RRB Technician Grade III Official Paper (Held On: 10 March, 2026 Shift 2) (Previous Year Paper)

Shortcut Trick

For half-yearly compounding, the rate is halved (10%) and the time is doubled (2 periods).

Effective rate for 2 successive cycles of 10% = 10 + 10 + (10 × 10)/100 = 21%.

Total Amount = 121% of 42,000.

⇒ 42,000 × 1.21 = 50,820.

∴ The correct answer is ₹50,820.

Alternate Method

Given:

Principal (P) = ₹42,000

Annual Rate (R) = 20%

Time (T) = 1 year

Since the interest is compounded half yearly:

⇒ Adjusted Rate (r) = R ÷ 2 = 20% ÷ 2 = 10% per half year

⇒ Number of periods (n) = T × 2 = 1 × 2 = 2

Formula Used:

Amount (A) = P(1 + r/100)ⁿ

Calculations:

⇒ A = 42,000 × (1 + 10/100)²

⇒ A = 42,000 × (110/100) × (110/100)

⇒ A = 42,000 × 1.1 × 1.1

⇒ A = 42,000 × 1.21

⇒ A = 50,820

∴ The correct answer is ₹50,820.

Additional Information

Compound Interest Half-Yearly Rule

When interest is compounded half-yearly, the annual rate is divided by 2 and the time (in years) is multiplied by 2 to get the effective number of conversion periods.

Quarterly Compounding

For quarterly compounding, the rate is divided by 4 and the time is multiplied by 4. The formula becomes A = P(1 + R/400)^4T.

Successive Percentage Change

Compound interest is an application of successive percentage change. For two periods with rate r, the effective rate is (r + r + r²/100)%.

Shortcut Trick

Net rate of filling = (1/14 − 1/63) = (9 − 2) / 126 = 7 / 126 = 1/18 tank/min.

Time to fill the full tank = 18 min.

Time to fill 1/3^rd of the tank = 18 × 1/3 = 6 min.

∴ The correct answer is 6 min.

Alternate Method

Given:

Time taken by Pipe A (Filling) = 14 min

Time taken by Pipe B (Emptying) = 63 min

Let the Total Capacity of the tank be the LCM of 14 and 63.

Calculations:

⇒ Combined efficiency of both pipes = 9 − 2 = 7 units/min

⇒ Work to be completed = 1/3 × Total Capacity

⇒ Work = 1/3 × 126 = 42 units

⇒ Time taken = Work ÷ Efficiency

⇒ Time taken = 42 ÷ 7 = 6 min

∴ The correct answer is 6 min.

Additional Information

Pipes and Cisterns Efficiency

Filling pipes have positive efficiency, while emptying pipes (leakage) have negative efficiency. Net work is the algebraic sum of efficiencies.

Time-Work Relation

Total Work = Efficiency × Time. If work is constant, Efficiency is inversely proportional to the time taken.

Fractional Work

To find the time for a fraction of work, multiply the Total Time taken for full work by that fraction.

Shortcut Trick

Consider the original number as 100%.

An increase of 110% makes the final percentage = 100% + 110% = 210%.

Given: 210% → 3570

Dividing by 210, we get 1% → 3570 ÷ 210 = 17.

Original number (100%) → 17 × 100 = 1700.

∴ The correct answer is 1700.

Alternate Method

Given:

Percentage increase = 110%

Resulting value = 3570

Formula Used:

Final Value = Original Value × (1 + Percentage Increase / 100)

Calculations:

⇒ Let the original number be x.

⇒ x + (110 / 100) × x = 3570

⇒ x + 1.1x = 3570

⇒ 2.1x = 3570

⇒ x = 3570 / 2.1

⇒ x = 35700 / 21

⇒ x = 1700

∴ The correct answer is 1700.

Additional Information

Percentage Increase Formula

New Value = Original Value × (100 + Increase%) ÷ 100. This is the standard method for growth calculations.

Reverse Percentage

To find the original value when the increased value is known, divide the final value by the multiplying factor (e.g., 2.1 for 110% increase).

Fractional Equivalence

An increase of 110% is equivalent to multiplying the number by 21/10 or 2.1. This helps in fast mental math.

Shortcut Trick

The ratio of the first two terms 2x : 3x simplifies to 2 : 3.

For proportion, the ratio of the next two terms 12 : y must also be 2 : 3.

Comparing the parts: 2 units = 12 ⇒ 1 unit = 6.

Therefore, y = 3 units = 3 × 6 = 18.

∴ The correct answer is 18.

Alternate Method

Given: The terms are 2x, 3x, 12, and y is the fourth proportional.

Formula Used: If a, b, c, d are in proportion, then a/b = c/d.

⇒ 2x ÷ 3x = 12 ÷ y

⇒ 2 ÷ 3 = 12 ÷ y

⇒ 2 × y = 12 × 3

⇒ 2y = 36

⇒ y = 36 ÷ 2

⇒ y = 18

∴ The correct answer is 18.

Additional Information

Mean Proportional

The mean proportional between two numbers a and b is calculated as √(a × b).

Third Proportional

If a, b, c are in continued proportion, then a/b = b/c, where c = b² ÷ a is the third proportional.

Product Rule

In any proportion a : b :: c : d, the product of extremes (a × d) is always equal to the product of means (b × c).

Shortcut Trick

The Mode of a data set is the value that appears most frequently.

By scanning the given numbers: 44 appears 3 times (at positions 2, 6, and 15).

Other frequent values like 43, 53, 55, and 48 appear only 2 times each.

Since 44 has the highest count, it is the mode.

∴ The correct answer is 44.

Alternate Method

Given: Data set = {43, 44, 43, 53, 42, 44, 46, 47, 55, 50, 52, 55, 40, 45, 44, 53, 48, 48}

Formula Used: Mode = The observation with the Maximum Frequency.

Calculations:

⇒ Count of 40 = 1

⇒ Count of 42 = 1

⇒ Count of 43 = 2

⇒ Count of 44 = 3

⇒ Count of 45 = 1

⇒ Count of 46 = 1

⇒ Count of 47 = 1

⇒ Count of 48 = 2

⇒ Count of 50 = 1

⇒ Count of 52 = 1

⇒ Count of 53 = 2

⇒ Count of 55 = 2

⇒ The maximum frequency is 3, which corresponds to the value 44.

∴ The correct answer is 44.

Additional Information

Mean (Arithmetic Average)

It is the sum of all observations divided by the total number of observations: Mean = (∑x_i) ÷ n.

Median

The middle value of the data when arranged in ascending or descending order. For n = 18 (even), it is the average of the 9^th and 10^th terms.

Empirical Relationship

In a moderately asymmetrical distribution, the relationship is: Mode = 3 × Median − 2 × Mean.

Range

The difference between the maximum and minimum values in the data set: Range = Max − Min.

Shortcut Trick

Total Profit Percentage = [ (100 + Profit%) / (100 − Less Weight%) − 1 ] × 100

Total Profit % = [ (100 + 15) / (100 − 14) − 1 ] × 100

Total Profit % = [ 115 / 86 − 1 ] × 100 = (29 / 86) × 100

Total Profit % ≈ 33.72%

Rounding to the nearest whole number, we get 34%.

∴ The correct answer is 34%.

Alternate Method

Given:

Claimed Profit = 15%

Dishonest Weight reduction = 14%

Calculations:

⇒ Let the Cost Price (CP) of 1000g spice be ₹1000.

⇒ Trader sells at 15% profit, so Selling Price (SP) = 1000 + 15% of 1000 = ₹1150.

⇒ Trader uses 14% less weight, so Actual weight given = 1000 − 14% of 1000 = 860g.

⇒ Cost Price for the trader for 860g = ₹860.

⇒ Total Profit = SP − Actual CP = 1150 − 860 = ₹290.

⇒ Total Profit % = (Profit / Actual CP) × 100

⇒ Total Profit % = (290 / 860) × 100 = 2900 / 86 ≈ 33.7209%.

⇒ Rounding to the nearest whole number → 34%.

∴ The correct answer is 34%.

Additional Information

Dishonest Dealer Concept

Profit % = [ (True Weight − Faulty Weight) / Faulty Weight ] × 100. When both profit margin and faulty weight are involved, use the ratio of multipliers.

Successive Percentage for Profit

Total Gain % = [ (1 + g₁/100)(1 + g₂/100) − 1 ] × 100, where g₁ is the price gain and g₂ is the gain from weight manipulation.

Effective Weight Gain

If a trader cheats by x% while buying and y% while selling using faulty weights, the net profit is calculated based on the actual quantity handled relative to the stated price.

Shortcut Trick

Total spend on Food and Education = 20% + 20% = 40% → Remaining = 60% (3/5)

Remaining after Shopping/Transport = 60% × (100% − 40%) = 36% (9/25)

Remaining after Medical = 36% × (100% − 20%) = 28.8% (36/125)

28.8% of Income = ₹23,040

Total Income = 23,040 ÷ 0.288 = 80,000

∴ The correct answer is ₹80,000.

Alternate Method

Given:

Expenditure on Food = 20% | Education = 20%

Expenditure on Shopping & Transport = 40% of the Remaining

Expenditure on Medical = 20% of the Rest

Final Savings = ₹23,040

Let the Monthly Income be ₹x.

⇒ Total spend on Food & Education = (20% + 20%) of x = 0.40x

⇒ Remaining after initial spend = x − 0.40x = 0.60x

⇒ Spend on Shopping & Transport = 40% of 0.60x = 0.24x

⇒ Remaining after shopping = 0.60x − 0.24x = 0.36x

⇒ Spend on Medical = 20% of 0.36x = 0.072x

⇒ Final Savings = 0.36x − 0.072x = 0.288x

⇒ 0.288x = 23,040

⇒ x = 23,040 ÷ 0.288

⇒ x = 80,000

∴ The correct answer is ₹80,000.

Additional Information

Successive Percentage Formula

When spending occurs on the Remaining amount, the net remaining fraction is calculated by multiplying (100 − r_n)% for each step.

Multiplier Method

Final Value = Initial Value × (1 − r₁/100) × (1 − r₂/100) × (1 − r₃/100). This avoids step-by-step subtraction.

Direct Base Comparison

If multiple expenditures are of the Total Income, they are added directly. If they are of the Remaining, they are calculated successively.

Shortcut Trick

Formula: LCM = Product ÷ HCF

Substitute the values: LCM = 98,304 ÷ 128

Shortcut (Unit Digit Method): The unit digit of the product is 4 and the HCF is 8. Since 8 × 8 = 64, the unit digit of the LCM must be 8.

Checking options: Only 768 ends with 8.

∴ The correct answer is 768.

Alternate Method

Given:

Product of two numbers = 98,304

HCF (Highest Common Factor) = 128

Formula Used:

Product of two numbers = HCF × LCM

Calculations:

⇒ 98,304 = 128 × LCM

⇒ LCM = 98,304 ÷ 128

⇒ LCM = 768

∴ The correct answer is 768.

Additional Information

Fundamental Relation

For any two numbers x and y, their Product = HCF(x, y) × LCM(x, y). This specific identity does not hold true for three or more numbers.

HCF and LCM of Fractions

LCM of Fractions = (LCM of Numerators) ÷ (HCF of Denominators). HCF of Fractions = (HCF of Numerators) ÷ (LCM of Denominators).

Divisibility Rule

The LCM of any set of numbers is always exactly divisible by the HCF of that same set of numbers.

Shortcut Trick

Use the Options Elimination method for the Father's age.

If Father's present age is 40 years, then Son's present age = 60 − 40 = 20 years.

5 years ago: Father = 40 − 5 = 35 years and Son = 20 − 5 = 15 years.

Check the Product of numerical values: 35 × 15 = 525.

This matches the given condition exactly.

∴ The correct answer is 40 years.

Alternate Method

Given: Sum of present ages of Father (F) and Son (S) = 60 years.

Formula Used: (F − n) × (S − n) = Product of ages n years ago.

⇒ Let Father's present age = F years.

⇒ Son's present age = (60 − F) years.

⇒ According to the question (5 years ago): (F − 5) × ((60 − F) − 5) = 525

⇒ (F − 5) × (55 − F) = 525

⇒ 55F − F² − 275 + 5F = 525

⇒ 60F − F² − 275 = 525

⇒ F² − 60F + 800 = 0

⇒ F² − 40F − 20F + 800 = 0

⇒ F(F − 40) − 20(F − 40) = 0

⇒ (F − 40)(F − 20) = 0

⇒ F = 40 or F = 20. (Since Father must be older than Son, F = 40)

∴ The correct answer is 40 years.

