SSC CGL 2022 Tier-I Official Paper (Held On : 06 Dec 2022 Shift 3) (Previous Year Paper)

The logic followed here is :- 

1. First letter ⇒ (3) is added to the position value according to English alphabetic series.
2. Second letter ⇒ (7) is subtracted from the position value according to English alphabetic series.
3. Third letter ⇒ (18) is subtracted from the position value according to English alphabetic series.

​So,

Hence, "DCB" is the correct answer.

Given:

• The image must not to be rotated

Hence, the correct answer is "Option - (3)".

Logic :

1. First letter →  (4) is added to the position value according to the English alphabetic series diagonally.
2. Second letter → (3) is added to the position value according to the English alphabetic series diagonally.
3. Third and fourth letter → (2) is added to the position value according to the English alphabetic series diagonally.

​​So,

• ​OILS: UNLS

​

​And,

• PAIN : PKDT

​​

​Similarly,

• QUIK: ?

​

Hence, "MKXU" is the correct answer.

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

Conclusions:

I. Some javascript are roblox → Does not follow (As all javascript are binary, all binary are minecraft and all roblox are minecraft. So, it is possible but not definitely true)

II. Some javascript are minecraft → Follow (As all javascript are binary and all binary are minecraft. So, as whole javascript comes in binary which whole comes in minecraft therefore it is definitely true)

III. All javascript are minecraft → Follow (As all javascript are binary and all binary are minecraft. So, as whole javascript comes in binary which whole comes in minecraft therefore it is definitely true)

∴ Here, Only conclusions II and III follow.

Hence, the correct answer is "Only conclusions II and III follow".

The figure that will replace the question mark (?) in the following figure series is shown below:

1) For '

' element : 

• Rotating in 90° the anti-clockwise direction
• Also alternately flipping of the arrows vertically.

​​

Final series is :

Hence, ‘option - (1)’ is the correct answer.

The correct mirror image of the given combination when mirror is placed at AB is as shown below:

Hence, the correct answer is "Option - (3)".

The logic followed here is :- (1) is subtracted from the position value according to English alphabetic series.

​So,​

Hence, the correct answer is "XSNI".

Given statement : 10 - 4 ÷ 20 + 16 × 8 = 16

According to the question, after interchanging of numbers and signs :

So,

• Option - (1) : 10 and 16, × and ÷

16 - 4 × 20 + 10 ÷ 8 = 16

16 - 4 × 20 + 1.25 = 16

16 + 1.25 - 80 = 16

17.25 - 80 = 16

-62.75 ≠ 16 (LHS ≠ RHS)

• Option - (2) : 4 and 8, × and -

10 × 8 ÷ 20 + 16 - 4 = 16

10 × 0.4 + 16 - 4 = 16

4 + 16 - 4 = 16

20 - 4 = 16

16 = 16 (LHS = RHS)

• Option - (3) : 4 and 20, + and ÷

10 - 20 + 4 ÷ 16 × 8 = 16

10 - 20 + 0.25 × 8 = 16

10 + 2 - 20 = 16

12 - 20 = 16

-8 ≠ 16 (LHS ≠ RHS)

• Option - (4) : 10 and 20, + and ×

20 - 4 ÷ 10 × 16 + 8 = 16

20 - 0.4 × 16 + 8 = 16

20 + 8 - 6.4 = 16

28 - 6.4 = 16

21.4 ≠ 16 (LHS ≠ RHS)

Hence, "Option - (2)" is the correct answer.

The logic followed here is : (2^nd number × 5) + 2 = 1^st number

So,

• 47 : 9

​9 × 5 + 2 = 47

45 + 2 = 47

47 = 47 (LHS = RHS)

And,

• 92 : 18​

​18 × 5 + 2 = 92

90 + 2 = 92

92 = 92 (LHS = RHS)

Similarly,

• ? : 14​

​14 × 5 + 2 = ?

70 + 2 = ?

72 = ? 

Hence, "72" is the correct answer.

Given statement : [{(32 × 20) - (2 ÷ 3)} + (2 - 4)] ÷ 3

According to the question, after interchanging the signs :

​⇒[{(32 - 20) + (2 × 3)} ÷ (2 + 4)] × 3

⇒[{12 + 6} ÷ 6] × 3

⇒[18 ÷ 6] × 3

⇒3 × 3

⇒9

Hence, "9" is the correct answer.

The figure that will replace the question mark (?) in the following figure series is shown below:

1) For '

' element : 

2) For '

' element : 

3) For '

' element : 

4) For '

' element : 

5) For '

' element : Rotating in the anti-clockwise direction.

​​

Final series is :

Hence, ‘option - (1)’ is the correct answer.

The logic followed here is : Multiplied by a number series starting with (1) and added (2) subsequently.

Hence, "60" is the correct answer.

According to the question, after Interchanging the given two numbers :

• 1 and 7

So,

• Option - (1) : 7 × 2 + 1 - 5 = 6

1 × 2 + 7 - 5 = 6

2 + 7 - 5 = 6

9 - 5 = 6

4 ≠ 6 (LHS ≠ RHS)

• Option - (2) : 1 × 2 - 7 + 8 ÷ 2 = 16

7 × 2 - 1 + 8 ÷ 2 = 16

7 × 2 - 1 + 4 = 16

14 + 4 - 1 = 16

18 - 1 = 16

17 ≠ 16 (LHS ≠ RHS)

• Option - (3) : 7 + 1 × 2 - 6 ÷ 3 = 13

1 + 7 × 2 - 6 ÷ 3 = 13

1 + 7 × 2 - 2 = 13

1 + 14 - 2 = 13

15 - 2 = 13

13 = 13 (LHS = RHS)

• Option - (4) : 7 + 6 ÷ 3 - 5 × 1 = - 10

1 + 6 ÷ 3 - 5 × 7 = - 10

1 + 2 - 5 × 7 = - 10

1 + 2 - 15 = -10

3 - 15 = -10

-12 ≠ -10 (LHS ≠ RHS)

Hence, "Option - (3)" is the correct answer.

The logic followed here is : (1^st number × 2^nd number) = 3^rd number

So,

• Option - (1) : (18, 3, 54)

​18 × 3 = 54

54 = 54 (LHS = RHS)

• Option - (2) : (14, 5, 70)

​14 × 5 = 70

70 = 70 (LHS = RHS)

• Option - (3) : (22, 7, 154)

​22 × 7 = 154

154 = 154 (LHS = RHS)

• Option - (4) : (26, 5, 132)

​26 × 5 = 132

130 ≠ 132 (LHS ≠ RHS)

Hence, "Option - (4)" is the correct answer.

Given :

Statement given : D # E # F & G @ H # I

So,

• D # E → E is the daughter of D.

• E # F → F is the daughter of E.

