SSC CGL 2021 Tier-I (Held On : 20 April 2022 Shift 3) (Previous Year Paper)

The Venn diagram that best represents the relationship among Electronic gadgets, Mobile phones, Table, Laptops is shown below:

Laptops and Mobile Phones are Electronic gadgets.

Table is a piece of furniture with a flat top and one or more legs, providing a level surface for eating, writing, or working at.

Hence, 'option 2' is the correct answer.

The logic followed here is:

Logic: Number of alphabet in the word is used as code.

34225 means ‘Red fort is in Delhi’

And,

5248 means ‘Delhi is very polluted’

Similarly,

‘Pollution level is changing’ means '?'

Hence, the correct answer is "9528".

Let total marks be 'x'.

Vaidehi got 92% marks in an examination and Kavyanjali got 78% marks in the same examination.

Vaidehi got 92% of x = 92/100 × x

and

Kavyanjali got 78% of x = 78/100 × x  

If the difference between the marks obtained by Vaidehi and Kavyanjali is 105.

92/100 × x - 78/100 × x = 105

⇒ (92 - 78)x / 100 = 105

⇒ 14x = 105 × 100

⇒ x = 10500/14

⇒ x = 750

The marks obtained by Vaidehi in the examination = 92/100 × x

= 92/100 × 750

= 690

Hence, '690' is the correct answer.

BODMAS Rule:

If the signs ‘–’ and ‘×’ are interchanged, then the following equation that can be correctly balanced is:

1. 42 ÷ 6 – 18 + 30 × 7 = 146

L.H.S = 42 ÷ 6 × 18 + 30 - 7 

= 7 × 18 + 30 - 7 

= 126 + 30 - 7

= 156 - 7

= 149 ≠ R.H.S

2. 64 ÷ 8 – 22 × 3 + 30 = 103

L.H.S = 64 ÷ 8 × 22 - 3 + 30

= 8 × 22 - 3 + 30

= 176 - 3 + 30

= 206 - 3

= 203 ≠ R.H.S

3. 119 ÷ 17 – 6 + 24 × 34 = 32

L.H.S = 119 ÷ 17 × 6 + 24 - 34

= 7 × 6 + 24 - 34

= 42 + 24 - 34

= 66 - 34

= 32 = R.H.S

4. 36 + 14 × 18 ÷ 3 – 12 = 72

L.H.S = 36 + 14 - 18 ÷ 3 × 12 

= 36 + 14 - 6 × 12 

= 36 + 14 - 72

= 50 - 72

= -22 ≠ R.H.S

Hence, '119 ÷ 17 – 6 + 24 × 34 = 32' is the correct answer.

The pattern followed here is:

Hence, the correct answer is "127".

Six students, Abhay, Bhanu, Charu, David, Ekta and Farheen, are the top six rank holders in a school.

1. Bhanu is on the fourth rank.

_____ > _____ > _____ > Bhanu > _____ > _____

2. Ekta is at the lowest rank among all the given students. 

_____ > _____ > _____ > Bhanu > _____ > Ekta

3. There are two students between the ranks of Ekta and David.

_____ > _____ > David > Bhanu > _____ > Ekta

4. The rank of Bhanu is between the ranks of Charu and David.

_____ > _____ > David > Bhanu > Charu > Ekta

5. The rank of Abhay is immediately below the rank of Farheen.

Farheen > Abhay > David > Bhanu > Charu > Ekta

Clearly, Charu is on the fifth rank.

Hence, 'Charu' is the correct answer.

The least possible Venn diagram is:          

Conclusions:

I. Some keys are springs. → Does not follow (As, No key is a chain and All springs are chains → No key is spring)

II. No key is a spring. → Follow (As, No key is a chain and All springs are chains → No key is spring)

III. No lock is a spring. → Does not follow (it is possible but not definite)

Hence, Only conclusion II follows.

The image embedded in given figure is as shown below:

Hence, 'option 3' is the correct answer.

Given sequence: L G _ Q _ G P _ L _ P Q _ G _ Q

1. P, L, P, G, L, Q → L G P Q - L G P P - L G P Q - L G Q Q
2. P, L, Q, G, L, P → L G P Q - L G P Q - L G P Q - L G P Q
3. P, Q, L, G, P, L → L G P Q - Q G P L - L G P Q - P G L Q
4. P, P, L, G, Q, L → L G P Q - P G P L - L G P Q - Q G L Q

• In option 2, LGPQ is repeated.

Hence, 'P, L, Q, G, L, P' is the correct answer.

The faces that will be opposite to each other when the paper is folded to form a cube is shown below:

 → The dice cannot be formed as 2 and 1 the faces opposite to each other. 2.

 → The dice cannot be formed as 5 and 3 the faces opposite to each other. 3.

 → The dice can be formed. 4.

 → The dice cannot be formed as 2 and 1 the faces opposite to each other.

 

Hence, 'option 3' is the correct answer.

The logic followed here is:

Logic: 1^st and 3^rd letters are coded according to the equivalent opposite letter of a particular letter of the word and 2^nd letter is moved two step forward to get the code.

Similarly,

Hence, 'PHJ' is the correct answer.

The paper when unfolded will appear as shown below:

Hence, 'option 3' is the correct answer.

The logic followed here is:

Hence, '221' is the correct answer.

Alternate Method The logic followed here is:

2³ + 1 = 9

3³ + 2 = 29

4³ + 3 = 67

5³ + 4 = 129

6³ + 5 = 221

7₃ + 6 = 349

Hence, '221' is the correct answer.

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

5. Training
6. Tranquil
7. Transfer
8. Transient
9. Transit

Hence, '5, 1, 3, 4, 2' is the correct answer.

The logic followed here is:

• Row 1: (12 - 2) × 7 = 10 × 7 = 70
• Row 2: (15 - 2) × 6 = 13 × 6 = 78

Similarly,

• Row 3: (11 - 2) × 9 = 9 × 9 = 81

Hence, '81' is the correct answer.

The logic followed here is:

• ​

Similarly,

Hence, 'SGWWFS' is the correct answer.

The logic followed here is:

Hence, 'jreq, LUIV' is the correct answer.

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

Case I: Piyush is Nephew of Ranjana.

Case II: Piyush is son of Ranjana. 

In the options, Nephew is given, so it is clear that Piyush is the Nephew of Ranjana.

Hence, ‘Nephew’ is the correct answer.

The logic followed here is:

Similarly,

Hence, '224418362446' is the correct answer.

The pattern followed here is:

Second number = (First Number)² + (First number + 1)² 

For 14 ∶ 421 →

14² + (14 + 1)²

= 14² + 15²

= 196 + 225 

= 421

(1): 11 ∶ 265 →

112 + (11 + 1)2

= 112 + 122

= 121 + 144

= 265 = second number

(2): 18 ∶ 611 →

182 + (18 + 1)2

= 182 + 192

= 324 + 361

= 685 ≠ 611

(3): 16 ∶ 421 →

162 + (16 + 1)2

= 162 + 172

= 256 + 289

= 545 ≠ 421

(4): 13 ∶ 310 →

132 + (13 + 1)2

= 132 + 142

= 169 + 196

= 365 ≠ 310

Hence, the correct answer is "Option(1)".

