RRB Technician Grade III Official Paper (Held On: 10 March, 2026 Shift 1) (Previous Year Paper)

Shortcut Trick

Effective percentage increase in Class A = 20 + 10 + (20 × 10) ÷ 100 = 32%.

Percentage increase in Class B = 20%.

Total Increase = (Initial A × 32% + Initial B × 20%) ÷ Total Initial

Total Increase = (25 × 32 + 30 × 20) ÷ 55 = (800 + 600) ÷ 55 = 1400 ÷ 55 ≈ 25.45%.

∴ The correct answer is 25.45%.

Alternate Method

Given: Initial students in Class A = 25, Class B = 30.

⇒ Total Initial Students = 25 + 30 = 55

⇒ Students in Class A after 1^st year = 25 + (20% of 25) = 25 + 5 = 30

⇒ Students in Class A after 2^nd year = 30 + (10% of 30) = 30 + 3 = 33

⇒ Students in Class B after 2 years = 30 + (20% of 30) = 30 + 6 = 36

⇒ Total Final Students = 33 + 36 = 69

⇒ Overall Increase = 69 − 55 = 14

⇒ Overall % Increase = (14 ÷ 55) × 100 = 25.4545...

∴ The correct answer is 25.45%.

Additional Information

Successive Percentage Increase

If a value increases by x% and then by y%, the net increase is given by [x + y + (xy ÷ 100)]%.

Weighted Average Percentage

The overall percentage change of a combined group is the sum of (individual change × its weight) divided by the total weight.

Base Value in Percentage

Always identify the original total as the base for calculating the overall percentage increase: (Change ÷ Initial Total) × 100.

Shortcut Trick

SP of X = 700 + 630 = 1330

Successive discounts of 60% and 20% ⇒ Net multiplier = 0.40 × 0.80 = 0.32

⇒ 32% of MP = 1330

For Article Y, a 76% discount ⇒ SP = (100 − 76)% = 24% of MP

SP of Y = (1330 ÷ 32) × 24 = 1330 × (3/4) = 997.5

Profit percentage on Y = [(997.5 − 700) ÷ 700] × 100 = 42.5%

∴ The correct answer is 42.5%.

Alternate Method

Given:

Cost Price (CP) of X and Y = ₹700 each

Profit on X = ₹630

Successive discounts on X = 60% and 20%

Single discount on Y = 76%

Formula Used:

Selling Price (SP) = Cost Price (CP) + Profit

Net Discount for successive d₁ and d₂ = [d₁ + d₂ − (d₁ × d₂ ÷ 100)]%

SP = Marked Price (MP) × (100 − Discount%) ÷ 100

Calculations:

⇒ SP of X = 700 + 630 = 1330

⇒ Effective discount on X = 60 + 20 − (60 × 20 ÷ 100) = 80 − 12 = 68%

⇒ SP of X = MP × (100 − 68) ÷ 100 = 0.32 × MP

⇒ 0.32 × MP = 1330 ⇒ MP = 1330 ÷ 0.32 = 4156.25

⇒ SP of Y = 4156.25 × (100 − 76) ÷ 100 = 4156.25 × 0.24 = 997.5

⇒ Profit on Y = 997.5 − 700 = 297.5

⇒ Profit % on Y = (297.5 ÷ 700) × 100 = 42.5%

∴ The correct answer is 42.5%.

Additional Information

Successive Discount Shortcut

Instead of calculating net %, use multiplying factors: Total multiplier = (1 − d₁/100) × (1 − d₂/100) × ... × (1 − d_n/100).

CP-MP-Profit-Discount Relation

The ratio of Marked Price to Cost Price is given by: MP/CP = (100 + Profit%) / (100 − Discount%).

Markup Percentage

The percentage by which the cost price is increased to set the marked price: Markup % = [(MP − CP) ÷ CP] × 100.

Shortcut Trick

Total interest percentage for 3 years at 6% per annum = 3 × 6% = 18%.

Given that 18% of the Principal = ₹720.

Principal (100%) = (720 ÷ 18) × 100.

Principal = 40 × 100 = ₹4000.

∴ The correct answer is ₹4000.

Alternate Method

Given:

Simple Interest (SI) = ₹720

Time (T) = 3 years

Rate (R) = 6% per annum

Formula Used:

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

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

Calculations:

⇒ 720 = (P × 6 × 3) ÷ 100

⇒ 720 = (P × 18) ÷ 100

⇒ P = (720 × 100) ÷ 18

⇒ P = 40 × 100

⇒ P = ₹4000

∴ The correct answer is ₹4000.

Additional Information

Total Amount Formula

The total amount (A) is the sum of the Principal and Simple Interest: A = P + SI = P(1 + RT/100).

Calculating Principal from Amount

If the total amount is given instead of interest, use: P = (A × 100) ÷ (100 + RT).

Effective Rate of Interest

For Simple Interest, the effective rate for the entire period is simply the product of the annual rate and the number of years (R × T).

Shortcut Trick

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

By scanning the given numbers, we find the following frequencies:

Numbers 40, 41, and 46 each appear 2 times.

The number 54 appears 3 times.

All other numbers appear only once.

∴ The correct answer is 54.

Alternate Method

Given Data: 47, 43, 50, 46, 46, 40, 54, 54, 40, 54, 42, 48, 44, 55, 52, 45, 41, 41

Method: Create a Frequency Distribution to identify the observation with the maximum count.

Calculations for frequency of each unique observation:

⇒ Number 40: 2 occurrences

⇒ Number 41: 2 occurrences

⇒ Number 46: 2 occurrences

⇒ Number 54: 3 occurrences

⇒ Others (47, 43, 50, 42, 48, 44, 55, 52, 45): 1 occurrence each

⇒ Since 54 has the highest count (3), it is the mode of the data.

∴ The correct answer is 54.

Additional Information

Empirical Formula

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

Types of Mode

Data with one mode is unimodal, two modes is bimodal, and more than two modes is multimodal.

Median for Ungrouped Data

First, arrange data in ascending order. If n is odd, Median = ((n+1)/2)^th term. If n is even, Median is the average of the (n/2)^th and (n/2 + 1)^th terms.

Shortcut Trick

For two successive decreases of a% and b%, the overall percentage decrease is calculated as:

Net Decrease = [a + b − (a × b)/100]%

Given a = 12 and b = 8:

Net Decrease = [12 + 8 − (12 × 8)/100]% = [20 − 0.96]% = 19.04%

∴ The correct answer is 19.04%.

Alternate Method

Given:

First decrease = 12%

Second decrease = 8%

Formula Used:

Net Change = Initial Value × (1 − r₁/100) × (1 − r₂/100)

⇒ Let the initial quantity be 100.

⇒ Value after first decrease = 100 − (12% of 100) = 88.

⇒ Value after second decrease = 88 − (8% of 88).

⇒ Calculation: 88 − (8 × 88)/100 = 88 − 7.04 = 80.96.

⇒ Total percentage decrease = 100 − 80.96 = 19.04%.

∴ The correct answer is 19.04%.

Additional Information

Successive Percentage Change

If a value is changed by x% and then by y%, the net percentage change is (x + y + xy/100)%. Use positive for increase and negative for decrease.

Single Equivalent Discount

For two successive discounts of d₁% and d₂%, the single equivalent discount is (d₁ + d₂ − d₁d₂/100)%.

Base Change Rule

Percentage changes are always calculated on the preceding value in a successive series, not the original initial value.

Shortcut Trick

Ratio of Quantity = 8 : 12 ⇒ 2 : 3

Since cost is directly proportional to quantity:

2 units = ₹136

1 unit = 136 ÷ 2 = ₹68

3 units = 68 × 3 = ₹204

∴ The correct answer is ₹204.

Alternate Method

Given:

Total cost of 8 pens = ₹136

Required number of pens = 12

Formula Used:

This problem follows the Unitary Method where we first find the value of one unit.

Cost of N items = (Cost of 1 item) × N

⇒ Cost of 1 pen = Total Cost ÷ Total Pens

⇒ Cost of 1 pen = 136 ÷ 8

⇒ Cost of 1 pen = ₹17

⇒ Now, cost of 12 pens = (Cost of 1 pen) × 12

⇒ Cost of 12 pens = 17 × 12

⇒ Cost of 12 pens = ₹204

∴ The correct answer is ₹204.

Additional Information

Unitary Method

A technique where we find the value of a single unit first to calculate the value of any required number of units.

Direct Proportion

Two quantities are in direct proportion if an increase in one leads to a corresponding increase in the other. Formula: x₁/y₁ = x₂/y₂.

Inverse Proportion

When an increase in one quantity causes a decrease in the other (e.g., more men taking less time). Formula: x₁ × y₁ = x₂ × y₂.

Shortcut Trick

Present Sum of ages = Sum 5 years ago + (2 × 5) = 30 + 10 = 40 years.

Present Difference of ages = 26 years (Given).

Nidhi's age (younger) = (Sum − Difference) ÷ 2 = (40 − 26) ÷ 2 = 14 ÷ 2 = 7 years.

∴ The correct answer is 7 years.

Alternate Method

Given: Difference (Ayushi − Nidhi) = 26 years. Sum (5 years ago) = 30 years.

Let Ayushi's age be A and Nidhi's age be N.

⇒ A − N = 26 ----(Equation 1)

Five years ago, their ages were (A − 5) and (N − 5).

⇒ (A − 5) + (N − 5) = 30

⇒ A + N − 10 = 30

⇒ A + N = 40 ----(Equation 2)

Subtracting Equation 1 from Equation 2:

⇒ (A + N) − (A − N) = 40 − 26

⇒ 2N = 14

⇒ N = 7

∴ The correct answer is 7 years.

Additional Information

Constant Difference Rule

The age difference between two individuals always remains constant, whether calculated in the past, present, or future.

Age Sum Property

If there are 'n' persons, their total sum of ages increases or decreases by (n × t) years over a time period of 't' years.

Sum and Difference Formula

When the Sum (S) and Difference (D) of two numbers are known: Larger number = (S + D) ÷ 2 and Smaller number = (S − D) ÷ 2.

Shortcut Trick

Use the direct formula for Total Surface Area (TSA) of a hollow cylinder:

TSA = 2π(R + r)(h + R − r)

Given: External radius (R) = 7 cm, Internal radius (r) = 6 cm, Height (h) = 20 cm.

TSA = 2 × (22/7) × (7 + 6) × (20 + 7 − 6)

TSA = 2 × (22/7) × 13 × 21 = 44 × 13 × 3 = 1716 cm².

∴ The correct answer is 1716 cm².

