RRB Group D Previous Year Paper (Held on: 27 Sept 2022 Shift 3)

The logic followed here is:

Thus, the number which will replace question mark in the series is 10.4.

Hence, the correct answer is Option (1).

Shortcut Trick

Area of a rhombus = 1/2 × Product of diagonals

Area = 1/2 × 24.6 × 28.6

Area = 12.3 × 28.6 = 351.78 cm²

∴ The correct answer is 351.78 cm².

Alternate Method

Given:

Diagonal 1 (d₁) = 24.6 cm

Diagonal 2 (d₂) = 28.6 cm

Formula Used:

Area of Rhombus = (d₁ × d₂) ÷ 2

Calculations:

⇒ Area = (24.6 × 28.6) / 2

⇒ Area = 703.56 / 2

⇒ Area = 351.78 cm²

∴ The correct answer is 351.78 cm².

Additional Information

Properties of Rhombus

All four sides are equal and diagonals bisect each other at right angles (90°).

Perimeter of Rhombus

Perimeter = 4 × side. Side can be found using diagonals: side = 1/2 × √(d₁² + d₂²).

Alternative Area Formula

Area = Base × Height, where base is any side and height is the perpendicular distance between opposite sides.

Shortcut Trick

Units digit = 6

Tens digit = 6 ÷ 3 = 2

Original number = 26

Required value = 26 × 2 = 52

∴ The correct answer is 52.

Alternate Method

Given:

Units digit (y) = 6

Tens digit (x) = y ÷ 3

Calculation:

⇒ x = 6 ÷ 3 = 2

⇒ Original number = 10x + y = 10(2) + 6 = 26

⇒ Twice the number = 2 × 26 = 52

∴ The correct answer is 52.

Additional Information

Standard Form of a 2-Digit Number

A 2-digit number with tens digit x and units digit y is represented as 10x + y.

Interchanging Digits

When digits are reversed, the new number is 10y + x. The difference between the original and reversed number is always divisible by 9.

Sum of Digits

The sum of the digits is simply x + y. The original number minus the sum of its digits is always a multiple of 9.

Shortcut Trick

Total Frequency (Σf) = 3 + 5 + 6 + 2 + 1 + 3 = 20

Sum of products (Σfx) = (5×3) + (10×5) + (15×6) + (20×2) + (25×1) + (30×3)

Σfx = 15 + 50 + 90 + 40 + 25 + 90 = 310

Arithmetic Mean = Σfx ÷ Σf = 310 ÷ 20 = 15.5

∴ The correct answer is 15.5.

Alternate Method

Given:

Observations (x) = {5, 10, 15, 20, 25, 30}

Frequencies (f) = {3, 5, 6, 2, 1, 3}

Formula Used:

Arithmetic Mean (x̄) = Σfx ÷ Σf

Calculations:

⇒ Σf = 3 + 5 + 6 + 2 + 1 + 3 = 20

⇒ Σfx = (5 × 3) + (10 × 5) + (15 × 6) + (20 × 2) + (25 × 1) + (30 × 3)

⇒ Σfx = 15 + 50 + 90 + 40 + 25 + 90

⇒ Σfx = 310

⇒ x̄ = 310 ÷ 20

⇒ x̄ = 15.5

∴ The correct answer is 15.5.

Additional Information

Arithmetic Mean for Discrete Series

The mean is calculated by dividing the sum of the products of values and their frequencies (Σfx) by the total number of observations (Σf).

Assumed Mean Method

Mean = A + (Σfd ÷ Σf), where A is the assumed mean and d = x − A. This simplifies calculations for larger datasets.

Properties of Mean

The algebraic sum of the deviations of a set of observations from their arithmetic mean is always zero, i.e., Σ(x − x̄) = 0.

The correct answer is " copper (ll) chloride ".

Key Points

• When copper oxide is added to dilute hydrochloric acid it results in the formation of copper chloride.
• The soluble copper chloride is blue-green in color and changes the color of the solution to blue-green.
• It is a double displacement reaction.

The correct answer is " ZnO" .

Key Points

• Reduction is also termed as the removal of oxygen.
• In the given reaction, ZnO looses an oxygen atom and is getting reduced to Zn.
• This is a displacement reaction.

Shortcut Trick

The given mathematical expression directly matches an algebraic identity.

x² + y² + z² − 2xy − 2xz + 2yz = (x − y − z)²

Substitute the given variable values into the term (x − y − z):

⇒ (b + c) − (c − a) − (a − b) = b + c − c + a − a + b = 2b

⇒ (2b)² = 4b²

∴ The correct answer is 4b².

Alternate Method

Given:

⇒ x = b + c

⇒ y = c − a

⇒ z = a − b

Formula Used:

⇒ (p + q + r)² = p² + q² + r² + 2pq + 2qr + 2pr

Calculation:

⇒ Let us evaluate the given expression: x² + y² + z² − 2xy − 2xz + 2yz

⇒ Comparing this with our standard formula, we take p = x, q = −y, and r = −z.

⇒ This simplifies our original expression cleanly to: (x − y − z)²

⇒ Next, we substitute the given expressions for x, y, and z into (x − y − z):

⇒ x − y − z = (b + c) − (c − a) − (a − b)

⇒ Distribute the negative signs and open the brackets:

⇒ x − y − z = b + c − c + a − a + b

⇒ Cancel out the opposite terms (+c − c) and (+a − a) and group together:

⇒ x − y − z = 2b

⇒ Finally, substitute this result back into the squared identity expression:

⇒ (x − y − z)² = (2b)² = 4b²

∴ The correct answer is 4b².

Additional Information

Trinomial Expansion Identity

The fundamental identity for the square of a sum of three variables is represented as (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca.

Sign Variations in Expansion

By assigning negative signs to the variables, we derive adapted identities depending on the requirements. For instance, (a − b − c)² = a² + b² + c² − 2ab + 2bc − 2ca.

Important Cubic Identity

a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca). Notably, if a + b + c = 0, the equation cleanly reduces to a single relation: a³ + b³ + c³ = 3abc.

The correct answer is " one-half of the revolution of the coil ".

Key Points

• The direction of the induced current in a square-shaped coil changes once per half revolution of the coil.
• The direction of the current in the coil remains unchanged.
• The current induced by the magnetic field composed of copper wire is rotated in order to rotate the magnetic field.
• At each turn, the direction of the induced current changes, and because it is a vector quantity**, it changes every half revolution.**
• Because the direction of the relative motion of the coil and the magnetic field changes every half cycle, the direction of the induced current changes every half rotation.

The correct answer is "Around 110".

Key Points

• The F1 generation is the first one. 100% yellow, then yellow(Y) is dominant trait and green(y) is recessive trait.
• F2 generation results from the crossing of F1 individuals (Yy).

      

• We receive 3 yellow plants and 1 green plant in the F2 generation.
• Therefore, there are three yellow and one green in this F2 generation.
• We pronounce it as 3:1.
• So, No. of green plants in pea plants are, $500\times{1\over4}=125$
• As the above value is an experimental one. So, around 110 pea plants are green in actuality.

The correct answer is 'Tropical Deciduous Forests'.

Key Points

• The most widespread forests  in india is Tropical Deciduous Forests.
• This forests are called Moonson Forest.
• There are two types of Deciduous Forests: Moist Deciduous Forests - Mixture of Trees and Grasses ; 2. Dry Deciduous Forests
• As the name goes, tropical deciduous forests are found along the Equator belt between the tropic of Cancer and tropic of Capricorn.

The correct answer is " Slaked lime ".

Key Points

• Bleaching powder is a pale yellowish powder existing with a strong smell of chlorine.
• It is synthesized by the action of chlorine gas on dry slaked lime ( Ca(OH)2).
• Ca(OH)₂ + Cl₂ → Ca(OCl2)+ H₂O

The correct answer is 'Mirabai Chanu'.

