SSC CGL 2022 Tier-I Official Paper (Held On : 09 Dec 2022 Shift 4) (Previous Year Paper)

The logic followed here is:

 

Logic: 

1) Addition of + 3 in all the consonants.

2) Subtraction of - 1 in all the Vowels.

3) First and last letters are interchanged.

1. 'FLOAT' is written as 'WONZI'

And,

2. 'TRIBE' is written as 'DUHEW'

Similarly,

3. 'GARDEN' is written as '?'

Hence, the correct answer is "QZUGDJ".

The pattern followed here is:

Logic: Addition of given number = 85.

1. (21, 33, 31)

⇒ 21 + 33 + 31 = 85

2. (42, 17, 26)

⇒ 42 + 17 + 26 = 85

Let's check each option one by one:

Option 1) (36, 14, 35)

⇒ 36 + 14 + 35 = 85. (Correct)

Option 2) (46, 19, 24)

⇒ 46 + 19 + 24 = 89 ≠ 85. (Incorrect)

Option 3) (35, 27, 29)

⇒ 35 + 27 + 29 = 91 ≠ 85. (Incorrect)

Option 4) (33, 39, 20)

⇒ 33 + 39 + 20 = 92 ≠ 85. (Incorrect)

Hence, the correct answer is "(36, 14, 35)".

The pattern followed here is:

We have to find after interchanging the given two signs and numbers which of the following equation will be correct.

Given: + and –, 6 and 3

Let's check each option one by one.

Option 1) 8 – 3 × 4 ÷ 6 + 9 = 18

Given: + and –, 6 and 3

After interchanging the given two signs and numbers we get the;

⇒ 8 + 6 × 4 ÷ 3 - 9 = 18

⇒ 8 + 6 × 1.33 - 9 = 18

⇒ 8 + 8 - 9 = 18

⇒ 16 - 9 = 18

⇒ 7 ≠ 18

⇒ LHS ≠ RHS

Option 2) 8 + 4 × 6 ÷ 3 – 9 = 15

Given: + and –, 6 and 3

After interchanging the given two signs and numbers we get the;

⇒ 8 - 4 × 3 ÷ 6 + 9 = 15

⇒ 8 - 4 × 0.5 + 9 = 15

⇒ 8 - 2 + 9 = 15

⇒ 17 - 2 = 15

⇒ 15 = 15

⇒ LHS = RHS

Option 3) 9 ÷ 3 × 4 – 6 + 7 = 10

Given: + and –, 6 and 3

After interchanging the given two signs and numbers we get the;

⇒ 9 ÷ 6 × 4 + 3 - 7 = 10

⇒ 1.5 × 4 + 3 - 7 = 10

⇒ 6 + 3 - 7 = 10

⇒ 9 - 7 = 10

⇒ 2 ≠ 10

⇒ LHS ≠ RHS

Option 4) 6 ÷ 2 × 8 + 3 – 1 = 20

Given: + and –, 6 and 3

After interchanging the given two signs and numbers we get the;

⇒ 3 ÷ 2 × 8 - 6 + 1 = 20

⇒ 1.5 × 8 - 6 + 1 = 20

⇒ 12 - 6 + 1 = 20

⇒ 13 - 6 = 20

⇒ 7 ≠ 20

⇒ LHS ≠ RHS

Hence, the correct answer is "Option 2".

The pattern followed here is:

Logic: The given word is synonyms of each other.

1. Pain : Agony → Agony means great pain or suffering from pain. Both are synonyms of each other.

Similarly,

2. Anger : Rage → Rage means a feeling of violent anger that is difficult to control. Both are synonyms of each other.

Hence, the correct answer is "Rage".

Additional Information 

The pattern followed here is:

Logic: First number + 10 = Second number and Second number + 20 = Third number.

Let's check each option one by one:

Option 1) (18 –28 –48 )

⇒ 18 + 10 = 28 and 28 + 20 = 48. (Correct)

Option 2) (17 –27 –47 )

⇒ 17 + 10 = 27 and 27 + 20 = 47. (Correct)

Option 3) (15 –25 –49 )

⇒ 15 + 10 = 25 and 25 + 20 = 45 ≠ 49. (Incorrect)

Option 4) (13 –23 –43 )

⇒ 13 + 10 = 23 and 23 + 20 = 43. (Correct)

Hence, the correct answer is "(15 - 25 - 49)".

The pattern followed here is:

Interchanging the given two numbers to find which of the following equation will be correct.

Let's check each option one by one:

Option 1) 9 × 3 – 4 ÷ 2 + 1 = 10

Given: 1 and 9

After interchanging the numbers we get the:

⇒ 1 × 3 – 4 ÷ 2 + 9 = 10

⇒ 1 × 3 - 2 + 9 = 10

⇒ 3 - 2 + 9 = 10

⇒ 12 - 2 = 10

⇒ 10 = 10

⇒ LHS = RHS

Option 2) 1 + 9 × 3 – 4 ÷ 2 = 18

Given: 1 and 9

After interchanging the numbers we get the:

⇒ 9 + 1 × 3 – 4 ÷ 2 = 18

⇒ 9 + 1 × 3 - 2 = 18

⇒ 9 + 3 - 2 = 18

⇒ 12 - 2 = 18

⇒ 10 ≠ 18

⇒ LHS ≠ RHS

Option 3) 9 × 2 + 1 – 8 ÷ 4 = 12

Given: 1 and 9

After interchanging the numbers we get the:

⇒ 1 × 2 + 9 – 8 ÷ 4 = 12

⇒ 1 × 2 + 9 - 2 = 12

⇒ 2 + 9 - 2 = 12

⇒ 11 - 2 = 12

⇒ 9 ≠ 12

⇒ LHS ≠ RHS

Option 4) 9 – 1 + 6 ÷ 2 × 3 = 7

Given: 1 and 9

After interchanging the numbers we get the:

⇒ 1 – 9 + 6 ÷ 2 × 3 = 7

⇒ 1 - 9 + 3 × 3 = 7

⇒ 1 - 9 + 9 = 7

⇒ 10 - 9 = 7

⇒ 1 ≠ 7

⇒ LHS ≠ RHS

Hence, the correct answer is "Option 1".

The pattern followed here is:

The total number of odd days is listed below:

Therefore 2015 will be the repeated calendar of 2009.

Shortcut Trick

So, 2008 + 1 = 2009 + 6 = 2015

So, 2015 is the same as the calendar for the year 2009

Hence, the correct answer is "2015".

The pattern followed here is:

Given:

Hence, the correct answer is "Option 2".

The correct mirror image of the given figure when the mirror is held on the PQ side.

Detailed Explanation:

• The correct mirror image of the given figure is "

" the third letter of the given image is 'A' and the correct mirror image is "A" but it is wrongly given in option 3 so option 3 is eliminated.

• The correct mirror image of the given figure is "

" the sixth number of the given image is '8' and the correct mirror image is "8" but it is wrongly given in options 1 & 4 so options 1 & 4 are eliminated.

 

Hence, the correct answer is "Option 2".

Additional Information

1. In the mirror image left side and right side changed vice-versa where the top and bottom remain the same.

2. In a real given mirror image the farthest number or alphabet appears closest in the mirror image.

The pattern followed here is:

Logic: No two opposite faces are adjacent to each other.

Let's find which number is faced opposite to face having '3'.

