SSC CGL 2022 Tier-I Official Paper (Held On : 08 Dec 2022 Shift 2) (Previous Year Paper)

We apply the options to the given equation by using the BODMAS rule:

Option 1) 8, 5 and -, +

8 - 44 + 5 × 25 = 203

8 - 44 + 125 = 203

133 - 44 = 203

89 ≠ 203

Option 2) 8, 44 and -, +

5 - 8 + 44 × 25 = 203

5 - 8 + 1100 = 203

1105 - 8 = 203

1097 ≠ 203

Option 3) 8, 5 and -, ×

8 + 44 × 5 - 25 = 203

8 + 220 - 25 = 203

228 - 25 = 203

203 = 203

Option 4) 8, 44 and -, ×

5 + 8 × 44 - 25 = 203

5 + 352 - 25 = 203

357 - 25 = 203

332 ≠ 203

Hence, the correct answer is "Option 3".

The pattern followed in the given sets is:

(Second term × Third term) × 2 = First term

Now we apply the pattern on the given sets:

(60, 5, 6)

(5 × 6) × 2 = 60

30 × 2 = 60

60 = 60

(48, 8, 3)

(8 × 3) × 2 = 48

24 × 2 = 48

48 = 48

Now we apply the pattern on the given options:

Option 1) (17, 3, 4)

(3 × 4) × 2 = 17

12 × 2 = 17

24 ≠ 17

Option 2) (25, 4, 6)

(4 × 6) × 2 = 25

24 × 2 = 25

48 ≠ 25

Option 3) (24, 4, 3)

(4 × 3) × 2 = 24

12 × 2 = 24

24 = 24

Option 4) (25, 4, 3)

(4 × 3) × 2 = 25

12 × 2 = 25

24 ≠ 25

So, (24, 4, 3) follows the same pattern as the given sets.

Hence, the correct answer is "Option 3".

Given:

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Now we substitute the values in the given equation:

C # D @ E % Z & L # M - N * P means C is the sister of D, D is the brother of E, E is the father of Z, Z is the son of L, L is the sister of M, M is the daughter of N, N is the wife of P.

Now we draw a chart by using the following symbols:

   

From the chart we can see that Z is the grandson of N.

Hence, the correct answer is "Option 4".

The pattern followed in the given sets is:

(Second term + Third term)² = First term

Now we apply the pattern on the given sets:

(144, 9, 3)

(9 + 3)² = 144

(12)2 = 144

144 = 144

(225, 7, 8)

(7 + 8)2 = 225

(15)2 = 225

225 = 225

Now we apply the pattern on the given options:

Option 1) (168, 5, 8)

(5 + 8)2 = 168

(13)2 = 168

169 ≠ 168

Option 2) (64, 5, 3) 

(5 + 3)2 = 64

(8)2 = 64

64 = 64

Option 3) (81, 6, 2)

(6 + 2)2 = 81

(8)2 = 81

64 ≠ 81

Option 4) (120, 7, 4)

(7 + 4)2 = 120

(11)2 = 120

121 ≠ 120

So, (64, 5, 3) follows the same pattern as the given sets.

Hence, the correct answer is "Option 2".

The pattern followed in the given series is:

The figures rotate in clockwise direction in the form of a triad.

Next step:

           

Now we consider the given options:

Option 1) 

The head of arrow is in opposite direction.

Option 2) 

Positioning of star and triangle is incorrect.

Option 3) 

The pattern is correct.

Option 4) 

The positioning of triangle is incorrect.

Hence, the correct answer is "Option 3".

The pattern followed in the given series is:

The figure moves in clockwise direction pointing towards the corners.

The letters are in alphabetical order where every third letter is skipped, the letters move in clockwise direction

Next step: 

The alphabets will be L and M. M will be on top left & L will be on bottom right.

The head of the figure will point towards bottom right corner.

                      

Now we consider the given options:

Option 1) 

The positions of alphabets is interchanged.

Option 2) 

The position of alphabets & the figure is incorrect.

Option 3) 

The positioning of the figure is incorrect.

Option 4) 

The pattern is correct.

Hence, the correct answer is "Option 4".

Statements:

All pens are plates.

All plates are trays.

Some trays are boxes.

Venn diagram:

Conclusions:

I. Some boxes are plates→ Does not follow → There is no information that connects plates & boxes thus the conclusion does not follow.

II. Some trays are pens→ Follow → We have 'all pens are plates' and 'all plates are trays' which means some trays are pens and the Some trays are pens is definitely follow.

III. Some boxes are pens→ Does not follow → There is no information that connects pens & boxes thus the conclusion does not follow.

IV. No tray is a plate → Does not follow → Since all plates are trays, definitely some trays are plates therefore the conclusion does not follow.

So, only conclusion II follows.

Hence, the correct answer is "Option 1".

The pattern followed in the given series is:

The difference between the terms is consecutive odd numbers.

So, 94 will complete the series.

Hence, the correct answer is "Option 1".

According to the English alphabets & their positional values:

The pattern followed in the given codes is:

First & third alphabets are coded as the positional value of opposite +2.

Second alphabet is coded as the positional value of opposite -2.

     

and 

Similarly,

Hence, the correct answer is "Option 3".

Given:

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Now we substitute the values in the given options:

Option 1) A ÷ C + B means,

A is the brother of C, C is the mother of B.

Here A is the brother of C so A cannot be the paternal uncle of C.

Option 2) A ÷ B + C means,

A is the brother of B, B is the mother of C.

Here A is maternal uncle of C so A cannot be the paternal uncle of C.

Option 3) A + B ÷ C means,

A is the mother of B, B is the brother of C.

Here A is the mother of C so A cannot be the paternal uncle of C.

Option 4) A ÷ B × C means,

A is the brother of B, B is the father of C.

Here A is the paternal uncle of C.

Hence, the correct answer is "Option 4".

The faces opposite to each other are shown below:

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

So, the opposite face are:

• 1 ⇔ 4
• 2 ⇔ 6
• 3 ⇔ 5

Now we consider the given options:

 Cannot be formed as a cube from the given sheet (As 3 is opposite to 5)

Cannot be formed as a cube from the given sheet (As 4 is opposite to 1)

Can be formed as a cube from the given sheet (As 2,4,5 are one of the number of each opposite pairs.)

Can be formed as a cube from the given sheet (As 1,5,6 are one of the number of each opposite pairs.)

So, only C and D will be formed.

Hence, the correct answer is "Option 1".

According to the English alphabets & their positional values:

The pattern followed in the given options is:

Option 1) 

Option 2) 

Option 3) 

Option 4) 

So, QLA does not belong to the group.

Hence, the correct answer is "Option 3".

The mirror image of the given pattern is:

Now we consider the given options:

Option 1) 

The pattern is correct.

Option 2) 

The shaded part in the circle is incorrect.

Option 3) 

Head of the arrow is in opposite direction.

Option 4) 

Triangle is positioned incorrectly.

Hence, the correct answer is "Option 3".

According to the English alphabets & their positional values:

The pattern followed in the given codes is:

First & last alphabets are interchanged, rest of the alphabets are increased by +2.

Similarly,

Hence, the correct answer is "Option 1".

The pattern followed in the given set is:

Commence : Begin → Commence is the synonym of begin.

Similarly,

Belief is the synonym of doctrine.

Hence, the correct answer is "Option 1".

