The option figure in which the given figure is embedded is option ‘1’.

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

The logic is:

Hence, ‘S’ is the correct answer.

The logic is:

Hence, ‘73’ is the correct answer.

The correct mirror image of the given figure when the mirror is placed on the right of the figure is shown below:

Hence, option 1 is the correct answer.

Faculty : Teachers → The faculty is described as a group of teachers. (Collective noun)

Similarly,

1. Ants ; Flock → A group of ants is called a colony. So this is not an example of collective noun.

2. Colony : Wolves → A group of wolves is called a pack. So this is not an example of collective noun.

3. Galaxy : Apartments → A group of apartments is called a block. So this is not an example of collective noun.

4. Fleet : Trucks → A group of trucks is called a fleet. So this is an example of collective noun.

Hence, ‘Fleet : Trucks’ is the correct answer.

‘Cardiologist’ is related to ‘heart’ → A cardiologist is a doctor with special training and skill in finding, treating and preventing diseases of the heart and blood vessels.

Similarly,

‘Neurologist’ → A neurologist is a medical doctor who specializes in treating diseases of the nervous system. The nervous system is made of two parts: the central and peripheral nervous system. It includes the brain and spinal cord.

Hence, the correct answer is brain.

 

P + R × T - Q ÷ S + U → P is brother of R, R is father of T, T is mother of Q, Q is son of S, S is brother of U.

S is son-in-law of R.

Hence, ‘Son-in-law’ is the correct answer.

When the transparent sheet of paper is folded at the dotted line, it will appear as shown below:

Hence, option 2 is the correct answer.

The logic is:

Difference of squares of the digits of first number is equals to the second number.

1. 28 - 60 → (8² - 2²) = 60

2. 42 - 12 → (4² - 2²) = 12

3. 39 - 72 → (9² - 3²) = 72

4. 57 - 38 → (7² - 5²) ≠ 38

Hence, ’57 - 38’ is the correct answer.

The logic is:

Hence, ‘XXZ’ is the correct answer.

The logic is:

Similarly,

Hence, ‘XTHRHN’ is the correct answer.

The logic is:

1. ADGJ

2. BEHK

3. JMPS

4. FHKM

Hence, ‘FHKM’ is the odd one out.

Given series:

The first two figures consist a triangle, then the next two figure has square inside it, so the next figure will definitely have pentagon inside it.

Again the arrows present in the images at odd place is same. Again the arrow at even place is same. So, the next image in the series is at odd place, so the arrow will be similar to that in the 1^st and 3^rd image.

All these conditions is satisfied by option 1.

Hence, option 1 is the correct answer.

1. Hexagon → Six sides
2. Nonagon → Nine sides
3. Pentagon → Five sides
4. Heptagon → Seven sides
5. Octagon → Eight sides

Arranging the following polygons in ascending order:

3. Pentagon → Five sides
4. Hexagon → Six sides
5. Heptagon → Seven sides
6. Octagon → Eight sides
7. Nonagon → Nine sides

Hence, the correct logical and meaningful order is ‘3 - 1 - 4 - 5 - 2’.

The logic is:

Similarly,

Hence, ‘40424’ is the correct answer.

The logic is:

1^st row: 5² + 4² = 41

3^rd row: 9² + 11² = 202

Similarly,

2^nd row: 15² + 6² = 261

Hence, ‘261’ is the correct answer.

1. • and ×

Given equation: 36 ÷ 2 × 12 + 3 - 6 = 24

After interchanging ‘+ and ×’, we get:

L.H.S = 36 ÷ 2 + 12 × 3 - 6

= 18 + 36 - 6

= 54 - 6

= 48 ≠ R.H.S

2. ÷ and ×

Given equation: 36 ÷ 2 × 12 + 3 - 6 = 24

After interchanging ‘÷ and ×’, we get:

L.H.S = 36 × 2 ÷ 12 + 3 - 6

= 6 + 3 - 6

= 3 ≠ R.H.S

3. • and ÷

Given equation: 36 ÷ 2 × 12 + 3 - 6 = 24

After interchanging ‘- and ÷’, we get:

L.H.S = 36 - 2 × 12 + 3 ÷ 6

= 36 - 24 + 0.5

= 12.5 ≠ R.H.S

4) × and -

Given equation: 36 ÷ 2 × 12 + 3 - 6 = 24

After interchanging ‘× and -’, we get:

L.H.S = 36 ÷ 2 - 12 + 3 × 6

= 18 - 12 + 18

= 36 - 12

= 24 = R.H.S

Hence, ‘× and -‘ is the correct answer.

The logic is:

1st row: 13 - 7 = 6

2nd row: 22 - 4 = 18

Similarly,

? - 15 = 7

? = 7 + 15

? = 22

Hence, ‘22’ is the correct answer.

Let us consider that the student answered ‘a’ answers correctly.

Now, the student attempted only 80% of the questions (given)

80% of 80 = 64

So, number of incorrect questions = (64 - a)

Also, the student scored 32 marks (given)

So, according to question:

a - (64 - a) = 32

→ a - 64 + a = 32

→ 2a = 96

→ a = 48

Hence, the student answered 48 questions correctly.

The least possible venn diagram is:

I. No dog is an elephant. → True

II. No lion is a dog. → False

III. Some elephants are dogs. → False

Hence, only conclusion I follows.

The number of triangles in the given figure is shown below:

Hence, the figure has total 10 triangles.

‘<’ is present in 3^rd dice.

The common symbol in 2^nd and 3^rd dice is ‘%’.

So, keeping ‘%’ constant and then rotating in clockwise direction, we get the symbols opposite to each other:

 

So, ‘@’ is opposite to ‘<’.

Hence, ‘@’ is the correct answer.

The require number will lie in the intersection of triangle and circle.