We are taking both dice figures (1) & (2) together.

Here, 5 and 3 are common in both dice and 6 is opposite to 1.

Now, we are taking dice figures (2) & (3) together.

 

Here, 5 is common in both the dice and the rotated clockwise direction.

Opposite pairs:

⇒ 6 ⇔ 1

⇒ 4 ⇔ 3

⇒ 5 ⇔ 2

Clearly, 3 is the opposite of 4.

Hence, the correct answer is "4".

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

P ÷ Q means that A is the mother of B

P - Q means that A is the father of B

P × Q means that A is the sister of B

Given relation code:

Find: Which expression shows that B is the daughter of A?

Let's check each option one by one.

Option 1) B – A ÷ C

B is the Father of A, and A is the Mother of C.

Here, B is the father of A. (Incorrect)

Option 2) A – B ÷ C

A is the Father of B and B is the Mother of C.

Here, B is the daughter of A. (Correct)

Option 3) A – B - C

A is the Father of B and B is the Father of C.

Here, B is the son of A. (Incorrect)

Option 4) C ÷ B – A

C is the Mother of B and B is the Father of A.

Here, B is the Son of C. (Incorrect)

Hence, the correct answer is "Option 2".

The pattern followed here is:

The least possible Venn diagram is shown below according to the given statements:

Statements:

Some watches are projectors.

All projectors are lamps.

Some heaters are watches.

Conclusions:

(I) Some heaters are projectors → Does not follow → As some watches are projectors, all projectors are lamps and some heaters are watches. So some heaters are projectors is possible but not definitely followed.

(II) Some lamps are heaters → Does not follow ​→ As some watches are projectors, all projectors are lamps and some heaters are watches. So some lamps are heaters is possible but not definitely followed.

(III) All watches are lamps → Does not follow ​→ As some watches are projectors and all projectors are lamps. So all watches are lamps is possible but not definitely followed.

So, None of the conclusions follow.

Hence, the correct answer is "Option 2".

The pattern followed here is: 

Paint : Art → Paint is the art or skill of producing things such as paintings, designs, etc.

Similarly,

Prose : Literature → Prose is a style of written or spoken language that follows the natural flow of speech and is related to Literature.

Hence, the correct answer is "Literature".

Additional Information

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

Given: 

A # B means ‘A is the brother of B’

A @ B means ‘A is the son of B’

A & B means ‘A is the wife of B’

A % B means ‘A is the mother of B’

 

Find how is B related to E.

Given:- A % B & C # D @ E

Clearly, B is the "Daughter-in-law" of E.

Hence, the correct answer is "Daughter-in-law".

The pattern followed here is:

1. The middle figure is rotated 90º clockwise direction in every next figure.

2. New letters formed and shifted in the clockwise direction in every next figure. 

Hence, the correct answer is "Option 4".

The logic followed here is

 15 - 11 = 4

 4 - 12 = - 8

• 8 - 13 = - 21
• 21 - 14 = - 35

- 35 - 15 = - 50 

Hence, the correct answer is - 50.

The position of the English alphabet letters is given below:

Hence, the correct answer is "QDIP".

The pattern followed here is:

Logic: (Third number - Second number) ÷ 3 = First number.

In (4, 21, 33) →

⇒ (33 - 21) ÷ 3

⇒ 12 ÷ 3 = 4

In (10, 111, 141)

⇒ (111 - 141) ÷ 3

⇒ 30 ÷ 3 = 10

Let's check each option one by one:

Option 1) (13, 45, 97)

⇒ (97 - 45) ÷ 3

⇒ 52 ÷ 3 = 17.33 ≠ 13. (Incorrect)

Option 2) (25, 53, 128)

⇒ (128 - 53) ÷ 3

⇒ 75 ÷ 3 = 25 = 25. (Correct)

Option 3) (65, 60, 244)

⇒ (244 - 60) ÷ 3

⇒ 184 ÷ 3 = 61.33 ≠ 65. (Incorrect)

Option 4) (24, 52, 128)

⇒ (128 - 52) ÷ 3

⇒ 76 ÷ 3 = 25.33 ≠ 24. (Incorrect)

Hence, the correct answer is "(25, 53, 128)".

The position of the English alphabet letters is given below:

Hence, the correct answer is "WJO".

The logic followed here is:

Logic: 

Step 1: First letter + 8 = Third letter

Step 2: Second and Third letters are opposite positional alphabets.

Let's check each option one by one:

Option 1) DOL

Option 2) MIP

Option 3) ARI

Option 4) NEV

Hence, the correct answer is "MIP".

The position of the English alphabet letters is given below:

1) 'TEACHER' is written as 'AETCREH'

And,

2) ‘STUDENT’ is written as ‘UTSDTNE’

Similarly,

3) 'CLASSES' be written as '?'

Hence, the correct answer is "ALCSSES".

The pattern followed here is:

Find which two signs should be interchanged to make the given equation correct.

Let's check each option one by one.

Option 1) − and ×

Given: 81 × 9 + 15 ÷ 3 – 4 = 50

After interchanging the signs we get the;

⇒ 81 - 9 + 15 ÷ 3 × 4 = 50

⇒ 81 - 9 + 3 × 4 = 50

⇒ 81 - 9 + 12 = 50

⇒ 93 - 9 = 50

⇒ 84 ≠ 50

⇒ LHS ≠ RHS

Option 2) + and −

Given: 81 × 9 + 15 ÷ 3 – 4 = 50

After interchanging the signs we get the;

⇒ 81 × 9 - 15 ÷ 3 + 4 = 50

⇒ 81 × 9 - 5 + 4 = 50

⇒ 729 - 5 + 4 = 50

⇒ 733 - 5 = 50

⇒ 728 ≠ 50

⇒ LHS ≠ RHS

Option 3) ÷ and −

Given: 81 × 9 + 15 ÷ 3 – 4 = 50

After interchanging the signs we get the;

⇒ 81 × 9 + 15 - 3 ÷ 4 = 50

⇒ 81 × 9 + 15 - 0.75 = 50

⇒ 729 + 15 - 0.75 = 50

⇒ 744 - 0.75 = 50

⇒ 743.25 ≠ 50

⇒ LHS ≠ RHS

Option 4) ÷ and ×

Given: 81 × 9 + 15 ÷ 3 – 4 = 50

After interchanging the signs we get the;

⇒ 81 ÷ 9 + 15 × 3 – 4 = 50

⇒ 9 + 15 × 3 - 4 = 50

⇒ 9 + 45 - 4 = 50

⇒ 54 - 4 = 50

⇒ 50 = 50

⇒ LHS = RHS

Hence, the correct answer is "Option 4".

The pattern followed here is:

The least possible Venn diagram is shown below according to the given statements.

Statements:

I. Some A are W.

II. Some W are C.

Conclusions:

I. All A are C → Does not follow → As Some A are W and Some W are C. So All A are C is possible but not definitely followed.

II. No C is W → Does not follow → As Some W are C. So No C is W is definitely not followed.

III. No W is A → Does not follow → As Some A are W. So No W is A is definitely not followed.

So, Neither conclusion follows.

Hence, the correct answer is "Option 1".

The pattern followed here is:

Logic: (First number)² - First number = Second number.

1. 12 : 132

⇒ (12)² → 144 - 12 = 132

And,

2. 20 : 380

⇒ (20)2 → 400 - 20 = 380

Similarly,

3. 2 : ?