According to given conditions, 

1) Shashank walks a certain distance, say X m, towards the west, then turns left and walks 35 m.

2. Then turns right and walks 18 m.

3) From there, he turns left and walks 9 m.

4. Then turns left again and walks 60 m.

 

5. He turns left and walks 9 m.

6. Then turns left again and walks 12 m.

7) He finally turns right and walks 35 m to reach a point that is exactly 10 m to the east of the starting point

Final, direction and distance figure.

⇒ EF - DC - HG - IJ = AB

⇒ 60 - 18 - 12 - 10 = 20 AB

Clearly, the value of X is 20 m.

Hence, '20' is the correct answer.

The logic followed here is as follows:

1. WTP → W - 3 = T; T - 4 = P
2. VSQ → V - 3 = S; S - 2 = Q
3. PMK → P - 3 = M; M - 2 = K
4. MJH → M - 3 = J; J - 2 = H

Hence, 'WTP' is different among them.

BODMAS Rule:

Given equation: 95 ÷ 19 × 3 – 7 + 29 = 13

1. • and ×

L.H.S =  95 ÷ 19 + 3 – 7 × 29

= 5 + 3 – 7 × 29

= 5 + 3 - 203

= 8 - 203

= -195 ≠ R.H.S

2. – and ×

L.H.S =  95 ÷ 19 - 3 × 7 + 29 

= 5 - 3 × 7 + 29

= 5 - 21 + 29

= 34 - 21

= 13 = R.H.S

3. – and +

L.H.S = 95 ÷ 19 × 3 + 7 - 29

= 5 × 3 + 7 - 29

= 15 + 7 - 29

= 22 - 29

= -7 ≠ R.H.S

4. ÷ and –

L.H.S = 95 - 19 × 3 ÷ 7 + 29 

= 95 - 19 × 0.42 + 29

= 95 - 7.98 + 29

= 124 - 7.98

= 116.02 ≠ R.H.S

Hence, '– and ×' is the correct answer.

The correct mirror image of the given figure when the mirror is placed at 'AB' is shown below:

Hence, 'option 4' is the correct answer.

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

If you look at the series closely, the shaded area is shifting inside in each step, and looking at the options figure we can say that the shaded portion will be in the second last position ie. before the circle in the centre.

Hence, 'option 2' is the correct answer.

The correct answer is Rs. 30,000.

Key Points

• In the union budget of India  2021-22, a proposal to enhance the Contingency Fund of India from ₹500 crore to ₹30,000 crores was introduced through a Finance Bill.
• The government has also altered the spending norms of the Contingency Fund of India.
• An amount equivalent to 40% of the Fund corpus shall be placed at the disposal of the Secretary, Ministry of Finance, Department of Expenditure for the purpose of meeting unforeseen expenditures like natural disasters and other emergencies.
• The remaining Contingency Fund releases exceeding the given limit shall be made with the approval of the Secretary to the Government of India, Department of Economic Affairs, after the approval of the Secretary to the Government of India, Department of Expenditure.

Additional Information

• The contingency fund was made to tackle unnatural conditions like disasters and other unforeseen emergency expenses.
• The fund is held by the Department of Economic Affairs on behalf of the President of India and it can be operated by executive action.
• The fund can be increased through a Finance Bill when Parliament is in session or through an ordinance if the house is not in session.
• The Contingency Fund of India is established under Article 267(1) of the Indian Constitution under the Contingency Fund Of India Act 1950.

The correct answer is Sambalpur.

Key Points

• It is located in Sambalpur, Odisha and is a composite dam and reservoir.
• The height of the Hirakund dam is 60.96 mt and the length is 25.8 km.
• Its present capacity is 347.5 MW with -
• Power House I (Burla): 2 x 49.5 MW, 3 x 37.5 MW, 2 x 32 MW
• PowerHouse II (Chiplima): 3 x 24 MW
• Pandit Jawaharlal Nehru laid its first batch of concrete on 12 April 1948.​

Additional Information 

• In 1952 Majumdar committee was formed to oversee the technical feasibility of this project.
• Cattle Island is located in one of the extreme points of the Hirakud Reservoir.

Important Points 

• Bhakra Dam is located on the Sutlej River.
• Krishna Raja Sagar Dam, Kallanai Dam, Mettur Dam, Kabini Dam and Amravati Dams are located on the Kaveri River.
• Maneri Damn, Joshiyara (Bhali) Dam, Koteshwar Dam, and Tehri Dam are located on the Bhagirathi River.

Important river water dispute

The correct answer is Phool Dei.

Key Points

• Phool Dei festival is the harvest festival celebrated in Uttarakhand.
• ​​Phool Dei is an auspicious folk festival celebrated to welcome the spring season.
• The festival is celebrated on the first day of the Hindu month, Chaitra.
• During this festival young girls gather together and go to every house in their village/towns with plates full of rice, jaggery, coconut, green leaves, and flowers and sing their folk songs.

Additional Information

• A pudding made from jaggery is the key dish of the festival.
• This festival showcases the inner connection between the communities living in the hills

The correct answer is Mumbai City FC.

Key Points

• Mumbai City FC on 10 October 2020 announced the appointment of a new Head Coach, Sergio Lobera, before the Hero Indian Super League (Hero ISL) 2020/21 season.
• He replaced Jorge Costa.
• He belongs to Spain and has coached different leagues around the world in a career spanning almost 25 years
• Lobera has managed clubs in Spain, Morocco, and India and also had a spell as assistant manager of FC Barcelona in 2012.
• Before joining the Mumbai team he was the coach of Goa Football Club.

Additional Information

• The team Mumbai City Football Club was formed in 2014 and owned by Ranbir Kapoor, Vimal Parekh, and the City Football group.
• Des Buckingham is the current coach of Mumbai City FC.
• Indian Super League 2020-21 was won by Mumbai City FC

The correct answer is 6-17 October 1949.

Key Points

• The Tenth Session of the Constituent Assembly was held from 6-17 October 1949.
• There was a total of 11 sessions of the Constituent Assembly.
• The last session of the Constituent Assembly was held from 24 to 26 November and on 26 Constitution of India (with 395 articles, 8 schedules, and 22 parts) was signed and accepted by all.
• The Assembly met once again on 24 January 1950, when the members appended their signatures to the Constitution of India but it was not a proper session.

