ICAR Technician 28 Feb 2022 Shift 3 (Previous Year Paper)

The correct answer is a mixture.

Key Points

• Soil is a mixture of organic matter, minerals, gases, liquids, and organisms that together support life.
• A mixture is a substance that consists of two or more elements or compounds not chemically combined together.
• For Example - Air is a mixture of gases like oxygen, nitrogen, argon, CO2, and water vapour.
• Homogeneous and Heterogeneous are types of mixtures.
• The layer of soil on Earth's surface is called the pedosphere.
• The study of soil is called **pedology.**​

Additional Information

• A Compound is when two or more element atoms combine together by either giving up electrons or sharing them.
• An atom is the smallest unit of matter.
• Matter is anything that takes up space and volume, atoms consist of a nucleus of protons and neutrons surrounded by electrons.
• An element is a pure substance that can not be separated into a simpler substance by physical or chemical means.

The correct answer is 22.

Key Points

• Bhasha Sangam is an initiative of the Ministry of Education under Ek Bharat Shreshtha Bharat, to teach basic everyday usage in 22 Indian languages.
• Ek Bharat Shreshtha Bharat was announced by the Hon'ble Prime Minister on 31^st October 2015 on the occasion of the 140^th birth anniversary of Sardar Vallabhbhai Patel.
• Through this innovative measure, the knowledge of culture.
• Languages are included in the Eighth Schedule of the Indian Constitution.

Important Points

• Languages included in this initiative are:-

1. Assamese
2. Bengali
3. Bodo
4. Dogri
5. Gujarati
6. Hindi
7. Kannad
8. Kashmiri
9. Konkani
10. Maithili
11. Malayalam
12. Manipuri
13. Marathi
14. Nepali
15. Odia
16. Punjabi
17. Sanskrit
18. Santhali
19. Sindhi
20. Tamil
21. Telugu and
22. Urdu.

The correct answer is Call your Cop.

Key Points

• The Call you Cop mobile app was launched by Nagaland Police.
• The mobile app holds high-security features that make it difficult to hack.
• App has phone numbers of Police Officers of Nagaland Police.
• This app will help to connect the citizens with the Nagaland Police easily.
• This app was launched by DGP T. John Longkumer at **Police headquarters in Kohima.**​

Additional Information

• Nagaland (As of 2022)
• ​Number of Districts - 16
• Number of Lok Sabha seats - 1
• Number of Rajya Sabha - 1
• Formed - 1^st December 1963.
• State Bird - Blyth's Tragopan

The correct answer is 1980.

Key Points

• All the countries began their development strategies in a very similar fashion.
• India made an announcement of its first 5-year plan in the year 1951.
• However, Pakistan made its 5-year plan announcement in the year 1956.
• China, on the other hand, made this announcement in the year 1953.
• Up until the 1980s, all 3 countries had very similar growth rates and **per capita incomes.**​

Additional Information

• The GDP growth rate improved (from an average of 2.9% in1970s to 5.6% in the 1980s) during the 1980s.
• July 1991, India has taken a series of measures to structure the economy and improve the BOP position.
• The new economic policy introduced changes in several areas.
• ​Liberalisation.
• Extending Privatisation.
• Globalization of the economy.
• Which are known as 'LPG' (Liberalisation, Privatisation, and Globalization).

The correct answer is Nitrogen.

Key Points

• The chemical composition of Vinegar is CH₃COOH.
• The common name is Vinegar.
• It is often used in food.
• Acetic acid is vinegar's chemical name.
• It is also used for the treatment of wounds.
• Involvements of chemical composition in vinegar are Carbon, Hydrogen, and **Oxygen.**​

Additional Information

• Nitrogen is present in all living things, including the human body and plants.
• Nitrogen gas is used in food storage to keep packaged or bulk food fresh.
• It is also used in the making of electronic parts, for industrial purposes and has many other useful applications.

The correct answer is Minerals.

Key Points

• Minerals are a micronutrient.
• Minerals
• Minerals are essential for life processes.
• Human mineral intake represents only about 0.3 % of our total intake of nutrients.
• Without minerals, we wouldn't be able to use the other 99.7 percent of foods.
• Plants passively or actively absorb minerals through roots procedures.

Important Points

• The essentials of an element for a plant are divided into two broad categories on the basis of their quantitative requirements.
• Macronutrients
• It is generally present in plant tissues in large amounts (in excess of 10 mmole Kg –1 of dry matter).
• The macronutrients are:

• Micronutrients
• Micronutrients or trace elements are needed in very small amounts (less than 10 mmole Kg –1 of dry matter).
• The Micronutrients are:

Additional Information

The correct answer is It took place in 1559.

Key Points

• The Battle of Chausa took place in June 1539.
• Battle of Chausa
• This battle was fought between Humayun and Farid Khan (Sher Shah Suri) on 26^th June 1539.
• Chausa is located in Buxar, Bihar.
• Humayun was defeated in this battle.
• After this battle, Farid Khan accepted the title of Sher Shah (Farīd al-Dīn Shēr Shah) as emperor.​

Additional Information

• Bahar Khan Lohani conferred the title of 'Sher Khan' on Farid Khan because he killed the tiger with one stroke of the sword.
• In 1540, Shershah Suri defeated Humayun in the battle of Kannauj (Bilgram) and after winning the battle, he ruled Agra and Delhi.​
• The sequence of the four major battles fought by the Humanyn is as follows:-

1. Battle of Devra - 1532 AD
2. Battle of Chausa - 1539 AD
3. Battle of Kannauj - 1540 AD
4. Battle of Sirhind - 1555 AD

The correct answer is HDFC Bank.

Key Points

• HDFC Bank has been voted ‘India’s Best Bank for SMEs’ at the Asia Money Bank Awards 2021.
• According to Asia money, HDFC Bank's wholesale banking department, the MSME(micro, small, and medium enterprises) business became the second-largest revenue contributor last year(2020).
• SME Loans (Small and Medium Enterprises) Loans that meet the financial needs of the Small and Medium Enterprises (SME) segment are known as SME loans.
• DBS Bank has been voted 'India's Best International Bank 2021' by Asiamoney.

Important Points

• Asia Money
• It is a financial publication established in 1989.
• It provides reports and analyses of the financial and investment markets.
• It issues reports to capital issuers, borrowers, institutional investors, and senior corporate and government monetary decision-makers with business interests in the Asia Pacific.
• Enterprises shall be classified as Micro, Small or Medium enterprises according to Investment in Plant & Machinery/equipment and Annual Turnover.
• Micro enterprises
• Investment in Plant and Machinery or Equipment: Not more than Rs.1 crore
• Annual Turnover; not more than Rs. 5 crore
• Small enterprise
• Investment in Plant and Machinery or Equipment: Not more than Rs.10 crore
• Annual Turnover; not more than Rs. 50 crore
• Medium enterprise
• Investment in Plant and Machinery or Equipment: Not more than Rs.50 crore
• Annual Turnover; not more than Rs. 250 crore

Additional Information

The correct answer is Ittira Davis.

Key Points

• In December 2021, Ittira Davis was appointed as new MD and CEO of Ujjivan Small Finance Bank.
• The Bank's board had appointed Ittira Davis as MD and CEO for a term of three years, but RBI has approved her appointment as MD and CEO of Ujjivan SFB for a tenure of one year with effect from January 14.
• Lttira Davis has an experience of a stint as the Chief Operating Officer of Ujjivan SFB.
• He joined Ujjivan in March 2015 and led the institution's transition from a Microfinance Institution (MFI) to an SFB.
• He took the place of Nitin Chugh, who had resigned as MD and CEO.

Important Points

• Sudha Suresh
• Recently(April 2022), ​She has been appointed as an independent director of Ujjivan Small Finance Bank.
• Prior to this, she has also held the positions of Managing Director and CEO (2017 - 2018) and Chief Financial Officer (2008 - 2017) of Ujjivan Financial Services Limited.
• Samit Kumar Ghosh
• ​He was the former Managing Director and CEO, and Founder of Ujjivan Small Finance Bank Ltd.
• Nitin Chugh
• ​Recently, Nitin Chugh has been appointed by the State Bank of India (SBI) as the Deputy Managing Director (DMD) to run the digital banking operations.
• He has also been the CEO of Ujjivan Small Finance Bank in the past.

Additional Information

• Small Finance Bank
• It has been registered as a Public Limited Company under the Companies Act, 2013 and It gets licensed under the Banking Regulation Act, 1949.
• The purpose of setting up of small finance banks is to supply credit to small business units, financial inclusion, lending to small and marginal farmers, micro and small scale industries, and other unorganised sector institutions.
• Ujjivan Small Finance Bank 
• Founded:- 1 February 2017
• ​Headquarters:- Bengaluru
• Founder:- Samit Ghosh
• MD and CEO:- Lttira Davis

The correct answer is Ashleigh Barty.

Key Points

• Ashleigh Barty won the title WTA player of the year 2021 for the second time.
• Prior to this, she also earned the WTA Player of the Year award in 2019.
• The WTA Player of the Year award is awarded by the World Tennis Association to women tennis players who have performed well in the year.

Important Points

• WTA (Women's Tennis Association) Year award 2021(Women)
• **Emma Raducanu (**Britain) won the 'WTA Newcomer of the Year award 2021'.
• Raducanu won the US Open 2021 women's singles title, her first Grand Slam championship.
• Barbora Krejcikova (Czech Republic) was named the WTA Most Improved Player of the Year 2021.
• Carla Suarez Navarro(Spain) won the title WTA Comeback Player of the Year award 2021.

Additional Information

• ATP (Association of Tennis Professionals) Awards 2021(Men)
• ATP Player of the year - Novak Djokovic
• Fans’ Favourite Award (Singles) - Roger Federer
• Ron Bookman Media Excellence Award - Prajwal Hegde (India)
• ATP Masters 1000 Tournament of the Year -  BNP Paribas Open (Indian Wells)

The correct answer is Odisha.

Key Points

• BrahMos Missile
• It is a medium-range supersonic cruise missile.
• It has been tested from the Integrated Test Range (ITR) at chandipur.
• Its speed is closer to 3 Mach and ranges up to 290 KM.
• It works on the fire and forget principle.
• The name BrahMos is derived from the Brahmaputra of India and the Moskva river of Russia.
• It is developed by BrahMos Aerospace (a joint venture of India’s DRDO and Russia’s Mashinostroyenia).
• India signs a deal with the Philipines to sell BrahMos missile.

Additional Information

• A cruise missile is an unmanned self-propelled (till the time of impact) guided vehicle that flies within the earth’s atmosphere and uses jet engine technology.
• Subsonic cruise missile travels at a speed lesser than that of sound. (0.8 Mach)
• Supersonic cruise missile travels at a speed of around 2-3 Mach.
• A hypersonic cruise missile travels at a speed of more than 5 Mach.

The correct answer Quit India Movement.