• F & G → F is the mother of G.

• G @ H → G is the husband of H.

• H # I → I is the daughter of H.

​

Thus, according to the final family tree G is the maternal grandson of E i.e Daughter’s son.

Hence, "Daughter’s son" is the correct answer.

Given :

So,

• Option - (1) : P - R × Q → P is the mother of R who is the brother of Q.

Here, Q's relationship with P cannot be determined, as Q can be either son or daughter of P.

• Option - (2) : Q - P × R → Q is the mother of P who is the brother of R.

Here, Q is the mother of P.

• Option - (3) : P - Q - R → P is the mother of Q who is the mother of R.

Here, Q is the daughter of P.

• Option - (4) : P - Q × R → P is the mother of Q who is the brother of R.

​

Here, Q is the son of P.

Thus, out of all options only option - (4) shows that Q is the son of P.

Hence, "Option - (4)" is the correct answer.

According to the sequence in the dictionary:

4. Oblate
5. Oblige
6. Oblique
7. Oblivion
8. Oblong

Hence, the correct option is "Oblique".

The logic followed here is :- 

• First and second letter - Difference between the position value according to English alphabetic series is (1).
• Second and third letter - Difference between the position value according to English alphabetic series is (3).

​So,

• Option - (1) : IJM

​

• Option - (2) : ABE

​

• Option - (3) : XYA

​

• Option - (4) : EFI

​

Thus, out of all the options 'XYA' is the odd one out as the difference between first and the second letter and difference between second and third letter is (1) and (2) respectively.

​Hence, "XYA" is the correct answer.

The pattern followed here is : (Square of 1^st number + 6) = 2^nd number

So,

• Option - (1) : 20 - 406

​$(20)^2$ + 6 = 406

400 + 6 = 406

406 = 406 (LHS = RHS)

• Option - (2) : 14 - 200

​$(14)^2$ + 6 = 200

196 + 6 = 200

202 ≠ 200 (LHS ≠ RHS)

• Option - (3) : 10 - 106

​$(10)^2$ + 6 = 106

100 + 6 = 106

106 = 106 (LHS = RHS)

• Option - (4) : 12 - 150

​$(12)^2$ + 6 = 150

144 + 6 = 150

150 = 150 (LHS = RHS)

Hence, "Option - (2)" is the correct answer.

The logic followed here is :

• The area where cricket is played is called pitch.

Similarly,

• The area where badminton is played is called court.

Hence, "Court" is the correct answer.

Additional Information

1. Track ⇒ The area where athletics is played is called track.
2. Field ⇒ The area where various sports takes place.
3. Pool ⇒ The area where swimming is done is called pool.

The logic followed here is : Square of 1^st number - (1^st number × 2) = 2^nd number

So,

• 12 : 120

​$(12)^2$ - (12 × 2) = 120

144 - 24 = 120

120 = 120 (LHS = RHS)

​And,

• 20 : 360

​$(20)^2$ - (20 × 2) = 360

400 - 40 = 360

360 = 360 (LHS = RHS)

Similarly,

• 3 : ?

​$(3)^2$ - (3 × 2) = ?

9 - 6 = ?

3 = ?

​Hence, "3" is the correct answer.

The logic followed here is :-  (3) is added to the position value according to English alphabetic series diagonally.

​So,

• 'GLAND’ is written as 'GQDOJ'

​

And,

• 'MEANS' is written as 'VQDHP'

​

​Similarly,

• 'CYSTS' is written as

​

Hence, "VWVBF" is the correct answer.

The faces opposite to each other are shown below:

⇒ As the alternate position becomes opposite to each other and the remaining two faces become opposite to each other.

So, the opposite pairs are:

• 1 → 3
• 2 → 4
• 5 → 6

Therefore, according to the question :

• The given figure A →  

⇒  Cannot be formed as a cube from the given sheet (As 5 and 6 are one of the opposite pairs)

• The given figure B →  

⇒ Cannot be formed as a cube from the given sheet (As 5 and 6 are one of the opposite pairs)

• The given figure C →  

⇒  can be formed as a cube from the given sheet (As 2, 3 and 6 are one numbers of each opposite pairs)

• The given figure D →  

⇒  Can be formed as a cube from the given sheet (As 1, 4 and 6 are one numbers of each opposite pairs)

​Here, Only figure C and D can be formed as a cube from the given sheet.

Hence, "Only C and D"  is the correct answer.

The logic followed here is : The number of letter in a word ⇒ Code 

Given :

• ‘birds are flying’ is coded as '536'

​birds → It is a five letters word, therefore the code is ⇒ 5

are → It is a three letters word, therefore the code is ⇒ 3

flying → It is a six letters word, therefore the code is ⇒ 6

• ‘the nest is big’ is coded as '3423'

​the → It is a three letters word, therefore the code is ⇒ 3

nest → It is a four letters word, therefore the code is ⇒ 4

is → It is a two letters word, therefore the code is ⇒ 2

big → It is a three letters word, therefore the code is ⇒ 3

And,

• ‘close the door’ is coded as '534'

​close → It is a five letters word, therefore the code is ⇒ 5

the → It is a three letters word, therefore the code is ⇒ 3

door → It is a four letters word, therefore the code is ⇒ 4

Similarly,

• ‘complete your work’ is coded as

​​complete → It is a eight letters word, therefore the code is ⇒ 8

your ​close → It is a four letters word, therefore the code is ⇒ 4

work ​close → It is a five letters word, therefore the code is ⇒ 4

Thus, code for ‘complete your work’ is '844'.

Hence, "844" is the correct answer.

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

Conclusions:

I. No D is A → Follow (As no A is D. So, as direct negative or cross relation is given therefore it is definitely true)

II. Some E are D → Follow (As all D are E. So, as whole D comes in E therefore it is definitely true)

∴ Here, Both conclusions I and II follows.

Hence, the correct answer is "Both conclusions I and II follows".

The correct answer is 2.

Key Points

• The Summer and Winter Olympic Games organized by the International Olympic Committee happen every four years.
• The Summer Olympics, more popularly known as just The Olympics is a much larger event with around 206 countries participating.
• The Winter Olympics is an event on a somewhat smaller scale, with around 91 countries participating.
• Even though these events occur every four years, the summer and winter events are arranged so that there's an Olympic event every two years.
• The Summer Olympics are held every leap year like 2016, 2020, and 2024.
• The Winter Olympics are held 2 years after the leap year like 2014, 2018, and 2022.

Additional Information

• The 2020 Olympics held in Tokyo in 2021.
• The 2022 Winter Olympics in China.
• In December 2020, it was announced that breakdancing, sport climbing, skateboarding and surfing were included as additional sports on the program for the 2024 Olympic Games in Paris
• Upcoming Olympics Events and host countries
• Paris- 2024 Summer Olympics
• Milan–Cortina d'Ampezzo - 2026 Winter Olympics
• Los Angeles - 2028 Summer Olympics
• Brisbane - 2032 Summer Olympics

The correct answer is 1757.