Additional Information

• The idea of a constituent assembly was first proposed in 1934 by M.N. Roy.
• The Assembly first met on 9 December 1946 but was boycotted by Muslim League.
• Under the Cabinet Mission plan of 1946, elections were held for the formation of the constituent assembly
• The Constitution of India was drafted by the Constituent Assembly.
• It was implemented under the Cabinet Mission Plan on 16 May 1946.
• The total time taken to form our constitution was 2 years, 11 months, and 18 days.
• And the money spent on framing the constitution: was Rs.64 lakhs.
• As our constitution was completed and accepted on the 26 of November 1949.
• We celebrate it as Consitution Day.

The correct answer is Kummattikali.

Key Points

• Kummattikali also known as Kummatti is a colourful mask dance of Kerala prevalent in Thrissur District, Palakkad District, and parts of South Malabar.
• It is mainly performed on the occasion of the Onam festival(the New year of the Malayalam calendar).
• During the Onam festival, people perform this dance while visiting the houses of various people across the town and the cities to collect small gifts and entertain people while walking across the streets.
• The dancers wear heavily painted colourful wooden masks depicting the faces of Krishna, Narada, Kiratha, and Darika.
• These masks are usually made out of saprophyte, jack fruit tree.

Additional Information

• The themes of Kummattikali are mostly taken from the stories of Ramayana, Darika Vadham, the story of Shiva, and folk tales like Manjan Nayare Pattu.

The correct answer is Forms covalent compounds.

Key Points

• Beryllium is a silvery-white metal which is relatively soft and has a low density.
• Beryllium is used in gears and cogs, particularly in the aviation industry.
• It has a very high melting point making it useful in nuclear work as well as having ceramic applications.
• Beryllium is used as an alloying agent in the production of beryllium copper.

Additional Information

• Beryllium and its compounds are toxic and carcinogenic which makes it unsuitable for humans.
• The beryllium content on Earth's crust is 2.6 ppm and in soil 6 ppm.
• Beryllium is the only alkaline earth metal that does not react with water because the size of beryllium is small and the ionization energy of this alkaline earth metal is high when compared to other elements in the group.

The correct answer is velocity.

Key Points 

CONCEPT:

• Velocity/ speed: The rate of change of displacement/distance of a body is called the velocity/speed of that body.
• Velocity is a vector quantity that has both magnitudes as well as direction.
• Acceleration: The rate of change of velocity is called the acceleration of the body.
• Acceleration is also a vector quantity.
• The slope of any velocity-time graph gives an acceleration of the body.
• Displacement: The minimum path length between two points is called displacement.
• Distance: The total path length between two points is called distance.

Explanation:

From the above explanation, we can see that, 

• Acceleration is the rate of change of velocity
• In other words, it is the ratio of change in velocity and time taken
• It has SI units of m/s2 and is also the derivative of velocity

Additional Information

The correct answer is sulphuric, or hydrochloric.

Key Points

• Acid drain cleaners commonly contain a high concentration of sulfuric acid(H₂SO₄) or hydrochloric (HCL) acid, and typically come in liquid form.
• They work by a process of acid hydrolysis to dissolve the fats and solids that create a drain blockage.
• Acid hydrolysis is the process of breaking down larger molecules into smaller molecules by adding acid and water to a substance.
• They work well on hair and grease, and they can dissolve small pieces of paper and other solids in the drain.

Additional Information

• There are four basic types of chemical drain cleaners: acid, caustic, oxidizing, and enzymatic.

The correct answer is 5 NTU.

Key Points

• According to  the Bureau of Indian Standards (BIS)(IS -10500 ∶ 2012), the maximum permissible limit of turbidity in potable water should be 5NTU
• Turbidity levels should be maintained between 1 and 5 NTU to keep the desirable drinking water quality as per the Bureau of Indian Standards.
• Turbidity in water means Turbidity is the measure of relative clarity of a liquid.
• Turbidity is produced by suspended particulate matter like plankton, inorganic and organic matter.

  Additional Information

• Turbidity is measured in NTU: Nephelometric Turbidity Units.
• The instrument used for measuring it is called a nephelometer or turbidimeter.
• It measures the intensity of light scattered at 90 degrees as a beam of light passes through a water sample.
• It is essential to eliminate the turbidity of water in order to effectively disinfect it for drinking purposes.

The correct answer is Varahamihira.

  Key Points

• Varahamihira, also known as Varaha or Mihira, was an Indian philosopher, astronomer, and mathematician.
• He was born in 505 A.D. in a Brahmin family at Kapittha, a village near Ujjain.
• He proposed that the Moon and planets are lustrous not because of their own light but due to sunlight.
• He was one of the nine gems of the court of legendary ruler Yashodharman Vikramaditya of Malwa.
• Varahamihira’s main work is the book Pancha Siddhantika (Treatise on the five Astronomical Canons gives us information about older Indian texts which are now lost).

Additional Information

• His important works also include Brahma Samhita which covers wide-ranging subjects of human interest, including astrology, planetary movements, eclipses, rainfall, rainfall, clouds even domestic relations, gems, pearls, and rituals.
• Varahamihira’s mathematical work included the discovery of trigonometric formulas.
• He was inspired by the great mathematician and astrologer of India Aryabhatta.
• Nagarjuna- Indian Buddhist Philosopher
• Banabhatta- Sanskrit prose writer and poet of India
• Amalananda- Sanskrit Scholar

The correct answer is It is conducted by the Ministry of Health and Family Welfare.

Key Points

• Swachh Survekshan - Urban is conducted by the Ministry of Housing and Urban Affairs.
• It was introduced by MoHUA in 2016 as a competitive framework to encourage cities to improve the status of urban sanitation while encouraging large-scale citizen participation.
• It is the world's largest urban sanitation and cleanliness survey.
• It was first conducted in 2016 in 73 cities having a population of one million or more and the capital city of States/ UTs and Mysore had bagged the award of being the cleanest city in India.​

Additional Information

• The Prime Minister launched the Swachh Bharat Mission Urban 2.0 on 1^st October 2021, with a vision of Garbage Free Cities.
• The theme of Swachh Survekshan 2023 is ‘Waste to Wealth’, which is aligned with SBMU 2.0’s commitment to promoting circularity in waste management.
• The ranking of Wards within the cities is also being promoted through Swachh Survekshan 2023.
• In the Cleanest City category of the Swachh Survekshan Results 2021, Indore city got 1^st place in the country, while Surat and Vijayawada got second and third place respectively.

The correct answer is Humidity.

Key Points  

• The rate of evaporation decreases with an increase in Humidity.
• The rate of evaporation mainly depends on these factors:
• Temperature: The rate of evaporation increases with an increase in temperature.
• Surface area: The rate of evaporation increases with an increase in surface area.
• Humidity: The amount of water vapour present in the air is called humidity.
• The rate of evaporation decreases with an increase in humidity.
• Wind speed: Evaporation increases with an increase in wind speed.

Additional Information

• Ambient Temperature:
• ​Ambient temperature is the air temperature of any object or environment where equipment is stored.
• Evaporation:
• It is the process of changing of a liquid into water vapour below its boiling point.

The correct answer is Chandragupta Maurya.