Key Points

• Weightlifter Mirabai Chanu won the BBC Indian Sportswoman of the Year award for 2021 on 29 March 2022.
• Chanu became the first Indian weightlifter to win a silver medal at an Olympic Games when she finished second in the 49kg category in Tokyo.
• Chanu won the gold medal in the 48kg division at the 2017 World Championships in Anaheim and followed up with Commonwealth Games gold in 2018.

Additional Information

• Saina Nehwal is related to Badminton.
• Saina Nehwal is an Indian professional badminton player.
• She won the Bronze medal in Olympic Games in London in 2012.
• She became the first Indian to win two singles gold medals in Commonwealth Games (2010 and 2018).
• She is honoured with awards:
• Arjuna Award in 2009
• Major Dhyan Chand Khel Ratna in 2010
• Padma Shri in 2010
• Padma Bhushan in 2016
• Radha Prakash Yadav is an Indian cricketer. She plays for Mumbai, Baroda and West Zone.
• PV Sindhu
• The full name of her is Pusarla Venkata Sindhu .She is a world-famous Indian female badminton player.
• PV Sindhu  won two consecutive medals at the Olympics- silver in the 2016 Rio Games and bronze at the 2020 Tokyo Olympics.
• In the year 2021 she won a  a silver at the Badminton World Federation (BWF).
• She was also the winner of the Syed Modi International Badminton title in January 2022.
• She became the first Indian to win a historic Gold at the Badminton World Championships in 2019.
• She also won the inaugural BBC Indian Sportswoman of the year award in 2019 following a public vote.

The correct answer is Both the gene for blood type A and the gene for blood type B are dominant.

Key Points

• The gene for blood group A is dominant and the gene for blood group B is recessive.
• This means that if a person has one gene for blood group A and one gene for blood group B, they will express the A phenotype because the A gene is dominant.
• The child having blood type AB indicates that they inherited one gene for A from the father and one gene for B from the mother.
• The A and B genes are codominant, which means that when they are both present, they both express themselves equally.
• If both the gene for blood type A and the gene for blood type B are inactive, then the person would have blood type O, which is the absence of A and B antigens on the surface of red blood cells.
• If both the gene for blood type A and the gene for blood type B are dominant, then a person with that genotype would have blood type AB, which is the codominant expression of both antigens.

Additional Information

• Blood type is determined by the presence or absence of specific antigens on the surface of red blood cells.
• The ABO blood group system is based on the presence or absence of two antigens: A and B.
• A person with blood type A has the A antigen on their red blood cells, while a person with blood type B has the B antigen.
• A person with blood type AB has both A and B antigens, while a person with blood type O has neither A nor B antigens.
• The Rh factor is another antigen that can be present on red blood cells, and its presence or absence determines whether a person is Rh positive or Rh negative.

Diagram is drawn following the given conditions:

1. Rehana is to the left of Arjun, who is seated exactly in front of Somil. 

   

2. Arjun exchanges his position with Somil.

   

After Arjun exchanges his position with Somil, Rehana is now in North-West direction of Arjun.

Hence, the correct answer is Option (3).

Diagram is drawn as per the given statements:

1. Bharti starts from point T and drives 7 km towards north.
2. Then she turns right, drives 4 km, turns right and drives 10 km.
3. She then turns right and walks 6 km.
4. She takes a final right turn, drives 3 km and stops at point E.

​ 

Bharti should drive 2 km in East direction to reach point T from point E.

Hence, the correct answer is Option (2).

Students who can speak all three languages i.e. Sanskrit, Hindi and English are highlighted below:

Thus, there are 17 such students.

Hence, the correct answer is Option (2).

The correct answer is Avoiding meat and animal products.

Key Points

• Avoiding meat and animal products would have the greatest impact on our environment because:
• Animal agriculture contributes significantly to greenhouse gas emissions, deforestation, and water pollution.
• It takes much more resources, such as land, water, and energy, to produce animal products compared to plant-based foods.
• By avoiding meat and animal products, we can reduce the demand for factory farming and encourage more sustainable and ethical farming practices.
• Plant-based diets have been shown to be healthier for humans and can also reduce the risk of chronic diseases.

Additional Information

• Stopping food wastage:
• This is also an important step to reduce the environmental impact of our food system.
• Food waste not only wastes resources and contributes to greenhouse gas emissions, but it also means that people are going hungry while food is being thrown away.
• However, it may not have as significant an impact as avoiding meat and animal products.
• Avoiding processed foods:
• ​Processed foods often contain unhealthy additives and preservatives, and they also require more resources to produce and package.
• However, this option may not have as significant an impact on the environment as avoiding meat and animal products.
• Growing your own food:
• ​This can be a great way to reduce the environmental impact of our food system, as it reduces the need for transportation and packaging.
• However, it may not be feasible for everyone and may not have as significant an impact as avoiding meat and animal products.

The pattern followed here is:

Thus, the pattern which logically completes the series is ZUH 35.

Hence, the correct answer is Option (4).

Shortcut Trick

For an investment to double itself under Simple Interest, the total interest earned must be equal to 100% of the Principal.

Shortcut formula for time period ($T$) to double: $T = 100 ÷ R$

Given Rate ($R$) = 4% per annum.

⇒ $T = 100 ÷ 4 = 25$ years.

∴ The correct answer is 25 years.

Alternate Method

Given:

⇒ Rate of Interest ($R$) = 4% p.a.

⇒ Let Principal ($P$) = $x$

⇒ Amount ($A$) = $2x$ (since the investment doubles)

Formula Used:

⇒ Simple Interest ($SI$) = Amount ($A$) − Principal ($P$)

⇒ $SI = (P \times R \times T) ÷ 100$

Calculations:

⇒ $SI = 2x − x = x$

⇒ $x = (x \times 4 \times T) ÷ 100$

⇒ $1 = (4 \times T) ÷ 100$

⇒ $4 \times T = 100$

⇒ $T = 100 ÷ 4$

⇒ $T = 25$ years.

∴ The correct answer is 25 years.

Additional Information

Tripling an Investment (Simple Interest)

To triple an investment ($A = 3P$), the interest earned must be $200\%$ of the principal. The time required is $T = 200 ÷ R$.

Rule of 72 (Compound Interest)

In Compound Interest, the approximate time to double an investment is found by dividing 72 by the annual interest rate ($T ≈ 72 ÷ R$).

Simple Interest Formula

The basic formula for Simple Interest is $I = P \times R \times T$, where $R$ is expressed as a decimal or fraction ($R/100$).

Decoding the information:

 

Given: 

3 ÷ 8 × 2 + 1 - 5

After interchanging: 

3 - 8 ÷ 2 × 1 + 5

= 3 - 4 × 1 +5

= 3 - 4 + 5

= 8 - 4

= 4

Hence, the correct answer is "4".

The correct answer is Stomata.

Key Points

• The pores present on the leaves of plants is stomota.
• Stomata are mostly found in green aerial parts of plants, such as stem and particularly on the leaves.
• Stomata play a very important role as it provides direct pathways between leaves and the air.
• Carbon dioxide enters through the stomata while water and oxygen exit through it.

Additional Information Some of the main functions of stomata are as follows:

 i. It opens and closes the pores in the leaves for an exchange of gases.

 ii. It allows the plant to take in carbon dioxide and give out oxygen for photosynthesis.

 Iii. Depending on the weather conditions, it closes or opens its pores to keep the moisture content developed.

Given: (Left) 72238 83745 52227 72253 63776 (Right)

Now, 1 is subtracted from the first digit of each number,

 

Now, the first digit numbers that will completely divisible with 3 is: 62238 and 62253.