We interchange the signs in the given options & solve them using BODMAS rule:

Option 1) 20 × 12 + 6 - 16 ÷ 8 = 20

20 - 12 + 6 × 16 ÷ 8 = 20

20 - 12 + 6 × 2 = 20

20 - 12 + 12 = 20

32 - 12 = 20

20 = 20

Option 2) 16 × 91 ÷ 13 + 5 - 3 = 24

16 - 91 ÷ 13 + 5 × 3 = 24

16 - 7 + 5 × 3 = 24

16 - 7 + 15 = 24

31 - 7 = 24

24 = 24 

Option 3) 20 × 8 - 4 ÷ 2 + 3 = 7

20 - 8 × 4 ÷ 2 + 3 = 7

20 - 8 × 2 + 3 = 7

20 - 16 + 3 = 7

23 - 16 = 7

7 = 7

Option 4) 4 × 5 - 36 ÷ 6 + 10 = 10

4 - 5 × 36 ÷ 6 + 10 = 10

4 - 5 × 6 + 10 = 10

4 - 30 + 10 = 10

14 - 30 = 10

-16 ≠ 10

Hence, the correct answer is "Option 4".

We apply the options to the given equation by using the BODMAS rule:

Option 1) 5 and 49, ÷ and -

686 ÷ 14 × 5 + 400 - 49 = 596

49 × 5 + 400 - 49 = 596

245 + 400 - 49 = 596

645 - 49 = 596

596 = 596

Option 2) 686 and 5, + and -

5 + 14 × 49 - 400 ÷ 686 = 596

5 + 14 × 49 - 0.5 = 596

5 + 686 - 0.5 = 596

691 - 0.5 = 596

690.5 ≠ 596

Option 3) 14 and 49, × and -

686 × 49 - 14 + 400 ÷ 5 = 596

686 × 49 - 14 + 80 = 596

33614 - 14 + 80 = 596

33694 - 14 = 596

33680 ≠ 596

Option 4) 14 and 5, + and ×

686 - 5 + 49 × 400 ÷ 14 = 596

686 - 5 + 49 × 28.5 = 596

686 - 5 + 1399 = 596

2085 - 5 = 596

2080 ≠ 596

Hence, the correct answer is "Option 1".

The correct order according to the English dictionary is:

4. Devocalize
5. Devolution
6. Devotement
7. Devotional
8. Devoutness

So, Devoutness will come at ‘Fifth’ position.

Hence, the correct answer is "Option 3".

The pattern followed in the given sets is:

(First term + Third term) ÷ 2 = Second term

Now we apply the pattern on the given sets:

(34, 23, 12)

(34 + 12) ÷ 2 = 23

46 ÷ 2 = 23

23 = 23

(71, 37, 3)

(71 + 3) ÷ 2 = 37

74 ÷ 2 = 37

37 = 37 

Now we apply the pattern on the given options:

Option 1) (75, 50, 20)

(75 + 20) ÷ 2 = 50

95 ÷ 2 = 50

47.5 ≠ 50

Option 2) (20, 65, 50)

(20 + 50) ÷ 2 = 65

70 ÷ 2 = 65

35 ≠ 65

Option 3) (19, 55, 91)

(19 + 91) ÷ 2 = 55

110 ÷ 2 = 55

55 = 55

Option 4) (14, 12, 16)

(14 + 16) ÷ 2 = 12

30 ÷ 2 = 12

15 ≠ 12

Hence, the correct answer is "Option 3".

Given figure:

Now we consider the given figures:

Option 1) The figure can be embedded.

Option 2) The figure cannot be embedded completely.

Option 3) The figure cannot be embedded completely.

Option 4) The figure cannot be embedded completely.

Hence, the correct answer is "Option 1".

The pattern followed in the given options is:

(First term)² - 1 = Second term

Now we consider the given options:

Option 1) 9 - 80

(9)2 - 1 = 80

81 - 1 = 80

80 = 80

Option 2) 7 - 52

(7)2 - 1 = 52

49 - 1 = 52

48 ≠ 52

Option 3) 5 - 24

(5)2 - 1 = 24

25 - 1 = 24

24 = 24

Option 4) 11 - 120

(11)2 - 1 = 120

121 - 1 = 120

120 = 120

So, 7 - 52 is the odd one.

Hence, the correct answer is "Option 2".

According to the English alphabets & their positional values:

The pattern followed in the given series is:

First alphabet is increased by +3, second is decreased by -6, third is increased by +9 and fourth is decreased by -12.

So, FODU will complete the series.

Hence, the correct answer is "Option 2".

According to the English alphabets & their positional values:

The pattern followed in the given series is:

The difference between the alphabets is -2.

So, E will complete the series.

Hence, the correct answer is "Option 4".

The pattern followed in the given set is:

Road : Concrete → Road is made up of concrete.

Similarly,

Wall is made up of brick.

Hence, the correct answer is "Option 4".

Statements:

I. Some Z are F.

II. All G are Z.

Venn diagram:

Conclusion:

I. No F is G → Does not follow → We don't have any information about G and F so the conclusion cannot be drawn.

II. Some G are not F → Does not follow → We don't have any information about G and F so the conclusion cannot be drawn.

III. No F is Z → Does not follow → We have 'some Z are F' which means some F are Z thus the conclusion does not follow.

So, neither conclusion follows.

Hence, the correct answer is "Option 4".

The correct answer is i - b, ii - a, iii - c, iv - d

Key Points

• The cell physically expands, duplicates organelles, and creates the molecular building blocks it will require in subsequent stages during G1phase, also known as the first gap phase.
• The cell creates a full copy of the DNA in its nucleus during the S phase. Moreover, it makes duplicates of the centrosome, a structure that organises microtubules. During M phase, the centrosomes aid in the separation of DNA.
• The second gap phase, also known as the G2 phase, is a time of increased cell growth, protein and organelle production, and content reorganisation in preparation for mitosis.
• The cell's nuclear DNA condenses into its visible chromosomes during mitosis and is separated by the mitotic spindle, a specialised microtubule-based structure.

Additional Information

• Stages of the cell cycle
• ​A cell must grow, replicate its genetic material (DNA), and physically split into two daughter cells before it may divide.
• The cell cycle is a structured, predictable series of actions that cells take to complete these goals.
• Because the two daughter cells can restart the entire process from the beginning after each cycle, the cell cycle is a cycle rather than a linear pathway.

The correct answer is I =$Q \over t$

Key Points

• Current, charge and time
• How much electric charge is transferred each second defines electrical current, giving the following relationship:
  

Current= charge/ time

Rearranging this we get,

charge= current * time

• The symbol for charge is Q, it is measured in coulombs (C).
• The symbol for current is I, it is measured in amperes (A).
• The symbol for time is t, it is measured in seconds (t).
• This gives the relation:
  

Q= I t*

Additional Information

• Current
• A stream of charged particles, like electrons or ions, moving through an electrical conductor or space is called an electric current.
• The net rate of electric charge flow through a surface or into a control volume is how it is measured.
• Depending on the conductor, the moving particles are referred to as charge carriers, which can be any one of a number of different types of particles.
• Charge
• A charge in physics can be any of a number of different quantities, such as the color charge in quantum chromodynamics or the electric charge in electromagnetism.
• The time-invariant generators of a symmetry group, and more specifically, the generators that commute with the Hamiltonian, are represented by charges.
• The charge invariance corresponds to the vanishing commutator [Q,H]=0, where H is the Hamiltonian because charges are frequently denoted by the letter Q.
• As a result, charges and conserved quantum numbers are linked; these are the eigenvalues q of the generator Q.
• Time
• From the past through the present and into the future, time is the ongoing sequence of existence and events that appears to be irreversible.
• It is a component quantity of various measurements used to order events, compare their durations or intervals, and quantify rates of change of quantities in material reality or conscious experience.
• Along with the three spatial dimensions, time is frequently referred to as a fourth dimension.