• Chandragupta Maurya fought against Seleucus in the Seleucid-Maurya War.

Key Points

• ​In 305 BCE, Emperor Chandragupta Maurya led a series of campaigns to retake the satrapies left behind by Alexander the Great when he returned westward.
• Seleucus I fought to defend these territories, but both sides made peace in 303 BCE.
• Seleucus lost the Seleucid-Mauryan War, and the two rulers reconciled with a peace treaty.

Additional Information 

• In 305 BCE, Seleucus I tried to reconquer the northwestern parts of India in order to claim them for the growing Seleucid Empire.
• Little is known of the campaign in which Chandragupta fought with Seleucus over the Indus Valley and the region of Gandhara—a very wealthy kingdom that had submitted decades earlier to Alexander the Great.
• The Greeks offered a Macedonian princess for marriage to Chandragupta, and several territories, including the satrapies of Paropamisade (modern-day Kamboja and Gandhara), Arachosia (modern-day Kandahar), and Gedrosia (modern-day Balochistan).
• In return, Chandragupta sent 500 war elephants, a military asset that would play a decisive role in Seleucus’ victory against western Hellenistic kings at the Battle of Ipsus in 301 BCE.

The correct answer is Pallavi Joshi.

Key Points 

• The 67^th National Film Awards were announced on March 22, 2021, in New Delhi.

​

Additional Information

• National Film Awards
• They were started in 1954.
• The awards are given in three sections – Feature Films, Non-Feature Films, and Best Writing on Cinema.
• The National Awards aim to encourage good films in all languages.
• They are given by the Directorate of Film Festivals (DFF).
• The Directorate of Film Festivals
• It was set up in 1973 for the promotion of Indian films and cultural exchange.
• It is under the Ministry of Information & Broadcasting.
• Ministry of Information & Broadcasting
• Minister- Shri Prakash Javadekar
• Secretary-  Amit Khare

The correct answer is Ramesh Pokhriyal.

Key Points

• Ramesh Pokhriyal Nishank has been awarded the Vatayan Lifetime Achievement Award 2020.
• Owing to his contribution to the field of literature the organization has awarded him with the worthy Award.
• He has presented the award through an online ceremony held in the presence of Mira Kaushik, Chairman Vatayan.

Additional Information 

• Ramesh Pokhriyal:
• Ramesh Pokhriyal is the then (Nov-2020) Minister of Education (formerly Human Resource Development Ministry) in the Union Cabinet.
• He was born in the year 1959 and hails from the state of Uttarakhand (Pauri Garhwal).
• He was elected as MLA from the Karnaprayag constituency in the year 1991 (in erstwhile Uttar Pardesh).
• He also served as the Chief Minister of Uttarakhand during the 2009-2011 period.
• He is presently MP in Lok Sabha from Haridwar constituency and also Minister of Education.
• He is also known by his pen name 'Nishank'.

Other honours felicitated upon Nishank:

• Sahitya Bharti Award, Bharat Samman Award, and Sathitya Gaurav Samman.
• Good Governance Award by Dubai
• Himal Gaurav Samman by Nepal.

Important Points 

• Current Education Minister: Dharmendra Pradhan
• Piyush Goyal has portfolios such as:
• Minister of Textiles
• Minister of Commerce and Industry and Minister of Consumer Affairs
• Food and Public Distribution

The correct answer is Muzaffarpur.

Key Points

• Shahid Khudiram Bose was the youngest revolutionary freedom fighter who opposed British Raj in India.
• He was born on  3 December 1889, in the Midnapore district of  West Bengal.
• Khudiram Bose was sent to Muzaffarpur Jail under the charge of conspiracy to kill the Chief Presidency Magistrate DH Kingsford of Muzaffarpur, also known as the Muzaffarpur Conspiracy.
• He was executed there on August 11, 1908, at the age of 18 years which made him the youngest martyr of the Indian Independence Movement.

Additional Information

• Khudiram, along with Prafulla Chaki, attempted to assassinate a British judge, Magistrate Douglas Kingsford, by throwing bombs at the carriage, carriage they suspected the man was in but unfortunately, Magistrate Kingsford was seated in a different carriage, and the throwing of bombs resulted in the deaths of two British women.
• At the time of his hanging, Khudiram was 18 years, 8 months, and 11 days.
• Prafulla Chaki shot himself just before the arrest.

The Correct Answer is Uttar Pradesh.

Key Points

• The Ken River is one of the major rivers of the Bundelkhand region of central India.
• It flows across Madhya Pradesh and Uttar Pradesh.
• It is a tributary of the Yamuna.

Additional Information 

• These are the rivers with their tributaries:

The correct answer is First Amendment

Key Points

• The first amendment of the Indian Constitution was done in 1951.
• Constitution (First Amendment) Act, 1951 amended Articles 15, 19, 85, 87, 174, 176, 341, 342, 372 and 376.
• Nehru administration made a provision limiting Article 19(1)(a) of the Constitution of India against "abuse of freedom of speech and expression.
• ​The right of citizens of India to practice any profession or to carry on any occupation, trade, or business conferred by article 19(1)(g) is subject to reasonable restrictions that the laws of the State may impose, in the interests of the general public.​

Additional Information

• There is a total of 106 amendments made in the Indian constitution, till now.
• The provision for amendments is given in article 368 of the Indian Constitution.
• Dr . Bhimrao Ramji Ambedkar is known as the father of the Indian Constitution.
• 42^nd amendment is also known as the mini-constitution of India.
• Amendments in our Indian constitution are adopted from the constitution of South Africa.

Important Points

The correct answer is 112.

Key Points

• In the Union Budget 2021-22 intensified strategy to improve nutritional outcomes across 112 aspirational districts is proposed under Mission Poshan 2.0.
• To strengthen nutritional content, delivery, outreach, and outcome, the Supplementary Nutrition Programme and the Poshan Abhiyaan were merged to launch Mission Poshan 2.0.
• Poshan 2.0 scheme is an umbrella scheme covering the Integrated Child Development Services.
• Implementation of this program will be done by the Ministry of Women and Child Development in collaboration with the Ministry of Health and Family Welfare and the Ministry of Education under the aegis of Early Childhood Care and Education (ECCE) within the National Education Policy (NEP)

Additional Information

• Poshan Abhiyan is the Indian government’s flagship programme to improve nutritional outcomes for children, pregnant women and lactating mothers.
• Its first phase was launched in March 2018 by the Prime Minister in Jhunjhunu, Rajasthan.
• Malnutrition refers to deficiencies, excesses or imbalances in a person’s intake of nutrients.

The correct answer is 16%.

Key Points

• According to Economic Survey 2020-21 during the first half of the financial year 2020-21, the services sector contracted by almost 16 percent.
• This decline was led by a sharp contraction in all sub-sectors particularly Trade, hotels, transport, communication & services related to broadcasting, which contracted by 31.5 percent in H1 FY 2020-21.
• The services sector accounts for over 54% of India’s GVA and nearly four-fifths of the total FDI inflow into India.
• The Indian start-up ecosystem has been progressing well amidst the COVID-19 pandemic, being home to 38 unicorns - adding a record number of 12 start-ups to the unicorn list last year.

