ICAR Technician 28 Feb 2022 Shift 1 (Previous Year Paper)

The correct answer is J.C. Bose.

Key Points

• Jagadish Chandra Bose (J.C. Bose) discovered the ultra-short radio waves.
• Shortwave radio is radio transmission using shortwave (SW) radio frequencies.
• Radio waves have the lowest energies, longest wavelengths, and lowest frequencies of any type of electromagnetic (EM) radiation.

 Important Points

• Jagadish Chandra Bose
• He invented a Crescograph that could detect very small motions within plant tissues or measure the growth in plants.
• He proved plants have life.
• He is known for his work on the wireless transmission of information using microwaves.
• Bose is known as the Father of wireless telecommunication.

Additional Information

• J.J. Thomson (1856-1940)
• He was awarded the Nobel prize in Physics in 1906 for his work on the discovery of electrons.
• ​J.J. Thomson was the first one to propose a model for the structure of an atom.
• Isaac Newton
• Sir Isaac Newton was a British mathematician, and one of the greatest mathematicians.
• His discoveries in optics (construction of white light) and mathematics (calculus),
• He is known for formulating three laws of motion which are Fundamentals of modern physics.
• He discovered the gravitational forces.
• Albert Einstein 
• Albert Einstein was a German-born theoretical physicist.
• Albert Einstein was awarded to Nobel Prize in Physics in 1921 "for his discovery of the law of the photoelectric effect.
• He also gives the theory of relativity.

The correct answer is 1539.

Key Points

• Farid Khan was honoured with the title of Sher Shah in 1539.
• In 1539, he defeated Humayun in the battle of Chausa(26 June 1539) and after that, he assumed the title Sher Shah as emperor.
• 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.

Important Points

• Sher Shah
• He was the founder of the Suri dynasty, which had its capital at Sasaram.
• He issued the coin Rupia as a currency and fixed standard weights and measures all over the empire.
• He also introduced Kabuliyat and Patta as the agricultural equipment of Settlement.
• He built the Grand Trunk Road that runs from Calcutta to Peshawar.
• He built Purana Quila in Delhi.
• He was buried in Sasaram.
• Administrative Unit and its head in Suri Dynasty.

The correct answer is an Indo-Aryan language.

Key Points

• The biggest language family of India is the Indo-Aryan language.
• Indo-Aryan languages, also called Indic languages.
• It is a subgroup of the Indo-Iranian branch of the Indo-European language family.
• Around 74% of Indians speak a language related to the Indo-Aryan language family.

Important Points

• Indo-Aryan languages are generally divided into three major divisions: 
• Old Indo-Aryan group- Vedic Sanskrit was born around this period.
• Sanskrit is one of the official languages of India and is known as a classical language of the country and is also considered to be the Mother of all Languages
• Middle Indo-Aryan:- Prakit language was developed during this period.
• Some languages have originated from Prakrit like Pali, Ardha Magadhi, and Apabharnsha.
• New (Modern) Indo-Aryan:- This group of languages is mainly spoken in northern and western India.
• The Indo-Aryan language family includes mostly all languages of northern and western India such as Hindi, Bengali, Marathi, Gujarati, Punjabi, Sindhi, Rajasthani, Assamese, Oriya, Pahari, Bihari, Kashmiri, Urdu, and Sanskrit.

Additional Information

• Mon-Khmer languages:
• Mon-Khmer languages have three types Khmer (Cambodian), Mon (a minority language in Myanmar and Thailand), and Vietnamese.
• North Dravidian language
• ​It is part of the Dravidian language group.
• In this group languages Brahui(Spoken in Baluchistan), Malto(spoken in tribal areas of Bengal and Odisha), and Kurukh(spoken in Bengal, Odhisha, Bihar, and Madhya Pradesh).
• Telugu is a Dravidian group language, numerically the largest of all Dravidian languages.
• Malayalam is the smallest and youngest of the Dravidian group.

The correct answer is State with the lowest literacy - Uttar Pradesh.

Key Points

• According to the census 2011 state with the lowest literacy rate- is Bihar, not Uttar Pradesh.
• ​According to the census 2011, Uttar Pradesh is the most populous state.

Additional Information

• Census 2011 was the 15th Census of India.
• The slogan of Census 2011 was 'Our Census our Future'.
• Total population of census 2011 - 1,210,193,422.
• The largest census in human history ever conducted by any Government till now.
• The Census of  2011 covered.
• ​640 Districts.
• 5924 Sub-districts.
• 5767 Tehsils.
• 7742 Towns.
• 6,49,481 Villages.
• ​According to the Census 2011
• The UT (Union Territory)  with the smallest area was Lakshadweep (30 km²).
• The state with the largest area was Rajasthan (3,42,239  km²).
• The state with the highest literacy rate was Kerala (94%).
• The first census of Independent India - 1951.

The correct answer is Nitrogen.

Key Points

• Baking powder is sodium bicarbonate, which is NaHCO₃.
• Nitrogen is not present in baking powder.
• Elements involved in baking powder are - Hydrogen, Carbon, Sodium, and Oxygen.
• Baking soda is used as Antacids, making bread soft, etc.

Additional Information

• Plaster of Paris is known as Gypsum which is prepared from the chemical compound, Calcium sulphate dihydrate (CaSO₄.2H₂O)
• Acetic acid which is known as ethanoic acid is an acidic, colourless liquid and organic compound with the chemical formula (CH₃COOH)

The correct answer is pensioners.

Key Points

• Union Minister launched Jitendra Singh Unique face recognition for pensioners.
• This face recognition technique of giving life certificates is a historic and far-reaching reform since it will touch the lives of not only 68 lakh central government pensioners but also those under EPFO and state government.
• Face recognition technology will benefit pensioners across the country in providing Life Certificate by simply using the mobile app.

Additional Information

• In December 2021, Union Minister Nitin Gadkari inaugurated and laid the foundation stone of the National Highway projects of 240 Km worth 9,119 crore rupees in Meerut and Muzaffarnagar, Uttar Pradesh.
• In October 2021, Union Minister of State (Independent Charge ) Earth Sciences Dr. Jitendra Singh officially launched India's First Manned Ocean Mission called the "Samudrayan project" in Chennai.

The correct answer is Circular - ayastra.

• Circular - ayastra: This option is an incorrect match.
• The term "ayastra" does not correspond to the shape of Dravida temples.
• While circular-shaped temples do exist, they are not typically associated with Dravida temple architecture.

Key Points

• Temples architecture of India is divided in **three broad categories.**​
• Nagara - Northern Style
• Dravida - South Style
• Vesara - mixed style
• ​Dravidian Style
• ​In the Southern Dravidian style, the tower or Gopuram consists of progressive smaller stories of pavilions.
• These temples were square in plan and pyramidal in shape.
• Included porches (Mandapams) and pillared halls and contained tanks or wells for water to be used for sacred purposes.

Additional Information

• Square - kuta:
• ​This refers to the square-shaped Dravida temple, which is characterized by having a square base or plan.
• The sanctum sanctorum or the inner shrine of the temple is usually located at the center.
• Rectangular - shala:
• This refers to the rectangular-shaped Dravida temple.
• Similar to the square shape, the rectangular shape of the temple signifies a structure with a rectangular base or plan.
• The inner shrine is typically placed at the center of the temple.
• Elliptical - gaja-prishta:
• ​This refers to the elliptical-shaped Dravida temple.
• The term "gaja-prishta" translates to "back of an elephant," and it describes the shape resembling the curve of an elephant's back.
• The temple has an elliptical or curved outline, with the inner sanctum located at the center.

The correct answer is Rs. 5,250 crores.

Key Points

• Madhya Pradesh CM launched three solar power projects.
• CM said power generation will start in March 2023.
• 3 solar plants of 1500 MW capacity in Shajapur, Agar, and Neemuch.
• 2 units of 550 MW (350+200) in Agar.
• 3 units of 450 MW (105+220+125) in Shajapur.
• 3 units of 500 MW (170+160+170) in Neemuch.
• 5300 MW is MP's daily solar power generation.