Hence, 3 married ladies are teachers.

The logic is:

2809 : 53 → 53² : 53

Similarly,

1024 : ? → 32² : 32

Hence, ‘32’ is the correct answer.

Monitor, Headphone, Mouse, Keyboard, Printer, Scanner, Speaker → These are devices and are physical in nature.

Windows → It is an operating system and not physical in nature.

Hence, ‘windows’ is the odd one out.

The Correct Answer is Option 1 i.e Bimbisara.

Bimbisara:

• Bimbisara was the founder of the Haryanka Dynasty.
• He was a contemporary of Buddha.
• He started the policy of annexation and expansion.
• He was succeeded by his son Ajatashatru.

Ajatashatru:

• He followed a more serious and aggressive policy of expansion.
• He won the long war with the Kosala in the form of a marriage alliance with the daughter of the Kosala king.
• Ajatashatru was succeeded by Udayin. Udayin built the city Patilputra (Patna).
• The Last King Nagadashak was killed by his own minister Sisunaga, who started the Sisunaga Dynasty.

The Correct Answer is Option 1 i.e Lai Haraoba.

The Correct Answer is Option 2 i.e Begumpet Airport.

Wing India 2020:

• It is a four-day civil aviation business exhibition and air show event held in Begumpet Airport, Hyderabad from March 12th to 15th.
• the event is being organized by the Federation of Indian Chambers of Commerce and Industry (FICCI) along with the Ministry of  Civil Aviation and Airports Authority of India.
• The theme of the year 2020 is "Flying for All" and the focus is on the new business acquisition, investments, policy formation and regional connectivity in the civil aviation industry.

The Correct Answer is Option 3 i.e Kodagu.

The Correct Answer is Option 2 i.e Gujarat.

Khadi and Village Industries Commission (KVIC):

• It is a statutory body formed by the Government of India under the KVIC Act of 1956.
• It is formed as an apex body under the Ministry of Micro, Small and Medium Enterprises.
• It aims to nurse employment and economic uplift in rural India and also aid in the development of Khadi and Village industries in rural India in conjunction with other agencies involved in rural development.
• KVIC opened its first silk processing plant in Gujarat to boost patola saree production.

The Correct Answer is Option 1 i.e Kerala.

• A two day organized Global Investors Meet – ASCEND 2020 was held in Kochi, Kerala from January 9-10.
• ASCEND 2020 showcased 18 mega projects with over Rs 100 crore investment each.
• The meet was inaugurated by the Chief Minister of Kerala Shri Pinarayi Vijayan and was organized by states Department of Industries.
• Chief Minister of Kerala- Pinarayi Vijayan.
• Governor of Kerala- Arif Mohammad Khan.

The Correct Answer is Option 1 i.e Thymus.

The Correct Answer is option 3 i.e Acetic acid.

Key Points

• Acetic acid is a major component of vinegar.
• Its concentration determines the strength of vinegar.
• It is a colorless liquid with a strong, corrosive pungent odor.
• Acetic acid is widely used in the food industry as vinegar and as an acidity regulator
• A lactic acid buildup around muscles.
• Nitric acid is used in the production of ammonium nitrate for fertilizers.
• ​Citric acid is naturally found in citrus fruits, most concentrated in lemons.

The Correct Answer is Option 2 i.e England.

• The Hastings International Chess Congress is an annual chess tournament which takes place in Hastings, England.
• India's P. Magesh Chandran wins the title in the 95th edition of the prestigious Hastings International Chess Congress.
• The tournament was held from 28th December 2019 to 5th January 2020.
• He defeated Indian GM Deep Sengupta in the eighth round to win the title.

The Correct Answer is Option 3 i. 1911.

The Gateway of India:

• It was built to commemorate the visit of King George V and Queen Mary to India.
• It was built during British rule and was inaugurated on December 4th, 1924.
• The Foundation stone was laid by the then governor of Mumbai, Sir George Sydenham Clarke on march 31st 1911.
• Architecture is built in Indo- Saracenic style.
• George Wittet was the chief architect of the Gateway of India.

The Correct Answer is Option 4 i.e Arunachal Pradesh.

• The Arunachal Pradesh Legislative Assembly (APLA) adopted a new logo, reflecting the distinct identity and culture of the state.
• The new logo consists of the national emblem and foxtail orchid (Rhynchostylis Retusa), the state flower.
• The national emblem in the logo signifies the federal system of the Indian Constitution.
• The flower foxtail orchid represents the state, its people and culture of the state.
• The blue color of the logo denotes the autonomy of the assembly secretariat.

Note: State’s Legislative Assemblies of Tripura, Meghalaya, and Mizoram use the National Emblem of India.

The correct answer is Snail.

Key Points

• Vertebrates are animals with an internal backbone or spinal column
• Animals that are not vertebrates are called invertebrates. Eg, Silverfish, spider, silkworm
• Vertebrates include mammals, birds, fish, amphibians, and reptiles.
• Snails are invertebrates.

The Correct Answer is Option 1 i.e Lodhi Gardens.

The Correct Answer is Option 4 i.e Narendra Modi.

The Correct Answer is Option 3 i.e Dhanananda.

Key Points

Nanda Dynasty:

• It was founded by Mahapadma Nanda around 467 BC.
• He Accepted the title of Ekrat meaning the sole sovereign who destroyed all other ruling princes.
• Dhanananda was the last ruler of the Nanda Dynasty, was overthrown by Chandra Gupta Maurya.
• During Dhanananda, Alexander attacked India and another King Porus fought against Alexander on the banks of river Jhelum i.e Battle of Hydaspes.

The Correct Answer is Option 3 i.e Bangladesh.