Additional Information

Constant Age Difference

The difference between the ages of two people always remains constant, regardless of how many years pass.

Sum of Ages over Time

If the sum of ages of 'n' people is S, then 't' years later, the sum will be S + (n × t).

Quadratic Roots for Ages

In age problems involving products, a quadratic equation is formed. Always pick the logically consistent root (e.g., father > son).

Logic followed here is.

Logic: Convert the mathematical statement into a linear equation to find the original number, and then calculate the sum of its digits.

Step 1: Finding the original number

Let the original number be x.

According to the given condition, one-third of the number increased by 25 results in 100.

The equation is: (x / 3) + 25 = 100

Subtracting 25 from both sides:

x / 3 = 100 − 25

x / 3 = 75

Multiplying by 3 to find x:

x = 75 × 3

x = 225

 

Step 2: Calculating the sum of the digits

The original number obtained is 225.

Sum of the digits = 2 + 2 + 5 = 9

 

Therefore, the sum of the digits of the original number is 9.

Shortcut Trick

A number is divisible by 55 if it is divisible by both 5 and 11.

Applying the divisibility rule of 11: Sum of alternate digits difference must be 0 or 11.

⇒ (9 + 7) − (A + B) = 16 − (A + B)

For the number to be divisible by 11, 16 − (A + B) = 11 (since 16 − (A + B) = 0 gives A + B = 16, which doesn't satisfy single-digit constraints for both cases of B).

⇒ A + B = 16 − 11 = 5

∴ The correct answer is 5.

Alternate Method

Given: The number 9A7B is divisible by 55.

Method: A number is divisible by 55 if it is divisible by its co-prime factors 5 and 11.

Step 1: Divisibility by 5

⇒ The last digit B must be 0 or 5.

Step 2: Divisibility by 11

⇒ (Sum of digits at odd positions) − (Sum of digits at even positions)

⇒ (B + A) − (7 + 9) must be 0 or a multiple of 11.

⇒ (A + B) − 16 = 0 or −11 or 11...

Step 3: Finding A + B

⇒ Case 1: (A + B) − 16 = 0 ⇒ A + B = 16.

If B = 0, A = 16 (Not possible, A is a digit). If B = 5, A = 11 (Not possible, A is a digit).

⇒ Case 2: (A + B) − 16 = −11 ⇒ A + B = 5.

If B = 0, A = 5 (Valid). If B = 5, A = 0 (Valid).

∴ The correct answer is 5.

Additional Information

Divisibility by 11

A number is divisible by 11 if the absolute difference between the sum of digits at odd places and even places is either 0 or a multiple of 11.

Composite Divisibility Rule

If a number is divisible by two co-prime numbers x and y, then it is also divisible by their product (x × y).

Divisibility by 5

A number is divisible by 5 if and only if its unit digit is either 0 or 5.

Shortcut Trick

Present Sum of ages = Sum 5 years ago + (2 × 5) = 83 + 10 = 93 years.

The Difference between ages always remains constant, so Difference = 21 years.

Since Ayushi is older, Nidhi's age (younger) = (Sum − Difference) ÷ 2.

Nidhi's present age = (93 − 21) ÷ 2 = 72 ÷ 2 = 36 years.

∴ The correct answer is 36 years.

Alternate Method

Given:

Let the present age of Ayushi be A and Nidhi be N.

Ayushi − Nidhi = 21 years → (Eq. 1)

Sum of ages 5 years ago = 83 years.

Calculations:

⇒ (A − 5) + (N − 5) = 83

⇒ A + N − 10 = 83

⇒ A + N = 83 + 10 = 93 → (Eq. 2)

⇒ From (Eq. 1), A = N + 21. Substitute this in (Eq. 2):

⇒ (N + 21) + N = 93

⇒ 2N + 21 = 93

⇒ 2N = 93 − 21 = 72

⇒ N = 72 ÷ 2 = 36

∴ The correct answer is 36 years.

Additional Information

Constant Age Difference

The difference between the ages of two individuals remains the same at any point in time (past, present, or future).

Sum and Difference Formula

If the sum (S) and difference (D) of two numbers are known, the larger number is (S + D) ÷ 2 and the smaller number is (S − D) ÷ 2.

Time Shifting for Sums

If the sum of ages of 'n' people is 'X' years 't' years ago, their present sum will be X + (n × t).

Shortcut Trick

Speed of first train = 180 / 12 = 15 m/s.

Relative Speed = (Total Length) ÷ Time = (180 + 120) / 15 = 300 / 15 = 20 m/s.

In same direction, Relative Speed = Speed₂ − Speed₁.

Speed of second train = 20 + 15 = 35 m/s.

Convert to km/h: 35 × (18/5) = 126 km/h.

∴ The correct answer is 126 km/h.

Alternate Method

Given:

Length of first train (L₁) = 180 m

Time to pass a stationary pole = 12 sec

Length of second train (L₂) = 120 m

Time to cross each other in same direction = 15 sec

Calculations:

⇒ Speed of first train (v₁) = Length / Time

⇒ v₁ = 180 / 12 = 15 m/s

⇒ Total distance covered during crossing = L₁ + L₂

⇒ Total Distance = 180 + 120 = 300 m

⇒ Relative Speed (v₂ − v₁) = Total Distance / Time of crossing

⇒ v₂ − 15 = 300 / 15

⇒ v₂ − 15 = 20

⇒ v₂ = 20 + 15 = 35 m/s

⇒ Speed of second train in km/h = 35 × (18/5)

⇒ Speed = 7 × 18 = 126 km/h

∴ The correct answer is 126 km/h.

Additional Information

Relative Speed Concept

When two objects move in the same direction, relative speed is the difference of their speeds (V₁ − V₂). If they move in opposite directions, it is the sum (V₁ + V₂).

Unit Conversion

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

Crossing Distance

When a train crosses another train, the total distance covered is always the sum of the lengths of both trains, regardless of the direction of motion.

Shortcut Trick

Solve the Numerator and Denominator independently using the BODMAS rule.

Numerator: (108 ÷ 12) − 12 + 7 ⇒ 9 − 12 + 7 = 4

Denominator: (60 ÷ 10) − (24 ÷ 12) ⇒ 6 − 2 = 4

Simplified result = 4 ÷ 4 = 1

∴ The correct answer is 1.

Alternate Method

Given:

Expression: ((120 − 12) ÷ (24 ÷ 2) − 12 + 7) / (5 × 12 ÷ 10 − (6 × 4) ÷ 12)

Formula Used:

The BODMAS rule defines the order of operations: Brackets, Of, Division, Multiplication, Addition, and Subtraction.

Calculations:

⇒ Numerator = (120 − 12) ÷ (24 ÷ 2) − 12 + 7

⇒ Numerator = 108 ÷ 12 − 12 + 7

⇒ Numerator = 9 − 12 + 7 = 4

⇒ Denominator = (5 × 12 ÷ 10) − (6 × 4 ÷ 12)

⇒ Denominator = (60 ÷ 10) − (24 ÷ 12)

⇒ Denominator = 6 − 2 = 4

⇒ Final Expression = 4 / 4 = 1

∴ The correct answer is 1.

Additional Information

BODMAS Rule

Always solve Brackets first (round (), then curly {}, then square []), followed by Division and Multiplication (from left to right), and finally Addition and Subtraction.

Fraction Simplification

When simplifying complex fractions, treat the Numerator and Denominator as separate expressions and simplify them completely before performing the final division.

Operator Precedence

Multiplication and division hold the same level of precedence; if both appear, solve the expression starting from the left side to the right side.

Shortcut Trick

Area of an equilateral triangle = (√3/4) × side² ⇒ 900 = (√3/4) × a².

side (a) = √(3600 / √3) = 60 / 3^1/4.

Perimeter = 3 × side = 3 × (60 / 3^1/4) = 60 × 3^{1 − 1/4} = 60 × 3^3/4.

Since 3^3/4 = (3³)^1/4 = 27^1/4 = ⁴√27.

∴ The correct answer is 60⁴√27 m.

Alternate Method

Given: Area of the equilateral triangle = 900 m².

Formula Used: Area = (√3 / 4) × a² | Perimeter = 3a (where a = side).

Calculations:

⇒ 900 = (√3 / 4) × a²

⇒ a² = (900 × 4) / √3

⇒ a² = 3600 / 3^1/2

⇒ a = √(3600 / 3^1/2) = 60 / (3^1/2)^1/2

⇒ a = 60 / 3^1/4

⇒ Perimeter = 3 × a = 3 × (60 / 3^1/4)

⇒ Perimeter = 60 × (3¹ / 3^1/4) = 60 × 3^{(1 − 1/4)}

⇒ Perimeter = 60 × 3^3/4 = 60 × (3³)^1/4

⇒ Perimeter = 60⁴√27 m.

∴ The correct answer is 60⁴√27 m.

Additional Information

Height of Equilateral Triangle

The altitude or height (h) is given by h = (√3 / 2) × a.

Inradius and Circumradius

For an equilateral triangle, Inradius (r) = a / (2√3) and Circumradius (R) = a / √3.

Area in terms of Height

The Area can also be calculated using its height as Area = h² / √3.

Properties

In an equilateral triangle, all three internal angles are equal to 60°, and the centroid, orthocenter, circumcenter, and incenter all coincide.

Shortcut Trick

A number divisible by both 6 and 9 must be a multiple of their Least Common Multiple (LCM).

LCM of 6 and 9 ⇒ 18.

Check the given options for multiples of 18: 18, 36, 54, 72, 90...

45 is not a multiple of 18 as 18 × 2.5 = 45.

∴ The correct answer is 45.

Alternate Method

To be divisible by both 6 and 9, a number must be even (divisible by 2), its sum of digits must be divisible by 3 (for 6), and its sum of digits must be divisible by 9.

Given Numbers: 54, 45, 90, 72

Step 1: Check Divisibility by 6 (Divisible by 2 and 3)

⇒ 54: Ends in 4 (Even) and 5 + 4 = 9 (Divisible by 3) → Divisible by 6.

⇒ 45: Ends in 5 (Odd) → NOT divisible by 6.

⇒ 90: Ends in 0 (Even) and 9 + 0 = 9 (Divisible by 3) → Divisible by 6.

⇒ 72: Ends in 2 (Even) and 7 + 2 = 9 (Divisible by 3) → Divisible by 6.

Step 2: Check Divisibility by 9 (Sum of digits divisible by 9)

⇒ 54: 5 + 4 = 9 → Divisible by 9.

⇒ 45: 4 + 5 = 9 → Divisible by 9.

⇒ 90: 9 + 0 = 9 → Divisible by 9.

⇒ 72: 7 + 2 = 9 → Divisible by 9.

Since 45 is divisible by 9 but NOT by 6, it is the correct choice.

∴ The correct answer is 45.

Additional Information

Divisibility Rule of 6

A number is divisible by 6 if it is simultaneously divisible by both 2 (last digit is 0, 2, 4, 6, or 8) and 3 (sum of digits is a multiple of 3).

Divisibility Rule of 9

A number is divisible by 9 if the sum of its digits is completely divisible by 9.

LCM Method for Combined Divisibility

To check divisibility by multiple numbers, find the LCM of those numbers; any common multiple must be divisible by this LCM.

Shortcut Trick

Solve the given expression using the BODMAS rule by simplifying inner parts first.

Simplify Parentheses and Division: (24 ÷ 3) = 8 and (20 ÷ 2) = 10.

Solve Curly Braces: {76 × 8 + 2 × 18} = {608 + 36} = 644.

Complete the Square Brackets and final Multiplication: [644 ÷ 644] × 10 = 1 × 10 = 10.

∴ The correct answer is 10.

Alternate Method

Given: [{76 × (24 ÷ 3) + 2 × 18} ÷ 644] × (20)/(2)

Formula Used: BODMAS Rule (Brackets, Of, Division, Multiplication, Addition, Subtraction)

⇒ [{76 × 8 + 2 × 18} ÷ 644] × 10

⇒ [{608 + 36} ÷ 644] × 10

⇒ [644 ÷ 644] × 10

⇒ 1 × 10

⇒ 10

∴ The correct answer is 10.

Additional Information

BODMAS Rule

The BODMAS rule dictates the order of operations: Brackets, Of (powers/roots), Division, Multiplication, Addition, and Subtraction.

Bracket Hierarchy

Always solve expressions in the order of Circular Brackets ( ), then Curly Braces { }, and finally Square Brackets [ ].