Additional Information

• Madhya Pradesh
• Number of Districts - 52
• Rajya Sabha seats - 11
• Lok Sabha seats - 29
• Chief Minister - Shivraj Singh Chauhan

The correct answer is Option 4.

Key Points

• Federal Bank in December 2021 launched an exclusive feature-rich scheme named ‘Mahila Mitra Plus’ for women.
• It is designed to make financial planning and investments easy for women.
• It includes: 
• Exclusive preferential interest rates on housing loans.
• Processing fee waivers for home loans.
• Complimentary and customised insurance cover.
• Women are also encouraged to open savings accounts in the names of their minor children through the provision of Two free FedFirst SB accounts in the name of minor children, with no minimum balance stipulation.

Additional Information

• Federal Bank is an Indian private sector bank.
• Headquarters: Kochi.
• Established: April 23, 1931.
• Tagline: Your Perfect Banking Partner.
• Federal Bank has been awarded the ‘Most Effective Bank-Fintech Partnership: Agile and Adaptable’ at the IBSi-Global Fintech Innovation Awards 2021.
• Federal Bank has launched “FEDDY”, an Artificial Intelligence-powered virtual assistant to help consumers with banking-related queries anytime.

The correct answer is Ajaz Patel.

Key Points

• On December 4, 2021, Left-arm spinner Ajaz Patel created history by taking all 10 wickets in an innings in the 2^nd test match against India.
• He plays for New Zealand
• He finished with figures of 10/119.
• He became the third bowler after Jim Laker and Anil Kumble to achieve the feat.

Additional Information

• Anil Kumble’s record had come against Pakistan in a Test match at Delhi’s Feroz Shah Kotla in 1999.
• Ravichandran Ashwin on 6 March 2022 surpassed the legendary Kapil Dev's haul of 434 wickets to become India's second-highest wicket-taker in Test cricket.
• Anil Kumble tops the charts with 619 wickets.

The correct answer is Argentina.

Key Points

• Argentina beat Germany in the final and won Men’s Junior Hockey World Cup 2021
• It was held at the Kalinga Stadium in Bhubaneswar, Orissa (India) from 24 November to 5 December 2021.
• The hosts India were the defending champions but lost to Germany in the semifinals.
• France and the India finished third and fourth, respectively.

The correct answer is Option 4.

Key Points

• 1kWh= 3.6 × 10⁶ J

<sup>

Additional Information</sup>

• Energy
• It is a scalar quantity.
• It is defined as the capacity of doing work.
• Its SI unit is Joule.

The correct answer is Sanjiv Mehta.

Key Points

• Sanjiv Mehta, previously Chairman and Managing Director of Hindustan Unilever Limited (HUL) was appointed as Federation of Indian Chambers of Commerce and Industry (FICCI) President.
• He is also a Director on the Board of Indian School of Business.

Additional Information

• Federation of Indian Chamber of Commerce and Industry (FICCI):
• FICCI is a non-governmental trade association.
• It was established in the year 1927.
• It was founded by Ghanshyam Das Birla and Purshottamdas Thakurdas.
• Its headquarters is in New Delhi.

The correct answer is Atal Bihari Vajpayee.

Key Points

• The 10th five-year plan (2002-07) was headed by former Prime Minister Late Atal Behari Vajpayee.
• It aimed to achieve an 8 per cent annual GDP growth.
• It also aimed to reduce the poverty rate by 5% by 2007.
• It also included reducing gender gaps in literacy and wage rates by at least 50% by 2007.
• The 11th five-year Plan (2007–2012)  was headed by former Prime Minister Manmohan Singh.
• It focused on rapid and inclusive growth.
• It aimed at increasing the growth rate in agriculture, industry, and services to 4%, 10%, and 9% respectively.
• Note: The Planning Commission has been dissolved and there is no 13th five-year plan.

Additional Information

• National Development Council (NDC) is the apex body that gives approval to five-year plans in India.
• It was established on 6th August 1952.
• The Prime minister is serving as the chairman of the National Development Council.
• The headquarters of the National Development Council (NDC) is in New Delhi.
• The first five-year plan is the only five-year plan adopted without the consent of the National Development Council.
• The first meeting of the NDC was chaired by Jawaharlal Nehru in November 1952.

The correct answer is the Civil Disobedience Movement.

Key Points 

• Civil Disobedience Movement:
• The movement started with the breaking of the Salt law at Dandi in March 1930.
• The Dandi March started from Gandhiji's ashram in Sabarmati to the Gujarati coastal town of Dandi.
• The movement was called off in the year 1931, and Gandhiji decided to enter into the Gandhi-Irwin Pact.
• The terms of the Gandhi-Irwin Pact included calling off of the movement, releasing of all prisoners, and manufacturing salt along the coast.
• The failure of the Second Round Table and the new repression cycle adopted by the British led to the relaunching of the Civil Disobedience Movement in 1932.
• When Gandhiji returned to India, Nehru, and Ghaffar Khan both were in jail, Congress was declared illegal.
• Thus, Gandhiji decided to relaunch the movement in 1932. But, it lost its momentum in 1934.

Important Points 

• The Salt March was notable for at least three reasons.
• It was this event that first brought Mahatma Gandhi to world attention.
• The march was widely covered by the European and American press.
• It was the first nationalist activity in which women participated in large numbers.
• Kamaladevi was herself one of the numerous women who courted arrest by breaking the salt or liquor laws.
• The socialist activist Kamaladevi Chattopadhyay had persuaded Gandhiji not to restrict the protests to men alone
• And perhaps most significant, it was the Salt March which forced upon the British the realization that their Raj would not last forever, and that they would have to devolve some power to the Indians.

The correct answer is 80.

Key Points

• On March 12, 1930, Mahatma Gandhi was informed about the plan of action.
• According to his plan, Gandhi, along with a band of 78 members (approximately 80) of Sabarmati Asharam, was to march from his headquarters in Ahmedabad through the villages of Gujrat for 240 miles.
• On reaching the coast of Dandi on 6th April 1930, the salt law was to be violated by collecting salt from the beach.
• The historic march, marking the launch of the Civil Disobedience Movement, began on March 12, and Gandhiji Broke the salt law at Dandi on April 6.
• Dandi March was organised in the year 1930 from Sabarmati Ashram in Ahmedabad to Dandi, under the leadership of Gandhi.
• It is also known as the Salt Satyagraha or Civil Disobedience Movement.
• Lord Irwin was viceroy during the launch of the Civil Disobedience Movement.
• Sarojini Naidu was among the leaders who accompanied Mahatma Gandhi during the Dandi March.

The correct answer is 66.6 %.

Key Points

• ZyCoV-D vaccine for 2022 is India's first indigenously developed plasma DNA vaccine against COVID-19.
• ZyCoV-D is also the world's first Plasmid DNA vaccine authorized for COVID-19.
• The vaccine is developed by Zydus Cadila using a non-replicating and non-integrating plasmid carrying the novel coronavirus.
• This vaccine is fully authorized by the DCGI (Drug Controller General of India).
• It will be started in 7 Indian states.
• ​Bihar
• Jharkhand
• Maharashtra
• Punjab
• Tamil Nadu
• Uttar Pradesh
• West Bengal
• It will be applied to the 12-18 years age group.
• It is 66.6% effective.
• It is a DNA plasmid-based Vaccine.

The correct answer is Eastern Hills.

Key Points

• The Purvanchal Range, or Eastern Mountains, is a sub-mountain range of the Himalayas in North-East India.
• The Purvanchal comprises :
• The Patkai Hills
• The Naga Hills
• Manipur Hills
• The Mizo Hills
• The Patkai Hills:
• The Patkai Hills are the hills on India's north-eastern border with Burma or Myanmar.
• The Naga Hills:
• The Naga Hills lie on the border of India and Burma.
• The Mizo Hills:
• Mizo Hills is a mountain range in southeastern Mizoram state.
• Mizo Hills form part of the north Arakan Yoma system.
• The range is an eastern extension of the Himalayan Range System, in northeastern India.
• It bends sharply to the south beyond the Dihang River gorge and spreads along the eastern boundary of India with Myanmar.
• The Purvanchal includes the hill of the Patkai hills, Naga Hills, Mizo Hills, and Manipur hills.

