SSC CGL 2022 Tier-I Official Paper (Held On : 09 Dec 2022 Shift 1) (Previous Year Paper)

Statements:

All Cadbury are Nestle.

No Nestle is Amul.

Some Cadbury are Britannia.

Conclusions:

I. Some Nestle are Britannia → Follow → Since 'All Cadbury are Nestle' & 'Some Cadbury are Britannia.' means that Britannia is a subset of Nestle so the conclusion follows.

II. No Amul is a Cadbury → Follow → Since 'All Cadbury are Nestle' & 'No Nestle is Amul..' means that 'No Cadbury is Amul.' and vice versa so the conclusion follows.

III .No Amul is Britannia. → Does not follow → Since 'Some Cadbury are Britannia.' & 'No Nestle is Amul..' means that No Amul is Britannia.' only a possibility not necessary so the conclusion does not follows.

Hence, Only conclusions I and II follow.

A paper is folded and cut when unfolded appear to look like as shown below :

Hence, 'Option 2' is the correct answer.

The logic followed here is :

Option 1 : 5 - 23

Option 2 : 11 - 123

Option 3 : 9 - 83

Option 4 : 7 - 51

Hence,'5 - 23' is the correct answer.

The logic followed here is:

Hence, '425' is the correct answer.

The logic followed here is:

Hence, '567' is the correct answer.

The pattern followed here is:

'Bull' is a male 'Cow'.

Similarly, let us check each option,

1. 'Doe' is a female 'deer'.
2. **'**Goose" is a female 'Gander'.
3. 'Mare' is a female 'Horse'.
4. 'Hen' is a female 'Cock'.

Hence, 'Goose' is the correct answer.

The logic followed here is:

Option 1 : UVZ

Option 2 : MNP

Option 3 : XYA

Option 4 : QRT

The logic in Option 1 is different from others.

Hence, 'Option1' is the correct answer.

The logic followed here is :

The Number of Vertices is reducing by one in each successive step.

Similarly, In next step number of vertices will be 2.

Hence, 'Option 4' is the correct answer.

Given:

A is the paternal uncle of B

Firstly, Draw family tree for each option,

Option 1 : A – C + B

A – C  means that A is the brother of C

C + B means that C is the father of B

Here, A is the paternal uncle of B. So it will be the answer.

Option 2 : A – B + C

A – B  means that A is the brother of B

B + C means that B is the father of C

Option 3 : A + B – C

A + B means that A is the father of B

B – C  means that B is the brother of C

Option 4 : A – B × C

A – B  means that A is the brother of B

B × C means that B is the mother of C

Hence, 'Option 1' is the correct answer.

Alphabet and their place vale:

The logic followed here is:

Logic: Letters are increasing and decreasing in prime number order.

​

So, 'FU' will complete the series.

Hence, 'FU' is the correct answer.

Statements:

I. All T are G.

II. All G are H.

The least possible Venn diagram for the given statement is:

Conclusions:

I. All T are H → follow → Since All T are G and All G are H it means All T are H  is definitely true.

II. No G is T → Does not follow → Since All T are G it means some G must be exists with T, hence the conclusion is definitely wrong.

III. All H are G → Does not follow → Since All G are H, so it can be possible but not definitely true.

Hence, Only conclusion I follows.

BODMAS Table:

Given Equation: 40 - 220 ÷ 2 × 2 + 20 = 200.

Now,  Interchanging two signs and two numbers,

1) 20 and 40, + and -

Given equation: 40 - 220 ÷ 2 × 2 + 20 = 200

After interchanging ‘20 and 40, + and -’, we get:

L.H.S = 20 + 220 ÷ 2 × 2 - 40

= 20 + 110 × 2 - 40

= 20 + 220 - 40

= 240 - 40

= 200 =  R.H.S.

2. 2 and 20, + and ×

Given equation: 40 - 220 ÷ 2 × 2 + 20 = 200

After interchanging ‘2 and 20, + and ×’, we get:

L.H.S = 40 - 220 ÷ 20 × 20 + 2

= 40 - 11 × 20 + 2

= 40 - 220 + 2

= 42 - 220

= -178 ≠ R.H.S.

3. 220 and 40,+ and ×

Given equation:40 - 220 ÷ 2 × 2 + 20 = 200

After interchanging ‘220 and 40,+ and ×’, we get:

L.H.S = 220 - 40 ÷ 2 + 2 × 20

= 220 - 20 + 2 × 20

= 220 - 20 + 40

= 260 - 20

= 240  ≠ R.H.S.

4) 220 and 20, ÷ and ×

Given equation:40 - 220 ÷ 2 × 2 + 20 = 200

After interchanging ‘220 and 20, ÷ and ×’, we get:

L.H.S = 40 - 20 × 2 ÷ 2 + 220 

= 40 - 20 × 1 + 220 

= 40 – 20 + 220

= 260 - 20

= 240 ≠ R.H.S.

Hence, "Option 1" is the correct answer.

BODMAS Table:

Given Equation: 7 * 9 + 3 ÷ 7 − 3 = 25.

Now, interchanging the given symbols in the options.

1)  ÷ and *

Given equation: 7 * 9 + 3 ÷ 7 − 3 = 25

After interchanging ‘÷ and *’, we get:

L.H.S = 7 ÷ 9 + 3 * 7 − 3

= 0.78 + 3 * 7 - 3

= 0.78 + 21 - 3

= 21.78 - 3

= 18.78 ≠ R.H.S

2) ÷ and +

Given equation: 7 * 9 + 3 ÷ 7 − 3 = 25

After interchanging ‘÷ and +’, we get:

L.H.S = 7 * 9 ÷ 3 + 7 − 3 

= 7 * 3 + 7 − 3 

= 21 + 7 - 3 

= 28 - 3

= 25 = R.H.S

3) + and *

Given equation: 7 * 9 + 3 ÷ 7 − 3 = 25

After interchanging ‘+ and *’, we get:

L.H.S = 7 + 9 * 3 ÷ 7 − 3 = 25

= 7 + 3.857 - 3

= 10.857 - 3

= 7.857 ≠ R.H.S

4) + and −

Given equation: 7 * 9 + 3 ÷ 7 − 3 = 25

After interchanging ‘+ and -’, we get:

L.H.S = 7 * 9 - 3 ÷ 7 + 3

= 7 * 9 - 0.42 + 3

= 63 - 0.42 + 3

= 66 - 0.42

= 65.58 ≠ R.H.S

Hence, the correct answer is "Option 2".

BODMAS Table:

Given Equations :

I. 7 - 5 × 4 + 8 ÷ 2

II. 7 × 3 - 8 ÷ 2 + 4 

after Interchanging, × and -, 4 and 2 , we get 

I. 7 × 5 - 2 + 8 ÷ 4 

=  7 × 5 - 2 + 2

= 35 - 2 + 2

= 37 - 2 = 35.