Division and Multiplication

While Division and Multiplication are often grouped, they must be performed strictly as per the priority order or left-to-right to avoid calculation errors.

Shortcut Trick

Total Work = LCM(12, 15, 20) = 60 units. Efficiencies: X = 5, Y = 4, Z = 3.

If a pipe is closed 3 hr before completion, add its potential work to the total work.

New Total Work = 60 + (3 × 3) = 69 units.

Total time = 69 ÷ (5 + 4 + 3) = 69 ÷ 12 = 5.75 hr.

5.75 hr = 5 hr + (0.75 × 60) min = 5 hr 45 min.

∴ The correct answer is 5 hr 45 min.

Alternate Method

Given: Pipes X, Y, and Z fill the tank in 12 hr, 15 hr, and 20 hr respectively.

Method: LCM efficiency approach. Let the total time to fill the tank be t hr.

⇒ Work done by X in t hr = 5 × t

⇒ Work done by Y in t hr = 4 × t

⇒ Work done by Z in (t − 3) hr = 3 × (t − 3)

⇒ Total work = 5t + 4t + 3(t − 3) = 60

⇒ 9t + 3t − 9 = 60

⇒ 12t = 69

⇒ t = 69 ÷ 12 = 5.75 hr

⇒ t = 5 hr + (0.75 × 60) min = 5 hr 45 min

∴ The correct answer is 5 hr 45 min.

Additional Information

Negative Efficiency

For outlet pipes or leakage, efficiency is taken as negative. Total efficiency = Σ(Inlet Efficiency) − Σ(Outlet Efficiency).

Combined Time Formula

If pipe A fills in x hr and pipe B fills in y hr, the total time taken when both are open is (x × y) ÷ (x + y).

Leaving Early Case

If a pipe stops n hours before completion, its work for n hours is added to the total work to find the total time directly.

Shortcut Trick

Total interest percentage = Rate × Time = 8% × 3 = 24%.

Given that 24% of the Principal is equal to ₹720.

⇒ 1% of Principal = 720 ÷ 24 = 30.

⇒ 100% (Sum of money) = 30 × 100 = ₹3000.

∴ The correct answer is ₹3000.

Alternate Method

Given:

Simple Interest (SI) = ₹720

Time (T) = 3 years

Rate (R) = 8% p.a.

Formula Used:

Simple Interest (SI) = (P × R × T) ÷ 100

⇒ 720 = (P × 8 × 3) ÷ 100

⇒ 720 = (P × 24) ÷ 100

⇒ P = (720 × 100) ÷ 24

⇒ P = 30 × 100

⇒ P = ₹3000

∴ The correct answer is ₹3000.

Additional Information

Total Amount

The total sum of money payable at the end of the time period is A = P + SI = P(1 + RT/100).

Simple Interest Characteristics

Unlike compound interest, simple interest is always calculated only on the initial Principal amount, meaning the interest remains constant for every year if the rate is fixed.

Rate of Interest Formula

To find the annual rate when other variables are known: R = (SI × 100) ÷ (P × T).

Shortcut Trick

Assume c = 0 as there are 3 variables and only 2 equations.

From the given: a + b = 13 and ab = 4.

The expression reduces to ab(a + b) + 0 + 0 + 0.

Substituting the values: 4 × 13 = 52.

∴ The correct answer is 52.

Alternate Method

Given: a + b + c = 13 and ab + bc + ca = 4.

From a + b + c = 13, we can derive:

⇒ a + b = 13 − c

⇒ b + c = 13 − a

⇒ c + a = 13 − b

Now, substitute these into the required expression ab(a + b) + bc(b + c) + ca(c + a) + 3abc:

⇒ ab(13 − c) + bc(13 − a) + ca(13 − b) + 3abc

⇒ 13ab − abc + 13bc − abc + 13ca − abc + 3abc

⇒ 13(ab + bc + ca) − 3abc + 3abc

⇒ 13(4)

⇒ 52

∴ The correct answer is 52.

Additional Information

Square of a Trinomial

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca). This identity relates the sum of terms, sum of squares, and sum of products taken two at a time.

Sum of Cubes Identity

a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca). If a + b + c = 0, then a³ + b³ + c³ = 3abc.

Value Assumption Method

In competitive exams, if an expression is symmetric and there are more variables than equations, setting one variable to 0 or 1 often leads to a quick solution.

Shortcut Trick

Use the Empirical Relation for moderately skewed distributions: Mode = 3 × Median − 2 × Mean.

Substitute the given values: Mode = 3 × 61 − 2 × 47.

Mode = 183 − 94 = 89.

∴ The correct answer is 89.

Alternate Method

Given:

Mean of the data = 47

Median of the data = 61

Formula Used:

The Empirical Relationship between the three measures of central tendency is:

Mode = 3(Median) − 2(Mean)

Calculations:

⇒ Mode = 3 × 61 − 2 × 47

⇒ Mode = 183 − 94

⇒ Mode = 89

∴ The correct answer is 89.

Additional Information

Symmetrical Distribution

In a perfectly symmetrical distribution (like the Normal Distribution), the values of Mean, Median, and Mode are all identical (Mean = Median = Mode).

Positively Skewed Distribution

In a positively skewed (right-skewed) distribution, the values generally follow the order: Mean > Median > Mode.

Negatively Skewed Distribution

In a negatively skewed (left-skewed) distribution, the values generally follow the order: Mean < Median < Mode.

Karl Pearson Formula

The empirical relation can also be expressed as: (Mean − Mode) = 3 × (Mean − Median), which is useful for calculating the coefficient of skewness.

Shortcut Trick

Selling Price (SP) of article X = 800 + 901 = ₹1701.

Effective multiplier for successive discounts of 73% and 37% = 0.27 × 0.63 = 0.1701.

Marked Price (MP) = 1701 ÷ 0.1701 = ₹10000.

SP of article Y = 10000 × (100% − 89%) = 10000 × 0.11 = ₹1100.

Profit % on Y = ((1100 − 800) ÷ 800) × 100 = 3/8 × 100 = 37.5%.

∴ The correct answer is 37.50%.

Alternate Method

Given:

Cost Price (CP) of X and Y = ₹800 each.

Profit on article X = ₹901.

Successive discounts on X = 73% and 37%.

Single discount on Y = 89%.

Formula Used:

SP = CP + Profit

SP = MP × (100 − D₁)% × (100 − D₂)%

Profit % = [(SP − CP) ÷ CP] × 100

Calculations:

⇒ SP of X = CP + Profit = 800 + 901 = ₹1701

⇒ Let Marked Price be MP.

⇒ 1701 = MP × (100 − 73)/100 × (100 − 37)/100

⇒ 1701 = MP × 27/100 × 63/100

⇒ 1701 = MP × 1701 / 10000

⇒ MP = ₹10000

⇒ For article Y, SP = MP × (100 − 89)/100

⇒ SP of Y = 10000 × 11/100 = ₹1100

⇒ Profit on Y = 1100 − 800 = ₹300

⇒ Profit % = (300 ÷ 800) × 100

⇒ Profit % = 3/8 × 100 = 37.5%

∴ The correct answer is 37.50%.

Additional Information

Successive Discount Formula

A single discount equivalent to two successive discounts d₁% and d₂% is given by (d₁ + d₂ − d₁×d₂/100)%.

Relationship between MP and CP

MP/CP = (100 + Profit%)/(100 − Discount%). This is useful when profit and discount percentages are directly given.

Percentage to Fraction Conversion

Important standard fractions: 1/8 = 12.5%, 3/8 = 37.5%, 5/8 = 62.5%, and 7/8 = 87.5%.

Shortcut Trick

Since cot θ = 1/√3 and 0° < θ < 90°, the value of θ is 60°.

Substitute θ = 60° into the given expression (treating the term as cos² θ to match the options).

Expression = (2 − sin² 60°) / (1 + cos² 60°) + cosec² 60° − sec² 60°

⇒ (2 − 3/4) / (1 + 1/4) + 4/3 − 4 = (5/4) / (5/4) + 4/3 − 4 = 1 + 4/3 − 4 = −5/3.

∴ The correct answer is − 5/3.

Alternate Method

Given: cot θ = 1/√3 where 0° < θ < 90°.

Formula Used:

If cot θ = 1/√3, then θ = 60°.

Fundamental values: sin 60° = √3/2, cos 60° = 1/2, cosec 60° = 2/√3, sec 60° = 2.

Calculations:

⇒ sin² 60° = (√3/2)² = 3/4

⇒ cos² 60° = (1/2)² = 1/4

⇒ cosec² 60° = (2/√3)² = 4/3

⇒ sec² 60° = 2² = 4

⇒ First Term = (2 − 3/4) / (1 + 1/4) = (5/4) / (5/4) = 1

⇒ Expression = 1 + 4/3 − 4

⇒ Expression = −3 + 4/3

⇒ Expression = (−9 + 4) / 3 = −5/3

∴ The correct answer is − 5/3.

Additional Information

Reciprocal Trigonometric Identities

cosec θ = 1/sin θ | sec θ = 1/cos θ | cot θ = 1/tan θ

Pythagorean Identities

sin² θ + cos² θ = 1 | 1 + tan² θ = sec² θ | 1 + cot² θ = cosec² θ

Double Angle Identity for Cosine

cos 2θ = cos² θ − sin² θ = 2cos² θ − 1 = 1 − 2sin² θ

Values at 60°

sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3

Shortcut Trick

Sum of parallel sides = (2 × Area) ÷ Height

Sum = (2 × 988) ÷ 52 = 1976 ÷ 52 = 38 cm

Larger side = (Sum + Difference) ÷ 2 = (38 + 9) ÷ 2 = 23.5 cm

Smaller side = (Sum − Difference) ÷ 2 = (38 − 9) ÷ 2 = 14.5 cm

∴ The correct answer is 23.5, 14.5 cm.

Alternate Method

Given:

Area of trapezium = 988 cm²

Perpendicular distance (h) = 52 cm

Difference between parallel sides (a − b) = 9 cm

Formula Used:

Area = 1/2 × (Sum of parallel sides) × (Distance between them)

Calculations:

⇒ Let the parallel sides be a and b.

⇒ a − b = 9 --- (Equation 1)

⇒ Area = 1/2 × (a + b) × 52 = 988

⇒ (a + b) × 26 = 988

⇒ a + b = 988 ÷ 26

⇒ a + b = 38 --- (Equation 2)

⇒ Adding Eq. 1 and Eq. 2: 2a = 47

⇒ a = 47 ÷ 2 = 23.5 cm

⇒ Subtracting Eq. 1 from Eq. 2: 2b = 29

⇒ b = 29 ÷ 2 = 14.5 cm

∴ The correct answer is 23.5, 14.5 cm.

Additional Information

Area of Trapezium

The area is calculated as Area = 1/2 × (Sum of parallel sides) × height.

Median of a Trapezium

The segment connecting the midpoints of the non-parallel sides. Length = (a + b) ÷ 2.

Isosceles Trapezium

A trapezium where the non-parallel sides are equal and the base angles are equal.

Perimeter of Trapezium

The sum of all four sides: P = a + b + c + d.

Shortcut Trick

Identify the Pythagorean triplet ratio 3 : 4 : 5.

Given Side = 15 (3 × 5) and Diagonal = 25 (5 × 5).

Other Side = 4 × 5 = 20 m.

Perimeter = 2 × (Length + Breadth) = 2 × (15 + 20) = 70 m.

∴ The correct answer is 70 m.

Alternate Method

Given:

Diagonal (d) of rectangular park = 25 m

One side (l) = 15 m

Formula Used:

Pythagorean Theorem: d² = l² + b²

Perimeter = 2 × (l + b)

Calculations:

⇒ 25² = 15² + b²

⇒ 625 = 225 + b²

⇒ b² = 625 − 225

⇒ b² = 400

⇒ b = √400 = 20 m

⇒ Perimeter = 2 × (15 + 20)

⇒ Perimeter = 2 × 35 = 70 m

∴ The correct answer is 70 m.

Additional Information

Area of Rectangle

The area is the product of its length and breadth: Area = l × b.

Properties of Rectangles

All internal angles are 90° and diagonals are equal in length and bisect each other.

Common Pythagorean Triplets

Frequently used triplets in exams include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25).

Diagonal of a Square

For a square of side 'a', the diagonal is always a√2.

The logic followed here is based on comparing the letters and their corresponding codes in the given words to find the unique digit for each letter.