So, there are 'Two' numbers that have its first digit is fully divisible by 3.

Hence, the correct answer is "Option 4".

Statement: G ≥ B > D > P < H = C

Conclusions:

1. P < C → True (G ≥ B > D > P < H = C, As H is greater than P and H is equal to C, thus C is also greater than P.)

2. G > P → True (G ≥ B > D > P < H = C, G, B and D all are greater than P.)

3. D < C → False (G ≥ B > D > P < H = C, There is no definite relationship between D and C, so we cannot say D < C.)

4. B > P → True (G ≥ B > D > P < H = C, G, B and D all are greater than P.)

Three out of four conclusions are correct. Only conclusion 3 is incorrect.

Hence, the correct answer is Option (3).

Such symbols are there which are immediately preceded by a letter and also immediately followed by a number are shown below:

Letters → Symbols → Number

Thus, there are 5 such patterns:

1. Z $ 6
2. M @ 4
3. X & 7
4. L # 3
5. Z * 5

 

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

Shortcut Trick

Check the sum of the base terms: (y − z) + (z − x) + (x − y) = 0.

When the sum of three terms is zero, then the sum of their cubes is equal to three times the product of those terms.

Result = 3 × (y − z) × (z − x) × (x − y).

∴ The correct answer is 3(x − y)(y − z)(z − x).

Alternate Method

Given:

(y − z)³ + (z − x)³ + (x − y)³

Formula Used:

The algebraic identity: a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca)

If a + b + c = 0, then a³ + b³ + c³ = 3abc.

Calculations:

⇒ Let a = (y − z), b = (z − x), and c = (x − y). ⇒ Calculate the sum: a + b + c = (y − z) + (z − x) + (x − y). ⇒ a + b + c = y − z + z − x + x − y = 0. ⇒ Since a + b + c = 0, then a³ + b³ + c³ = 3abc. ⇒ (y − z)³ + (z − x)³ + (x − y)³ = 3(y − z)(z − x)(x − y).

∴ The correct answer is 3(x − y)(y − z)(z − x).

Additional Information

Sum of Cubes Identity

The general identity is a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca). It is vital for factoring cubic expressions.

Special Case: a + b + c = 0

When the sum of the variables is zero, the identity simplifies to a³ + b³ + c³ = 3abc, which is frequently tested in banking and standard exams.

Standard Cube Expansion

The expansion of (x − y)³ is x³ − 3x²y + 3xy² − y³ or x³ − y³ − 3xy(x − y).

The correct answer is transpiration.

Key Points

• Plants produce carbon dioxide and water as respiratory waste products.
• Plants get rid of excess water by transpiration and guttation.
• Transpiration is the evaporative loss of water from plant leaves through stomata.

​

Additional Information 1. Evaporation is the process by which a liquid turns into a gas.

2. Transportation is defined as  the movement of goods and persons from place to place and the various means by which such movement is accomplished. 3. Translocation is defined as a genetic change in which a piece of one chromosome breaks off and attaches to another chromosome.

Code in certain language:

 

Conditions:

 

Given digits: 231578

Condition 1 → Does not follow (As one of the last two numbers is even).

Condition 2 → Does not follow (As one of the last two numbers is odd).

Thus, '231578' will be coded as 'DLTPAW'.

Hence, the correct answer is Option (3).

The correct answer is Recycling.

Key Points

• The most effective way to reduce waste generation is by recycling.
• This is because recycling helps in reducing the amount of waste that goes into landfills and helps conserve natural resources.
• Here are some of the reasons why recycling is the most effective way to reduce waste generation:
• Reduces the amount of waste going to landfills:
• ​Recycling helps in reducing the amount of waste that goes into landfills.
• This is because when we recycle, we are reusing materials that would otherwise end up in landfills.
• Conserves natural resources:
• Recycling helps conserve natural resources like timber, water, and minerals.
• By recycling materials like paper, plastic, and metal, we are reducing the need to extract new resources from the environment.
• Reduces greenhouse gas emissions:
• Recycling also helps in reducing greenhouse gas emissions.
• This is because when we recycle materials, we are using less energy than we would if we were to extract new resources from the environment.
• Creates jobs:
• Recycling also creates jobs in the recycling industry.
• This is because recycling requires people to collect, sort, and process materials, and these jobs help stimulate the economy.

Additional Information

• Garbage:
• Garbage is the waste that is produced from households, businesses, and industries.
• Garbage is typically sent to landfills or incinerators for disposal.
• While incineration can help in reducing the volume of waste, it can also lead to air pollution.
• Making compost:
• ​Composting is the process of decomposing organic waste like food scraps and yard waste into a nutrient-rich soil amendment.
• While composting can help in reducing the amount of waste going to landfills, it is not a solution for all types of waste.
• Combustion:
• ​Combustion refers to the process of burning waste to generate energy.
• While combustion can help in reducing the volume of waste going to landfills, it can also lead to air pollution and greenhouse gas emissions.

The correct answer is "Sakya".

Key Points

• Around 2500 years ago, Gautama Buddha was born.
• Buddha was a native of Sakya Gana, a small "Gana." He was born a Kshatriya.
• At a young age, he set off from home to learn more about life and its hardships and to investigate them.
• Enlightenment refers to this period of Buddha Shakyamuni's life.
• He later attained enlightenment and was given the name Buddha, which means "Wise Man." The Buddha will travel to Sarnath, which is close to Varanasi, next.
• He first instructed his followers in Sarnath.

Shortcut Trick

Total Capacity (LCM of 60, 75, 50) = 300 units.

Efficiency of Pipe A = 300 ÷ 60 = 5; Pipe B = 300 ÷ 75 = 4.

Combined Efficiency (A + B − C) = 300 ÷ 50 = 6.

Efficiency of Emptying Pipe C = (5 + 4) − 6 = 3.

Time taken by Pipe C = 300 ÷ 3 = 100 min.

∴ The correct answer is 100 min.

Alternate Method

Given: Time for Pipe A = 60 min, Time for Pipe B = 75 min, Time for (A + B − C) = 50 min.

Formula Used: Efficiency = Total Work ÷ Time Taken

⇒ Let efficiency of emptying pipe be C.

⇒ Efficiency(A) + Efficiency(B) − Efficiency(C) = Combined Efficiency

⇒ 5 + 4 − C = 6

⇒ 9 − C = 6

⇒ C = 3 units/min

⇒ Time to empty full cistern = Total Capacity ÷ Efficiency(C)

⇒ Time = 300 ÷ 3 = 100 min

∴ The correct answer is 100 min.

Additional Information

Inlet and Outlet Pipes

Inlet pipes add to the volume (positive efficiency) while outlet or leak pipes remove volume (negative efficiency).

Total Work Concept

For pipes and cisterns, the LCM of individual times is taken as the total capacity of the tank to simplify calculations.

Time and Efficiency Relation

Time taken is inversely proportional to efficiency: Time = Work ÷ Efficiency.

The correct answer is "2017".

Key Points

• The 12th Five-Year Plan of the Government of India (2012–17) served as the last Five-Year Plan for the nation.
• Its main theme was “Faster, More Inclusive and Sustainable Growth”
• The Twelfth Five-Year Plan of the Government of India was expected to achieve a growth rate of 9%, however, on December 27, 2012, the National Development Council approved an 8% growth rate instead.
• In the 12th Five-Year Plan, the government wants to see a 10% decrease in poverty.
• The strategy aims to enhance national infrastructure initiatives while avoiding bottlenecks of any form.
• In the 12th five-year plan, the planning commission's report seeks to increase private infrastructure investment by up to $1 trillion and lower the government's subsidy burden from 2% of GDP to 1.5% of GDP.
• The plan's subsidies will be transferred in cash via the Unique Identification Number.