The correct answer is Violin.

Key Points

• Lalita and Nandini
• ​Globally renowned violin maestros Dr. M. LALITHA and M. NANDINI have been hailed as the current generation's "QUEENS OF VIOLIN - Weekend."
• Popularly referred to as the "Viola Sisters," these exceptional musicians have mesmerised audiences with their captivating music and have earned recognition as the "ONLY FEMALE DUO IN ASIA to perform the World music, South Indian Classical, Fusion, and Western Classical music - Le Mauricien."
• "Music runs in their blood, they must have played music even while they were in their mother's womb," said famous music critic Subbudu. Lalitha and Nandini are the torchbearers of a special musical history, but they have also developed their own unique styles.

Additional Information

• A mandolin is a stringed instrument from the same family as the lute that is often played with a pick.
• Although there are also variations with 4 string, 5 string, 6 string, 5 double string (10 strings), 6 (12 strings), and even 4 triple string (12 strings) courses, the most typical configuration has four courses of doubled strings tuned in unison, providing a total of 8 strings.
• A stringed instrument known as the sarod is utilised in Hindustani music on the Indian subcontinent. It is one of the most well-known and renowned instruments, along with the sitar. It is renowned for having a rich, substantial, inward tone.
• The woodwind family of instruments, which includes the flute, is used in classical music. Flutes are aerophones, which means they produce music by vibrating a column of air, like all woodwind instruments do.

The correct answer is Manipuri

Key Points

• Guru Nileshwar Mukharjee
• ​Guru Nileshwar Mukharjee is a well-known Manipuri dancer from India who has achieved international fame.
• The Manipuri community is well aware of Guru Nileshwar Mukharjee, who has made significant contributions to the field of Manipuri dance. Guru Nileshwar Mukharjee gave a Ras performance in Sylhet in 1920, and Rabindranath Tagore was so moved that he summoned him to Shantiniketan.
• In the history of Manipuri dance, this event marked a turning point. Within a decade, it had transcended national boundaries and made Guru Nileshwar Mukharjee's name well-known around the world.

Additional Information

• Manipuri Dance
• One of the eight principal Indian classical dance styles, the Manipuri Dance is also known as the Manipuri Raas Leela (Meitei: Jagoi Raas, Raas Jagoi) and is a product of the state of Manipur.
• The dancing style is characterised by delicate, calm body motions and compassionate gaze, and it is infused with the spiritual themes of Madhura Raas of Radha-Krishna.
• Whether a dancer is Hindu or not, their expressions are generally calm and convey Bhakti Rasa, or the emotion of devotion.

The correct answer is Pattadakal Dance Festival 

Key Points

• Government of Karnataka
• Pattadakal Dance Festival is organised by the Government of Karnataka, usually in January, to celebrate the grandeur of temples at a UNESCO heritage site.
• ​The Government of Karnataka, often known as the Karnataka Government or just the Karnataka Government and formerly known as the Government of Mysore, is a democratically elected state body whose governor serves as the state's ceremonial head.
• On the chief minister's recommendation, the governor, who is in office for five years, names his council of ministers.
• The chief minister and his council of ministers, who are given extensive legislative authority, manage the government on a daily basis even while the governor continues to serve as the state's ceremonial leader.

Additional Information

• Pattadakal Dance Festival 
• ​A little village in Karnataka called Pattadakallu, sometimes known as Pattadakal, is a prime example of Indian architecture.
• This village, which sits on the banks of the Malaprabha River, has been designated a UNESCO World Heritage Site.
• The word "Pattadakallu" means "site of coronation." As the name implies, this location served as the site of the Chalukya dynasty's coronation ceremonies, giving it significant historical and cultural value.

The correct answer is Second

Key Points

• Mahalanobis Model
• ​Marxist economists Grigory Feldman from the Soviet Union and Prasanta Chandra Mahalanobis from India separately developed the Feldman-Mahalanobis model in 1953.
• Mahalanobis ended up serving as the Second Five Year Plan's primary economist and was the focus of many of the country's most contentious economic discussions.
• The core of the strategy is a change in industrial investment patterns towards the development of a domestic consumption goods sector.

Additional Information

• Long-term expansion of the nation's consumer goods production capacity depends on the capital goods sector having a high adequate capacity.
• The division between the two different categories of goods helped consumers better comprehend the trade-off between levels of immediate and future consumption and provided a clearer explanation of Marx's theories in Das Kapital.
• Feldman, an economist for the GOSPLAN planning committee at the time, first proposed these concepts in 1928, outlining the theoretical justifications for a two-department growth strategy.

The correct answer is Back Court

Key Points

• Basketball
• ​Basketball is a team sport in which two teams, usually each with five players, compete against one another on a rectangular court with the main goal of preventing the opposing team.
• From shooting through their own hoop while putting the ball through the defender's hoop, which is a basket with a diameter of 18 inches (46 cm) mounted 10 feet (3.048 m) high to a backboard at each end of the court.
• Timed play is stopped following a foul, and the player who committed the foul or who was chosen to attempt a technical foul is awarded one, two, or three one-point free throws.
• The team with the most points at the conclusion of regulation play wins, but if the score is tied when regulation play ends, extra time (overtime) must be played.

Additional Information

• Players move the ball forward by either passing it to a teammate or bouncing it while running or walking (dribbling), both of which require a high level of ability.
• Players can use a variety of shots on offence, such as layups, jump shots, and dunks. On defence, they can steal the ball from a player who is dribbling, intercept passes, or block shots.
• Finally, either the offence or the defence can grab rebounds, which are missed shots that rebound off the rim or backboard.

The correct answer is Atlantic Ocean

Key Points

• Atlantic Ocean
• ​At a size of roughly 106,460,000 km2, the Atlantic Ocean is the second-largest of the world's five oceans (41,100,000 sq mi).
• It covers around 29% of the sea surface area and about 20% of the planet's surface.
• The "Old World" of Africa, Europe, and Asia is known to be distinguished from the "New World" of the Americas in the perspective of Europe.
• The Atlantic Ocean has been crucial to the growth of human society, globalisation, and the history of numerous states since it divides Africa, Europe, and Asia from the Americas.

Additional Information

• The Atlantic Ocean served as the epicentre of the Columbian Exchange and the namesake slave trade from the 16th through the 19th centuries, and it also periodically served as the scene of naval conflicts.
• While there haven't been any major military conflicts in the Atlantic in the current 21st century, the ocean still plays an important role in international trade.
• Both these naval battles and growing trade from regional American powers like the United States and Brazil increased in intensity during the 20th century.
• The Atlantic Ocean is located in an elongated, S-shaped basin that runs between North and South America to the west and Europe and Africa to the east.

The correct answer is Brahmaputra.

Key Points

• Brahmaputra
• ​The Brahmaputra is a trans-boundary river that passes through Tibet, Bangladesh, and Northeastern India.
• Other names for it include Luit in Assamese, the Yarlung Tsangpo in Tibetan, the Siang/Dihang River in Arunachali, and the Jamuna River in Bangla.
• It is the 15th longest river in the world and the 9th largest by discharge.
• The Yarlung Tsangpo River has its source in the Manasarovar Lake region, close to Mount Kailash, in Burang County, Tibet, on the northern face of the Himalayas.