The correct answer is Kathak.

Key Points

• Kathak
• This word is derived from Katha (Sanskrit Word).
• They were originally a cast of storytellers in north India (Uttar Pradesh) who used to sing gestures and songs in temples.
• This dance was spread in India during the Bhakti Movement in the 16th-17th centuries.
• Famous Kathak dancers – Birju Maharaj, Lachu Maharaj, Sambu Maharaj, Shovna Narayan, Sitara Devi.

Additional Information

• Classical dance in India​

The correct answer is 2024.

Key Points

• Minister of State for Personnel, Public Grievances, and Pensions Jitendra Singh said, "India will have nine nuclear reactors By 2024" in Rajaya Sabha during the question hour.

Additional Information 

• The nuclear energy program was launched in India under the leadership of Dr.  Homi Jehangir Bhabha (Father of Atomic Energy Research).
• India’s first nuclear power reactor was named Apsara and is located in **Tarapur, Maharashtra.**​

The correct answer is Shivalik Small Finance Bank.

Key Points

• Shivalik Small Finance Bank (SSFB) signed a partnership agreement with fintech firm Indiagold to launch a loan against digital gold.
• This collaboration will provide instant loans to customers up to Rs 60000 against digital gold.​

Additional Information 

• Shivalik Small Finance Bank
• Headquarters – Noida, UP
• MD & CEO – Anshul Swami
• Capital Small Finance Bank
• Headquarters – Jalandhar, Punjab
• MD & CEO – Sarvjit Singh Samra
• AU Small Finance Bank
• Headquarters – Jaipur, Rajasthan
• MD & CEO – Sanjay Agarwal
• Ujjivan Small Finance Bank
• Headquarters – Bengaluru, Karnataka
• MD & CEO – Ittira Davis

The correct answer is Thanjavur.

Key Points

• India’s first digital food museum
• It has been established by FCI (Food Corporation of India) and VITM (Visvesvaraya Industrial and Technological Museum).
• Thanjavur is also known as the cultural capital of Tamil Nadu.
• It is inaugurated through a video conference by the Union Minister of Consumer Affairs Mr. Piyush Goyal.
• FCI was established on 14th January 1965
• Chairman and MD is Atish Chandra (March 2022).

The correct answer is Russia.

Key Points 

• Russian Tennis Federation (Daniil Medvedev) won the Davis Cup 2021 finals title defeating Croatia.
• Devis cup 2021 was held in Madrid (Spain), Innsbruck (Austria), and Turin (Italy).

Additional Information

• Davis Cup
• This is the international tennis men’s competition.
• It is managed and run by International Tennis Federation (ITF).
• It was founded by Dwight Davis in 1900 and began as a challenge match between the USA and the British Isles.
• USA has won the Davis Cup 32 times and Australia has won 28 times.

The correct answer is Vijaywada.

Key Points

• The Chinese scholar Xuan Zhang studied Buddhist scriptures in Vijaywada.
• Xuan Zang stayed in Bezawada (the old name of Vijayawada) for a couple of years.
• He stayed here to copy and study the Abhidhammapitakam, the last of the three pitakas constituting the Pali canon, the scriptures of Theravada Buddhism.

Important Points

• Xuan Zhang 
• Xuan Zhang(also spelled Hsuan Tsang or Hiuen Tsang) is also known as The prince of the travelers.
• He traveled to India in the year 629–645 CE.
• He came to India in 629 AD during the reign of Harshavardhana (Vardhana dynasty).
• He came to India to study the fountainhead of Buddhism.
• He described the relationship between Indian Buddhism and Chinese Buddhism.
• He came to India during the early Tang Dynasty (China).​

Additional Information

• Around 640 C.E, Xuan Zang travelled to Andhradesam to visit the Viharas at Amaravathi and Nagarjunakonda.
• He also stayed at Amaravathi for a couple of years to study the Abhidhammapitak.
• He wrote a book Si-yu-ki: Buddhist Records of the Western World in which he described Buddhism and the journey of India and its ruler.

The correct answer is Three.

Key Points

• Lttira Davis was appointed as new MD and CEO of Ujjivan Small Finance Bank for three years.
• 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.
• 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.

Additional Information

• Ujjivan Small Finance Bank 
• Founded:- 1 February 2017
• ​Headquarters:- Bengaluru
• Founder:- Samit Ghosh
• MD and CEO:- Lttira Davis

Shortcut Trick

Find the slant height (L) of the cone: L = √(24² + 7²) = 25 cm.

The radius (r) of the sphere tangent to the slant sides and the base is given by r = (R × H) / (L + R).

Calculation: r = (7 × 24) / (25 + 7) = 168 / 32 = 5.25 cm = 21/4 cm.

Ratio of volumes = [(4/3)πr³] / [(1/3)πR²H] = 4r³ / (R²H) = [4 × (21/4)³] / (7² × 24) = 63/128.

∴ The correct answer is 63 ∶ 128.

Alternate Method

Given:

Base radius of conical vessel (R) = 7 cm

Vertical height of conical vessel (H) = 24 cm

The sphere is placed such that its highest point is at the level of the cone's base.

Formula Used:

Volume of Sphere (V_s) = (4/3) × π × r³

Volume of Cone (V_c) = (1/3) × π × R² × H

Slant height (L) = √(R² + H²)

Calculations:

⇒ Slant height L = √(7² + 24²) = √(49 + 576) = 25 cm

⇒ Let the radius of the sphere be r. The center of the sphere is at a distance r from the base and the slant sides.

⇒ From similar triangles or geometry of incenter: r / (H − r) = R / L

⇒ r / (24 − r) = 7 / 25

⇒ 25r = 168 − 7r

⇒ 32r = 168 ⇒ r = 168/32 = 5.25 = 21/4 cm

⇒ V_s = (4/3) × π × (21/4)³ = (4/3) × π × (9261/64) = 3087π / 16 cm³

⇒ V_c = (1/3) × π × 7² × 24 = 8 × 49 × π = 392π cm³

⇒ Ratio V_s : V_c = (3087π / 16) / 392π

⇒ Ratio = 3087 / (16 × 392) = 441 / (16 × 56) = 63 / (16 × 8) = 63 / 128

∴ The correct answer is 63 ∶ 128.

Additional Information

Inradius of Cone Cross-section

The radius of the largest sphere that can be fit inside a cone is r = (R × H) / (L + R), which corresponds to the inradius of the triangular cross-section.

Volume Ratios

When a sphere is inscribed in a cylinder of the same radius and height (h = 2r), the volume of the sphere is 2/3 the volume of the cylinder.

Slant Height Property

In any right circular cone, the relationship between slant height L, vertical height H, and base radius R is always L² = H² + R².

Given:

cos2 θ + 3 = 3 (cot2 θ + sin2 θ), 0° < θ < 90°

Formula used:

sin²(θ) + cos2(θ) = 1

Calculation:

⇒ cos2θ + 3 = 3 (cot2θ + sin2θ)

⇒ cos2θ + 3 = 3 (cot2θ + (1 - cos2θ))

⇒ cos2θ + 3 - 3 + 3 × cos2θ = 3 × cot2θ  

⇒ 4 × cos2θ = 3 × cot2θ 

⇒ 4 × cos2θ = 3 ×$\frac{\text {cos}^2 (θ)}{\text {sin}^2 (θ)}$

⇒ 4 × sin2θ = 3

⇒ sinθ = $\frac{√3}{{2}}$

Since $\text {sin} (60^{∘}) = \frac{√{3}}{2}$

∴ $θ = 60^{∘}$

Now, on substituting the value the respective variable in the required expression, we get,

⇒ ? = (2 cos (60^∘) + 3 sin (60∘))

⇒ ? = (2 × $\frac{1}{2}$ + 3 × $\frac{√{3}}{2}$)

⇒ $?= \frac{2+3√{3}}{2}$

∴ The required value is (2 cos (θ) + 3 sin (θ)) is $\frac{2+3√{3}}{2}$.