⇒ (2)2 → 4 - 2 = 2

Hence, the correct answer is "2".

The pattern followed here is:

1. Middle figure straight line rotated 45º clockwise direction, and one line is added after every rotation in the next figure.

2. In every next figure one dot is added in the anti-clockwise direction.

Hence, the correct answer is "Option 1".

The correct answer is Sultan Muhammad Ghori.

Key Points

• The best-known Chahamana ruler was Prithviraj III (1168- 1192), who defeated an Afghan ruler named Sultan Muhammad Ghori in 1191, but lost to him the very next year, in 1192.
• Mohammad Ghori’s invasion (1175 AD-1206 AD):
• Mohammad Ghori invaded India and laid the foundation of Muslim domination in India.
• He is often considered the founder of Muslim rule in India.
• Battle of Tarain:
• In the first battle of Tarain was fought in 1191 the Ghori Forces were completely rooted out by Prithviraj Chauhan.
• The second battle of Tarain was fought in 1192 in which Prithviraj Chauhan was defeated by Ghori.

Additional Information Prithviraja III

• Prithviraja III was the best-known Chahamana ruler.
• From 1177 to 1192 CE, Prithviraja III, sometimes referred to as Prithviraj Chauhan or Rai Pithora, ruled Sapadalaksha.
• He belonged to the Chauhan (Chahamana) dynasty, and his capital was Ajmer in modern-day Rajasthan. In 1177 CE, when Prithviraj ascended the throne as a boy, he inherited a kingdom that ran from Thanesar in the north to Jahazpur (Mewar) in the south.
• He sought to enlarge this realm by engaging in military operations against nearby countries, most notably by defeating the Chandelas.

The correct answer is Odissi.

Key Points

• Sharmila Biswas is an accomplished Odissi dancer in addition to being a Kathak and Manipuri dancer.
• She started learning Odissi dance in the late 1970s under the guidance of Guru Kelucharan Mohapatra, a renowned exponent of the dance form.
• She went on to become one of his foremost disciples and performed with his troupe, Srjan, for many years.
• She has also choreographed and performed her own Odissi productions, such as "Sita Haran," "Sufi and the Mystic," and "Abhimanyu Vadh."
• She has performed Odissi extensively in India and abroad, including at the World Dance Festival in Germany and the Festival of India in the USSR.
• She has received several awards and honours for her contributions to Odissi dance, including the Odisha State Sangeet Natak Akademi Award and the Nritya Choodamani award.
• Padma Shri, 2011.
• Sangeet Natak Akademi Award, 2005.
• National Nritya Shiromani Award, 1994.
• Nandita Kripalini Award, 1992.
• Kalidas Samman, 2008.
• Nritya Choodamani, 2000.
• She has also been honoured with the title of "Jewel of India" by the Rotary Club of India in recognition of her contributions to Indian classical dance.​

Additional Information

The correct answer is France.

Key Points

• The 2024 Olympic Games are hosted by France's capital, Paris, and the 2028 US city of Los Angeles.
• Breaking, the official name for competitive breakdancing, will become an Olympic sport for the first time at the 2024 Paris Olympics.
• The athletic style of dance form emerged in the United States in the 1970s.
• Breakdancing is among 4 sports that have got the green light to be part of the Paris Olympic Games in 2024.
• Surfing, skateboarding and sport climbing were also approved for Paris 2024, but all three events will also be a part of the Tokyo Olympics in 2021.
• The move comes after the International Olympic Committee's new rule which allows host cities to pick and propose new games that are popular in the region.
• The Paris 2024 games will be fully gender equal with 50% female and male participation, as compared with 48.8% female participation in the Tokyo games.​​

Additional Information

• In ancient times it was organized in 1896 in Athens, the capital of Greece.
• It was named Olympic because of the sport played on Mount Olympia.
• Pierre de Kuvarthen is considered the father of the modern Olympic Games.
• The International Olympic Committee was founded in 1894 by Pierre de Coubertin. It is headquartered in Lausanne, Switzerland.
• The Olympic Games motto is "Citius-Altius-Fortius".​
• The five rings on the Olympic emblem are of five colors, red, blue, green, yellow, and black.
• These rings represent five continents. (North and South America is considered a continent.)

The correct answer is Kanjar.

Key Points

• Chakri Dance
• Chakri dance is one of the famous folk dances that are performed by the women of Rajasthan.
• Chakri is the dance of the tribe called Kanjars.
• Popular in area of Kota and Baran district of Rajasthan.
• The unique factor of the Chakri dance is that this is purely performed by the women of Rajasthan.

Additional Information

List of folk dances from different states:-

The correct answer is Federalism.

Key Points

• Among the given options term 'Federalism' is not mentioned in the Preamble of the constitution of India.
• The Preamble is called the introduction letter of the Indian Constitution.
• In a Constitution, it presents the intention of its framers, the history behind its creation, and the core values and principles of the nation.
• The ideals behind the Preamble to India’s Constitution were laid down by Jawaharlal Nehru’s Objectives Resolution, adopted by the Constituent Assembly on January 22, 1947.​

Additional Information

Keywords in Preamble

• We, the people of India: It indicates the ultimate sovereignty of the people of India.
• Sovereignty means the independent authority of the State, not being subject to the control of any other State or external power.
• Sovereign: The term means that India has its own independent authority and it is not a dominion of any other external power.
• In the country, the legislature has the power to make laws that are subject to certain limitations.
• Socialist: The term means the achievement of socialist ends through democratic means.
• It holds faith in a mixed economy where both the private and public sectors co-exist side by side.
• It was added in the Preamble by 42^nd Amendment, 1976.
• Secular: The term means that all the religions in India get equal respect, protection, and support from the state.
• The preamble was amended by the 42^nd Constitutional Amendment Act in 1976, in which three new words Socialist, Secular, and Integrity were added.
• It was incorporated in the Preamble by the 42^nd Constitutional Amendment, 1976.
• Democratic: The term implies that the Constitution of India has an established form of Constitution that gets its authority from the will of the people expressed in an election.
• Republic: The term indicates that the head of the state is elected by the people.
• In India, the President of India is the elected head of the state.
• The preamble was amended by the 42^nd Constitutional Amendment Act in 1976, in which three new words Socialist, Secular, and Integrity were added.
• It secures justice, liberty, equality to all citizens of India and promotes fraternity amongst the people.
• It has been widely regarded as soul of the constitution.

The correct answer is 5, state government.

State Finance Commissions (SFCs):

• The State Finance Commission (SFC) is created by the 73^rd and 74^th Constitutional Amendments to manage State level fiscal relations in India.
• Under Article 243-I of the Constitution of India, the governor of a state is required to constitute a Finance Commission every five years.
• It decides the resource allocation between the state government and the Panchayati Raj Institutions.
• It allocates resources to Panchayati Raj institutions at all three levels in terms of taxes, duties and levies to be collected by the state and the local bodies.
• Article 243Y:- SFC shall also review the financial position of the Municipalities and Panchayats and make recommendations to the Governor about the principles that should govern the distribution of tax collected by the state.
• It also recommends about:
• Grants-in-aid to Panchayati Raj Institutions from the consolidated fund of a state.
• Ways to improve the financial position of the Panchayati Raj Institutions.

The correct answer is carbon credit trading schemes.