Key Points

• The East India Company forces led by Robert Clive engaged Siraj-Ud-Daulah in the Battle of Plassey.
• Siraj-Ud-Daulah was the Nawab of Bengal.
• Siraj was enraged by the widespread abuse of trade privileges by EIC authorities.
• The battle of Plassey in 1757 was brought on by EIC's continued wrongdoing against Siraj-Ud-Daulah.
• About 160 kilometers north of Calcutta, close to the little settlement of Plassey (Palashi), the forces came together on the banks of the Bhagirathi-Hooghly River.
• On June 23, 1757, the Battle of Plassey was fought.
• Robert Clive's army of 3,000 troops destroyed Siraj-Ud-army Daulah's of 50,000 warriors, 40 cannons, and 10 war elephants.
• In 11 hours, the battle was over, and Siraj-Ud-Daulah fled after losing.

Additional Information 

• The Battle of Plassey was fought in north-eastern India on 23 June 1757.
• Troops of the British East India Company, led by Robert Clive, came up against the forces of Siraj-ud-Daulah, the last Nawab of Bengal, and his French allies.
• Clive's victory eventually led to the British becoming the greatest economic and military power in India.
• The Battle of Plassey had political significance for it laid the foundation of the British Empire in India.; it has been rightly regarded as the starting point of British rule in India.
• Result of Battle of Plassey-
• As a result of this victory, Mir Jafar became the Nawab of Bengal. He gave large sums of money plus the zamindari of 24 Parganas to the English.
• English rivals, the French were ousted.

The correct answer is Ulaanbaatar, Mongolia.

Key Points

• The 35^th Asian Wrestling Championships 2022 was held in Ulaanbaatar, Mongolia.
• Indian wrestlers secured a total of 17 medals, which includes 1 Gold, 5 Silver, and 11 Bronze medals.
• Ravi Kumar Dahiya is the only Gold medalist representing India, in the men’s 57kg freestyle category, defeating Kazakhstan’s Rakhat Kalzhan on technical superiority
• Ravi Kumar evolved the first Indian to win 3 Gold medals in Asian Wrestling Championships which includes 2020 in New Delhi, India, 2021 in Almaty, Kazakhstan and 2022 in Ulaanbaatar, Mongolia respectively.

Additional Information

• Ravi Kumar Dahiya won the silver medal in 57kg men’s freestyle wrestling at the Tokyo Olympics.
• The Asian Wrestling Championships 2023 will be held in Astana, Kazakhstan.
• India will be hosting the Asian Wrestling Championships for the 9^th time.
• The 1^st edition of the Asian Wrestling Championship was held in 1979 in Jalandhar, India.

The correct answer is social welfare benefits.

Key Points

• Shri Manoj Joshi, Secretary, Ministry of Housing and Urban Affairs (MoHUA) launched the ‘SVANidhi se Samriddhi’ program.
• ‘SVANidhi se Samriddhi’, an further program of PMSVANidhi was launched on 4^th January 2021 in 125 cities in Phase 1, covering approximately 35 Lakh Street vendors.
• MoHUA has been executing Prime Minister Street Vendors AtmaNirbhar Nidhi (PM SVANidhi), a Central Sector Scheme since 1^st June 2020. The scheme strives to provide an affordable working capital loan to Street Vendors
• ‘‘SVANidhi se Samriddhi’ program was started to provide social safety benefits to street vendors for their holistic development and socio-economic upliftment.

Additional Information

• Union Minister of MoHUA - Shri Hardeep Singh Puri
• Other Important schemes launched by Central and States Govt.
• AYURSWASTHYA Yojana - - Ministry of Ayush
• Ladli Laxmi 2.0 - Madhya Pradesh
• Friends of Library - Tamil Nadu
• One District One Sport - Uttar Pradesh

The correct answer is Uttarakhand.

Key PointsConiferous trees

• In mountainous areas, the decrease in temperature with increasing altitude leads to the corresponding change in natural vegetation.
• As such, there is a succession of natural vegetation belts in the same order as we see from the tropical to the tundra region.
• The wet temperate type of forest is found between a height of 1000 and 2000 metres.
• Evergreen broad-leaf trees, such as oaks and chestnuts predominate.
• Between 1500 and 3000 metres, temperate forests containing coniferous trees, like pine, deodar, silver fir, spruce and cedar, are found.
• These forests cover mostly the southern slopes of the Himalayas, places having high altitudes in southern and northeast India.
• At higher elevations, temperate grasslands are common.
• At high altitudes, generally, more than 3,600 metres above sea level, temperate forests and grasslands give way to the Alpine vegetation.
• Silver fir, junipers, pines and birches are the common trees of these forests.
• However, they get progressively stunted as they approach the snowline. Ultimately, through shrubs and scrubs, they merge into the Alpine grasslands.
• These are used extensively for grazing by nomadic tribes, like the Gujjars and the Bakarwals.
• At higher altitudes, mosses and lichens form part of tundra vegetation.

The correct answer is Both I and II.

Key Points

• Indicators are the substances that are used to identify whether the given substance is an acid or a base.
• There are two types of indicators:
• Natural: These indicators are obtained from natural sources like plants.
• Turmeric, litmus, china rose petals, etc., are some of the naturally occurring indicators. Hence, option 4 is correct.
• The most commonly used natural indicator is litmus.
• It is extracted from lichens.
• It is available in the form of a solution, or in the form of strips of paper, known as litmus paper.
• Generally, it is available as red and blue litmus paper.
• Man-made: Phenolphthalein, Methyl orange are some of the man-made indicators.

Important Points

• The indicators change their color when added to a solution containing an acidic or a basic substance.
• Turmeric is a natural indicator. It contains the substance curcumin which changes color with pH.
• When a substance is basic, the color is red, whereas when the substance is acidic, the color of turmeric turns yellow to brown.
• Films containing aqueous hibiscus extract were developed as a natural pH indicator.
• Natural polymers (chitosan, gelatine, and starch) were used for the film production.
• Great color changes from pink (pH = 2.0) to yellow (pH = 13.0) were observed.
• Acids turn blue litmus red. Bases do not change the color of blue litmus.
• Acids do not change the color of red litmus. Bases turn red litmus blue.

Additional Information 

Some indicators and their color change are given in the table below:

The correct answer is Both I and II. 

Key Points

• Namaste is a Central Govt. Scheme of the Ministry of Social Justice and Empowerment (MoSJE) as a joint initiative of the MoSJE and the Ministry of Housing and Urban Affairs (MoHUA).
• NAMASTE emphasises the safety and dignity of sanitation workers in urban India.
• This scheme aims to provide sustainable livelihood and enhancing their occupational safety through capacity building and improved access to safety gear and machines.