Additional Information

• The river travels through southern Tibet before plunging into the Mountains in the Yarlung Tsangpo Great Canyon and entering Arunachal Pradesh.
• It enters India close to the Arunachal Pradesh settlement of Gelling and travels through Bangladesh as the Jamuna before flowing southwest through the Assam Valley (not to be confused with the Yamuna of India).
• It combines with the Ganges, also known as the Padma in Bangladesh, in the enormous Ganges Delta, where it then becomes the Meghna and eventually empties into the Bay of Bengal.

The correct answer is Sultan Iltutmish 

Key Points

• Minhaj-i-Siraj 
• ​Minhaj-al-Din Siraj-al-Din Abu Amr Othman ibn Minhaj al-Siraj Juzjani, also known as Muhammad Juzjani (born 1193), was a Persian historian who lived in the 13th century.
• He was born in the Ghur district. Juzjani moved to Ucch and then to Delhi in 1227.
• The main historian for the Mamluk Sultanate of Delhi in northern India was Juzjani. and discussed the Ghurid dynasty.
• Also, he penned the Tabaqat-i Nasiri (1260 CE) for Delhi's Sultan Nasiruddin Mahmud Shah. After 1266, he passed away.

Additional Information

• Sultan Iltutmish 
• ​The third Mamluk king to oversee the erstwhile Ghurid lands in northern India was Shams ud-Din Iltutmish.
• He is regarded as the de facto founder of the Delhi Sultanate because he was the first Muslim ruler to do so from Delhi.
• Iltutmish was sold into slavery as a young child and spent his early years in Bukhara and Ghazni working for many owners.

The correct answer is 4-10

Key Points

• Joint Liability Group
• The National Bank for Agriculture and Rural Development (NABARD), a rural development organisation in India.
• It has developed the idea of Joint Liability Groups in 2014 to offer institutional finance to small farmers.
• In order to obtain a loan from a bank without providing any collateral, a Joint Liability Group is made up of four to ten individuals from the same town or locality who share a similar socioeconomic background.

Additional Information

• JLGs can be promoted by Business Facilitators/Correspondents, NGOs, Farmers Clubs, Farmers Federations, Panchayati Raj Institutions, Agricultural Universities, Bank Branches, PACS, Cooperative Societies, Individuals, mFIs, and Many Others.
• A bank may choose to provide funding to a JLG either directly to the group or to an individual member. In either scenario, the loan balance must be repaid in full by all JLG members.

The correct answer is Parliament

Key Points

• Parliament
• ​The Republic of India's highest legislative body is the Indian Parliament.
• The Rajya Sabha (Council of States) and the Lok Sabha are the two houses that make up India's bicameral legislature (House of the People).
• The president has full authority to call a session of either house of Parliament, prorogue it, or dissolve the Lok Sabha in his capacity as head of the legislature.
• Only on the advice of the prime minister and his Union Council of Ministers can the president utilise these powers.

Additional Information

• The members of the Rajya Sabha are chosen by the members of all state legislative assemblies via proportional representation, whereas the members of the Lok Sabha are chosen by the Indian population directly through voting in single-member districts.
• The authorised membership of the Parliament is 543 in the Lok Sabha and 245 in the Rajya Sabha, including 12 candidates with experience in a variety of literary, artistic, scientific, and social service disciplines. In New Delhi's Sansad Bhavan, the Parliament convenes.

The correct answer is West Bengal.

Key Points

• West Bengal​
• ​India's eastern state is called West Bengal. With nearly 91 million people living in an area of 88,752 km2,
• it is located along the Bay of Bengal (34,267 sq mi). West Bengal is the eighth-most populous country subdivision in the world, the fourth-most populous state in India, and the thirteenth-largest state in terms of area.
• Its borders with Bangladesh in the east, Nepal and Bhutan in the north, and Bhutan are all parts of the Bengal area of the Indian subcontinent.
• It also has boundaries with the states of Assam, Jharkhand, Bihar, Sikkim, and Odisha in India. Kolkata, India's third-largest metropolis and seventh-largest city by population, serves as the state capital.

Additional Information

• The early history of the region was marked by a series of Indian empires, internal strife, and a struggle for supremacy between Buddhism and Hinduism. Several significant Janapadas were located in ancient Bengal, but the earliest cities date to the Vedic era.
• The Vangas, Mauryans, and Guptas were only a few of the ancient pan-Indian dynasties that ruled over this area.
• The Gaua Kingdom, the Pala Empire, and the Sena Empire all had their capitals in the Gaua fortress.

The correct answer is Motion.

Key Points

• Motion is the process of moving or being moved, as well as a change in the position of an item.
• The term "body in motion" refers to when a body's location in relation to its environment changes.
• Motion is mathematically characterised in terms of displacement, distance, velocity, acceleration, and speed as well as the observer's frame of reference and the measurement of the change in the body's position with respect to that frame as time passes.
• Kinematics is the field of physics that studies forces and their effects on motion, while dynamics studies the motion of objects without regard to their source.

Additional Information

• Speed
• The distance travelled in relation to the time it took to travel that distance is how speed is defined. As speed simply has a direction and no magnitude, it is a scalar quantity.
• Velocity
• In kinematics, the area of classical mechanics that deals with the motion of bodies, velocity is a fundamental idea. A physical vector quantity called velocity must have both a magnitude and a direction in order to be defined.
• Distance
• Distance is the scientific term used to describe how far an object has moved. A scalar quantity, distance. The International System of Units defines a metre as the distance unit.

The correct answer is India.

Key Points

• The FIFA U-17 Women's World Cup, an international women's youth football competition played every two years between the under-17 national teams of FIFA member organisations, reached its seventh iteration in 2022.
• The first competition took place in 2008. India served as the host nation for the competition; however, the 2020 edition was postponed owing to the COVID-19 epidemic.
• After the 2017 FIFA U-17 World Cup for men, it was India's second time hosting a FIFA event, and it was also the country's first FIFA women's football event.

Additional Information

• An international association football competition for female players under the age of 17 is known as the FIFA U-17 Women's World Cup.
• Federation Internationale de Football Association is in charge of organising it (FIFA).
• Since 2008, the competition has been held in even-numbered years.
• Spain is the reigning champion after taking home a second championship in 2022 in India.

The correct answer is 1906.

Key Points

• The Muslim League was established on December 30, 1906, at Dacca, under the direction of Aga Khan, the Nawab of Dhaka, and Nawab Mohsin-ul-Mulk.
• When British India was divided in 1947, a political party was the driving force behind the movement asking for the creation of a separate Muslim nation out of concern that Hindus would take over an independent India.
• It was established to protect Indian Muslims' rights.
• In 1913, Muhammad Ali Jinnah joined the league.
• The league was terminated on August 14th, 1947. In Pakistan, Bangladesh, and India, it is still present in political parties under different guises.

Additional Information The objectives of the league were:

• To inspire loyalty to the British government among Muslims.
• To protect Muslims' political rights and to express such rights to the government.
• To combat the animosity Muslims have towards other communities.

The correct answer is Subhash Chandra Bose.