Key Points

• Energy Conservation (Amendment) Bill, 2022
• ​The Energy Conservation (Amendment) Bill, 2022 was introduced by the government in the Lok Sabha.
• The bill provides for mandatory minimum use of non-fossil sources, including green hydrogen and green ammonia, for feedstock, setting up a carbon trading market, and bringing large residential buildings under energy conservation.
• This bill seeks to amend the Energy Conservation Act, 2001.
• The Act promotes energy efficiency and conservation. It provides for the regulation of energy consumed by appliances, household appliances, buildings, and industries.

Additional Information

• Energy Conservation Act, 2001
• ​The Energy Conservation Act, 2001 was enacted to provide for efficient use of energy and its conservation and matters connected therewith, the Act provides for the establishment and incorporation of the Bureau of Energy Efficiency (BEE).

The correct answer is Three.

Key Points

• Parliament has passed the Delhi Municipal Corporation (Amendment) Bill, 2022.
• The Bill seeks to further amend the Delhi Municipal Corporation Act, 1957 and proposes to unify the three existing Municipal Corporations of Delhi into one Municipal Corporation of Delhi.
• It also limits the number of councillors from 272 to the maximum of 250.​

Additional Information

• The bill envisages bringing about greater transparency, improved governance and more efficient delivery of civic service for the people of Delhi.
• At present, the three corporations in Delhi, North, South and East Delhi Municipal Corporations, have a total of 272 seats.
• While North and South corporations have 104 wards each, the East Corporation has 64 wards.

The correct answer is The ribosomes of a polysome translate the mRNA into proteins.

Key Points

• Ribosomes of a Polysome 
• The cellular organelles known as ribosomes are made up of proteins and rRNA.
• Before they bind to mRNA, they are present free in the cytoplasm.
• Multiple ribosomes that are joined to the mRNA and contain developing polypeptide chains are known as polysomes.
• A polysome is made up of all the ribosomes that translate a single mRNA into many polypeptides (Proteins) at once.
• The coding portion of mRNA, which is transported by multiple ribosomes, forms a polysome.
• The presence of so much mRNA in cells is due to the ability of multiple ribosomes to participate in an mRNA's function.

Additional Information

• Prokaryotic, eukaryotic, and membrane-bound polrsomes all have different structures of multiple ribosomes also known as polyribosomes.
• Bacterial polysomes are found in double-row formations in prokaryotic cells, and ribosomes are in touch with one another within smaller subunits.
• Planar double row polysomes and densely packed 3D helices, which are similar to that of prokaryotic polysomes, are also present in eukaryotic cells.
• Polysomal profiling is a method for determining the level of gene expression by monitoring polysome activity.
• Translation is stopped using polysomal profiling, and sucrose gradient is then used to separate the resultant cells so that they may be collected by centrifugation.

The correct answer is Proportional income tax.

Key Points

• Proportional income tax can act as an automatic stabilizer in the economy because it changes in response to changes in income levels.
• This means that as income levels rise or fall, the tax burden for individuals and households changes proportionally, which can help to stabilize the economy during periods of economic growth or recession.
• During periods of economic growth, when income levels rise, the government collects more revenue from income taxes.
• This increased revenue can be used to fund government spending programs or to reduce the budget deficit.
• The government can also use this increased revenue to reduce taxes or to provide tax incentives, which can further stimulate economic growth.

Additional Information

• Professional tax
• ​ is a tax levied by some state governments in India on individuals who earn an income through professions, trades, callings or employment.
• It is a form of tax that is collected by the state government, and the rates and rules may vary from state to state.
• The professional tax is usually deducted by the employer from the salary of the employee and then remitted to the state government.
• The amount of professional tax that an individual has to pay depends on the monthly or annual income earned, as well as the state in which the individual is employed.
• The purpose of the professional tax is to help state governments generate revenue for their various development projects and public services.
• It is a mandatory tax that must be paid by individuals who are liable to pay it, and failure to do so can result in penalties and fines.
• Wealth tax:
• ​A tax on an individual's net wealth or assets.
• A form of direct taxation aimed at reducing wealth inequality.
• Levied annually based on the value of an individual's assets at a particular point in time.
• Can be applied to a range of assets including cash, property, investments, and personal possessions.
• The tax rate is usually a percentage of the net value of these assets and can vary depending on the level of wealth.
• Used by some countries to generate revenue and fund public services
• Capital gains tax:​
• Capital gains tax is a tax on the profit earned from the sale of an asset, such as stocks, real estate, or artwork.
• It is a form of direct tax that is imposed by the government on the capital gains realized by an individual or a business.
• When a person or a business sells an asset, the capital gain is the difference between the selling price of the asset and the purchase price or the cost basis of the asset.
• The capital gains tax is then calculated based on the capital gain realized.
• The capital gains tax rate can vary depending on the asset, the length of time the asset was held, and the taxpayer's income level.

The correct answer is South America.

Key Points

• Amazon River is the largest river in the world.
• The Amazon basin is the largest drainage basin in the world.
• The Amazon River is the second-longest river (Only to the Nile), in the world.
• Amazon river in South America is the largest river in the world while Nile river in Africa is the longest river in the world.
• It is the longest and largest river flows through South America.
• The Amazon river originates in the Andes Mountains of Peru.
• It flows through the countries of Peru, Bolivia, Venezuela, Colombia, Ecuador, and Brazil.
• Amazon river discharges its water into the Atlantic Ocean.
• There are no bridges across the entire width of the Amazon river.
• Most part of the Amazon river flows through Brazil.

Mistake Points

• The world’s largest river- Amazon (South America)
• The world’s longest river- Nile (Africa)

Additional InformationSouth America 

• South America is the fourth largest continent in the world.
• The area of South America is 17.84 million sq. km and stands at fourth in terms of area.
• There are 14 countries in this continent.
• Brazil is the largest country and Suriname is the smallest country in this country.
• The longest mountain (fold) chain in the world 'Andes' is located in this country.
• The largest river of the world, river Amazon flows here and along with its tributaries drains Peru, Bolivia, Columbia, Ecuador, Venezuela, and Brazil.
• Amazon forms the world's most diverse Biological hotspot here, the rain forests of Amazon also called 'the lungs of the Earth'.
• The world's highest navigable lake, Lake Titicaca is located here between Peru and Bolivia.
• Amazon, Orinoco, Parana, Uruguay, and Paraguay are the major river flowing through this continent.

Important Points

• Following is the area-wise rank of seven continents:
• Asia> Africa> North America> South America> Antarctica> Europe > Australia.

The correct answer is Bhadrapada.

Key Points

• Krishna Janmashtami celebrated this year on August 30.
• The festival marks the birth of Lord Krishna and is celebrated on the eighth day of the dark fortnight in the month of Bhadrapada (August-September) in India.
• It is also known as Krishna Janmashtami or Gokulashtami.
• According to Hindu mythology, Krishna, the human incarnation of Lord Vishnu, was born on this day to destroy Mathura's demon king, Kansa, the brother of Krishna's virtuous mother, Devaki.

Additional Information

• As per the Vikram Samvat, the following are the months in a calendar year :
• Chaitra
• Vaishak
• Jyastha
• Aashad
• Shravan
• Bhadrapad
• Ashwin
• Kartik
• Agrahayna
• Pausa
• Magh
• Phalgun

The correct answer is the Intermediate goods method.