Archaeologist R. Nagaswamy

• He was honored at the Silver Jubilee International Conference of Art held in Dhaka by Bangladesh Culture Minister K.M. Khalid.
• He was honored for the contribution of Nagaswamy in art, archaeology, history, and culture.
• Ramachandran Nagaswamy is an Indian historian, archeologist, and epigraphist who is known for his work on temple inscriptions and art history of Tamil Nadu.
• He was awarded India's third highest civilian award the Padma Bhushan in 2018.

The Correct Answer is Option 1 i.e Seventh.

Brand Finance Nation Ranking:

• India has jumped from 9th position to 7th position in the Brand Finance Nation ranking of 2019.
• While the US, China, and Germany continued to be the top three nations respectively even in 2019.
• Brand Finance is an independent brand business valuation consultancy.
• It advises branded organizations or those with tangible assets, on how to maximize their value through effective management, of their brand and other intangible assets. It is headquartered in London, United Kingdom.

The Correct Answer is Option 1 i.e Ananthagiri Hill.

Borra

The correct answer is Sulfenic acid.

• When we cut an onion, the synthase enzyme converts the amino acid sulfoxides of the onion into Sulfenic Acid.
• Citric acid is found naturally in citrus fruits, especially in lemons.
• Nitric acid, also known as aqua fortis and "spirit of nitre" is a highly corrosive mineral acid.
• Sulfuric acid, also known as vitriol, is a mineral acid composed of the elements sulfur, oxygen, and hydrogen.
• Note: A sulfenic acid is an organosulfur compound and oxoacid with the general formula RSOH.

The Correct Answer is Option 4 i.e Gujarat.

The Correct Answer is Option 1 i.e Coulomb's Law.

Key Points 

The Correct Answer is Option 1 i.e Fouaad Mirza.

Fouaad Mirza:

• He is an Indian equestrian.
• He won silver medals in both the individual eventing and the team eventing at the 2018 Asian Games.
• He became the first Indian since 1982 to win a medal at an individual event in the sport at the Asian Games.
• He received the Arjuna Award in 2019.
• He became the first Indian equestrian to qualify for Olympics 2020 in the last 20 years.

The Correct Answer is Option 4 i.e IIT Kanpur.

'Industry 4.0':

• ‘Industry 4.0’ commonly referred to as the fourth industrial revolution, is a name given to the current trend of automation, interconnectivity and data exchange in manufacturing technologies to increase productivity.
• Industry 4.0 is a complex Cyber-Physical Systems which synergizes production with digital technologies, the Internet of Things, Artificial Intelligence, Big Data & Analytics, Machine Learning, and Cloud Computing.
• A Pilot Project for ushering in 'Industry 4.0' in the country, is launched for implementation at the Modern Coach Factory (MCF), Raebareli.
• The Ministry of Railways and Department of Science & Technology have joined hands in partnership with IIT Kanpur for taking up a unique project on ‘Industry 4.0’ by launching a Pilot Project for implementation at Modern Coach Factory, Raebareli.

The Correct Answer is  Amitabh Bachchan.

 

Link: https://bit.ly/3lgE1Tt

The Correct Answer is Option 4 i.e celestial bodies.

Celestial bodies:

• The sun, the moon, and all those objects shining in the night sky are called celestial bodies.
• Some celestial bodies are very big and hot. They are made up of gases.
• They have their own heat and light, which they emit in large amounts.
• These celestial bodies are called stars. The sun is a star

Key Points

Planets:

• Some celestial bodies do not have their own heat and light.
• They are lit by the light of the stars. Such bodies are called planets.
• The word ‘planet’ comes from the Greek word “Planetai” which means ‘wanderers’.
• The earth on which we live is a planet.
• There are eight planets in our solar system.
• In order of their distance from the sun, they are Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune.
• It gets all its heat and light from the sun, which is our nearest star.
• If we look at the earth from a great distance, say the moon, it will appear to be shining just as the moon

Additional Information

Satellite:

• A Satellite is a celestial body that moves around the planets in the same way as the planets move around the sun.

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Asteroid:

• Apart from the stars, planets, and satellites, there are numerous tiny bodies that also move around the sun.
• These bodies are called asteroids.
• They are found between the orbits of Mars and Jupiter
• Meteoroids: A meteoroid is a small rock or particle of debris in our solar system and it burns up as it passes through the Earth's atmosphere, is known as a meteor.
• Constellations: Constellation is a group of stars that makes some imaginary shape or pattern in the sky.
• Asteroids: A belt of minor planets revolving around the sun between Mars & Jupiter.
• Celestial bodies: It is any natural body outside of the earth's atmosphere.Examples are the Moon, Sun, and the other planets of our solar system.
• The term night sky refers to the nighttime appearance of celestial objects like stars, planets, and the Moon, which are visible in a clear sky.

Shortcut Trick

For a quadrilateral circumscribed about a circle, the sum of opposite sides is equal.

Formula: AB + CD = BC + AD

Substitute the given values: 8 + 6 = 7 + AD

14 = 7 + AD ⇒ AD = 7 cm

∴ The correct answer is 7 cm.

Alternate Method

Given:

Length of side AB = 8 cm

Length of side BC = 7 cm

Length of side CD = 6 cm

When a circle is inscribed in a quadrilateral, the tangents drawn from an external point to the circle are equal in length.

Let the points of tangency on AB, BC, CD, and DA be P, Q, R, and S respectively.

⇒ AP = AS, BP = BQ, CQ = CR, and DR = DS

Adding these: (AP + BP) + (CR + DR) = (BQ + CQ) + (AS + DS)

⇒ AB + CD = BC + AD

⇒ 8 + 6 = 7 + AD

⇒ 14 = 7 + AD

⇒ AD = 14 − 7

⇒ AD = 7 cm

∴ The correct answer is 7 cm.

Additional Information

Pitot Theorem

It states that in a tangential quadrilateral, the two sums of opposite sides are equal. This is a necessary and sufficient condition for a circle to be inscribed.