II. 7 - 3 × 8 ÷ 4 + 2

= 7 - 3 × 2 + 2

= 7 - 6 + 2

= 9 - 6 = 3.

Hence, The correct answer is '35 and 3'.

The correct mirror image of the given figure when the mirror is placed at line AB will be: 

 

Hence, "Option 1" is the correct answer.

If today is Friday, And, it will repeat after every 7 days.

Now, lets say today is Friday, so after every 7 days it will occur again.

1 week = 7 days.

So, 55 days = 55/7 = 7 week and 6 odd days.

Now, After 55 days = Friday + 6 = Thursday.

Hence, the correct answer is 'Option 2'.

The logic followed here is:

Logic: First number + Third number = Second number

(19, 23, 4)   → 19 + 4 =23.

(13, 27, 14) → 13 + 14 = 27.

Now, checking the options:

Option 1 : (25, 26, 2)

⇒ 25 + 2 = 27 ≠ 26.

The logic does not apply here.

Option 2 : (25, 27, 4)

⇒ 25 + 4 = 29 ≠ 27.

The logic does not apply here.

Option 3 : (25, 27, 2)

⇒ 25 + 2 = 27.

The logic apply here.

Option 4 : (25, 26, 4)

⇒ 25 + 4 = 29 ≠ 26.

Hence, 'Option 3' is the correct answer.

Family Chart:

Code and their meaning**:**

 

Given expression: Z * A $ B – C * D % E

1) Z * A means ‘Z is the sister of A'

2) A $ B means ‘A is the son of B’

3) B – C means ‘B is the mother of C’

4) C * D means ‘C is the sister of D'

5) D % E means ‘D is the husband of E’

Draw family tree accordingly,

So, Z is the 'Husband’s sister' of E.

Hence, 'Option 3' is the correct answer.

The image embedded is:

 

Hence, the correct answer is "Option 3".

The logic followed here is :

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

​For, (182, 170, 97)

For, (156, 128, 92)

Apply the logic on each options,

Option 1 : (156, 194, 79)

 

The logic does not apply here.

Option 2 : (121, 102, 61)

 

The logic does not apply here.

Option 3 : (98, 74, 54)

 

The logic does not apply here.

Option 4 : (138, 130, 73)

 

The logic apply here.

Hence , the correct answer is 'Option 4'.

Alphabet table:

The logic followed here is:

Here, 'ZVCQ' will complete the series.

hence, the correct answer is  ZVCQ .

The logic followed here is:

For, ‘ANIMISM’ is written as ‘INAMMSI’.

For, ‘EAGERLY’ is written as ‘GAEEYLR’.

Similarly, for 'FULFILL'

So, ‘FULFILL’ is written as ‘LUFFLLI’,

Hence, 'LUFFLLI' is the correct answer.

The pattern followed here is:-

Logic: Profession and the tool commonly used by that profession

Now,

Carpenter : Hammer → Carpenters use hammers to build and repair wooden structures.

Similarly,

Fisherman : Net → A fisherman uses a net to catch fish.

Hence, the correct answer is "Option 1".

Additional InformationButcher : Meat → A butcher uses a knife to cut meat.

Mechanic : Car → A mechanic fixes and maintains cars with some tools.

Nurse : Patient → A nurse takes care of patients.

The pattern followed here is :

• ​Pattern of B: In each consecutive box, the direction of the letter B changes in an anticlockwise manner at 90o and shifted from left to mid to right circularly. Thus, in the following box, the B will face toward right.

​          

• Pattern of Arrow: In each consecutive box, the direction of the arrow changes in an clockwise manner at 45^o. Thus, in the following box, the arrow will face** toward left**.

          

• Pattern of <: In each consecutive box, Symbol ^ flipped anti-clockwise at 90o at same position in next figure and then shifted from mid to right to left circularly. Thus, in the following box, the < will face upwards.

​          

So,

The complete series will be: 

Hence, 'Option 4' is the correct answer.

The logic followed here is :

For, ‘FELLOW’ is written as ‘NGHYQN’

For, ‘PURPLE’ is written as ‘TWRGNR’

Similarly, for 'SUPRISED'

So, 'SUPRISED' will be written as ‘TRWUFGUK’.

Hence, 'TRWUFGUK' is the correct answer.

The correct answer is Warehousing.

Key Points 

• Auxiliaries to trade refers to the factors that are essential for bringing the goods from the place of their production to the place of their consumption. In simple words, these are essential services and functions that assist the business.
• Warehousing: Warehousing refers to the process of stocking of goods so that it can be delivered to customers effectively. It is essential in maintaining the supply of the product, as per the demand in the market.
• Warehouses act as storage centers for goods, thereby helping the business with a source of goods storage and also assist in keeping the products safe throughout the entire process of reaching from producer to consumer.

The correct answer is Rohini Bhate 

Key Points

• The Nrityabharati Kathak Dance Academy was founded by Kathak dancer Rohini Bhate in 1947.
• Rohini Bhate was a Bharatnatyam and Kathak dancer. She was born in Patna and went to school in Pune .
• She advised Khairagarh University and Lalit Kala Kendra, University of Pune, on the teaching of kathak and other dance forms.
• She received the Kalidas Samman in 2001, the Maharashtra Rajya Puraskar in 1977, and the Sangeet Natak Akademi Award in 1979, and the Maharashtra Gaurav Puraskar in 1990.
• She became a Fellow of the Sangeet Natak Akademi in 2006 and received the Akademi Ratna Award in 2007.

The correct answer is Tripura.

Key Points

• In an attempt to ensure the social security of over 7,000 tea garden workers of Tripura, the state government has announced a special scheme named "Mukhyamantri Chaa Srami Kalyan Prakalpa".
• The scheme assures housing, rations, and financial support to the tea garden workers.
• A sum of Rs 85 crore would be spent from the government's exchequer for the implementation of the scheme.

Important Points 

• The key components of the scheme are as follows:
• Provide land or home to farmers for constructing house
• To allot land of defunct tea garden on lease basis through Cooperative
• To provide drinking water, electricity, shelter, education and healthcare amenities to each and every family
• Priority group ration card to be given
• Promote their children to send them in pre-primary to Higher Secondary schools
• Social pension to all eligible families
• Social allowance
• Maternal and Child Health Care assistance to be given
• Ensure health insurance as per their ability
• To provide supportive appliances for handicapped persons
• To monitor and coordinate with the managers for providing an eco-friendly environment
• To be brought under PMAY (Pradhan Mantri Awas Yojana).

The correct answer is Africa.