Alternate Method

Given:

External radius (R) = 7 cm

Internal radius (r) = 6 cm

Height (h) = 20 cm

π = 22/7

Step-by-step Calculation:

⇒ Outer Curved Surface Area = 2 × π × R × h = 2 × (22/7) × 7 × 20 = 880 cm²

⇒ Inner Curved Surface Area = 2 × π × r × h = 2 × (22/7) × 6 × 20 = 5280/7 cm²

⇒ Area of two circular ends (rings) = 2 × π × (R² − r²) = 2 × (22/7) × (7² − 6²)

⇒ Area of ends = 2 × (22/7) × (49 − 36) = 2 × (22/7) × 13 = 572/7 cm²

⇒ Total Surface Area = Outer CSA + Inner CSA + Area of Ends

⇒ TSA = 880 + (5280/7) + (572/7) = 880 + (5852/7)

⇒ TSA = 880 + 836 = 1716 cm²

∴ The correct answer is 1716 cm².

Additional Information

Volume of Hollow Cylinder

The amount of material used to make the pipe is given by the volume formula: V = πh(R² − r²).

Lateral Surface Area (LSA)

For a hollow cylinder, LSA is the sum of the outer and inner curved surfaces: LSA = 2πh(R + r).

Cross-sectional Area

The area of the ring formed at the base is A = π(R² − r²).

Shortcut Trick

Given cotθ = 1/√3. Since cot 60° = 1/√3, we have θ = 60°.

Substitute θ = 60° into the expression (treating cos2, cosec2, sec2 as squared terms):

Value = (2 − sin²60°) / (1 + cos²60°) + (cosec²60° + sec²60°)

Value = (2 − 3/4) / (1 + 1/4) + (4/3 + 4) = (5/4) / (5/4) + 16/3 = 1 + 16/3 = 19/3.

∴ The correct answer is 19/3.

Alternate Method

Given: cotθ = 1/√3 and 0° < θ < 90°.

Formula Used: In a right-angled triangle, cotθ = Adjacent / Opposite. Trigonometric ratios: sinθ = Opp/Hyp, cosθ = Adj/Hyp, cosecθ = Hyp/Opp, secθ = Hyp/Adj.

⇒ θ = 60°

⇒ sinθ = √3/2 ⇒ sin²θ = 3/4

⇒ cosθ = 1/2 ⇒ cos²θ = 1/4

⇒ cosecθ = 2/√3 ⇒ cosec²θ = 4/3

⇒ secθ = 2 ⇒ sec²θ = 4

⇒ Expression = (2 − 3/4) / (1 + 1/4) + (4/3 + 4)

⇒ Expression = (5/4) / (5/4) + (4/3 + 12/3)

⇒ Expression = 1 + 16/3

⇒ Expression = (3 + 16) / 3 = 19/3

∴ The correct answer is 19/3.

Additional Information

Trigonometric Table Values

For 60°: sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3, cot 60° = 1/√3, sec 60° = 2, cosec 60° = 2/√3.

Pythagorean Identities

Fundamental identities include: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ.

Double Angle Formulas

cos 2θ = 2 cos²θ − 1 = 1 − 2 sin²θ and sin 2θ = 2 sinθ cosθ. Note that in simplified text, cos2θ often represents cos²θ depending on context.

Shortcut Trick

Identify factors of 6 whose sum is 5: these are 3 and 2.

⇒ √x = √3 + √2

The expression (x − 1) / √x simplifies to √x − 1/√x.

Since (√3)² − (√2)² = 1, the reciprocal 1/√x is √3 − √2.

⇒ Result = (√3 + √2) − (√3 − √2) = 2√2.

∴ The correct answer is 2√2.

Alternate Method

Given: x = 5 + 2√6

Formula Used: (a + b)² = a² + b² + 2ab and (a + b)(a − b) = a² − b²

⇒ x = 3 + 2 + 2√(3 × 2)

⇒ x = (√3)² + (√2)² + 2 × √3 × √2

⇒ x = (√3 + √2)²

⇒ √x = √3 + √2

⇒ (x − 1) / √x = (x / √x) − (1 / √x) = √x − (1 / √x)

⇒ 1 / √x = 1 / (√3 + √2)

⇒ Rationalizing the denominator: 1/(√3 + √2) × (√3 − √2)/(√3 − √2) = √3 − √2

⇒ (x − 1) / √x = (√3 + √2) − (√3 − √2) = 2√2

∴ The correct answer is 2√2.

Additional Information

Square Root of Surds

For a surd of the form a + 2√b, if we find two numbers p and q such that p + q = a and pq = b, then √(a + 2√b) = √p + √q.

Rationalization of Denominators

To rationalize 1/(√a + √b), multiply both numerator and denominator by its conjugate (√a − √b).

Conjugate Rule

If x = a + √b and a² − b = 1, then the reciprocal 1/x is simply a − √b.

Shortcut Trick

In a partnership, Profit Ratio = (Capital × Time) Ratio.

Profit Ratio of Aarju and Bimla = (55000 × 12) : (Capital_B × 9) = 8 : 9.

Cancelling 9 on both sides: 55000 × 12 / Capital_B = 8.

Capital_B = (55000 × 12) ÷ 8 = 55000 × 1.5 = 82,500.

∴ The correct answer is ₹82,500.

Alternate Method

Given Data:

Capital of Aarju (C_A) = ₹55,000

Investment Period of Aarju (T_A) = 12 months

Investment Period of Bimla (T_B) = 12 − 3 = 9 months

Profit Ratio (P_A : P_B) = 8 : 9

Formula Used:

P₁ ÷ P₂ = (C₁ × T₁) ÷ (C₂ × T₂)

Calculations:

⇒ 8 ÷ 9 = (55000 × 12) ÷ (C_B × 9)

⇒ 8 ÷ 9 = (55000 × 12) ÷ (9 × C_B)

⇒ 8 × 9 × C_B = 55000 × 12 × 9

⇒ 8 × C_B = 55000 × 12

⇒ C_B = (55000 × 12) ÷ 8

⇒ C_B = 55000 × (3 ÷ 2)

⇒ C_B = 55000 × 1.5

⇒ C_B = 82500

∴ The correct answer is ₹82,500.

Additional Information

Partnership Profit Sharing

In a simple partnership, profits are shared in the ratio of investments if the time period is the same. In a compound partnership, profits are shared in the ratio of the product of capital and time.

Working and Sleeping Partners

A working partner manages the business and may receive an additional share or salary from the profit, whereas a sleeping partner only invests money.

Capital-Time Inverse Relation

For a fixed amount of profit, the capital invested is inversely proportional to the time: C₁ × T₁ = C₂ × T₂.

Shortcut Trick

Evaluate the square roots directly by recognizing perfect squares:

√225 = 15

√0.04 = 0.2 (since √4 = 2 and 2 decimal places become 1)

√4.84 = 2.2 (since √484 = 22 and 2 decimal places become 1)

Required value = 15 + 0.2 − 2.2 = 13

∴ The correct answer is 13.

Alternate Method

Given: √225 + √0.04 − √4.84

Method: Convert decimals to fractions to simplify finding the square roots.

⇒ √225 = 15

⇒ √0.04 = √(4 ÷ 100) = 2 ÷ 10 = 0.2

⇒ √4.84 = √(484 ÷ 100) = 22 ÷ 10 = 2.2

⇒ Now, substitute these values back into the expression:

⇒ 15 + 0.2 − 2.2

⇒ 15.2 − 2.2 = 13

∴ The correct answer is 13.

Additional Information

Square Roots of Decimals

The number of decimal places in the square root is always half the number of decimal places in the original number (e.g., √0.0001 = 0.01).

Perfect Squares to Remember

Memorizing squares like 15² = 225, 22² = 484, and 25² = 625 is essential for fast calculation in competitive exams.

BODMAS Rule

Always follow the BODMAS/VBODMAS priority (Brackets, Orders, Division, Multiplication, Addition, Subtraction) for simplification problems.

Shortcut Trick

To find the constant number to be added to a, b, c, and d to make them proportional, use the formula:

x = |(ad − bc) ÷ ((a + d) − (b + c))|

Here, a = 2, b = 3, c = 11, d = 13.

x = |(2 × 13 − 3 × 11) ÷ ((2 + 13) − (3 + 11))| = |(26 − 33) ÷ (15 − 14)| = |−7 ÷ 1| = 7.

∴ The correct answer is 7.

Alternate Method

Given: Numbers are 2, 3, 11, and 13. Let the number to be added be x.

Formula Used: If a, b, c, d are in proportion, then (a + x) ÷ (b + x) = (c + x) ÷ (d + x).

⇒ (2 + x) ÷ (3 + x) = (11 + x) ÷ (13 + x)

⇒ (2 + x)(13 + x) = (11 + x)(3 + x)

⇒ 26 + 13x + 2x + x² = 33 + 3x + 11x + x²

⇒ 26 + 15x = 33 + 14x

⇒ 15x − 14x = 33 − 26

⇒ x = 7

∴ The correct answer is 7.

Additional Information

Fourth Proportional

If four numbers a, b, c, and d are in proportion (a:b = c:d), then d is called the fourth proportional and d = (b × c) ÷ a.

Mean Proportional

The mean proportional between two numbers a and b is given by the formula √(a × b).

Third Proportional

If a, b, and c are in continued proportion (a:b = b:c), then c is the third proportional and c = b² ÷ a.

Shortcut Trick

Total age increase for Juhi and Surbhi in 5 years = 5 × 2 = 10 years.

Present Sum of their ages = 66 + 10 = 76 years.

Given Difference in ages = 24 years (Juhi is older).

Surbhi's age (Younger) = (Sum − Difference) ÷ 2 = (76 − 24) ÷ 2 = 52 ÷ 2 = 26 years.

∴ The correct answer is 26 years.

Alternate Method

Given:

Difference between Juhi (J) and Surbhi (S) = 24 years ⇒ J − S = 24

Sum of ages 5 years ago = 66 years.

Formula Used:

Sum of ages after n years = Past Sum + (Number of people × n)

Calculations:

⇒ (J − 5) + (S − 5) = 66

⇒ J + S − 10 = 66

⇒ J + S = 76 ----(Equation 1)

⇒ J − S = 24 ----(Equation 2)

Subtracting Equation 2 from Equation 1:

⇒ (J + S) − (J − S) = 76 − 24

⇒ 2S = 52

⇒ S = 52 ÷ 2 = 26

∴ The correct answer is 26 years.

Additional Information

Constant Age Difference

The numerical difference between the ages of two people always remains the same, regardless of how many years pass or how many years ago it was.

Sum of Ages Rule

If the sum of ages of 'n' persons is 'S' at present, then 'x' years ago, the sum was S − (n × x), and 'x' years hence, it will be S + (n × x).

Finding Individual Values

Given Sum (S) and Difference (D) of two values: Larger = (S + D) ÷ 2 and Smaller = (S − D) ÷ 2.

Shortcut Trick

To find the value of 27³ − 22³, check the unit digit of the expression.