Given:

KIDS is coded as '1248'

SILK is coded as '2941'

 

Therefore, the code for 'L' is 9.

Hence, the correct answer is "Option 2".

Given:

FG 151 : EH 101

MN 262 : LO 212

UV 379 : ?

Hence, the correct answer is 'Option 2'.

The logic followed here is:

The given number series is a multiplication series where each subsequent term is obtained by multiplying the previous term by a constant factor of 4.

Thus, the missing term in the series is 5120.

Hence, the correct answer is 'Option 1';

The logic followed here is

Logic: Second letter = First letter + 3; Third letter = Second letter − 2 (based on the alphabetical positions of the letters).

Option 1)  JMK,

Option 2) GFT

This letter-cluster does not follow the rule.

Option 3) DGE,

Option 4) PSQ,

Therefore, the letter-cluster that is different from the others is GFT.

Hence, the correct answer is 'Option 2'.

The logic followed in this pattern is a constant alphabetical shift where each letter of the first group is moved 10 positions backward (or 16 positions forward) in the alphabetical sequence to obtain the corresponding letter in the second group.

For 'GDA' related to 'WTQ':

For 'URO' related to 'KHE':

Applying the same logic to 'AXU':

Thus, the pair AXU-QNK follows the same pattern.

Hence, the correct answer is 'Option 3'.

Given mapping:

The expression is: 4 B 5 D 20 A 2 C 17 = ? D 3

Let the missing value (?) be represented as x.

By substituting the values in the equation:

= 4 × 5 − 20 ÷ 2 + 17 = x − 3

= 4 × 5 - 10 + 17 + 3 = x

= 20 - 10 + 17 + 3 = x

= 40 - 10 = x

= x = 30

Therefore, the value that will come in place of the question mark is 30.

Hence, the correct answer is "Option 1".

Given Word:

LARKISH

Rule:

• Each vowel is changed to the next letter in the English alphabet.
• Each consonant is changed to the previous letter in the English alphabet.

<br>

Transformation:

New group of letters:

K, B, Q, J, J, R, G

Letter frequencies:

J → 2 times

K, B, Q, R, G → 1 time each

Therefore, only 1 letter appears exactly twice.

Hence, the correct answer is "Option 2".

Logic followed here is the Comparison Method of coding, where we identify common letters and digits between the given words.

Given:

CURL is coded as '4386'

LURE is coded as '5846'

 

Therefore, the code for 'E' is 5.

Hence, the correct answer is "Option 4".

1. Assume 7 people are sitting around a circular table facing the centre. In this arrangement, Left is Clockwise and Right is Counter-clockwise. Let's place Q at position 1. U sits third to the left of Q, so we move clockwise: 1 → 2 → 3 → 4. U is at position 4.

2. Q is the immediate neighbour of both T and W. This means T and W must be at positions 2 and 7. W sits second to the left of R (meaning R is second to the right of W). If W is at pos 2, R would be at pos 7 (occupied by T). So W must be at pos 7 and T at pos 2. Moving counter-clockwise from W(7): 7 → 6 → 5. So R is at pos 5.

3. E is not an immediate neighbour of U. The neighbours of U(4) are pos 3 and pos 5(R). Therefore, E cannot be at pos 3. E must be at pos 6. The remaining person, Y, sits at pos 3.

To find the number of people between W and Y from the right of Y: Counting right (counter-clockwise) from Y(3) to W(7), the people are U(4), R(5), and E(6). Thus, there are 3 people

between them.

<br>

Hence, the correct answer is "Option 2".

Given Series:

(Left) 2 4 9 6 7 1 4 6 3 9 5 2 4 8 6 1 3 2 7 6 2 5 8 1 (Right)

Required pattern:

Odd Digit → Odd Digit → Even Digit

Checking the series:

(Left) 2 4 9 7 1 4 4 6 3 3 9 5 5 9 5 2 2 4 8 6 1 3 2 7 6 2 5 8 1 (Right)

Odd digits satisfying the condition:

→ 1 in (7 1 4)

→ 5 in (9 5 2)

→ 3 in (1 3 2)

Therefore, there are 3 such odd digits.

Hence, the correct answer is "Option 2 — Three".

The movement of Chandan from the starting Point A can be traced as follows:

• Chandan starts from Point A and drives 11 km towards the East.
• He takes a left turn and drives 8 km towards the North.
• He takes another left turn and drives 13 km towards the West.
• He takes another left turn and drives 13 km towards the South.
• He takes a final left turn and drives 2 km towards the East and stops at Point P.

Step-by-step Calculation:

Chandan is currently 5 km South of Point A. To reach Point A again, he must drive towards the North.

Therefore, he should drive 5 km to the north.

Hence, the correct answer is "Option 2".

Given:

Let us first decode the given symbols and then draw a family tree.

 

On checking: A + B @ H # K ÷ T

1. A + B - 'A + B' means A is the daughter of B.

2. B @ H - 'B @ H' means B is the father of H.

3. H # K - 'H # K' means H is the mother of K.

4. K ÷ T- 'K ÷ T' means K is the wife of T.

From the family tree, H is the mother of K (T's wife) and A is the sister of H. Therefore, A is the Wife's mother's sister of T.

Hence, the correct answer is "Option 3".

The logic followed here is

Logic: First letter + 4 = Second letter and Second letter − 3 = Third letter (based on their alphabetical positions).

Option 1) JNK,

Option 2) GKH,

Option 3) QUR,

Option 4) INJ

This letter cluster does not follow the (+4, −3) pattern.

Therefore, the letter cluster that is different from the others is INJ.

Hence, the correct answer is 'Option 4'.

The logic followed here is:

Position of Nehal from the top = 22^nd

Position of Nehal from the bottom = 10^th

To find the total number of students in the class, the formula used is:

Total number of students = (Position from the top + Position from the bottom) − 1

= (22 + 10) − 1

= 32 − 1

= 31

Hence, the correct answer is 'Option 2'.

Given Word:

BESWARM

Rule:

• Change each letter to the next letter in the English alphabetical order.
• Then arrange all the new letters in alphabetical order.

<br>

Transformation:

<br>

New letters:

C, F, T, X, B, S, N

Arranged alphabetically:

B, C, F, N, S, T, X

Counting from the right:

Therefore, the 4th letter from the right is N.

Hence, the correct answer is "Option 4 — N".

Logic followed here is the tracking of movement based on the cardinal directions (North, South, East, and West) and calculating the final position relative to the starting point.

Step-by-step movement from Point A:

1. Sunny starts from Point A and drives 5 km towards the north. 2. He then takes a left turn, drives 6 km, turns left and drives 7 km. 3. He then takes a left turn and drives 8 km. 4. He takes a final left turn, drives 2 km and stops at Point P.

Calculating the final coordinates relative to Point A (0,0):

Therefore, Sunny should drive 2 km to the West to reach Point A again.

Hence, the correct answer is "Option 3".

1. Since all persons face the centre: Right = Counterclockwise and Left = Clockwise. Fix F at position 1. E sits third to the left of F (clockwise direction): F(1) → 2 → 3 → E(4).

2. F sits second to the left of D. This means D is positioned such that F is two places clockwise from it: D → skip 1 → F(1). Therefore, D is at position 5: D(5) → pos(6) → F(1).

3. Only J sits between F and G. Since F is at position 1 and position 5 is occupied by D, the sequence F-J-G must occupy positions 1, 2, and 3. Thus, J is at position 2 and G is at position 3.

4. K sits in the remaining position 6. We verify that K(6) is not an immediate neighbor of E(4), which is correct as E is flanked by G(3) and D(5).

<br>

To find how many people sit between K and G when counted from the right of G, we count in a counterclockwise direction from G(3) to K(6): G(3) → J(2) → F(1) → K(6). The people between them are J and F, which totals Two.

<br>

Hence, the correct answer is 'Option 4'.

The logic followed here is:

Logic: Each term in the series is obtained by multiplying the previous term by 2.

 

The missing number in the series is 416.

Hence, the correct answer is "Option 2".

Given:

1) J is kept second from the bottom.

2) Only three boxes are kept between J and I.

3) K is kept at one of the positions below J.

 

4) Only G is kept between I and L.

5) H is kept at one of the positions above I.

Final Arrangement:

 

The box kept between O (position 3) and K (position 1) is J (position 2).

Thus, only one box is kept between O and K.

Hence, the correct answer is 'Option 3'.

Logic followed here is based on the alphabetical positions of the letters in each cluster.

For DZA:

For MJK:

For EAB:

For CYZ:

The pattern followed by DZA, EAB, and CYZ is (−4, +1), whereas MJK follows the pattern (−3, +1).

Therefore, MJK is the odd one out.

Hence, the correct answer is "Option 2".

Sign Mapping:

 

For the given equation:

5 B 2 D 10 A 2 C 12 = ? D 3 →

Let the value of the question mark be x.

By substituting the signs into the equation, we get:

= 5 × 2 − 10 ÷ 2 + 12 = x − 3

= 5 × 2 - 5 + 12 + 3 = x

= 10 - 5 + 12 + 3 = x

= 25 - 5 = x

= x = 20

Therefore, the value that will come in place of the question mark is 20.

Hence, the correct answer is "Option 4".

Given Series:

(Left) 7 3 2 5 1 4 6 8 3 4 9 7 5 4 2 6 1 7 9 (Right)

Required pattern:

Even Digit → Digit → Odd Digit

Checking the series:

(Left) 7 3 2 5 1 4 6 8 3 4 9 7 5 4 2 6 1 7 9 (Right)

Digits satisfying the condition:

→ 5 in (2 5 1)

→ 8 in (6 8 3)

→ 9 in (4 9 7)

→ 6 in (2 6 1)

→ 1 in (6 1 7)

Therefore, there are 5 such digits.

Hence, the correct answer is "Option 1 — Five".

Logic followed here is.

The movement of Sanjay from the starting Point Y to the final Point Z is calculated by determining the net displacement in the Horizontal (East-West) and Vertical (North-South) directions.

Movement Analysis:

<br>

Calculation of Displacement:

Total Horizontal Displacement (East-West):

− 32 − 66 + 45 = − 53 km (which means 53 km West from Point Y).

Total Vertical Displacement (North-South):

− 51 + 14 + 37 = 0 km.

<br>

Shortest distance to reach Point Y from Point Z:

Position of Point Z relative to Point Y is 53 km West.

To reach Point Y from Point Z, he must travel in the opposite direction (East).

Distance = 53 km

Direction = East

<br>

Therefore, he should drive 53 km towards East to reach Point Y again.

The logic followed in this pattern is a series of alphabetical shifts where each letter is moved backward by a decreasing value (−3, −2, −1).

For 'ZRD' related to 'WPC':

For 'WET' related to 'TCS':

Applying the same logic to 'YDG':

Thus, YDG is related to VBF.

Hence, the correct answer is 'Option 1'.

The logic followed here is based on the positions of the letters in the English Alphabetical order (A=1, B=2, ..., Z=26).

Logic: Second letter = First letter + 3; Third letter = Second letter − 2.

Option 1)  RUS,

Option 2) KNL,

Option 3) NQO,

Option 4) MHG,

This letter-cluster does not follow the rule.

Therefore, the letter-cluster that is different from the others is MHG.

Hence, the correct answer is 'Option 4'.

The correct answer is Only (A) and (D).

Key Points

• Statement A refers to High melting and boiling points. Ionic compounds are composed of a giant 3D crystal lattice where ions are held together by exceptionally strong electrostatic forces of attraction. Overcoming these forces to change the state requires a massive amount of thermal energy. Hence, statement A is correct.
• Statement B discusses solubility in non-polar solvents. Ionic compounds are polar and strictly follow the like dissolves like principle. While they dissolve readily in polar solvents like water (due to ion-dipole interactions), they are generally insoluble in non-polar solvents such as benzene, ether, or kerosene. Hence, statement B is incorrect.
• Statement C describes electrical conductivity in the solid state. In solid form, the ions are locked in fixed positions within the crystalline structure and are not free to move. Electricity is only conducted in molten or aqueous states where the lattice breaks down, allowing ions to migrate. Hence, statement C is incorrect.
• Statement D highlights that these compounds are brittle. When a mechanical force is applied, layers of the lattice shift, bringing ions of the same charge into close proximity. The resulting strong electrostatic repulsion causes the crystal to shatter easily. Hence, statement D is correct.