Key Points

• Quit India Movement
• The slogan ‘Quit India’ was coined by socialist congress leader Yusuf Meher Ali.
• This movement was launched because of the failure of the Cripps Offer and public discontent against wartime hardships.
• It was started on 8^th August 1942 by Mahatma Gandhi.
• It is also known as August Movement.
• The main slogan of this Movement is “Quit India and do or die” which was given by Gandhiji at Gowalia Tank Maidan (August Kranti Maidan).
• Aruna Asaf Ali operated underground radio and magazine and hoisted a flag at Gowalia tank. She is popularly known as the Grand old lady.

Additional Information

• Civil disobedient movement
• It was started in January 1930 by the Indian national congress in the Lahore session.
• Dandi march began on 12 march 1930 from Sabarmati Ashram and reached Dandi on 6 April and Gandhiji broke the salt law.
• Non-cooperation movement
• It was started in September 1920 by Indian National Congress in the Calcutta session.
• It aims to refuse to cooperate with the government, surrender titles (Kaiser-i-Hind – Gandhiji and Knighthoods – Rabindranath Tagore), and boycott of Govt. schools, offices, and courts.
• It was suspended in 1922 by Mahatma Gandhi due to the Chauri-Chaura incident.
• Khilafat Movement was started by two brothers Shoukat Ali and Mohammad Ali against the British Government.

The correct answer is Ujjain.

Key Points

• Solar Power Project
• The CM of Madhya Pradesh Shivraj Singh Chouhan laid foundation stones for setting up three solar power plants of 1500 Megawatt capacity in shajapur, agar, and neemuch at a cost of Rs 5250 Cr.
• Madhya Pradesh aims to produce 50% electricity through renewable energy.

Additional Information

• Madhya Pradesh (As of April 2022)
• Capital – Bhopal
• Governor – Mangubhai C. Patel (March 2022)
• Chief Minister – Shivraj Singh Chouhan
• Loksabha & Rajya Sabha Seats – 29 & 11
• Vidhan Sabha  – 230
• Important Places/ tourist sites – Khajuraho Temple, Mahakaleshwar (Ujjain), Bhimbetka cave, Gwalior Fort, Bandhavgarh, Pench, and Kanha National Park, and Sanchi Stupa.

The correct answer is Gomateshwara.

Key Points

• Gomateshwara
• It is also known as Bahubali.
• It is an important pilgrimage site of Jainism.
• According to Jainism, Gomateshwara is the son of Rishabhdev.
• It is located in Shravanabelagola, Karnataka.
• It was built in 981 AD by Chamundaraya (Western Ganga Dynasty)
• It is carved out of a single block of granite and looms on the top of Vindhya Giri hill.

Additional Information

• Lepakshi Nandi is located in lepakshi, Andhra Pradesh and it was built by Viranna and Virupanna.
• Bhojeshwar Temple is located in Bhojpur, Madhya Pradesh and was built by Paramara Raja Bhoja.
• Buddha Statue of Hyderabad is standing on the rock of Gibraltar in the middle of Hussain Sagar Lake.

The correct answer is PV Sindhu.

Key Points

• PV Sindhu is an Indian badminton player.
• She was born on 5^th July 1995 in Hyderabad.
• She has received the Arjuna Award (2013), Major Dhayan Chand Khel Ratna (2016), Padma Shri (2015), and Padma Bhushan (2020).
• She has won the bronze medal in the 2020 Tokyo Olympics and the silver medal in the 2016 Rio Olympics.

Additional Information

• Summer Olympic
• It is also known as the Games of the Olympiad.
• It was managed and run by International Olympic Committee (IOC) which was founded by Baron Pierre de Coubertin
• The first Summer Olympics was held in Athens in 1896.
• ​Sakshi Malik is an Indian freestyle wrestler and She won the bronze medal (58 categories) at the Rio 2016 Olympics.
• MIthali Raj is an Indian Cricket player.
• ​Deepika Kumari is an Indian Archer player.

The correct answer is Cadila Healthcare.

Key Points

• ZyCoV-D is a DNA plasmid-based COVID-19 vaccine developed by Indian pharmaceutical company Cadila Healthcare, with support from the Biotechnology Industry Research Assistance Council.
• It is approved for emergency use in India.

Additional Information

• Cipla Limited is an Indian multinational pharmaceutical company, headquartered in Mumbai, which primarily develops medicines to treat respiratory.
• Dr. Reddy's Laboratories is an Indian multinational pharmaceutical company located in Hyderabad, made  Indian Drugs and Pharmaceuticals Limited.
• Lupin Limited is an Indian multinational pharmaceutical company based in Mumbai, Maharashtra, focus areas include paediatrics, cardiovascular, anti-infectives, diabetology, asthma and anti-tuberculosis.

The correct answer is Delhi.

Key Points

• National Institute of Kathak Dance, also known as Kathak Kendra.
• It is the dance institution for the Indian classical dance form of Kathak.
• Also, a unit of the Sangeet Natak Akademi, India's National Academy of Music, Dance and Drama, was situated in New Delhi in 1964.​

Additional Information

• The Jawahar Kala Kendra is a multi-arts centre located in the city of Jaipur, India with the purpose of preserving Rajasthani arts and crafts.
• Krishi Vigyan Kendra and Banaras Hindu University, formerly Central Hindu College, is a collegiate, research university located in Varanasi, Uttar Pradesh, India.
• The National Botanical Research Institute (NBRI) is a research institute of CSIR in Lucknow, Uttar Pradesh, India.

The correct answer is 96.2%.

Key Points

• With 96.2 per cent literacy, Kerala has the most literate state in the country.
• Andhra Pradesh bottom with a rate of 66.4 per cent, showed a report based on National Statistical Office (NSO) survey.
• Kerala implemented social reform in the early **20^th century that allowed access to education for lower castes and women.**​

Additional Information

• TOP 5 STATE LITERACY RATE

The correct answer is New Zealand.

Key Points

• Ajaz Patel becomes just the third bowler in the history of men's Test cricket to take all 10 wickets in an innings (Jim Laker in 1956 and Anil Kumble in 1999).
• Ajaz Yunus Patel is a New Zealand cricketer who plays for Central Districts in domestic cricket.
• Patel is a slow left-arm orthodox spin bowler. He made his international debut for the New Zealand cricket team in October 2018.

Important Points 

• Kumble achieved this against Pakistan in the second Test played in Delhi in February 1999.

The correct answer is Centrum Housing Finance Limited.

Key Points

• Bank of Baroda (BoB) has signed a memorandum of understanding (MoU) with Centrum Housing Finance (CHFL) for sourcing and financing housing loan borrowers under the co-lending model.
• Centrum Housing Finance Limited (CHFL) is a housing finance company catering to 'middle class' India and providing financial inclusion to Lower and Middle Income (LMI) families by providing them with a long term housing finance.

Additional Information  

• Bajaj Finance Limited is a subsidiary of Bajaj Finserv, an Indian non-banking financial company headquartered in the city of Pune, India.
• Founder: Rahul Bajaj.
• Tata Capital Limited is a financial and investment service provider in India.
• ​CEO- Rajiv Sabharwal
• Headquarters-Mumbai
• Founded-2007.
• Muthoot Finance Ltd. is an Indian financial corporation and the largest gold loan NBFC in the country.
• Headquarters: Kochi
• Founder: Mathai George Muthoot.
• Founded: 1939.

The correct answer is 130.

Key Points

• The full form of UNDP is the United Nations Development Programme.
• Its headquarters are in New York City.
• It was established in 1965.
• Head of UNDP is Achim Steiner​

Additional Information

• The idea of Human Development Index (HDI) was firstly proposed by Pakistani Economist Mahbub-Ul-Haq.
• The first report of HDI was released in 1990.
• HDI measured on 3 Dimension:-
• Sta n dard of Living
• Health
• Education

Important Points

• Norway was in first place.
• Niger was in last place.
• The total number of countries participating in it is 189.
• India’s neighbouring country which performance is better than India.
• Sri Lanka - 76
• Mald  ives - 101

The correct answer is Oxidation reaction.

Key Points Corrosion

• Corrosion is a natural phenomenon that occurs due to the chemical reaction of metal in presence of Liquid or air.
• In which pure metal is converted into disintegrated metal.
• Corrosion occurs due to the loss of electrons.

Additional Information

• Corrosion is a Redox reaction.
• A redox reaction is expressed as a transfer of electrons between two reactants which participate in it.
• It is a combination of Reduction and Oxidation reactions.

Oxidation:- ​​

• An increase in oxidation state or loss of electrons is known as Oxidation.

Reduction:-

• A decrease in oxidation state or gain of electrons is known as Reduction.

The correct answer is He was born in 1855.

Key Points

• He was popularly known as Punjab Kesar, the Lions of Punjab.
• He was born in the Dhudike district of Punjab on 28^th January 1865.
• Along with the Bal Gangadhar Tilak and Bipin Chandra Pal, he formed the Lal, Bal, Pal of extremist leaders.
• A peaceful rally led by Lajpat Rai was started to protest the arrival of members of the Simon Commission in Lahore.
• Superintendent of Police Scott ordered his officers to “Lathi Charge” the demonstrators.
• Lala Lajpat Rai was severely injured and died as a result of the altercation.
• Bhagat Singh planned the killing of Scott in relation to the death of Lala Lajpat Rai.

Additional Information

• Some of the Important books authored by Lala Lajpat Rai is as follows:-
• The Story of my Deportation
• The problem of national education in India.
• Unhappy India
• England’s debt to India
• The message of The Bhagavad Gita
• The United States of America: A Hindu’s Impressions
• The depressed classes

The correct answer is Anil Menon.

Key Points

• NASA
• The full form of NASA is the National Aeronautics and Space Administration.
• It was formed in 1958.
• Its headquarters are in Washington DC.
• The administrator of NASA is Bell Nelson.
• Anil Menon
• Anil Menon SpaceX flight surgeon selected for NASA as it plans the first human mission to the moon after 50 years.
• Anil Menon is among the 10 latest trainee astronauts who will join NASA.

Additional Information

• ISRO (As of 2022)
• Its full form is Indian Space Research Organization.
• It was formed in 1969.
• Its headquarters are in Bengaluru.
• Chairman of ISRO is S. Somanath.
• JAXA - Japan Aerospace Exploration Agency.
• CNSA - China National Space Administration.
• ESA - The European Space Agency.
• CNES - The National Center for Space Studies.
• Roscosmos is a Russian Space Agency.

The correct answer is Vesara.

Key Points

• Hoysaleswara temple is located in the Halebidu so it is also known as the Halebidu temple.
• It was built by Vishnuvardhan.
• Hoysaleswara temple is dedicated to Nataraja (Lord Shiva).​

Additional Information 

• Mainly 3 types of Indian temple architecture

Nagara

• The Nagara style is mainly associated with the northern parts of India.
• The Nagara type style is mainly situated in between the Himalayas and Vindhyas region means the northern parts of India.

Dravida

• Dravida style advanced in the Southern region of India.
• It is situated between the Krishna and Kaveri River.

Vesara

• Vesara style is the combination of Nagara and Dravida styles.
• Vesara style is mainly situated between the Krishna and Vindhyas region.

Shortcut Trick

Convert mixed fraction Volume: 56 4/7 = (56 × 7 + 4) / 7 = 396/7 cm³.