Additional Information

• The Department of Economic Affairs, Ministry of Finance presents the Survey in the Parliament every year, just before the Union Budget.
• It is prepared under the guidance of the Chief Economic Adviser of India.
• The Economic survey 2020-21 was dedicated to the Corona Warriors.
• According to the Economic survey 2020-21, India’s real GDP recorded an 11.0% growth in FY2021-22 and nominal GDP grew by 15.4% – the highest since independence.
• India expected to have a Current Account Surplus of 2% of GDP in FY21, a historic high after 17 years.

The correct answer is 88^th.

Key Points

• The division of powers between the Union and the States is in the Seventh Schedule.
• Seventh Schedule
• It contains three lists:
• Union List,
• State List and
• Concurrent List
• 7^th Schedule of Indian Constitution – Union List had 97 subjects.
• Now, it has 100 subjects.
• 88^th Amendment added a new subject in the Union List called ‘taxes on services.’
• The state list has 61 subjects. Earlier, it had 66 items.
• The 69^th Constitutional Amendment Act of 1991 made special provisions in relation to National Capital.
• Laws cannot be made by the Delhi government on three subjects under State List named as – Public Order, Police & Land.
• The concurrent list has 52 subjects enumerated under it.

Additional Information

• 85^th Amendment:
• Reservation in promotion for Scheduled Caste tribe candidates in government services.
• 87^th Amendment: 
• ​Extend the usage of 2001 national census population figures for statewide distribution of parliamentary seats.
• 88^th Amendment (2003):
• ​Provision of tax on services.
• 91^st Amendment (2003) :
• Amendment in party change system, recognition of merger of an entire party only.
• 56^th Amendment:
• 56^nd Amendment Act, 1987 of the Indian Constitution conferred Statehood on Goa and forms a new Union Territory of Daman and Diu.
• Goa thus became the 25^th State of the Indian Republic.
• 62^nd Amendment:
• The 62^nd Amendment Act extends by 10 years the reservation of seats for the Scheduled Castes and Scheduled Tribes in the Lok Sabha and State Assemblies as well as nomination representatives of the Anglo-Indian community.

Important Points 

The correct answer is Red Soil.

Key Points

• Red soil develops on crystalline igneous rocks in areas of low rainfall in the eastern and southern parts of the Deccan Plateau and also along the piedmont zone of the western ghats.
• The soil develops a reddish color due to a wide diffusion of iron in crystalline and metamorphic rocks.
• Red soils denote the largest soil group of India, covering an area of about 350,000 sq. km (10.6% of India's area) across the Peninsula.
• It contains thin organic and organic-mineral layers of highly leached soil resting on a red layer of alluvium.
• The red soils, with the proper use of fertilizers and irrigation techniques, give a good yield of cotton, wheat, rice, pulses, millets, tobacco, oil seeds, potatoes, and fruits

Additional Information

• Approximately 10.6% of the total geographical area of India is covered by red soil, including
• ​Tamil Nadu, parts of Karnataka, southeastern Maharashtra, eastern Andhra Pradesh, and Madhya Pradesh, Orissa, Chhattisgarh, Chota Nagpur (Jharkhand), south Bihar, West Bengal (Birbhum and Bankura), Uttar Pradesh (Mirzapur, Jhansi, Banda, and Hamirpur), Aravallis, eastern Rajasthan, Assam, Nagaland, Manipur, Mizoram, Tripura, and Meghalaya
• Alluvial Soil:
• ​Mostly available soil in India (about 43%) covers an area of 143 sq. km, wheat, maize, sugarcane, pulses, oilseed, etc. are mainly cultivated.
• Arid Soil:
• ​Also known as Desert soil, it accounts for over 4.42 % of the total land area of the country.
• These soils are saline in nature and in certain regions the salt content is so high that common salt is obtained by evaporating water.
• Black Soil:
• ​Black soil is also known as “Regur Soil” or the “Black Cotton Soil”.

The correct answer is Uzbekistan.

Key Points

• The 2023 Men's Boxing World Championships will be held in Uzbekistan's capital Tashkent.
• AIBA President Umar Kremlev confirmed Tashkent as the host city during a visit to Uzbekistan
• This is the 22^nd edition of the Championship, the first time to be hosted by Uzbekistan.
• It is held every 2 years.
• The previous edition of  AIBA Men's World Boxing Championships 2021 was held in Belgrade, Serbia

Additional Information

• Association Internationale de Boxe Amateur (AIBA), is an independent sports organization that sanctions amateur (Olympic-style) boxing matches and awards world and subordinate championships.
• Alongside the Olympic boxing program, they are the highest level of competition for the sport.
• India won only 1 Bronze Medal AIBA Men's World Boxing Championships 2021 by Akash Kumar.

The correct answer is Justice NV Ramana.

Key Points

• Justice NV Ramana took the oath as the 48^th Chief Justice of India on 24 April 2021 at Rashtrapati Bhawan.
• In March 2021, then Chief Justice SA Bobde of the Supreme Court recommended him as his successor.
• Nuthalapati Venkata Ramana was born on 27 August 1957 at Ponnavaram, Andhra Pradesh, India.
• Previously, he was a judge on the Supreme Court of India, chief justice of Delhi High Court and the acting chief justice of Andhra Pradesh High Court.

Additional Information

• Justice DY Chandrachud was appointed as the acting chairman of the National Legal Services Authority.
• Earlier this post was occupied by Chief Justice of India UU Lalit.
• Justice DY Chandrachud, a Supreme Court judge, is a former Chief Justice of the Allahabad High Court and a former judge of the Bombay High Court.
• Justice Chandrachud will take over as the 50^th Chief Justice of India in November 2022, succeeding UU Lalit.
• National Legal Services Authority (NALSA)
• The National Legal Services Authority (NALSA) has been constituted under the Legal Services Authorities Act, 1987 to monitor and evaluate the implementation of legal aid programs.
• To lay down policies and principles for providing legal services under this Act.

Calculation:

The expenditure for the years 2017 and 2018 taken together = 300 + 325 = 625 Cr.

The income for the years 2015 and 2016 taken together = 280 + 325 = 605 Cr.

Now, the difference = 625 - 605 = 20 Cr.

∴ The difference between the expenditure for the years 2017 and 2018 taken together and the income for the years 2015 and 2016 taken together is 20 Cr.