Key Points

• The Nile is a major north-flowing river in northeastern Africa and is commonly considered the longest river in the world.
• With a length of 6,853 km (4,258 miles), the Nile is an "international" river as its drainage basin covers 11 countries, namely, Tanzania, Uganda, Rwanda, Burundi, the Democratic Republic of the Congo, Kenya, Ethiopia, Eritrea, South Sudan, Republic of the Sudan and Egypt.
• In particular, the Nile is the primary water source of Egypt and Sudan.
• The River Nile has two major tributaries, the White Nile and the Blue Nile.
• The White Nile is considered to be the headwaters and primary stream of the Nile itself.
• The Blue Nile, however, is the source of most of the water and silt.

 

          

Additional Information

The correct answer is Be with Yoga, Be at Home!

Key Points

• Taking cognizance of the pandemic situation in 2021, the Ministry of Ayush, GoI decided the message of the International Day of Yoga 2021 celebration will be "Be with Yoga, Be at Home!".
• Owing to the tireless efforts of Indian Prime Minister Shri Narendra Modi, the Ministry of Ayush launched the "Be With Yoga, Be At Home" video campaign in an effort to unite the world's citizens when they were at home and to propagate the traditional yoga philosophy for the benefit of both physical and mental health.
• The Ministry of Ayush promoted traditional Yoga and encouraged people to get ready for and participate actively in International Day of Yoga through the "Be With Yoga, Be At Home" video campaign.
• The campaign encouraged engagement on MyGov and a number of social media sites, including Facebook, Twitter, and Instagram.
• The campaign entailed submitting a video of the participant performing one Yogasana for up to one minute, along with or without a brief reflection from the participant.

Additional Information

• Following its establishment at the UN General Assembly in 2014, the International Day of Yoga is now observed yearly on June 21 throughout the world.
• Yoga is a collection of physical, mental, and spiritual exercises that have their roots in ancient India. It is practised majorly to control and still the mind.
• In his 2014 speech to the UN, Indian Prime Minister Narendra Modi proposed the date of June 21 since it is the longest day of the year in the Northern Hemisphere and has significant meaning in many cultures.

The correct answer is Subsidies payment.

Key Points

• A subsidy is a benefit provided to a person, company, or institution, typically by the government.
• It can be either direct subsidies (like cash payments) or indirect subsidies (such as tax breaks).
• A government payment made directly or indirectly to people or businesses is known as a subsidy.
• According to economic theory, subsidies can be used to counteract market failures and boost economic efficiency.
• Loans and advances by the central government, expenditures on the acquisition of land, and investment in shares are all capital expenditures.

Additional Information

• Government capital expenditures are funds used to develop buildings, machinery, equipment, educational and healthcare facilities, etc.
• It covers the costs incurred by the government to make investments that will yield earnings or dividends in the future and to acquire fixed assets like land.  Repaying a debt is a capital expenditure because it reduces obligation in addition to creating assets.
• By enhancing production facilities and increasing operational effectiveness, capital expenditures that result in the formation of assets are long-term in nature and enable the economy to create income for many years.
• Additionally, it raises labour force participation, assesses the state of the economy, and increases its potential for future growth.

The correct answer is Kalbelia.

Key Points

• Padma Shri awardee, Gulabo Sapera is a Kalbelia dancer of Rajasthan.
• The Kalbelia are the tribe of snake charmers located at the Thar Desert in Rajasthan.
• Both men and women can perform the Kalbelia dance, which is an essential component of their culture.
• The Kalbelia dance is also known as "Sapera Dance" and "Snake Charmer Dance".
• Beginning in 2010, Rajasthan's Kalbelia dance and songs were added to UNESCO's representative list of the Intangible Cultural Heritage of Humanity.
• At the Kalbelia dance, men play a variety of traditional instruments as women dance.

Additional Information

• A prominent form of folk dance in the northern Indian state of Rajasthan is Bhavai.
• During the Bhavai performance, the male or female performers balance several earthen pots or brass/metal pitchers while dancing dexterously, pirouetting and then swaying with the soles of their feet perched on the top of glass bottles, on the edge of a sword, on the rim of a brass/metal thali or plate and on the broken glass.
• The Kamar tribes perform the Tertali or Terah Taali traditional dance. Two or three ladies typically perform it while seated on the ground.
• The Bhil tribe performs the Ghoomar or Ghumar, a traditional Rajasthani folk dance. The dance is mostly done by veiled ladies wearing the ghaghara, which are long, flowing dresses.

The correct answer is 10°N to 30°N latitude.

Key Points

• The region of the Ganga Brahmaputra Basin lies in 10°N to 30°N latitude.
• The Ganga Brahmaputra Basin is located in the subtropical regions of the Indian subcontinent.
• The Ganga Brahmaputra Basin is the basin created by the Ganga and Brahmaputra rivers and their tributaries.
• The key characteristics of this basin are its rich plains, the Himalayas, and the Sunderbans delta.
• The annual monsoon season, which brings rain from June to September, replenishes this area as well with its torrential downpours.

Additional Information

• A river basin is a geographical region that is drained by a river and its tributaries. Eventually, the water that has been drained produces a bay, river, lake, or any other form of water body.
• For the purposes of developing and planning for water resources, river basins are considered to be the primary hydrological unit.
• There are 12 primary river basins and they are, the Brahmaputra River Basin, Ganga River Basin, Cauvery River Basin, Indus River Basin, Krishna River Basin, Godavari River Basin, Mandovizuari River Basin, Mahanadi River Basin, Narmada River Basin, Tapi River Basin, Mahi River Basin, and Pamba River Basin.
• The catchment area of each river is at least 20,000 square km and the catchment area for these rivers as a whole is 25.3 lakh square km.

The correct answer is Haryana.

Key Points

• The state government of Haryana launched the Matrushakti Udaymita Scheme to provide support to women entrepreneurs.
• The Haryana "Matrushakti Udyamita Scheme" was unveiled by Chief Minister Manohar Lal Khattar to assist women in starting their own businesses.
• The main goal of this scheme is to support female entrepreneurs in all types of businesses, trades, and enterprises.
• In addition, he announced the "Sushma Swaraj Award", which comes with a 5 lakh rupee prize.
• Under the Matrushakti Udyamita Scheme, anyone who wants to start their own business, trade, or venture will be able to access soft loans from financial institutions, according to the government.

Additional Information

• The Central Government and State Governments of India created Women Empowerment Schemes to empower women.
• Providing equal access to resources, opportunities, decision-making authority, and capacity building for women of all ages is referred to as "women empowerment".
• Education, skill development, more work options, etc. can all contribute to the empowerment of women.
• Women's access to education, financial aid, and employment possibilities are the main goals of women empowerment schemes. These initiatives also seek to advance the health and happiness of women.