 Additional Information

• Nature of Ionic Bonding:
• Formed through the complete transfer of valence electrons from a metal to a non-metal.
• Involves the formation of cations (positive) and anions (negative).
• Prominent examples include Sodium Chloride (NaCl) and Magnesium Oxide (MgO).
• Solubility and Hydration:
• Solubility depends on the balance between Lattice Energy and Hydration Energy.
• Water has a high dielectric constant, which effectively weakens the attraction between ions, facilitating dissolution.
• Crystalline Structure:
• Ionic solids exist as regular geometric patterns rather than discrete molecules.
• The specific coordination number of an ion depends on the relative sizes of the cation and anion.

The correct answer is Both (i) and (iii).

Key Points

• Statement (i) refers to the phenomenon where white light, upon entering a glass prism, splits into its constituent spectrum of seven colors. This occurs because different colors of light possess different wavelengths and travel at different speeds within the glass, causing them to bend at different angles. Hence, statement (i) is correct.
• Statement (ii) claims that the speed of light remains constant. In physics, the speed of light is only a constant (approximately 3 × 10⁸ m/s) in a vacuum. When light transitions from a rarer medium like air to a denser medium like glass, its velocity must change according to the optical density of the material. Hence, statement (ii) is incorrect.
• Statement (iii) correctly identifies that the speed of light decreases. Glass has a higher refractive index (approximately 1.5) compared to air (approximately 1.0003). As per the relation v = c/n, where v is the velocity in the medium and n is the refractive index, a higher index leads to a lower velocity. Hence, statement (iii) is correct.

 Additional Information

• Refraction and Dispersion:
• Refraction is the bending of light as it passes from one transparent substance into another. This bending is caused by the change in speed.
• The Angle of Deviation depends on the wavelength; Violet light deviates the most because it has the shortest wavelength and slows down the most, while Red light deviates the least.
• Refractive Index (n):
• It is a dimensionless number that describes how fast light travels through the material.
• The Absolute Refractive Index is calculated as the ratio of the speed of light in a vacuum to the speed of light in the medium.
• For optical glass, the index typically ranges from 1.5 to 1.9 depending on the density and composition.

The correct answer is Both statements I and II are true.

Key Points

• Olfactory indicators are specific substances whose smell or odour varies significantly depending on whether they are placed in an acidic or basic medium. The term 'olfactory' is derived from the sense of smell, making these indicators unique compared to visual indicators like Litmus paper or Methyl orange. Hence, statement I is correct.
• Onion serves as a reliable olfactory indicator in laboratory settings; when finely chopped onion is treated with a basic solution like Sodium Hydroxide (NaOH), its characteristic pungent smell is neutralized and cannot be detected. However, in an acidic medium like Hydrochloric Acid (HCl), the smell persists. Hence, statement II is correct.
• Vanilla extract is another common example used to distinguish between acids and bases. The pleasant, sweet smell of vanilla disappears when it is added to a basic solution, whereas the smell remains unchanged when added to an acidic solution. Hence, statement II is correct.

 Additional Information

• Clove Oil:
• Like onion and vanilla, Clove oil is a prominent olfactory indicator. Its distinct smell is lost in the presence of a base but is retained in the presence of an acid.
• Educational Utility:
• Olfactory indicators are extremely important for visually impaired students in chemistry laboratories. Since these students cannot observe color changes in indicators like Phenolphthalein, they rely on their sense of smell to perform acid-base titrations.
• Chemical Reaction:
• The change in odour occurs because the hydroxide ions (OH−) in a base react with the organic compounds responsible for the smell in these indicators, effectively neutralizing the volatile molecules. Acids (H+) typically do not destroy these specific aromatic compounds.

The correct answer is A molecule of an element consists of only one type of atom bonded together.

Key Points

• Molecules of an element are defined as groups of two or more atoms of the same chemical element held together by chemical bonds. For instance, an Oxygen molecule (O₂) contains only oxygen atoms. Hence, statement 2 is correct.
• Molecules that contain two or more different types of atoms are classified as compounds (e.g., CO₂), not molecules of a single element. Hence, statement 1 is incorrect.
• Many elements naturally exist as molecules rather than isolated atoms to achieve chemical stability. Examples include Nitrogen (N₂), Hydrogen (H₂), and Phosphorus (P₄). Hence, statement 3 is incorrect.
• While some elements like Noble Gases exist as monoatomic entities, many other elements exist as diatomic or polyatomic molecules in their natural state. Hence, statement 4 is incorrect.

 Additional Information

• Understanding Atomicity:
• Monoatomic: Elements that are stable as single atoms, such as Helium (He), Neon (Ne), and Argon (Ar).
• Diatomic: Molecules composed of exactly two atoms of the same element. This group includes Halogens like Fluorine (F₂) and Iodine (I₂).
• Polyatomic: Elements consisting of three or more atoms. Examples include Ozone (O₃) and Sulfur, which often forms ring-shaped S₈ molecules.
• Chemical Bonding in Elements:
• Atoms within a molecule of an element are typically joined by covalent bonds.
• Since the atoms are identical, they have the same electronegativity, resulting in non-polar covalent bonds.
• This configuration allows elements to fill their valence shells and reach a lower potential energy state.

The correct answer is 11.11 m/s.

Key Points

• The question asks for the conversion of the final speed from kilometers per hour (km/h) to meters per second (m/s).
• The given final speed of the train is 40 km/h. The train starts from rest, meaning its initial velocity (u) is 0.
• To convert a speed value from km/h to m/s, we multiply the value by the conversion factor 5/18.
• This factor is derived because 1 kilometer is equal to 1000 meters and 1 hour is equal to 3600 seconds (60 minutes × 60 seconds). Therefore, 1 km/h = 1000 / 3600 m/s = 5/18 m/s.
• Calculation: 40 × (5/18) = 200 / 18.
• Performing the division: 200 ÷ 18 = 11.111..., which is rounded to 11.11 m/s.
• Although the time is given as 10 minutes (which is 600 seconds), it is not required for this specific unit conversion of the final speed, but it would be necessary if calculating acceleration.

 Additional Information

• Units of Measurement:
• Meters per second (m/s) is the SI unit (International System of Units) for velocity and speed, used globally in scientific calculations.
• Kilometers per hour (km/h) is a common unit used for vehicle speeds and transport systems in most countries.
• Motion Parameters:
• Acceleration (a): Defined as the rate of change of velocity. Formula: a = (v − u) / t.
• In this context, the acceleration of the train is (11.11 − 0) / 600, resulting in approximately 0.0185 m/s².
• Uniform Acceleration: The calculation assumes the train speeds up at a constant rate over the 10-minute duration.

The correct answer is 126 W.

Key Points

• Power is defined as the rate at which work is done or energy is transformed per unit of time.
• The formula for calculating power is Power (P) = Work Done (W) / Time (t).
• Work Done in this context is the product of the Force (weight of the girl) and the Displacement (height climbed) in the direction of the force.
• According to the given data, the girl exerts a force equivalent to her weight, which is 420 N, to move a distance of 6 m.
• The total Work Done is calculated as: Work = 420 N × 6 m = 2520 Joules.
• The time taken to perform this specific amount of work is 20 seconds.
• By applying the values to the power formula: Power = 2520 J / 20 s = 126 Watts.

 Additional Information

• Watt (W):
• It is the SI unit of power, representing one Joule per second (1 J/s).
• The unit is named after the Scottish inventor James Watt for his contributions to the development of the steam engine.
• Scalar Quantity:
• Power is a scalar quantity because it only has magnitude and no specific direction.
• It describes the efficiency or speed of an energy conversion process.
• Dimensional Formula:
• The dimensional representation of power is [M¹L²T⁻³].
• This is derived from Work [M¹L²T⁻²] divided by Time [T].

The correct answer is C > A > B.

Key Points

• In chemistry, a displacement reaction occurs when a more reactive element removes a less reactive element from its compound or salt solution.
• The statement 'A displaces B' logically concludes that A is more chemically active than B. In terms of the reactivity series, A must be positioned higher than B to facilitate this reaction. This establishes the relationship A > B.
• The statement 'C displaces A' signifies that C possesses a higher oxidation potential or greater reactivity than A. Therefore, C is capable of replacing A in a chemical bond, establishing the relationship C > A.
• By synthesizing these two logical inequalities (C > A and A > B), we derive the complete sequence of reactivity as C > A > B.
• This hierarchy is a fundamental aspect of inorganic chemistry, where elements are ranked based on their ability to lose or gain electrons during redox reactions.

 Additional Information

• The Reactivity Series:
• It is an analytical arrangement of metals in descending order of their chemical reactivity.
• Metals like Potassium (K) and Sodium (Na) are at the top because they react vigorously with water and oxygen even at room temperature.
• Metals like Gold (Au), Silver (Ag), and Platinum (Pt) are at the bottom because they are highly stable and resistant to oxidation.
• The ability of a metal to displace Hydrogen from acids is a key metric used to determine its position relative to other elements in the series.
• This principle is essential in extractive metallurgy, specifically in carbon reduction or electrolytic reduction processes used to obtain pure metals from ores.

The correct answer is Cell wall made of cellulose.

Key Points

• The cell wall is a rigid, semi-permeable protective layer found on the outside of the plasma membrane in plant cells, fungi, and certain prokaryotes.
• In a hypotonic environment, the concentration of solutes outside the cell is lower than inside; consequently, water enters the cell through the process of osmosis.
• As water fills the vacuole, it exerts an internal force known as turgor pressure against the cell wall.
• The cellulose microfibrils within the cell wall provide immense tensile strength, allowing the wall to exert an equal and opposite wall pressure.
• This mechanical resistance prevents the cytoplast from expanding to the point of lysis or bursting, which would otherwise occur in animal cells lacking this structure.
• The resulting state, called turgidity, is essential for maintaining the upright posture of non-woody plants and structural stability.

 Additional Information

• Cellulose:
• It is a polysaccharide consisting of a linear chain of several hundred to many thousands of β(1→4) linked D-glucose units.
• It is the primary structural component of the primary cell wall of green plants and is the most abundant organic polymer on Earth.
• Vacuole:
• A large membrane-bound organelle that occupies a significant volume of the plant cell.
• It stores water, enzymes, and waste products and plays a critical role in generating the osmotic gradient that drives water entry.
• Cell Membrane:
• Also known as the plasma membrane, it is a thin phospholipid bilayer that is selectively permeable.
• While it manages the transport of molecules, it is flexible and fragile; without the cell wall, it cannot withstand high internal pressure.
• Cytoplasm:
• The thick solution that fills each cell and is enclosed by the cell membrane, containing cytosol, organelles, and various particles.
• It serves as the medium for metabolic reactions but does not provide mechanical resistance against osmotic bursting.

The correct answer is Both the statements are true.

Key Points

• Statement I: Most carbon compounds are covalent in nature, meaning they are formed by the sharing of electrons between atoms rather than the complete transfer of electrons. Because electricity requires the movement of mobile charge carriers like free electrons or ions, and covalent compounds do not dissociate into ions in solution or molten states, they are typically poor conductors of electricity. While exceptions like graphite exist due to its unique structure, the vast majority of organic carbon compounds are insulators. Hence, statement I is correct.
• Statement II: Carbon compounds generally consist of molecules held together by relatively weak intermolecular forces, such as Van der Waals forces or London dispersion forces. In contrast, ionic compounds are held together by extremely strong electrostatic forces of attraction between oppositely charged ions in a crystal lattice. Because significantly less thermal energy is required to overcome the weak attractions between covalent carbon molecules compared to the strong bonds in ionic compounds, carbon compounds exhibit much lower melting and boiling points. Hence, statement II is correct.

 Additional Information

• Bonding and Conductivity:
• Carbon's Tetravalency: Carbon has four valence electrons, allowing it to form four covalent bonds with other atoms.
• Non-Electrolytes: Most carbon compounds do not conduct because they lack delocalized electrons. Graphite is an exception because each carbon atom is bonded to only three others in a hexagonal layer, leaving one electron per atom free to move.
• Physical Properties:
• Ionic Compounds: Examples like Sodium Chloride (NaCl) have very high melting points (801°C) because of the strong Coulombic attractions.
• Carbon Compounds: Examples like Methane (CH₄) or Ethanol have very low melting points and often exist as gases or liquids at room temperature due to weak intermolecular bonding.
• Catenation: This is carbon's unique ability to form long chains and structures, leading to a diverse range of compounds with varying physical properties, yet most maintain their covalent characteristics.

The correct answer is 0 m/s.