Key Points

• On January 23, 1897, Subhash Chandra Bose was born in Cuttack, Orissa Division, Bengal Province.
• The 23rd of January is observed as "Parakram Diwas," the anniversary of Netaji's birth.
• Subhash Chandra Bose is more commonly referred to as "Netaji."
• Vivekananda's teachings had a significant impact on him, and he regarded him as his spiritual teacher.
• The "Political Guru of Subhash Chandra Bose" was Chittaranjan Das.
• At the Tripuri Session in 1939 and the Haripura Session in 1938, he was the first INC President to be elected.
• In 1939, he created  the Forward Bloc.

Additional Information

• Chittaranjan Das
• ​Chittaranjan Das, also known as Deshbandhu (Friend of the Nation), was an Indian freedom fighter, political activist, and lawyer during the Indian Independence Movement.
• He also founded and served as the leader of the Swaraj Party in undivided Bengal during the time of British Colonial rule in India. Chittaranjan Das passed away on June 16, 1925.
• He goes by the initials C. R. Das. In addition to several articles and essays, he was closely affiliated with a number of literary clubs and composed poetry.

The correct answer is Both I and II.

Key Points

• Shakas were of Scythian ethnic descent. With the Great Yueh Chi tribe's (Chinese Tribe's) expulsion of the Shakas from the plains of Syr Darya (Jaxartes) in the second century BCE, the Shakas moved into northwest India. Aspirants should be aware that the terms "Sakas," "Shakas," and "Indo-Scythians" are interchangeable and have the same meaning.
• In the middle of the third century CE, the Deccan was the place of genesis of the ancient Indian dynasty known as the Vakataka.
• Their territory is thought to have spanned the Tungabhadra River in the south, the Arabian Sea in the west, and the easternmost reaches of Chhattisgarh, as well as the southernmost reaches of Malwa and Gujarat in the north.
• They were the most significant Satavahana successors in the Deccan and lived at the same time as the Guptas in northern India.

Additional Information

• Shishunaga Dynasty
• Shishunaga Dynasty, established in Magadha by Shishunaga, an Amatya, overthrew the Haryanka Dynasty.
• He organised the populace's uprising against the Haryanka Dynasty, conquered Magadha, and established Patliputra as its capital.
• The son of a Licchavi king of Vaishali was Shishunaga. In addition to Sindh, Karachi, Lahore, Herat, Multan, Kandahar, and Vellore, Shishunaga extended his realm to include modern-day Jaipur in Rajasthan.
• Even the Shishunaga Dynasty expanded to Madurai and Cochin in the south, Murshidabad in the east, and Mand in the west. Kakavarna, also known as Kalashoka, and his ten sons succeeded Shishunaga.
• The throne of this nation was afterwards taken by the Nanda Empire.

The correct answer is Harsh V Shringla.

Key Points

• Harsh Vardhan Shringla, who was born in 1962, will serve as India's chief coordinator for the G20 in 2023.
• Prior to this, he held positions as Thailand's ambassador, Bangladesh's high commissioner, and India's foreign secretary.
• Harsh Vardhan Shringla attended Mayo College in Ajmer before earning his degree from St. Stephen's College at Delhi University.
• Prior to joining the Indian Foreign Service, he worked in both the public and commercial sectors of India.

Additional Information

• G20
• The Group of Twenty, or G20, is an intergovernmental organisation made up of the European Union and 19 other nations (EU).
• It strives to address important global economic concerns like international financial stability, reducing greenhouse gas emissions, and sustainable development.
• The majority of the world's greatest economies, both industrialised and developing countries, make up the G20; together, they represent two-thirds of the world's population, 60% of its land area, and around 80% of the world's gross world product (GWP).

The correct answer is Lavender.

Key Points

• With the Council of Scientific and Industrial Research's "aroma mission," the Ministry of Science and Technology's Purple Revolution or Lavender Revolution seeks to advance the domestic aromatic crop-based agro-economy (CSIR).
• The goal aims to boost farmers' incomes and encourage large-scale commercial lavender farming.
• The major product is lavender oil, which goes for at least Rs. 10,000 per litre.
• Medicines, incense sticks, soaps, and air fresheners are some more popular goods.
• Lavender growing is particularly cost-effective because it generates income right away.

Additional Information

• By switching from imported aromatics to domestic ones, it seeks to advance the domestic aromatic crop-based agro-economy.
• The objective included providing free lavender seedlings to first-time growers, and paying those who had previously grown lavender Rs. 5–6 per plant.
• Lavender growing has started in the Ramban district of Jammu and Kashmir as part of the CSIR-Aroma Indian Institute of Integrative Medicine (IIIM) Programme.
• Almost all 20 districts of Jammu and Kashmir have lavender farms.

The correct answer is 82.

Key Points

• Chemical element lead has the atomic number 82 and the letter Pb for its symbol.
• It is a dense hefty metal that is heavier than the majority of everyday materials.
• Lead is pliable and soft, and it also has a low melting point.
• Lead has a tinge of blue when it is first cut, and it is bright and grey. When exposed to air, it tarnishes to a drab grey tone.
• Three of lead's isotopes are ends of significant nuclear decay chains of heavier elements, and lead has the highest atomic number of any stable element. Even trace levels of lead are harmful, especially for young infants.

Additional Information

The correct answer is Article 368.

Key Points

• Part XX (Article 368) of the Indian Constitution outlines the constitutional amendment process.
• This process protects the Constitution of India's sanctity and serves as a check on the arbitrary authority of the Indian Parliament.
• Article 368 does not clearly forth the legislative process to be followed at various points of the change.
• The Indian Constitution's Article 368 deals directly with constitutional amendments.

Additional Information

• Article 365
• ​According to Article 365, the President may declare that a situation has arisen in which the State's government cannot be carried out in accordance with the provisions of this Constitution if any State has failed to comply with or give effect to any directives provided in the exercise of the Union's executive power under any of the provisions of this Constitution.
• Article 360
• The Indian Constitution's Part XVIII (Articles 352-360) discusses the Emergency Provisions. Article 360 specifically addresses the current financial crisis. The Indian Constitution contains three different categories of emergency provisions. The remaining two are:
• National Emergency - Article 352
• Emergency resulting from a breakdown of the state's constitutional machinery - President's Rule - Article 356
• Article 355
• "It shall be the duty of the Union to defend every State from external invasion and internal unrest and to ensure that the government of every State is carried on in conformity with the provisions of this Constitution," states Article 355 of the Constitution.

The correct answer is Grand trunk road.

Key Points

• Sher Shah Suri, a king of the 16th century, constructed the Grand Trunk Road.
• It was referred to as "Sadak e Azam."
• The road was essential for inland trade since it connected India to Central Asia.
• This street's previous names include Uttarapath, Badshahi Sarak, and Sarak e Azam.
• The longest and oldest significant road in Asia.
• 3,710 kilometers make up this road's length.
• The west end of this route is in Kabul, Afghanistan, and the east end is in Teknaf, Bangladesh.

Additional Information

• Sher Shah Suri
• Sher Shah Suri (1472, or 1486 - 22 May 1545), born Farid Khan, was the founder of the Sur Empire in India, with its capital in Sasaram in modern-day Bihar.
• He reduced the weight of silver coins to 178 grams and gave the money the Sanskrit name rupee, which is an ancient synonym for silver.
• In the year 1540 CE, Sher Shah seized power over the Mughal Empire. His son Islam Shah succeeded him after his untimely death in 1545 CE.