The correct answer is Bhagat Singh.

Key Points

• Bhagat Singh stated during his trial that “he did not wish to glorify the cult of the bomb and pistol but wanted a revolution in society”.
• During the early 20th century Indian independence movement, Bhagat Singh was a revolutionary hero.
• He participated in two well-known attacks on British authority, one on a local police head and the other on the Delhi Central Legislative Assembly.
• Bhagat Singh, a fierce critic of British rule in India, was hanged in 1931 at the age of 23 for engaging in activities deemed criminal.

Additional Information

• Surya Sen, an Indian revolutionary best known for organising the 1930 Chittagong Armory Raid, played a significant role in the campaign for Indian independence from British control in India.
• A revolutionary from India, Sachindra Nath Sanyal, was a co-founder of the Hindustan Republican Army, which was established to engage in military resistance against the British Empire in India.
• A journalist and revolutionary from India, Barindra Kumar Ghosh was a founding member of the revolutionary Jugantar Bengali weekly.

The correct answer is Bombay Presidency.

Key Points

• Abhinav Bharat Society was established in the Bombay Presidency.
• Vinayak Damodar Savarkar and his brother Ganesh Damodar Savarkar created the secret Indian Independence organisation known as the Abhinav Bharat Society (Young India Society) in 1904.
• When Vinayak Savarkar was still a student at Fergusson College in Pune, it was first established at Nasik as the "Mitra Mela".
• The organisation then expanded to several hundred revolutionaries and political activists, with branches in different regions of India and also to London when Savarkar travelled to London to study law.
• Following a few assassinations of British officials by the Abhinav Bharat Society, the Savarkar brothers were found guilty and sentenced to prison, and the organisation was declared disbanded in 1952.

Additional Information

• Calcutta's development as a Presidency city began in 1699 by the East India Company. In 1715, the British finished constructing the Old Fort, and in 1717, the Mughal emperor Farrukh-siyar granted the East India Company freedom of trade in exchange for a 3,000 rupee yearly payment.
• The United Provinces of British India, also referred to simply as the United Provinces, was a British Indian province that was created on January 3, 1921, as a result of the renaming of the United Provinces of Agra and Oudh.
• Punjab Province was a British Raj province in colonial India, with Lahore serving as its capital. Following the Second Anglo-Sikh War, the British East India Company designated much of the Punjab region a province under British Colonial Authority in April 1849.

The correct answer is antiparticle.

Key Points

• The British physicist named Paul Dirac was known to have introduced the concept of antiparticle in 1930.
• The antiparticle is a subatomic particle that has the same mass as an ordinary matter particle but an opposing electric charge and magnetic moment.
• For example, the negatively charged electron's antiparticle is the positron, which has a positive charge.
• Paul Dirac asserted that all matter has an antimatter-like counterpart when he was formulating the quantum mechanical theory. If a particle is charged, it and its antiparticle should have opposing charges.
• Carl Anderson identified a positively charged particle in 1932 by examining the cosmic ray particle trails in a cloud chamber. This particle appeared to have an electron-like mass.
• Anderson's particle, which was given the name "positron", was the first antiparticle to be experimentally proved.

Additional Information

• Atoms are desorbed from a hot surface through a physical process called thermal ionisation, sometimes referred to as surface ionisation or contact ionisation, which results in the creation of ions.
• High-energy charged particles, x-rays, and gamma rays are all components of cosmic radiation, which stars create. The earth receives secondary radiation that is created when charged particles interact with the atmosphere.
• The nuclear model is a term that refers to Rutherford's atomic model. The protons and neutrons, which make up almost all of the mass of the nuclear atom, are found in the nucleus, which lies at the centre of the atom. Since they surround the nucleus, the electrons take up the majority of the atom's volume.

The correct answer is Kalaimamani.

Key Points

• The Bharatnatyam dancer, Priyadarshini Govind was awarded the Kalaimamani in 1998.
• The highest civilian honour in the state of Tamil Nadu is called the Kalaimamani. The Tamil Nadu Iyal Isai Nataka Mandram (literature, music, and theatre), a division of the Directorate of Art and Culture, Government of Tamil Nadu, presents these honours to acknowledge the accomplishments of artists in the state.
• Priyadarsini has learned this art form from the best after receiving instruction from two legends, Padma Bhushan Smt. Kalanidhi Naryanan and Kalaimamani S.K. Rajarathnam Pillai.
• Priyadarsini has emerged as a spokesperson for Smt. Kalanidhi's padam repertoire due to her natural knack for abhinaya and her love and devotion to her craft.

Additional Information

• A one-time prize for performing arts in India given to honour Rabindranath Tagore's 150th birthday, the Sangeet Natak Akademi Tagore Ratna is similar to the Sangeet Natak Akademi Fellowship.
• An artist from any of the Indian traditional dance disciplines is given the Nritya Choodamani Award by Sri Krishna Gana Sabha each year, which is a highly prized and renowned honour.
• The Madhya Pradesh government in India annually bestows the Kalidas Samman, a prize for the arts.

The correct answer is Habeas corpus.​

Key Points

• The special order of the courts, Habeas corpus, means “arrested person should be presented before the court".
• Habeas corpus is a legal remedy that allows a person to inform a court about an illegitimate arrest or imprisonment and ask that the court compel the person's custodian—typically a prison official—to bring the prisoner to court so that the validity of the arrest can be determined.
• William Blackstone referred to the writ of habeas corpus as a "great and efficient writ in all manner of illrgal confinement".
• Habeas corpus is a summons that has the same legal effect as a court order; it is directed to the custodian and requests that a prisoner be brought before the court together with documentation of the custodian's authority so that the court can evaluate if the custodian has the right to hold the prisoner

 Additional Information

• .A judicial remedy known as a mandamus is an order from a court to any government, subordinate court, corporation, or public authority directing them to perform (or refrain from performing) a particular act that they are legally required to perform (or refrain from performing), and which is in the nature of a public duty and, in some cases, a statutory duty.
• Certiorari is a legal procedure used to ask for judicial review of a ruling made by a lower court or government body.
• A writ of prerogative known as a quo warranto demands that the person to whom it is addressed provide proof of their legal capacity to exercise a right, power, or franchise.

The correct answer is proven misbehaviour or incapacity.

Key Points

• A judge of the Supreme Court can be removed only on the grounds of proven misbehaviour or incapacity.
• An order of the President must be passed after an address in each House of Parliament, supported by a majority of the members present and voting and by a majority of not less than two-thirds of those present and voting, and presented to the President in the same Session for such removal on the ground of such proven misbehavior or incapacity.
• Otherwise, a judge of the Supreme Court cannot be removed from office. The practice of law in any court or before any other body in India is prohibited for anyone who has served as a judge of the Supreme Court.