 Alternate Method

 Calculation:

⇒ cos2θ + 3 = 3 (cot2θ + sin2θ)

⇒ cos2θ + 3 = 3cot2θ + 3sin2θ

⇒ cos2θ + 3 - 3sin2θ = 3cot²θ 

⇒ cos²θ + 3(1 - sin²θ) = 3cot²θ 

⇒ 4cos²θ = 3cos²θ/sin²θ 

⇒ sin²θ = 3/4

⇒ sinθ = √ 3/2

⇒ θ = 60^° 

Now, substituting θ = 60^° in the given expression

⇒ (2 cos (60∘) + 3 sin (60∘))

⇒ (2 × $\frac{1}{2}$ + 3 × $\frac{√{3}}{2}$)

⇒ $\frac{2+3√{3}}{2}$

∴ The correct answer is $\frac{2+3√{3}}{2}$

Given:

The ratio of the investment of A, B, and C = 5 : 6 : 4

Concept Used:

Investment ratio = Profit Ratio 

When time is constant

Calculation:

Let the total profit be P

According to the question

The profit remaining = P - 0.3P = 0.7P

The profit of A = ${0.7P\ × {5\over 15}}\ +\ 0.3P\ =\ {8P\over 15}$

The share of B and C = ${0.7P\ × {10\over 15}}\ =\ {7P\over 15}$

According to the question  

⇒ ${8P\over 15}\ -\ {7P\over 15}\ =\ 20000$

⇒ P = 15 × 20000

⇒ The total profit = 300000

⇒ The profit of B = 0.7P × ${6\over 15}$ 

⇒ 0.7 × 300000 × ${6\over 15}$ = 84000

∴ The profit of B is 84000.

Given:

The income of A = Rs. 35600

Formula used:

${GM\ =\ {GM\over TM}\ × 100}$ Where, GM = Gain marks, and TM = Total marks

Calculation:

Let the savings increase by X per cent

According to the question

The savings of A = 35600 × ${1\over 5}$ = 7120

The expenditure of A = 35600 - 7120 = 28480

The new income = 35600 × ${5\over 4}$ = 44500

The expenditure of A = 28480 × ${13\over 10}$ = 37024

The new savings of A = 44500 - 37024 = 7476

The saving percentage increase = ${7476\ -\ 7120\over 7120}\ \times 100\ =\ {0.05}\ \times 100\ =\ 5\%$

The monthly saving increased by = 7476 - 7120 = 356

∴ The required result will be Rs. 356.

Shortcut Trick

Use the algebraic identity: a³ + b³ = (a + b)³ − 3ab(a + b).

Here, let a = 2x and b = 3y, so a + b = 2x + 3y = 5/2 and ab = (2x)(3y) = 6xy = 6(1/6) = 1.

8x³ + 27y³ = (5/2)³ − 3(1)(5/2) = 125/8 − 15/2.

8x³ + 27y³ = 125/8 − 60/8 = 65/8.

∴ The correct answer is 65/8.

Alternate Method

Given:

2x + 3y = 5/2

xy = 1/6

Formula Used:

(a + b)³ = a³ + b³ + 3ab(a + b)

Calculations:

Taking the cube on both sides of the equation 2x + 3y = 5/2:

⇒ (2x + 3y)³ = (5/2)³

⇒ (2x)³ + (3y)³ + 3(2x)(3y)(2x + 3y) = 125/8

⇒ 8x³ + 27y³ + 18xy(2x + 3y) = 125/8

Substitute the values of xy = 1/6 and 2x + 3y = 5/2:

⇒ 8x³ + 27y³ + 18 × (1/6) × (5/2) = 125/8

⇒ 8x³ + 27y³ + 3 × (5/2) = 125/8

⇒ 8x³ + 27y³ + 15/2 = 125/8

⇒ 8x³ + 27y³ = 125/8 − 15/2

To subtract, make the denominators equal:

⇒ 8x³ + 27y³ = 125/8 − (15 × 4) / (2 × 4)

⇒ 8x³ + 27y³ = 125/8 − 60/8

⇒ 8x³ + 27y³ = 65/8

∴ The correct answer is 65/8.

Additional Information

Sum of Cubes Identity

a³ + b³ = (a + b)(a² − ab + b²) or a³ + b³ = (a + b)³ − 3ab(a + b).

Difference of Cubes Identity

a³ − b³ = (a − b)(a² + ab + b²) or a³ − b³ = (a − b)³ + 3ab(a − b).

Square of Binomial

(a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b².

Given:

The cost of pen is 64% less than the cost price of notebook.

Calculation:

Let the cost price of price of the pen be P and cost price of notebook be N

According to the question

⇒ P = 0.36N     ----(1)

⇒ 5P + 3N = 120

⇒ (5 × 0.36N) + 3N = 120

⇒ 1.8N + 3N = 120

⇒ 4.8N = 120

⇒ N = 25 and P = 9

The cost price of 8 pens and 5 notebooks = (9 × 8) + (5 × 25) = 72 + 125 = Rs. 197

∴ The required result will be Rs. 197.

Given:

Abhi sell A at profit and loss respectively = 25% & 15%

Abhi sell B at loss and profit respectively = 10% & 25%

Formula used:

P = SP - CP

L = CP - SP     Where, L = Loss, P = Profit, CP = Cost price, and SP = Selling price

Calculation:

Let the cost price of A and B is A and B respectively

According to the question

⇒ 0.85A + 1.25B = A + B

⇒ 85A + 125B = 100A + 100B

⇒ 15A = 25B

⇒ 3A = 5B     ----(1)

⇒ 1.25A + 0.9B = A + B + 114

By multiplying by 100 both sides

⇒ 125A + 90B = 100A + 100B + 11400

⇒ 125A + 18 × 5B = 100A + 20 × 5B + 11400

⇒ 125A + (18 × 3A) = 100A + (20 × 3A) + 11400

⇒ 125A + 54A = 100A + 60A + 11400

⇒ 179A = 160A + 11400

⇒ 19A = 11400

The cost price of A = 600

The cost price of B = 360

The selling price of A in first case = 0.85 × 600 = 510

The selling price of B in first case = 1.25 × 360 = 450

The required difference = 510 - 450 = 60

∴ The required result will be 60.

Given:

The motorboat can go 8.4 km downstream and 4.8 km upstream = 1 hour

Formula used:

${S\ =\ {D\over T}}$     Where, S = Speed, D = Distance, and T = Time

Calculation:

Let the upstream speed and downstream speed of the boat be U and D respectively

According to the question

⇒ ${8.4\over D}\ +\ {4.8\over U}\ =\ 1$     ----(1)

⇒ ${17.5\over D}\ +\ {9\over U}\ =\ 2$     ----(2)

By solving the equation (1) and (2)

⇒ ${84U\ +\ 48D\over 10}\ =\ {175U\ +\ 90D\over 20}$

⇒ 6D = 7U     ----(3)

By solving the equation (1) and (3)

⇒ D = 14 km/h, U = 12 km/h

The speed of motorboat in still water = ${14\ +\ 12\over 2}\ =\ {13\ km/hr}$

The time is taken by boat to cover 31.2 km in still water = ${31.2\over 13}\ =\ 2.4\ hours$

∴ The required result will be 2.4 hours.

Given:

The speed of man from point A to B = 50 km/hr

The speed of man from B to C = 75 km/hr

Formula used:

${Average\ speed\ =\ {Total\ distance\over Total\ time}}$

Calculation:

Let the distance between A to B and B to C be 2d and 3d respectively

So, The distance between A to C = 5d

The total distance = 2d + 3d + 5d = 10d

The total time = $2d\over 50$ + ${3d\over 75}$ + $5d\over x$ 

⇒ ${6dx\ +\ 6dx\ +\ 750d\over 150x}\ =\ {12dx+750d\over 150x}$

Now, According to the question 

⇒ ${61{11\over 49}}\ =\ {10d\over {d(12x + 750)\over 150x}}$

⇒ ${3000\over 49}\ =\ {10\ \times 150x\over (12x+750)}$

⇒ 49x = 24x + 1500

⇒ x = ${1500\over 25}\ =\ 60\ km/hr$

∴ The required result will be 60 km/hr.