Area of Tangential Quadrilateral

The area (A) can be calculated using the inradius (r) and semi-perimeter (s): A = r × s, where s = (a + b + c + d) ÷ 2.

Cyclic Quadrilateral

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

Tangents from External Point

The lengths of tangents drawn from an external point to a circle are always equal.

Shortcut Trick

Apply Componendo and Dividendo on the given equation: (sec θ − tan θ) / (sec θ + tan θ) = 3/5.

⇒ (sec θ) / (−tan θ) = (5 + 3) / (3 − 5) = 8 / −2 = −4 ⇒ sec θ / tan θ = 4 ⇒ 1/sin θ = 4 ⇒ sin θ = 1/4.

Then cosec θ = 4 and cot θ = √(cosec²θ − 1) = √15.

Expression = (4 + √15) / (4 − √15) = (4 + √15)² / (16 − 15) = 16 + 15 + 8√15 = 31 + 8√15.

∴ The correct answer is 31 + 8√15.

Alternate Method

Given: (sec θ − tan θ) / (sec θ + tan θ) = 3/5

Formula: (a/b) = (c/d) ⇒ (a+b)/(a-b) = (c+d)/(c-d)

⇒ By cross multiplication: 5(sec θ − tan θ) = 3(sec θ + tan θ)

⇒ 5 sec θ − 5 tan θ = 3 sec θ + 3 tan θ

⇒ 2 sec θ = 8 tan θ ⇒ sec θ / tan θ = 4

⇒ (1/cos θ) / (sin θ/cos θ) = 4 ⇒ 1/sin θ = 4 ⇒ cosec θ = 4

⇒ sin θ = 1/4, so cos θ = √(1 − (1/4)²) = √15 / 4

⇒ cot θ = cos θ / sin θ = (√15 / 4) / (1/4) = √15

⇒ Required value = (4 + √15) / (4 − √15)

⇒ Rationalizing: [(4 + √15) × (4 + √15)] / [(4 − √15) × (4 + √15)]

⇒ (16 + 15 + 8√15) / (16 − 15) = 31 + 8√15

∴ The correct answer is 31 + 8√15.

Additional Information

Componendo and Dividendo

If a/b = c/d, then (a+b)/(a-b) = (c+d)/(c-d). This is useful for simplifying ratios involving trigonometric functions.

Trigonometric Identities

sec²θ − tan²θ = 1 and cosec²θ − cot²θ = 1. These identities are fundamental for converting between functions.

Rationalization

To remove radicals from the denominator, multiply the numerator and denominator by the conjugate (a + √b → a − √b).

Shortcut Trick

If x + 1/x = k, then x² + 1/x² = k² − 2 and x⁶ + 1/x⁶ = (x² + 1/x²)³ − 3(x² + 1/x²).

From x² + 3x + 1 = 0, dividing by x gives x + 1/x = −3.

Substituting k = −3: x² + 1/x² = (−3)² − 2 = 7.

Now, x⁶ + 1/x⁶ = 7³ − 3(7) = 343 − 21 = 322.

∴ The correct answer is 322.

Alternate Method

Given: The quadratic equation x² + 3x + 1 = 0.

Formula Used: (a + b)² = a² + b² + 2ab and (a + b)³ = a³ + b³ + 3ab(a + b).

Step 1: Simplify the given equation.

⇒ x² + 3x + 1 = 0

Dividing the entire equation by x:

⇒ x + 3 + 1/x = 0

⇒ x + 1/x = −3

Step 2: Find the value of x² + 1/x².

Squaring both sides:

⇒ (x + 1/x)² = (−3)²

⇒ x² + 1/x² + 2(x)(1/x) = 9

⇒ x² + 1/x² + 2 = 9

⇒ x² + 1/x² = 7

Step 3: Find the value of x⁶ + 1/x⁶.

Cubing both sides of the result from Step 2:

⇒ (x² + 1/x²)³ = 7³

⇒ (x²)³ + (1/x²)³ + 3(x²)(1/x²)(x² + 1/x²) = 343

⇒ x⁶ + 1/x⁶ + 3(1)(7) = 343

⇒ x⁶ + 1/x⁶ + 21 = 343

⇒ x⁶ + 1/x⁶ = 343 − 21

⇒ x⁶ + 1/x⁶ = 322

∴ The correct answer is 322.

Additional Information

Algebraic Power Relations

If x + 1/x = n, then x² + 1/x² = n² − 2 and x³ + 1/x³ = n³ − 3n. These can be nested to find higher powers like x⁴ or x⁶.

Sum and Product of Roots

For a quadratic equation ax² + bx + c = 0, the sum of roots is −b/a and the product of roots is c/a.

Expansion Identities

(a + b)³ = a³ + 3a²b + 3ab² + b³ | (a − b)³ = a³ − 3a²b + 3ab² − b³.

Shortcut Trick Test standard values for θ: If sinθ = 1/3, then cos2θ = 1 − 1/9 = 8/9. Substitute in equation: 2(1/3) + 15(8/9) = 2/3 + 40/3 = 42/3 = 14 (Too high). Test sinθ = 3/5 (from 3-4-5 triplet): 2(3/5) + 15(16/25) = 6/5 + 48/5 = 54/5 = 10.8 (Too high). Test sinθ = 1/5: 2(1/5) + 15(24/25) = 2/5 + 72/5 = 74/5 = 14.8 (Too high). Solving the quadratic 15sin2θ − 2sinθ − 8 = 0 gives sinθ = 4/5. For sinθ = 4/5, tanθ = 4/3, cosθ = 3/5, secθ = 5/3 → 4/3 + 3/5 + 5/3 = 3 + 3/5 = 3(3/5). ∴ The correct answer is 3(3)/(5). Alternate Method Given: 2sinθ + 15cos2θ = 7 Formula Used: cos2θ = 1 − sin2θ