Use formula (2/3)πr³ = 396/7 ⇒ (2/3) × (22/7) × r³ = 396/7.

Simplify: r³ = (396/44) × 3 = 9 × 3 = 27 ⇒ Radius (r) = 3 cm.

Direct formula for TSA: 3πr² = 3 × (22/7) × 3² = 3 × 22/7 × 9 = 594/7 cm².

∴ The correct answer is 594/7 cm².

Alternate Method

Given:

Volume of solid hemisphere (V) = 396/7 cm³

Value of π = 22/7

Formula Used:

Volume (V) = (2/3) × π × r³

Total Surface Area (TSA) = 3 × π × r²

Calculations:

⇒ (2/3) × (22/7) × r³ = 396/7

⇒ (44/21) × r³ = 396/7

⇒ r³ = (396/7) × (21/44)

⇒ r³ = 9 × 3 = 27

⇒ r = √[3]27 = 3 cm

⇒ TSA = 3 × (22/7) × (3)²

⇒ TSA = 3 × (22/7) × 9

⇒ TSA = (66 × 9) / 7 = 594/7 cm²

∴ The correct answer is 594/7 cm².

Additional Information

Curved Surface Area (CSA)

For a hemisphere, the curved surface area is half that of a sphere: CSA = 2 × π × r².

Volume of a Sphere

A hemisphere is exactly half of a sphere. Volume of sphere = (4/3) × π × r³.

Total Surface Area of Sphere

Unlike a solid hemisphere which has a flat circular base, a sphere only has a curved surface: TSA (Sphere) = 4 × π × r².

Given:

The base of a right pyramid is a square, whose diagonal is 20 cm.

The height of the pyramid is 20 cm.

Formula used:

The lateral surface area of the pyramid = \(\frac{1}{2}\) × p × l

Where, 

p, is the perimeter of the base of the pyramid

l, is the slant height of the square pyramid

The slant height of the square pyramid = \(\sqrt{(\frac{a}{2})^2 + h^2}\)

Where, 

a, is the side length of the square base

h, is the height of the square pyramid

The perimeter of the square = 4a

The diagonal length for a square = √2a

Where, 

a, is the side length of the square

Calculation:

Accoording to the question, the required image is:

The diagonal length for a square,

⇒ d = √2a

⇒ 20 = √2a

⇒ a = 10√2 cm

The perimeter of the square = 4 × 10√2 = 40√2 cm

The slant height of the square pyramid,

⇒ l = \(\sqrt{(\frac{a}{2})^2 + h^2}\)

⇒ l = \(\sqrt{(\frac{10√{2}}{2})^2 + 20^2}\)

⇒ l = \(\sqrt{50+400}\)

⇒ l = 15√2 cm

The lateral surface area of the pyramid = \(\frac{1}{2}\) × 40√2 × 15√2 = 600 cm²

∴ 600 cm² is the lateral surface area of the square pyramid.

Shortcut Trick

According to the question, the required figure is:

In Δ ABC,

BC2 = AB2 + AC2

⇒ BC2 = 62 + 82

⇒ BC2 = 36 + 64

⇒ BC2 = 100

⇒ c = BC = 10 cm

Now, 

x = \(\frac{abc}{a^2 + b^2 + ab}\)

⇒ x = \(\frac{6 \times 8 \times 10}{6^2 + 8^2 + 48} = \frac{120}{37}\) cm

Alternate MethodGiven:

In ΔABC, ∠A = 90°, AB = 6 cm, AC = 8 cm.

A square DEFG is drawn inside the triangle with side FG along with BC, E and D on sides AB and AC, respectively.

Calculation:

According to the question, the required figure is:

In Δ ABC,

BC² = AB² + AC²

⇒ BC² = 6² + 8²

⇒ BC2 = 36 + 64

⇒ BC2 = 100

⇒ BC = 10 cm

Now, 

In ΔEBF,

tan(α) = \(\frac{a}{FB}\)    ----(1)

In ΔCDG,

tan(α) = \(\frac{CG}{a}\)    ----(2)

In ΔABC,

tan(α) = \(\frac{8}{6}\)       ----(3)

From equation (1) and (3), we get,

⇒ FB = \(\frac{6a}{8}\)

From equation (2) and (3), we get,

⇒ CG = \(\frac{8a}{6}\)

Now, 

BC = CG + a + FB

On substituting the value of the respective variable, we get,

⇒ 10 = \(\frac{8a}{6}\) + a + \(\frac{6a}{8}\)

⇒ 10 = \(\frac{64a + 48a + 36a}{48}\)

⇒ a = \(\frac{480}{148}\) = \(\frac{120}{37}\)

∴ The side length of the required square (in cm) is \(\frac{120}{37}\).

Given:

The average age of A, B and C, 7 years ago, was 42 years and the average age of B and C, 9 years ago, was 36 years.

Calculation:

Since the average age of A, B and C, 7 years ago, was 42 years,

Therefore, 

The value of the present average age of A, B and C = 42 + 7 = 49 years

The sum of ages of A, B and C = 49 × 3 = 147 years

Since the average age of B and C, 9 years ago, was 36 years,

Therefore, 

The value of the present average age of B and C = 36 + 9 = 45 years

The sum of ages of B and C = 45 × 2 = 90 years

Now, 

The age of A = 147 - 90 = 57 years

∴ The present age (in years) of A is 57.

Additional Information

Since the average age of A and B, n years ago, was T years,

Therefore, 

The value of the present average age of A and B = (T + n) years

Shortcut Trick

Express 34 − 24√2 as a perfect square: 18 + 16 − 2(3√2)(4) = (3√2 − 4)².

The fourth root reduces to a square root: ⁴√(34 − 24√2) = √(3√2 − 4).

Multiply with the second term: √(3√2 − 4) × √(3√2 + 4).

⇒ √[(3√2)² − 4²] = √(18 − 16) = √2.

∴ The correct answer is √2.

Alternate Method

Given:

Expression: ⁴√(34 − 24√2) × √(4 + 3√2)

Formula Used:

(a − b)² = a² + b² − 2ab

(a − b)(a + b) = a² − b²

Calculation:

⇒ First, simplify the expression inside the fourth root: 34 − 24√2

⇒ Split 34 into 18 and 16, and express 24√2 as 2 × 3√2 × 4

⇒ 34 − 24√2 = 18 + 16 − 2(3√2)(4)

⇒ 34 − 24√2 = (3√2)² + (4)² − 2(3√2)(4)

⇒ Using the formula, it can be expressed as a perfect square:

⇒ 34 − 24√2 = (3√2 − 4)²

⇒ Taking the fourth root: [ (3√2 − 4)² ]^1/4 = (3√2 − 4)^2/4 = √(3√2 − 4)

⇒ Note: 3√2 = √18 and 4 = √16. Since 3√2 > 4, the principal root remains positive.

⇒ Now substitute this simplified term back into the given expression:

⇒ √(3√2 − 4) × √(3√2 + 4)

⇒ Combine both parts under a single square root:

⇒ √[ (3√2 − 4)(3√2 + 4) ]

⇒ Apply the difference of squares formula:

⇒ √[ (3√2)² − (4)² ]

⇒ √(18 − 16)

⇒ √2

∴ The correct answer is √2.

Additional Information

Square Root of a Surd

To find the square root of a binomial surd like a − √b, attempt to express it as a perfect square (x − y)². Always ensure x > y so the principal square root remains a positive real number.

Difference of Squares

The algebraic identity (a − b)(a + b) = a² − b² is highly effective for simplifying the product of conjugate surds by completely eliminating the square root terms.

Rules of Indices

The fractional powers of surds follow the rule: ^m√(aⁿ) = a^n/m. In this problem, taking the fourth root of a perfect square naturally reduces the expression to a square root.

Shortcut Trick

Interest for 2.5 years = 19,116 − 16,956 = ₹2,160

Interest per year = 2,160 ÷ 2.5 = ₹864

Principal = 16,956 − (864 × 4) = 16,956 − 3,456 = ₹13,500

Rate of interest = (864 ÷ 13,500) × 100 = 6.4%

∴ The correct answer is 6.4%.

Alternate Method

Given:

Amount after 4 years (A₁) = ₹16,956

Amount after 6.5 years (A₂) = ₹19,116

Formula Used:

Amount (A) = Principal (P) + Simple Interest (SI)

SI = (P × R × T) ÷ 100

Calculations:

⇒ Interest for (6.5 − 4) years = 19,116 − 16,956

⇒ Interest for 2.5 years = ₹2,160

⇒ Interest for 1 year = 2,160 ÷ 2.5 = ₹864

⇒ Interest for 4 years = 864 × 4 = ₹3,456

⇒ Principal (P) = A₁ − SI for 4 years

⇒ P = 16,956 − 3,456 = ₹13,500

⇒ Rate of Interest (R) = (SI for 1 year × 100) ÷ P

⇒ R = (864 × 100) ÷ 13,500

⇒ R = 864 ÷ 135 = 6.4%

∴ The correct answer is 6.4%.

Additional Information

Simple Interest Properties

In simple interest, the interest remains constant for every time period as it is always calculated on the original Principal amount.

Direct Rate Calculation

Rate R = (Interest × 100) ÷ (P × T). If T = 1 year, then R = (Interest for 1 year ÷ P) × 100.

Amount Formula

Total Amount A = P (1 + (R × T) ÷ 100), where P is principal, R is rate per annum, and T is time in years.

Given:

A computer is available for ₹76,800 cash or ₹23,840 cash down payment and three equal half-yearly instalments.

The interest charged is 20% p.a., compounded semi-annually.

Formula used:

D = \(\frac{A}{R} × (1 - \frac{1}{(1+R)^n})\)

Where, 

D, is the amount due (or amount on which interest is to be applied)

A, is the value of the instalment

R, is the rate of interest applied

n, is the number of instalments

Calculation:

The price of the computer = ₹76,800

The down payment is done for buying the computer = ₹23,840

The remaining amount (which is paid in three equal instalments) = ₹76,800 - ₹23,840 = ₹52,960

The rate of interest applied annually = 20% p.a.

The rate of interest applied semi-annually = 10% p.a.

The number of instalments, n = 3

Now, 

By substituting the value of the respective variable in the required formula, we get,

⇒ ₹52,960 = \(\frac{A}{10\text{%}} × (1 - \frac{1}{(1+\text{10%})^3})\)

⇒ ₹52,960 = \(\frac{A}{0.1} × (1 - \frac{1}{(1+0.1)^3})\)

⇒ ₹52,960 = \(\frac{A}{0.1} × (1 - \frac{1}{(1.1)^3})\)

⇒ ₹5296 = A × \( (1 - \frac{1000}{(1331)})\)

⇒ A = ₹21296

The total amount paid in three instalments = ₹21296 × 3 = ₹63888

The interest paid = ₹63888 - ₹52960 = ₹10928

∴ ₹10928 is the interest charged in the instalment scheme.

Given:

(\(1.0\bar{5} \div 0.\overline{95}\)) × \(0.4\overline{09}\)

Formula used:

Recurring decimal to fraction conversion:

x.abc = \(\frac{abc - ab}{900}\) (Example for notation)

Calculation:

(\(1.0\bar{5} \div 0.\overline{95}\)) × \(0.4\overline{09}\) = ?