Shortcut Trick

Divisor m = HCF of (5398 − 4546) and (6889 − 5398).

m = HCF(852, 1491) = 213.

n = Remainder of (4546 ÷ 213) = 73.

Required value = 2m − 3n = 2(213) − 3(73).

426 − 219 = 207.

∴ The correct answer is 207.

Alternate Method

Given: Numbers are 4546, 5398, and 6889. When divided by m, each leaves a remainder n.

Formula: If numbers A, B, and C leave the same remainder when divided by m, then m is the HCF of (B − A) and (C − B).

⇒ First difference = 5398 − 4546 = 852

⇒ Second difference = 6889 − 5398 = 1491

⇒ Find HCF(852, 1491):

⇒ 852 = 2 × 2 × 3 × 71 = 4 × 213

⇒ 1491 = 3 × 7 × 71 = 7 × 213

⇒ m = 213

⇒ Now, find the remainder n:

⇒ n = 4546 ÷ 213

⇒ 4546 = (213 × 21) + 73

⇒ n = 73

⇒ Value of 2m − 3n = 2(213) − 3(73)

⇒ 426 − 219 = 207

∴ The correct answer is 207.

Additional Information

HCF of Differences Property

When multiple numbers are divided by a common divisor leaving the same remainder, that divisor must be a factor of the differences between those numbers.

Finding the Constant Remainder

Once the divisor m is identified using HCF, the remainder n can be found by applying the formula n = Number mod m.

General Euclidean Algorithm

For large numbers, HCF(a, b) can be found quickly using the property HCF(a, b) = HCF(b, a mod b).

Given:

p varies inversely as the square of q

Formula used:

$p\ =\ {k\over q^2}$     Where k = integer

Calculation:

According to the question

⇒ ${1.2\ =\ {k\over 4.5^2}}$

⇒ k = 1.2 × 20.25 = 24.3

When p = 0.3

⇒ 0.3 = ${24.3\over q^2}$

⇒ q² = 81

The possible value of q = ± 9

∴ The required result will be 9.

Given:

The expenditure of companies A and B in 2018 = 2 : 3

The profit percentage of A in 2018 = 45%

The profit percentage of B in 2018 = 40%

Formula used:

$Profit\%\ =\ { (Income - Expenditure) \over Expenditure} × 100$

Calculation:

The expenditure of companies A and B is 2e and 3e respectively and the income of A is X

The income of B = 355 - X

The profit of A = X - 2e 

The profit of B = 355 - X - 3e

The profit percentage of A

⇒ 45 = ${X\ -\ 2e\over 2e}\ × 100$

⇒ 90e = 100X - 200e

⇒ 290e = 100X

⇒ 29e = 10X     ----(1)

The profit percentage of B

⇒ 40 = ${355\ -\ X\ -\ 3e\over 3e}\ × 100$

⇒ 120e = 35500 - 100X - 300e

⇒ 12e = 3550 - 10X - 30e     ----(2)

By equation (1) and (2)

⇒ 12e = 3550 - 29e - 30e

⇒ 59e + 12e = 3550

⇒ 71e = 3550

⇒ e = 50

The expenditure of A and B in 2018 = 2e + 3e = 5e = 5 × 50 = 250

∴ The required result will be 250.

Shortcut Trick

In a cyclic quadrilateral ABCD, if diagonal AC bisects diagonal BD, the product of the sides meeting at the endpoints of the bisecting diagonal are equal.

Property: AB × BC = AD × CD

⇒ 21 × 16.8 = AD × 14

⇒ AD = (21 × 16.8) ÷ 14

⇒ AD = 1.5 × 16.8 = 25.2 cm

∴ The correct answer is 25.2 cm.

Alternate Method

Given: AB = 21 cm, BC = 16.8 cm, CD = 14 cm, and diagonal AC bisects BD.

Formula Used: Area of Δ = 1/2 × side₁ × side₂ × sin(included angle)

Let the diagonals intersect at E. Since AC bisects BD, BE = ED.

⇒ Area(ΔABE) = Area(ΔADE) (Triangles with same altitude from A and equal bases BE = ED)

⇒ Area(ΔCBE) = Area(ΔCDE) (Triangles with same altitude from C and equal bases BE = ED)

⇒ Area(ΔABC) = Area(ΔADC)

⇒ 1/2 × AB × BC × sin(∠B) = 1/2 × AD × CD × sin(∠D)

In a cyclic quadrilateral, opposite angles are supplementary: ∠B + ∠D = 180°.

⇒ sin(∠B) = sin(180° − ∠D) = sin(∠D)

⇒ AB × BC = AD × CD

⇒ 21 × 16.8 = AD × 14

⇒ 352.8 = AD × 14

⇒ AD = 352.8 ÷ 14

⇒ AD = 25.2 cm

∴ The correct answer is 25.2 cm.

Additional Information

Ptolemy's Theorem

For any cyclic quadrilateral, the product of the diagonals is equal to the sum of the products of the opposite sides: AC × BD = (AB × CD) + (BC × AD).

Diagonal Ratio Property

If the diagonals of a cyclic quadrilateral ABCD intersect at E, then AE/EC = (AB × AD) / (CB × CD) and BE/ED = (BA × BC) / (DA × DC).

Area of Cyclic Quadrilateral

Brahmagupta's Formula: Area = √((s−a)(s−b)(s−c)(s−d)), where s is the semi-perimeter (a+b+c+d)/2.

Given:

The time = ${3{1\over 2}}$ years

The amount = Rs. 16160

Formula used:

${SI\ =\ {P\ × R\ × T\over 100}}$     

Where, SI = Simple interest, P = Principal, R = Rate of interest, and T = Time period

Amount = Principal + SI

Calculation:

Let the principle be Rs. P and the rate of interest be R%

According to the question,

Amount = Principal + SI

⇒ 16160 = P + ${P\ × R\ × 3.5\over 100}$

⇒ 1616000 = 100P + 3.5PR

⇒ 1616000 = P × (100 + 3.5R)     ----(1)

Again, Amount = Principal + SI

⇒ 20000 = P + ${P\ × R\ × 7.5\over 100}$

⇒ 2000000 = P × (100 + 7.5R)     ----(2)

Now, dividing equation (1) by equation (2) we get,

⇒ ${101\over 125} =\ {100\ +\ 3.5R\over 100\ +\ 7.5\ R}$

⇒ 12500 + 437.5R = 10100 + 757.5R

⇒ 757.5R - 437.5R = 12500 - 10100

⇒ 320R = 2400

R% = 7.5% 

Now substituting the value of R% in equation (2) we get,

2000000 = P × (100 + 7.5R)

⇒ 2000000 = P × (100 + 7.5 × 7.5)

⇒ 2000000 = P × (100 + 56.25)

⇒ 2000000 = P × (156.25)

⇒ P = Rs. 12800

SI = \({12800\ ×~ 2 × 7.5\ × 13\over 5\ × 100} \)

⇒ ${12800\ × 15\ × 13\over 5\ × 100} =\ {4992}$

∴ The required result will be Rs. 4992.

Given:

The cost price of the motorcycle = Rs. 135000

Formula used:

${A\ =\ {x\over {(1\ +\ {R\over 100})}}}\ +\ {x\over {(1\ +\ {R\over 100})^2}}\ +\ {x\over {(1\ +\ {R\over 100})^3}}\ +...$     

Where, A = Amount, R = Rate, x = Instalment

Calculation:

Let the total interest paid to be X

According to the question

⇒ ${135000\ -\ 23880 =\ {x\over {(1\ +\ {15\over 100})}}}\ +\ {x\over {(1\ +\ {15\over 100})^2}}\ +\ {x\over {(1\ +\ {15\over 100})^3}}\$

⇒ ${111120 =\ {20x\over {23}}}\ +\ {400x\over {529}}\ +\ {8000x\over12167}\$

⇒ ${111120 =\ {529\ × 20x\ +\ 23\ ×\ 400x\ +\ 8000x \over 12167}}$

⇒ ${111120 =\ {10580x\ +\ 9200x\ +\ 8000x \over 12167}}$

⇒ ${111120 =\ {27780x \over 12167}}$

⇒ The each instalment = 12167 × 4 = 48668

The total amount paid = 23880 + (3 × 48668) = 23880 + 146004 = 169884

The total interest = 169884 - 135000 = 34884

∴ The required result result will be 34884.