Unit digit of 27³ = unit digit of 7³ = 7 × 7 × 7 = 343 → 3

Unit digit of 22³ = unit digit of 2³ = 2 × 2 × 2 = 8 → 8

Unit digit of the result = (3 − 8) → Borrow 1 → (13 − 8) = 5

Among the given options, only 9035 ends with the digit 5.

∴ The correct answer is 9035.

Alternate Method

Given:

First number (a) = 27

Second number (b) = 22

Formula Used:

a³ − b³ = (a − b)(a² + ab + b²)

Calculations:

⇒ 27³ − 22³ = (27 − 22) × (27² + 27 × 22 + 22²)

⇒ 27³ − 22³ = 5 × (729 + 594 + 484)

⇒ 27³ − 22³ = 5 × (1807)

⇒ 27³ − 22³ = 9035

∴ The correct answer is 9035.

Additional Information

Sum of Cubes Identity

The sum of two cubes is factored as: a³ + b³ = (a + b)(a² − ab + b²).

Difference of Squares

One of the most used algebraic identities: a² − b² = (a − b)(a + b).

Cube of a Difference

The whole cube of a binomial difference: (a − b)³ = a³ − 3a²b + 3ab² − b³.

Shortcut Trick

Use the direct formula for one filling tap (x) and one emptying tap (y):

Net time taken = (x × y) ÷ (y − x)

Where x = 11 hr and y = 22 hr.

Time = (11 × 22) ÷ (22 − 11) = (11 × 22) ÷ 11 = 22 hr.

∴ The correct answer is 22 hr.

Alternate Method

Given:

Time taken by filling tap = 11 hr

Time taken by emptying tap = 22 hr

Formula Used:

Total Work = LCM of individual times.

Efficiency = Total Work ÷ Time.

Calculations:

⇒ Efficiency of Filling Tap = 22 ÷ 11 = +2 units/hr

⇒ Efficiency of Emptying Tap = 22 ÷ 22 = −1 units/hr (negative because it empties)

⇒ Net efficiency when both taps are open = 2 − 1 = 1 unit/hr

⇒ Time required to fill the tank = Total Capacity ÷ Net Efficiency

⇒ Time = 22 ÷ 1 = 22 hr

∴ The correct answer is 22 hr.

Additional Information

Inlet and Outlet Pipes

An Inlet pipe fills a tank (positive work), while an Outlet pipe empties it (negative work). Net work is the algebraic sum of their efficiencies.

Time and Efficiency Relation

Time taken to complete a task is inversely proportional to the efficiency. If a tap fills a tank in 'n' hours, its work done in 1 hour is 1/n.

Capacity Calculation

If the flow rate (liters/hr) is given, Capacity = Flow Rate × Time taken to fill the tank.

Shortcut Trick

The empirical relation between measures of central tendency is Mode = 3 × Median − 2 × Mean.

Substitute the given values: Mode = 3(30) − 2(20).

Mode = 90 − 40 = 50.

∴ The correct answer is 50.

Alternate Method

Given:

Mean of the data = 20

Median of the data = 30

Formula Used:

For a moderately skewed distribution, the empirical formula is:

Mode = 3 × Median − 2 × Mean

Calculations:

⇒ Mode = 3 × 30 − 2 × 20

⇒ Mode = 90 − 40

⇒ Mode = 50

∴ The correct answer is 50.

Additional Information

Empirical Relation Applicability

The formula Mode = 3Median − 2Mean is an approximation used specifically for moderately skewed frequency distributions.

Symmetric Distribution

In a perfectly symmetrical distribution (like the Normal distribution), the Mean, Median, and Mode are all equal (Mean = Median = Mode).

Positively Skewed Distribution

For a positively skewed distribution (tail to the right), the relation is generally Mean > Median > Mode.

Negatively Skewed Distribution

For a negatively skewed distribution (tail to the left), the relation is generally Mean < Median < Mode.

Shortcut Trick

Simplify brackets first: (5/6 + 2/6) = 7/6 and (0.4 + 0.3) = 0.7 = 7/10.

Apply division and multiplication: (7/6 × 10/7) × 1/2 = 10/6 × 1/2 = 5/6.

Complete addition and subtraction: 5/6 + 1/2 − 1/6 = 4/6 + 3/6 = 7/6.

∴ The correct answer is 7/6.

Alternate Method

Given Expression: ((5/6) + (1/3)) ÷ ((2/5) + 0.3) × 0.5 + (1/2) − (1/6)

Method Used: Apply the BODMAS rule to solve the expression in the correct sequence.

⇒ Step 1 (Solve Brackets): (5/6 + 2/6) ÷ (2/5 + 3/10) × 0.5 + 1/2 − 1/6

⇒ Step 2: (7/6) ÷ (4/10 + 3/10) × 0.5 + 1/2 − 1/6

⇒ Step 3: (7/6) ÷ (7/10) × 0.5 + 1/2 − 1/6

⇒ Step 4 (Division & Multiplication): (7/6 × 10/7) × 1/2 + 1/2 − 1/6

⇒ Step 5: (10/6) × (1/2) + 1/2 − 1/6

⇒ Step 6: 5/6 + 1/2 − 1/6

⇒ Step 7 (Addition & Subtraction): (5/6 − 1/6) + 1/2 = 4/6 + 1/2

⇒ Final Step: 2/3 + 1/2 = (4 + 3)/6 = 7/6

∴ The correct answer is 7/6.

Additional Information

BODMAS / VBODMAS Rule

It defines the order of operations: Vinculum, Brackets, Of, Division, Multiplication, Addition, and Subtraction.

Reciprocal in Division

To divide by a fraction, multiply by its reciprocal: (a/b) ÷ (c/d) = (a/b) × (d/c).

Fraction to Decimal Conversion

Common conversions like 1/2 = 0.5, 2/5 = 0.4, and 1/3 ≈ 0.33 help in speeding up simplification steps.

Shortcut Trick

For a semi-circle, the relationship between perimeter (P) and radius (r) is simplified as:

P = r × (36/7)

Given radius (r) = 21 cm

P = 21 × (36/7) ⇒ 3 × 36 = 108 cm

∴ The correct answer is 108 cm.

Alternate Method

Given:

Radius (r) of the semi-circle = 21 cm

Value of π = 22/7

Formula Used:

Perimeter of semi-circle = Length of Curved Arc + Diameter

Perimeter = πr + 2r

Calculations:

⇒ Length of curved arc = π × r

⇒ Arc length = (22/7) × 21

⇒ Arc length = 22 × 3 = 66 cm

⇒ Diameter of semi-circle = 2 × r

⇒ Diameter = 2 × 21 = 42 cm

⇒ Total Perimeter = 66 + 42

⇒ Total Perimeter = 108 cm

∴ The correct answer is 108 cm.

Additional Information

Area of Semi-circle

The space occupied by the semi-circle is half the area of a full circle: Area = (1/2) × π × r².

Perimeter of a Quadrant

For a quarter-circle (quadrant), the perimeter is calculated as (πr/2) + 2r, which simplifies to 25r/7 when π = 22/7.

Circumference vs Perimeter

Circumference usually refers to the distance around a full circle (2πr), while perimeter of a sector or semi-circle includes the straight boundaries (radii or diameter).

Shortcut Trick

LCM of 12, 15, and 20 is 60 units (Total Work).

Efficiencies: A = 5, B = 4, C = −3 (outlet).

Work done by A + B in 5 hr = (5 + 4) × 5 = 45 units.

Remaining work = 60 − 45 = 15 units.

Combined efficiency (A + B + C) = 5 + 4 − 3 = 6 units/hr.

Total time = 5 + (15 ÷ 6) = 5 + 2.5 = 7.5 hr.

∴ The correct answer is 7.5 hr.

Alternate Method

Given:

Time taken by Pipe A = 12 hr

Time taken by Pipe B = 15 hr

Time taken by Outlet C = 20 hr

Calculations:

⇒ Total capacity of the tank (LCM of 12, 15, 20) = 60 units

⇒ Efficiency of Pipe A = 60 ÷ 12 = 5 units/hr

⇒ Efficiency of Pipe B = 60 ÷ 15 = 4 units/hr

⇒ Efficiency of Outlet C = 60 ÷ 20 = −3 units/hr (negative because it empties)

⇒ Work done by A and B in 5 hours = (5 + 4) × 5 = 45 units

⇒ Remaining work = 60 − 45 = 15 units

⇒ Now Pipe C is also opened, so net efficiency = 5 + 4 − 3 = 6 units/hr

⇒ Time to fill remaining tank = 15 ÷ 6 = 2.5 hr

⇒ Total time = Initial time + Remaining time = 5 + 2.5 = 7.5 hr

∴ The correct answer is 7.5 hr.

Additional Information

Negative Efficiency

In Pipes and Cisterns, an outlet pipe (leak) is always assigned a negative efficiency because it reduces the total work done by the inlet pipes.

Efficiency Formula

Efficiency = Total Work ÷ Time. Total work is usually taken as the Least Common Multiple (LCM) of the individual time periods.

Work and Wages

If multiple pipes or workers are involved, the total wages are distributed in the ratio of the work done by each, which is (Efficiency × Time).

Shortcut Trick

One-way Distance = [ (Product of Speeds) ÷ (Sum of Speeds) ] × Total Time

⇒ Distance = [ (48 × 60) ÷ (48 + 60) ] × 9

⇒ Distance = [ 2880 ÷ 108 ] × 9

⇒ Distance = 26.666... × 9 = 240 km

∴ The correct answer is 240 km.

Alternate Method

Given: Speed A → B = 48 km/h, Speed B → A = 60 km/h, Total Time = 9 hr.

Formula: Speed is inversely proportional to Time when Distance is constant.

⇒ Ratio of Speeds (A→B : B→A) = 48 : 60 = 4 : 5

⇒ Ratio of Time taken (A→B : B→A) = 5 : 4

⇒ Total time units = 5 + 4 = 9 units

⇒ 9 units = 9 hours ⇒ 1 unit = 1 hour

⇒ Time taken from City A to City B = 5 × 1 = 5 hours

⇒ Distance = Speed × Time = 48 km/h × 5 hr = 240 km

∴ The correct answer is 240 km.

Additional Information

Average Speed Formula

When a person travels equal distances at speeds x and y, the Average Speed for the whole journey is (2xy) ÷ (x + y).

Total Distance vs. Displacement

In this journey, the total distance covered is 2D (480 km), while the displacement is 0 as the traveller returns to the starting point.

Conversion Factor

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

Shortcut Trick

Effective quarterly rate = 12% ÷ 4 = 3%.

Number of quarters in 6 months = 6 ÷ 3 = 2.

Successive percentage change for 2 periods = 3 + 3 + (3 × 3) ÷ 100 = 6.09%.