⇒ \(1.0\bar{5} = \frac{105 - 10}{90} = \frac{95}{90} = \frac{19}{18}\)

⇒ \(0.\overline{95} = \frac{95}{99}\)

⇒ \(0.4\overline{09} = \frac{409 - 4}{990} = \frac{405}{990} = \frac{9}{22}\)

⇒ ? = (\(\frac{19}{18} \div \frac{95}{99}\)) × \(\frac{9}{22}\)

⇒ ? = (\(\frac{19}{18} \times \frac{99}{95}\)) × \(\frac{9}{22}\)

⇒ ? = (\(\frac{1}{2} \times \frac{11}{5}\)) × \(\frac{9}{22}\)

⇒ ? = \(\frac{11}{10} \times \frac{9}{22}\)

⇒ ? = \(\frac{9}{20}\)

⇒ ? = 0.45

∴ The correct answer is option (1).

Given: 

The tangent at a point P, on a circle with centre O, intersects the diameter BA produced at Q.

∠QPB = 103o

Concept used:

The angle subtended by diameter on any point on the circumference of a circle is 90°.

The angle between a tangent and a chord is equal to the angle in the alternate segment.

Calculation:

According to the question, the required figure is:

 

As we know,

The angle subtended by diameter on any point on the circumference of a circle is 90°.

So, 

∠BPA = 90°

Now, 

∠BPQ = ∠BPA + ∠APQ

⇒ 103° = 90° + ∠APQ

⇒ ∠APQ = 13°

As per the alternate segment theorem.

∠PBQ = ∠APQ = 13°

In ΔBPA,

∠PBA + ∠BAP + ∠APB = 180°

⇒ 13° + ∠BAP + 90° = 180°

⇒ ∠BAP + 103° = 180°

⇒ ∠BAP = 77°

Now,

∠BAP + ∠PAQ = 180°

⇒ 77° + ∠PAQ = 180°

⇒ ∠PAQ = 103°

∴ The measure of ∠PAQ is 103°.

Given:

X and Y start their journey at the same time from place A to B and B to A, respectively.

After meeting each other on the way, X and Y complete their remaining journey in 5\({ {4} \over 9}\) hours and p hours, respectively.

The speed of X is 28\({{4} \over 7}\)% more than that of Y.

Formula used:

\(\frac{S_x}{S_y} = \sqrt{\frac{T_{y(AC)}}{T_{x(CB)}}}\)

Where, 

S_x and S_y are the speed of trains X and Y respectively.

T_y(AC) and T_x(CB) are the time required by train Y and train X to complete the remaining journey after they meet at point C.

Calculation:

According to the question, the required figure is:

Let the speed of Y = 700 kmph

The speed of X = (100% + 28\({{4} \over 7}\)%) of 700 = 900 kmph

The time required by X to complete the journey from C to B = 5\(\frac{4}{9}\) = \(\frac{49}{9}\) hours

The time required by Y to complete the journey from C to A = P hours

Now, 

⇒ \(\frac{900}{700} = \sqrt{\frac{P}{\frac{49}{9}}}\)

⇒ \(\frac{9}{7} = \sqrt{\frac{9P}{49}}\)

Take square on both sides, 

⇒ \(\frac{81}{49} = {\frac{9P}{49}}\)

⇒ P = 9 hours

∴  The value of P (in hours) is 9.

Given:

A invests  ₹3 lakhs more than B.

B invests his capital for 7\({{1} \over 2}\) months, while A invests his capital for 2\({{1} \over 2}\) more months than B.

The share of B in the total profit of ₹6.25 lakhs is ₹1.25 lakhs less than the share of A,

Calculation:

Let the amount invested by B = ₹x lakhs

The amount invested by A = ₹(x + 3) lakhs

The time of investment of B =  7\({{1} \over 2}\) = 7.5 months

The time of investment of A =  (7\({{1} \over 2}\) + 2\({{1} \over 2}\)) = 10 months

The required ratio for the capital invested = (₹x × 7.5) : (₹(x + 3) × 10)

Now, 

The total profit earned by A and B = ₹6.25 lakhs

Let 'P_A' and 'P_B' be the profit earned by A and B.

So, 

⇒ P_A + P_B = ₹6.25 lakhs    ----(1)

The share of B in the total profit is ₹1.25 lakhs less than the share of A,

Therefore,

⇒ PA - PB = ₹1.25 lakhs    ----(2)

By solving the equation (1) and (2), we get,

P_A = ₹3.75 lakhs and P_B = ₹2.50 lakhs

Now, 

As we know, the ratio in which profit is divided is the same as the ratio of capital invested (product of amount invested and period of investment).

⇒ (₹x × 7.5) : (₹(x + 3) × 10) = 2.5 : 3.75

⇒ (x × 7.5) : (10x + 30) = 2.5 : 3.75

It can be written as:

⇒ \(\frac{7.5x}{10x + 30} = \frac{2.5}{3.75}\)

⇒ \(\frac{7.5x}{10x + 30} = \frac{2}{3}\)

⇒ 22.5x = 20x + 60

⇒ 2.5x = 60

⇒ x = 24

∴ The amount invested by B is ₹24 lakhs.

Shortcut Trick

Multiply 2x + 1/(3x) = 5 by 3/2 to get the terms for cubing.

⇒ 3x + 1/(2x) = 15/2

Here, a = 3x, b = 1/(2x), so their product ab = 3/2.

Apply the shortcut formula: a³ + b³ = (a + b)³ − 3ab(a + b)

⇒ Value = (15/2)³ − 3 × (3/2) × (15/2) = 3375/8 − 135/4 = 3105/8

∴ The correct answer is 3105/8.

Alternate Method

Given:

2x + 1/(3x) = 5

Formula Used:

(a + b)³ = a³ + b³ + 3ab(a + b)

Calculation:

⇒ Multiply the given equation by 3/2 to adjust the coefficients.

⇒ (3/2) × [2x + 1/(3x)] = (3/2) × 5

⇒ 3x + 1/(2x) = 15/2

⇒ Cubing both sides of the equation:

⇒ (3x + 1/(2x))³ = (15/2)³

⇒ (3x)³ + (1/(2x))³ + 3 × (3x) × (1/(2x)) × (3x + 1/(2x)) = 3375/8

⇒ 27x³ + 1/(8x³) + (9/2) × (15/2) = 3375/8

⇒ 27x³ + 1/(8x³) + 135/4 = 3375/8

⇒ 27x³ + 1/(8x³) = 3375/8 − 270/8

⇒ 27x³ + 1/(8x³) = 3105/8

∴ The correct answer is 3105/8.

Additional Information

Sum of Cubes Formula

The algebraic expansion a³ + b³ = (a + b)(a² − ab + b²) is widely used for factorization. It can also be expressed directly for value calculation as a³ + b³ = (a + b)³ − 3ab(a + b).

Difference of Cubes Formula

Similarly, the difference of cubes is given by a³ − b³ = (a − b)(a² + ab + b²) or equivalently a³ − b³ = (a − b)³ + 3ab(a − b).

Reciprocal Term Expansions

For an equation of the form x + 1/x = k, we can quickly determine x³ + 1/x³ = k³ − 3k. When the terms have different coefficients like ax + 1/(bx) = k, the identity adjusts to k³ − 3(a/b)k.

Given: 

A shopkeeper sells, by mistake, an article at 32% less than its original selling price and finds that he gets ₹67.20 less when compared to the original selling price.

If he gets a profit of 40% by selling the article at its original selling price,

Formula used:

C.P = \(\frac{\text {S.P}}{(100\text {% + P%})}\)

Where, 

C.P, is the cost price of the article

S.P, is the selling price of the article

P%, is the profit (in per cent)

Calculation:

Let the original selling price of the article = ₹x

The mistaken selling price of the article = (100% - 32%) of ₹x = ₹0.68x

Since the shopkeeper sells, by mistake, an article at 32% less than its original selling price and finds that he gets ₹67.20 less when compared to the original selling price,

Therefore, 

⇒ ₹x - ₹0.68x = ₹67.20

⇒ ₹0.32x = ₹67.20

⇒ x = ₹210

The original selling price of the article = ₹210

The cost price of the article to gain 40% gain = \(\frac{₹210}{(100\text {%} + 40\text {%})}\) = ₹150

∴ ₹150 is the required cost price of the article.

Given:

The number of female employees in office B = 250

The number of female employees in office C = 400

The number of female employees in office D = 550

The number of male employees in office A = 250

The number of male employees in office C = 500

Calculation:

The number of female employees in office B who got a promotion = 60% of 250 = 150

The number of female employees in office C who got a promotion = 55% of 400 = 220

The number of female employees in office D who got a promotion = 30% of 550 = 165

The number of male employees in office A who did not got a promotion = (100% - 42%) of 250 = 145

The number of male employees in office C who did not got a promotion = (100% - 45%) of 500 = 275

The total number of female employees in offices B, C and D who got a promotion = 150 + 220 + 165 = 535

The total number of male employees in offices A and C who did not get a promotion = 145 + 275 = 420

The required percentage = \(\frac{535-420}{420} \) ≈ 27.4%

∴ The total number of female employees in offices B, C and D who got a promotion is 27.4% more than the total number of male employees in offices A and C who did not get a promotion.

Shortcut Trick

Let the cost of a car be 100 units, then the cost of a motorcycle is 100 − 60% = 40 units.

Ratio of the initial total cost of 4 cars to 5 motorcycles = (4 × 100) : (5 × 40) = 400 : 200 = 2 : 1.

Net Percentage Change = [(Ratio₁ × Change₁) + (Ratio₂ × Change₂)] ÷ Total Ratio

Net Percentage Change = [(2 × −10) + (1 × 40)] ÷ (2 + 1) = (−20 + 40) ÷ 3 = +20 ÷ 3 = +6 ²⁄₃%.

The positive sign denotes an overall increase in the total cost.

∴ The correct answer is 6 ²⁄₃% increase.

Alternate Method

Given:

⇒ Cost of a motorcycle is 60% less than the cost of a car.

⇒ Quantity analyzed: 4 cars and 5 motorcycles.

⇒ Cost Change: Car cost decreases by 10%, Motorcycle cost increases by 40%.

Formula Used:

⇒ Percentage Change = (Net Change ÷ Initial Total Value) × 100

Cost Comparison (4 Cars & 5 Motorcycles)

Initial Cost: 600

New Cost: 640

• 40 Increase

% Increase = (40 ÷ 600) × 100 = 6 2/3%

Calculations:

⇒ Let the initial cost of 1 car = 100 units.

⇒ Initial cost of 1 motorcycle = 100 − (60% of 100) = 40 units.

⇒ Initial total cost = (4 × 100) + (5 × 40) = 400 + 200 = 600 units.

⇒ New cost of 1 car = 100 − 10% = 90 units.

⇒ New cost of 1 motorcycle = 40 + (40% of 40) = 40 + 16 = 56 units.

⇒ New total cost = (4 × 90) + (5 × 56) = 360 + 280 = 640 units.

⇒ Overall change in cost = 640 − 600 = +40 units.