Given:

The number of students who appeared in 2017 and 2019 from school B = 250 and 350 respectively

The number of students passed in 2018 and 2020 from school A = 240 and 270 respectively

The number of students passed in 2018 and 2020 from school C = 250 and 340 respectively

Formula used:

${GM\% =\ {GM\over TM}}\ \times 100$     Where, GM = Gain marks, TM = Total marks

Calculation:

The students appeared in 2017 and 2019 from school B = 250 + 350 = 600

The total students passed in 2018 and 2020 from school A and C = 240 + 270 + 250 + 340 = 1100

The required percentage = ${1100\ -\ 600\over 1100}\ \times 100\ =\ 45.45\%$

∴ The required result will be 45.45%.

Given:

Radius of cylindrical bucket (r) = 21 cm

Height of cylindrical bucket (h) = 24 cm

Length of rectangular tub (l) = 66 cm

Breadth of rectangular tub (b) = x cm

Height of rectangular tub (h) = 12 cm

Formula Used:

Volume of cylinder = πr²h, 

Volume of rectangular cuboid = lbh, Where l =  length, b= breadth, h = height 

Calculation:

In the figure, a cylindrical bucket full of water is emptied into a rectangular tub.

According to the question, 

Volume of water in cylindrical bucket = Volume of water in rectangular tub

⇒ $π r^2 h$ = lbh

⇒ $\frac{22}{7} × 21^2 × 24$ = 66 × x × 12

⇒ 21 × 24 = x × 12

⇒ x = 42 cm

Breadth of rectangular tub = 42 cm

∴ The value of x is 42 cm.

Additional InformationRight circular cylinder:

• Curved/Lateral surface area of cylinder = $2π r h$
• Total surface area of cylinder = $2π r (h+r)$

Cuboid:

• Curved/Lateral surface area of cylinder = $2(l + b)h$
• Total surface area of cylinder = $2(lb + bh + hl)$

 

Mistake PointsDifference between curved surface area and total surface area: Total surface area refers to area including the base and the curved surface or lateral surface area refers to the area of only curved part excluding it's base.

Given:

A alone can complete the work = 40 days

Formula used:

W = E × T     Where, W = Work, E = Efficiency, and T = Time

Calculation:

Let the C work for D days

According to the question

The efficiency of A = ${1\over 40}$

The efficiency of B = ${1\over B}$ = ${1\over 40}\ \times {4\over 5}\ =\ {1\over 50}$

The efficiency of C = ${1\over C}$= ${1\over 50}\ \times {8\over 5}\ =\ {4\over 125}$

A and B work for 8 days and when C join B and C work for all remaining days

⇒ ${8\over 40}\ +\ {8\ +\ D\over 50}\ +\ {4D\over 125}\ =\ {1}$

⇒ ${1\over 5}\ +\ {8\ +\ D\over 50}\ +\ {4D\over 125}\ =\ {1}$

⇒ ${200\ +\ 160\ +\ 20D\ +\ 32D\over 1000}\ =\ 1$

⇒ 20D + 32D = 1000 - 360

⇒ 52D = 640

⇒ 13D = 160

The required total number of days to complete the work = ${160\over 13}\ +\ 8\ =\ {160\ +\ 104\over 13}\ =\ {264\over 13}\ =\ {20{4\over 13}}\ days$

∴ The required result will be ${20{4\over 13}}\ days.$

Given:

459xy is divisible by 3, 7 and 11

Concept Used:

Divisibility rule of 3 = If sum of all digit of number is divisible by 3 then whole number is divisible by 3

Divisibility rule of 11 = Sum of odd place digit - sum of even place digit = 0 or multiple of 11

Calculation:

According to the question, 459xy is divisible by 3, 7 and 11

Sum of all digit = 4 + 5 + 9 + x + y = 18 + x + y

⇒ x + y = 0 or multiple of 3     ----(1)

To check the divisibility rule of 11

⇒ 4 + 9 + y - 5 - x = 0 or multiple of 11

If, 8 + y - x = 0

⇒ x - y = 8     ----(2)

In equation (2) only x = 9 and y = 1 will give result 8, but (x + y = 9 + 1) will not satisfy equation (1) because 10 is not a multiple of 3.

And, 8 + y - x = 11

⇒ y - x = 3     ----(3)

In equation (3) only x = 6 and y = 9 will give result 3, and (x + y = 6 + 9) will satisfy equation (1) because 15 is a multiple of 3.

Also checking (459xy = 45969) is divisible by 3, 7 and 11

Now, substituting the value of x = 6 and y = 9 in (x2 - y2 + 3xy), we get

⇒ (x2 - y2 + 3xy) = (6² - 9² + 3 × 6 × 9)

⇒ 36 - 81 + 162

⇒ 117

Now from the above given options only 13 is a multiple of 117.

∴ (x² - y² + 3xy) is a multiple of 13.

Shortcut Trick

The extra time taken to cross the platform instead of the pole is directly due to the length of the platform.

Extra time = 51 sec − 27 sec = 24 sec.

Speed in m/s = 42 × (5/18) = 35/3 m/s.

Length of platform = Speed × Extra time = (35/3) × 24 = 35 × 8 = 280 m.

∴ The correct answer is 280 m.

Alternate Method

Given:

Speed of the train = 42 km/h

Time to cross electric pole (T₁) = 27 sec

Time to cross platform (T₂) = 51 sec

Formula Used:

Distance = Speed × Time

Speed (m/s) = Speed (km/h) × (5/18)

⇒ Speed in m/s = 42 × (5/18) = 35/3 m/s

⇒ Length of train (L) = Speed × Time to cross pole

⇒ L = (35/3) × 27 = 35 × 9 = 315 m

⇒ Total distance to cross platform = Length of train + Length of platform (P)

⇒ L + P = Speed × Time to cross platform

⇒ 315 + P = (35/3) × 51

⇒ 315 + P = 35 × 17

⇒ 315 + P = 595

⇒ P = 595 − 315 = 280 m

∴ The correct answer is 280 m.

Additional Information

Unit Conversion

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

Train Crossing a Point Object

When a train crosses an electric pole, a tree, or a stationary person, the distance covered is equal to the length of the train.

Train Crossing a Large Object

When a train crosses a platform, a bridge, or another stationary train, the distance covered is equal to the Sum of the lengths of the train and the object.

Given:

In ∆ ABC, ∠C = 90°

M and N are the mid points of sides AB and AC, respectively.

CM and BN intersect at a right angle at D.

BC = 9 cm.

CONCEPT :

1. As we know, the intersection point of the medians are called the centroid and the centroid cuts the median in the ratio 2 : 1.

2. B.P.T Theorem  (If a line joins mid point of two lines then that line is parallel and half of the third side )

3. Similarity A.A  (If two angles of a triangle are equal to two angles of another triangle then these two triangles are similar to each other)

Calculation:

According to question,

BD : DN = 2x : x  (By concept 1)

Now Δ CDB ∼ Δ NCB (by concept 3)

⇒ BN/BC = BC/DB

⇒ 3x/9 = 9/2x

⇒ 6x² = 81

⇒ x = √81/√6 

⇒ x = 9/√6

Now,

BN = 3x (BN = BD + DN)

⇒ BN = 3 × (9/√6)

⇒ BN = 9√6/2

The length (in cm) of BN is 9√6/2

Alternate MethodGiven:

In ∆ ABC, ∠C = 90°

M and N are the mid points of sides AB and AC, respectively.

CM and BN intersect at a right angle at D.

BC = 9 cm,

Formula used:

As we know, the intersection point of the medians are called the centroid and the centroid cuts the median in the ratio 2 : 1.

As we know, the circumcenter lies on the mid-point of the hypotenuse of the right angel triangle.