Total Amount = 106.09% of 20,000 = 200 × 106.09 = 21218.

∴ The correct answer is ₹21218.

Alternate Method

Given:

Principal (P) = ₹20,000

Annual Rate (R) = 12%

Time (T) = 6 months

Formula Used:

For quarterly compounding:

Rate (r) = R ÷ 4 per quarter

Number of periods (n) = T × 4 (where T is in years) or Total Quarters

Amount (A) = P × (1 + r/100)ⁿ

Calculations:

⇒ Quarterly rate (r) = 12% ÷ 4 = 3%

⇒ Time (n) = 6 months = 2 quarters

⇒ A = 20,000 × (1 + 3/100)²

⇒ A = 20,000 × (1.03)²

⇒ A = 20,000 × 1.0609

⇒ A = 21,218

∴ The correct answer is ₹21218.

Additional Information

Compound Interest Compounding Frequencies

If interest is compounded Half-yearly, rate becomes R/2 and time becomes 2T. If Quarterly, rate becomes R/4 and time becomes 4T.

Effective Rate of Interest

It is the actual annual interest rate when compounding occurs more than once a year. Formula: (1 + r/n)ⁿ − 1, where n is the number of compounding periods.

Difference between CI and SI

For 2 years/periods, the difference (D) between Compound Interest and Simple Interest is given by D = P × (r/100)².

Shortcut Trick

For a regular polygon, the number of sides (n) = 360° ÷ exterior angle.

n = 360° ÷ 40° = 9 sides

The number of diagonals in a polygon of n sides is given by [n × (n − 3)] ÷ 2.

Diagonals = [9 × (9 − 3)] ÷ 2 = [9 × 6] ÷ 2 = 27

∴ The correct answer is 27 diagonals.

Alternate Method

Given:

Measure of each exterior angle of a regular polygon = 40°

Formula Used:

1. Number of sides (n) = 360° ÷ Exterior angle
2. Number of diagonals = [n × (n − 3)] ÷ 2

Calculations:

⇒ Let the number of sides of the regular polygon be n.

⇒ We know that the sum of all exterior angles of any polygon is 360°.

⇒ Since it is a regular polygon, each exterior angle is equal.

⇒ n × 40° = 360°

⇒ n = 360° ÷ 40°

⇒ n = 9 (The polygon is a nonagon).

⇒ Number of diagonals = [n × (n − 3)] ÷ 2

⇒ Diagonals = [9 × (9 − 3)] ÷ 2

⇒ Diagonals = [9 × 6] ÷ 2

⇒ Diagonals = 54 ÷ 2 = 27

∴ The correct answer is 27 diagonals.

Additional Information

Interior and Exterior Angle Relationship

In any regular polygon, the sum of an interior angle and its adjacent exterior angle is always 180°.

Sum of Interior Angles

The sum of all interior angles of a polygon with n sides is calculated as (n − 2) × 180°.

Each Interior Angle of Regular Polygon

The measure of each interior angle in a regular polygon is [(n − 2) × 180°] ÷ n.

Shortcut Trick

When two articles are sold at the same Selling Price (SP) with equal Profit% and Loss% (x%), there is always a net loss of (x²/100)%.

Net Loss % = (45 × 45) ÷ 100 = 20.25%.

Total SP = ₹42,000 + ₹42,000 = ₹84,000.

Loss Amount = [Loss% ÷ (100 − Loss%)] × Total SP = [20.25 ÷ 79.75] × 84,000.

Loss Amount ≈ 0.2539 × 84,000 ≈ ₹21,329.

∴ The correct answer is ₹21,329 loss.

Alternate Method

Given: Selling Price of each article = ₹42,000, Profit on first = 45%, Loss on second = 45%.

Formula Used: Cost Price (CP) = [SP × 100] ÷ (100 ± P/L%)

⇒ CP of first article = (42,000 × 100) ÷ 145 ≈ ₹28,965.52

⇒ CP of second article = (42,000 × 100) ÷ 55 ≈ ₹76,363.64

⇒ Total CP = 28,965.52 + 76,363.64 = ₹1,05,329.16

⇒ Total SP = 42,000 + 42,000 = ₹84,000

⇒ Overall Loss = 1,05,329.16 − 84,000 = ₹21,329.16

⇒ Nearest integer = ₹21,329 loss.

∴ The correct answer is ₹21,329 loss.

Additional Information

Net Percentage Change

When a value is increased by x% and then decreased by x%, the net change is always a decrease of x²/100 %.

Same CP Case

If two articles are bought for the same Cost Price and sold at x% profit and x% loss respectively, there is No Profit, No Loss in the whole transaction.

Successive Profit/Loss

For successive changes of a% and b%, net change % = [a + b + (ab/100)]%, where positive values are for profit/increase and negative for loss/decrease.

Shortcut Trick

Using the Ratio Method: 15 notebooks : ₹240 = x notebooks : ₹992

Reducing the first ratio: 1 : (240 ÷ 15) → 1 : 16

This means every 1 notebook costs ₹16.

Number of notebooks (x) = 992 ÷ 16 = 62

∴ The correct answer is 62 notebooks.

Alternate Method

Given:

Initial quantity = 15 notebooks

Initial cost = ₹240

Available amount = ₹992

Formula Used:

Cost of 1 unit = Total Cost ÷ Total Quantity

Required Quantity = Available Amount ÷ Cost of 1 unit

Calculations:

⇒ Cost of 15 notebooks = ₹240

⇒ Cost of 1 notebook = 240 ÷ 15

⇒ Cost of 1 notebook = ₹16

⇒ Number of notebooks for ₹992 = 992 ÷ 16

⇒ Number of notebooks = 62

∴ The correct answer is 62 notebooks.

Additional Information

Unitary Method

A technique where we first find the value of a single unit and then multiply it by the required number of units to find the total value.

Direct Proportion

When two quantities increase or decrease at the same rate, they are in direct proportion. Formula: x₁/y₁ = x₂/y₂.

Cost Calculation

Total Cost = Unit Price × Quantity. Conversely, Quantity = Total Cost ÷ Unit Price.

Given:

Based on the statement provided, we can identify the following relationships:

 

Let us break down the relations step-by-step to determine how G is related to L:

1. G is the mother of Y - G is female and belongs to the generation above Y.

2. Y is the sister of V - Y and V are siblings (both children of G). Since G is the mother of Y, G is also the mother of V.

3. L is the husband of V - This indicates L (male) and V (female) are a married couple.

4. S is the father of L - S is the parent of L in the generation above.

Since G is the mother of V, and V is the wife of L, it follows that G is the mother of L’s wife.

Therefore, G is the Wife's mother of L.

Hence, the correct answer is 'Option 3'.

Given Word:

DEARTHS

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:

E, F, B, S, U, I, T

Arranged alphabetically:

B, E, F, I, S, T, U

Counting from the right:

<br>

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

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

Given:

Seven boxes (A, B, C, D, E, F, G) are kept one over the other from 1 (bottom) to 7 (top).

1. Only two boxes are kept below D.

2. Only three boxes are kept between E and D.

Position Box

7 (Top) E

6  

5  

4  

3 D

2  

1 (Bottom)  

 

3. C is kept above A but below G (G > C > A).

4. B is kept above F but below A (A > B > F).

Combining the above points, we get the relative order: G > C > A > B > F. These five boxes must fit into the remaining five positions (6, 5, 4, 2, and 1).

Final Arrangement:

7 E

6 G

5 C

4 A

3 D

2 B

1 F

 

Thus, the box kept immediately above A is C.

Hence, the correct answer is 'Option 3'.

Given:

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

 

On checking the expression: F − G × H ÷ I − J ÷ K

1. F − G - 'F − G' means 'F is the wife of G' (So, G is the husband of F).

2. G × H - 'G × H' means 'G is the brother of H'.

3. H ÷ I - 'H ÷ I' means 'H is the son of I' (This means G is also the son of I).

4. I − J - 'I − J' means 'I is the wife of J' (So, J is the father of G and H).

5. J ÷ K - 'J ÷ K' means 'J is the son of K'.

Therefore, J is the son of K, and G is the son of J. This means G is the Son's son of K.

Hence, the correct answer is "Option 4".

Given:

Seven boxes A, B, C, D, E, F, and G are kept one over the other from 1 (bottom) to 7 (top).

1) B is kept third from the top.

2) Only three boxes are kept between D and B.

3) A is kept above B.

 

4) A is kept below E. (Since A is at position 6, E must be at position 7).

5) C is kept immediately below G.

6) C is kept above F.

Final Arrangement:

 

Box A is at position 6 and box C is at position 3. The boxes between them are B (5) and G (4).

Thus, two boxes are kept between C and A.

Hence, the correct answer is 'Option 1'.

Sign Transformation:

 

Given Equation: 48 − 96 × 12 ÷ 4 + 32

By replacing the signs according to the table above, we get the new equation:

= 48 + 96 ÷ 12 × 4 − 32

= 48 + 8 × 4 - 32

= 48 + 32 - 32

= 48

Therefore, the value that comes in place of the question mark is 48.

Hence, the correct answer is "Option 3".

Given Series:

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

Perfect Squares:

1, 4, 9   (1 is also a perfect square)

Required pattern:

Perfect Square → Even Digit → Even Digit

Checking the series:

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

Even digits satisfying the condition:

→ 2 in (9 2 6)

→ 2 in (1 2 6)

→ 8 in (4 8 2)

→ 8 in (9 8 6)

→ 2 in (4 2 2)

Therefore, there are 5 such even digits.

Hence, the correct answer is "Option 3 — Five".

Given:

TYRE is coded as '8526'

BYTE is coded as '5986'

 

Hence, the correct answer is "Option 1".

The logic followed here is

Logic: Second letter-cluster = (Each letter of the first letter-cluster) + 2.

Option 1)  OV - QX,

Option 2) KU - MT,

This pair does not follow the rule.

Option 3) DK - FM

Option 4) HO - JQ,

Therefore, the letter-cluster pair that is different from the others is KU - MT.

Hence, the correct answer is 'Option 2'.

Given Series:

(Left) D J L O E A H O A F A G E B G A N R S H X (Right)

Required pattern:

Consonant → Consonant → Vowel

Checking the series:

(Left) D J L O O E A H O A F A G E B B G A A N R S H X (Right)

Consonants satisfying the condition:

→ L in (J L O)

→ G in (B G A)

Therefore, there are 2 such consonants.

Hence, the correct answer is "Option 3 — Two".

The logic followed in this pattern is a consistent alphabetical shift where each letter is moved forward by two positions in the English alphabet.

For 'VRHM' related to 'XTJO':

For 'DZPU' related to 'FBRW':

Applying the same logic to 'JVFA':

Thus, JVFA is related to LXHC.

Hence, the correct answer is 'Option 2'.

