delta PQR)} = \frac{144}{49}$ 

 AB = 12 cm, BC = 7 cm,  AC = 9 cm

Concept:

The ratio of the areas of two similar triangles is equal to the ratio of the squares of the corresponding sides.

$\text{Area of (ΔABC)}\over{\text{Area of (ΔPQR)}}$ = $({{AC}\over{PR}})^2$

Calculation:

$\text{Area of (ΔABC)}\over{\text{Area of (ΔPQR)}}$ = $({{AC}\over{PR}})^2$ 

⇒ $144\over49$ = $({{AC}\over{PR}})^2$ 

⇒ $AC\over PR$ = $\sqrt{144\over49}$ = $12\over7$

⇒ $9\over PR$ = $12\over7$

⇒ PR = ${9\times 7}\over12$ = $21\over4$ cm

∴ PR = $21\over4$ cm

Given:

Sales of cars by dealer D in May 2018 = 600 

Sales of cars by dealer D in June 2018 = 780 

Calculation:

Percentage increase of sales in June 2018 over May 2018 = (780 - 600) ÷ 600 × 100 

⇒ 30%

The percentage increase of sales of cars by dealer D in July 2018 = Percentage increase of sales in June 2018 over May 2018

⇒ (July sales - June sales) ÷ June sales × 100 = 30%

⇒ (July sales - 780) ÷ 780 × 100 = 30 

⇒ July sales - 780 = 3 × 78 

∴ July sales = 234 + 780 = 1014

Shortcut Trick Percentage increase of sales in June 2018 over May 2018 = 30%

July sales =  June sales × (100 + 30)% 

⇒ 780 × 130% 

⇒ 780 × 130/100 = 1014

Confusion Points Please read the question correctly, the data of July is given separately in the question.

Shortcut Trick

Profit % = 21%.

The relation between Cost Price (CP) and Selling Price (SP) depends on the profit earned.

Multiplier for SP = CP × (100 + Profit%) ÷ 100.

Ratio CP : SP = 100 : (100 + 21) = 100 : 121.

Note: The discount on the printed price is irrelevant for the ratio of CP and SP when profit is given.

∴ The correct answer is 100 : 121.

Alternate Method

Given → Profit % = 21%, Discount % = 21%. Formulas: Profit = SP − CP | SP = CP × (100 + Profit%) ÷ 100 | Ratio = CP : SP.

⇒ Let the Cost Price (CP) of the book be 100 units. ⇒ Since Profit % = 21%, the Selling Price (SP) = 100 + 21% of 100. ⇒ SP = 100 + 21 = 121 units. ⇒ Ratio of CP : SP = 100 : 121. ∴ The correct answer is 100 : 121.

Additional Information

Relation between CP and MP

The ratio of Cost Price to Marked Price is given by CP : MP = (100 − Discount%) : (100 + Profit%).

Profit and Selling Price

Selling Price is always calculated on Cost Price to determine profit, regardless of the discount offered on the Marked Price.

Formula used:

Percentage of sales over production = Sales/ Production × 100

Calculation:

Percentage of sales over production in 2015 = 1000/1250 × 100 = 80%

Percentage of sales over production in 2016 = 1290/1400 × 100 = 92.14%

Percentage of sales over production in 2017 = 1100/1450 × 100 = 75.86%

Percentage of sales over production in 2018 = 1450/1500 × 100 = 96.67%

Percentage of sales over production in 2019 = 1390/1600 × 100 = 86.875%

∴ Clearly, in 2016 and 2018 sales are more than 90% of the respective production.

Given:

When x is subtracted from each of 19, 28, 55 and 91, the numbers so obtained in this order are in proportion

Calculation:

As per the question; 

(19 - x)/(28 - x) = (55 - x)/(91 - x)

⇒ (19 - x) × (91 - x) = (28 - x) × (55 - x)

⇒ 1729 - 19x - 91x + x² = 1540 - 28x - 55x + x²

⇒ 189 = 27x

∴ x = 7

Shortcut Trick

Radius (r) = 25 ÷ 2 = 12.5 cm

Half of the chord (a) = 21 ÷ 2 = 10.5 cm

Using the Pythagorean relation for the perpendicular distance (d):

⇒ d = √(r² − a²)

⇒ d = √(12.5² − 10.5²)

⇒ d = √((12.5 − 10.5) × (12.5 + 10.5)) = √(2 × 23) = √46 cm

∴ The correct answer is √46 cm.