Additional Information

• NAMASTE aims to achieve the following outcomes:
• Zero death in sanitation work in India.
• All sanitation work is to be performed by skilled workers.
• No sanitation workers come in immediate contact with human faecal matter.
• Sanitation workers are collectivized into SHGs and are authorised to run sanitation enterprises.
• All Sewer and Septic tank sanitation workers have access to alternative livelihoods.
• Strengthened supervisory and monitoring procedures at national, state and ULB levels to ensure enforcement and monitoring of safe sanitation work.
• Increased awareness amongst sanitation services seekers to seek services from registered and skilled sanitation workers.

The correct answer is Tanvi Shah.

Key Points

• Tanvi Shah is the first female from India to win a Grammy award.
• She hails from Tamil Nadu and was born on 1 December 1985.
• She has sung in Telugu, Tamil, Hindi, Spanish, Portuguese, and Arabic.
• She shared the Grammy Award for Best Song written for visual media with AR Rehman and Gulzar at the 52^nd Grammy Awards for the song Jai Ho for writing Spanish lyrics for the song.
• Other awards won by her are the BMI award in 2009, World Sound Track Award in 2009.

Additional Information

• The Grammy Award is an award presented by the US Recording Academy to recognize outstanding achievements in the music industry of the United States.
• On May 4, 1959, the very first Grammy Award ceremony was held to respect and honour the musical accomplishments of performers for the year 1958.
• The first Indian to receive a Grammy Award was Ravi Shankar in 1968 for 'West Meets East in the category of Best Chamber Music Performance

The correct answer is 763.5.

• The total number of literate persons in a given age group is expressed as a percentage of the total population in that age group.
• The adult literacy rate measures literacy among persons aged 15 years and above, and the youth literacy rate measures literacy among persons aged 15 to 24 years.
• Many families, especially in rural areas believe that having a male child is better than having a baby girl.
• So the male child gets all the benefits.
• India's literacy rate is at 74.04%.
• According to the census 2011, the number of literates in India is 763.5 million. Hence, option 4 is correct.
• The highest literacy rate is of Kerala with a 94.00 ​% literacy rate whereas Bihar is the least literate state in India, with a literacy of 61.80 %.

The correct answer is To confer certain unique phenotypic characteristics to bacteria.

Key Points

• Plasmid
• The plasmid is an extrachromosomal genetic element used in genetic engineering that occurs in many bacterial strains.
• Plasmids are circular deoxyribonucleic acid (DNA) molecules that replicate independently of the bacterial chromosome.
• They are not essential for the bacterium but may confer a selective advantage.
• One class of plasmids, carcinogenic (or Col ) factors, determines the production of proteins called colicins, which have antibiotic activity and can kill other bacteria.
• Another class of plasmids, R factors, confers upon bacteria resistance to antibiotics.
• Some Col factors and R factors can transfer themselves from one cell to another and thus are capable of spreading rapidly through a bacterial population.
• A plasmid that is attached to the cell membrane or integrated into the bacterial chromosome is called an episome.

Additional Information

• DNA molecule is packaged into thread-like structures called chromosomes.
• They are present in the nucleus of each cell.
• Each chromosome is made up of DNA tightly coiled around many times around proteins called histones that support its structure.
• Chromosomes are made up of a DNA and protein complex found in the nucleus of eukaryotic cells.
• It is made of proteins and a single molecule of deoxyribonucleic acid (DNA).
• The DNA and proteins together form chromatin fibres, which gets further condensed into chromosomes.
• This condensation or packaging of the DNA helps to accommodate long DNA molecules inside the nucleolus.

The correct option is 1989.

Key Points

• The Election Commission was a single-member body when it was set up in 1950 till October 1989, with only the Chief Election Commissioner as its only member.
• After the Election Commissioner Amendment Act 1989, it has been made a multi-member body.
• Since October 1989 up to today, it is a three-member commission with one Chief Election Commissioner and two election commissioners.
• Prompted by the Supreme Court's observation in S. S. Dhanoa's case and also in the wake of certain controversial decisions taken by the CEC resulting in a serious confrontation between the Commission and the Government of India, the latter decided to provide for a Multi-member Election Commission.

Additional Information

• The Election Commission of India (Articles 324 to 329) was formed on 25 January 1950;
• The body administers elections to the Lok Sabha, Rajya Sabha, state Legislative Assemblies, state legislative Councils, and the offices of the President and Vice President of the country.
• Chief Election Commissioner of India is Gyanesh Kumar

The correct answer is Rashtrakutas.

Key Points

• The Rashtrakuta Dynasty ruled parts of South India from the 8^th to the 10^th century CE.
• The Rashtrakutas were originally known to be the feudatories of the Western Chalukyas of Vatapi.
• They ruled from Manyakheta in Karnataka.
• Krishna III was the last greatest in the Rashtrakuta dynasty.
• Krishna III defeated Parantaka Chola in the battle of Takkolam.
• Dantidurg was the founder of the Rashtrakuta dynasty, he defeated Chalukyas of Badam.

Additional Information

• Chera
• The Cheras controlled the central and northern parts of Kerala and the Kongu region of Tamil Nadu
• Vanji was the capital of the Chera Kingdom.
• The Cheras, Cholas and Pandyas were the most powerful three kingdoms to have ruled during the Sangam Age,
• Pala
• The Pala empire was founded by Gopala
• The Palas were staunch supporters of Mahayana Buddhism.
• The Pala period is also known as a ‘Golden Era’ in Bengali history.
• Gurjara-Pratihara
• The kingdom was founded by Nagabhatta I.
• Bhoja was the Pratihara dynasty's greatest emperor and the actual founder of the empire.
• The Pratiharas who ruled over Kannauj.

The correct answer is Mrinalini Sarabhai.

Key Points

• Prominent Kathakali classical dancer Mrinalini Sarabhai was awarded the maiden 'Nishagandhi Puraskaram' of Kerala Tourism.
• The annual award, containing a purse of Rs. 1.5 lakh, citation and statuette, is to honour eminent personalities in the fields of dance and music.
• She was educated at Shantiniketan under the supervision of Rabindranath Tagore.
• She was honoured with Padma Bhushan in 1992 and Padma Shri in 1965.
• She also received the Kerala Sangeetha Nataka Akademi Fellowship in 1988.