The logic followed here is:

Thus, the number that will replace question mark in the series is 335.

Hence, the correct answer is Option (3).

The correct answer is "1996".

Key Points

• The Company was first incorporated under the name "L.N. On January 10, 1996, Finco Gems Private Limited was established with the RoC as a private limited company under the Companies Act, of 1956.
• The RoC issued a new certificate of incorporation for the Company on May 24, 2005, by the name change to Au Financiers (India) Private Limited to reflect the Company's varied financial activity.
• The Company was converted into a public limited company and given the name "Au Financiers (India) Limited" thanks to a special resolution accepted by the Shareholders at the EGM held on January 10, 2013.
• The Company has been given permission to establish an SFB, according to a letter from the RBI dated October 7, 2015.
• By its letter dated December 20, 2016, the RBI subsequently gave the Company its final clearance to create an SFB.
• The name of the Company was changed to "AU Small Finance Bank Limited" and a new certificate of incorporation was issued by the RoC on April 13, 2017, by the Company being created as an SFB.
• Mr. Sanjay Agarwal is the Managing Director & CEO of the Bank.

Drawing sitting arrangement as per the given information:

1. A, B, C, D, E and F are sitting around a circular table facing the centre.

2. B is sitting second to the right of D.

3. C is sitting third to the right of A.
4. A is an immediate neighbour of both E and F. D sits to the immediate right of E.

Therefore, C is sitting to the immediate left of B.

Hence, the correct answer is Option (2).

Given statement: M, N, O, P, Q and R live on six different floors of the same building. The lowermost floor in the building is numbered 1, the floor above it, number 2 and so on till the topmost floor is numbered 6.

1. Q lives on an odd-numbered floor but not on the lowermost floor. 

 

2. Only one person lives between Q and N.

 

3. R lives immediately above Q.

 

4. Only two people live between O and Q. 

 

5. P lives immediately above R. 

Case 1b will be eliminated here as there is N just above R. Case 2 will also be eliminated at this step because there is no floor above R for P to live.

 

Thus, the position of M will be 2^nd floor.

 

Hence, the correct answer is Option (3).

Shortcut Trick

To convert a recurring decimal to a fraction directly:

Numerator = (Complete number without decimal) − (Non-recurring part) = 331 − 3 = 328

Denominator = (Place 9 for each recurring digit after decimal) = 99

Fraction = Numerator / Denominator = 328/99

∴ The correct answer is 328/99.

Alternate Method

Given:

The decimal number is 3.313131...

Method Used:

Let x be the given recurring decimal. Multiply x by 100 since there are 2 repeating digits, then subtract the original equation.

Calculations:

⇒ Let x = 3.313131...   — (1)

⇒ Multiplying both sides by 100:

⇒ 100x = 331.313131...   — (2)

⇒ Subtracting (1) from (2):

⇒ 100x − x = 331.313131... − 3.313131...

⇒ 99x = 328

⇒ x = 328 / 99

∴ The correct answer is 328/99.

Additional Information

Pure Recurring Decimal

A decimal where all digits after the decimal point are repeated is called a pure recurring decimal. Convert by writing the repeated sequence divided by an equal number of 9s.

Mixed Recurring Decimal

A decimal where some digits after the decimal point do not repeat, followed by repeating digits. Convert by subtracting the non-repeating part from the entire number, divided by 9s for repeating digits and 0s for non-repeating digits.

Terminating Decimal

A decimal that ends after a finite number of digits. Its fractional equivalent always has a denominator whose prime factors consist only of 2 and/or 5.

Given statement: Five friends live on five different floors of the same building. The lowermost floor in the building is numbered 1, the floor above it, number 2 and so on till the topmost floor is numbered 5.

1. Only Meera lives on the floor between Shrishti and Richa. No one lives above Shrishti.

 

2. Anshul lives on the floor immediately below Richa. 

 

3. Chandan lives on the first floor.

 

Therefore, Anshul lives on the second floor.

Hence, the correct answer is Option (2).

The correct answer is "A, B, and C".

Key Points

• The criteria for a language to be a classical language are:

1. Exceptional antiquity of its earliest records or history, which dates back between 1500 and 2000 years.
2. A body of prehistoric writings or literature that has been preserved for future speakers as a priceless cultural legacy.
3. The literary heritage must be distinctive and distinct, and it must not borrow from literature from another language group.
4. The literature and language of the classical era are distinct from those of the modern era, and there may be a break between the classical era's language and its offshoots.

• So, comparing the properties listed above which are the real criteria to that in question we get,
• First property → True.
• Second property→ True.
• Third property→ True.
• Forth property→ False.

Shortcut Trick

If tanθ + cotθ = k, then tan²θ + cot²θ = k² − 2.

Given: tanθ + cotθ = 14

Applying the trick: tan²θ + cot²θ = 14² − 2 = 196 − 2 = 194.

Required value: 7 × (tan²θ + cot²θ) = 7 × 194 = 1358.

∴ The correct answer is 1358.

Alternate Method

Given: tanθ + cotθ = 14

Formula Used: (a + b)² = a² + b² + 2ab and tanθ × cotθ = 1

⇒ Square both sides: (tanθ + cotθ)² = 14²

⇒ tan²θ + cot²θ + 2(tanθ × cotθ) = 196

⇒ tan²θ + cot²θ + 2(1) = 196

⇒ tan²θ + cot²θ = 196 − 2 = 194

⇒ Multiply by 7: 7 × (tan²θ + cot²θ) = 7 × 194

⇒ 7tan²θ + 7cot²θ = 1358

∴ The correct answer is 1358.

Additional Information

Reciprocal Trigonometric Ratios

tanθ and cotθ are reciprocals, meaning tanθ × cotθ = 1. Similarly, sinθ × cosecθ = 1 and cosθ × secθ = 1.

Algebraic Identity for Squares

For any variable x where x + 1/x = n, the value of x² + 1/x² is always n² − 2. Here, x represents tanθ and 1/x represents cotθ.

Trigonometric Identities

1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ. These can be used to convert square terms into other trigonometric functions if required.

The correct answer is 'Battery'.

Key Points

• Battery helps to maintain potential difference across a conductor.
• Potential difference is measured in volts (V )and is also known as voltage.
• It is also defined as the amount of work, that has been done in moving a unit charge from one point to another.

Additional Information

• A plug key is an electrical component that acts as a switch. To turn on the circuit, a little metallic key is put into a metallic base, and the circuit is shut off if it is detached from its base.
• Resistance is a measure of the opposition to current flow in an electrical circuit. Resistance is measured in ohms, symbolized by the Greek letter omega (Ω).
• A voltmeter is a device that measures voltages of direct current or alternating current on a scale commonly in volts, millivolts, or kilovolts. Many voltmeters are digital that give readings as numerical displays.

The correct answer is 'white powder of magnesium oxide' .

Key Points

• When magnesium ribbon is burnt, it reacts with the oxygen present in the air to form magnesium oxide.
• Magnesium oxide is made up of Mg⁺² and O^-2  is connected to each other by an ionic oxide.
• The color of magnesium is white.
• The reaction is 2Mg+O₂=2MgO.

Shortcut Trick

Initial Pay = ₹125

Final Pay = ₹145

Increase = 145 − 125 = 20

Percentage Increase = (Increase ÷ Initial Pay) × 100 = (20 ÷ 125) × 100

Percentage Increase = 1/6.25 × 100 = 16%

∴ The correct answer is 16%.