⇒ 2sinθ + 15(1 − sin2θ) = 7 ⇒ 2sinθ + 15 − 15sin2θ = 7 ⇒ 15sin2θ − 2sinθ − 8 = 0 ⇒ 15sin2θ − 12sinθ + 10sinθ − 8 = 0 ⇒ 3sinθ(5sinθ − 4) + 2(5sinθ − 4) = 0 ⇒ (3sinθ + 2)(5sinθ − 4) = 0 ⇒ sinθ = 4/5 (since 0° < θ < 90°) ⇒ tanθ = 4/3, cosθ = 3/5, secθ = 5/3 ⇒ tanθ + cosθ + secθ = 4/3 + 3/5 + 5/3 = 9/3 + 3/5 = 3 + 3/5 = 18/5 = 3(3/5) ∴ The correct answer is 3(3)/(5). Additional Information Trigonometric Identities Fundamental identities include sin2θ + cos2θ = 1, 1 + tan2θ = sec2θ, and 1 + cot2θ = cosec2θ. Pythagorean Triplets Common triplets like (3, 4, 5), (5, 12, 13), and (8, 15, 17) help in identifying side ratios quickly when one trigonometric ratio is known. Quadratic Roots in Trig When solving asin2θ + bsinθ + c = 0, always check the range (e.g., −1 to 1 for sine) and the given quadrant.

If x is the mean proportional between 12.8 and 64.8, then

12.8 : x : : x : 64.8

⇒ 12.8/x = x/64.8

⇒ x² = 12.8 × 64.8

⇒ x = √[16 × 0.8 × 0.8 × 81]

⇒ x = 4 × 0.8 × 9

If y is the third proportional to 38.4 and 57.6, then

38.4 : 57.6 : : 57.6 : y

⇒ 38.4/57.6 = 57.6/y

⇒ y = (57.6 × 57.6)/38.4

⇒ y = 86.4

Now,

2x : y = 2 × 4 × 0.8 × 9 : 86.4 = 2 : 3

Short Trick :

Mean Proportional x = √12.8 × 64.8 = 28.8

Third proportional y = (57.6 × 57.6)/38.4 = 86.4

Now,

2x : y = 2 × 28.8 : 86.4 = 2 : 3

Shortcut Trick

Efficiency ratio A : B : C = 100 : 125 : (125 × 1.28) = 100 : 125 : 160 = 20 : 25 : 32.

Total Work = Efficiency of A × Days = 20 × 40 = 800 units.

5 days work of (A+B+C) = 5 × (20 + 25 + 32) = 5 × 77 = 385 units.

Remaining work = 800 − 385 = 415 units.

Time taken by B alone = 415 ÷ 25 = 16.6 days = 16(3/5) days.

∴ The correct answer is 16(3/5) days.

Alternate Method

Given: A completes work in 40 days. Efficiency of B is 25% more than A. Efficiency of C is 28% more than B.

⇒ Efficiency of A = 20 (assumed to make B = 25 and C = 32)

⇒ Total Work = 40 × 20 = 800 units

⇒ Combined Efficiency (A+B+C) = 20 + 25 + 32 = 77 units/day

⇒ Work done in 5 days = 77 × 5 = 385 units

⇒ Remaining Work = 800 − 385 = 415 units

⇒ Time taken by B = Remaining Work ÷ Efficiency of B

⇒ Time = 415 ÷ 25 = 83 ÷ 5 = 16(3/5) days

∴ The correct answer is 16(3/5) days.

Additional Information

Efficiency and Time Relationship

Efficiency is inversely proportional to time taken. If A is x% more efficient than B, Efficiency Ratio A:B = (100+x):100.

Total Work Formula

Total Work = Efficiency × Time. This remains constant for a specific piece of work regardless of the number of people working.

Work Done by Group

When people work together, their efficiencies are additive. Combined Efficiency = E₁ + E₂ + ... + E_n.

Given:

If x4 + x2y2 + y4 = 21 and x2 + xy + y2 = 7

Formula used:

x4 + x2y2 + y4 = (x2 – xy + y2) (x2 + xy + y2)

Calculation:

x⁴ + x²y² + y⁴ = 21 and

x² + xy + y² = 7      --- (1)

x4 + x2y2 + y4 = (x2 – xy + y2) (x2 + xy + y2)

⇒ 21 = (x² - xy + y²) × 7

⇒ (x² - xy + y²) = 21/7

⇒ (x² - xy + y²) = 3      --- (2)

equation (1) from equation (2)

2xy = 4

⇒ xy = 2

⇒ (xy)² = 4

From equation (1)

x² + y² = 7 – 2

⇒ x² + y² = 5

Now,

(1/x² + 1/y²)

⇒ (x² + y²)/(xy)²

⇒ 5/4

∴ The value will be 5/4.

Shortcut Trick

Let Cost Price (CP) be 100.

Marked Price (MP) = 100 × 1.40 = 140.

Successive discounts of 20% and 25% result in a net multiplier of 0.80 × 0.75 = 0.6.

Selling Price (SP) = 140 × 0.6 = 84.

Loss = CP − SP = 100 − 84 = 16 units. Given 16 units = Rs. 140.

CP = (140 ÷ 16) × 100 = 8.75 × 100 = Rs. 875.

∴ The correct answer is Rs. 875.

Alternate Method

Given: Markup = 40%, Discounts = 20% and 25%, Total Loss = Rs. 140.

Formulas: MP = CP × (100 + Markup%) ÷ 100 | Net Discount% = a + b − (ab ÷ 100) | SP = MP × (100 − D%) ÷ 100

⇒ Effective Discount = 20 + 25 − (20 × 25 ÷ 100) = 45 − 5 = 40%.