The correct answer is 30.9%

Key Points

• According to the "Rangarajan Report on Poverty", the poverty ratio has declined from 39.6% in 2009-10 to 30.9% in 2011-12 in rural India and from 35.1% to 26.4% in urban India.
• The all-India ratio was 29.5%. In rural India, 260.5 million individuals were below poverty and in urban India 102.5 million were under poverty. Totally, 363 million were below poverty in 2011-12.
• The Methodology recommended by the Expert Group (Rangarajan) for estimation of poverty is as follows:
• The poverty line should be based on certain normative levels of adequate nourishment, clothing, house rent, conveyance and education, and a behaviorally determined level of other non-food expenses.

Additional Information

• Poverty 
• When a person lacks the means of support and necessities for a given level of living, they are said to be in a state or condition of poverty.
• Several social, economic, and political factors can contribute to or be a result of poverty.
• There are two major ways to assess poverty in statistics and economics: absolute poverty compares income to the amount required to cover basic requirements like food, clothes, and shelter; relative poverty assesses how well a person is doing in comparison to others in the same situation.

The correct answer is zooplankton.

Key Points

• Although there may be some fish that consume phytoplanktons, the main consumers in a lake environment are zooplanktons. Zooplankton is thus the dominant second trophic level in a lake environment.
• The majority of zooplankton species are tiny, microscopic organisms. a single-celled protist that primarily consumes bacteria, for instance. Nonetheless, some zooplankton species can grow fairly large such as jellyfish, arrow worms, etc.
• They are the main consumers because they consume producers like phytoplankton, which are then consumed by small fish (secondary consumers).

Additional Information

• Food Chain
• A food chain demonstrates the relationship of feeding between several creatures in a specific habitat or ecosystem.
• The energy flow from the sun to producers, from producers to consumers, and from consumers to decomposers like fungi is demonstrated by a food chain.
• They also demonstrate how animals rely on other living things for nourishment.

Given:

AB = PQ

∠B = ∠Q

Formula used:

SAS rule is the two side of the triangle is equal and the angle made by these side is also equal

Calculations:

According to the question,

In ∆ABC and ∆PQR

⇒ AB = PQ

⇒ ∠B = ∠Q

So, angle B is made the side AB and BC and angle Q is made up by the side PQ and QR

So, BC must be equal to QR to make these two triangle congruent by SAS rule

⇒ Hence, Option (1) is correct

Given:

The principal amount becomes 3 times in 10 years

Formula Used:

Interest = $\frac{PRT}{100}$

Calculations:

The principal amount becomes 3 times in 10 years

⇒ Interest earned in 10 years = 3P - P = 2P

According to the formula,

⇒ Interest = $\frac{PRT}{100}$

⇒ 2P = $\frac{P\times R \times 10}{100}$

⇒ R = 20%

⇒ Hence, The yearly rate of the simple interest is 20%

Given:

Candidates passed from institute C = 8

Candidates passed from institute D = 26

Candidates appeared from institute B = 24

Candidates appeared from institute E = 20

Formula used:

Percentage = $\frac{\text{candidate passed}}{\text{candidate appeared}}\times100$

Calculations:

⇒ Total candidate passed in institution C and D = 8 + 26 = 34

⇒ Total candidate appeared in institution B and E = 24 + 20 = 44

According to the formula,

⇒ Percentage = $\frac{\text{34}}{\text{44}}\times100$ = 77%

⇒ Hence, The percentage is 77%

Shortcut Trick

Relative speed in opposite directions = 48 + 60 = 108 km/hr.

Convert relative speed to m/s = 108 × (5/18) = 30 m/s.

Length of the slower train = Relative Speed × Time taken.

Length = 30 × 12 = 360 metres.

∴ The correct answer is 360 metres.

Alternate Method

Slower Train (L)

48 km/hr

Faster Train

60 km/hr

Man

Relative Speed = 108 km/hr

Given:

Speed of slower train (S₁) = 48 km/hr

Speed of faster train (S₂) = 60 km/hr

Time taken to cross the man (T) = 12 seconds

Formula Used:

Relative Speed (Opposite Direction) = S₁ + S₂

Distance = Speed × Time

Conversion: 1 km/hr = 5/18 m/s

Calculations:

⇒ Relative Speed = 48 + 60 = 108 km/hr

⇒ Relative Speed in m/s = 108 × (5/18) = 30 m/s

⇒ The man sitting in the faster train represents a point object moving at 60 km/hr in the opposite direction.

⇒ The distance covered to cross the man is exactly equal to the length of the slower train.

⇒ Length of slower train = 30 m/s × 12 s

⇒ Length of slower train = 360 metres

∴ The correct answer is 360 metres.

Additional Information

Relative Speed

When two bodies move in the same direction, relative speed = |S₁ − S₂|. When they move in opposite directions, relative speed = S₁ + S₂.

Crossing a Point Object

When a train crosses a pole, tree, or a standing/moving man, the total distance covered is equal to the length of the crossing train itself.

Speed Conversion Rule

To convert km/hr to m/s, multiply by 5/18. To convert m/s to km/hr, multiply by 18/5.

Given:

Shopkeeper declares a discount of 15%

Selling price = Rs. 1445

Formula used:

SP = MP × (100 - Discount)%

Calculations:

According to the formula,

⇒ SP = MP × (100 - 15)%

⇒ 1445 = MP × 85%

⇒ MP = $\frac{1445}{85}\times100$ = 1700

⇒ Hence, The marked price of the chair is Rs. 1700

Given:

Ajay and Vijay work on the same machine to cut diamonds but on alternate hour basis

Ajay can complete the work in 6 hours

Vijay completes it in 16 hours

Formula used:

Total work = Efficiency × Time

Calculations:

⇒ Total work = LCM of 6 and 16 = 48

⇒ Efficiency of Ajay = 48 / 6 = 8 

⇒ Efficiency of Vijay = 48 / 16 = 3

As they work in alternative basis

⇒ In 2 hours = 8 + 3 = 11 work is done

⇒ 8 hours = 44 work

Remaining 44 work has to be done by Ajay

⇒ Time = 4 / 8 = 0.5 hours = 30 minutes 

⇒ Total time taken by Ajay and Vjay to complete the work on alternative basis = 8h + 30min

⇒ Hence, They take 8h 30 min to complete the work

Given:

∠A = 55º

 AB = DE

AC = DF

∠E = 85°

∠F = 40°

Formula used:

Sum of angles of the triangle = 180º

Calculations:

In ∆DEF

⇒ ∠D + ∠E + ∠F = 180

⇒ ∠D = 180 - 85 - 40 = 55°

So, In ∆ABC and ∆DEF

⇒ AB = DE

⇒ AC = DF

⇒ and ∠A = ∠D = 55º

So, It is congruent by SAS rule

⇒ Hence, Option 1 is correct

Given:

John drives 250 km at 50 km/h

He drives 350 km at 70 km/h

Formula used:

Speed = Distance / Time

Calculations:

⇒ Time taken to cover first distance = 250 / 50 = 5 h

⇒ Time taken to cover second distance = 350 / 70 = 5 h

Average speed is 

⇒ Average speed = (250 + 350) / (5 + 5) = 600 / 10 = 60 km/h

⇒ Hence, His average speed is 60 km/h

Given:

y = 1 + √3 + √4

Formula used:

(x + y)² = x² + y² +2xy

Calculations:

⇒ y = 1 + √3 + √4,

⇒ y = 1 + √3 + 2 = 3 + √3

⇒ y² = (3 + √3)²

⇒ y² = 9 +3 + 6√3 = 12 + 6√3

According to the question,

⇒ 2y4 - 8y3 - 6y2 + 28y - 84

⇒ 2y²(y² - 4y - 3) + 28(3 + √3) - 84

⇒ 2y2(y2 - 4y - 3 + 4 - 4) + 84 + 28√3 - 84

⇒ 2y2(y2 - 4y + 4 - 7) + 28√3

⇒ 2y2{(y-2)² - 7} + 28√3

Now put the value of y and y²

⇒ 2(12 + 6√3){(3 + √3 -2)² - 7} + 28√3

⇒ (24 + 12√3){(1 + √3)2 - 7} + 28√3

⇒ (24 + 12√3){1 + 3 + 2√3 - 7} + 28√3

⇒ (24 + 12√3){2√3 - 3} + 28√3

⇒ 48√3 - 72 + 72 - 36√3 + 28√3

⇒ 40√3

⇒ Hence, Option 1 is correct

Shortcut Trick

Concept Used:

A common tangent is a line that touches both circles.