Alternate Method

Given:

Original Hourly Wage (Old Pay) = ₹125

Increased Hourly Wage (New Pay) = ₹145

Formula Used:

Percentage Increase = [(New Value − Old Value) ÷ Old Value] × 100

Calculations:

⇒ Increase in pay = 145 − 125 = ₹20

⇒ Percentage increase = (20 ÷ 125) × 100

⇒ Percentage increase = (4 ÷ 25) × 100

⇒ Percentage increase = 4 × 4 = 16%

∴ The correct answer is 16%.

Additional Information

Percentage Change Formula

Used to find the relative change: Percentage Change = (Difference ÷ Original Value) × 100.

Base Value Concept

In percentage increase or decrease problems, the Initial Value or Original Value is always taken as the 100% base.

Successive Percentage Change

If a value is increased by x% and then y%, the effective change is [x + y + (xy ÷ 100)]%.

Shortcut Trick Ratio a : b = 2 : 3 and a + b = 145 Value of 5 parts = 145 ⇒ 1 part = 29 a = 2 × 29 = 58 and b = 3 × 29 = 87 Mean proportional c = √(a × b) = √(58 × 87) = 29√6 c ≈ 29 × 2.449 ≈ 71.02 ∴ The correct answer is 71.

Alternate Method Given: Sum of two parts a + b = 145 Ratio a : b = 2 : 3 Proportion a : c :: c : b where c > 0 Formula Used: Mean proportional c between a and b is given by c2 = ab.

Calculations: ⇒ a = (2 / (2 + 3)) × 145 ⇒ a = (2 / 5) × 145 = 2 × 29 = 58 ⇒ b = (3 / 5) × 145 = 3 × 29 = 87 ⇒ a : c = c : b ⇒ c / a = b / c ⇒ c2 = a × b ⇒ c2 = 58 × 87 ⇒ c2 = 5046 ⇒ c = √5046 ⇒ Since 712 = 5041 and 722 = 5184 ⇒ c ≈ 71.035 Rounding to the nearest integer: ∴ The correct answer is 71.

Additional Information Mean Proportional If three quantities a, b, and c are in continued proportion such that a : c = c : b, then c is the mean proportional and c = √(ab). Third Proportional In the continued proportion a : b = b : c, the term c is called the third proportional to a and b, calculated as c = b2 / a. Fourth Proportional If four quantities a, b, c, and d are in proportion (a : b = c : d), then d is the fourth proportional, calculated as d = (b × c) / a.

Given:

The given pie chart depicts the number of chairs manufactured of four brands (A, B, X and Y).

The total number of chairs manufactured of the four brands together is 200.

The total manufacturing cost of chairs of Brand A was ₹14,400.

 Calculation: 

The total number o chairs manufactured of chairs of Brand A = ${36\over 100}\times 200 = 72$

The total manufacturing cost of 72 chairs of Brand A = 14400.

The total manufacturing cost of one chair of Brand A = ${14400\over 72} =200$.

Hence, 'option 1' is correct.

The correct answer is 'South'.

Key Points

• Right-hand rule says -the magnetic field made by a current in straight conductor curls around the conductor in a ring. You can find it by pointing your right thumb in the direction of the conducting straight wire and curling your fingers. Your fingers will be curled in the same direction as the magnetic field around the conductor.
• Direction of the magnetic field produced by the straight conductor carrying current in the east direction is along the South direction at a point directly below the conductor.

The logic followed here is:

The sum of first and last number is divisible by the second number.

first number + third number = divisible by second number

1. (11, 117, 2)

11 + 2 = 13

117 ÷ 13 = 9

2. (9, 56, 5)

9 + 5 = 14

56 ÷ 14 = 4 

Now, let us check each options:

1. (13, 169, 2)

13 + 2 = 15 

169 is not exactly divisible by 15.

2. (12, 108, 6)

12 + 6 = 18

108 ÷ 18 = 6 

Thus, 108 is exactly divisible by 18.

3. (7, 46, 1)

7 + 1 = 8

46 is not exactly divisible by 8.

4. (10, 98, 2)

10 + 2 = 12

98 is not exactly divisible by 12.

Hence, the correct answer is Option (2).

The correct answer is Lactic acid and pyruvate.

Key Points

• Respiration is a metabolic process that occurs in all organisms.
• It is a biochemical process that occurs within the cells of organisms.
• In this process, the energy (ATP-Adenosine triphosphate) is produced by the breakdown of glucose which is further used by cells to perform various functions.
• In this step, enzymes split a molecule of glucose into two molecules of pyruvate, which releases energy that is transferred to ATP.
• Lactic acid  fermentation occurs in some strains of bacteria and in skeletal muscle and produces lactic acid .
• During the reaction ,NaDH is oxidized to NaD⁺.

The correct answer is "Average propensity to save".

Key Points

• The average propensity to save (APS) is a macroeconomic term referring to the percentage of income that is saved rather than spent on current items.
• Also known as the savings ratio, it is typically stated as a proportion of the total disposable income of the household (income minus taxes).
• The opposite of APS is the average consumption propensity (APC). APS is also associated with the marginal propensity to save (MPS), which measures the proportion of a change in income that is saved as opposed to consuming.
• A population's APS is a key economic indicator. From a personal finance standpoint, the present savings rate of a population can be correlated with ageing-related habits, such as retirement savings.
• Savings indicate the decision to forego current spending in favour of future consumption, which frees up the real economic resources required for productive investment and the production of capital goods.
• The greater a society's APS, the greater its citizens' preference for investing in the future over consumption in the present. Directly propelling sustainable economic expansion is an investment in productive capital goods.

The correct answer is Zn (s) + 2AgNO3 (aq) → Zn(NO3)2 (aq) + 2Ag(s).

Key Points

• The chemical formula of zinc is  Zn.
• The chemical formula of silver nitrate is AgNO₃.
• The chemical formula of zinc nitrate is Zn(NO₃)₂ .
• And the chemical formula of silver is Ag.
• The reaction will be:  Zn (s) + 2AgNO3 (aq) → Zn(NO3)2 (aq) + 2Ag(s).

The correct answer is 100 W, 20 Ω.

Key Points

• Given in the question:
• Potential difference across the circuit (V) = 20 V
• Current in the circuit (I) = 5A
• Using the formula P = V x I,
• P = 20V x 5A = 100 W
• Thus, the power dissipated in the circuit is 100 W.
• Now, using Ohm's Law, R_net = V / I , we get,
• Rnet = 20/5 = 4 Ω
• This gives us the net resistance across the circuit.
• As, it is given that the bulb are identical, their resistance will be same (say R).
• Since the bulbs are connected in parallel combination,
• 1/R_net = 1/R + 1/R +1/R + 1/R + 1/R
• 1/R_net = 5/R
• 1/4 = 5/R (as Rnet = 4 Ω)
• R = 5 x 4
• R = 20 Ω
• Thus, the resistance of each bulb is 20 Ω.
• Therefore, the power dissipated in the circuit and the resistance of each bulb are 100W and 20 Ω respectively.

Additional Information

• Ohm's Law:
• Ohm's law states that the electric current flowing through a conductor is directly proportional to the potential difference across its ends, provided the physical state of the conductor remains unchanged.
• If I be the current flowing through the conductor and V the potential difference across its ends, then $V {\displaystyle \propto } I$
• Hence, V = I x R , where, R is the constant of proportionality termed as Resistance of the conductor.
• Power:
• Energy is the word for the ability to perform tasks. Power is the energy used to complete a task in a given amount of time.
• P is used to represent it.
• These three formulas are used to calculate power in any electrical circuit.
• P = V x I
• P = I² x R
• P = V² / R
• Resistance:
• The amount of obstruction to current passage in an electrical circuit is measured as resistance. Ohms, or the Greek letter omega (Ω), are used to measure resistance (R).
• There are mainly two ways of combination of resistance:
• Series Combination:
• When two or more resistances are connected end to end and the same current passes through each one successively, they are said to be connected in series. In this instance, the sum of the individual resistances found in the series combination determines the equivalent or total resistance.
• In terms of mathematics, the equivalent resistance (R) of any number of resistances connected in series (R1, R2, R3,...) is as follows:
• R = R1 + R2 + R3 + …
• Parallel Combination:
• When two or more resistances are connected between two places and have distinct current directions, they are referred to as being in parallel connections. In these circuits, the current splits off and recombines at the point where the branches meet.
• The equivalent resistance (R) of any number of parallel resistances (R1, R2, R3,...) can be found mathematically as follows:
• 1/R = 1/R1 + 1/R2 + 1/R3 + …

Given:

Heat Produced =200 J.