The logic followed here:

Only two people sit between T and Q when counted from the left of Q.

Only three people sit between W and U when counted from the right of U.

E sits to the immediate right of W.

Y is an immediate neighbour of U as well as Q.

 

So, 2 peoples sit between R and U when counted from the left of R.

Hence, the correct answer is option (4).

The logic followed in this pattern is based on alphabetical shifts according to the positional values of the letters.

For 'KNG' related to 'MLF':

For 'GRN' related to 'IPM':

Applying the same logic to 'JSN':

The logic requires a +2, −2, −1 shift. Only the pair JSN : LQM satisfies this pattern.

Hence, the correct answer is 'Option 4'.

Given:

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

 

On checking: S ÷ E + W × T − N

1. S ÷ E - 'S ÷ E' means 'S is the father of E'.

2. E + W - 'E + W' means 'E is the sister of W'.

3. W × T - 'W × T' means 'W is the wife of T'.

4. T − N - 'T − N' means 'T is the brother of N'.

Therefore, S is the father of W, and W is the wife of T (who is the brother of N). Thus, S is the father of N's brother's wife.

Hence, the correct answer is "Option 2".

The logic followed here is

Logic: First letter − 3 = Third letter and Second letter − 3 = Fourth letter.

Option 1) QR - MP,

This pair does not follow the rule.

Option 2) KO - HL,

Option 3) XB - UY

Option 4) DI - AF,

Therefore, the letter-cluster pair that is different from the others is QR - MP.

Hence, the correct answer is 'Option 1'.

The logic followed here is

Logic: In each pair, the first letter of the second cluster is the 4th letter after the first letter of the first cluster, and the second letter of the second cluster is the 4th letter after the second letter of the first cluster.

Option 1)  UX−XD,

This pair does not follow the rule.

Option 2) FI−JM,

Option 3) DG−HK,

Option 4) SV−WZ,

Therefore, the letter-cluster pair that is different from the others is UX−XD.

Hence, the correct answer is 'Option 1'.

The logic followed to determine the final position is based on calculating the net displacement in the North-South and East-West directions.

Step-by-step Movement tracking for Anu:

1. Anu starts from Point A and drives 3 km towards the north. 2. He then takes a right turn, drives 2 km, turns right and drives 5 km. 3. He then takes a right turn and drives 4 km. 4. He takes a final right turn, drives 2 km and stops at Point P.

To go from a point in the West back to the starting point, he must drive towards the East.

Therefore, he should drive 2 km to the east.

Hence, the correct answer is "Option 3".

The logic followed here is:

The given number series follows an alternating addition pattern as shown below:

The logic is to add 7 and 9 alternatively to the previous number.

Thus, the missing term in the series is 86.

Hence, the correct answer is 'Option 4';

Given Word:

PANDEMIC

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.

Transformation:

New group of letters:

O, B, M, C, F, L, J, B

Letter frequencies:

B → 2 times

O, M, C, F, L, J → 1 time each

Therefore, the letter appearing exactly twice is B.

Hence, the correct answer is "Option 3 — B".

Logic followed here is.

Logic: The code for the word is determined by comparing the common letters between the words and identifying the corresponding unique digit assigned to the remaining letter.

Given:

DUKE is coded as '8123'

DESK is coded as '2314'

For

 

Therefore, the code for 'S' is 4.

Hence, the correct answer is "Option 2".

Logic followed here is.

Logic: (1st letter + 2 = 2nd letter) and (2nd letter − 3 = 3rd letter).

For

KHG→

<br>

For

MOL →

<br>

For

NPM →

<br>

For

GIF →

 

Therefore, the letter-cluster KHG is the odd one out because it does not follow the (+2, −3) pattern common to the others.

Hence, the correct answer is "Option 1".

1. Since all persons face the centre: right = counter-clockwise (CCW), left = clockwise (CW). Fix E at position 1. Only two people are seated between E and F when counted from the right of E: E(1) → pos(2) → pos(3) → F(4).

2. Only two people are seated between F and D from the left of D (CW): D → pos(6) → pos(5) → F(4). Thus, D is at pos(7).

3. Only three people are seated between E and B from the left of B (CW): B → pos(4) → pos(3) → pos(2) → E(1). B is at pos(5).

4. C is to the immediate left of G (CW). Available positions are 2, 3, and 6. If G is at pos(3), immediate left is pos(2), which is vacant. So G(3) and C(2).

5. The remaining person A occupies pos(6). Final arrangement: E(1), C(2), G(3), F(4), B(5), A(6), D(7).

<br>

To find the number of people between A and F when counted from the right of F (CCW): F(4) → B(5) → A(6). Only B sits between them. Thus, 1 person is seated between A and F.

<br>

Hence, the correct answer is 'Option 2'.

The logic followed in this pattern is a series of alphabetical shifts where each letter is moved forward by a fixed number of positions.

For 'EHLF' related to 'NQUO':

For 'HKOI' related to 'QTXR':

Applying the same logic to 'JMQK':

Thus, JMQK is related to SVZT.

Hence, the correct answer is 'Option 4'.

The logic followed in this number series is based on adding the squares starting from 1 to 6 to get the next term.

 

Therefore, the number that should come in place of the question mark is 97.

Hence, the correct answer is "Option 2".

The mapping of letters to operators is as follows:

 

For the given equation:

147 A 3 C 51 D 16 B 3 = ? →

By substituting the operators in place of the letters, we get the following expression:

= 147 ÷ 3 + 51 − 16 × 3

= 49 + 51 - 16 × 3

= 49 + 51 - 48

= 100 - 48

= 52

Therefore, the value that will come in place of the question mark is 52.

Hence, the correct answer is "Option 2".

The correct answer is Only (ii).

Key Points

• Statement (i) claims the speed of sound decreases with temperature. However, the speed of sound in a gas is directly proportional to the square root of the absolute temperature, as shown by the formula $v = \sqrt{\gamma RT/M}$. As temperature increases, molecules move faster, transferring the vibrational energy more quickly. Hence, statement (i) is incorrect.
• Statement (ii) mentions that sound travels faster in solids than in gases. Sound is a mechanical wave that propagates through the interaction of particles. Solids have a much higher elastic modulus (stiffness) and closer atomic arrangement than gases, which allows the disturbance to travel at significantly higher velocities. Hence, statement (ii) is correct.
• Statement (iii) suggests the speed is independent of medium properties. In reality, the speed of sound is strictly determined by the elasticity and density of the medium through which it travels. It varies from approximately 343 m/s in air to over 5000 m/s in steel. Hence, statement (iii) is incorrect.

 Additional Information

• Phase of Matter Comparison:
• In solids, the atoms are tightly bound by strong intermolecular forces, leading to high bulk modulus values and faster transmission.
• In liquids, the speed is generally higher than in gases but lower than in solids due to moderate compressibility.
• In gases, the particles are far apart, resulting in the slowest speed of sound among the three states.
• Medium Properties:
• Elasticity: This is the primary factor; more elastic (stiffer) materials allow sound to travel faster.
• Density: While higher density generally decreases speed ($v \propto 1/\sqrt{\rho}$), the massive increase in elasticity when moving from gas to solid outweighs the density increase.
• Humidity: Increased humidity in air slightly increases the speed because water vapor reduces the overall density of the air.

The correct answer is 0.5 A.

Key Points

• In a Series Connection, resistors are connected end-to-end such that the same current flows through each resistor sequentially. The Equivalent Resistance (R_eq) is calculated by the algebraic sum of individual resistances: R_eq = R₁ + R₂ + ... + R_n.
• For the given circuit, the two resistors are 5 Ω and 15 Ω. Therefore, the total resistance is calculated as: R_total = 5 Ω + 15 Ω = 20 Ω.
• Ohm's Law defines the relationship between voltage (V), current (I), and resistance (R) in an ideal conductor, expressed by the formula: V = I × R.
• To find the current flowing through the circuit, we rearrange the formula to I = V / R. Substituting the given values: I = 10 V / 20 Ω.
• Performing the division, the current I = 0.5 A. This indicates that a charge of 0.5 Coulombs passes through any point in the circuit every second.

 Additional Information

• Series Circuit Characteristics:
• The Current (I) remains uniform at every point in a series circuit; thus, 0.5 A flows through both the 5 Ω and 15 Ω resistors.
• The Voltage (V) across the battery is divided among the resistors. Using V = IR, the voltage drop across the 5 Ω resistor is 2.5 V, and across the 15 Ω resistor, it is 7.5 V, totaling the source 10 V.
• If one component in a series circuit fails or breaks, the entire circuit is interrupted, and current stops flowing completely.
• Comparison with Parallel Circuits:
• In a Parallel Connection, the reciprocal of the total resistance is the sum of the reciprocals of individual resistances: 1/R_p = 1/R₁ + 1/R₂.
• Parallel circuits maintain the same voltage across all branches, but the total current is the sum of currents through each branch.
• Units of Measurement:
• Resistance is measured in Ohms (Ω), named after Georg Simon Ohm.
• Electric Current is measured in Amperes (A), representing the rate of flow of electric charge.
• Potential Difference is measured in Volts (V), representing the work done to move a unit charge between two points.

The correct answer is A–2, B–1, C–3, D–4.

Key Points

• Sodium (Na) possesses an atomic number of 11, which indicates it has 11 electrons in its neutral state. According to the Bohr–Bury rules, these are distributed as 2 in the K-shell, 8 in the L-shell, and the remaining 1 in the M-shell. Hence, Sodium (Na) matches with 2, 8, 1.
• Magnesium (Mg) has an atomic number of 12. Its electron configuration is determined by filling the shells in increasing order of energy, resulting in 2 electrons in the K-shell, 8 in the L-shell, and 2 in the M-shell. Hence, Magnesium (Mg) matches with 2, 8, 2.
• Aluminium (Al) is the element with atomic number 13. Following the principle of maximum capacity and shell order, it houses 2 electrons in its innermost shell, 8 in the second, and 3 in the third shell. Hence, Aluminium (Al) matches with 2, 8, 3.
• Calcium (Ca) has an atomic number of 20. While the M-shell can hold up to 18 electrons, the octet rule dictates that the outermost shell cannot have more than 8 electrons; thus, after 8 electrons fill the M-shell, the next 2 enter the N-shell. Hence, Calcium (Ca) matches with 2, 8, 8, 2.

 Additional Information

• Electronic Configuration Principles:
• 2n² Formula: This rule determines the maximum number of electrons a shell can hold, where n represents the orbit number or energy level (K=1, L=2, M=3, N=4).
• Shell Capacities: Based on the formula, the K-shell holds 2, L-shell holds 8, M-shell holds 18, and N-shell holds 32 electrons.
• Valence Shell Stability: The outermost shell of any atom cannot accommodate more than 8 electrons, ensuring chemical stability as seen in Noble Gases.
• Step-wise Filling: Electrons do not occupy a new shell until the inner shells are completely filled, creating a structured electronic arrangement.
• Valency: The number of electrons in the outermost shell (valence electrons) determines the valency and reactivity of the element.