Additional Information

• Madhavi Mudgal is an Indian classical dancer known for the Odissi dance style.
• Shobana Chandrakumar Pillai is an Indian actress and Bharatanatyam dance
• Uday Shankar was a dancer and choreographer from India who adapted Western theatrical techniques to conventional Hindu dance.
• Some other famous artists of different classical dance
• Bharatanatyam - Sonal Mansingh, Bala Saraswati, Narthaki Natraj, CV Chandrasekhar, Leela Samson, Mrinalini Sarabhai, Padma Subramanyam, Rukmini Devi, Yamini Krishnamurti.
• Kathak - Shobhna Narayan, Bharti Gupta, Birju Maharaj, Durga Das, Gopi Krishna, Sambhu Maharaj, Sitara Devi.
• Kuchipudi - Raja, Kaushalaya Reddy, Siddhendra Yogi, Josyula Seetharamaiah, Vempathi Chinna Sathyam, Yamini Krishnamurthy, Ragini Devi.
• Mohiniattam - Protima Bedi, Smitha Rajan, Sunanda Nair, Narayana Panikker, Gopika Verma, Pallavi

The correct answer is Sattriya.

Key Points

Sattriya dance

• Satriya Nritya is the classical dance form of Assam which represents the Satriya culture, the basis of the religious and cultural fabric of Assam.
• It is a dance-drama performance art with origins in the Krishna-centered Vaishnavism monasteries of Assam and attributed to the 15^th-century Bhakti movement scholar and saint Mahapurush Srimanta Sankardev.
• Dance historian Dr. Sunil Kothari has recently been bestowed with the Madhabdev Award by the Government of Assam for popularising the Sattriya dance.
• Sattriya was given the status of classical dance in the year 2000 by the Sangeet Natak Akademi.

Additional Information

The correct answer is Uday Shankar.

Key Points

• Uday Shankar was an Indian dancer and choreographer.
• He is well renowned for infusing elements of Indian classical, folk, and tribal dance into a fusion dance style by adopting European theatrical techniques to Indian classical dance.
• He received the Sangeet Natak Akademi Fellowship in 1962 for his lifetime achievements.
• He received the Padma Vibhushan from the Indian government, in 1971.

Additional Information

• Rajee Narayan:
• Bharatanatyam dancer, pianist, and composer Rajee Narayan resided in Mumbai, India.
• Sangeeta Shastra Mala is a book she published that covers the fundamentals of Carnatic music.
• Devjani Chaliha:
• In addition to teaching at Rabindra Bharati University, Debjani Chaliha is a Manipuri dancer.

The correct answer is 14.

Key Points 

• On the 26^th of August 2009, the Indian Parliament passed a historic act known as the ‘Right to Education Act’ or RTE Act but the RTE Act was implemented in India with effect from 1^st April 2010.
• It is an act that aims to provide free and compulsory education to all children between the age of 6 to 14 years in India under Article 21A of the Indian Constitution.

Additional Information

Features of the RTE Act - 2009 under Article 21A: 

• Prohibition of private tuition by teachers to equalize educational opportunities to all.
• Reservation of  25% seats for children belonging to economically weaker sections in private schools.
• 45 hours including preparation hours as the minimum number of working hours per week for the teacher.
• Not to deploy teachers in non-academic work except elections, population census, and disaster relief duties.
• Establishment of primary and upper primary schools within the distance of 1km and 3kms respectively from child's house.

The correct answer is Speak about elaborate surgical procedures.

Key Points

• One of the most well-known mathematicians from India is Aryabhata.
• In Kusumapura, Pataliputra, he was born in the Gupta period, or during the Gupta Dynasty's control, in 475 CE.
• He was renowned for having an exceptional understanding of the subject of astronomy.
• Aryabhata properly estimated pi's value to the nearest two decimal places, which is 3.14.
• Rightfully insisting that the earth revolves around the sun on a daily basis was Aryabhatta.
• Additionally, he believed that the apparent movement of the stars was produced by the relative motion brought on by the earth's rotation.
• Aryabhata used scientific experiments to explain lunar and solar eclipses.
• He said that the moonshine and planets are caused by sunlight that is reflected.
• He described eclipses in terms of shadows that are cast on the planet.

Additional Information

• Sushruta (c. 7^th or 6^th century BCE) was a physician in ancient India known today as the “Father of Indian Surgery” and “Father of Plastic Surgery” for inventing and developing surgical procedures.
• He spoke about elaborate surgical procedures.
• The definition of an ideal surgeon according to the great surgeon Sushruta is “A person who possesses courage and presence of mind, a hand free from perspiration, tremor less grip of sharp and good instruments and who carries his operations to the success and advantage of his patient who has entrusted his life to the surgeon. The surgeon should respect this absolute surrender and treat his patient as his own son.”

The correct answer is Africa.

Key Points

• The Sahara Desert is located in Africa.
• It is the largest desert in the world.

Additional Information

• All northern Africa is filled with this dessert.
• The total area of the Desert is somewhere about 3,320,000 square miles.
• The Desert expands and contracts, so the area of the Desert varies.
• In the west, the Desert is bordered by the Atlantic Ocean.
• The Atlas Mountains and the Mediterranean Sea border the Desert in the north.
• The Red Sea bounds the Mountain in the East.
• Atop the African shield, the Sahara sits.
• Denuded pre-Cambrian rocks make the African shield.
• Mesozoic deposits cover the Sahara Desert.

The correct answer is Moneylenders.

Key Points

• Today's institutional framework for rural banking comprises of a number of multi-agency entities, including
• commercial banks,
• Regional Rural Banks,
• co-operatives, and
• land development banks.
• Self-Help Groups (SHGs) have recently appeared to close the gap in the established credit system.

Additional Information

• Regional rural banks:
• In 1975, the Regional Rural Banks (RRBs) were founded.
• It was founded in accordance with the terms of the 1976 Regional Rural Banks Act and the Ordinance published on September 26, 1975.
• RRBs are financial organizations that guarantee sufficient credit for rural industries like agriculture.
• Commercial banks:
• A commercial bank is a type of financial organization that offers its clients services including loans, CDs, savings accounts, overdraft protection, etc.
• Cooperative banks:
• A co-operative bank is a diminutive financial institution whose owners and clients are its members.
• They are registered under the States Cooperative Societies Act and are governed by the Reserve Bank of India (RBI).
• Land development banks:
• The LDB's primary goals are to advance agricultural development and productivity as well as land use.
• The LDB provides long-term finance to members directly through its branches.

The correct answer is AG Tansely.

Key Points

• An early proponent of the field of ecology is the English botanist AG Tansely.
• In biology, he introduced the idea of an ecosystem.
• Tansley was knighted in 1950 and elected a Member of the Royal Society in 1915.
• Regular Tansley Reviews are published by The New Phytologist, and the New Phytologist Trust bestows the Tansley Award in his honor.

Additional Information

• Lindeman :
• Ecologist Raymond Laurel Lindeman was a pioneer.
• His graduation work on the subject of trophic dynamics is regarded as a foundational study in the field of ecosystem ecology.
• Turreson:
• Turreson was a pioneer in ecological genetics and an evolutionary botanist from Sweden.