Key Points

• Expenditure method 
• The expenditure method is a system for calculating gross domestic product (GDP) that combines consumption, investment, government spending, and net exports.
• It is the most common way to estimate GDP. It says everything that the private sector, including consumers and private firms, and government spending within the borders of a particular country, must add up to the total value of all finished goods and services produced over a certain period of time.
• This method produces nominal GDP, which must then be adjusted for inflation to result in the real GDP.
• Product Method:
• The product or value-added method is a way of computing the national income of a country.
• This system is also known as the output or inventory method. This method calculates national income by adding value to a product at every stage of its production.
• Income Method:
• In the income method, the national income is measured by adding up the pretax income generated by the individuals and companies in the economy.
• It consists of income from wages, rent of buildings and land, interest on capital, profits, etc.

Additional Information

• The expenditure on construction is calculated either by finding out the total money spent on new construction or by adding up the value of inputs used in construction.
• The expenditure on material inputs consists of expenditure on cement, steel, bricks, wood, fixtures and fittings. Besides, factor income payments to labour and capital are to be added with the expenditure on inputs to find out the value of construction.

The correct answer is Rajasthan.

Key Points

• Sambhar Lake is India’s largest inland salt lake located near Jaipur, Rajasthan.
• The lake is surrounded on all sides by the Aravali hills.
• Sambhar Lake has been designated as a Ramsar site in 1990 because the wetland is a key wintering area for tens of thousands of flamingos and other birds that migrate from northern Asia.
• The specialized algae and bacteria growing in the lake provide striking watercolours and support the lake ecology that, in turn, sustains the migrating waterfowl.

Additional Information

The correct answer is Palk Strait.

Key Points

A strait is a narrow channel of water that joins two water bodies and separates two land masses.

• Palk Strait
• The strait is between southeastern India and northern Sri Lanka.
• The southwestern portion of the strait is also called Palk Bay.
• The strait is named after Robert Palk, who was a governor of Madras during the Company Raj period.
• The strait is 40 to 85 miles (64 to 137 km) wide, 85 miles long, and less than 330 feet (100 meters) deep.
• It receives several rivers, including the Vaigai (India), and it contains many islands on the Sri Lankan side.
• The port of Jaffna, the commercial center for northern Sri Lanka, lies on the strait.

Additional Information

Important Points

The Correct Answer is Bhakra Dam.

Key Points

• Bhakra Dam is the highest straight gravity dam in India.
• It is around 226 m high and 518 m long.
• Bhakra dam is constructed on Satluj river and located in Himachal Pradesh and Punjab border near Nangal city.

Additional Information

The correct answer is NH₄OH.

Key Points

• The chemical formula for ammonium hydroxide is NH₄OH.
• Ammonium hydroxide is a colourless liquid with a strong, pungent odour.
• It is made by dissolving ammonia gas in water, and the resulting solution contains around 30-35% ammonia by weight.
• Uses:
• Ammonium hydroxide is commonly used as a cleaning agent in household and industrial applications.
• It is an effective degreaser and can help remove stains and dirt from surfaces.
• Ammonium hydroxide is also used in the manufacturing of various products, such as fertilizers, plastics, textiles, and pharmaceuticals.
• It is used in the food industry as a leavening agent and pH regulator.
• It is sometimes used to adjust the acidity of foods, such as baked goods and cheeses, and to prevent the browning of certain fruits and vegetables.
• While ammonium hydroxide is generally considered safe for its intended uses, it can be hazardous if mishandled or used improperly.
• The fumes can be irritating to the eyes, nose, and throat, and prolonged exposure can cause respiratory problems.

Additional Information

• Important elements and their day-to-day uses:
• Sodium chloride (NaCl) - Also known as table salt, sodium chloride is commonly used as a seasoning for food.
• Calcium carbonate (CaCO₃) - Calcium carbonate is used in a variety of products, including toothpaste, antacids, and baking powder.
• Sodium bicarbonate (NaHCO₃) - Also known as baking soda, sodium bicarbonate is used as a leavening agent in baking, as well as for cleaning and deodorizing.
• Hydrogen peroxide (H₂O₂) - Hydrogen peroxide is used as a disinfectant and for bleaching and whitening various materials, such as hair and teeth.
• Ammonium nitrate (NH₄NO₃) - Ammonium nitrate is used as a fertilizer in agriculture and as an ingredient in explosives.
• Ethanol (C₂H₅OH) - Ethanol is used as a fuel, a solvent, and as an ingredient in alcoholic beverages.
• Acetic acid (CH3COOH) - Acetic acid is used as a food preservative, as well as in the production of vinegar and other chemicals.
• Sodium hypochlorite (NaClO) - Sodium hypochlorite is used as a disinfectant and for bleaching various materials, such as textiles and paper products.
• Citric acid (C₆H₈O₇) - Citric acid is used as a food preservative and flavouring agent, as well as in cleaning products and pharmaceuticals.
• Sodium carbonate (Na₂CO₃) - Also known as washing soda, sodium carbonate is used as a cleaning agent and water softener, as well as in the production of glass and other chemicals.

The famous Bolan Pass is situated in  Mehrgarh.

Key Points 

• Mehrgarh is a Neolithic site.
• It is located near the Bolan Basin of Balochistan in Pakistan.
• It is one of the earliest sites known.
• It shows evidence of farming and herding done by man in very early times.
• Archaeological evidence suggests that Neolithic culture existed in Mehergarh as early as 7000 BC.
• Mehrgarh in Bolan Pass was probably one of the places where people learnt to grow barley and wheat, and rear sheep and goats for the first time in this area.
• Other finds at Mehrgarh include remains of square or rectangular houses.
• Each house had four or more compartments, some of which may have been used for storage.

Additional Information

• Name of the major cities are given below:
• Mohenjodaro (Sind)-It is located on the right bank of the Indus.
• Kalibangan (Rajasthan)-It was on the banks of the river Ghaggar river.
• Chanhudaro - It lies on the left bank of the Indus in the south of Mohenjodaro.
• Lothal(Gujarat) - It is located at the head of the Gulf of Cambay.
• Surkotada (Gujarat) - It is at the head of the Rann of Kutch.
• Banawali (Haryana) - It was situated on the banks of the now-extinct Sarasvati.
• Dholavira (Gujarat) - It excavated is in the Kutch district.
• Social features-
• Indus valley civilization is the first urbanization in India.
• It has a well-planned drainage system, grid pattern, and town planning.
• They have possessed equality in society.
• Religious Facts-
• Matridevi or Shakti is the Mother goddess.
• Yoni worship and Nature worship existed.
• They worshipped trees like Peepal.
• People also worshipped Fire worship called Havan Kund.
• Pashupati Mahadeva is known as the lord of Animals.
• The people of Indus Valley Civilization worshipped Animal worship like Unicorn and ox.
• Economic facts-
• Indus valley civilization is based on agriculture.
• Trade and commerce flourished in this period.
• A dockyard has been found at Lothal.
• There were export and import.
• Production of cotton was there.
• There was also the unit of measurement.
• Weights and measures of truth existed in Harappan culture was seen at Lothal.
• The weights were made of limestone, steatite, etc. and were usually cubical in shape.

The correct answer is 78%.

Key Points 

Nitrogen and Oxygen are the main components in the Earth's atmosphere by volume as these two gases together account for 99% of the atmosphere.