The correct answer is Resistance offered by air to a moving object.

Key Points

• The Universal Law of Gravitation, formulated by Sir Isaac Newton, states that every particle of matter in the universe attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
• The falling of apples from a tree is caused by the gravitational attraction between the Earth and the apple, pulling the object toward the center of the planet.
• The weight of objects on Earth is a direct measure of the gravitational force exerted by the Earth on that object, calculated using the relationship Weight = Mass × gravity.
• The formation of tides in the Earth's oceans is a result of the gravitational pull exerted by the Moon and the Sun, which creates tidal bulges as the Earth rotates.
• Air resistance, or drag, is a frictional force exerted by air molecules against the motion of an object. This is a contact force dependent on the object's velocity, surface area, and shape, and is not explained by the principles of universal gravitation.

 Additional Information

• Gravitational Constant (G)
• It is a universal constant with a value of approximately $6.674 \times 10^{-11} \text{ N m}^{2} \text{ kg}^{-2}$.
• Unlike the acceleration due to gravity (g), G remains constant throughout the universe.
• Air Resistance (Fluid Friction)
• It acts in the direction opposite to the relative motion of the object.
• In a vacuum, where air is absent, air resistance is zero, allowing objects of different masses to fall at the same rate.
• It is a non-conservative force because the energy spent overcoming it is dissipated as heat.
• Tidal Forces
• Tides are influenced more by the Moon than the Sun because of the Moon's proximity to Earth, despite the Sun's much larger mass.

The correct answer is 0 m.

Key Points

• The field is a square with a side length of 11 m. The total perimeter of the field is 4 × 11 = 44 m.
• The farmer completes one full round of the boundary in 20 seconds.
• The total time spent walking is 3 minutes and 20 seconds. To calculate this in seconds: (3 × 60) + 20 = 200 seconds.
• The number of rounds completed is determined by dividing the total time by the time per round. Calculation: 200 / 20 = 10 full rounds.
• Since the farmer completes exactly 10 rounds, he returns precisely to the starting point on the boundary of the square field.
• Displacement is a vector quantity defined as the shortest straight-line distance between the initial position and the final position. Because the initial and final positions are the same, the displacement is zero. Hence, the magnitude of displacement is 0 m.

 Additional Information

• Distance vs. Displacement:
• Distance is the total path length traveled, which in this case is 10 rounds × 44 m = 440 m. Unlike displacement, it is a scalar quantity and cannot be zero if motion has occurred.
• Displacement depends only on the endpoints of the motion, not the path taken.
• Vector Properties:
• If the farmer had stopped after 10.5 rounds (210 seconds), his displacement would be the diagonal of the square, calculated using the Pythagoras theorem as √(11² + 11²), which is approximately 15.56 m.
• In circular or closed-loop motion, displacement is always zero if the object completes an integer number of revolutions or circuits.

The correct answer is 1.33.

Key Points

• The Refractive Index (n) of a material is a dimensionless physical quantity that indicates the factor by which the speed of light is reduced inside the medium compared to its speed in a vacuum.
• The mathematical relationship is expressed by the formula n = c / v, where c represents the speed of light in a vacuum and v represents the speed of light in the given medium.
• The speed of light in a vacuum (c) is a universal physical constant valued at approximately 3 × 10⁸ m/s (more precisely 299,792,458 m/s).
• Given that the speed of light in the medium is 2.25 × 10⁸ m/s, the calculation is performed as follows: n = (3 × 10⁸) / (2.25 × 10⁸).
• The 10⁸ terms in the numerator and denominator cancel out, leaving the ratio 3 / 2.25. Converting the decimal, 2.25 is equivalent to 9/4, making the equation 3 / (9/4), which simplifies to 12/9 or 1.33.
• A refractive index of 1.33 is a significant scientific value as it identifies the medium as liquid water at standard temperature and pressure.

 Additional Information

• Optical Density and Refraction:
• The refractive index is a measure of the optical density of a medium; a higher index implies light travels slower and bends more toward the normal when entering from a vacuum.
• Snell's Law (n₁ sinθ₁ = n₂ sinθ₂) utilizes these indices to calculate the change in direction of light at an interface.
• The Absolute Refractive Index of a vacuum is exactly 1, while for air, it is approximately 1.0003, often rounded to 1 for basic physics calculations.
• Other common materials include Crown Glass (n ≈ 1.52), Flint Glass (n ≈ 1.66), and Diamond, which has a very high index of 2.42, contributing to its extreme dispersion and brilliance.
• The refractive index can also depend on the wavelength of light, a phenomenon known as dispersion, which causes the separation of white light into a spectrum when passing through a prism.

The correct answer is A reaction where atoms or groups in a molecule are replaced by another atom or group..

Key Points

• A substitution reaction is a fundamental chemical process where one atom or a functional group in a molecule is replaced by a different atom or group without changing the basic carbon skeleton.
• These reactions are primary characteristics of saturated hydrocarbons (alkanes) and aromatic compounds (like benzene) because these molecules lack multiple bonds where addition could occur.
• In alkanes, substitution usually involves the replacement of hydrogen atoms by halogens (Chlorine, Bromine, or Iodine) in the presence of ultraviolet (UV) light or high temperatures.
• The general representation of a simple substitution is A-B + C → A-C + B, where C displaces B from the original molecule.
• For example, in the chlorination of methane (CH₄), a chlorine atom replaces a hydrogen atom to produce chloromethane (CH₃Cl) and hydrogen chloride (HCl).

 Additional Information

• Decomposition Reaction:
• Involves a single chemical compound breaking down into two or more simpler substances or elements.
• Usually requires an energy input in the form of heat (thermal), light (photolysis), or electricity (electrolysis).
• Combination Reaction:
• Occurs when two or more reactants (elements or compounds) unite to form a single, more complex product.
• Commonly referred to as a synthesis reaction.
• Hydrogenation:
• A specific type of addition reaction where hydrogen (H₂) is added across the double or triple bonds of unsaturated hydrocarbons.
• Typically requires catalysts like Nickel (Ni), Palladium (Pd), or Platinum (Pt).

The correct answer is 1H and 2H.

Key Points

• Isotopes are defined as atoms of the same chemical element that have the same atomic number (number of protons) but different mass numbers (sum of protons and neutrons).
• The pair ¹H and ²H represents the isotopes of Hydrogen. Both possess an atomic number of 1, confirming they belong to the same element.
• ¹H, commonly known as Protium, is the most abundant isotope of hydrogen, consisting of one proton and zero neutrons, giving it a mass number of 1.
• ²H, known as Deuterium or heavy hydrogen, consists of one proton and one neutron, resulting in a mass number of 2.
• Because isotopes have the same number of protons and electrons, they exhibit nearly identical chemical properties but vary in physical properties such as mass and density.

 Additional Information

• Isobars:
• These are atoms of different elements that have the same mass number but different atomic numbers.
• The pair ¹⁶O (Oxygen, atomic number 8) and ¹⁶F (Fluorine, atomic number 9) are isobars because they share the same mass of 16.
• Similarly, ⁴⁰Ca (Calcium, atomic number 20) and ⁴⁰Ar (Argon, atomic number 18) are isobars with a mass number of 40.
• Isotones:
• These are atoms of different elements that possess the same number of neutrons. For example, ¹²C and ¹⁴N are not isotones because Carbon-12 has 6 neutrons and Nitrogen-14 has 7 neutrons.
• Atomic Notation:
• In the notation ^AX, X is the element symbol, A is the mass number, and the atomic number (Z) is often written as a subscript preceding the symbol.

The correct answer is 90 omega.

Key Points

• The Electrical Resistance of a conductor is determined by the formula R = ρ × (L / A), where ρ (rho) represents the Resistivity of the material, L is the Length, and A is the Cross-sectional Area.
• When a wire is stretched, its physical dimensions change, but its Total Volume remains constant. The volume (V) is calculated as the product of the cross-sectional area and the length (V = A × L).
• In this scenario, the wire is stretched to three times its original length (L' = 3L). Because the volume is constant (V = A × L = A' × 3L), the new cross-sectional area (A') must decrease to one-third of the original area (A' = A / 3).
• Substituting the new dimensions into the resistance formula, the new resistance (R') is calculated as: R' = ρ × (3L / (A / 3)). This simplifies to R' = 9 × (ρ × L / A), which is 9 × R.
• Given the initial resistance is 10 omega, the new resistance becomes 9 × 10 = 90 omega. This demonstrates that for a constant volume, resistance varies with the Square of the Length (R ∝ L²).

 Additional Information

• Resistivity (ρ):
• It is an Intrinsic Property of the material, meaning it depends only on the nature of the substance and its Temperature, not on the shape or size.
• While Resistance increases nine-fold during stretching, the Resistivity remains unchanged as long as the material and temperature are constant.
• Factors Affecting Resistance:
• Length: Doubling the length by adding another wire (without stretching) only doubles the resistance because the area remains constant.
• Area: Increasing the thickness (area) reduces resistance, which is why Thicker Wires are used for high-current applications to minimize power loss.
• Temperature: For most Metallic Conductors, resistance increases linearly with temperature due to increased electron-ion collisions.

The correct answer is Living cells with thin walls, capable of division in young tissues.

Key Points

• Parenchyma is the most fundamental and abundant type of simple permanent tissue, forming the major part of the plant ground tissue system in roots, stems, and leaves.
• These are living cells at maturity, characterized by a functional protoplast, a prominent nucleus, and a large central vacuole used for storage and maintaining turgor pressure.
• The cells possess thin primary cell walls primarily composed of cellulose and hemicellulose, which lack lignification, allowing for flexibility and metabolic activity.
• A critical feature of parenchyma is its meristematic potential; in young tissues or upon injury, these cells can regain the ability for cell division, facilitating wound healing and regeneration.
• Structurally, they are often isodiametric (spherical or polygonal) with thin walls and prominent intercellular spaces that enable efficient gas exchange within the plant body.

 Additional Information

• Collenchyma (Option 1):
• These are living cells with asymmetric primary wall thickenings, specifically at the corners, due to pectin and cellulose deposition.
• They provide mechanical support and flexibility, allowing young plant organs like the petiole to bend without breaking.
• Sclerenchyma (Option 3):
• These cells are typically dead at maturity and characterized by extremely thick secondary walls reinforced with lignin.
• They provide the primary structural rigidity and mechanical strength to mature plant parts like seeds and vascular bundles.
• Epidermis (Option 4):
• This is the outermost single layer of cells covering the entire plant body.
• It performs a guarding function, protecting against mechanical injury and regulating transpiration through specialized structures like stomata.