The correct answer is Six.

Key Points

• Cations and anions are combined to form ionic molecules. \
• In sodium chloride, the cation is sodium (Na+) and the anion is chloride (Cl-).
• The anion (Cl-) and the cation (Na+) in the crystal structure alternate back and forth in a predictable pattern.
• The cubic lattice of sodium ions and a related cubic lattice of chlorine ions are entangled in the crystal structure of sodium chloride.
• Thirteen sodium ions and fourteen chloride ions make up each unit cell of sodium chloride.
• Eight unit cells share an ion at a corner, two unit cells share one at the face's center, and four unit cells share one at the edge.
• Each sodium ion is surrounded by six chloride ions, and each chloride ion is surrounded by six sodium ions.
• The sodium chloride unit cell contains four sodium ions and four chloride ions.

Additional Information

• The symmetrical three-dimensional structural groupings of atoms, ions, or molecules inside a crystalline solid as points are known as the crystal lattice.
• It can be characterized as the spatial geometrical arrangement of the atoms, ions, or molecules that make up the crystalline solid.
• A crystal lattice is created by substituting a representative point for each group of atoms.

The correct answer is external sector.

Key Points

• The balance of payments summarizes an economy's international economic dealings.
• These transactions consist of transfer payments as well as the export and import of products, services, and financial assets (like foreign aid).
• The current account, the capital account, and the financial account are the three primary subcategories of the BOP.
• The influx and outflow of goods and services into a nation are represented by the current account.
• All overseas capital transfers are noted in the capital account.
• International monetary flows associated with investments in companies, properties, bonds, and stocks are recorded in the financial account.
• The BOP should be 0 when the current account is in balance with the total of the capital and finance accounts, but this rarely happens.

Additional Information

• The balance of payments deficit is the shortfall when autonomous foreign exchange payments exceed autonomous foreign exchange receipts.
• Foreign exchange transactions that are done independently of the state's balance of payments and for a person's own benefit are referred to as autonomous transactions.

The correct answer is Soil erosion.

Key Points

• ​Decomposition literally means "to break down."
• Decomposition is a metabolic process that consumes complex chemicals as raw materials, processes them, and then transforms them into simpler compounds.
• Decomposers, which include bacteria, fungi, and a few other microorganisms, are those who start the decomposition process.
• To stay alive, they consume dead organisms.
• Dead and decaying animals and plants are used as the raw materials for the breakdown process.
• That results in the production of nutrients, carbon dioxide, water, and other things.
• Raw elements like dead plants, animals, and their remains are referred to as detritus.
• These microbes, collectively referred to as saprophytes, then digest this debris.
• Nutrients are released into the water and soak into the soil through a process called leaching.
• The process of converting complicated molecules into simpler ones is known as catabolism.
• Humification is the process by which dark-colored humus develops on the soil.

Additional Information

• Soil erosion:
• ​The wearing away of the topsoil layer is known as soil erosion.
• It contributes to soil degradation.
• The dynamic action of erosive agents, such as water, ice, snow, air, plants, and animals, is what drives this natural process.
• It has caused a sharp rise in river pollutants and sedimentation, clogging up the water bodies and causing a fall in the number of aquatic creatures.
• Degraded areas lose their ability to hold water, which causes floods.

The correct answer is Piyush Goyal.

Key Points

• ​SETU, or Supporting Entrepreneurs in Transformation and Upskilling, is a Ministry of Commerce & Industry project in India.
• The program will provide mentorship and support in a variety of areas, including fundraising, market access, and commercialization, for start-ups in India by connecting them to investors and start-up ecosystem experts in the US.
• The interaction will be supported by the mentorship site offered by Startup India's MAARG program, which is a one-stop shop for finding solutions for businesses in India.
• Its full form is **Mentorship, Advice, Assistance, Resilience, and Growth.**​

Additional Information

• Amit Shah :
• Indian politician Amit Anil Chandra Shah has been the home affairs minister since 2019.
• Yogi Adityanath:
• Yogi Adityanath, a politician and Hindu monk from India, is currently Uttar Pradesh's 21^st chief minister.
• Nitin Gadkari:
• The Minister for Road Transport & Highways in the Indian Government is Nitin Jairam Gadkari, a Maharashtrian politician.
• Piyush Goyal:
• Piyush Goyal is the Minister of Consumer Affairs, Food & Public Distribution

The correct answer is Pratihara.

Key Points

• Raja Nagabhata I Pratihara was the founder of Bhinmal branch of Pratiharas.

Additional Information

• Raja Nagabhata I Pratihara
• He was the founder of the Bhinmal branch of Pratihara.
• He formed a triple alliance with Jaysimha & Bappa Rawal to defeat Arabs in the Battle of Rajasthan
• The strongest of the Gurjara-Prathira branch was the one at Bhinmal, under king Vyaghramukh.
• The Gurjar clan, which ruled at Bhinmal was known as Chapas, means excelled in archery or strong bowmen.
• As per the records of Heun Tsang, the famous astronomer and mathematician Bramhagupta was in the court of Vyaghramukha.

Shortcut Trick

A central angle of 90° represents exactly 1/4^th of a full circle (360°).

Total circle area = π × r² = 3.14 × 10² = 314 m².

Area of sector = 314 ÷ 4 = 78.5 m².

∴ The correct answer is 78.5 m².

Alternate Method

Given:

Radius (r) = 10 m

Central Angle (θ) = 90°

Formula Used:

Area of Sector = (θ / 360°) × πr²

Calculations:

⇒ Area = (90 / 360) × 3.14 × 10 × 10

⇒ Area = (1 / 4) × 3.14 × 100

⇒ Area = 0.25 × 314

⇒ Area = 78.5 m²

∴ The correct answer is 78.5 m².

Additional Information

Length of Arc

The distance along the curved part of the sector is given by l = (θ/360) × 2πr.

Perimeter of a Sector

The total boundary of a sector includes the arc length and two radii: Perimeter = l + 2r.

Relationship between Area and Arc Length

The area of a sector can also be calculated using arc length (l): Area = (1/2) × l × r.

Given:

AB = 20 cm, PB = 12 cm and CP = 8 cm

Concept used:

If two chords intersect in a circle, then the products of the measures of the segments of the chords are equal.

AE × EC = DE × EB

Calculation:

According to the concept,

AP × PB = CP × PD

⇒ (AB - PB) × 12 = 8 × PD

⇒ (20 - 12) × 12 = 8 × PD

⇒ 8× 12 = 8 × PD

⇒ PD = 12

∴ The measure of PD is 12 cm.

Given:

Marked price = Rs. 3000

Discounts = 15% and 15%

The cost of transportation = Rs. 250

Required profit = 20%

Formula used:

C.P. = [(100 - Discount%)/100] × M.P.