Here, the common tangents to the two circles are shown in the figure.

For two separate circles, there are:

⇒ 2 direct common tangents + 2 transverse common tangents

∴ The number of common tangents is 4.

Alternate Method

Given: Two circles are non-intersecting, non-touching, and one does not lie inside the other.

Formula: Let C₁ and C₂ be the centers and r₁ and r₂ be the radii. If C₁C₂ > r₁ + r₂, then 4 common tangents exist.

Calculations:

⇒ Direct Common Tangents (DCT): These are tangents where both circles lie on the same side of the line. There are exactly 2 such tangents (top and bottom).

⇒ Transverse Common Tangents (TCT): These are tangents where the circles lie on opposite sides of the line. There are exactly 2 such tangents (crossing in the middle).

⇒ Total Tangents = Count(DCT) + Count(TCT)

⇒ Total Tangents = 2 + 2 = 4

∴ The correct answer is 4.

Additional Information

Touching Externally

If the distance between centers d = r₁ + r₂, there are 3 common tangents (2 direct and 1 at the point of contact).

Intersecting Circles

If |r₁ − r₂| < d < r₁ + r₂, there are 2 common tangents (only direct common tangents).

Touching Internally

If the distance between centers d = |r₁ − r₂|, there is only 1 common tangent.

One Inside Another

If d < |r₁ − r₂|, there are 0 common tangents as one circle is completely enclosed by the other.

Given:

A person having bought goods for Rs. 400 sells half of it at a gain of 5%

He has to gain 25% on the whole

Formula used:

Percentage = $\frac{Profit}{CP}×100$

Calculations:

As he sell half of it at a gain of 5%

Rs. 200 at a profit of 5%

⇒ SP = 200 + 5% of 200 = 200 + 10 = Rs. 210

Now he has to gain 25% on the whole

⇒ Gain = 400 + 25% of 400 = 400 + 100 = Rs. 500

⇒ Profit he has to earned more = 500 - 210 = Rs. 290

⇒ Profit = 290 - 200 = 90

According to the formula,

⇒ Percentage = $\frac{90}{200}×100$= 45%

⇒ Hence, He has to sell the remaining part at 45%

Shortcut Trick

1^st ​Half of 400 = 200, sells at 5% profit, CP = 200 SP = 210

Final profit on 400, 25% so CP = 400 and SP = 500

So, the 2^nd half CP = 200 and SP = (500 - 210) = 290

Profit% = 90/200 × 100 = 45%

Shortcut Trick

Given cot A = 12/5, which represents Base/Perpendicular ratio.

Using the Pythagorean triplet (5, 12, 13), the Hypotenuse is 13.

sin A = Perpendicular / Hypotenuse = 5 / 13.

∴ The correct answer is 5/13.

Alternate Method

Given: cot A = 12/5

Formula Used: cot A = Base (B) / Perpendicular (P); sin A = Perpendicular (P) / Hypotenuse (H); H² = P² + B²

⇒ Let Perpendicular (P) = 5k and Base (B) = 12k

⇒ Hypotenuse (H) = √(P² + B²)

⇒ H = √((5k)² + (12k)²) = √(25k² + 144k²) = √(169k²)

⇒ H = 13k

⇒ sin A = P / H = 5k / 13k = 5/13

∴ The correct answer is 5/13.

Additional Information

Trigonometric Ratios

Standard ratios: sin = P/H, cos = B/H, tan = P/B, cot = B/P, sec = H/B, cosec = H/P.

Pythagorean Identity

Fundamental identity: sin²A + cos²A = 1 and 1 + cot²A = cosec²A.

Common Triplets

Frequently used right-angled triangle sides: (3,4,5), (5,12,13), (8,15,17), (7,24,25).

Shortcut Trick

Use the trigonometric property: tan θ × tan(90° − θ) = 1.

Here, θ = 6° and (90° − θ) = 84°, so tan 6° × tan 84° = 1.

The standard value of tan 45° = 1.

The product is 1 × 1 = 1.

∴ The correct answer is 1.

Alternate Method

Given: The expression is tan 6° × tan 45° × tan 84°.

Formula Used:

1. Complementary Angle: tan(90° − θ) = cot θ
2. Reciprocal Identity: tan θ × cot θ = 1
3. Standard Value: tan 45° = 1

Calculations:

⇒ tan 6° × tan 45° × tan 84°

⇒ tan 6° × 1 × tan(90° − 6°)

⇒ tan 6° × cot 6°

⇒ tan 6° × (1 ÷ tan 6°)

⇒ 1

∴ The correct answer is 1.

Additional Information

Complementary Angle Properties

If A + B = 90°, then sin A = cos B, tan A = cot B, and sec A = cosec B. Also, tan A × tan B = 1.

Trigonometric Identities

Standard identities include sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ.

Standard Values of tan θ

tan 0° = 0, tan 30° = 1/√3, tan 45° = 1, tan 60° = √3, and tan 90° is undefined (∞).

Given:

data shown below:

Calculations:

Change in the number of persons who have taken the insurance policy in the year 2014 to that of in year 2011

= 72 - 54

= 18

Required percentage

= 18 × 100/54

= 100/3

= 33.33%

Hence, The Required value is 33.33%.

Given:

 $x = {{\sqrt5 - \sqrt4} \over {\sqrt5 + \sqrt4}}$ and $y = {{\sqrt5 + \sqrt4} \over {\sqrt5 - \sqrt4}}$

Calculations:

According to question,

 $x = {{\sqrt5 - \sqrt4} \over {\sqrt5 + \sqrt4}}$ 

⇒x = 9 - 4√5

⇒x² = 161 - 72√5

and $y = {{\sqrt5 + \sqrt4} \over {\sqrt5 - \sqrt4}}$ 

⇒y = 9 + 4√5

⇒y² = 161 + 72√5

Now, 

 $x^2 - xy + y^2 \over x^2 +xy + y^2$

= (161 - 72√5 - 1 + 161 + 72√5)/(161 - 72√5 + 1 + 161 + 72√5)

= (322 - 1)/(322 + 1)

= 321/323

Hence, The Required value is 321/323.