Registance=2 ohm.

Formula used:

Amount of heat produced $H=\frac{V^2t}{R}$

[Where V is the amount of Voltage produced; t is the time required ;R is the resistance]

Calculation:

$200=\frac{\mathrm{V}^2 \times 1}{2}\ \Rightarrow \mathrm{V}^2=400 \\Rightarrow \mathrm{V}=20 \mathrm{v}$    

Hence the correct answer is 20 volt.

The correct answer is '6 cm'.

Key Points

• In optics, the relationship between the distance of the image (v), the distance of the object (u), and the focal length (f) of the lens is given by the formula known as the Lens formula. The Lens formula is applicable for convex as well as concave lenses. These lenses have negligible thickness. The formula is as follows:

$\Large{1\over f}= {1\over v}-{1\over u}$

Where f is focus of lens, v is distance of image and u is distance of object.

• An given object of height 3 cm is placed at a distance 30 cm in front of a convex lens of focal length 20 cm. then Image of object formed at distance(v) of,

 $\Large{1\over v}= {1\over f}+{1\over u}$                

$\large v={f\times u\over f+u}$                        

$\large v={20\times (-30)\over 20+(-30)} = 60 cm$

• The relationship between size of image and object w.r.t. distance of object is as follows:

$\large{Height : of:image\over Height:of:object}={v\over- u}$

$\large{Height : of:image\over 3}={60\over -30}$

$\large{Height : of:image}=-2\times 3= -6cm:$

where negative sign indicate nature of image.

• Hence , the height of image will be at 6cm.

Additional Information

• Sign convention in the case of the concave mirror:
• Since, the object is always placed in front of the mirror hence the sign of the object is taken as negative.
• Since, the center of curvature and focus lie in front of the concave mirror, signs of the radius of curvature and focal length are taken as negative in the case of the concave mirror.
• When an image is formed in front of the mirror, the distance of the image is taken as – (negative), and when the image is formed behind the mirror, the distance of the image is taken as + (positive).
• Height of image is taken as positive in the case of the erect image and taken as negative in the case of the inverted image.
• Sign convention in the case of a convex mirror:
• Since, the object is always placed in front of the mirror hence the sign of the object is taken as negative.
• Since, the center of curvature and focus lies behind the convex mirror, signs of the radius of curvature and focal length are taken as + (positive) in the case of the convex mirror.
• In the case of a convex mirror, images are always formed behind the mirror, thus the distance of the image is taken as positive.
• In the case of a convex mirror, always an erect image is formed, thus the height of the image is taken as positive.

The correct answer is "2002".

Key Points

• Article 21-A was added to India's Constitution by the Constitution (Eighty-Sixth Amendment) Act of 2002.
• It says that all children between the ages of 6 and 14 have the right to a free and mandatory education in a way that the State can decide by law.
• The Right of Children to Free and Compulsory Education (RTE) Act, 2009, which is the result of Article 21-A, says that every child has the right to full-time elementary education of satisfactory and equal quality in a formal school that meets certain essential norms and standards.
• On April 1, 2010, Article 21-A and the RTE Act became law. The words "free and mandatory" are in the name of the RTE Act.
• "Free education" means that no child, other than one whose parents have put him or her in a school that isn't paid for by the government, should have to pay any kind of fee, charge, or expense that might keep him or her from getting an elementary education and finishing it.
• "Compulsory education" means that the government and local authorities have to make sure that all kids between the ages of 6 and 14 can get into elementary school, go to school, and finish it.
• With this, India has moved toward a rights-based framework that makes it a legal requirement for the Central and State Governments to implement this basic child right, as written in Article 21A of the Constitution and as stated in the RTE Act.

The correct answer is "Technological Interventions for Tribal Empowerment (TITE)".

Key Points

• Tribal Sub Plan Technological Interventions for Tribal Empowerment (TITE):
• The department's grant-in-aid scheme for the Tribal Sub Plan, TITE aims to improve the living conditions of scheduled tribes and give them more power through sustainable science and technology (S&T) activities and applications.
• By creating an action research project, S&T applications and interventions should meet the following goals:
• To encourage research, development, and adaptation of science and technology to improve the lives of tribal groups.
• Start of S&T-based programmes in major socioeconomic sectors that are tailored to the location.
• To keep traditional artisan skills alive by building on local innovation and knowledge systems and using a participatory research approach to make local resource management plans.
• To look into and promote alternative ways to make a living in places where people are losing their jobs, focusing on innovative community-based approaches and technological options to increase the impact on the ground.
• To improve the equipment and machinery that is already there and to bring in new techniques, systems, and a full set of technology.
• Widening the effect of the above S&T-based activities by encouraging the replication and scaling up of successful approaches that lead to empowering tribal populations and building their technical skills.

Shortcut Trick

Target average for 11 innings = 33 runs.

Total deficiency in first 7 innings = 7 × (33 − 28) = 7 × 5 = 35 runs.

Total deficiency in next 3 innings = 3 × (33 − 32) = 3 × 1 = 3 runs.

Score needed in 11^th inning = Target average + Total deficiency = 33 + (35 + 3) = 71.

∴ The correct answer is 71 runs.

Alternate Method

Given: Average of first 7 innings = 28 runs, Average of next 3 innings = 32 runs, Overall target average for 11 innings = 33 runs.

Formula Used: Total Sum = Average × Number of innings.

⇒ Total runs in first 7 innings = 7 × 28 = 196

⇒ Total runs in next 3 innings = 3 × 32 = 96

⇒ Total runs in first 10 innings = 196 + 96 = 292

⇒ Total runs required for 11 innings = 11 × 33 = 363

⇒ Score needed in 11^th inning = Total runs required − Runs scored in 10 innings

⇒ Score needed in 11^th inning = 363 − 292 = 71

∴ The correct answer is 71 runs.

Additional Information

Weighted Average

When different groups have different averages, the combined average is given by: Average_total = (n₁×A₁ + n₂×A₂ + ...) ÷ (n₁ + n₂ + ...).

Change in Average

If a new observation is added, the new score = (New Average × New Total Count) − (Old Average × Old Total Count).

Deviation Rule

The sum of deviations of all observations from their arithmetic mean is always zero (Σ(x_i − mean) = 0).

Concept:

Concave Mirror and Graph Analysis:

• The relationship between object distance (u), image distance (v), and focal length (f) of a concave mirror is given by the mirror formula:
• Mirror Formula: $\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$
• When plotting a graph of $\frac{1}{v}$ against $\frac{1}{u}$, the intercepts can provide the focal length.
• The x-intercept represents $\frac{1}{f}$ and the y-intercept also represents $\frac{1}{f}$.
• SI Unit: The SI unit of focal length is meter (m), but in this context, we use centimeters (cm).
• Dimensional Formula: The dimensional formula for focal length is [L].