• Nitrogen(N2): A colourless and tasteless gas, accounts for approx 78% of the atmosphere. It is a major component of chlorophyll and amino acids, the building blocks of proteins.
• Oxygen(O2): A colourless, odourless and tasteless gas, accounts for approx 21% of the atmosphere and makes life possible.

 

Additional Information

The atmospheric composition of gases are as follow

The Correct answer is 1905​.

Key Points

• Partition of ​Bengal took place in 1905.
• Lord Curzon was the one who divided it.
• On October 16, 1905, the division of Bengal took effect.
• The British used the partition of Bengal to carry out their divide and rule agenda in India.
• One of the most prominent anti-partition movements was the Swadeshi movement.
• In 1911, the Bengal Partition was abolished. Lord Hardinge II canceled it
• Bengal was divided for the second time in 1947 as part of India's split.

Additional Information

Anti partition Movement:

• Opposing the decision of dividing the state, people came out on the streets of Bengal in protest.
• People showcased the Hindu-Muslim unity in various processions.
• The day of partition was observed as 'National Day of Mourning'.
• People came out on streets singing 'Bande Mataram' and 'Amar Sonar Bangla'.
• On the day of Raksha Bandhan Hindus and the Muslims tied rakhis on the wrists of each other as a symbol of Brotherhood.

The correct answer is 1905.

Key Points

• Shyamji Krishna Varma started the India house in London, in 1905.
• India House was a student residence at Cromwell Avenue in Highgate, North London.
• Shyamji Krishna Varma is an Indian patriot, lawyer, and, journalist.
• Founder of the Indian Home Rule Society, India House, and The Indian Sociologist in London.
• He founded the India House and The Indian Sociologist In 1905.
• The Indian Sociologist was an Indian nationalist journal.
• Shyamji inaugurated a new organization called The Indian Home Rule Society, On 18 February 1905.

The correct answer is Calcutta.

Key Points

• West Bengal CM Mamata Banerjee launched the ‘Duare Ration’ (ration at doorstep) scheme on 16 November 2021.
• It will benefit around 10 crore people in the state.
• She also inaugurated a WhatsApp chatbot for the Food and Supplies Department of the state government and a mobile application, ‘Khadya Sathi: Amar Ration Mobile App’ to help people apply for ration cards
• ​CM Banerjee said that the government has decided to increase the commission for ration dealers from Rs 75 to Rs 150 per quintal of food grain.
• The scheme was announced before the West Bengal Assembly Elections held in March-April 2021.
• She said that the government would provide around 21,000 ration dealers with financial assistance of Rs 1 lakh each to purchase vehicles for delivering ration to people.
• Vehicles carrying foodgrain will be delivering rations to the doorsteps of the beneficiaries once the scheme is implemented.
• The government started the project on a pilot basis with as many as 3,000 ration dealers in the state from September 2021.
• The Calcutta High Court on September 2022 declared the 'Duare Ration Scheme' (ration at the doorstep) of the West Bengal government ultra vires to the National Food Security Act, 2013 and is legally void.

Important Points

• The National Food Security Act (NFSA 2013) is also known as the Right to Food Act.
• It provides food and nutritional security life at affordable prices i.e subsidized food grains to approximately two-thirds of India's 1.2 billion people.
• The act was enacted by Manmohan Singh's Indian National Congress-led UPA (United Progressive Alliance) Government.
• The act was introduced in the Parliament on 22^nd December 2011, promulgated a presidential ordinance on 5^th July 2013, signed on 10^th September 2013, and enacted into law on 12^th September 2013.
• The act includes Midday Meal Scheme, Public Distribution System (PDS), Integrated Child Development Services Scheme, and also recognizes maternity entitlements.
• The Midday Meal Scheme and the Integrated Child Development Services Scheme are for the benefit of every eligible individual whereas the PDS will reach about two-thirds of the population i.e 50% in urban areas and 75% in rural areas.

The correct answer is Belgium.

Key Points

• In the Tokyo Olympics, the Indian women’s hockey team scripted history by qualifying for the Olympic Games semifinals for the first time, on 2 August 2021.
• This comes a day after, Indian Men’s Hockey team entered the Olympic semifinals after four decades beating Great Britain 3-1.
• However, the men's team lost to Belgium in the semifinals.

Additional Information

• TOKYO OLYMPICS
• The 2020 Olympics was held in Tokyo from 23 July to 8 August 2021.
• The International Olympics Committee postponed the games due to Covid-19.
• This was the second time When Japan will host the Olympics. (First in 1964)
• Miraitowa is the official mascot of the 2020 Summer Olympics.
• The mascot is created by Japanese artist Ryo Taniguchi.
• The International Olympic Committee (IOC)
• IOC is a not-for-profit independent international organization.
• Created- 23 June 1894
• The first Olympic Games of the modern era were conducted in April 1896.
• The Olympic Charter is the codification of the Fundamental Principles of Olympism.
• President- Thomas Bach
• Olympic Motto- CITIUS – ALTIUS - FORTIUS meaning  (FASTER - HIGHER - STRONGER -COMMUNITER)
• The 138^th session of the International Olympic Committee met between 20-21 July 2021.
• At this meeting, the International Olympic Committee decided to change the Olympic motto 'Citius, Altius, Fortius' to 'Citius, Altius, Fortius, Communiter'.

The correct answer is 74.04 percent.

Key Points

• According to the 2011 census, the literacy rate in India is 74%.
• To be more precise the literacy rate was 74.04%.
• The literacy rate of the male population and the female population was 82.14 % and 65.46% respectively.
• The literacy rate in the 2001 census was 64.8%.
• The 2011 Census was the 15^th Indian Census with the motto 'Our Census, Our future'.
• The Census was conducted in two phases, the House Listing being the first and Population Enumeration was the second.
• It was the first time when the biometric information was collected.

Additional Information

Census 2011

• The Indian Census is the most credible source of information on Demography (Population characteristics), Economic Activity, Literacy and Education, Housing Household Amenities, Urbanisation, Fertility and Mortality, etc.
• Census 2011 was the 15^th National Census of the Country.
• Census was introduced in India during the era of Lord Mayo in the year 1872. It came into force in 1881.
• This census was conducted under the guidance of Mr. C. Chandramouli.
• The slogan of census 2011 was ‘Our Census, Our future’.
• The population of the nation as per the provisional figures of Census 2011 was 1210.19 million of which 623.7 million (51.54%) were males and 586.46 million (48.46%) were females. Uttar Pradesh is the most populous state in India its population is more than the population of Brazil.

The correct answer is Sarangi.

Key Points

• Ustad Sultan Khan was an Indian Sarangi player.
• A sarangi is a bowed stringed instrument with a skin-covered resonator.
• Ustad Sultan Khan was also a classical vocalist belonging to Sikar Gharana.
• He was awarded the Padma Bhushan in 2010.​

Additional Information

Calculation:

The difference between the number of boys in schools M and Q

⇒ (34 - 8)% × 6000 = 1560

The sum of the number of boys in schools J, K, and L

⇒ (12 + 6 + 22)% × 6000 = 2400

∴ Only II follows.

Shortcut Trick

Identify the remainder of each term when divided by 18.

Remainder of 19¹⁹ ÷ 18 → (18 + 1)¹⁹ ÷ 18 leaves remainder 1¹⁹ = 1.