⇒ Percentage Increase = (40 ÷ 600) × 100 = 400 ÷ 60 = 20 ÷ 3 = 6 ²⁄₃%.

∴ The correct answer is 6 ²⁄₃% increase.

Additional Information

Weighted Average in Percentages

When a whole is divided into distinct parts, the net percentage change is the weighted average of the percentage changes of its components: Net % = (n₁x₁ + n₂x₂) ÷ (n₁ + n₂).

Percentage Change

Percentage change must always be calculated strictly with respect to the initial base value: % Change = (Final Value − Initial Value) ÷ Initial Value × 100.

Successive Percentage Change

If a single value undergoes two successive percentage changes a% and b%, the effective net change is given by the shortcut formula: (a + b + ab ÷ 100)%.

Given:

A pole of height 4 m is fixed on the top of a tower.

The angle of elevation of the top of the pole as observed from a point on the ground is 60o.

The angle of depression of the same point from the top of the tower is 45°.

Formula used:

For the right-angled triangle.

tan(θ) = \(\frac{p}{b}\)

Calculation:

According to the question, the required figure is:

In ΔABC,

tan(60^∘) = \(\frac{4+h}{b}\)   

⇒ √3 = \(\frac{4+h}{b}\)        ----(1)

In ΔDBC,

tan(45∘) = \(\frac{h}{b}\)

⇒ 1 = \(\frac{h}{b}\)

⇒ h = b              ----(2)

By using the equation (1) and (2), 

⇒ √3 = \(\frac{4+h}{h}\)

⇒ √3h = 4 + h

⇒ h × (√3 - 1) = 4

⇒ h = \(\frac{4}{√{3} -1} × \frac{√{3} +1}{√{3} +1}\)

⇒ h = \(\frac{4(√{3} +1)}{(√{3})^2 -1} \)

⇒ h = \(\frac{4(√{3} +1)}{(3 -1)} \)

⇒ h = 2 × (√3 + 1) m

∴ 2(√3 + 1) is the required height (in m) of the tower.

Given:

A dealer offers 35% discount on the marked price of an article and still gains 30%.

The cost of the article increases by 25%,

Formula used:

D% = \(\frac{M.P-S.P}{M.P}\) × 100

Where, 

D, is the discount per cent

M.P and S.P are the mark price and selling price respectively.

P% = \(\frac{S.P-C.P}{C.P}\) × 100

Where, 

S.P and C.P are the selling price and cost price respectively.

Calculation:

Let the mark price of the article = ₹200

Since the dealer offers 35% discount on the marked price of an article,

Therefore,

The selling price of the article = (100% - 35%) of ₹200 = ₹130

Since the dealer earns 30% profit,

Therefore, 

The cost price of the article = \(\frac{₹130}{(100\text {%} + 30\text {%})}\) = ₹100

Now, 

The increased cost price of the article = (100% + 25%) of ₹100 = ₹125

Since the dealer wants to earned 24% profit,

Therefore, 

The new selling price = (100% + 24%) of ₹125 = ₹155

Now, 

The mark price of the article remains the same = ₹200

The discount per cent = \(\frac{₹200-₹155}{₹200}\) × 100 = 22.5%

∴ 22.5% discount should he give now on the same marked prize so as to earn 24% profit on the new cost.

Given:

Let x be the least number which when divided by 16, 24, 30, 36 and 45, the remainder in each case is 4,

x is divisible by 28.

The HCF of x and 3193 is y.

Calculation:

The L.C.M of 16, 24, 30, 36 and 45 = 720

The least number which when divided by 16, 24, 30, 36 and 45, the remainder in each case is 4 = L.C.M of (16, 24, 30, 36 and 45) + 4 = 720n + 4

The required number which divisible by 28 = (720n + 4) ÷ 28

For n = 1,

The required number = (720(1) + 4) ÷ 28 = (25 × 28) + 24

For n = 2,

The required number = (720(2) + 4) ÷ 28 = (51 × 28) + 16

So,

For n = 4,

The required number = (720(4) + 4) ÷ 28 = (103 × 28)

Therefore, 

The required number, x = 2884

Now, 

The prime factorisation of 2884 = 2 × 2 × 7 × 103

The prime factorisation of 3193 = 31 × 103

The H.C.F of (2884 and 3193), y = 103

According to the question,

The sum of the digits of 103 = 1 + 0 + 3 = 4

∴ The required value of y is 4.

Additional InformationThe least number which when divided by a, b, and c, the remainder in each case is k = L.C.M of (a, b, c) + k

Given:

The distance = 100 km,

Train A takes 30 minutes or 0.5 hours less time than train B.

The speed of train B is 10 kmph less than that of train A

Formula used:

Distance = Speed × Time

Calculation:

Let 's' and 't' be the usual time to travel the required distance.

The time is taken by train B = t hr

The time is taken by train A = (t - 0.5) hr

The speed of the train A = s kmph

The speed of the train B = (s - 10) kmph

The required distance to be travelled = 100 km

Now, 

⇒ s × (t - 0.5) = 100    ----(1)

⇒ (s - 10) × t = 100    ----(2)

On equating the above equations, we get,

⇒ st - 0.5s = st - 10t

⇒ - 0.5s = - 10t

⇒ t = 0.05s    ----(3)

Now, substitute the equation (3) in equation (1), we get,

⇒ s × (0.05s - 0.5) = 100

Multiply the above equation 100 on both sides of the equation,

⇒ 100 × s × (0.05s - 0.5) = 100 × 100

⇒ 5s² - 50s = 10,000

⇒ 5s2 - 50s - 10,000 = 0

⇒ s2 - 10s - 2000 = 0

⇒ s2 - 50s + 40s - 2000 = 0

⇒ s(s - 50) + 40(s - 50) = 0

⇒ (s + 40)(s - 50) = 0

We get,

s = -40, 50

Since the negative value for speed can not be possible, 

Therefore, 

The speed of the train A = 50 kmph

∴ The speed of train A is 50 kmph.

Shortcut Trick

Ratio of the cost of a Shirt (S) to a Trouser (T) = 80 : 100 = 4 : 5.

Cost difference = (3S + 5T) − (5S + 2T) = 37 units − 30 units = 7 units.

7 units = ₹1,260 ⇒ 1 unit = ₹180.

Required cost = 4S + 3T = 4 × 4 + 3 × 5 = 31 units = 31 × 180 = ₹5,580.

∴ The correct answer is ₹5,580.

Alternate Method

Given:

The cost of a shirt is 20% less than the cost of a trouser.

The difference between the cost of (3 shirts + 5 trousers) and (5 shirts + 2 trousers) is ₹1,260.

Calculations:

⇒ Let the cost of a trouser (T) be 5x.

⇒ The cost of a shirt (S) is 20% less = 5x − (20/100 × 5x) = 4x.

⇒ Cost of 3 shirts and 5 trousers = 3(4x) + 5(5x) = 12x + 25x = 37x.

⇒ Cost of 5 shirts and 2 trousers = 5(4x) + 2(5x) = 20x + 10x = 30x.

⇒ Given difference = 37x − 30x = 7x.

⇒ 7x = 1,260 ⇒ x = 1,260 ÷ 7 = 180.

⇒ We need to find the cost of 4 shirts and 3 trousers.

⇒ Required cost = 4(4x) + 3(5x) = 16x + 15x = 31x.

⇒ Required cost = 31 × 180 = 5,580.

∴ The correct answer is ₹5,580.

Additional Information

Ratio & Proportion Basics

When one quantity is given as a percentage of another, converting the percentage directly to a ratio simplifies calculations. For example, 20% less means 80% or 4/5 of the base value.

Linear Equations Setup

Word problems often translate into linear equations. Assigning variables to a common multiplier (like 4x and 5x) prevents fractions and maintains integer relationships.

Unitary Method

Finding the value of 1 base unit first (x = 180) allows easily calculating the value of any other required combination simply by multiplying the unit factor with the target equation.

Given:

A takes 6 more days than B to complete a certain work.

B starts the work and A took over after 4 days and completed the remaining work in 12 days.

Calculation:

The time is taken by B to complete the work = t days

The time is taken by A to complete the work = (t + 6) days

The total work to be done by A and B = L.C.M of (t, t + 6) = t × (t + 6) units

The efficiency of A = \(\frac{t(t + 6)}{(t+6)}\) = t

The efficiency of B = \(\frac{t(t + 6)}{(t)}\) = (t + 6)

Since B starts the work and A took over after 4 days,

Therefore, B worked for first 4 days.

The amount of work done by B in 4 days = 4 × (t + 6) units

The remaining work done by A in 12 days = 12 × t units

Now,

⇒ 4 × (t + 6) + 12t = t × (t + 6)

⇒ 4t + 24 + 12t = t² + 6t

⇒ 24 + 16t = t2 + 6t

⇒ t2 - 10t - 24 = 0

⇒ t2 - 12t + 2t - 24 = 0

⇒ t(t - 12) + 2(t - 12) = 0

⇒ (t + 2)(t - 12) = 0

We get, t = -2, 12

The total work to be done by A and B = 12 × (12 + 6) = 216 units

The efficiency of A = 12

The efficiency of B = 18

The 62.5% of the original work = \(\frac{5}{8}\) × 216 = 135 units

The combined efficiency of A and B = 12 + 18 = 30

Let 'TAB' is the time taken by A and B together to complete 62.5% of the original work.

⇒ TAB = \(\frac{135}{30}\)

⇒ TAB = 4\(\frac{15}{30}\) = 4\(\frac{1}{2}\) days

∴ A and B together can complete 62.5% of the original work in 4\(\frac{1}{2}\) days.

Given:

R varies directly as S and inversely as T.

When R = \({ {2} \over 3}\) and T = \({ {5} \over 7}\), S = \({ {4} \over 21}\).

Calculation:

According to the question,

R ∝ S and R ∝ \(\frac{1}{T}\)

Therefore, 

R ∝ \(\frac{S}{T}\)

⇒ R = \(\frac{kS}{T}\)

Now, on substituting the value of the respective variable in the above formula, we get,

⇒ \(\frac{2}{3}\) = \(k\frac{\frac{4}{21}}{\frac{5}{7}}\)

⇒ \(\frac{2}{3}\) = \(k\frac{4}{15}\)

⇒ k = \(\frac{5}{2}\)

Now, 

⇒ R = \(\frac{kS}{T}\)

The above-obtained formula can be written as:

⇒ S = \(\frac{RT}{k}\)

⇒ S = \(\frac{\frac{3}{4} \times \frac{1}{9}}{\frac{5}{2}}\)

⇒ S = \(\frac{\frac{1}{12}}{\frac{5}{2}}\)

⇒ S = \(\frac{1}{30}\)

∴ The required value of S is \(\frac{1}{30}\).