Additional Information

• ​The process for removal judges is outlined in Article 124(4) of the Indian Constitution. The President must issue an order before removing a Supreme Court Judge from office.
• The consent of both Houses of Parliament is required for such an order. It needs a particular majority of the members who are present and voting within the same session.
• The criteria for removal are proven misconduct or incompetence. After giving notice to the Speaker or Chairman of the Lok Sabha and Rajya Sabha, respectively, it comes to light.
• A three-judge panel conducts a thorough examination. They present the report during the following joint session of both Houses of Parliament.

The correct answer is Vasco Da Gama.

Key Points

• In order to reach India and establish a maritime passage from Europe to the East, the Portuguese explorer, Vasco da Gama set off from Lisbon in the year 1497.
• The sea route to India was found by Vasco da Gama in 1498.
• His expedition made multiple stops in Africa until arriving at the trading town of Calicut, India, in May 1498, after sailing down the western coast of Africa and circumnavigating the Cape of Good Hope.

Additional Information

• One of the men who helped establish British supremacy in India was Clive Lord, the first British governor of the Bengal Presidency.
• By opening up new trade routes and creating connections amongst various populations, Henry the sailor contributed to creating the modern world.
• The early 17th century sea explorer and navigator Captain Hudson was most remembered for his discoveries of what is now Canada and parts of the northeastern United States.

The correct answer is Himachal Pradesh.

Key Points

• Himachal Pradesh is the first "smoke-free state" in India, and this historic achievement was made possible by the effective execution of the "Himachal Grihini Suvidha Scheme" of the Himachal Pradesh government and the "Pradhan Mantri Ujjwala Yojana" (PMUY) of the Central Government.
• The programme, which aimed to provide 8 crores deposit-free LPG connections to low-income households, was introduced by Prime Minister Narendra Modi on May 1, 2016, starting in the Uttar Pradesh district of Ballia. The goal was reached in September 2019.
• On August 10, 2021, Ujjwala 2.0 was launched on a pan-India basis to install another one crore LPG connections and cover the remaining homes covered by PMUY.
• A crore LPG connections under Ujjwala 2.0 have been provided as of January 31, 2022, exceeding the government's goal.

Additional Information

• India's first "Carbon Neutral Panchayat" is Palli in Jammu & Kashmir.
• Due to the successful execution of the "Himachal Grihini Suvidha Scheme" of the Himachal Pradesh Government and the "Pradhan Mantri Ujjwala Yojana (PMUY)" of the Central Government, Himachal Pradesh is the first "smoke-free state" in India.
• Diu is the first Union Territory of India to use only solar energy. In just three years, the Union Territory (UT) of Diu has become the first renewable energy surplus region in the entire country.
• Goa has succeeded in providing 100% Functional Home Tap Connections (FHTCs) in rural areas, making it the first "Har Ghar Jal" state in the nation.

The correct answer is 11 m.

Key Points

• Soccer is another name for football. The competition's rules will determine whether natural or artificial surfaces can be used for matches. Artificial surfaces are green in colour.
• At a distance of 16.5 metres from the interior of each goalpost, two lines are drawn perpendicular to the goal line. A line drawn parallel to the goal line connects these lines, which extend 16.5m into the playing field. The penalty area is the region enclosed by these lines and the goal line.
• A penalty mark is placed 11 metres from and equally spaced from the goalposts in each penalty area.
• Outside of the penalty area, a circle with a radius of 9.15m is drawn with its centre at each penalty mark.

Additional Information

• Association football also referred to as football or soccer, is a team sport that is played between two teams of 11 players each.
• During the game, the players mostly use their feet to kick the ball around a rectangular surface known as a pitch.
• The purpose of the game is to score more goals than your opponent's team by moving the ball into a goalpost that is being defended by your opponent's team.
• The game has often been divided into two 45-minute halves for a total game time of 90 minutes.

The correct answer is Square or rectangular.

Key Points

• Square or rectangular type of house remains were found in Mehrgarh site of Harappan civilization.
• In the Kacchi Plain in Balochistan, Pakistan, is the Neolithic archaeological site known as Mehrgarh.
• Early occupants of Mehrgarh built their square or rectangular homes out of local mud, kept their grain in granaries, made their own tools out of copper ore found nearby, and lined their big baskets with bitumen.
• In addition to raising sheep, goats, and cattle, they also grew dates, jujubes, einkorn wheat, and six-row barley.

Additional Information

• The Indus Valley Civilization, commonly referred to as the Indus Civilization, was a Bronze Age civilization in the northwest of South Asia that flourished between 2600 and 1900 BCE and was active from 3300 to 1300 BCE.
• The Indus Civilization is frequently referred to as Harappan after its type site Harappa, which was the first to be discovered and excavated in what was then the Punjab province of British India and is now Punjab, Pakistan, early in the 20th century.
• The Early Harappan and Late Harappan cultures existed side by side in the same region.
• Neolithic cultures, of which Mehrgarh in Balochistan, Pakistan, is the oldest and best-known, were the source of the early Harappan cultures. To set the Harappan Civilization apart from the older ones, it is frequently referred to as Mature Harappan.

The correct answer is Alfred Werner.

Key Points

• In 1893, the Swiss chemist, Alfred Werner, was the first to understand the molecular structures of inorganic substances – chemical compounds that do not contain carbon.
• Werner asserts that the attraction forces come uniformly from the centre of the atom rather than being associated with any particular directions in the atom. He clarified the structures of substances by assigning a new coordination number to an atom's bonds.
• The octahedral structure of transition metal complexes was proposed by Alfred Werner, who was awarded the Nobel Prize in Chemistry in 1913.
• Modern coordination chemistry's foundation was created by Werner and he was the first to receive the Nobel prize for inorganic chemistry.

Additional Information

• William Ramsay, who won the 1904 Nobel Prize in Chemistry, contributed to the creation of the noble gases as a new group in the periodic table.
• In 1943, George de Hevesy, a radiochemist from Hungary, won the Nobel Prize in Chemistry for discovering the radiotracer concept.
• John Dalton was an English chemist, physicist, and meteorologist best known for his work on human optics and for introducing the atomic theory to chemistry.

The correct answer is Chlorine.

Key Points

• Calcium hypochlorite is the chemical term for bleaching powder and its chemical formula is $CaOCl_2$.
• Bleaching powder is a pale yellowish powder that dissolves in water and has a strong chlorine odour.
• Chlorine gas, which is a byproduct of the chlor-alkali process, reacts with dry slaked lime to create bleaching powder.
• Thus, chlorine gas is used for the manufacturing of bleaching powder.
• Bleaching powder has its applications as a bleaching agent, oxidizing agent, and disinfectant.