⇒ Let Cost Price be x. Then Marked Price = 1.4x.

⇒ Selling Price = 1.4x × (1 − 0.40) = 1.4x × 0.6 = 0.84x.

⇒ Loss = CP − SP ⇒ 140 = x − 0.84x.

⇒ 140 = 0.16x ⇒ x = 140 ÷ 0.16 = 875.

∴ The correct answer is Rs. 875.

Additional Information

Marked Price & Discount

Marked Price is the price listed on the product. Discount is always calculated on the Marked Price to arrive at the Selling Price.

Successive Discounts

When two discounts d₁ and d₂ are given, the equivalent single discount is [d₁ + d₂ − (d₁ × d₂ ÷ 100)]%.

From the following figure

∠SQP = ∠OQP = ∠SRO

OQ = OS (Given)

∠QPR = ∠QSR = ∠OSR 

∴ We can say that,

ΔPOQ∼ΔSOR

PQ/SR = OQ/OR

OQ/OR = x/14

Given, OQ = OS

OS/OR = x/14     --- (1)

As we know,

ΔPOS∼ΔQOR

OS/OR = PS/QR

OS/OR = 25.2/16.8      --- (2)

From equation (1) and equation (2)

x/14 = 25.2/16.8

x = (25.2 × 14)/16.8

∴ x = 21 cm

.

Concept used:

Length of perpendicular bisector = length of median, AD = √3/2 × a

Calculation:

⇒ 4√3 = √3/2 × a

⇒ a = 4√3 × 2/√3

⇒ a = 8 cm

So perimeter of triangle = 8 × 3 = 24 cm

Given:

We have to calculate the value of, cosec (85° + θ) – sec(5° – θ) – tan (55° + θ) + cot(35° – θ)

Calculation:

cosec (85° + θ) – sec(5° – θ) – tan (55° + θ) + cot(35° – θ)

= cosec (85 + θ) – cosec [90 – (5 – θ)] – tan (55 + θ) + tan [90 – (35 – θ)]

= cosec (85 + θ) – cosec (85 + θ) – tan (55 + θ) + tan (55 + θ)

= 0

Given:

$\left( {18 \div 2;of\frac{1}{4}} \right); \times ;\left( {\frac{2}{3} \div \frac{3}{4}; \times ;\frac{5}{8}} \right) \div \left( {\frac{2}{3} \div \frac{3}{4}of\frac{3}{4}} \right)$

Concept Used:

$\Rightarrow \left( {18 \div 2;of\frac{1}{4}} \right); \times ;\left( {\frac{2}{3} \div \frac{3}{4}; \times ;\frac{5}{8}} \right) \div \left( {\frac{2}{3} \div \frac{3}{4}of\frac{3}{4}} \right)$

$\Rightarrow \left( {18 \div 2 \times \frac{1}{4}} \right); \times ;\left( {\frac{2}{3}{\rm{;}} \times {\rm{;}}\frac{4}{3}; \times ;\frac{5}{8}} \right) \div \left( {\frac{2}{3} \div \frac{3}{{4}}} \times \frac{3}{4} \right)$

$\Rightarrow \left( {18 \div \frac{1}{2}} \right); \times ;\left( {\frac{5}{9}} \right) \div \left( {\frac{2}{3} \div \frac{9}{{16}}} \right)$

$\Rightarrow \left( {18 \times 2} \right); \times ;\left( {\frac{5}{9}} \right) \div \left( {\frac{2}{3} \times \frac{16}{{9}}} \right)$

$\Rightarrow \left( {18 \times 2} \right); \times ;\left( {\frac{5}{9}} \right) \div \left( {\frac{32}{27} {}{{}}} \right)$

$\Rightarrow \left( {36} \right); \times ;\left( {\frac{5}{9}} \right) \times \left( {\frac{27}{{32}}} \right)$

$\Rightarrow \left( {\frac{135}{8}} \right)$

$\Rightarrow {\rm{;}}16\frac{7}{8}$

Let the speed of B be 5 unit

Speed of A = 5 × (6/5) = 6 unit

Time taken by A after meeting = 5/2 hrs

Time taken by B after meeting = x hrs

As we know,

(Speed of A/Speed of B)² = Time taken by B/Time taken by A

⇒ (6/5)² = x/(5/2)

⇒ 36/25 = 2x/5

⇒ x = 18/5

⇒ x = $3\frac{3}{5}$

GIven:

(a + b – c)³ + (a – b + c)³ – 8a³

Concept used:

Trial and error

Calculation:

Let us use the trial and error method to obtain the solution.

Let a = b = c = 1

Then,

(a + b – c)3 + (a – b + c)3 – 8a3 = 1 + 1 – 8 = (-6)

Now, lets check option (1):

3a(a + b – c)(a – b + c) = 3 × (1) × (1) ≠ (-6)

Now, lets check option (2):

6a(a + b – c) (a – b + c) = 6 × (1) × (1) ≠ (-6)

Checking option 3^rd.

6a(a – b + c) (c – a – b) = 6 × 1 × (-1) = (-6)

Hence, option 3 is right.

∴ The solution of the question is 6a(a – b + c) (c – a – b). 

Alternate Method(a + b - c)³ + (a - b + c)³ - 8a³

⇒ (a + b - c)3 + (a - b + c)3 - (2a)3

Let (a + b - c) = x, (a - b + c) = y, -2a = z

x + y + z = a + b - c + a - b + c - 2a

x + y + z = 0

We know that,

If, x + y + z = 0, then x³ + y³ + z³ = 3xyz

(a + b - c)³ + (a - b + c)³ + (-2a)³ = 3(a + b - c) (a - b + c) (-2a)

⇒ -6a (a + b - c) (a - b + c)

⇒ 6a(-a - b + c)(a - b + c)

⇒ 6a (a - b + c) (c - a - b)

Shortcut Trick

Let initial cost price be 100.