Given:

The production of sugar in 2015 = 300 million kg

The production of sugar in 2017 = 420 million kg

The production of sugar in 2019 = 450 million kg

The export of sugar in 2015 = 200 million kg

The export of sugar in 2016 = 280 million kg

The export of sugar in 2018 = 260 million kg

Calculation:

The average production of sugar in 2015, 2017 and 2019 = \(\frac{300+420+450}{3}\) = \(\frac{1170}{3}\) = 390 million kg

The total export of sugar in 2015, 2016 and 2018 = 200 + 280 + 260 = 740 million kg

The required percentage = \(\frac{390}{740}\) × 100 ≈ 52.7%

∴ The average production of sugar in 2015, 2017 and 2019 is approximately 52.7% of the total export of sugar in 2015, 2016 and 2018.

Given:

The average weight of some persons in a group was 68 kg.

5 persons with an average weight 73.4 kg joined the group.

8 persons with an average weight 65.5 kg left the group,

The average weight of the persons in the group increased by 1.175 kg.

Formula used:

Average weight = \(\frac{\text {Sum of weight of 'n' people}}{\text {n}}\)

Calculation:

Let 'n' be the initial number of persons in the group.

The total weight of 'n' person in the group = (68 × n) kg 

The total weight of the 5 person who joined the group = 73.4 × 5 = 367 kg

The total weight of the 8 person who left the group = 65.5 × 8 = 524 kg

Since the average weight of the persons in the group increased by 1.175 kg,

Therefore,

⇒ \(\frac{68n + 367 - 524}{n + 5 - 8}\) = 68 + 1.175

⇒ \(\frac{68n - 157}{n - 3}\) = 69.175

⇒ 68n - 157 = 69.175 × (n - 3)

⇒ 68n - 157 = 69.175n - 207.525

⇒ 69.175n - 68n = 207.525 - 157

⇒ 1.175n = 50.525

⇒ n = 43

∴ The initial number of persons in the group is 43.

Given:

A is 40% more efficient than B,

C is 25% less efficient than B.

Working together, they can complete the work in 20 days.

Formula used:

Efficiency × Time = Work

Calculation:

According to the question,

The efficiency of A, E_A = (100% + 40%) of E_B = 1.4E_B

The efficiency of C, E_C = (100% - 25%) of E_B = 0.75E_B

Let the working efficiency of B = 1 unit a day

Therefore, 

The efficiency of A, E_A = 1.4 units a day

The efficiency of C, E_C = 0.75 a day

If they worked together, they can complete the work in 20 days,

The total amount of work done in 20 days,

⇒ (E_A + E_B + E_C) × 20

⇒ (1.4 + 1 + 0.75) × 20

⇒ (3.15) × 20

⇒ 63 units

Now, 

The two-third of the total work = \(\frac{2}{3}\) × 63 = 42 units

The total time required to complete two-thirds of the work by B and C = \(\frac{42}{1 + 0.75}\) = \(\frac{42}{1.75}\) = 24 days

∴ B and C together can complete two-thirds of the same work in 24 days.

The correct answer is Photosynthesis in plants.

Concept:

Chemical reaction:

• A process in which one or more substances, the reactants, are converted to one or more different substances, the products.
• Substances are either chemical elements or compounds.
• A chemical reaction rearranges the constituent atoms of the reactants to create different substances as products.

Explanation:

Endothermic reaction:

• A chemical reaction in which the reactants absorb energy and get converted into products.

Photosynthesis in plants: 

• It occurs in green plants in the presence of sunlight and chlorophyll.
• The reactants carbon dioxide and water are converted into glucose and oxygen.
• CO₂ + H₂O + Sunlight(Heat) → C₆H₁₂O₆ + O₂
• **_​**Solar energy is converted into chemical energy.
• Since energy is being consumed in photosynthesis, it is termed an endothermic reaction.

Additional Information​ 

Exothermic reaction:

• A chemical reaction in which heat energy is released.
• Examples: 
• Burning of natural gas
• Conversion of vegetable matter into compost
• ​Burning of magnesium ribbon in the air.

The correct answer is 0.25 g per gram of water.

Concept:

• Solution:  A homogenous mixture of two or more substances. It has two components solute and solvent.
• Solute: The substance that is dissolved in the solvent.
• Solvent: The substance in which solute is dissolved.

For example, in a salt solution, salt is solute and water is solvent. 

Explanation:

Solubility: 

The maximum amount of a solute that can be dissolved in 100 g of a solvent at a specified temperature is known as the solubility of that solute in that solvent (at that temperature).

Given: Mass of sugar (solute) = 62.5 g

Mass of water (solvent) = 250g 

Solubility = (Mass of solute / mass of solvent) × 100 

= \(\frac{62.5}{250} × 100\)

= 25 %  or 25/100 = 0.25 g per gram of water

The correct answer is 

Concept:

Deforestation: It refers to the cutting of trees on a large scale.

Explanation:

Chipko movement:

• Also called Chipko andolan, it was a nonviolent social and ecological movement by rural villagers, particularly women, in India in the 1970s.
• It aimed at protecting trees and forests slated for government-backed logging.
• The movement originated in the Himalayan region of Uttarakhand in 1973 and quickly spread throughout the Indian Himalayas.
• The Hindi word chipko means “to hug” or “to cling to” and reflects the demonstrators’ primary tactic of embracing trees to impede loggers.
• This movement was led by Sundarlal Bahunguna.

The correct answer is All somatic human cells have the same amount of DNA in their nuclei.

Concept:

DNA: 

• It stands for Deoxyribonucleic Acid.
• It is the genetic material found in most organisms and is inherited from one generation to another.
• It carries all the information about the anatomy (internal structure), physiology ( working of the body organs), and morphology (external appearance) of an organism.
• It is found inside the nucleus of the cell.

Chromosome:

• These are the X-shaped structures visible only when the cell is in the dividing phase.
• They are made up of DNA. DNA condenses to form the chromosomes.

Explanation:

DNA in the Human body:

• There are two types of cells, somatic and germ cells.
• Somatic cells are the cells that make up the whole body while germ cells are the gametes that take part in reproduction.
• Somatic cells have 46 chromosomes while gametes have half of them i.e. 23 chromosomes.
• Some cells like RBCs ( red blood cells) lack nuclei.

Key Points

The correct answer is 9/4.

Concept:

Resistance in series:

• When resistors are connected end-to-end in a circuit, they are said to be in "series."
• R = R1 + R2 + R3 +....+ R_n
• Current remains the same across all the resistors connected in series.

Resistance in parallel: 

• When both ends of resistors are connected, they are said to be in parallel.
• 1/R = 1/R1 + 1/R2 = 1/R3 ....
• The potential difference across all the resistors connected in parallel remains the same.

Electric Power:

• The energy consumed per unit of time or the rate of doing work is called power.
• Power = work done / time taken = Vq / t = VI = V2/R = I2R

Explanation:

Resistors A (6 Ω), and B (12 Ω) are connected in parallel.

Equivalent resistance, 1/R1 = 1/6 + 1/12  = 2+1/ 12 = 3/12 = 1/4 Ω 

R1 = 4 Ω 

Now, the resistors R1, C, and D are in series.

R2 = 4 + 6 + 12  = 22 Ω 

Equivalent current, I in the circuit = V/R2 = 11 / 22 =  1/ 2   A.

Potential difference across A = IR1 = 4/2 = 2 V

PC = I2C = 1/2 × 1/2 × 6  = 3/2 Watt       (since current remains same in series combination)

PA = V2/A = 2 × 2  × 1/ 6 = 2/3 Watt  (since voltage remains the same in parallel combination)

The ratio (P_C/P_A) = 3/2 × 2/3 = (3 × 3) / (2 × 2) = 9/4.

The correct answer is 30 cm; 2.0 cm.

Concept:

Mirror formula:\(\frac{1}{f} = \frac{1}{v} + \frac{1}{u}\), where f = focal length, v = image distance and u = object distance.

Magnification: m = \(\frac{Hi}{Ho} = \frac{-v}{u}\) , where Hi = height of image, Ho = height of object, v = image distance, and u = object distance.

Concave mirror: 

• It is part of a whole spherical mirror that reflects light from the hollow part.
• It is a converging mirror because it converges the incident rays at a point.

Explanation:

Given, u = -15 cm        f = -30 cm              h = 1 cm 

\(\frac{1}{f} = \frac{1}{v} + \frac{1}{u}\)

\(\frac{1}{-30} = \frac{1}{v} + \frac{1}{-15}\)

\(\frac{1}{-30} = \frac{1}{v} - \frac{1}{15}\)

\(\frac{1}{v} = \frac{1}{-30} + \frac{1}{15} \)

\(\frac{1}{v} = \frac{-1 + 2}{30} \)

\(\frac{1}{v} = \frac{1}{30} \)

v = 30 cm

m = \(\frac{Hi}{Ho} = \frac{-v}{u}\) = \(\frac{Hi}{2} = \frac{-(30)}{-15}\) 

Hi = 2 cm

Following New Cartesian Sign Convention, the image is formed at v = 30 cm and its height HI = 2 cm.

The correct answer is 'A' and 'B' are metals; and 'A' is more reactive than 'B'.

Concept:

Chemical reaction:

• It is a process in which one or more substances, the reactants, are converted to one or more different substances, the products.
• A chemical reaction rearranges the constituent atoms of the reactants to create different substances as products.

Bases:

• They are bitter in taste and soapy to touch.
• They turn red litmus blue.
• They have a pH value greater than 7.

Explanation:

Solution is basic because it turned red litmus blue.

• When a substance burns in the air, it reacts with oxygen and forms oxides.
• Oxides when mixed with water form hydroxides.
• A substance that turns red litmus blue is basic in nature.
• Metal oxides and hydroxides are basic in nature.
• So, elements A and B must be metals.
• Some reactive metals react with dilute acids like HCl to form salts and hydrogen gas is released in the process.
• Thus, A will be more reactive than B since it reacts with dilute acid while B does not.

The correct answer is viruses.

Concept:

Disease: 

• Disfunctioning of one or more organs or systems of the body, characterized by various signs and symptoms is called a disease.

Types of diseases:

• Infectious diseases: 
• ​These diseases spread from one person to another easily.
• They are caused by germs such as bacteria, viruses, protozoa, etc.
• Example: Common cold, typhoid, cholera, etc.
• Non-infectious diseases: 
• These diseases are caused due to nutritional deficiency, sedentary lifestyle, or genetic defects.
• These diseases do not spread from one person to another.
• Example: Night blindness, colour blindness, etc.

Explanation:

Jaundice is a non-infectious disease wherein the liver is affected resulting in a yellowish discolouration of the skin and the white space of the eye.

Jaundice can be caused by a problem in any of the three phases of bilirubin production.

• Before the production of bilirubin, unconjugated jaundice is caused due to increased levels of bilirubin
• During the production of bilirubin, jaundice can be caused by:
• Viruses, including Hepatitis A, chronic Hepatitis B and C, and Epstein-Barr virus infection (infectious mononucleosis).
• Alcohol.
• Autoimmune disorders.
• Rare genetic metabolic defects.
• Medicines, including acetaminophen toxicity, penicillins, oral contraceptives, etc.
• After bilirubin is produced, jaundice may be caused by obstruction (blockage) of the bile ducts from:
• Gallstones.
• Inflammation (swelling) of the gallbladder.
• Gallbladder cancer.
• Pancreatic tumour.