The correct answer is 384 J.

Key Points

• Kinetic Energy is defined as the energy possessed by an object due to its motion. It represents the work needed to accelerate a body of a given mass from rest to its stated velocity.
• The fundamental formula for calculating Translational Kinetic Energy is given by: $K.E. = \frac{1}{2} mv^2$.
• From the problem statement, we have the following data:
• Mass (m) = 12 kg.
• Constant Speed (v) = 8 m/s.
• By substituting these values into the energy equation: $K.E. = \frac{1}{2} \times 12 \times (8)^2$.
• Calculating the square of the velocity first: 8 × 8 = 64.
• Continuing the calculation: 0.5 × 12 × 64 = 6 × 64 = 384.
• The resulting value for the kinetic energy is 384 Joules (J).

 Additional Information

• Relationship with Variables:
• Mass: Kinetic energy is linearly proportional to the mass. If the mass of the object is doubled, the energy doubles.
• Velocity: Kinetic energy is proportional to the square of the velocity. This means if the speed is doubled, the energy increases by a factor of four.
• SI Units: The standard International System of Units (SI) unit for energy is the Joule (J). One Joule is defined as 1 kg·m²/s².
• Scalar Quantity: Kinetic energy is a scalar quantity, meaning it only has magnitude and no direction. Unlike velocity or momentum, the direction of travel does not affect the total energy value.
• Work-Energy Theorem: This theorem states that the net work done on an object is equal to the change in its kinetic energy.

The correct answer is Uterus.

Key Points

• The uterus, also known as the womb, is a hollow, thick-walled, pear-shaped muscular organ located in the female pelvic cavity between the bladder and the rectum.
• After fertilisation occurs in the fallopian tube, the resulting zygote undergoes multiple cell divisions to become a blastocyst, which then travels to the uterus for implantation.
• Implantation typically occurs 6 to 10 days after fertilisation, where the embryo embeds itself into the endometrium, the vascularized inner lining of the uterine wall.
• The uterus is the primary site for gestation, providing a protective environment and necessary physiological support for the embryo to develop into a fetus over a period of approximately 280 days or 40 weeks.
• During pregnancy, the placenta develops within the uterus to facilitate the exchange of oxygen, nutrients, and antibodies from the mother to the developing fetus while removing metabolic waste products.
• The muscular layer of the uterus, the myometrium, possesses the unique ability to expand significantly during pregnancy and contract powerfully during parturition (childbirth).

 Additional Information

• Fallopian Tube
• These are narrow tubes that serve as the pathway for the ovum to travel from the ovary to the uterus and are the specific site where sperm meets the egg for fertilisation, usually in the ampulla region.
• Ovary
• The primary female reproductive organs (gonads) responsible for the production of oocytes (eggs) and the secretion of key reproductive hormones, including estrogen and progesterone.
• Vagina
• An elastic, muscular canal that connects the cervix to the outside of the body; it serves as the birth canal during delivery and the site for the deposition of semen during intercourse.

The correct answer is -1.25 D.

Key Points

• Myopia, commonly referred to as nearsightedness, is a vision condition where close objects appear clear, but distant objects are blurry. This occurs when the eyeball is too long or the cornea is too curved, causing light to focus in front of the retina.
• To correct myopia, a concave lens (diverging lens) is required. This lens diverges incoming light rays so that they appear to originate from the person's actual far point, effectively moving the image onto the retina.
• The lens formula is expressed as: 1/f = 1/v − 1/u. For a myopic eye to see distant objects, the object is assumed to be at infinity (u = −∞), and the image must be formed at the person's far point (v = −80 cm or −0.8 m).
• Substituting these values into the formula: 1/f = 1/(−80) − 1/(−∞). Since 1/∞ is 0, we find that the focal length (f) is equal to the far point, which is −80 cm.
• The Power (P) of a lens is the reciprocal of its focal length in meters (P = 1/f). Converting the focal length, f = −0.8 m. Therefore, P = 1 / (−0.8) = −1.25 Dioptres (D).

 Additional Information

• Concave Lens Properties:
• These lenses are thinner at the center and thicker at the edges.
• They always possess a negative focal length, which results in a negative power value in prescriptions.
• Dioptre (D):
• It is the standard unit of measurement for the optical power of a lens or curved mirror.
• A power of +1.25 D would indicate a convex lens, used for correcting Hypermetropia (farsightedness), where the image forms behind the retina.

The correct answer is Moon revolving around the Earth in a circular orbit.

Key Points

• Uniform circular motion is defined as the motion of an object traveling at a constant speed along a circular path.
• In this type of motion, while the magnitude of velocity (speed) remains unchanged, the direction of motion changes continuously at every point on the path.
• The Moon revolving around the Earth is a prime example because it maintains a nearly constant speed as it covers its orbit, driven by the Earth’s gravitational pull.
• The gravitational force acts as the centripetal force, which is always directed toward the center of the orbit, perpendicular to the object's velocity.
• Since the direction changes at every instant, the object is said to have centripetal acceleration, even though the speed is uniform.
• This motion is also periodic, meaning the Moon takes the same amount of time to complete one full revolution around the Earth.

 Additional Information

• Non-Uniform Circular Motion:
• A roller coaster on a loop or a yo-yo in a vertical circle are examples of non-uniform motion because gravity increases the speed while descending and decreases it while ascending.
• A racing car slowing down on a curved track involves tangential acceleration (or deceleration), which changes the magnitude of velocity, disqualifying it from being uniform.
• In uniform circular motion, the angular velocity remains constant, whereas in non-uniform motion, it fluctuates over time.

The correct answer is The direction of motion changes continuously.

Key Points

• Velocity is defined as a vector quantity, which means it possesses both magnitude (speed) and a specific direction of motion.
• Acceleration is the rate of change of velocity over time. For an object to accelerate, there must be a change in its speed, its direction, or both.
• In the case of an athlete running on a circular path, even if the speed remains constant, the direction of their motion is forced to change at every infinitesimal point along the curve to maintain the circular trajectory.
• Since the direction is changing continuously, the velocity vector is not constant. This change in velocity per unit of time constitutes accelerated motion.
• This specific form of motion is known as Uniform Circular Motion, where the magnitude of velocity (speed) is fixed, but the motion is still categorized as accelerated due to the directional shift.

 Additional Information

• Centripetal Acceleration:
• The acceleration acting on the athlete is directed towards the center of the circular path and is perpendicular to the velocity vector.
• The formula for this acceleration is a = v²/r, where v is the constant speed and r is the radius of the track.
• Tangential Velocity:
• At any given moment, the athlete's velocity vector is tangent to the circle. If the athlete were to suddenly lose grip or stop turning, they would move in a straight line along this tangent.
• Centripetal Force:
• According to Newton's Second Law (F = ma), this acceleration requires a centripetal force acting towards the center, which in this case is provided by the friction between the athlete's shoes and the track.

The correct answer is Buoyancy.

Key Points

• Aerenchyma is a specialized modification of parenchyma tissue that contains exceptionally large intercellular air spaces or cavities.
• In aquatic plants, known as hydrophytes (such as Lotus, Water Hyacinth, and Hydrilla), these air cavities provide buoyancy, allowing the plant parts to float on or remain suspended in water.
• The analogy follows a tissue-to-function relationship: just as Chlorenchyma (parenchyma containing chloroplasts) is responsible for photosynthesis, Aerenchyma is primarily responsible for providing upthrust or buoyancy.
• Beyond buoyancy, these air channels facilitate the efficient internal diffusion of gases like oxygen and carbon dioxide from the aerial parts to the submerged roots, which are often in hypoxic (low oxygen) environments.
• The structure of Aerenchyma consists of thin-walled cells arranged in a way that creates a honeycomb-like framework, maximizing air storage while minimizing weight.

 Additional Information

• Food Storage
• This is the fundamental role of simple parenchyma cells found in the cortex of roots and the mesophyll of stems.
• Storage tissues often contain leucoplasts or amyloplasts to store starch, as seen in potato tubers.
• Mechanical Strength
• This is the primary function of Sclerenchyma and Collenchyma.
• Sclerenchyma consists of dead cells with thick lignified walls, providing rigidity to the plant body.
• Water Absorption
• The root hair zone of the epiblema (epidermis of the root) is specifically designed for the absorption of water and minerals from the soil.
• This process occurs via osmosis and is aided by the xylem for upward conduction.

The correct answer is Because these metals cannot be reduced by carbon or hydrogen.

Key Points

• Metals positioned at the top of the Reactivity Series, such as Potassium (K), Sodium (Na), Calcium (Ca), Magnesium (Mg), and Aluminum (Al), are extremely reactive and possess a very high Chemical Affinity for oxygen.
• Because these metals are stronger reducing agents than Carbon or Hydrogen, they cannot be extracted from their oxides by heating with coke or hydrogen gas; the carbon is unable to break the powerful Ionic Bonds between the metal and oxygen.
• The Electrolytic Reduction method is required to provide sufficient energy to decouple these stable compounds. During electrolysis of their molten salts (like Molten Sodium Chloride), the metal ions are discharged at the Cathode.
• At the negative electrode (Cathode), metal cations gain electrons to become neutral atoms. For example, the reduction of sodium ions is represented as: Na⁺ + e⁻ → Na.
• This process ensures the production of high-purity metals which would otherwise be impossible to obtain through standard Pyrometallurgical techniques used for medium-reactivity metals like Zinc or Iron.

 Additional Information

• Native State:
• Highly reactive metals never occur in the Native State (free form) in nature due to their tendency to react with Oxygen, Moisture, and Carbon Dioxide. Only noble metals like Gold (Au) and Platinum (Pt) are found in their native form.
• Hall-Heroult Process:
• This is a specific industrial application for Aluminum extraction where Alumina (Al₂O₃) is dissolved in molten Cryolite (Na₃AlF₆) to lower the melting point from 2050°C to about 950°C, enhancing electrical conductivity.
• Volatility and Melting Points:
• The oxides of these metals (like MgO or Al₂O₃) are not volatile; they have extremely high Melting Points and Lattice Enthalpies, making them highly stable solids that resist thermal decomposition.

The correct answer is 63 J.

Key Points

• In physics, Work Done is defined as the product of the component of the force in the direction of the displacement and the magnitude of this displacement.
• The mathematical expression for work done is W = F × s × cos(θ), where F represents the magnitude of the constant force, s is the displacement, and θ is the angle between the force and the displacement vector.
• According to the problem, a constant force of 21 N moves an object through a distance of 3 m in its own direction. This implies that the force and displacement are parallel, resulting in an angle θ = 0°.
• Since the value of cos(0°) is exactly 1, the formula simplifies to the direct product of force and displacement: W = 21 N × 3 m.
• Performing the multiplication, we get 21 × 3 = 63. The unit of measurement for work in the International System of Units (SI) is the Joule (J), which is equivalent to one Newton-meter (N·m).