Initial Selling Price = 100 × 1.25 = 125.

New Cost Price = 100 × 1.20 = 120.

New Selling Price = 125 × 1.10 = 137.5.

New Profit = 137.5 − 120 = 17.5.

Profit% = (17.5 ÷ 120) × 100 ≈ 14.6%.

∴ The correct answer is 14.6%.

Alternate Method

Given: Initial markup = 25%, CP increase = 20%, SP increase = 10%.

Formulas: Profit = SP − CP | Profit% = (Profit ÷ CP) × 100

⇒ Let Cost Price (CP₁) = 100.

⇒ Selling Price (SP₁) = 100 + 25% of 100 = 125.

⇒ New Cost Price (CP₂) = 100 + 20% of 100 = 120.

⇒ New Selling Price (SP₂) = 125 + 10% of 125 = 125 + 12.5 = 137.5.

⇒ Profit = 137.5 − 120 = 17.5.

⇒ Profit Percentage = (17.5 ÷ 120) × 100 = 1750 ÷ 120 = 14.583%.

⇒ Rounding to one decimal place, we get 14.6%.

∴ The correct answer is 14.6%.

Additional Information

Base Value for Profit

Profit is always calculated on the actual cost price unless specified otherwise. Formula: Profit% = (Profit ÷ CP) × 100.

Markup and Selling Price

Marked Price is the price at which SP is fixed initially before any subsequent changes or discounts.

Formula Used:

When two Δ are similar,

Area of Δ 1/Area of Δ 2 = (side of Δ 1)²/(side of Δ 2)² 

Calculation:

(area of ΔABC) : (are of ΔDEF) = (AL)² : (DM)²

(AL)² : (DM)² = 9 : 25

AL/DM = 3/5

Let AL = 3 and DM = 5

∴ (DM + AL)/(DM – AL) = (5 + 3)/(5 – 3) = 8/2 = 4.

Shortcut Trick

For the first 2 years, the effective Compound Interest rate is 10 + 10 + (10 × 10)/100 = 21%.

For the fractional 1/3 year, the rate is 10 × 1/3 = 10/3%.

Total effective rate = 21 + 10/3 + (21 × 10/3)/100 = 21 + 3.333 + 0.7 = 25.033...% (or exactly 751/30%).

Sum (P) = (CI × 100) / Effective Rate = (1201.60 × 3000) / 751 = Rs. 4,800.

∴ The correct answer is Rs. 4,800.

Alternate Method

Given: Rate (R) = 10%, Time (T) = 2 1/3 years, Compound Interest (CI) = Rs. 1,201.60.

Formula Used: Amount (A) = P(1 + R/100)ⁿ × [1 + (Fractional Time × R)/100]

⇒ Amount (A) = P(1 + 10/100)² × [1 + (1/3 × 10)/100]

⇒ A = P × (1.1)² × (1 + 1/30)

⇒ A = P × 1.21 × (31/30)

⇒ A = P × (121/100) × (31/30) = P × (3751/3000)

⇒ CI = A − P = P(3751/3000 − 1) = P(751/3000)

⇒ 1201.60 = 751P / 3000

⇒ P = (1201.60 × 3000) / 751

⇒ P = 1.6 × 3000 = 4800

∴ The correct answer is Rs. 4,800.

Additional Information

Effective Rate of CI

For two successive rates x% and y%, the combined rate is x + y + (xy/100)%.

CI for Fractional Years

When time is T = n + a/b years, Amount = P(1 + R/100)ⁿ[1 + (a/b × R)/100].

Difference between CI and SI

For 2 years, CI − SI = PR² / 100². For 3 years, CI − SI = [PR²(300 + R)] / 100³.

When 200 is divided by a positive integer x, the remainder is 8. So,

200 – 8 = 192

As we know,

192 is divisible by (more than 8) 12, 16, 24, 32, 48, 64, 96 and 192.

So, we can say x = 8.

Mistake point: 

We can't consider x = 1, 2, 3, 4, 6, and 8 as remainder is 8. So, x > 8.

Let the fifth number be x, then

Average of first four number is = 3x

According to the question

4 × 3x + x = 85.8 × 5

⇒ 12x + x = 429

⇒ 13x = 429

⇒ x = 429/13

∴ x = 33

Let internal and external radius of the track be r m and R m respectively.

Internal circumference of the track = 2πr

2 πr = 440

⇒ 2 × (22/7) × r = 440

⇒ r = 70 m

External circumference of the track = 2πR

2 πR = 506

⇒ 2 × (22/7) × R = 506

⇒ R = 506 × (7/44)

⇒ R = 80.5 m

Area of track = π (R² – r²) = (22/7) × (80.5² – 70²) = 22/7 × 150.5 × 10.5 = 4966.5 m²

Cost of levelling 1 m² = Rs. 6

∴ Cost of levelling 4966.5 m² = 4966.5 × 6 = Rs. 29,799

$33;\frac{1}{3}\% ; = ;33; + ;\frac{1}{3}$

As the capacity of both vessels is the same the average percentage of orange juice, in the new vessel = (35 + 33 + 1/3)/2 = 34 + 1/6 = $34\frac{1}{6}\%$

Percentage of apple juice, in new vessel = 100% - $34\frac{1}{6}\% ; = ;65\frac{5}{6}\%$

Number of engineers recruited by companies A and B in 2015 and 2018 = 132 + 118 + 148 + 112 = 510

Number of engineers recruited by companies C and D in 2014 and 2018 = 85 + 105 + 105 + 125 = 420

Required ratio = 510 : 420 = 17 : 14

Number of engineers recruited by company A in 2014 to 2017 = 120 + 132 + 128 + 140 = 520

Number of engineers recruited by all four companies in 2019 = 150 + 118 + 110 + 122 = 500

Required percentage = [(520 – 500)/500] × 100 = 4%

Total number of engineers recruited by company B = 90 + 118 + 98 + 106 + 112 + 118 = 642

Average number of engineers recruited by company B = 642/6 = 107

There are three years 2014, 2015 and 2016 the number of engineers recruited by company D is less than the average number of engineers recruited by B in the given six years.