Additional Information

• Nitrogen:
• With its application in fertilizers, nitric acid, nylon, dyes, and explosives, nitrogen is crucial to the Cl.
• Sulphur:
• One of the most significant components used as an industrial raw material is sulphuric acid, which is the principal derivative of sulphur.
• Sulfur is also used to make fertilisers, batteries, detergents, fungicides, matches, and pyrotechnics.
• Oxygen:
• The majority of combustion processes require oxygen, an element that belongs to the periodic table's chalcogen group.

The correct answer is Dhaman.

Key Points

• The dance form, Dhaman is associated with the state of Himachal Pradesh.
• The Nati dance is unique to the Sirmaur, Kullu, and Shimla areas of Himachal Pradesh. The dance was the "most people attending and participating in a Nati dance session at a single place," according to Guinness World Records.
• Himachal Pradesh also features additional dance styles as Kinnauri, Thoda, Jhora, Jhali, Chharhi, Dhaman, Chhapeli, Mahasu, Dangi, Chamba, Thali, Jhainta, Daf, and Stick dance.

Additional Information

• One of the most well-known dance styles from Jammu & Kashmir is rauf. During Ramzan and Eid in the spring, women perform rauf dance.
• A prominent dancing style among the tribes living in Jharkhand and the neighbouring state of Bihar is phagua and is performed during the festival of Holi.
• Maharashtra and Goa are known for their renowned folk dance known as Koli Dance created by the Kolis of Mumbai. All of the Koli festivals are always celebrated with Koli dance, which reflects the rhythm of the ocean waves.

The correct answer is Maize, Cucumber, Fig and Melon.

Key Points

• Plants having both the male and female reproductive parts present are called monoecious plants.
• All gametes, male and female, are produced by monoecious plants. Androecium, which refers to the male portion, is made up of stamens, and gynoecium, which refers to the female portion, is made up of the pistil.
• Some examples of monoecious plants are Maize, Cucumber, Chara, Fig, Melon, and Coconut.
• It's commonly known as a hermaphrodite or bisexual flower when a flower possesses both male and female reproductive organs.
• It's known as unisexual when a flower just has male or female reproductive organs. Both staminate and pistillate flowers can be found on a single plant in a monoecious species.

Additional Information

• Dioecious Plant:
• Dioecious plants refers to plants that have male and female reproductive parts on different plants.
• A dioecious plant is one that has either female or exclusively male reproductive organs in its flowers.
• Because they only have one sex organ, dioecious plants are referred to as imperfect flowers.
• Examples:
• papaya, willow, date palm, poplar, etc.

The correct answer is 1981.

Key Points

• The annual growth rate percentage population in India gradually started declining in the year 1981.
• ​India has raised the average marriage age, improved women's access to education, and enhanced their quality of life. As a result, the birth rate has decreased since 1981.
• Other factors that influenced the declining birth rate are implementation of family planning program, awareness and knowledge of the benefits of nuclear family, and increasing literacy rate among states.
• India's growth rate is very rapid despite a decline in its population. India's population increased dramatically between 1951 and 1981.
• The years from 1921 to 1951 are referred to as the period of stable population growth in India. This is brought on by decreased death rates and enhanced sanitation and healthcare systems.

Additional Information

• Increases in a population's or a dispersed group's size are known as population growth. In reality, there is a 1.1% annual increase in the number of people on earth, or about 83 million people.
• Because natural resources are overused for human progress, population growth and rising consumption are major contributors to environmental problems including biodiversity loss and climate change.
• The Sustainable Development Goals, which aim to raise living standards globally, lessen society's negative effects on the environment, and promote human welfare, are the main focus of international policy aimed at lessening the effects of human population expansion.

The correct answer is Option 3. 

Key Points

• Gay-Lussac's Law of Gaseous Volumes, also known as Gay-Lussac's Law or the Law of Combining Volumes of Gases, was formulated by the French chemist Joseph Louis Gay-Lussac in the year 1808.
• This law states that the ratios of volumes of gases involved in a chemical reaction can be expressed in small whole numbers, provided all gases are at the same temperature and pressure.
• It is an extension of the earlier work of Joseph Louis Gay-Lussac and Amedeo Avogadro on the relationship between the volume of a gas and the number of gas particles (Avogadro's Law).
• Gay-Lussac's Law is often used in the context of reactions involving gaseous substances, particularly in stoichiometry calculations.

Concept Used:

Calculation:

cos60 = 1/2

cos240 = cos(180 + 60) = -cos60 = - 1/2

cos360 = cos(360 - 0) = cos0 = 1

Now according to the question, 

⇒ (1/2)³ - (-1/2)³ -  (1)³

⇒ 1/8 + 1/8 - 1 = 2/8 - 1 = -3/4

∴ The value of the given expression is -3/4.

Given: 

x + y + z = 0

Concept used: 

Let the value of x = 1 and y = 1.

Put any value of x and y which satisfy the equation and you get the value of z.

So, the value of z will be -2. 

Then put the value in the second given equation and you will get the answer.

Calculation:

Put the value of x, y and z in the second equation, i.e

$\frac{x^2}{yz}+\frac{y^2}{zx}+\frac{z^2}{xy}$

⇒ ${1\over -2} + {1\over -2} + {4\over 1}$

⇒ - 0.5 - 0.5 + 4

⇒ -1 + 4

⇒ 3​

Therefore, the value of the given equation is 3.

Given:

P got 35% more votes than Q

R got 15% more votes than Q.

P overtook R by 2,412 votes.

90% voters voted and no invalid or illegal votes were cast

Concept used:

Let the total votes of Q be 100x

P =  × 100x + 100x = 135x

R =  × 100x + 100x = 115x

Calculation:

135x - 115x = 2412

20x = 2412

x = 120.6

So, the vote obtained by Q = 100x = 100 × 120.6 = 12060

Vote obtained by P = 135x = 135 × 120.6 = 16281

Vote obtained by R = 115x = 115 × 120.6 = 13869

Total votes obtained by all of them = 12060 + 16281 + 13869 = 42210

Now, according to question, only 90% of the voters casted votes, therefore:

⇒ $90 \over 100$ × y = 42210 ⇒ y = $42210 \times 100 \over 90$ ⇒ y = 46900 (y = total votes)

Therefore, the number of voters in the voting list are 46900.

The correct answer is (11a.m.)

Key Points -

Suppose they meet after x hours after 8 a.m.

Distance covered by A after x hours = 25x km.

Distance covered by B after (x-1) hours =30(x-1) km.