Key Points

• Average velocity is defined as the total displacement divided by the total time taken for the motion.
• Displacement is a vector quantity that measures the shortest straight-line distance between the initial position and the final position of an object.
• In this problem, Reshma swims in a 120 m long pool and covers a total distance of 240 m. This indicates she swims from the starting point to the other end (120 m) and returns to the original starting point (another 120 m).
• Because she returns to her starting position, her final position is the same as her initial position. Therefore, her total displacement is 0 m.
• The time taken is given as 1 minute, which is equal to 60 seconds.
• Calculating the average velocity using the formula: Average Velocity = Total Displacement / Total Time = 0 m / 60 s = 0 m/s.

 Additional Information

• Average Speed:
• Unlike velocity, speed is a scalar quantity and is calculated using the total distance traveled rather than displacement.
• In this scenario, the total distance is 240 m.
• The average speed would be 240 m / 60 s = 4 m/s.
• Scalar vs Vector Quantities:
• Distance and speed are scalar quantities as they only have magnitude.
• Displacement and velocity are vector quantities as they possess both magnitude and direction.
• Whenever an object completes a round trip and returns to its starting point, its net displacement is always zero, regardless of the distance covered.

The correct answer is Condoms help prevent the transmission of sexually transmitted diseases to some extent.

Key Points

• Condoms are recognized as barrier contraceptives that establish a physical wall between partners, preventing the exchange of bodily fluids like semen and vaginal secretions.
• They are primarily recommended because they significantly reduce the risk of transmitting Sexually Transmitted Infections (STIs), including HIV/AIDS, Gonorrhea, Chlamydia, and Syphilis.
• In addition to infection control, they serve as a highly effective tool for family planning by preventing sperm from entering the uterus, thereby avoiding fertilization and unwanted pregnancy.
• The phrase 'to some extent' is vital because protection is not absolute; mechanical failure (breakage or slippage) or contact with infected skin areas not covered by the condom can still lead to transmission.

 Additional Information

• Methodology and Limitations:
• Contraceptive Efficacy: While they are very effective, they do not completely eliminate risks; according to the World Health Organization (WHO), consistent and correct use is required to reach maximum safety levels.
• Fertility: Condoms are used to prevent conception and do not increase fertility in either males or females; in fact, they are the opposite of fertility treatments.
• Legal Status: Usage is promoted for public health and reproductive rights rather than being a mandatory law for population balance.
• Specific Protection: They are particularly effective against STIs spread through discharge, but slightly less effective against those spread through skin-to-skin contact, such as Human Papillomavirus (HPV) or Herpes.
• Materials: Most are made from Latex, though Polyurethane or Polyisoprene options exist for those with allergies.

The correct answer is They contain different types of cells working together for transport.

Key Points

• Permanent tissues are groups of cells that have lost the ability to divide and have attained a definite shape, size, and specialized function through differentiation.
• These tissues are termed complex because they are composed of more than one type of cell that coordinate together to perform a common vital function, unlike simple tissues which consist of only one cell type. Hence, the statement that they contain different types of cells working together is correct.
• Xylem and phloem are the primary conducting tissues or vascular tissues in plants, acting as a plumbing system for nutrient and water movement.
• Xylem is responsible for the upward conduction of water and dissolved minerals from the soil to the leaves and other plant parts.
• Phloem facilitates the process of translocation, which involves the transport of organic nutrients (food) synthesized during photosynthesis from the leaves to storage organs and growing regions.

 Additional Information

• Constituents of Xylem:
• Tracheids and Vessels: Tubular structures that conduct water; they are dead cells at maturity with thick lignified walls.
• Xylem Parenchyma: The only living component of xylem, involved in food storage and lateral conduction of water.
• Xylem Fibres: Provide mechanical support to the plant body.
• Constituents of Phloem:
• Sieve Tubes: Living cells with perforated end walls called sieve plates, lacking a nucleus at maturity.
• Companion Cells: Specialized parenchymatous cells that help regulate the pressure gradient in sieve tubes.
• Phloem Parenchyma and Phloem Fibres: Used for storage and structural support respectively.
• Vascular Bundles: In the plant body, xylem and phloem are usually organized together into vascular bundles, a defining characteristic of Tracheophytes.

The correct answer is Salt, carbon dioxide and water.

Key Points

• When an acid reacts with a metal hydrogen carbonate (commonly referred to as a bicarbonate), it undergoes a chemical transformation that consistently produces three specific substances: a salt, carbon dioxide, and water.
• The general chemical word equation for this reaction is: Acid + Metal Hydrogen Carbonate → Salt + Carbon Dioxide + Water.
• A classic scientific example of this reaction is the interaction between Hydrochloric acid (HCl) and Sodium hydrogen carbonate (NaHCO₃). The balanced molecular equation is HCl + NaHCO₃ → NaCl + H₂O + CO₂.
• The formation of Carbon dioxide (CO₂) is visually indicated by brisk effervescence as the gas bubbles through the liquid mixture.
• The salt produced during the reaction depends on the specific metal present in the bicarbonate and the anion provided by the acid; for example, using Sulfuric acid would produce a sulfate salt.
• This reaction is biologically and chemically significant because the hydrogen ions (H⁺) from the acid combine with the bicarbonate ions (HCO₃⁻) to form carbonic acid, which is highly unstable and immediately decomposes into water and gas.

 Additional Information

• Metal Carbonates:
• Like hydrogen carbonates, metal carbonates (such as Calcium carbonate or Sodium carbonate) react with acids to produce salt, water, and carbon dioxide.
• Confirmatory Test for Carbon Dioxide:
• The gas evolved can be identified by passing it through limewater (calcium hydroxide). The limewater turns milky due to the precipitation of insoluble Calcium carbonate (CaCO₃).
• Practical Applications:
• Baking: In cooking, Sodium bicarbonate reacts with acidic ingredients (like lemon juice or vinegar) to release CO₂, which makes dough or batter rise.
• Fire Safety: Soda-acid fire extinguishers utilize this reaction to generate a large volume of CO₂ gas to extinguish fires by displacing oxygen.
• Analysis of Incorrect Options:
• Hydrogen gas is released when acids react with reactive metals (like Zinc or Magnesium), not carbonates.
• Oxygen gas is typically produced during the decomposition of certain oxides or peroxides, not in acid-bicarbonate reactions.
• Ammonia (NH₃) is produced when ammonium salts react with strong bases or alkalis.

The correct answer is Electromagnet.

Key Points

• A solenoid is defined as a long coil containing a large number of close turns of insulated copper wire wrapped in the shape of a cylinder. When an electric current passes through it, it creates a magnetic field that is nearly uniform inside the coil.
• The magnetic field produced inside a solenoid is directly proportional to the current flowing through it and the number of turns per unit length. This field is strong enough to magnetize materials with high magnetic permeability.
• When a piece of soft iron is placed inside the core of a current-carrying solenoid, the iron becomes strongly magnetized. This arrangement is specifically known as an electromagnet.
• The magnetism of an electromagnet is temporary and can be controlled; the core loses its magnetic properties as soon as the circuit is broken and the current stops flowing.
• The polarity of an electromagnet can be reversed by changing the direction of the current, making it highly versatile for industrial applications like electric bells, cranes, and magnetic resonance imaging (MRI) machines.

 Additional Information

• Permanent Magnets:
• Unlike electromagnets, these are made from hard magnetic materials like steel or alloys like Alnico. They retain their dipole alignment even after the magnetizing field is removed.
• Toroid:
• A toroid is a solenoid that has been bent into a circular shape (like a doughnut). The magnetic field is entirely confined within the interior space of the ring, with zero field outside the ideal toroid.
• Natural Magnets:
• These are substances found in nature, such as Magnetite (an oxide of iron, Fe₃O₄), also known as lodestone. They possess permanent magnetism but are generally too weak and brittle for practical engineering use.

The correct answer is Secretion and protection.

Key Points

• Epithelial tissue is one of the four basic types of animal tissues that acts as a protective covering or lining for the body surface, internal organs, and cavities.
• The primary function of protection is evident in the skin (stratified squamous epithelium), which provides a physical barrier against mechanical injury, pathogenic invasion, and dehydration.
• The secretion function is carried out by specialized cells known as glandular epithelium, which synthesize and release substances such as hormones, sweat, saliva, and digestive enzymes.
• These tissues are characterized by tightly packed cells with very little intercellular matrix, ensuring they form a continuous sheet that regulates the movement of materials between different compartments.
• Apart from secretion and protection, epithelial tissues in the small intestine and kidney tubules are specialized for absorption and filtration respectively.
• A distinguishing feature of all epithelial tissue is that it sits on a non-cellular basement membrane, which separates it from the underlying connective tissue.

 Additional Information

• Coordination:
• This is the fundamental role of Nervous Tissue, which uses neurons to transmit electrical signals and neurotransmitters to control body activities.
• Support:
• This is the primary function of Connective Tissue, which includes bone, cartilage, tendons, and ligaments that provide structural framework and bind organs together.
• Muscle contraction:
• This function is unique to Muscular Tissue, which contains specialized proteins like actin and myosin that facilitate locomotion and posture.
• Types of Epithelium:
• Simple Squamous: Facilitates diffusion in the alveoli of lungs.
• Cuboidal Epithelium: Found in kidney tubules for mechanical support and secretion.
• Ciliated Columnar: Found in the respiratory tract to move mucus via hair-like cilia.

The correct answer is Scattering of light is not prominent at such heights.

Key Points

• The sky appears blue to an observer on the ground due to a phenomenon called Rayleigh scattering, where sunlight interacts with the molecules of the Earth's atmosphere.
• As altitude increases, the density of the atmosphere decreases rapidly, leading to a significantly lower concentration of gas molecules and suspended particles (like dust or water droplets).
• At very high altitudes, such as those reached by supersonic jets or spacecraft, there are not enough particles to scatter the shorter wavelengths (blue and violet) of sunlight toward the observer's eyes.
• Because the light is not being redirected through scattering, the sky does not acquire a blue tint and instead appears dark or black, even in the presence of direct sunlight.
• The path of light simply passes through the thin upper atmosphere without hitting enough obstacles to create the diffuse sky radiation that we perceive as daylight color on the surface.

 Additional Information

• Rayleigh Scattering Law:
• This law states that the amount of scattering is inversely proportional to the fourth power of the wavelength (1/λ⁴). This explains why blue light (shorter wavelength) scatters much more than red light (longer wavelength).
• Atmospheric Absorption:
• While the atmosphere does absorb certain parts of the electromagnetic spectrum (like UV rays via the Ozone layer), the dark appearance of the high-altitude sky is not caused by the absorption of blue light, but rather the absence of its scattering.
• Tyndall Effect:
• This is the scattering of a beam of light by colloidal particles. In a clear sky, the gas molecules themselves act as the scattering agents for sunlight.
• Visual Perception in Space:
• In the vacuum of space, there is no medium at all. Consequently, the sky appears pitch black, and the Sun appears as a stark white disk, as there is no atmospheric filtration or scattering to change its perceived color.

The correct answer is 2 sec.

Key Points

• The frequency of the sound wave is provided as 2 kHz. To perform standard calculations, this must be converted into Hertz (Hz). Since 1 kHz = 1000 Hz, the frequency (f) is 2000 Hz.
• The wavelength (λ) is given as 45 cm. Converting this into the SI unit of meters (m) by dividing by 100 results in 0.45 m.
• The speed of the wave (v) is calculated using the fundamental wave equation: Speed = Frequency × Wavelength. Substituting the values: v = 2000 Hz × 0.45 m = 900 m/s.
• The total distance (d) the wave needs to travel is 1.8 km. Converting this to meters (1 km = 1000 m) gives a total distance of 1800 m.
• The time taken (t) is determined by the formula Time = Distance / Speed. Substituting the derived values: t = 1800 m / 900 m/s = 2 seconds.

 Additional Information

• Wave Terminology:
• Frequency: This represents the number of compressions or rarefactions that cross a fixed point per unit time. It is inversely proportional to the time period.
• Wavelength: The distance between two consecutive centers of compression or two consecutive centers of rarefaction in a longitudinal wave.
• Speed of Sound: The speed depends on the elasticity and density of the medium. Sound travels fastest in solids, slower in liquids, and slowest in gases.
• Nature of Sound: Sound waves are mechanical waves and cannot travel through a vacuum; they require a material medium for propagation.

The correct answer is Reduction and reuse of waste.