Remainder of 20 ÷ 18 → 20 = (18 × 1) + 2, so the remainder is 2.

Total remainder = (1 + 2) = 3.

∴ The correct answer is 3.

Alternate Method

Given:

Expression = 19¹⁹ + 20

Divisor = 18

Formula Used:

Property of remainders: (a + b) mod n = [(a mod n) + (b mod n)] mod n

Binomial remainder theorem: (n + 1)^k mod n = 1^k = 1

Calculations:

⇒ First, find the remainder for 19¹⁹:

⇒ 19 = (18 + 1)

⇒ 19¹⁹ = (18 + 1)¹⁹

⇒ By binomial expansion, all terms except the last one (1¹⁹) are multiples of 18.

⇒ 19¹⁹ ≡ 1¹⁹ (mod 18)

⇒ 19¹⁹ ≡ 1 (mod 18)

⇒ Now, find the remainder for 20:

⇒ 20 = (18 × 1) + 2

⇒ 20 ≡ 2 (mod 18)

⇒ Combining the results:

⇒ Remainder (19¹⁹ + 20) ÷ 18 = (1 + 2) mod 18

⇒ Remainder = 3

∴ The correct answer is 3.

Additional Information

Binomial Theorem for Remainders

Any expression in the form (ax + 1)ⁿ when divided by x will always leave a remainder of 1ⁿ = 1.

Negative Remainder Concept

If an expression is (ax − 1)ⁿ divided by x, the remainder is (−1)ⁿ. If n is even, remainder is 1; if n is odd, remainder is −1 (which is equivalent to x − 1).

Euler's Totient Theorem

If a and n are co-prime, then a^φ(n) ≡ 1 (mod n), where φ(n) is Euler's totient function representing the count of numbers up to n that are co-prime to n.

Given:

sin 75° + sin 15°

Concept used:

1. 2 × sin θ × cos θ = sin 2θ
2. sin (90° - θ) = cos θ
3. sin 30° = 1/2 4. sin2 θ + cos2 θ = 1

Calculation:

sin 75° + sin 15°

⇒ sin (90° - 15°) + sin 15°

⇒ cos 15° + sin 15°

⇒ sin 15° + cos 15° = k (let)

​According to the concept,

sin² 15° + cos² 15° = 1

⇒ (sin 15° + cos 15°)² - 2.sin 15°.cos 15° = 1

⇒ k² - sin (2 × 15)° = 1

⇒ k2 - sin 30° = 1

⇒ k2 - 1/2 = 1 

⇒ k² = 1 + 1/2

⇒ k² = 3/2

⇒ k = $\sqrt{\frac{3}{2}}$

∴ The value of sin 75° + sin 15° is $\sqrt{\frac{3}{2}}$.

Shortcut Trick

We know, sin C + sin D = $2 \frac{sin~(C+D)}{2}\frac{cos~(C-D)}{2}$

sin 75° + sin 15 = $2 sin~\frac{(75+15)}{2}cos\frac{(75-15)}{2}$ = 2 × sin 45° × cos 30° = 2 × $\frac{1}{\sqrt{2}}$ × $\frac{\sqrt{3}}{2}$ = $\sqrt{\frac{3}{2}}$

Calculation:

The difference between the number of seats won by party R and party Q in all the elections put together

⇒ (90 + 80 + 100 + 50 + 80 + 65) - (80 + 90 + 60 + 70 + 76 + 70)

⇒ 19

∴ The difference between the number of seats won by party R and party Q in all the elections put together is 19.

Calculation:

The ratio of the number of students joining university A to the number of students joining university F

⇒ 275 : 170 = 55 : 34

∴ The ratio of the number of students joining university A to the number of students joining university F is 55 : 34.

Given:

A grocery shop is offering a 10% discount on the purchase of Rs.500 and above. A 5% discount is given on the purchase of value above Rs.250 but below Rs.500. A discount of an additional 1% is given if payment is made instantly in cash. 

He bought 25 packets of biscuits and one packet is priced at Rs.30.

Concept used:

1. Final discount percentage after two successive discounts of A% and B% = $(A + B - {AB \over 100})\%$ 2. Selling price = Marked Price × (1 - Discount%)

Calculation:

Total billed price = 25 × 30 = Rs. 750

Since he paid in cash, he would get two consecutive discounts of 10% and 1%.

So, final discounts = 10 + 1 - (10 × 1)/100 = 10.9%

Now, he would have to pay = 750 × (1 - 10.9%) = Rs. 668.25

∴ He would have to pay Rs. 668.25.

Given:

The ratio of corresponding sides of two similar triangles$\sqrt{5} : \sqrt{7}$.

Concept used:

If two triangles are similar, then the ratio of the area of both triangles is proportional to the square of the ratio of their corresponding sides.

Calculation:

​According to the concept,

Now, the ratio of the area of the two triangles

⇒ $({\sqrt5})^2:({\sqrt7})^2$

⇒ 5 : 7

∴ The ratio of the area of the two triangles is 5 : 7.

Given:

cosec θ + cot θ = 2

Concept used:

1. cosec2 θ - cot2 θ = 1 2. a² - b² = (a + b)(a - b)

Calculation:

cosec θ + cot θ = 2      ....(1)

​According to the concept,

cosec² θ - cot2 θ = 1

⇒ (cosec θ - cot θ)(cosec θ + cot θ) = 1

⇒ 2(cosec θ - cot θ) = 1

⇒ (cosec θ - cot θ) = 1/2      ....(2)

Solving (1) and (2), cosec θ = $\frac{5}{4}$

∴ The value of cosec θ is $\frac{5}{4}$.

Given:

ΔABC is an isosceles triangle

AB = AC = 5 cm and BC = 8 cm

Concept Used:

In an isosceles triangle, the median to the base from the opposite vertex and the perpendicular bisector of the base are the same.

Calculation:

In ΔABC, AD is perpendicular bisector

⇒ BD = DC = BC/2 = 8/2 = 4 cm

In ΔADC, by Pythagoras Theorem,

AC² = DC² + AD²

⇒ AD² = 5² - 4² = 9

⇒ AD = 3 cm

Now, in ΔADC,

tan C = AD/DC = 3/4

Now, in ΔADB,

cot B = BD/AD = 4/3

⇒ tan C - cot B = (3/4) - (4/3)

⇒ tan C - cot B = (9 - 16)/12

⇒ tan C - cot B = (-7)/12

∴ The value of tan C - cot B is (-7)/12.

Alternate Method

Given:

In the given figure, ABC is an isosceles triangle with BC = 8 cm and AB = AC = 5 cm.

Concept used:

1. sin² α + cos2 α = 1
2. tan α = sin α ÷ cos α = 1/ cot α 3. In trigonometry, the Cosine Rule says that the square of the length of any side of a given triangle is equal to the sum of the squares of the length of the other sides minus twice the product of the other two sides multiplied by the cosine of angle included between them.