Given:

$\frac{1 \frac{1}{2}+1 \frac{3}{7} \div\left(1 \frac{3}{5} \text { of } 1 \frac{1}{4}\right) \times 2 \frac{1}{3}}{2 \frac{2}{3} \div \frac{4}{9} \times \frac{5}{6}+14}$

Concept used:

Calculation:

$\frac{1 \frac{1}{2}+1 \frac{3}{7} \div\left(1 \frac{3}{5} \text { of } 1 \frac{1}{4}\right) \times 2 \frac{1}{3}}{2 \frac{2}{3} \div \frac{4}{9} \times \frac{5}{6}+14}$

⇒ $\frac{\frac{3}{2}+\frac{10}{7} \div\left(\frac{8}{5} \text { of } \frac{5}{4}\right) \times \frac{7}{3}}{\frac{8}{3} \div \frac{4}{9} \times \frac{5}{6}+14}$

⇒ $\frac{\frac{3}{2}+\frac{10}{7} \div\left(2\right) \times \frac{7}{3}}{\frac{8}{3} \div \frac{4}{9} \times \frac{5}{6}+14}$

⇒ $\frac{\frac{3}{2}+\frac{5}{7} \times \frac{7}{3}}{6 \times \frac{5}{6}+14}$

⇒ $\frac{\frac{3}{2}+\frac{5}{3}}{5 +14}$

⇒ ${19 \over 6} \over 19$

⇒ $\frac{1}{6}$

∴ The simplified value is $\frac{1}{6}$.

Given:

A and B can do a work in 12 days and 18 days, respectively.

Concept used:

Total work = Work done per day (efficiency)× Total time taken 

Calculation:

LCM (12, 18) = 36

Let the total work be the LCM of the time taken by different workers.

Total work = 36 units

Efficiency of A = 36/12 = 3 units a day

Efficiency of B = 36/18 = 2 units a day

Let the efficiency of C be P units per day.

According to the question,

4 × (3 + 2) + 4 × (3 + P) = 36

⇒ 20 + 12 + 4P = 36

⇒ 4P = 36 - 32

⇒ 4P = 4

⇒ P =  1

Now, the time taken by C to complete 50% of the work 

⇒ (50% × 36)/1 = 18 days

∴ C alone will complete 50% of the same work in 18 days.

Given:

Divide Rs. 66,300 between A and B in such a way that the amount that A receives after 8 years is equal to the amount that B receives after 10 years.

Rate of interest = 10% p.a.

Concept used:

In case of the compound interest,

Amount = P(1 + R/100)^N

CI = P(1 + R/100)^N - P

where

P = Principal amount

R = Rate of interest per year

N = Time in years

Calculation:

Let the share of A be P.

So, the share of B = (66300 - P)

According to the question,

P(1 + 10/100)⁸ = (66300 - P)(1 + 10/100)¹⁰

⇒ P = (66300 - P)(1 + 10/100)10 - 8

⇒ P = (66300 - P)(1 + 10/100)2

⇒ P = (66300 - P)(1.1)2

⇒ P = (66300 - P) × 1.21

⇒ P = 80223 - 1.21P

⇒ 2.21P = 80233

⇒ P = 80233/2.21

⇒ P = 36300

Now, B's share = 66300 - 36300 = 30000

∴ A's and B's share are Rs. 36,300 and Rs. 30,000.

Shortcut TrickAmount of A after 8 years = Amount of B after 10 years

⇒ P_A[1 + R/100]⁸ = P_B[1 + 10/100]¹⁰

⇒ PA (11/10)⁸ = PB(11/10)¹⁰

⇒ PA/PB = (11/10)² = 121/100

⇒ PA + PB = 121 + 100 = 221

According to the question, 

⇒ 221 = 66,300

⇒ 1 = 66,300/221 = 300

⇒ P_A = 300 × 121 = 36,300

⇒ P_B = 300 × 100 = 30,000

Given:

A circle is circumscribed on a quadrilateral ABCD.

∠DAB = 100°, ∠ADB = 35° and ∠CDB = 40°

Concept used:

In the case of a cyclic quadrilateral, the sum of opposite angles of the quadrilateral is 180°.

The sum of all internal angles of a triangle is 180°.

Calculation:

Considering ΔDAB, 

∠DAB + ∠ADB + ∠DBA = 180°

⇒ 100° + 35° + ∠DBA = 180°

⇒ ∠DBA = 180° - 135° 

⇒ ∠DBA = 45°

⇒ ∠ABD = 45°

Since the circle is circumscribed on a quadrilateral ABCD, ABCD is a cyclic quadrilateral.

So, ∠ADC + ∠ABC = 180°

⇒ (∠ADB + ∠CDB) + (∠ABD + ∠DBC) = 180°

⇒ 35° + 40° + 45° + ∠DBC = 180°

⇒ ∠DBC = 180° - 120° 

⇒ ∠DBC = 60°

∴ The measure of ∠DBC is 60°.

Given:

0° < θ < 90°

Concept used:

cotθ = cosθ/sinθ

tanθ = sinθ/cosθ 

cosec²θ - cot²θ = 1

(A - B)² = (B - A)²

Calculation:

$\left(\frac{1-\cot θ}{1-\tan θ}\right)^{2}$ + 1

⇒ $({{1 - {cosθ \over sinθ}} \over {1 - {sinθ \over cosθ}}})^2$ + 1

⇒ $({ {{sinθ - cosθ} \over sinθ} \over {{cosθ - sinθ} \over cosθ}})^2$ + 1

⇒ $({ {{cosθ - sinθ} \over sinθ} \over {{cosθ - sinθ} \over cosθ}})^2$ + 1

⇒ $({cosθ \over sinθ})^2$ + 1

⇒ cot²θ + 1 = cosec²θ

∴ The value of $\left(\frac{1-\cot θ}{1-\tan θ}\right)^{2}$+ 1 is cosec2θ.

Calculation:

The average number of youth in the five states = (12 + 8 + 11.5 + 10 + 9) × 100000/5 = 1010000

∴ The average number of youth in the five states is 1010000.

Calculation:

The number of students obtained grade B = 1800 × 27% = 486

The number of students obtained grade C = 1800 × 24% = 432

Now, the number of more students = 486 - 432 = 54

∴ 54 more students have obtained grade B than those who have obtained grade C.

Shortcut Trick

If A + B = 90°, then sin² A + sin² B = 1 and cos² A + cos² B = 1.

In the numerator, 62° + 28° = 90° ⇒ 5(1) − 21 = −16.

In the denominator, 35° + 55° = 90° ⇒ 7(1) + 1 = 8.

Final Value = −16 ÷ 8 = −2.

∴ The correct answer is -2.

Alternate Method

Given: (5 cos² 62° + 5 cos² 28° − 21) ÷ (7 sin² 35° + 7 sin² 55° + 1)

Formula Used: cos θ = sin(90° − θ) and sin θ = cos(90° − θ); sin² θ + cos² θ = 1

⇒ Numerator = 5 cos² 62° + 5 cos²(90° − 62°) − 21

⇒ Numerator = 5(cos² 62° + sin² 62°) − 21

⇒ Numerator = 5(1) − 21 = −16

⇒ Denominator = 7 sin² 35° + 7 sin²(90° − 35°) + 1

⇒ Denominator = 7(sin² 35° + cos² 35°) + 1

⇒ Denominator = 7(1) + 1 = 8

⇒ Expression Value = −16 ÷ 8 = −2

∴ The correct answer is -2.