Symptoms : 

• ​The yellowness of the Skin and Eye Whites
• Dark Tinge in Urine
• Vomiting
• High Fever
• Lack of Hunger
• Abnormal Weight Loss
• Stomach Pain

The correct answer is the availability of food and access to money.

Concept:

Food security: It refers to the availability, accessibility, and affordability of food to all people at all times.

Explanation:

Food security has the following dimensions:

• Availability of Food: It means food production within the country, food imports, and the previous year’s stock stored in government granaries.
• Accessibility of Food: It means food is within reach of every person.
• Affordability of Food: It implies that an individual has enough money to buy sufficient, safe, and nutritious food to meet one’s dietary needs.

Key Points

• India's Efforts for Food Security during the Covid-19 Pandemic:
• High Production of Foodgrains: India registered an all-time high production of foodgrains at 305 million tonnes as well as in exports during 2020-21, contributing to global food security.
• KISAN RAIL: It was introduced to transport the essential commodities including perishable horticulture produce, milk, and dairy products from the production centers to the large urban markets.
• Pradhan Mantri Garib Kalyan Package: Under this scheme, free food grains were provided to 810 million beneficiaries and it has been further extended in which workers will benefit till November 2021.
• PM Kisan Scheme: More than Rs.1,37,000 Crore have been sent to the bank accounts of over 100 million farmers under this to provide income support to the farmers.

Additional Information

• Food and Agriculture Organization:
• FAO is a specialized agency of the United Nations that leads international efforts to defeat hunger.
• World Food Day is celebrated every year on 16^th October to mark the anniversary of the founding of the FAO in 1945.
• It is one of the UN food aid organizations based in Rome (Italy).
• Its sister bodies are the World Food Programme and the International Fund for Agricultural Development (IFAD).
• Initiatives Taken:
• Globally Important Agricultural Heritage Systems (GIAHS). Monitors the Desert Locust situation throughout the world.
• The Codex Alimentarius Commission or CAC is the body responsible for all matters regarding the implementation of the Joint FAO/WHO Food Standards Programme.
• The International Treaty on Plant Genetic Resources for Food and Agriculture.

The correct answer is options (ii) and (iv).

Concept:

Mixtures:

• A mixture is a substance that consists of two or more elements or compounds not chemically combined together.
• For Example - Air is a mixture of gases like oxygen, nitrogen, argon, CO2, and water vapour.

Explanation:

Types of Mixtures:

• Homogeneous Mixture:
• Those mixtures, in which the substances are completely mixed together and are indistinguishable from one another, are called homogeneous mixtures.
• For example Soda water, Soft drinks, lemonade, salt or sugar solution, etc.
• Heterogeneous Mixture:
• Those mixtures in which the substances remain separate and one substance is spread throughout the other substance as small particles, droplets, or bubbles are called heterogeneous mixtures.
• All the suspensions and colloids are heterogeneous mixtures.
• For Example: - Sugar and sand mixture, muddy river water, soap solution.

The correct answer is Sodium hydrogen carbonate

Concept:

Baking soda:

• The chemical name of baking soda is sodium hydrogen carbonate (NaHCO₃).
• It is a mild non-corrosive basic salt.

Explanation:

Preparation of Baking Soda:

• It is produced through the Solvay process by using sodium chloride, sodium chloride, water, and carbon dioxide as raw materials.

NaCl + H₂O + CO₂ + NH₃   → NH₄Cl + NaHCO₃ 

On heating Sodium hydrogen carbonate, the following reaction takes place:

2NaHCO₃ + heat → Na₂ CO₃ + H₂O + CO₂ 

Na2 CO3 is used to make washing soda that is used in the soap industry.                                                     

Uses of baking soda:

Baking soda undergoes a reaction with mild edible acid such as tartaric acid to manufacture baking powder.

• NaHCO3   +  H⁺ → CO₂  + H₂O + Sodium salt of acid

• This carbon dioxide produced makes bread or cake rise thereby making them soft and spongy.

Being alkaline it is also an active ingredient in antacids that acts by neutralizing the excess acid produced in the stomach.

It is also used in soda-acid fire extinguishers.

Thus, compound Z is  Sodium hydrogen carbonate​.

The correct sequence is Sublimation; dissolution in water; filtration; evaporation.

Concept:

• Sublimation: It is a process of conversion of a solid directly into vapour, without going into the liquid phase.
• Filtration: It is the process in which solid particles in a liquid or gaseous fluid are removed by the use of a filter medium that permits the fluid to pass through but retains the solid particles.
• Evaporation: It is the conversion of liquid into vapour below its boiling point.

Explanation:

• Camphor (ammonium chloride) on heating sublime that is directly converted into vapour. In contrast to this, chalk powder and common salt do not sublime.
• So, from a mixture, containing ammonium chloride (or camphor), common salt, and chalk powder, ammonium chloride (or camphor) can be easily separated by the process of sublimation.
• On cooling the vapour of ammonium chloride (or camphor) changes into solid.
• Common salt is soluble in water but sand is not soluble in water.
• When a mixture of common salt and chalk powder in water is passed through filter paper, chalk remains on the filter paper but other components of the mixture i.e., common salt and water pass through filter paper.
• From a solution of common salt in water, common salt can be obtained by evaporation.

So, the correct sequence will be Sublimation; dissolution in water; filtration; evaporation.

The correct answer is 1.12; towards.

Concept:

Refraction of Light: 

• The bending of the ray of light passing from one medium to the other medium is called refraction.

Refractive Index (n):

• n = sin i / sin r, where 'i' =  angle of incidence  and 'r' = angle of refraction.
• Also, n = c/v, where c = speed of light in a vacuum, and v = speed of light in a medium.

Explanation:​

Relative refractive index:

• It refers to the refractive index of one material medium with respect to another one.
• \(n_{21}= \frac{n_{2}}{n_{1}}\)

Refractive index of glass (dense flint) with respect to turpentine oil:

\(n= \frac{Refractive \ index \ of \ glass (dense flint)}{Refractiev \ index\ of\ Turpentine \ oil}= \frac{1.65}{1.47}=1.12\)

• Greater the refractive index, the denser the medium.
• Thus, glass (1.65) is denser than turpentine oil (1.47).
• When the ray of light moves from a denser to a rarer medium, the refracted ray moves away from the normal.
• When the ray of light moves from a rarer to a denser medium, the refracted ray moves towards the normal.
• Thus, when a light ray will move from turpentine oil to glass, it will move towards the normal.

The refractive index of glass (dense flint) with respect to turpentine oil is 1.12 and the light ray bends towards the normal in the glass.

The correct answer is Zero.

Concept:

Lens:

• The transparent curved surface which is used to refract the light and make an image of any object placed in front of it is called a lens.
• A convex lens can converge a beam of parallel rays to a point on the other side of the lens

Power of lens:

• It is equal to the reciprocal of the focal length of the lens/mirror in meters.
• Power (P) = 1/f, where f is the focal length of the lens/mirror.
• P = \(\frac{100}{f}\) where f is the focal length in centimeters.
• The unit of power of a lens is Dioptre when the focal length of the lens is taken in meters (m).
• The focal length of the concave lens is negative and hence its power is also negative.
• The focal length of the convex lens is positive and hence its power is also positive.

Lens formula: \(\frac{1}{f}=\frac{1}{v}-\frac{1}{u}\)  where f is focal length, v is image distance and u is object distance.

Explanation:

For first convex lens, L1

Power = \(\frac{1}{f}\)

f = \(\frac{1}{p}\) = \(\frac{100}{5}\) = 20 cm

Given: Object distance, u = -40 cm, Focal length, f = 20 cm

 v=?

\(\frac{1}{f}=\frac{1}{v}-\frac{1}{u}\)

\(\frac{1}{20} = \frac{1}{v} - \frac{1}{-40}\)

\(\frac{1}{20} = \frac{1}{v} + \frac{1}{40}\)

\(\frac{1}{20} - \frac{1}{40} = \frac{1}{v} \)

\(\frac{1}{v} = \frac{2-1}{40}\)

\(\frac{1}{v} = \frac{1}{40}\)

v = 40 cm

For second convex  lens, L2

f = \(\frac{1}{p}\) = \(\frac{100}{4}\) = 25 cm

Object distance, u = -60-(-40)= -60 + 40 = -20cm, focal length, f = 25cm

v = ?

\(\frac{1}{f}=\frac{1}{v}-\frac{1}{u}\)

\(\frac{1}{25} = \frac{1}{v} - \frac{1}{-20}\)

\(\frac{1}{25} = \frac{1}{v} + \frac{1}{20}\)

\(\frac{1}{25} - \frac{1}{20} = \frac{1}{v} \)

\(\frac{1}{v} = \frac{4-5}{100} = \frac{-1}{100} \)

v= -100 cm 

Distance between original object and final image = 40 + 60 - 100 = Zero.

The correct answer is beyond C.

Concept:

Spherical mirror:

• A spherical mirror is a mirror that has the shape of a piece cut out of a spherical surface.
• There are two types of spherical mirrors: concave, and convex.

Explanation:

Concave mirror:

• It is part of a whole spherical mirror that reflects light from the hollow part.
• It is a converging mirror because it converges the incident rays at a point.

When the object is between C (Centre of curvature) and F (focus):

Characteristics of Image Formed:

Position: Beyond C

Size: Magnified/ enlarged 

Nature: Real and inverted

Additional Information All six cases of image formation by the concave mirror:

Convex mirror: 

• It is part of a whole spherical mirror that reflects light from the bulged-out part.
• It diverges the incident rays.

The correct answer is Figure B.

Concept:

Hypermetropia:

• It is also known as far-sightedness.
• A person with hypermetropia can see faraway objects clearly but cannot see nearby objects clearly.

Explanation:

Cause of hypermetropia:

• Due to weakening of the ciliary muscles and/or the decreased flexibility of the lens.
• The focal length of the eye lens is too long.
• The eyeball has become too small. ​

(a) Near point of a Hypermetropic eye

(b)  Hypermetropic eye

(c) Correction for Hypermetropic eye

 

• The image of a distant object is formed behind the retina and not on the retina.
• The defect is corrected by using convex lenses.

Additional InformationMyopia (Near-sightedness):

• A person with Myopia can see nearby objects clearly.
• A person with myopia cannot see faraway objects clearly.
• The far point for the myopic eye is nearer than infinity.
• Cause:
• Excessive curvature of the eye lens
• Elongation of eyeball
• The image of a distant object is formed in front of the retina and not on the retina.
• Defected is corrected by using Concave lenses such that the lens will bring the image back onto the retina.

Presbyopia:

• The power of accommodation of the eye usually decreases with ageing.
• The ciliary muscles weaken and thereby the flexibility of the eye lens reduces.
• The near point moves away.
• Spectacles with convex lenses are recommended.

The correct answer is Soft tissues of plants and animals are not seen in fossils.

Concept:

Fossil:

• A fossil is the mineralized partial or complete form of an organism that has been preserved as a cast, impression, or mold.
• It gives physical evidence of ancient life and has provided the basis for the theory of evolution.

Explanation:

The correct answer is ultraviolet radiation.