Key Points

• The primary objective of switching to steel plates and reusable water bottles is to eliminate the generation of single-use plastic waste at its source.
• Reduction involves consuming less and avoiding items that create trash. By refusing disposable plastic cups, the school prevents thousands of pieces of non-biodegradable waste from entering landfills annually.
• Reuse refers to the practice of using an item multiple times before it is discarded or recycled. Unlike plastic plates which are thrown away after one meal, stainless steel utensils are designed for long-term durability and repeated use over several years.
• This action fosters a sustainable environment by lowering the demand for new plastic production, which is a highly energy-intensive process involving fossil fuels.
• Transitioning to reusable items is a practical application of the Zero Waste philosophy, focusing on resource conservation rather than just waste disposal management.

 Additional Information

• The Waste Management Hierarchy prioritizes strategies according to their environmental impact:
• Reduce: The most preferred method, emphasizing prevention and source reduction.
• Reuse: Extends the lifecycle of products, reducing the need for reprocessing.
• Recycle: Involves converting waste into raw materials; while beneficial, it is less efficient than reduction as it requires energy and transportation.
• Benefits of Steel over Plastic:
• Health: Steel is non-toxic and does not leach chemicals like BPA (Bisphenol A) or phthalates into food and water.
• Recyclability: Steel is 100% recyclable and can be repurposed indefinitely without losing its structural integrity, unlike many plastics that degrade during downcycling.
• Pollution Control: Reducing plastic usage directly combats microplastic pollution in water bodies and protects aquatic ecosystems.

The correct answer is Osmosis - because solvent molecules move from the dilute solution to the concentrated solution through the semipermeable membrane.

Key Points

• Osmosis is a fundamental biological and chemical process characterized by the spontaneous net movement of solvent molecules (usually water) through a semipermeable membrane.
• The movement occurs from a region of lower solute concentration (dilute solution/high water potential) to a region of higher solute concentration (concentrated solution/low water potential).
• A semipermeable membrane is a biological or synthetic material that allows only certain molecules, typically the solvent, to pass through while obstructing the passage of solute particles like sugar.
• In this experiment, the water molecules move toward the side with the higher sugar concentration to reach thermodynamic equilibrium, causing the solution level to rise in the concentrated beaker and fall in the dilute one.
• This process continues until the hydrostatic pressure generated by the difference in liquid levels equals the osmotic pressure of the solution.

 Additional Information

• Diffusion
• Involves the movement of solute particles from high concentration to low concentration.
• Unlike osmosis, it does not strictly require a semipermeable membrane to occur.
• Sedimentation
• A physical process where particles settle at the bottom of a liquid due to gravity.
• It is primarily used in water treatment and does not involve movement across membranes based on concentration.
• Evaporation
• A phase transition where molecules at the surface of a liquid gain enough energy to enter the gaseous state.
• While it can change liquid levels, it is not driven by solute gradients or membrane filtration.

The correct answer is 18, 8.

Key Points

• The maximum number of electrons that can be accommodated in any energy level or shell of an atom is calculated using the Bohr-Bury formula 2n², where n is the orbit number.
• For the M shell, which is the third energy level, the value of n is 3. Substituting this into the formula gives 2 × 3² = 18. Hence, the maximum capacity of the M shell is 18.
• Calcium (Ca) has an atomic number of 20, indicating it has 20 electrons in its neutral state to be distributed across energy shells.
• In the electronic distribution of Calcium, the K shell (n=1) holds 2 electrons and the L shell (n=2) holds 8 electrons, accounting for 10 electrons.
• Although the M shell can theoretically hold 18 electrons, the Bohr-Bury scheme and the Octet Rule dictate that the outermost shell cannot contain more than 8 electrons for stability.
• Therefore, the remaining 10 electrons of Calcium are split: 8 electrons fill the M shell, and the final 2 electrons enter the N shell. Hence, in Calcium, the M shell contains 8 electrons.

 Additional Information

• Electronic Shell Characteristics:
• K Shell (n=1):
• It is the closest shell to the nucleus.
• Maximum capacity is 2 × 1² = 2 electrons.
• L Shell (n=2):
• The second energy level with a maximum capacity of 2 × 2² = 8 electrons.
• N Shell (n=4):
• For Calcium, this is the valence shell containing 2 electrons.
• The theoretical maximum capacity for n=4 is 32 electrons.
• Chemical Properties:
• Because Calcium has 2 valence electrons, it belongs to Group 2 of the periodic table and is classified as an alkaline earth metal.

The correct answer is Transmit electrical impulses to other neurons or target organs.

Key Points

• An axon is a long, slender, cylindrical projection of a neuron that conducts electrical impulses known as action potentials away from the nerve cell body or soma.
• The primary role of the axon is to act as the primary transmission line of the nervous system, ensuring that information is delivered to specific target cells, including other neurons, muscle cells, or glandular cells.
• Transmission begins at the axon hillock, where summation of signals occurs, and ends at the axon terminals (synaptic bulbs), where neurotransmitters are released to cross the synaptic cleft.
• This process allows for rapid communication across long distances in the body, which is essential for sensory perception, motor coordination, and reflex actions.

 Additional Information

• Structural Components of Axons:
• Myelin Sheath: Many vertebrate axons are insulated by a myelin sheath, formed by Schwann cells in the Peripheral Nervous System (PNS) and oligodendrocytes in the Central Nervous System (CNS), which significantly increases conduction velocity.
• Nodes of Ranvier: These are unmyelinated gaps where voltage-gated sodium channels are concentrated, enabling saltatory conduction where the impulse jumps between nodes.
• Axoplasm and Axolemma: The cytoplasm (axoplasm) and the cell membrane (axolemma) of the axon maintain the electrochemical gradient necessary for signaling.
• Differentiation from Other Neuronal Parts:
• Dendrites: Unlike axons, dendrites are typically shorter, branched, and function to receive signals from other neurons rather than transmit them.
• Cell Body (Soma): Contains the nucleus and is responsible for protein synthesis and the metabolic maintenance of the entire neuron.

The correct answer is Water-bearing strata.

Key Points

• A dug well is a traditional, shallow excavation usually ranging from 2 to 5 meters in diameter and 5 to 20 meters in depth, designed to tap into groundwater near the surface.
• Water in these wells is collected from water-bearing strata, also known as unconfined aquifers, where the pore spaces between soil or rock particles are completely saturated with water.
• The water level in a dug well represents the local water table or phreatic surface, and the well is replenished as water moves from the surrounding permeable formation into the well cavity through gravity and hydrostatic pressure.
• These strata typically consist of alluvial deposits, sands, gravels, or fractured rocks that possess high hydraulic conductivity, allowing for the steady yield of water for irrigation or domestic use.
• The yield of a dug well is relatively low compared to deep wells, often limited by the recharge rate of the shallow saturated zone and seasonal fluctuations in the precipitation patterns.

 Additional Information

• Impervious Strata:
• These layers consist of materials like clay, shale, or compact crystalline rock that have very low permeability and do not allow water to flow through easily.
• In well construction, an impervious layer (aquiclude) often acts as the bottom boundary of a shallow well or the confining layer for a pressurized artesian aquifer.
• Deep Strata:
• Refers to geological formations located significantly below the surface, usually tapped by tube wells or borewells using mechanical drilling rigs.
• Dug wells rarely access deep strata because manual excavation becomes hazardous and structurally unstable beyond a certain depth.

The correct answer is Mass stays the same everywhere, but weight changes with location.

Key Points

• Mass is a measure of the amount of matter contained within an object and is considered an intrinsic property. It does not depend on the external environment or gravitational field, meaning it remains constant whether the object is on Earth, the Moon, or in deep space. Hence, statement 4 is correct.
• Weight is the gravitational force exerted on an object by a large body like a planet. It is calculated using the formula weight = mass × acceleration due to gravity (g). Since the value of g varies depending on the mass and radius of the celestial body, an object's weight changes based on its location. Hence, statement 4 is correct.
• Statement 1 suggests that both properties are variable, but since mass is a scalar quantity that is independent of position, this assertion is physically impossible. Hence, statement 1 is incorrect.
• Statement 2 claims that weight remains constant everywhere, which ignores the fundamental physics that weight is directly proportional to local gravity, which is not uniform throughout the universe. Hence, statement 2 is incorrect.
• Statement 3 incorrectly proposes that mass is the variable factor and weight is the constant, which contradicts the Law of Conservation of Mass and the principles of Newtonian mechanics. Hence, statement 3 is incorrect.

 Additional Information

• Scientific Units and Classification:
• Mass is a scalar quantity measured in Kilograms (kg) using a beam balance or a physical balance.
• Weight is a vector quantity measured in Newtons (N) using a spring balance, as it represents a force directed toward the center of a mass.
• Gravitational Variations:
• On the Moon, the gravity is approximately 1.62 m/s², which is roughly 1/6th of Earth's gravity (9.8 m/s²). Therefore, a person with a mass of 60 kg will always have a mass of 60 kg, but their weight will drop from 588 N on Earth to about 98 N on the Moon.
• Even on Earth, weight fluctuates slightly; it is higher at the Poles and lower at the Equator because the Earth is an oblate spheroid and rotation creates centrifugal effects.

The correct answer is -3.

Key Points

• The object distance (u) is given as −20 cm because, according to the New Cartesian Sign Convention, the distance of the object from the optical center is always measured against the direction of incident light.
• The focal length (f) for a convex lens (converging lens) is positive, thus f = +15 cm.
• The Lens Formula is used to determine the image distance (v): $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$.
• By substituting the known values into the formula: 1/v − 1/(−20) = 1/15, which simplifies to 1/v + 1/20 = 1/15.
• Rearranging the equation to solve for v: 1/v = 1/15 − 1/20. Taking the Least Common Multiple (LCM) of 15 and 20, which is 60, we get 1/v = (4 − 3) / 60 = 1/60.
• The image distance (v) is therefore +60 cm, indicating the image is formed on the opposite side of the lens.
• Magnification (m) for a spherical lens is calculated using the ratio of image distance to object distance: m = v / u.
• Substituting the calculated values: m = 60 / (−20) = −3.

 Additional Information

• Nature of the Image:
• The negative sign in magnification signifies that the image is real and inverted.
• A positive sign would have indicated a virtual and erect image.
• The magnitude of magnification (|m| = 3) indicates that the image is enlarged and is three times the height of the object.
• Convex Lens Properties:
• When an object is placed between F₁ and 2F₁ (as in this case, where 15 cm < 20 cm < 30 cm), the image is always formed beyond 2F₂.
• The height of the image (h_i) can be found using m = h_i / h_o. Here, −3 = h_i / 2.0, so h_i = −6.0 cm.

The correct answer is First (index) finger.

Key Points

• Fleming’s Left Hand Rule is a fundamental principle used in electromagnetism to find the direction of force experienced by a current-carrying conductor in a magnetic field.
• The First finger (also known as the Index finger or Forefinger) represents the direction of the Magnetic Field, typically directed from the North Pole to the South Pole.
• The Middle finger represents the direction of the Electric Current flowing through the wire (from positive to negative terminal).
• The Thumb represents the direction of the Force or the resultant Motion of the conductor.
• For this rule to be valid, the thumb, index finger, and middle finger must be held mutually perpendicular (at 90° angles) to each other.

 Additional Information

• This rule was developed by John Ambrose Fleming in the late 19th century to help visualize the interaction in electric machines.
• Electric Motors: The Left Hand Rule is primarily applicable to motors, where electrical energy is converted into mechanical energy through the interaction of current and magnetic fields.
• Right Hand Rule: Fleming also devised a Right Hand Rule for Electric Generators. In this rule, the index finger still represents the Magnetic Field, but the middle finger represents the Induced Current and the thumb represents the Motion of the conductor.
• Lorentz Force: The physical basis for this rule is the Lorentz Force equation, where the force is the cross-product of the velocity vector (or current) and the magnetic field vector.

The correct answer is pistil.

Key Points

• The pistil is the female reproductive part of a flower, typically located in the center, and is often referred to as the carpel when considering its individual structural units.
• It is fundamentally composed of three primary parts: the stigma, the style, and the ovary.
• The stigma is the sticky or feathery tip at the top of the pistil that serves the critical function of receiving pollen grains during the pollination process.
• The style is a tube-like stalk that connects the stigma to the ovary, acting as a structural support and a pathway for the growth of the pollen tube.
• The ovary is the basal, swollen portion of the pistil that contains ovules; after fertilization occurs, the ovules develop into seeds while the ovary matures into a fruit.
• Flowers containing only female reproductive parts are known as pistillate or unisexual flowers, whereas those containing both male and female parts are called bisexual flowers.