S.P. = C.P. × [(100 + Profit%)/100]

Where,

C.P. = Cost price;

S.P. = Selling price;

M.P. = Marked price

Calculation:

The cost price = 300 × [(100 - 15)/100] × [(100 - 15)/100]

⇒ 3000 × (85/100) × (85/100)

⇒ Rs. 2167.5

Total spending = 2167.5 + 250

⇒ Rs. 2417.5 

The selling price = 2417.5 × [(100 + 20)/100]

⇒ 2417.5 × (120/100)

⇒ Rs. 2901

∴ The required selling price is Rs. 2901.

Shortcut Trick

Speed of B = Distance gap ÷ Time gap = 50 m ÷ 5 s = 10 m/s.

Time taken by B to finish the race = 1000 m ÷ 10 m/s = 100 s.

Time taken by A = 100 s − 5 s = 95 s.

Speed of A = 1000 m ÷ 95 s = 200/19 m/s.

Speed Difference = Speed of A − Speed of B = 200/19 − 10 = (200 − 190)/19 = 10/19 m/s.

∴ The correct answer is 10/19 m/s.

Alternate Method

Given:

Total Race Distance = 1000 m

A beats B by distance = 50 m

A beats B by time = 5 s

This implies that when A finishes the race (covers 1000 m), B is 50 m behind the finish line and takes 5 seconds more to cover that distance.

⇒ Speed of B = Distance covered ÷ Time taken = 50 m ÷ 5 s = 10 m/s

⇒ Time taken by B to complete 1000 m = 1000 m ÷ 10 m/s = 100 s

⇒ Time taken by A to complete 1000 m = Time of B − Time gap = 100 s − 5 s = 95 s

⇒ Speed of A = Total Distance ÷ Time of A = 1000 m ÷ 95 s = 200/19 m/s

⇒ Difference in speeds = Speed of A − Speed of B = (200/19) − 10

⇒ Difference = (200 − 190) ÷ 19 = 10/19 m/s

∴ The correct answer is 10/19 m/s.

Additional Information

Fundamental Motion Formula

The core relationship in problems involving movement is: Speed = Distance ÷ Time.

Concepts of Beating in Races

If A beats B by 'd' meters, then when A covers L distance, B covers (L − d). If A beats B by 't' seconds, then B's total time is Time_A + t.

Relative Speed

When two objects move in the same direction, their relative speed is the difference of their individual speeds (V₁ − V₂).

Calculation:

Cost price of A1 = 350

Selling price of A4 = 550

So, ratio = 350 : 550

⇒ 7 : 11

∴ The required answer is 7 : 11.

Given:

The selling price of 28 items is equal to the cost price of 17 items.

Concept used:

Selling price = Cost price - Loss

Calculation:

Let cost price of each item be Rs. 1

So, cost price of 28 items = Rs. 28

Selling price of 28 items = Rs. 17

Loss = 28 - 17

⇒ Rs. 11

Loss % = (11/28) × 100

⇒ $39{2 \over7}$%

∴ The loss percentage is $39{2 \over7}$%.

Given:

The price of petrol is increased by 28%

Calculation:

Let the price of petrol before the increase be Rs. 100a per 1000 ml

New price = 100a × 128%

⇒ Rs. 128a

Now,

If the consumer wants to maintain the expenditure as before he will still buy petrol for Rs. 100a

Now,

In Rs. 100a he will get = (1000/128a) × 100a

⇒ 781.25 ml

Consumption decrease = 1000 ml - 781.25 ml

⇒ 218.75

% decrease = (218.75/1000) × 100

⇒ 21.875 ≈ 21.88%

∴ The required answer is 21.88%.

Given:

A sum of money amounts to Rs. 767 in 3 years, and to Rs. 806 in 4 years on simple interest at 6% annum.

Calculation:

Interest for 1 year = 806 - 767

⇒ 39

Interest in 3 years = 39 × 3

⇒ 117

So, sum = 767 - 117

⇒ 650

∴ The sum is Rs. 650.

Calculation:

Amount spent for cricket in degree = 72°

Amount spent for tennis in degree = 54°

Difference = 72° - 54°

⇒ 18°

Required % = (18°/54°) × 100

⇒ 33.33%

∴ The required answer is 33.33%.

Given:

cot 13° cot 27° cot 45° cot 63° cot 77°

Formulas used:

Calculation:

cot 13° cot 27° cot 45° cot 63° cot 77°

⇒ cot 13° × cot 27° × cot 45° × cot (90° - 27°) × cot (90° - 13°)

⇒ cot 13° × cot 27° × cot 45° × tan13° × tan 27°

⇒ 1 × 1 × 1 = 1

∴ The required result is 1.

Given:

The average of eight consecutive odd numbers is 48.

Concept used:

Average = Sum of elements/Number of elements

Calculation:

Let the eight consecutive odd number be x, (x + 2), (x + 4), (x + 6), (x + 8), (x + 10), (x + 12), (x + 14)

Average of eight consecutive odd numbers = 48

⇒ {x + (x + 2) + (x + 4) + (x + 6) + (x + 8) + (x + 10) + (x + 12) + (x + 14)}/8 = 48

⇒ (8x + 56) = 384

⇒ 8x = 328

⇒ x = 41

So, 5^th number = 41 + 8 = 49

7^th number = 41 + 12 = 53

Total = 49 + 53

⇒ 102

∴ The required answer is 102.

Shortcut Trick

The average of eight consecutive odd numbers is 48,

This means the position of "48" is in the middle 4.5^th, among the 8 consecutive odd numbers.

So, the numbers are 1^st = 41, 2^nd = 43, 3^rd = 45, 4^th = 47, (48) , 5^th = 49, 6^th = 51, 7^th = 53, 8^th = 55

The sum of 5^th and  7^th number (49 + 53) = 102

Given:

The sides of a triangle are in the ratio $1 \over3$ ∶ $1 \over5$ ∶ $1 \over6$.

Calculation:

LCM of 3, 5, 6 of 30

So, the ratio of sides = 30 × $1 \over3$ : 30 × $1 \over5$ : 30 × $1 \over6$

⇒ 10 : 6 : 5

So, 10 + 6 + 5 = 147

⇒ 21 = 147

⇒ 1 = 7

So, smallest side = 5 × 7

⇒ 35

∴ The length of the smallest side is 35 cm.

Given:

(sin A + cos A)(1 – sin A cos A)

Concept used:

a³ + b³ = (a + b)(a² + b² - ab)

sin2θ + cos2θ = 1

Calculation:

(sin A + cos A)(1 – sin A cos A)

⇒ (sin A + cos A)(sin² A + cos² A – sin A cos A)

⇒ sin³ A + cos³ A

∴ The required answer is sin3 A + cos3 A.