Alternate Method

Given:

Diameter of the circle = 25 cm

Length of the chord = 21 cm

Formula Used:

Radius (r) = Diameter ÷ 2

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

In a right-angled triangle, by Pythagoras theorem: Hypotenuse² = Base² + Perpendicular²

Calculations:

Let O be the centre of the circle and AC be the chord of length 21 cm.

Let OM be the perpendicular from the centre O to the chord AC.

⇒ Radius of the circle (OA) = 25 ÷ 2 = 12.5 cm

⇒ Since OM is perpendicular to AC, it bisects AC.

⇒ AM = MC = AC ÷ 2 = 21 ÷ 2 = 10.5 cm

In right-angled triangle OMA, by Pythagoras theorem:

⇒ OA² = OM² + AM²

⇒ 12.5² = OM² + 10.5²

⇒ 156.25 = OM² + 110.25

⇒ OM² = 156.25 − 110.25

⇒ OM² = 46

⇒ OM = √46 cm

Option Evaluation:

Option (1) √41 is incorrect because 12.5² − 10.5² = 46, not 41.

Option (2) √23 is incorrect because the square of the distance is 46, not 23.

Option (3) √56 is incorrect because 156.25 − 110.25 = 46, not 56.

Option (4) √46 is correct as it perfectly satisfies the Pythagorean relation OM = √46 cm.

∴ The correct answer is √46 cm.

Additional Information

Perpendicular Bisector Theorem

The perpendicular drawn from the centre of a circle to a chord always bisects the chord into two equal halves.

Pythagoras Theorem

In any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides: Hypotenuse² = Base² + Height².

Chord Length Formula

The length of a chord can be calculated as 2 × √(r² − d²), where r is the radius and d is the perpendicular distance from the centre.

Given:

Total no of male teachers in the city = 1650 + 1075 + 1280 + 1170 + 690 = 5865

Total no of female teachers in the city = 2375 + 2651 + 1520 + 1085 + 859 = 8490

Calculation:

∴ Male teachers : Female teachers = 5865 : 8490 

⇒ 391 : 566

Given:

Principal = Rs.8192

Time = $1 \frac{1}{4}$ years or 15 months 

Rate = 15% Per annum 

⇒ Rate as per 5-monthly basis = 5/12 × 15 = 25/4% 

Formulas used:

Compound Interest = Amount - Principal 

Amount = P × (1 + R/100)ⁿ

Calculation:

Time period = 3 (15 months/5)

Rate as per 5-monthly basis = 5/12 × 15 = 25/4% 

Compound Interest = 8192 × (1 + 25/4 ÷ 100)³ - Principal 

⇒ 8192 × (1 + 1/16)³ - 8192 = 8192 [ 17/16 × 17/16 × 17/16 - 1] 

⇒ 8192 [(4913 - 4096) ÷ 4096] 

⇒ 8192 × 817/4096

⇒ 1634 

∴ Compound Interest = Rs.1634

Alternate MethodBy Fraction Method 

Rate = 25/4% =  25/4× 100 = 1/16 

CI is calculated on Principal, so it will be the base (100%) for the interest. 

Rate % = Rate/100 = Interest/Principal 

⇒ Interest/Principal = 1/16 

Interest = Amount - Principal 

⇒ Amount = 1 + 16 = 17 

Time Period = 3 (5-monthly) 

$=\frac{Principal}{Amount}=\left(\frac{16}{17}\right)^{n}=\frac{16^3}{17^3}$

⇒ 8192/ Amount = 4096/4913

⇒ So, if 4096 = 8192,

So, Compound Interest (4913 - 4096) = 817; 

⇒ 817 = 817/4096 × 8192 = 1634

∴ Compound Interest = Rs.1634

Given:

XP = PQ = QY 

Ratio of time taken by boat to go from X to Q and Y to Q = 4 : 5

Formulas used:

Speed of the boat going downstream = (Speed of the boat + Speed of the current)

Speed of the boat going upstream = (Speed of the boat - Speed of the current)

Time = Distance/Speed 

Calculation:

Let XP = PQ = QY = d 

Let the speed of the boat = a km/h

Let the speed of the current = b km/h

As per the question:

2d/(a + b) ÷ d/(a - b) = 4/5 

⇒ 2(a - b)/(a + b) = 4/5 

⇒ 14b = 6a 

⇒ a/b = 7/3

Speed of the boat downstream = (7 + 3) = 10

Speed of the current = 3 

∴ The required ratio = 10 : 3

Given:

$\rm \frac{cosec \ θ}{cosec\ θ -1} + \frac{cosec \ θ}{cosec\ θ +1}- \tan^2 θ , 0^\circ < θ < 90^\circ$

Formulas used:

a² - b² = (a + b)(a - b)

sec2 θ - tan2 θ = 1

cosec² θ - cot² θ = 1  

⇒ cosec2 θ - 1 = cot2​ θ

Calculation:9

$\rm \frac{cosec \ θ}{cosec\ θ -1} + \frac{cosec \ θ}{cosec\ θ +1}- \tan^2 θ$

On taking LCM of (cosecθ - 1) and (cosecθ + 1)

⇒ [cosec θ × (cosec θ + 1) + cosec θ × (cosec θ - 1)]/(cosec θ - 1) (cosec θ + 1) - tan² θ 

⇒ [cosec θ × (cosec θ + 1) + cosec θ × (cosec θ - 1)]/(cosec² θ - 1)  - tan2 θ

⇒ cosec θ [ cosec θ + 1 + cosec θ - 1]/(cosec2 θ - 1) - tan2 θ

⇒ 2cosec² θ/cot2​ θ - tan2 θ

⇒ 2sin2 θ/cos2​ θ × sin2 θ - tan2 θ

⇒ 2sec2 θ - tan2 θ

⇒ sec2 θ + sec2 θ - tan2 θ

⇒ sec2 θ + 1

∴ The required result = sec2 θ + 1

Alternate Method By Value Putting Method

At θ = 30°

LHS:

$\rm \frac{cosec \ θ}{cosec\ θ -1} + \frac{cosec \ θ}{cosec\ θ +1}- \tan^2 θ$

⇒ 2/(2-1) + 2/(2 + 1) - 1/3

⇒ 2 + 2/3 - 1/3 

⇒ 2 + 1/3 = 7/3 

RHS:

Option 1: 1 - tan2 θ ≠ 7/3

Option 2: sec2 θ ≠ 7/3

Option 3: 2sec2 θ ≠ 7/3

Option 4: sec2 θ + 1

⇒ (2/√3)² + 1 

⇒ 4/3 + 1 = 7/3

∴ LHS = RHS

Given:

cot 25°cot 35°cot45°cot 55°cot 65°

Formulas used:

cot (90°- A). cot A = 1

tan (90°- A). tan A = 1

Calculation:

cot 25°cot 35°cot45°cot 55°cot 65°

⇒ cot 25° × cot 35° × cot45° × cot (90° - 35°) × cot (90° - 25°)    [ tan 45° = 1]

⇒ 1 × 1× 1 = 1 

∴ The required result is 1

Shortcut Trick

We can also look for pairs of angles which can give us sum equivalent to 90° and cancel them out directly.

⇒ 25° + 65° = 90° 

And 35° + 55° = 90° 

cot 45° will be equal to 1.

By doing so we will get 1 as our answer.

Given:

Income of A is 45% more than the income of B

Income of C is 60% less than the sum of the incomes of A and B

Income of D is 20% more than that of C

The difference between the incomes of B and D is Rs.13200

Calculation:

Let the income of B = 100 

⇒ Income of A = 100 × 145/100 = 145

Income of C = (100 + 145) × 40/100 = 98

Income of D = 98 × 120/100 = 117.6

The difference between income of D and B = 117.6 - 100 = 17.6

⇒ 17.6 = Rs.13200 

∴ Income of C = 13200 × 98/17.6 = Rs.73500

Given:

∠ADC = 78° and ∠BEC = 52°

Concept:

The sum of a pair of opposite angles is 180° of a cyclic quadrilateral. 

Calculation:

Now, In quadrilateral ABCD,

∠ADC + ∠ABC = 180° (Sum of opposite angles are 180°)

⇒ ∠ABC = 180° - 78° = 102° 

∠EBC = 180° - 102° = 78° [Linear Pair]

In ΔBEC,

∠BEC + ∠EBC + ∠BCE = 180° 

⇒ 52° + 78° + ∠BCE = 180° 

⇒ ∠BCE = 180° - 130° = 50°

⇒ ∠BCE = ∠DCF = 50° (Vertically opposite angle)

∠ADE + ∠FDC = 180° [Linear Pair]

⇒ ∠FDC = 180° - ∠ADE = 180° - 78° = 102°

In ΔDCF,

∠FDC + ∠DCF + ∠DFC = 180° 

102 + 50 + ∠DFC = 180° 

∠DFC = 180° - 152° = 28° 

∠DFC = ∠AFB = 28° 

∴ The measure of ∠AFB is 28°.