 Additional Information

• Stamens:
• These represent the male reproductive part of the flower, consisting of the anther (which produces pollen) and the filament (the stalk supporting the anther).
• Sepals:
• These are green, leaf-like structures that form the outermost whorl called the calyx, primarily serving to protect the flower bud during its early developmental stages.
• Petals:
• Modified leaves that constitute the corolla; they are usually brightly colored and scented to attract pollinators such as insects, bats, and birds for successful reproduction.

The correct answer is To combine desirable traits by crossing genetically dissimilar plants.

Key Points

• Hybridisation is a fundamental technique in crop improvement that involves the crossing of two or more plants with different genetic makeups to produce a progeny called a hybrid.
• The primary objective of this process is to aggregate superior traits from different parents—such as high yield, disease resistance, early maturity, and nutritional quality—into a single genotype.
• It utilizes the concept of genetic recombination, where the reshuffling of genes during meiosis and subsequent fertilization creates genotypic variations that do not exist in the original parental lines.
• Hybridisation often leads to Heterosis, also known as hybrid vigor, where the offspring exhibits enhanced physiological characteristics and biological efficiency compared to either parent.
• The process typically involves careful selection of parents, emasculation (removal of anthers), and controlled pollination to ensure the desired genetic transfer occurs.

 Additional Information

• Mutation Breeding:
• Unlike hybridisation, this method uses mutagens (like Gamma rays or chemicals) to induce sudden, inheritable changes in the DNA sequence to create new traits.
• Cloning and Micropropagation:
• This involves asexual reproduction to produce Somaclones. These plants are genetically identical to the parent, which is the opposite of the genetic diversity aimed for in hybridisation.
• Types of Hybridisation:
• Inter-varietal: Cross between two different varieties of the same species.
• Inter-specific: Cross between two different species of the same genus.
• Inter-generic: Cross between plants belonging to two different genera, often used to introduce hardiness or stress tolerance.

The correct answer is Store cell sap and help maintain turgor pressure.

Key Points

• In mature plant cells, the large central vacuole is a prominent membrane-bound organelle that can occupy up to 90% of the total cellular volume.
• The primary function of this vacuole is the storage of cell sap, which is a concentrated fluid containing water, sugars, amino acids, mineral salts, and sometimes pigments or waste products.
• By accumulating solutes, the vacuole facilitates the entry of water via osmosis, which generates turgor pressure against the cell wall. This internal pressure is critical for maintaining the structural rigidity and upright posture of non-woody plants.
• The vacuole is surrounded by a single semi-permeable membrane known as the tonoplast, which actively transports ions and other materials against concentration gradients.

 Additional Information

• Protein Synthesis:
• This process is primarily carried out by ribosomes and the rough endoplasmic reticulum (RER), rather than the vacuole.
• Cellular Respiration:
• The production of ATP (Adenosine Triphosphate) through the oxidation of glucose occurs in the mitochondria, often referred to as the powerhouse of the cell.
• Control of Cell Division and Heredity:
• The nucleus serves as the control center of the cell, housing chromosomes and DNA, which carry genetic information and regulate mitosis and meiosis.
• Comparison with Animal Cells:
• Unlike plant cells, animal cells contain several small, temporary vacuoles mainly used for storage, excretion, or endocytosis, and they do not contribute to structural turgidity.

The correct answer is Movement of food in the alimentary canal.

Key Points

• Unstriated muscles, also referred to as smooth muscles, are characterized by their lack of transverse bands or striations when observed under a microscope, giving them a smooth appearance.
• These muscles are involuntary, meaning their functional activities are not under the conscious control of the will but are regulated by the autonomic nervous system.
• The cells of smooth muscles are spindle-shaped (fusiform) and contain a single, centrally located nucleus.
• The alimentary canal utilizes these muscles to perform peristalsis, which is the coordinated, wave-like contraction and relaxation that propels bolus and chyme through the digestive tract.
• Beyond the digestive system, these muscles are also found in the walls of other hollow visceral organs like blood vessels, the urinary bladder, and the uterus, managing functions such as vasoconstriction and organ emptying.

 Additional Information

• Cardiac Muscle Tissue:
• Responsible for the rhythmic contraction of the heart.
• While it is involuntary like smooth muscle, it is striated and contains specialized intercalated discs that allow rapid transmission of electrical impulses for synchronized beating.
• Skeletal Muscle Tissue:
• Responsible for movements like waving the hand and moving the legs to walk.
• These muscles are both striated and voluntary, attached to the skeletal system via tendons and controlled by the somatic nervous system.
• They consist of long, cylindrical, multinucleated fibers that fatigue more easily compared to smooth or cardiac muscles.

The correct answer is the atom is mostly empty space.

Key Points

• In 1911, Ernest Rutherford conducted the alpha-particle scattering experiment by bombarding a very thin gold foil (approximately 1000 atoms thick) with high-energy alpha particles. Hence, the observation that most particles passed straight through the foil was central to his atomic model.
• The observation that a vast majority (nearly 99%) of the alpha particles traveled through the gold foil without any deflection or change in direction led to the conclusion that the atom is mostly empty space. Hence, the statement that the atom is mostly empty space is correct.
• Since these positively charged helium ions did not encounter any electrostatic repulsion or physical obstruction in their path, it indicated that the mass and positive charge of the atom were not spread out as previously suggested by the Plum Pudding Model. Hence, the conclusion regarding the empty nature of the atom is correct.

 Additional Information

• Rutherford's Atomic Model provided several other groundbreaking insights:
• The Nucleus: A very small fraction of particles were deflected by large angles, and 1 in 12,000 rebounded completely (180°), proving that the positive charge and nearly all the mass are concentrated in a tiny region called the nucleus.
• Nuclear Charge: Because alpha particles (He²⁺) are positive, their deflection by the center confirmed that the nucleus is positively charged, not negatively charged.
• Size Proportions: Rutherford estimated that the radius of the nucleus is about 10⁵ times smaller than the radius of the atom, reinforcing the empty space theory.
• Electron Orbits: The model proposed that electrons revolve around the nucleus in circular paths, similar to planets revolving around the sun.

The correct answer is One-fourth.

Key Points

• The acceleration due to gravity (g) on the surface of a planet is calculated using the formula g = GM/R², where G represents the Universal Gravitational Constant, M is the mass of the planet, and R is its radius.
• According to the question, the mass (M) of the Earth remains constant while the radius (R) is doubled, such that the new radius R' = 2R.
• By substituting the new radius into the gravitational formula, the new acceleration g' is expressed as g' = GM / (2R)². Squaring the term in the denominator results in g' = GM / 4R².
• Comparing the new equation to the original formula (g = GM/R²) shows that the new gravity g' is equal to (1/4) × g. This mathematically confirms that the gravitational pull becomes one-fourth of the original value.

 Additional Information

• Universal Gravitational Constant (G):
• First measured by Henry Cavendish in 1798, its value is approximately 6.674 × 10⁻¹¹ Nm²/kg². It is a scalar quantity and remains uniform throughout the universe.
• Factors Affecting 'g':
• Shape of the Earth: Earth is an oblate spheroid; the radius at the equator is about 21 km longer than at the poles. Consequently, g is maximum at the poles and minimum at the equator.
• Altitude and Depth: The value of g decreases as we move upwards from the surface (altitude) or downwards into the Earth (depth). At the center of the Earth, the value of g becomes zero.
• Rotation: The centrifugal force generated by Earth's rotation reduces the effective value of g, specifically at the equator.
• Inverse Square Law: Gravity follows an inverse square relationship with distance. If the distance from the center of mass doubles, the force of attraction is reduced to 2² or 4 times less.

The correct answer is 1 Ω.

Key Points

• Ohm's Law, formulated by the German physicist Georg Simon Ohm in 1827, defines the relationship between voltage, current, and resistance in an electrical circuit.
• The law states that the electric current (I) flowing through a conductor is directly proportional to the potential difference (V) across its ends, provided physical conditions like temperature remain constant.
• The mathematical formula for this relationship is V = I × R, which can be rearranged to find resistance (R) using R = V ÷ I.
• In this problem, the potential difference (V) is given as 1 Volt (1 V) and the current (I) is given as 1 Ampere (1 A).
• By substituting these values into the formula: R = 1 V ÷ 1 A = 1 Ω.
• Thus, the electrical resistance of the conductor is exactly 1 Ω, indicating that for every volt of potential applied, one ampere of current will flow.

 Additional Information

• Understanding Resistance (R):
• Definition: It is the property of a material to oppose the flow of electric charges (electrons).
• SI Unit: The unit of resistance is the Ohm (Ω). One ohm is defined as the resistance of a conductor such that a potential difference of 1 Volt causes a current of 1 Ampere to flow.
• Resistivity (ρ): While resistance depends on shape and size, resistivity is an intrinsic property of the material. Metals like Silver and Copper have very low resistivity.
• Factors Affecting Resistance:
• Length (L): Resistance is directly proportional to the length of the conductor. Doubling the length doubles the resistance.
• Cross-sectional Area (A): Resistance is inversely proportional to the area. A thicker wire has less resistance than a thinner wire of the same material.
• Temperature: In metallic conductors, resistance increases with an increase in temperature due to more frequent collisions between electrons and metal ions.

The correct answer is The pH of the solution becomes close to 7.

Key Points

• A neutralisation reaction occurs when an acid and a base react with each other to produce salt and water as the primary products.
• In this chemical process, the hydrogen ions (H⁺) from the acid combine with the hydroxide ions (OH⁻) from the base to form H₂O (water) molecules.
• The pH scale is a logarithmic measure of the acidity or alkalinity of a solution, ranging from 0 to 14; a pH of 7 signifies a perfectly neutral solution.
• As the acidic and basic properties cancel each other out during the reaction, the concentration of H⁺ and OH⁻ ions balances, causing the pH level to shift from extreme values toward the neutral mark of 7.
• Confirmation of neutralisation is best achieved by observing this pH shift using a pH meter or universal indicator, as it directly tracks the loss of acidic or basic identity.

 Additional Information

• Exothermic Nature:
• Most neutralisation reactions are exothermic, meaning they release heat energy into the surroundings.
• Consequently, the temperature of the solution typically rises rather than dropping drastically.
• Indicator Changes:
• Common indicators like Litmus change color based on pH; blue litmus turns red in acidic conditions, while red litmus turns blue in basic conditions.
• In a complete neutralisation of a strong acid and base, the solution will not turn blue to red, but rather stay at a point where the indicator reflects neutrality (like green for universal indicator).
• Strength of Reactants:
• The final pH depends on the strength of the reactants; a strong acid and strong base result in a pH of exactly 7.
• A strong acid reacting with a weak base results in an acidic salt (pH < 7), while a weak acid and strong base yield a basic salt (pH > 7).

The correct answer is 500 J.

Key Points

• The problem is based on the Law of Conservation of Mechanical Energy, which states that the total mechanical energy (sum of kinetic and potential energy) of an object remains constant if only conservative forces like gravity are acting on it.
• The mass (m) of the body is given as 10 kg and the initial velocity (u) with which it is thrown upwards is 10 m/s.
• At the moment of projection (ground level), the body possesses maximum Kinetic Energy (KE) and zero Potential Energy (PE). The formula for Kinetic Energy is $KE = \frac{1}{2}mv^{2}$.
• By substituting the given values, the initial Kinetic Energy is calculated as: 1/2 × 10 kg × (10 m/s)² = 1/2 × 10 × 100 = 500 Joules.
• As the body rises, its velocity decreases due to the acceleration due to gravity (g = 10 m/s²), and its Kinetic Energy is converted into Gravitational Potential Energy.
• At the maximum height, the vertical velocity of the body becomes zero, meaning its Kinetic Energy becomes 0 J.
• According to the conservation principle, the entire initial Kinetic Energy (500 J) must be converted into Potential Energy at this peak point. Hence, the Potential Energy at maximum height is 500 J.

 Additional Information

• Potential Energy (PE) formula:
• Potential energy can also be calculated using the formula PE = mgh, where h is the height above the reference level.
• Using the kinematic equation v² = u² − 2gh, and setting final velocity v = 0 at maximum height: 0 = 10² − 2 × 10 × h, which results in h = 5 meters.
• Placing this height back into the PE formula: 10 kg × 10 m/s² × 5 m = 500 J.
• Units and Measurements:
• The SI unit of energy is the Joule (J), named after the English physicist James Prescott Joule.
• One Joule is defined as the work done by a force of one Newton acting through a distance of one meter.
• In terms of base units, 1 J = 1 kg·m²/s².