Shortcut Trick

The ratio of circumferences of two circles is always equal to the ratio of their radii.

Given Ratio of Radii (Outer : Inner) = 5 : 4

Difference in ratio parts = 5 − 4 = 1 unit

This 1 unit represents the path width = 50 metres.

Inner radius = 4 units = 4 × 50 = 200 metres.

∴ The correct answer is 200 metres.

Alternate Method

Given:

Ratio of Outer Circumference to Inner Circumference = 5 : 4

Width of the circular path = 50 metres

Formula Used:

Circumference of a circle = 2 × π × radius

Width of circular path = Outer Radius (R) − Inner Radius (r)

Calculations:

⇒ (2 × π × R) ÷ (2 × π × r) = 5 ÷ 4

⇒ R ÷ r = 5 ÷ 4

⇒ R = 5r ÷ 4

⇒ Path Width = R − r = 50

⇒ (5r ÷ 4) − r = 50

⇒ (5r − 4r) ÷ 4 = 50

⇒ r ÷ 4 = 50

⇒ r = 50 × 4 = 200 metres

∴ The correct answer is 200 metres.

Additional Information

Area of a Circular Ring

The area of the path between two concentric circles is given by A = π × (R² − r²).

Relationship of Parameters

For any two circles, the ratio of their radii, diameters, and circumferences are all equal (R₁/R₂ = D₁/D₂ = C₁/C₂).

Area Ratio

The ratio of the areas of two circles is the square of the ratio of their radii (Area₁/Area₂ = (R₁/R₂)²).

Given:

2a + 3b = 10

ab = 3

Concept used:

(a + b)² = a² + b² + 2ab

Calculation:

2a + 3b = 10

⇒ (2a + 3b)² = 10²

⇒ 4a² + 9b2 + 12ab = 100

⇒ 4a2 + 9b2 + 36 = 100

⇒ 4a2 + 9b2 = 100 - 36

⇒ 4a2 + 9b2 = 64

∴ The required answer is 64.

Calculation:

Number of students who opted for other than statistics subject in school A = 450 × 80%

⇒ 360

Number of students who opted for other than statistics subject in school D = 400 × 85%

⇒ 340

Total = 360 + 340

⇒ 700

Average = 700/2

⇒ 350

∴ The required answer is 350.

Given:

4x - 7y = 11

xy = 8

Concept used:

(a + b)2 = a2 + b2 + 2ab

Calculation:

4x - 7y = 11

⇒ (4x - 7y)2 = 112

⇒ 16x2 + 49y2 - 56xy = 121

⇒ 16x2 + 49y2 - 448 = 121

⇒ 16x2 + 49y2 = 121 + 448

⇒ 16x2 + 49y2 = 569

∴ The required answer is 569.

Given:

The number is 16, 18, 36.

Calculation:

LCM of 16, 18, 36 = 144

As 144 = 12²

So, the answer is 144

∴ The required answer is 144.

Given:

sin θ = $1 \over 2$

Concept used:

Calculation:

sin θ = $1 \over 2$

⇒ sinθ = sin 30°

So, θ = 30°

3cos θ - 4 cos3 θ

⇒ 3 × $\sqrt3 \over 2$ - 4 × $3\sqrt3 \over 8$

⇒ ${3\sqrt3 \over 2} - \frac{3\sqrt3}{2}$

⇒ 0

∴ The required answer is 0.

Given:

The numbers are ${a^3 + b^3} \over {a - b}$, $a^2 - b^2 \over{a^2-ab+b^2}$

Concept used:

Mean proportion of a and b = $\sqrt{ab}$

Calculation:

$\sqrt{{{a^3 + b^3} \over {a - b}}\times{a^2 - b^2 \over{a^2-ab+b^2}}}$

⇒ $\sqrt{{{(a + b)(a^2-ab+b^2)} \over {a - b}}\times{(a+b)(a-b) \over{a^2-ab+b^2}}}$

⇒ $\sqrt{(a + b)^2}$

⇒ a + b

∴ The required answer is a + b.

Calculation:

When we divide 481 by 9 we get 53 as the quotient

Again,

When we divide 43 by 9 we get 4 as the quotient

So, total number which are divisible by 9 from 43 and 481 = 53 - 4

⇒ 49

∴ The required answer is 49.

Given:

A and B working separately can complete a piece of work in 10 and 16 days

Concept used:

Total work = efficiency of the worker × time taken by them

Calculation:

Let the total work be 80 units

So, the efficiency of A and B = 8 units/day, 5 units/day

Effective efficiency of the two = 13 units/day

According to the question,

They work on alternative days

So, in 2 days work completed = 13 units

In 12 days work completed = 78 units

Now,

The remaining 2 units will be completed by A in 2/8 days or 1/4 days

So, total time = 12 + 1/4 days

⇒ $12{{1} \over 4}$ days

∴ The work is completed in $12{{1} \over 4}$ days.

Given:

$({k - {1 \over k}})({k^2 + {1 \over k^2}})({k^4 + {1 \over k^4}})({k^8 + {1 \over k^8}})({k^{16} + {1 \over k^{16}}})({k^{32} + {1 \over k^{32}}})$

Concept used:

(a² - b²) = (a + b)(a - b)

Calculation:

$({k - {1 \over k}})({k^2 + {1 \over k^2}})({k^4 + {1 \over k^4}})({k^8 + {1 \over k^8}})({k^{16} + {1 \over k^{16}}})({k^{32} + {1 \over k^{32}}})$

⇒ $\frac{({k - {1 \over k}})({k^2 + {1 \over k^2}})({k^4 + {1 \over k^4}})({k^8 + {1 \over k^8}})({k^{16} + {1 \over k^{16}}})({k^{32} + {1 \over k^{32}}})({k + {1 \over k}})}{({k +{1 \over k}})}$    [By multiplying and dividing $k +{1 \over k}$]

⇒ $\frac{({k^2 - {1 \over k^2}})({k^2 + {1 \over k^2}})({k^4 + {1 \over k^4}})({k^8 + {1 \over k^8}})({k^{16} + {1 \over k^{16}}})({k^{32} + {1 \over k^{32}}})}{({k +{1 \over k}})}$ 

⇒ $\frac{({k^4 - {1 \over k^4}})({k^4 + {1 \over k^4}})({k^8 + {1 \over k^8}})({k^{16} + {1 \over k^{16}}})({k^{32} + {1 \over k^{32}}})}{({k +{1 \over k}})}$ 

⇒ $\frac{({k^8 -{1 \over k^8}})({k^8 + {1 \over k^8}})({k^{16} + {1 \over k^{16}}})({k^{32} + {1 \over k^{32}}})}{({k +{1 \over k}})}$