 

Calculation:

Given,

x-intercept = $\frac{1}{u} = 0.1$ cm^-1

y-intercept = $\frac{1}{v} = 0.11$ cm^-1

Using the intercepts to find the focal length (f),

⇒ $\frac{1}{f} = x\text{-intercept} = 0.1$

⇒ $f = \frac{1}{0.1} = 10$ cm

Similarly,

⇒ $\frac{1}{f} = y\text{-intercept} = 0.11$

⇒ $f = \frac{1}{0.11} \approx 9.09$ cm

Taking the average of both values,

⇒ $f = \frac{10 + 9.09}{2} \approx 9.55$ cm

∴ The focal length of the mirror is 9.55 cm.

Given: INTERNATIONAL 

Solution: 

Forward Pair - IN, NT

Backward Pair - LN, LO, NO, RT

Hence, six such pair of which has as many letters between them in the word as they have between them in the English alphabetical series.

Hence, 'option 4' is correct.

Shortcut Trick

In a parallelogram and its special cases (rectangle, rhombus, square), the diagonals always bisect each other.

In a kite, only the longer diagonal bisects the shorter diagonal at a 90° angle.

The shorter diagonal does not bisect the longer one; therefore, they do not bisect each other.

∴ The correct answer is Diagonals of a kite bisect each other.

Alternate Method

Given: Properties of diagonals for various quadrilaterals.

Formula Used: A parallelogram is a quadrilateral where opposite sides are parallel, ensuring diagonals bisect each other.

⇒ In the kite ABCD shown above, diagonal BD is bisected by AC at point E (BE = ED).

⇒ However, diagonal AC is not bisected by BD (AE ≠ EC).

⇒ For parallelograms, rectangles, and rhombuses, the intersection point E is the midpoint of both diagonals.

⇒ Thus, the statement regarding the kite is not necessarily true for both diagonals.

&there4) The correct answer is Diagonals of a kite bisect each other.

Given:

(17, 238, 7)

(23, 414, 9)

Solution:

the logic follows here is:

Now applying all options,

option 1:

option 2:

option 3:

option 4: 

Thus, the logic applied on set (27, 270, 5)

Hence, 'option 3' is correct.

The correct answer is "Shah Mal".

Key Points

• During the 1857 uprising, Shah Mal, a simple farmer from Bijrol village in Uttar Pradesh's Baghpat district (it used to be Meerut), led a fierce rebellion in Baraut pargana.
• On May 10, 1857, the Sepoy uprising began in Meerut. Shah Mal and other rebels from the area attacked a large group of merchants.
• In the end, he attacked all the way across the tehsil of Baraut, destroyed the boat bridge over the Yamuna in Baghpat, and looted its bazaar.
• Shah Mal showed his leadership by making alliances with people outside of his area and sending people to Delhi to ask the rebels there for help.
• Baghpat quickly became an important place for the sepoys from Delhi to get supplies.
• Shah Mal rallied the Chowrassee des, or 84 villages, in his area, and with the help of the freed prisoners from Meerut jail, he grew his rebel forces even more.
• Shah Mal set up an effective intelligence network and turned the bungalow of an English officer into a "hall of justice" where he settled local disputes and gave justice.
• This made it hard for the English to communicate with Meerut.
• Shah Mal died in a battle in Baraut in July 1857, when he was fighting against the rebels.

Given:

total number of cushions sold in store x-

in February= 196

in march= 114

total number of cushions sold in store z in march= 155

formula used:

percentage= ${value\over total: value}\times100$

calculation:

total  number of cushions sold in store x = 196+114= 310

total number of cushions sold in store z in march= 155

 total numbers of cushions sold more in store x than store z= 310-155= 155

percentage= ${155\over155}\times100$= 100%

Therefore, the cushions sold in store x in both months is 100% more than the cushions sold in store z in the month of march.

Shortcut Trick Sum of numbers = 48, HCF = 6 Sum of co-prime ratios (a + b) = 48 ÷ 6 = 8 Possible co-prime pairs (a, b) where a > b are (7, 1) and (5, 3). Pairs (6, 2) and (4, 4) are rejected as they are not co-prime. ∴ The correct answer is 2.

Alternate Method Given: Sum = 48, HCF = 6, Condition: x > y

⇒ Let the numbers be x = 6a and y = 6b, where a and b are co-prime numbers. ⇒ 6a + 6b = 48 ⇒ 6(a + b) = 48 ⇒ a + b = 8 ⇒ Pairs of (a, b) whose sum is 8 and are co-prime (HCF = 1): ⇒ Case 1: (7, 1) → x = 6×7 = 42, y = 6×1 = 6 (Valid as 42 > 6) ⇒ Case 2: (5, 3) → x = 6×5 = 30, y = 6×3 = 18 (Valid as 30 > 18) ⇒ Note: (6, 2) is not co-prime (HCF=2), and (4, 4) is not co-prime (HCF=4). ∴ The correct answer is 2.

Additional Information HCF and Sum Relation If the HCF of two numbers is H, the numbers can be represented as Ha and Hb, where a and b are co-prime. Their sum is H(a + b). Co-prime Numbers Two numbers are said to be co-prime if their Highest Common Factor (HCF) is 1. Example: (3, 5), (7, 8). HCF and LCM Relation Product of two numbers = HCF × LCM. Also, LCM is always exactly divisible by the HCF.

The logic follows here is - 

Thus,  'GESIO' will be written in HGVMT.

hence, the correct answer is 'option 2'.

Given: 9 + $5 @$ 2 6 3 % 2 & 8 # * 2 9 ! 4 3

According to given information,

• Sign $\rightarrow$ Sign  $\rightarrow$ Even number

 

So, there are 2 signs  (from left to right) that are immediately preceded by a sign and immediately followed by an even number.

Hence, 'option 4' is correct.

Shortcut Trick A quadratic equation is an equation of the second degree, meaning the highest power of the variable is 2. Scan for the highest exponent: Option 1 has x2 but it cancels out; Option 2 has degree 4; Option 3 has degree 3. In 5x2 + 4x − 2 = 7x, the highest power is 2 and the x2 term does not cancel. ∴ The correct answer is 5x2 + 4x − 2 = 7x.

Alternate Method Given: Four mathematical equations. Formula Used: The standard form of a quadratic equation is ax2 + bx + c = 0, where a ≠ 0. Step-by-step Evaluation: Case 1: 4x2 + 6x + 5 = 4x(x + 1) ⇒ 4x2 + 6x + 5 = 4x2 + 4x ⇒ 6x − 4x + 5 = 0 ⇒ 2x + 5 = 0 (This is a linear equation because degree = 1). Case 2: (x + 7)4 = 14 ⇒ The highest power of x is 4 (This is a quartic equation). Case 3: 3x + 7 = 7x3 ⇒ 7x3 − 3x − 7 = 0 (This is a cubic equation because degree = 3). Case 4: 5x2 + 4x − 2 = 7x ⇒ 5x2 − 3x − 2 = 0 ⇒ Here, the highest power of x is 2 and the coefficient of x2 is 5 (not zero). ⇒ This matches the standard form ax2 + bx + c = 0. ∴ The correct answer is 5x2 + 4x − 2 = 7x.

Additional Information Nature of Roots For a quadratic equation ax2 + bx + c = 0, the discriminant D = b2 − 4ac determines the roots. If D > 0, roots are real and distinct; if D = 0, roots are real and equal; if D < 0, roots are imaginary. Sum and Product of Roots If α and β are the roots, then Sum of roots (α + β) = −b/a and Product of roots (αβ) = c/a. Quadratic Formula The roots can be found using the formula x = (−b ± √(b2 − 4ac)) / 2a. Equation Degrees Linear equations have degree 1, Quadratic equations have degree 2, Cubic equations have degree 3, and Quartic equations have degree 4.

Shortcut Trick

Let CP_A be x and CP_B be (2540 − x).