Calculation:

​According to the concept (Cosine Rule),

cos B = $\frac {8^2 + 5^2 - 5^2}{2 \times 8 \times 5}$ = $\frac {4}{5}$      ....(1)

So, sin B

⇒ $\sqrt {1 - cos^2 B}$

⇒ $\sqrt {1 - ({\frac {4}{5}})^2}$

⇒ $\sqrt {1 - {\frac {16}{25}}}$

⇒ $\sqrt {{\frac {25 - 16}{25}}}$

⇒ $\sqrt {\frac {9}{25}}$

⇒ $\frac {3}{5}$      ....(2)

Now, tan C - cot B

⇒ tan B - cot B (∵ ABC is an isosceles triangle where AB = AC. So, ∠B = ∠C)

⇒ $\frac {sin B}{cos B} - \frac {cos B}{sin B}$

⇒ $\frac {sin^2 B - cos^2 B}{sin B \times cos B}$

⇒ $\frac {(\frac {3}{5})^2 - (\frac {4}{5})^2}{\frac {3}{5} \times \frac {4}{5}}$ (From 1 and 2)

⇒ $-\frac{7}{12}$

∴ The value of tan C - cot B is $-\frac{7}{12}$.

Given:

$\frac{3}{8}, \frac{5}{16}, \frac{7}{2}$

Concept used:

LCM = LCM of numerators ÷ HCF of denominators

Calculation:

LCM (3, 5, 7) = 105

HCF (8, 16, 2) = 2

Now, LCM of $\frac{3}{8}, \frac{5}{16}$ and $\frac{7}{2}$

⇒ LCM (3, 5, 7) ÷ HCF (8, 16, 2)

⇒ 105/2 = $52\frac{1}{2}$

∴ LCM of  $\frac{3}{8}, \frac{5}{16}$ and $\frac{7}{2}$ is ​$52\frac {1}{2}$.

Given:

In an election between two candidates, a candidate who gets 64% of the votes polled is elected by a majority of 252 votes.

Calculation:

Let the total number of votes be Q.

Losing candidate got = Q × (100 - 64)% = Q × 36%

​According to the question,

Q × (64 - 36)% = 252

⇒ Q × 28% = 252

⇒ Q = 252 ÷ 28%

⇒ Q = 900

∴ The total number of votes is 900.

Calculation:

Now, the average number of salesmen in C2, C4 and C5

⇒ (10 + 5 + 15)/3 = 10

∴ The average number of salesmen in C2, C4 and C5 is 10.

Given:

$y + \frac{1}{y} = 11$

Concept used:

1. (a - b)³ = a³ - 3ab(a - b) - b³
2. (a + b)² = (a - b)² + 4ab

Calculation: 

$y + \frac{1}{y} = 11$

⇒ $(y + \frac{1}{y})^2 = 11^2$

⇒ $(y - \frac{1}{y})^2 + 4 = 11^2$

⇒ $(y - \frac{1}{y})^2 = 121 - 4$

⇒ $(y - \frac{1}{y})^2 = 117$

⇒ $(y - \frac{1}{y}) = 3√{13}$      ....(1)

⇒ $(y - \frac{1}{y})^3 = 117^{2 \over 3}$

⇒ $y^3 - \frac{1}{y^{3}} - 3(y - \frac {1}{y}) = 117^{2 \over 3}$

⇒ $y^3 - \frac{1}{y^{3}} - 3 × {3√{13}} = (3√{13})^3$

⇒ $y^3 - \frac{1}{y^{3}} = 351√{13} + 9√{13}$

⇒ $y^3 - \frac{1}{y^{3}} = 360√{13}$

∴ The value of $y^3 - \frac{1}{y^{3}}$ is $360√{13}$.

Alternate Method If y + 1/y = p, then

y - 1/y = √(p² - 4) = q and y³ - 1/y³ = q³ + 3q

Now, y + 1/y = 11

y - 1/y = √(11² - 4) = √117 = 3√13

y3 - 1/y3 = (3√13)³ + 3 × 3√13

⇒ 27 × 13√13 + 9√13

⇒ 351√13 + 9√13 = 360√13

∴ y3 - 1/y3 = 360√13

Given: 

Diameter of each sphere = 6 cm

The height and diameter of the base of the cylinder are 90 cm and 4 cm respectively.

Concept used:

1. Volume of a cylinder = πR²H (where r = Radius of the base, H = height) 2. Volume of a sphere = ${4\pi R^3 \over 3}$ (R is the radius of the sphere)

Calculation:

Let the number of solid spheres be N.

Radius of each sphere = 6/2 = 3 cm

Radius of the base radius of the cylinder = 4/2 = 2 cm

​According to the question,

N × $\frac {4π × 3^3}{3}$ = π ×  2² × 90

⇒ N × ${4π × 3^2}$ = 4π × 90

⇒ N = 10

∴ The number of solid spheres, each of diameter 6 cm, that could be moulded to form a solid metal cylinder of height 90cm and diameter 4 cm is 10.

Shortcut Trick

Use the half-angle identity: cosec θ + cot θ = √((1 + cos θ) / (1 − cos θ)).

Given: b cos θ = a → cos θ = a/b.

Substitute cos θ in the formula: √((1 + a/b) / (1 − a/b)).

Simplify the fraction: √(((b + a)/b) / ((b − a)/b)) = √((b + a) / (b − a)).

∴ The correct answer is √(b+a)/(b-a).

Alternate Method

Given: b cos θ = a ⇒ cos θ = a/b

In a right-angled triangle, cos θ = Adjacent / Hypotenuse.

⇒ By Pythagoras Theorem: Perpendicular = √(Hypotenuse² − Base²) = √(b² − a²).

⇒ sin θ = Perpendicular / Hypotenuse = √(b² − a²) / b

⇒ cosec θ + cot θ = (1 / sin θ) + (cos θ / sin θ) = (1 + cos θ) / sin θ

⇒ cosec θ + cot θ = (1 + a/b) / (√(b² − a²) / b)

⇒ cosec θ + cot θ = ((b + a) / b) × (b / √(b² − a²))

⇒ cosec θ + cot θ = (b + a) / √((b + a)(b − a))

⇒ cosec θ + cot θ = √(b + a) / √(b − a) = √((b + a) / (b − a))

∴ The correct answer is √(b+a)/(b-a).

Additional Information

Trigonometric Identity

The relation cosec²θ − cot²θ = 1 can be factored as (cosecθ − cotθ)(cosecθ + cotθ) = 1.

Reciprocal Relation

cosec θ + cot θ is the reciprocal of cosec θ − cot θ.

Half-Angle Formula

The expression cosec θ + cot θ is always equal to cot(θ/2).

Pythagorean Triplet

For any right triangle with sides x, y and hypotenuse z, x² + y² = z².

Given:

$\left( x-\frac{1}{x}\right)^{2}$ = 12

Concept used:

1. (a - b)² = a² - 2ab + b²
2. (a + b)² = (a - b)² + 4ab

Calculation:

$( x-\frac{1}{x})^{2}$ = 12

⇒ $\left( x^2 - 2 + \frac{1}{x^2}\right) = 12$

⇒ $\left( x^2 + \frac{1}{x^2}\right) = 12 + 2$

⇒ ${x^2 + \frac {1}{x^2}} = 14$

⇒ $({x^2 + \frac {1}{x^2}})^2 = 14^2$ (Squaring both sides)

⇒ $({x^2 - \frac {1}{x^2}})^2 + 4 = 14^2$

⇒ $({x^2 - \frac {1}{x^2}})^2 = 196 - 4$

⇒ $({x^2 - \frac {1}{x^2}})^2 = 192$

⇒ $({x^2 - \frac {1}{x^2}}) = 8\sqrt{3}$

∴ The value of $({x^2 - \frac {1}{x^2}})$ is $8\sqrt{3}$.