Calculation:

According to the question, the required figure is:

As we know, the intersection point of the medians are called the centroid and the centroid cuts the median in the ratio 2 : 1.

Since CM and BN are median,

Therefore, 

CM : DM = 3x : x

BN : DN = 3x : x

As we know, the circumcenter lies on the mid-point of the hypotenuse of the right angel triangle.

Therefore, 

⇒ CM = AM = BM = 3x

So, 

⇒ AB = 6x

Let AC be 2a.

Since N is the mid-point on the line AC.

⇒ AN = CN = a

In Δ  ABC,

AB² = AC² + BC²  

⇒ 36x² = 4z² + 81    ----(1)

In Δ CDB,

BC² = CD² + BD²  

⇒ 81 = 4x² + 4y² 

⇒ 4x² + 4y² = 81    ----(2)

In Δ CNB,

BN² = CN² + BC²  

⇒ 9y² = z² + 81     ----(3)

On equating the equation (1) and (3), 

⇒ 36x² - 36y² = -243    ----(4)

On equating the equation (2) and (4), 

y = $√ \frac{27}{2}$ 

BN = 3y = 3 × $√ \frac {27}{2}$ = $\frac{9 √{6}}{2}$

∴ '$\frac{9√{6}}{2}$'  is the required answer.

⇒

Given:

The average of 26 numbers = 54.5

Formula used:

Average = Sum of all the terms/Total number of terms

Calculation:

Let the average of the remaining number be X

According to the question

The sum of all 26 numbers = 54.5 × 26 = 1417

The sum of all 15 and 12 numbers = (15 × 50.4)  + (12 × 62.5) = 756 + 750 = 1506

The 15th number = 1506 - 1417 = 89

When excluded the 15th number = 1417 - 89 = 1328

The average of the remaining numbers = ${1328\over 25}\ =\ 53.12$

∴ The required result will be 53.12.

Shortcut Trick

Market Price (MP) = 264 ÷ 0.88 = 300

Since 20% profit is earned at MP, CP = 300 ÷ 1.2 = 250

New CP = 80% of 250 = 200

New SP = 120% of 200 = 240

∴ The correct answer is Rs. 240.

Alternate Method

Given: Selling Price (SP₁) = Rs. 264, Discount = 12%, Profit (at MP) = 20%

Formula: MP = SP × 100 ÷ (100 − D), CP = MP × 100 ÷ (100 + P)

⇒ Market Price (MP) = 264 × (100 ÷ 88) = 264 ÷ 0.88 = Rs. 300

⇒ Given: If no discount, Profit = 20% ⇒ MP = 120% of CP

⇒ Original Cost Price (CP) = 300 × (100 ÷ 120) = 300 ÷ 1.2 = Rs. 250

⇒ New Cost Price = 250 − 20% of 250 = 250 − 50 = Rs. 200

⇒ New Selling Price (at 20% profit) = 200 × 1.2 = Rs. 240

∴ The correct answer is Rs. 240.

Additional Information

Relation between MP and CP

The direct ratio is MP ÷ CP = (100 + Profit%) ÷ (100 − Discount%).

Profit Percentage Formula

Profit % = [(SP − CP) ÷ CP] × 100. It is always calculated on the Cost Price unless specified.

Discount Calculation

Discount = MP − SP. Discount % is always calculated on the Market Price (MP).

Given:

The expression = 5 - 10 ÷ 10 of 5 - (12 - 15) × 4 ÷ 10 + 15 ÷ 30 × 2

Concept -

Calculation:

⇒ 5 - 10 ÷ 10 of 5 - (12 - 15) × 4 ÷ 10 + 15 ÷ 30 × 2

⇒ 5 - 10 ÷ 50 - (-3) × 4 ÷ 10 + 15 ÷ 30 × 2

⇒ 5 - ${1\over 5}$ + 3 × 0.4 + 0.5 × 2

⇒ 5 - ${1\over 5}$ + 1.2 + 1

⇒ 5 - ${1\over 5}$ + 2.2

⇒ ${25\ -\ 1\ +\ 11\over 5}\ =\ {35\over 5}\ =\ 7$

∴ The required result will be $7.$

Given:

Pipe A can fill the tank = 9 hours

Pipe B can fill the tank = 15 hours

Formula used:

W = E × T     Where, W = Work, E = Efficiency, and T = Time

Calculation:

Let the after T hours leakage be repaired

According to the question

One hours work of pipe A = ${1\over 9}$

One hours work of pipe B = ${1\over 15}$

One hour of work of leakage = ${2\over 45}$

The tank can be filled by A and B in one hour = ${1\over 9}\ +\ {1\over 15}\ =\ {10\ +\ 6\over 90}\ =\ {16\over 90}\ =\ {8\over 45}$

Both A and B can fill the tank = ${45\over 8}\ hours$

Tank filled with leakage = 6 hours

⇒ ${6\over 9}\ +\ {6\over 15}\ -\ {2T\over 45}\ =\ 1$

By solving the equation 

The time is taken to repair the leakage in the tank = T = ${1{1\over 2}}\ hours.$

∴ The required result will be ${1{1\over 2}}\ hours.$

The correct answer is Colloids are homogeneous mixtures.

Key Points

Concept:

Colloidal solution/ colloid:

• The type of solution in which the size of the solute particle is in the range of 1 to 100 nm, is called a colloid.
• A colloid is a heterogeneous system in which one substance is dispersed (dispersed phase or colloidal particles) in another substance which is called a dispersion medium.

Explanation:

Properties of colloids:

• A colloid appears to be homogeneous but actually, it is heterogeneous.
• The particles of most of the colloids cannot be seen even with a microscope.
• It scatters a beam of light passing through it (Tyndall effect).
• The size of the solute particle is in the range of 1 to 100 nm.
• Since they are heterogeneous in nature, centrifugation techniques can be used to separate the particles.

Additional Information

The correct answer is real and of same size.

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 at C (Centre of curvature):

Characteristics of Image Formed:

• Position of the image - At C
• Nature of image - Real & Inverted
• Size - Same as that of the object

Additional InformationConvex 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 Metal carbonates are insoluble in water. 

Concept:

Double displacement reaction: 

• It is a chemical reaction in which two compounds react and exchange their ions forming new products.
• This reaction often results in the formation of an insoluble compound called a precipitate.

Explanation:

When carbon dioxide gas is passed through an aqueous solution of calcium hydroxide, the following chemical reaction takes place: 

CO₂ + Ca(OH)₂ → CaCO₃ ↓ + H₂O

A white precipitate of calcium carbonates is formed, along with water.

This is an example of a double displacement reaction. 

The precipitate CaCO₃ (metal carbonate) is insoluble in water.

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....
• 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

Calculation:

 

Resistors A (r Ω), B (2r Ω), and C (3r Ω) are connected in parallel.

Equivalent resistance, 1/R1 = 1/r + 1/2r +1/3r = 6+3+2/ 6r = 11/6 r 

R1= 6r/11

Now, the resistors R1, D, F, and G are in series.

R2 = 6r/11 + r + 2r + 3r = (6r + 11r + 22r + 33r) / 11 = 72r / 11

Equivalent current, I in the circuit = V/R2 = $\frac{11E}{72r}$

Potential difference across C, V' = IR1 = $\frac{11E}{72r}$  ×  $\frac{6r}{11}$ = E / 12

PG = I2G = $(\frac{11E}{72r})^2$ × 3r = ^{$\frac{121E^2 × 3r}{72r ×72r}$}  = 121E²/ 72r × 24r  (since current remains same in series combination)

PC = V' 2/C =  $\frac{(\frac{E}{12})^2}{3r}$  =  E²/ 12 × 12 × 3r (since voltage remains same in parallel combination)

P_C/ P_G = $\frac{\frac{E^2}{12 × 12 × 3r}}{\frac{121 E^2}{72r × 24}}$ = $\frac{E^2 × 72r × 24}{12 × 12× 3r × 121 E^2}$ = 4/ 121

The ratio (PC/PG) = 4/121.

The correct answer is to perform the same function in both animals but is anatomically very different.