Number of engineers recruited by company B in 2014 and 2017 = 90 + 106 = 196

Number of engineers recruited by company C during 2015 to 2019 = 93 + 94 + 98 + 105 + 110 = 500

Required percentage = 196/500 × 100 = 39.2%

The correct answer is option 3 i.e. auspicious

The sentence refers to the beginning of a new business, this should ideally begin on a good occasion. Option 1 'occasional' refers to something that happens once in a while and thus cannot be a day to start a business. Option 2 is incorrect because and an 'ominous' day is a bad day to start a business. 'Audacious' refers to a person that is bold and cannot be applied to a day. So we left with auspicious which means showing or suggesting that future success is likely: PROPITIOUS

Thus, the correct answer is option 3.

The correct answer is option "2"

• The original sentence refers to the modern man acquiring more wealth in order to invest in pleasures, thus the term 'pleasures' acting as a noun should have an adjective before that describes it. This would be the adjective form of the verb 'sense', thus it would be 'sensory'.
• Correct Sentence :- The modern man is busy acquiring more and more wealth and designing ways to invest it in more sensory pleasures.

The correct answer is option 1 i.e. He asked me what I was doing.

• We make changes in tenses of reported speech only when the reporting verb is in the past tense.
• When the reported speech is an interrogative sentence we follow these steps-
• Reporting verb 'said' is changed into 'asked'.
• Change the sentence, given in the reported speech, from interrogative to assertive.
• Tense of reported speech is changed accordingly. ( in above sentence- 'are' changes into 'was')
• personal Pronouns of reported speech are changed as per subject and object of reporting verb. Here, the second person- 'You' will change into 'I' as per object - 'me' (hidden).
• By following these steps we get the final sentence - " He asked me what I was doing*."*

Thus, the correct answer is option 1.

The correct answer is option 2 i.e. degradation in values

Key Points

• In the given question 'degrade' is used as a verb which makes the sentence grammatically incorrect.
• The original sentence refers to a loss in values (degrade) but the verb 'degrade' needs to be replaced with the noun form 'degradation'.

Thus, the correct answer is option 2.

The correct answer is option 2 i.e. pitiful

Pathetic: Refers to someone or something that causes pity,  having a capacity to move one to either compassionate or contemptuous pity.

Pitiful:  exciting pitying contempt (as by meanness or inadequacy)

Thus, it is clear that pathetic and pitiful are synonymous words

Let's look at the meaning of other words:

Curious: Refers to someone who is eager to learn something

Insignificant: Refers to someone who is unimportant 

Dull: Refers to something that lacks shine or interest

Thus, the correct answer is option 2.

The correct answer is option 1 i.e. sad

Hilarious: Refers to someone/something that is funny and happy

Sad: Refers to someone or something that is unhappy

Thus, it is clear that hilarious and sad are opposite words

Let's look at the meaning of other words:

Blithe: Refers to casual indifference

Happy: Refers to someone who is content and pleased

Merry: Refers to someone who is happy and excited

Thus, the correct answer is option 1.

The correct answer is option 4 i.e. dependent

Key Points

The passage refers to machines and their relation with man, man cannot grow 'helpless on' machines - this would be the wrong preposition. one cannot be 'inferior on' someone, one is inferior 'to' someone. One can also not be 'subordinate on' someone, they are subordinate 'to' someone.

Thus, the correct answer is option 4. 

Additional Information 

• Helpless:- unable to defend oneself or to act without help.
• Inferior:- lower in rank, status, or quality.
• Subordinate:- lower in rank or position

The correct answer is option 3 i.e. 'Masters'

Key Points

• The context of the sentence suggests that machines, which were made to be servants of man, are now in a position of control or authority, hence the term "masters" fits appropriately.
• The term "masters" is used to convey the idea that machines now have a dominating role over humans, who have become heavily dependent on them.
• Thus, the correct answer is option 3.

The correct answer is option 4 i.e. 'upon'

Key Points

One cannot wait 'under' machines, one cannot wait 'into' machines and one cannot wait 'from' machines. Thus, one waits 'on' or 'upon' machines

Wait on/upon means to serve.

Thus, the correct answer is option 4.

The correct answer is option 4 i.e. 'given'

Key Points

Here the sentence is in the passive voice and we need the third form of the verb 'give', so 'given' is the only correct choice here. 

Thus, the correct answer is option 4

The correct answer is option 4 i.e. 'optimum'

Key Points

Here we need an adjective that will qualify our abstract noun temperature, as per the context of the passage we need a word which will mean an ideal or suitable temperature. Now let's look at the meaning of all the options.

• Optimum: Refers to something that is ideal, something that is best
• Optimist: a person who is inclined to be hopeful and to expect good outcomes: someone who is given to optimism
• Optional:  involving an option: not compulsory.
• Outdated: no longer current : OUTMODED.

After going through the meaning of all the options, 'optimum' is the best choice.

Thus, the correct answer is option 4.

The correct answer is option 3 i.e. emigrant

An 'emigrant' refers to someone who leaves their own country to settle in another

Let's look at the meaning of other words:

Native: Refers to someone who is born in a certain country and belongs to that country

Foreigner: Refers to someone who comes from a country that is not one's own

Tourist: Refers to someone who goes to another for travel 

Thus, the correct answer is option 3.