Additional Information

Complementary Angle Identities

For complementary angles where θ₁ + θ₂ = 90°, then sin θ₁ = cos θ₂, tan θ₁ = cot θ₂, and sec θ₁ = cosec θ₂.

Pythagorean Identities

The three fundamental identities are sin² θ + cos² θ = 1, 1 + tan² θ = sec² θ, and 1 + cot² θ = cosec² θ.

Simplified Results

If A + B = 90°, then sin² A + sin² B = 1 and cos² A + cos² B = 1 are frequently used to simplify fractions in exams.

Given:

A ladder of length 3.5 m just reaches the top of a wall.

The ladder makes an angle of 60° with the wall

Concept used:

Cosθ = Base/Hypotenuse

Calculation:

Let AC, BC, and AB be the length of the ladder, the height of the wall, and the distance between the foot of ladder and the wall.

ΔABC is a right-angled triangle at B. 

In the case of ΔABC,

cos 60° = BC/AC

⇒ 1/2 = BC/3.5

⇒ BC = 1.75

∴ The height of the wall is 1.75 m.

Given:

In △ABC, ∠A = 68°

Concept used:

  

ΔABC is a triangle with incenter I. 

So, ∠BIC = 90° + ${1 \over 2}$ × ∠BAC

Calculation:

  

According to the concept,

∠BIC = 90° + ${1 \over 2}$ × ∠BAC

⇒ ∠BIC = 90° + ${1 \over 2}$ × 68°

⇒ ∠BIC = 90° + 34°

⇒ ∠BIC = 124°

∴ The measure of ∠BIC is 124°.

Given:

In a right-angled triangle PQR, ∠Q = 90°.

A and B are the mid-points of PQ and PR, respectively.

PQ = 16 cm, QR = 30 cm and PR = 34 cm, 

Concept used:

Pythagoras Theorem

Calculation:

    

Considering ΔPQR,

PQ² + QR² = PR²

⇒ 16² + 30² = PR²

⇒ PR = $√ {16^2 + 30^2}$

⇒ PR = $√ {1156}$

⇒ PR = 34

According to the question,

PA = AQ = 16/2 = 8 cm

PB = BR = 34/2 = 17 cm

Now, ΔPAB is a right-angled triangle at A.

Considering ΔPAB,

PA2 + AB2 = PB2

⇒ 8² + AB2 = 17²

⇒ AB² = 289 - 64

⇒ AB = √225

⇒ AB = 15

Now, the perimeter of the trapezium ABRQ = 15 + 17 + 30 + 8 = 70 cm

∴ The perimeter of the trapezium ABRQ is 70 cm.

Given:

The length of the body diagonal of a cube is 8$\sqrt{3}$ cm.

Concept used:

Body diagonal of a cube = ​​$\sqrt{3}$A 

The volume of a cube = A³

(where A is the length of each side)

Calculation:

Let the length of each side of the cube be A.

According to the concept,

$\sqrt{3}$ × A = 8$\sqrt{3}$

⇒ A = 8

Now, the volume of the cube = 8³ = 512 cm³

∴ The volume of the cube is 512 cm³.

Given: 

The marks of one student got recorded as 95 in place of 57, as a result of which there was an increase in the average score by 0.95.

Concept used:

Total = Average × Number of entities

Calculation:

Let the original average and number of students be P and N.

If 57 was recorded, the original total = N × P

According to the question,

N × P - 57 + 95 = (P + 0.95) × N

⇒ NP + 38 = NP + 0.95N

⇒ N = 38/0.95

⇒ N = 40

∴ The number of students is 40.

Given:

The person bought a book at a 31% discount on its printed price.

If no discount was given, then he would have to pay Rs. 2,480 more. 

Concept used:

Selling price = Marked Price - Marked Price × Discount%

Calculation:

Let the printed price of the book be 100p.

After discount the selling price of the book = 100p - 100p × 31% = 69p      ....(1)

According to the question,

69p + 2480 = 100p

⇒ 100p - 69p = 2480

⇒ 31p = 2480

⇒ p = 2480/31

⇒ p = 80

Now, the price that he paid for the book

⇒ 69 × 80 (From 1) = Rs. 5520

∴ He paid Rs. 5520.

Shortcut Trick

In a cyclic quadrilateral where one side is the diameter of the circle:

∠PSR = 90° + ∠RPQ

Given ∠RPQ = 23°

∠PSR = 90° + 23° = 113°.

∴ The correct answer is 113°.

Alternate Method

Given: PQRS is a cyclic quadrilateral, PQ is the diameter, and ∠RPQ = 23°.

Formula Used: Angle in a semicircle is 90°; Sum of opposite angles in a cyclic quadrilateral is 180°.

⇒ Since PQ is the diameter, the angle subtended by it at the circumference is 90°.

⇒ ∴ ∠PRQ = 90°.

⇒ In ΔPQR, the sum of angles is 180°.

⇒ ∠PQR = 180° − (∠PRQ + ∠RPQ)

⇒ ∠PQR = 180° − (90° + 23°) = 180° − 113° = 67°.

⇒ In cyclic quadrilateral PQRS, opposite angles are supplementary.

⇒ ∠PSR + ∠PQR = 180°

⇒ ∠PSR = 180° − 67° = 113°.

∴ The correct answer is 113°.

Additional Information

Cyclic Quadrilateral Property

A quadrilateral is cyclic if all its vertices lie on a single circle. The sum of its opposite angles is always 180°.

Angle in a Semicircle

Thales's theorem states that if P, Q, and R are points on a circle where the line PQ is a diameter, then the angle ∠PRQ is a right angle (90°).

Angle at the Center

The angle subtended by an arc at the center of a circle is double the angle subtended by the same arc at any point on the remaining part of the circle.

Given:

5$\sqrt{3}$ + $\sqrt{75}$ = 17.32

Calculation:

5$\sqrt{3}$ + $\sqrt{75}$ = 17.32

⇒ 5$\sqrt{3}$ + $5\sqrt{3}$ = 17.32

⇒ $10\sqrt{3}$ = 17.32

⇒ $\sqrt{3}$ = 17.32/10

⇒ $\sqrt{3}$ = 1.732       ....(1)

Now,

14$\sqrt{3}$ + $\sqrt{108}$

⇒ 14$\sqrt{3}$ + $\sqrt{36 × 3}$

⇒ 14$\sqrt{3}$ + $6\sqrt{3}$

⇒ $20\sqrt{3}$

⇒ 20 × 1.732 (From 1)

⇒ 34.64

∴ The value of 14$\sqrt{3}$ + $\sqrt{108}$ is 34.64.

Given:

Three partners shared the profit in a business in the proportion of 9 ∶ 8 ∶ 11. They invested their capital for 4 months, 6 months, and 18 months, respectively. 