                      $Distance = Time * speed$

                         135  = 25x + 30(x-1)

                         135  = 25x + 30x - 30

                         135  = 55x - 30

                         135 + 30 = 55x 

                             165/55  = x

                           i.e , x = 3 hours 

          The two cars will meet after 3 hours ; 8a.m. + 3hours = 11 a.m. 

      HENCE ,THE CORRECT ANSWER IS 11 A.M.

Given:

Three successive discounts of 10% = single discount of ?

Formula used:

Successive discount = X + Y - $XY \over 100$

where X = first discount

Y = second discount

Calculation:

First discount of 10% and 10% = 10 + 10 - $100 \over 100$ = 20 -1 = 19%

Second final discount of 19% and 10% = 19 + 10 - $190 \over 100$ = 29 - 1.9 = 27.1%

∴ Three successive discounts of 10% is equivalent to a single discount of 27.1%.

Given:

PT is a tangent to a circle, whose centre is O.

OP = 17 cm

OT = 8 cm

Formula used:

Pythagoras Theorem: In a right-angle triangle, the hypotenuse is equal to - 

OP² = OT² + TP²

Calculation:

This, the length of the tangent TP is equal to:

TP2 = OP2 - OT2  

TP2 = (17)2 - (8)2  

TP2 = 289 - 64

TP2 = 225 

TP = √225 = 15

∴ The length of the tangent segment PT is equal to 15 cm.

Given:

96, 136 and 504

Concept used:

LCM, also referred to as Least Common Multiple for a number of integers is the smallest positive integer that is evenly divisible by all the numbers.

Calculation:

Prime factorization of the given numbers is: 

96 = 2 × 2 × 2 × 2 × 2 × 3

136 = 2 × 2 × 2 × 17

504 = 2 × 2 × 2 × 3 × 3 × 7

LCM (96, 136, 504) = 2 × 2 × 2 × 2 × 2 × 3 × 3 × 7 × 17

⇒ 34272 

∴ The LCM of 96, 136 and 504 is 34272.

Given:

Profit percentage = 20%

Used weight = 800 g

Real weight = 1000 g

Formula used:

Gain percentage = $(Selling: price - Cost : price) \over Cost : price$ × 100 

Calculation:

Selling Price of 800 g goods with 20% profit = $1000 (100 + 20) \over 100$ = Rs. 1200

Cost price of 800 g goods = Rs. 800 

Gain = Selling price - Cost price = 1200 - 800 = 400

Gain % = $400 \over 800$× 100 %

⇒ (0.5 × 100) % 

⇒ 50% 

∴ His real gain percentage is 50%.

Shortcut Trick

 

So, real gain% = (6 - 4)/4 × 100 = 50%

Given:

Circumference of first circle (C₁) = 264 cm

Circumference of second circle (C₂) = 396 cm 

Formula used:

Circumference = 2πr

Difference in radius, r₂ - r₁ = ${C_2 \over 2π} - {C_1 \over 2π}$

Calculation:

r2 - r1 = ${1\over 2π}$ (C2 - C1)

⇒ ${1\over 2π}$(396 - 264) cm 

⇒ (0.159 × 132) cm

⇒ 20.99 cm

~ 21 cm 

∴ The difference between the radii of two circles is 21 cm.

Given:

In triangle ABC,

$∠ B$ = 900 

$\angle C$ = 450

AC = $2\sqrt 2$ cm

Concept used:

In a right-angle triangle, 

cos θ = $base \over hypotenuse$ = $BC \over AC$

Calculation:

cos 450 = $BC \over 2√2$

$1 \over \sqrt 2$ = $BC \over 2√2$

BC = $2 \times √2 \over√2$ = 2 cm

∴ The length of BC is 2 cm.

Given:

Principal = Rs. 10200

Rate = 12.5 %

A = Rs. 19125

Time period = ?

Formula used:

Simple interest = $P \times R \times T \over 100$ 

where P = Principal amount

R = Rate

T = Time period

S.I. = A - P

Calculation:

S.I. = 19125 - 10200 = 8925

S.I.  = $10200 \times 12.5 \times T \over 100$

8925 = $10200 \times 12.5 \times T \over 100$

T = $8925 \times 100 \over 10200 \times 12.5$

⇒ 7 years

∴ The time in which a sum of Rs. 10200 amounts to Rs. 19125 at the rate of 12.5 percent per annum at simple interest is 7 years.

Given:

Import:

In Year 2015-16 = 738 lakhs rupees

In Year 2016-17 = 1200 lakhs rupees

In Year 2017-18 = 1600 lakhs rupees

In Year 2018-19 = 1537 lakhs rupees

In Year 2019-20 = 1310 lakhs rupees

Export:

In Year 2015-16 = 825 lakhs rupees

In Year 2016-17 = 1014 lakhs rupees

In Year 2017-18 = 1240 lakhs rupees

In Year 2018-19 = 1522 lakhs rupees

In Year 2019-20 = 1650 lakhs rupees

Calculation:

Difference in import and export in Year 2015-16 = 825 - 738 = 87 lakhs rupees

Difference in import and export in Year 2016-17 = 1200 - 1014 = 186 lakhs rupees

Difference in import and export in Year 2017-18 = 1600 - 1240 = 360 lakhs rupees

Difference in import and export in Year 2018-19 = 1537 - 1522 = 15 lakhs rupees

Difference in import and export in Year 2019-20 = 1650 - 1310 = 340 lakhs rupees

∴ The difference between import and export is minimum in the year 2018-19.

Given:

5 sin2A + 3 cos2 A = 4, 0<A<900

Concept used:

sin²θ + cos²θ = 1

Calculation:

2 sin2A + 3(sin2A + cos2 A) = 4

2 sin2A + 3 = 4

2 sin2A = 1

sin2A = $1 \over 2$

sin A = $1 \over √2$

A = 45°

Now, tan A = tan 45° = 1

∴ The value of tan A is equal to 1.

Given:

Total number of staff members = 120

Average weight = 62.5 kg

Reduced average weight = 250g = 0.25 kg

New average weight = 62.25 kg

Formula used:

Average = $Sum: of :weight :of :all :the :staff : members \over Total: no.: of :staff : members$ 

Let the weight of that one staff member who left be x,

and addition of weight of other staff be y.

Calculation:

The total weight of all the staff members in the hotel before one of them left is: 

120 staff members × 62.5 kg/person = 7500 kg

When one staff member leaves, the total weight of the remaining 119 staff members is: 

119 staff members × (62.5 kg - 0.25 kg)

⇒ 119 staff members × 62.25 kg/person 7407.75 kg

⇒ 7407.75 kg

The difference in weight before and after the staff member left is: 

7500 - 7407.75 = 92.25 kg

According to the problem statement, this difference in weight is equal to the weight of the staff member who left the hotel, which is x kg.