SP_A = 1.15x and SP_B = 0.75 × (2540 − x).

Given: SP_A − SP_B = 109 ⇒ 1.15x − [1905 − 0.75x] = 109.

1.9x = 2014 ⇒ x = 1060. So, CP_A = 1060 and CP_B = 1480.

Total SP = 1.15(1060) + 0.75(1480) = 1219 + 1110 = 2329.

Total Loss = 2540 − 2329 = 211. Loss % = (211 ÷ 2540) × 100 ≈ 8.3%.

∴ The correct answer is loss, 8.3%.

Alternate Method

Formulas: SP = CP × (100 ± P/L)% | Total Profit/Loss = Total SP − Total CP

⇒ Let Cost Price of A = A and Cost Price of B = B. ⇒ A + B = 2540 --- (i) ⇒ SP_A = 1.15A and SP_B = 0.75B. ⇒ 1.15A − 0.75B = 109 --- (ii) ⇒ Multiply (i) by 0.75: 0.75A + 0.75B = 1905. ⇒ Add to (ii): 1.9A = 2014 ⇒ A = 1060. ⇒ B = 2540 − 1060 = 1480. ⇒ Total SP = (1060 × 1.15) + (1480 × 0.75) = 1219 + 1110 = 2329. ⇒ Loss = 2540 − 2329 = 211. ⇒ Loss % = (211 ÷ 2540) × 100 = 8.307...% ≈ 8.3%.

∴ The correct answer is loss, 8.3%.

Additional Information

Overall Profit/Loss Percentage

It is calculated based on the Net Change in the value of the total transaction relative to the Total Cost Price.

Linear Equations in Profit and Loss

When two articles have different CP and different conditions on SP, representing them as x and (Total - x) simplifies the calculation to one variable.

Shortcut Trick

The difference between Total Surface Area (TSA) and Lateral Surface Area (LSA) of a cube is always the area of the two bases.

TSA − LSA = 2 × (Area of one face) = 2a²

2a² = 4050 ⇒ a² = 2025

a = √2025 = 45 units.

∴ The correct answer is 45 units.

Alternate Method

Given: Difference between Total Surface Area and Lateral Surface Area = 4050 units².

Formula Used:

⇒ Total Surface Area (TSA) of cube = 6a²

⇒ Lateral Surface Area (LSA) of cube = 4a²

Where 'a' is the side length of the cube.

Calculations:

⇒ TSA − LSA = 4050

⇒ 6a² − 4a² = 4050

⇒ 2a² = 4050

⇒ a² = 4050 ÷ 2

⇒ a² = 2025

⇒ a = √2025

⇒ a = 45 units

∴ The correct answer is 45 units.

Additional Information

Volume of Cube

The amount of space occupied by the cube, calculated as Volume = a³.

Diagonal of Cube

The longest rod that can be placed inside a cube is its diagonal, given by d = a√3.

Surface Area of Cuboid

For a cuboid with dimensions l, b, h: TSA = 2(lb + bh + hl) and LSA = 2h(l + b).

Given:

• Five friends live on five different floors of the same building.
• The lowermost floor in the building is numbered 1, the floor above it, number 2 and so on till the topmost floor is numbered 5 .

Solution:

(1. ) Namita lives on the floor which is second from the top. 

(2. ) Avdhesh lives on the floor immediately above Namita.

(3. ) Ritu lives on the lowermost floor.

(4.) Sanjana lives on one of the floor between Anand and Ritu. 

Hence, 'Sanjana'  lives on the fourth floor from the top.

Given number : 38572352

Condition:

• 1 is added to each even digit
• 2 is subtracted from each odd digit

After applying above condition,

Second from the left = 9

Fourth from the right = 3

 the sum of the digits which are second from the left and fourth from the right = 9 + 3 = 12

Hence, '12' is the correct answer.

Given:

Five friends, Hiba, Suhani, Pritam, Ahmed, and Rosy are sitting around a circular table facing the centre.

Solution:

(1. ) Rosy is second to the right of pritam.

(2.) Ahmed is second to the left of Pritam,

(3.) Rosy is second to the right of Pritam, and is an immediate neighbour to both Suhani and Ahmed.

(4.) Now , Hiba sits on only vacant seat.

Thus, Hiba is seated second to the left of Suhani.

Hence, the correct answer is 'option 2'.

Shortcut Trick

Total Work (LCM of 20, 30) = 60 units.

Efficiency: Navya = 3 units/day, Bhavya = 2 units/day.

Alternate pair (Bhavya + Navya): 2 + 3 = 5 units in 2 days.

Total time = (60 ÷ 5) × 2 = 12 × 2 = 24 days.

∴ The correct answer is 24 days.

Alternate Method

Given: Time taken by Navya = 20 days, Time taken by Bhavya = 30 days.

Formula Used: Total Work = LCM of individual days; Efficiency = Total Work ÷ Time.

Calculations:

⇒ Work done in Cycle 1 (Day 1 by Bhavya + Day 2 by Navya) = 2 + 3 = 5 units.

⇒ Time taken for 1 cycle = 2 days.

⇒ Number of cycles required = Total Work ÷ Work per cycle = 60 ÷ 5 = 12 cycles.

⇒ Total time = 12 cycles × 2 days/cycle = 24 days.

∴ The correct answer is 24 days.

Additional Information

Alternate Day Work

In alternate work, identify the starting person and calculate work done in one full rotation (cycle) to find the total duration.

Pipes and Cisterns

Similar to time and work, but outlet pipes have negative efficiency, reducing the total work done per cycle.

Work and Wages

Wages are always distributed in the ratio of work done by each person, which is Efficiency × Time.

Shortcut Trick

Let the Cost Price (CP) be 100%.

Loss = 6%, so Selling Price (SP) = 100% − 6% = 94%.

Given, 94% = ₹940.

Using the Unitary Method: 1% = 940 ÷ 94 = 10.

Therefore, 100% (CP) = 10 × 100 = ₹1,000.

∴ The correct answer is ₹1,000.

Alternate Method

Given: Selling Price (SP) = ₹940, Loss% = 6%.

Formula: CP = [SP × 100] ÷ [100 − Loss%]

⇒ CP = [940 × 100] ÷ [100 − 6]

⇒ CP = [940 × 100] ÷ 94

⇒ CP = 10 × 100

⇒ CP = ₹1,000

∴ The correct answer is ₹1,000.

Additional Information

Cost Price & Selling Price

The Cost Price is the price at which an item is bought. If SP is less than CP, there is a Loss. Formula: Loss = CP − SP.

Profit and Loss Percentage

Profit% and Loss% are always calculated on the Cost Price unless stated otherwise in the problem.

The correct answer is 2.4 m, 12 m and $\frac{1}{3}$ respectively.

Key Points

We know that f = -3.0 m (it is negative because the mirror is concave), v = -4.0 m (it is negative because the image is real and formed on the same side as the object).

Substitute these values into the mirror equation:

$\frac{1}{-3}$ = $\frac{1}{-4}$ + $\frac{1}{u}$

Solving this gives:

u = -12 m (it is negative because the object is on the same side as the light source).

Now we can find the magnification m. The magnification of a mirror is the ratio of the image height (h') to the object height (h), and it's also equal to -v/u.

Therefore, m = $\frac{h'}{h}$ = $\frac{-v}{u}$

We know that h' = 80 = 0.8cm  m (it is positive because the image is inverted).

Therefore, m = $\frac{0.8}{h}$ = $\frac{-4}{-12}$ =$\frac{1}{3}$

Solving this gives:

h = 0.8 * 3 = 2.4 m

Therefore, the object height is 2.4 m, the object distance is 12 m, and the magnification is $\frac{1}{3}$.