Concept:

Ozone layer:

• It is a layer of ozone molecules found in the stratosphere (the second layer of the atmosphere).
• Ozone is a molecule made up of three oxygen atoms, often referenced as O3.

Explanation:

Importance of Ozone layer:

• The presence of ozone in the stratosphere prevents about 99.5 percent of the sun’s harmful ultraviolet (UV) radiations from reaching the earth’s surface.
• Thus, protecting humans and other animals from its effect.

Harmful effects of UV Radiations:

• They lead to aging of the skin, cataracts, sunburn, skin cancer, the killing of many phytoplanktons, damage to fish productivity, etc.
• It has also been reported that plant proteins get easily affected by UV radiation which leads to the harmful mutation of cells.
• It also increases the evaporation of surface water through the stomata of the leaves and decreases the moisture content of the soil.

Additional InformationElectromagnetic spectrum:

​

X-rays:

• X-rays have higher energy and can pass through most objects, including the body.
• Medical x-rays are used to generate images of tissues and structures inside the body.

Gamma rays:

• Gamma rays have the smallest wavelengths and the most energy of any wave in the electromagnetic spectrum.
• They are produced by the hottest and most energetic objects in the universe, such as neutron stars and pulsars, supernova explosions, and regions around black holes.

Infrared rays:

• Infrared radiation is a type of radiant energy that's invisible to human eyes but that we can feel as heat.
• IR is a type of electromagnetic radiation produced when atoms absorb and then release energy.

The correct answer is 44%.

Concept:

Resistance (R): 

• Resistance is a measure of the opposition to current flow in an electrical circuit.
• It is measured in ohms, symbolized by the Greek letter omega (Ω).
• \(R = \frac{ρ l}{A}\) where ρ is resistivity, l is length, and A is the area of the cross-section of the wire.

Explanation: 

Suppose original length = L, Area of cross-section = A, radius of wire = r and Resistance = R

Since the wire is the same, the resistivity ρ will be the same.

After the increase in length, let us take new length = L', radius of wire = r' and new resistance = R'

L' = 120 L/ 100 = 6L/5 ( since there is 20% increase)

The volume of the wire remains the same ( given).

So, Volume of Original Wire = volume of new wire 

π r² L = π r ' ² L'

r²L = r' ² L'

r' ² = r² L /  L'

r' ² = 5 r²  / 6

Increase in resistance = \(\frac{\frac{\rho L'}{A'}-\frac{\rho L}{A}}{\frac{\rho L}{A}}\) × 100

\(=\frac{\frac{\rho L'}{\pi r^2} - \frac{\rho L}{\pi r^2}}{ \frac{\rho L}{\pi r^2} } \times 100\)

\(=\frac{\frac{6 \times 6 \rho L}{5 \times 5 \pi r^2} - \frac{\rho L}{\pi r^2}}{ \frac{\rho L}{\pi r^2} } \times 100\)

\(=\frac{\frac{36}{25}\frac{\rho L}{\pi r^2} - \frac{\rho L}{\pi r^2}}{ \frac{\rho L}{\pi r^2} } \times 100\)

\(=\frac{\frac{36 \rho L - 25 \rho L}{25\pi r^2}}{ \frac{\rho L}{\pi r^2} } \times 100\)

\(= \frac{11 \rho L}{25 \pi r^2} \times \frac{\pi r^2}{ \rho L} \times 100 = 44%\)

Thus, the increase in length is 44%.

The correct answer is statements (i), (ii), and (iii).

Concept:

• Magnetic field: The region around a magnet in which its force can be experienced.
• Solenoid:  A long, thin loop of wire, often wrapped around a metallic core, which produces a magnetic field when an electric current is passed through it.

Explanation:

Magnetic field due to a long solenoid carrying current I:

• The magnetic field around a solenoid is similar to that around a bar magnet.
• The magnetic field lines move from south to north inside the solenoid and north to south outside the solenoid. Thus, forming closed loops.
• Inside the solenoid, the magnetic field lines move parallel to each other. Thus, it is uniform inside the solenoid.
• The magnetic field is strongest inside the solenoid.

​

The correct answer is  'X' is a base and is an alkali.

Concept:

pH scale:

• A scale developed to measure the concentration of hydrogen ions in a solution is known as the pH scale where p in pH implies ‘potenz’ in German.
• pH scale ranges from 0 (highly acidic) to 14 (highly alkaline).
• The pH of a neutral solution is 7.
• An acidic solution is represented by a value less than 7 on the pH scale whereas the basic solution is represented by a value greater than 7 on the pH scale.

Explanation:

Bases:

• They are bitter in taste.
• They are slippery and soapy to touch.
• They turn red litmus blue.
• Bases soluble in water are called Alkalis.
• They have a pH greater than 7.

Thus, X is a base as well as an alkali.

Additional Information

Acids:

• They are sour in taste.
• They turn blue litmus red.
• They have a pH lower than 7.

The correct answer is options (i) and (iv).

Concept:

• Corrosion: The eating away of the surface of metals due to the action of air, moisture, or chemicals is called corrosion.

Explanation:

The rusting of iron can be prevented by painting, oiling, greasing, galvanizing, chrome plating, anodizing, or making alloys.

• Galvanization: It is a method of protecting steel and iron from rusting by coating them with a thin layer of zinc.
• Alloying: An alloy is a homogeneous mixture of two or more metals, or a metal and a non-metal. When iron is mixed with nickel and chromium, we get

stainless steel, which is hard and does not rust.

• Anodising: It is an electrolytic process in which a layer of oxide is formed on the metal surface making it resistant to further corrosion.
• Painting, oiling, and greasing form a protective film between air and the surface of iron protecting it from corrosion.

Additional InformationSoldering:

• It is a joining process used to join different types of metals together by melting solder.
• Solder is an alloy usually made of tin and lead.

Thermit reaction: 

• The displacement reaction between iron oxide and aluminum releasing a lot of heat is called the thermit reaction.
• It is used to join railway tracks or cracked machine parts.

Carbonization: 

• The dead plants and vegetation due to temperature and high pressure over hundreds of years slowly turned into coal.
• This slow conversion of dead plants and forests into coal is called the process of carbonization.

The correct answer is aerial, aquatic, and terrestrial.

Concept:

Biological classification:

• The classification is the process of arranging organisms, both living and extinct, into groups based on similar characteristics.
• The science of naming and classifying organisms is called taxonomy.
• The term is derived from the Greek word: taxis (“arrangement”) and nomos (“law”).

Explanation:

Aristotle's Classification System:

• Aristotle was the earliest to attempt a more scientific basis for classification.
• He used simple morphological characters to classify plants into trees, shrubs, and herbs.
• He also divided animals into two groups, those which had red blood and those that did not.
• He further subdivided animals into three categories aerial, aquatic, and terrestrial.​

Additional Information

Two Kingdom Classification System:

• It was given by Carolus Linnaeus.
• He divided organisms into two groups autotrophic plants and heterotrophic animals.

Three Kingdom Classification System:

• In the year 1866, Ernst Haeckel, classified living organisms into three kingdoms i.e. Plantae, Protista, and Animalia.
• The new kingdom Protista included all those organisms, which lack the capability of tissue differentiation. This group included algae, fungi, and Protozoa.

Four Kingdom Classification System:

• Copeland in the year 1956, introduced the kingdom-Monera.
• He categorized organisms into prokaryotes and eukaryotes.
• Bacteria are the sole members of Kingdom monera. They are prokaryotes.

Five Kingdom Classification System:

• In the year 1969, RH Whittaker proposed this system.
• He created a separate group for fungi.
• The primary criterion for classification here were:
• Cell structure
• Modes of nutrition
• Reproduction
• Thallus organization
• Phylogenetic relationship

The correct answer is V1 ∶ V2 ∶ V3 = 36 ∶ 16 ∶ 9.

Concept:

Ohm's Law:

• It states that the potential difference between two points is directly proportional to the current passing through the conductor, provided the temperature remains the same.
• V = IR, where V = potential difference between two points, I = current, R = Resistance.

Resistance:

• The property of a substance to resist the flow of current through it.
• R = ρ L / A, where ρ = resistivity, L = length of conductor, and A = area of cross section.

Explanation:

The current remains the same in series combination. 

Let the current flowing through the wire be I ampere. 

For Wire AC

Length = L, Radius = 2r, Area of cross- section = 4π r2

Resistance, R1 =  \(\frac{\rho L}{4\pi r^2}\)

Voltage, V1 = I R1 = \(\frac{I\rho L}{4\pi r^2}\)

For Wire CD

Length = L, Radius = 3r, Area of cross- section = 9π r2

Resistance, R1 =  \(\frac{\rho L}{9\pi r^2}\)

Voltage, V1 = I R1 = \(\frac{I\rho L}{9\pi r^2} \)

For Wire BD

Length = L, Radius = 4r, Area of cross- section = 16π r2

Resistance, R1 =  \(\frac{\rho L}{16\pi r^2}\)

Voltage, V1 = I R1 = \(\frac{I\rho L}{16\pi r^2} \)

The ratio V1: V2 : V3 = \(\frac{I\rho L}{4\pi r^2}\) :  \(\frac{I\rho L}{9\pi r^2} \)  :  \(\frac{I\rho L}{16\pi r^2} \) = 1/4 : 1/9 : 1/16 = 36 : 16 : 9. 

V1 ∶ V2 ∶ V3 = 36 ∶ 16 ∶ 9.

The correct answer is suberin.

Concept:

Cork: 

• As the plants grow older, the outer protective tissue (i.e., epidermis) undergoes certain changes.
• A strip of the secondary meristem, called cork cambium replaces the epidermis of the stem.

Explanation:

Characteristics of cork cells:

• The cells are rectangular and their protoplast is vacuolated and contains tannins and chloroplasts.
• Cells are dead and compactly arranged without intercellular spaces.
• The walls of cork cells are heavily thickened with an organic substance (a fatty substance), called suberin, which makes these cells impermeable to water and gases.

The function of cork cells:

• Cork is protective in function.
• Its cells prevent desiccation (loss of water from the plant body), infection, and mechanical injury.
• The cork cambium gives off new cells on both sides, thus, forming cork (phellem) on the outer side and the secondary cortex or phelloderm on the inner side.
• The layer of cells which is cut by cork cambium on the outer side ultimately becomes several layered thick cork (bark) of trees.

Additional Information Lignin: 

• Lignin is an organic polymer that is abundant in the cell walls of some specific cells like sclerenchyma.
• It has many biological functions such as water transport, mechanical support, and resistance to various stresses.

Cutin: 

• It is a waxy water-repellent substance found in the cuticle of plants and is made up of polymerized esters of fatty acids.

Cellulose:

• It is a polysaccharide made up of repeating units of glucose and is found in the cell wall of plants.

The correct answer is Both A and R are true and R is not the correct explanation of A.

Key Points

• In 1834, a Customs Union, known as Zollverein was formed at the initiative of Prussia which was the predecessor of Germany.
• By 1866 most of the German states became members of this union.
• It led to the establishment of free trade throughout the country.
• It helped in awakening the national interest among the people of Germany by fusing individual and provincial interests among them.
• The main aim was to manage tariffs and policies of the economy within their territories.