So, we can write: 

92.25 kg = x kg

∴ The weight of the staff member who left the hotel is 92.25 kg.

 

Shortcut Trick 

There are 120 staff members and their average weight 62.5 kg. Now, one member left and average weight decreases by 250 g

It means that member have the extra weight than average is (250 × 119) g = 29750 g = 29.75 kg

So, the staff member's weight is 62.5 + 29.75 = 92.25 kg

Given:

Time taken by all to complete the work = 4 days

Time taken by Anshu to complete the work = 9 days

Time taken by Swati to complete the work = 12 days

Concept used:

Anshu's 1 day work = $1 \over 9$

Swati's 1 day work = $1 \over 12$

Anshu's, Swati's and Rajni's one day work together = $1 \over 4$

Calculation:

Let's denote the fraction of the work completed per day by Anshu, Swati, and Rajni as "a", "s", and "r", respectively. 

When they work together, they complete the work in 4 days, so we can write: 

4(a + s + r) = 1 

We also know that Anshu alone will take 9 days to complete the work, so we can write: 

9a = 1 

Similarly, Swati alone will complete the work in 12 days, so we can write: 

12s = 1 

To find the time taken by Rajni alone to complete the work, we need to solve for "r". We can start by rearranging the first equation to get: 

a + s + r = $\frac{1}{4}$

Then, we can substitute the values of "a" and "s" from the second and third equations, respectively: 

$\frac{1}{9}$ + $\frac{1}{12}$ + r = $\frac{1}{4}$ 

Simplifying this equation, we get:

⇒$\frac{7}{36}$ + r = $\frac{1}{4}$

⇒r = $\frac{1}{4}-\frac{7}{36}$

⇒r = $\frac{1}{18}$

 ∴ Rajni alone will take 18 days to complete the work.

Given: 

$\frac{{(p - q)^3 + (q - r)^3 + (r - p)^3}}{{12(p - q)(q - r)(r - p)}}$

Formula used: 

If p - q = a, q - r = b and r - p = c,

then a + b + c = 0

and a³ + b³ + c³ = 3abc

Calculation:

$\frac{{(p - q)^3 + (q - r)^3 + (r - p)^3}}{{12(p - q)(q - r)(r - p)}}$

⇒ $3(p-q)(q-r)(r-p) \over 12(p-q)(q-r)(r-p)$

⇒ $3 \over 12$

⇒ $1 \over 4$ 

 ∴ The value of he given expression is $\bf 1 \over 4$.

Given:

Formula used:

Mean = $\Sigma fixi \over \Sigma fi$

where fi = frequency of particular value

xi = Mid value of the frequency class

Calculation: 

 

Mean = $1670 \over 50$ = 33.4 

 ∴ The mean of the given frequency distribution table is 33.4.

Given: 

Number of books sold by 7 shops:

Calculation:

Number of books sold by A and B together = 34 + 48 = 82

Number of books sold by E and F together = 64 + 110 = 174

Difference = 174 - 82 = 92

 ∴ The difference between number of books sold by A and B together and the number of books sold by E and F together is 92.

Given:

ABC is an isosceles triangle.

 $\angle ABC$ = 55°

AD is a median to BC.

Concept used:

As, AD is a median to BC, $\angle BDA$ = 90°

Calculation:

In the right-angle triangle BAD,

∠BDA + ∠BAD + ∠ABD = 180°

90° + ∠BAD + 55° = 180°

∠BAD = 35° 

 ∴ The measure of $\bf \angle BAD$ is 35°.

Given :

${\left( {a + b} \right)^3}$- $a^3$ + $b^3$ 

Formula used:

(a + b)³ = a³ + b³ + 3ab (a + b)

Calculation:

(a + b)3 = a3 + b3 + 3ab (a + b)

⇒ (a + b)3 - a3 - b3 = 3ab (a + b)

add 2b³ both side,

⇒ (a + b)3 - a3 - b3 + 2b3 = 3ab (a + b) + 2b3

⇒ (a + b)3 - a3 + b3 = 3ab (a + b) + 2b3

∴ The value of $\bf {\left( {a + b} \right)^3}$ - $\bf a^3$ + $\bf b^3$  is 3ab (a + b) + 2b3.

Given:

sin 28° sin 35° sin 45° sec 62° sec 55°

Concept used:

sec A° = cosec(90° - A°)

$cosec:\theta={1\over sin:\theta}$

sin 45° = ${1\over \sqrt 2}$

Calculation:

sin 28° sin 35° sin 45° sec 62° sec 55°

=> sin 28° sin 35° sin 45° cosec (90^o - 62°) cosec (90^o - 55°)

=> sin 28° sin 35° sin 45° cosec 28^o  cosec 35°

=> $\large sin 28° sin 35° sin 45° \over sin 28° sin 35°$

=> sin 45° = ${1\over \sqrt 2}$

∴ The value of sin 28° sin 35° sin 45° sec 62° sec 55° is $\bf \frac{1}{{\sqrt 2 }}$.

Shortcut Trick

The third proportional of two quantities X and Y is given by directly evaluating Y² ÷ X.

Here, X = a and Y = b⁴ / 4a.

Third proportional = (b⁴ / 4a)² ÷ a

= (b⁸ / 16a²) × (1 / a) = b⁸ / 16a³

∴ The correct answer is b⁸ / 16a³.

Alternate Method

Given:

First term = a

Second term = b⁴ / 4a

Formula Used:

If three terms p, q, and r are in continuous proportion, then p : q :: q : r, which gives p × r = q².

a First Term

:

b⁴ / 4a Mean Term

::

x Third Term

a × x = (b⁴ / 4a)²

Calculations:

Let the third proportional be x.

⇒ a : (b⁴ / 4a) :: (b⁴ / 4a) : x

⇒ Product of Extremes = Product of Means

⇒ a × x = (b⁴ / 4a) × (b⁴ / 4a)

⇒ a × x = (b⁴)² / (4a)²

⇒ a × x = b⁸ / 16a²

⇒ x = (b⁸ / 16a²) × (1 / a)

⇒ x = b⁸ / 16a³

∴ The correct answer is b⁸ / 16a³.

Additional Information

Mean Proportional

For any two numbers x and y, their mean proportional is given by the square root of their product: √(x × y).

Third Proportional

For two numbers x and y, the third proportional c satisfies the relation x : y :: y : c, giving the formula c = y² ÷ x.

Fourth Proportional

For three numbers a, b, and c, the fourth proportional d satisfies a : b :: c : d, yielding the result d = (b × c) ÷ a.

Compounded Ratio

When two or more ratios are multiplied together term by term (antecedent with antecedent and consequent with consequent), the resulting ratio is called a compounded ratio.