Concept:

**Evidence for evolution:**​

• Fossils
• Analogous and homologous organs
• Embryological evidence

Anatomy: It refers to the internal structure of an organism. 

Explanation:

Homologous organs:

• These organs are similar in origin and structure but perform different functions.
• For example, the Forelimbs of frogs, lizards, bats, and humans have the same basic structural plan but different functions.
• The wings help in flight whereas the human forearm helps in various activities.

Analogous organs:

• These organs have similar functions but are different in structure and origin.
• For instance, the wings of a bat and a bird are similar in function, but this similarity does not mean that these animals are more closely related.
• The wings of a bat are just the folds of skin that are stretched between its fingers whereas the wings of birds are modified forearms covered in feathers.

Thus, we can say that analogous organs perform the same function in both animals but are anatomically very different.

The correct answer is 4 - 1 - 3 - 5 - 2.

Concept:

The Human eye: 

• Humans have paired eyes that are located in sockets of the skull called orbits.
• The eyes are one of the sense organs that give the sensation of sight.

Explanation:

Structure of the human eye:

• The wall of the eyeball is composed of three layers- the outer sclera, middle choroid, and inner retina.
• The anterior portion of the sclera is called the cornea. It is transparent.
• The choroid becomes thick in the anterior part to form the ciliary body.
• The ciliary body continues forward to form a pigmented and opaque structure called the iris which is the visible coloured portion of the eye.
• A transparent crystalline lens is held in place by ligaments attached to the ciliary body behind the iris.
• In front of the lens, the aperture surrounded by the iris is called the pupil.
• The retina has two specialized cells- rods and cones that perceive colour and daylight vision.

The pathway of light rays (photons) entering our eye:

The correct answer is -100 cm; -4.0 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 = -25 cm        f = -20 cm              h = 1 cm 

$\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$

$\frac{1}{-20} = \frac{1}{v} + \frac{1}{-25}$

$\frac{1}{-20} = \frac{1}{v} - \frac{1}{25}$

$\frac{1}{v} = \frac{1}{-20} + \frac{1}{25}$

$\frac{1}{v} = \frac{-5 + 4}{100}$

$\frac{1}{v} = \frac{-1}{100}$

v = -100 cm

m = $\frac{Hi}{Ho} = \frac{-v}{u}$ = $\frac{Hi}{1} = \frac{-(-100)}{-25}$ = H_i = - 4 cm

Following New Cartesian Sign Convention, the image is formed at v = -100 cm and its height hI = -4 cm.

The correct answer is 20g. 

Concept:

• Solution: A solution has two components i.e., solute and solvent.
• Solute: The substance that gets dissolved in the solvent.
• Solvent: The substance in which solute is dissolved.

Mass by mass percentage of solution: (Mass of solute/mass of solute + mass of solvent ) × 100

Explanation:

Let the mass of sugar be x grams.  

Given: Percentage of sugar in water = 10 %

Mass of water = 180 g

Now, Mass percentage of sugar = (mass of sugar/mass of sugar + mass of water) × 100

⇒ 10 = [x / (x + 180)] × 100

⇒ (x + 180) = 10x

⇒ 9x = 180

⇒ x = 180/9 = 20 g 

Mass of sugar = x = 20g

The correct answer is 21%.

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).
• Ohms is named after George Simon Ohm (1784-1854), a German physicist who studied the relationship between voltage, current and resistance.
• $R = \frac{ρ l}{A}$ 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 decrease in length, let us take new length = L', radius of wire = r' and new resistance = R'

L' = 110 L/ 100 = 11 L/ 10( since there is 10% 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 = 10r2L /11 L

r' 2 = 10 r2  / 11

Decrease in resistance = $\frac{R'-R}{R} × 100$

$\rm \frac{\left(\frac{\rho L'}{A'}-\frac{\rho L{}}{A}\right)}{\frac{\rho L}{A}}\times100$

$\rm \frac{\left(\frac{\rho L'}{\pi r'^2}-\frac{\rho L{}}{\pi r^2}\right)}{\frac{\rho L}{\pi r^2}}\times100$

= $\rm \frac{\frac{\rho L'}{\pi r'^2 }-\frac{\rho L}{\pi r^2}}{\frac{\rho L}{\pi r^2}}\times 100$

= $\rm \frac{\frac{11 \times 11 L \rho}{10 \times 10\pi r^2 }-\frac{\rho L}{\pi r^2}}{\frac{\rho L}{\pi r^2}}\times 100$

= $\rm \frac{\frac{121 \rho L}{100\pi r^2 }-\frac{\rho L}{\pi r^2}}{\frac{\rho L}{\pi r^2}}\times 100$

= $\rm \frac{\frac{121 \rho L-100 \rho L}{100\pi r^2 }}{\frac{\rho L}{\pi r^2}}\times 100$

= $\rm \frac{21 \rho L}{100 \pi r^2}\times \frac{100}{\rho L}\times \pi r^2=21\%$

The resistance increases by 21%.

The correct answer is a plastic bag.

Concept:

Waste materials: 

• Waste materials are either liquid or solid in form, and their components may be either hazardous or inert in their effects on health and the environment.
• The term waste is typically applied to solid waste, sewage (wastewater), hazardous waste, and electronic waste.

Explanation:

Biodegradable waste: 

• Those substances that can be decomposed by the action of decomposers like bacteria and fungi are called biodegradable wastes.
• For example organic matter like human excreta, animal waste, dead bodies of plants and animals, leftover food items, etc.

​Non-biodegradable waste:

• Those substances that can not be decomposed by the action of decomposers like bacteria and fungi are called non-biodegradable wastes.
• For example plastic, rubber, glass, artificial clothes like nylon, rayon, electronic waste like batteries, etc.

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

Concept:

Atomic number: It is the number of electrons or protons found in an atom.

Electronic configuration:

• The arrangement of electrons in different shells around the nucleus.
• There are 7 shells K, L, M, N, and so on.
• K shell can accommodate only 2 electrons while others can accommodate 8 electrons.

Ionic compounds: The compounds formed by the complete transfer of electrons from one element to another.

Explanation:

Element X

• Atomic no. of X = 20 ( Calcium)
• Electronic configuration = 2, 8, 8, 2.
• Valency = +2, since it will lose 2 electrons

Element Y

• Atomic no. of  Y = 8 (oxygen)
• Electronic configuration = 2, 6.
• Valency = -2, since it will gain 2 electrons

Compound Z

• 2 electrons will be transferred from element X to element Y and an ionic bond is formed.

The formula for Z will be XY.

The compound will be Calcium oxide (CaO).

Therefore, statements (ii) and (iv) are correct.

The correct answer is V1 ∶ V2 ∶ V3 = 36 ∶ 9 ∶ 4.

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 = r, Area of cross- section = π r²

Resistance, R1 =  $\frac{\rho L}{\pi r^2}$

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

For Wire CD

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 BD

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

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

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

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

V1 ∶ V2 ∶ V3 = 36 ∶ 9 ∶ 4.

The correct answer is Both (ii) and (iii) are correct.

Concept:

Magnetic field (B):

• The area around a magnet where a magnetic force is experienced is called the magnetic field.
• It is a quantity that has both direction and magnitude, (i.e., Vector quantity).

Explanation:

Magnetic field due to a current-carrying coil:

• It is directly proportional to the current passing through the coil i.e., B ∝ I
• It is directly proportional to the number of turns of the coil  i.e., B ∝ n
• The direction of the magnetic field can be found with the help of the right-hand thumb rule.

Additional InformationRight-hand thumb rule: 

• According to Maxwell's right-hand thumb rule if we tend to hold a straight current-carrying wire in our right hand such that our thumb represents the direction of the current then the curls of our fingers would represent the direction of the magnetic field lines.

The correct answer is 1.08 : 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 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 kerosene with respect to water:

• n = refractive index of kerosene / refractive index of water = $\frac{1.44}{1.33}$ = 1.08
• Greater the refractive index, the denser the medium. Thus, kerosene (1.44) is denser than water (1.33).
• 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.

The refractive index of kerosene with respect to water is 1.08 and the light ray bends toward the normal in kerosene.