Concept used:

Profit is shared according to the capital invested.

Total investment = Invested Capital × Time period of the investment

Calculation:

Let the invested capital by them be P, Q, and R respectively.

According to the concept,

(P × 4) : (Q × 6) : (R × 18) = 9 : 8 : 11

⇒ 4P : 6Q : 18R = 9 : 8 : 11

Equating individual terms we get,

4P = 9

⇒ P = 9/4

Similarly, Q = 8/6 & R = 11/18

Now, we get,

P : Q : R = 9/4 : 8/6 : 11/18

⇒ P : Q : R = 9/4 × 36 : 8/6 × 36 : 11/18 × 36 

⇒ P : Q : R = 81 : 48 : 22

∴ The ratio of their capitals is 81 : 48 : 22.

Given:

If the article is sold at 70% of this price, then there is a loss of 10%.

Concept used:

Selling Price = Cost Price × (1 - Loss%)

Selling Price = Cost Price - Cost Price × Loss%

Profit% = ${(Selling\ Price\ -\ Cost\ Price) \over Cost\ Price} \times 100\%$

Calculation:

Let the article be originally sold for 100p.

If it is sold at 70% of this price, the selling price would be

⇒ 100p × 70% = 70p

Let the cost price of the article be Q.

According to the concept,

Q - Q × 10% = 70p

⇒ Q - 0.1Q = 70p

⇒ 0.9Q = 70p

⇒ Q = 70p/0.9

⇒ Q = 700p/9

Now, the profit% when it's sold at the original selling price = ${{100p\ -\ {700p \over 9}} \over {700p \over 9}} \times 100\%$

⇒ $\frac{200}{7}$%

∴ $\frac{200}{7}$% is the percentage profit when it is sold at the original selling price.

Shortcut TrickLet the article be originally sold for Rs. 900.

If it is sold at 70% of this price, the selling price would be

⇒ 900 × 70% = Rs. 630

So, the cost price of the article = 630 ÷ 0.9 = Rs. 700

Now, the profit% when it's sold at the original selling price = ${{900 - 700} \over 700} \times 100\%$

⇒ $\frac{200}{7}$%

∴ $\frac{200}{7}$% is the percentage profit when it is sold at the original selling price.

Given:

6, 8 and 15

Concept used:

LCM is the smallest common multiple of two or more numbers.

Calculation:

The smallest number that's exactly divisible by 6, 8, and 15 is the LCM (6, 8, 15).

LCM (6, 8, 15) = 120

120 = 23 × 5 × 3

So, to get the smallest perfect square number that is exactly divisible by 6, 8, and 15, we have to multiply 120 by 2, 5, and 3.

Now, the smallest perfect square number = 120 × 2 × 3 × 5 = 3600

∴ 3600 is the smallest number that is a perfect square and is divisible by each of the numbers 6, 8, and 15.

Given:

(4a - 3b) = 1, ab = $\frac{1}{2}$, where a > 0 and b > 0

Concept used:

(A + B)2 - (A - B)2 = 4AB

(A - B)² + 4AB = (A + B)²

(A - B)2 = A² - 2AB + B2

A³ + B³ = (A + B) (A² - AB + B²)

Calculation:

(4a - 3b) = 1

⇒ (4a - 3b)² = 1²

⇒ (4a - 3b)2  + 4 × 4a × 3b = 1 + 4 × 4a × 3b

⇒ (4a + 3b)2 = 1 + 48ab

⇒ (4a + 3b)2 = 1 + 48 × 1/2

⇒ (4a + 3b)2 = 1 + 24

⇒ (4a + 3b)2 = 25

⇒ (4a + 3b) = 5      ....(1)

(4a - 3b) = 1

⇒ (4a - 3b)2 = 12

⇒ (4a)² - 2 × 4a × 3b + (3b)² = 1

⇒ (4a)2 - 24ab + (3b)2 = 1

⇒ (4a)2 - 12ab + (3b)2 = 1 + 12ab

⇒ (4a)2 - 12ab + (3b)2 = 1 + 12 × 1/2

⇒ (4a)2 - 12ab + (3b)2 = 1 + 6

⇒ ((4a)2 - 12ab + (3b)2) = 7      ....(2)

Now,

(64a3 + 27b3)

⇒ (4a)3 + (3b)3

⇒ (4a + 3b) ((4a)2 - 12ab + (3b)2)

⇒ 5 × 7 (From 1 and 2) = 35

∴ The value of (64a3 + 27b3) is 35.

Given:

a + b + c = 6

a2 + b2 + c2 = 32

a3 + b3 + c3 = 189

Formula used:

a2 + b2 + c2 = (a + b + c)2 - 2(ab + bc + ca)

a3 + b3 + c3 - 3abc = (a + b + c) (a2 + b2 + c2 - (ab + bc + ca))

Calculation:

a2 + b2 + c2 = 32

⇒ (a + b + c)² - 2(ab + bc + ca) = 32

⇒ 6² - 2(ab + bc + ca) = 32

⇒ -2(ab + bc + ca) = 32 - 36

⇒ -2(ab + bc + ca) = -4

⇒ -(ab + bc + ca) = -4/2

⇒ (ab + bc + ca) = 2      ....(1)

a3 + b3 + c3 = 189

⇒ (a + b + c) (a2 + b2 + c2 - (ab + bc + ca)) + 3abc = 189

⇒ 6 × (32 - 2) + 3abc = 189 (From 1)

⇒ 6 × 30 + 3abc = 189

⇒ 3abc = 189 - 180

⇒ 4abc = (9 × 4)/3

⇒ 4abc = 12

∴ The value of 4abc is 12.

Given:

A is 25% more than B, and B is 40% less than C. 

C is 20% more than D.

Calculation:

Let D be 100p. (where p ≠ 0)

So,

C = 100p + 100p × 20% = 120p

B = 120p - 120p × 40% = 72p

A = 72p + 72p × 25% = 90p

Now, % less = ${{100p - 90p} \over 100} \times 100\%$ = 10%

∴ A is 10% less than D.

Given:

The distance between two towns is covered in 7 hours at a speed of 50 km/h.

Concept used:

Distance = Speed × Time

Calculation:

Distance between two towns = 7 × 50 = 350 km

Required new time for covering the distance = 7 - 2 = 5 hours

Required new speed = 350/5 = 70 kmph

Now, the increment in speed = 70 - 50 = 20 kmph

∴ By 20 kmph the speed should be increased so that 2 hours of travelling time will be saved.

Calculation:

The average of number car passed = (105 + 130 + 115 + 95 + 45) ÷ 5 = 98

Now, the hour(s) is the number of cars passed less than the average number of cars passed from 7 am to 12 noon

⇒ 6-7,  10 -11, 11-12

∴ The hour(s) is the number of cars passed less than the average number of cars passed from 7 am to 12 noon are  6-7,  10 -11, 11-12.