Alternate Method

We know, if x = $\frac{(p - q)}{(p + q)}$ and y = $\frac{(p + q)}{(p - q)}$ then (x + y) = $\frac{2\times (p^2 + q^2)}{(p^2 - q^2)}$

 $x = {{\sqrt5 - \sqrt4} \over {\sqrt5 + \sqrt4}}$ and $y = {{\sqrt5 + \sqrt4} \over {\sqrt5 - \sqrt4}}$  so, x + y = $\frac{2(\sqrt5^2 + \sqrt4^2)}{(5 - 4)}$ = 18 and xy = 1

So, $x^2 - xy + y^2 \over x^2 +xy + y^2$ = $\frac{(x + y)^2 - 3xy}{(x + y)^2 -xy}$ = $\frac{18^2 - 3}{18^2 -1}$ = 321/323

Shortcut Trick

Use the standard trigonometric identity relation: (sin θ + cos θ)² + (sin θ − cos θ)² = 2(sin² θ + cos² θ).

Since sin² θ + cos² θ = 1, we have (sin θ + cos θ)² + (sin θ − cos θ)² = 2.

Let (sin θ + cos θ) = x. Then x² + (4/5)² = 2.

x² = 2 − 16/25 = (50 − 16)/25 = 34/25 ⇒ x = √34/5.

∴ The correct answer is √34 / 5.

Alternate Method

Given: sin θ − cos θ = 4/5

Formula Used: (a + b)² + (a − b)² = 2(a² + b²) and sin² θ + cos² θ = 1

⇒ Squaring the given equation: (sin θ − cos θ)² = (4/5)²

⇒ sin² θ + cos² θ − 2 sin θ cos θ = 16/25

⇒ 1 − 2 sin θ cos θ = 16/25

⇒ 2 sin θ cos θ = 1 − 16/25 = 9/25

⇒ Now, consider (sin θ + cos θ)² = sin² θ + cos² θ + 2 sin θ cos θ

⇒ (sin θ + cos θ)² = 1 + 9/25 = 34/25

⇒ sin θ + cos θ = √(34/25) = √34 / 5

∴ The correct answer is √34 / 5.

Additional Information

Fundamental Identity

The most basic trigonometric identity is sin² θ + cos² θ = 1, which holds for all values of θ.

Algebraic-Trig Relation

If sin θ ± cos θ = k, then squaring both sides gives 1 ± 2 sin θ cos θ = k², which relates the sum/difference to the product.

Conversion Formula

The relationship between (sin θ + cos θ) and (sin θ − cos θ) is given by (sin θ + cos θ) = √(2 − (sin θ − cos θ)²).

Given:

Length of the rod = 12 cm

Diameter of the rod = 4 cm

Length of the wire = 12 m

Calculations:

According to question,

Volume of rod = Volume of wire

⇒π(r')²h' = πr²h

⇒r² = (r')²h'/h

⇒r² = (2)² × 12/1200

⇒r² = 0.04

⇒r = √0.04 = 0.2

So, thickness= diameter= 2 × 0.2 = 0.4 cm

Hence, The Required value is 0.4 cm.

Given:

1st Proportional(a) = 16

2nd Proportional(b) = 26

3rd Proportional(c) = 32

Formula Used:

4th Proportional(d) = (b x c) / a

Calculation:

According to the formula,

⇒ 4th Proportional(d) = (b x c) / a = (26 x 32)/16 = 52

∴ The fourth proportion is 52

Given:

tan (α + β) = √3, tan (α - β) = 1 

Calculations:

According to question,

tan (α + β) = √3

⇒ tan (α + β) = tan60°

⇒α + β = 60° ----------------------(1)

And, tan (α - β) = 1

⇒α - β = 45° -----------------------(2)

From equations (1) and (2), we get:

α = 105°/2

β = 15°/2

So, tan (6α) 

= tan(6 × 105°/2)

= tan315°

= tan(360° - 45°)

= - tan(45°)

= - 1

Hence, The Required value is -1.

Given:

 ${2p \over {p^2 - 5p + 1}} = {1 \over10}, p \ne 0$

Calculations:

According to question,

 ${2p \over {p^2 - 5p + 1}} = {1 \over10}, p \ne 0$,

⇒20p = p² - 5p + 1

⇒p² + 1 = 25p

⇒p + 1/p = 25

Hence, The Required value is 25.

Given:

In a division sum, the divisor is 10 times the quotient and 4 times the remainder. If the remainder is 45.

Concept used:

Dividend = Divisor × Quotient + Remainder

Calculation:

The divisor is four times the remainder

Remainder = 45

Divisor = 45 × 4

The Divisor is Ten times the Quotient 

⇒ 45 × 4 = 10 × Q

⇒ Q = (45 × 4) / 10

⇒ Q = 18

Now using the Basic Formulae

Dividend = 45 × 4 × 18 + 45

Dividend = 3285

Hence, The Required value is 3285.

Given:

 The table below shows the number of players participating in two games in six different schools.

 

Calculations:

No. of students participating in game L from School D and E

= 82 + 68

= 150

No. of students participating in game M from School H and I

= 86 + 108

= 194

Percentage required

= (194 - 150) × 100/194

= 44 × 100/194

= 22.68%

Hence, The Required value is 22.68%.

Calculations:

HCF of 222, 642 and 1062

= HCF of 6 × 37 , 6 × 107 and 6 × 177

= 6

Hence, The Required value is 6.

Given:

The table given below shows the cost price and profit percentage of 5 articles.

 

Calculations:

Selling price of P = 190% of 450 = 1.9 × 450 = 855

Selling price of Q = 200% of 250 = 500

Selling price of R = 160% of 600 = 960

Selling price of S = 170% of 730 = 1241

Selling price of T = 180% of 400 = 720

total Selling price of P and Q = 855 + 500 = 1355

Total Selling price of R, S and T= 960 + 1241 + 720 = 2921

Percentage required

= (1355) × 100/2921

= 1355 × 100/2921

= 46.38%

Hence, The Required value is 46.38%.

Shortcut Trick

Total distributed profit = Total profit × 90%.

A's share = Total profit × (90 / 100) × [9 / (9 + 5)].

29840 = Total profit × (9 / 10) × (9 / 14).

Total profit = 29840 × (140 / 81) = 4177600 / 81 = 51575.30...

∴ The correct answer is Rs. 51,575.

Alternate Method

Given:

Investment ratio of A and B = 9 : 5

Percentage of profit going to charity = 10%

A's profit share = Rs. 29,840

Formula Used:

Profit ratio = Investment ratio (when the time period is identical)

Calculations:

⇒ Let the total profit be P.

⇒ Profit distributed between A and B = P − 10% of P = 90% of P = 0.9P.

⇒ Investment ratio of A and B = 9 : 5

⇒ Total ratio parts = 9 + 5 = 14 parts.

⇒ A's profit share = (9 / 14) of the remaining profit

⇒ 29840 = (9 / 14) × 0.9P

⇒ 29840 = (8.1 / 14) × P

⇒ P = (29840 × 14) / 8.1

⇒ P = 417760 / 8.1 = 4177600 / 81

⇒ P = 51575.308...

⇒ Rounding to the nearest integer, Total Profit ≈ Rs. 51,575

∴ The correct answer is Rs. 51,575.

Additional Information

Sleeping Partner

A partner who only invests money but does not take part in the management of the business is known as a sleeping partner. Active partners receive a fixed salary or a certain percentage of the profit for managing the business before the remaining profit is distributed among all partners.

Compound Partnership

When investments are made for different time periods, it is called a compound partnership. The profit share ratio is the product of capital invested and the time period of investment (P₁ : P₂ = C₁×T₁ : C₂×T₂).

Joining Mid-way

If a partner joins x months after the start of a financial year, their investment time period for that year is (12 − x) months, which must be multiplied by their capital to determine their final profit share ratio compared to the primary investors.