 Additional Information

• Nature of Work:
• Scalar Quantity: Even though force and displacement are vector quantities, work is a scalar quantity because it is the dot product of those two vectors. It has magnitude but no direction.
• Dimensional Formula: The dimensional representation of work is [ML²T⁻²], which is the same as the dimensions for energy.
• The Joule (J):
• One Joule is defined as the amount of work done when a force of 1 Newton displaces a body through 1 meter in the direction of the applied force.
• In the CGS system, the unit of work is the erg. The relationship between the two is 1 Joule = 10⁷ ergs.
• Conditions for Zero Work:
• Work is zero if the displacement is zero (e.g., pushing against a stationary wall).
• Work is also zero if the force is perpendicular to the displacement (e.g., a person carrying a bucket horizontally, where the force of gravity is at 90° to the motion).

The correct answer is It ensures direct transfer of the male germ-cell to the ovary, preventing wastage of pollen grains.

Key Points

• The pollen tube is a specialized tubular structure produced by the male gametophyte (pollen grain) after it adheres to the stigma of a flower.
• The primary functional advantage of this tube is to facilitate Siphonogamy, a process where non-motile male gametes are transported directly to the female gametophyte without the need for external water.
• The tube germinates from the vegetative cell and grows through the tissues of the stigma and style to reach the ovules located within the ovary.
• This mechanism ensures that the male germ-cells are delivered precisely to the micropyle of the ovule, thereby minimizing the loss or wastage of gametes during the journey to the egg cell.
• The formation of the pollen tube is an evolutionary advancement in Angiosperms and Gymnosperms that allows for high fertilization efficiency and successful seed production even in dry environments.

 Additional Information

• Mechanism of Pollen Tube Growth:
• The growth of the pollen tube is chemotropic, meaning it is guided by chemical gradients of calcium, boron, and sugars secreted by the synergid cells.
• The pollen tube typically carries two male gametes; one is destined for the egg cell and the other for the central cell (polar nuclei) during double fertilization.
• Double Fertilization and Seed Development:
• Syngamy occurs when one male gamete fuses with the egg cell to form a diploid zygote, which eventually develops into the embryo.
• Triple Fusion occurs when the second male gamete fuses with the two polar nuclei to form the triploid endosperm, which serves as a nutrient-rich tissue for the developing seed.
• While the ovule develops into a seed, the ovary matures into a fruit, protecting the genetic material for future generations.
• Reproductive Structures:
• The stigma acts as a receptive surface that recognizes compatible pollen through pollen-pistil interaction.
• The style is the elongated stalk that connects the stigma to the ovary, providing the pathway for the pollen tube to elongate via apical growth.

The correct answer is Improved lifestyles leading to more disposable and non-biodegradable materials.

Key Points

• Modern societies have transitioned toward a high-consumption lifestyle, primarily driven by industrialization and increased purchasing power, resulting in a significant rise in per capita waste production.
• The convenience-driven market has led to the widespread adoption of single-use items and excessive packaging, which are mostly composed of plastics, polystyrene, and other synthetic polymers that do not decompose naturally.
• Unlike organic waste, non-biodegradable materials persist in the ecosystem for hundreds of years, leading to the clogging of drainage systems, soil degradation, and the accumulation of microplastics in the global food chain.
• The rapid obsolescence of technology contributes to the exponential growth of Electronic Waste (e-waste), which contains hazardous substances like lead, mercury, and cadmium, posing severe risks to public health and biodiversity.
• Current urbanization trends prioritize the Linear Economy model (take-make-dispose), which lacks efficient resource recovery mechanisms, making waste management a massive environmental challenge.

 Additional Information

• Biodegradable Waste:
• Consists of organic matter such as food scraps, paper, and agricultural residues.
• These substances are broken down into simpler molecules by microorganisms like bacteria and fungi through natural biological processes.
• Waste Management Hierarchy:
• Reduction: The most effective method, involving the limitation of waste at the source.
• Recycling: Processing used materials into new products to reduce the consumption of fresh raw materials.
• Composting: A controlled biological process that transforms organic waste into nutrient-rich humus for soil conditioning.
• Landfilling: The disposal of waste by burying it, which is the least preferred method due to leachate and methane emissions.

The correct answer is High-quality packaging.

Key Points

• High-quality packaging is a preventive measure and a management practice used to preserve agricultural produce. It acts as a protective barrier against external factors and is not a cause of grain storage loss.
• Storage losses are generally classified into biotic factors (living organisms) and abiotic factors (non-living environmental conditions).
• Packaging technologies, such as hermetic bags or metallic silos, are designed to inhibit the entry of moisture and pests, thereby maintaining the viability and nutritional quality of the grains.
• Scientifically, airtight packaging reduces the availability of oxygen, which prevents the survival of aerobic insects and suppresses the growth of aerobic fungi.
• Effective storage management can reduce post-harvest losses, which otherwise range from 10% to 25% of the total production in various agricultural systems due to poor storage conditions.

 Additional Information

• Biotic Factors:
• Rodents: Species like Rattus rattus cause massive losses by consuming grains and contaminating them with excreta, urine, and hair, which can transmit diseases like Leptospirosis.
• Fungi and Bacteria: Microorganisms thrive in stored grains, causing biochemical degradation, discoloration, and the production of dangerous mycotoxins (e.g., Aflatoxin).
• Abiotic Factors:
• Inadequate Moisture: High moisture content (typically above 14%) promotes the respiration rate of the grain, leading to internal heating and rapid microbial spoilage.
• Temperature: Elevated temperatures within a storage unit accelerate the metabolism of insect pests and speed up the deterioration of the grain's chemical composition.

The correct answer is H2SO4 and N2O5.

Key Points

• Sulfuric acid (H₂SO₄) is a strong mineral acid. A single molecule of H₂SO₄ consists of 2 atoms of Hydrogen, 1 atom of Sulfur, and 4 atoms of Oxygen. By summing these components (2 + 1 + 4), the total atomicity is determined to be 7 atoms per molecule.
• Dinitrogen pentoxide (N₂O₅) is a chemical compound that serves as a powerful oxidizer. One molecule of N₂O₅ contains 2 atoms of Nitrogen and 5 atoms of Oxygen. Adding these together (2 + 5) results in a total count of 7 atoms per molecule.
• Since both H₂SO₄ and N₂O₅ possess exactly 7 atoms in their molecular structure, they represent the pair with an identical total number of atoms.

 Additional Information

• The atomicity of the other provided pairs is as follows:
• NH₃ and CH₄: Ammonia (NH₃) has 4 atoms (1 Nitrogen + 3 Hydrogen), while Methane (CH₄) has 5 atoms (1 Carbon + 4 Hydrogen).
• C₂H₆O and PCl₃: Ethanol or Dimethyl ether (C₂H₆O) contains 9 atoms (2 Carbon + 6 Hydrogen + 1 Oxygen), whereas Phosphorus trichloride (PCl₃) contains only 4 atoms (1 Phosphorus + 3 Chlorine).
• CO₂ and H₂O₂: Carbon dioxide (CO₂) is a triatomic molecule with 3 atoms (1 Carbon + 2 Oxygen), while Hydrogen peroxide (H₂O₂) is a tetraatomic molecule with 4 atoms (2 Hydrogen + 2 Oxygen).

The correct answer is Plant cells have a rigid cell wall..

Key Points

• When cells are placed in pure water, which acts as a hypotonic solution, water molecules move into the cells through the process of osmosis to balance solute concentrations.
• Animal cells are surrounded only by a thin, flexible plasma membrane. As water enters, the internal osmotic pressure increases, causing the cell to swell until the membrane ruptures, a process known as osmotic lysis.
• Plant cells contain a tough, rigid cell wall located outside the plasma membrane, primarily composed of cellulose, which provides significant mechanical strength.
• As water enters the plant cell, the large central vacuole fills up and pushes the protoplast against the cell wall, generating turgor pressure.
• The rigid cell wall exerts an equal and opposite wall pressure against the expanding cytoplasm, which limits further water intake and prevents the cell from bursting.

 Additional Information

• Hypotonic Solution
• A solution where the solute concentration is lower than that inside the cell, leading to a net influx of water.
• Turgidity
• The state of being turgid helps plants maintain their structural integrity and keep leaves upright for maximum sunlight absorption.
• Plasmolysis
• Occurs when a plant cell is placed in a hypertonic solution, causing water to leave the cell and the cell membrane to shrink away from the cell wall.
• Cellulose
• A polysaccharide that forms the primary structural component of the green plant cell wall, providing the tensile strength required to withstand internal pressure.

The correct answer is Surface area.

Key Points

• Mitochondria are double-membrane-bound organelles known as the powerhouse of the cell because they are the primary sites for ATP (Adenosine Triphosphate) synthesis.
• The inner mitochondrial membrane is deeply folded into finger-like projections called cristae.
• The primary function of these folds is to vastly increase the surface area of the inner membrane within the limited volume of the organelle.
• This increased surface area allows for a much larger number of electron transport chain (ETC) complexes and ATP synthase enzymes to be embedded in the membrane.
• A greater concentration of these proteins enables more efficient oxidative phosphorylation, leading to a higher rate of ATP generation to meet the metabolic demands of the cell.
• The structure-to-function relationship of the cristae is a classic biological example of maximizing efficiency through geometric optimization.

 Additional Information

• Mitochondrial Anatomy:
• Outer Membrane: A smooth phospholipid bilayer containing porins that make it permeable to small molecules and ions.
• Intermembrane Space: The narrow region between the membranes where protons (H⁺) are accumulated to establish the proton motive force.
• Mitochondrial Matrix: The dense internal fluid containing circular DNA, 70S ribosomes, and enzymes for the TCA (Krebs) cycle.
• Functional Concepts:
• ATP Synthase: Often called the F₀-F₁ particle or oxysome, it utilizes the flow of protons back into the matrix to catalyze the conversion of ADP to ATP.
• Cell Wall Thickness: This term refers to the structural layer outside the plasma membrane in plants (cellulose), fungi (chitin), and bacteria (peptidoglycan); mitochondria do not possess cell walls.
• Metabolic Activity: Tissues with high energy requirements, such as cardiac muscle, contain mitochondria with a higher density of cristae to support rapid ATP production.

The correct answer is 18 u.

Key Points

• The molecular mass of a compound is the sum of the atomic masses of all the constituent atoms present in a single molecule of that substance.
• The chemical formula for water is H₂O, indicating it contains two atoms of Hydrogen and one atom of Oxygen.
• The atomic mass of Hydrogen (H) is approximately 1 u, and the atomic mass of Oxygen (O) is approximately 16 u.
• To find the total mass, the calculation is performed as: (2 × 1 u) + (1 × 16 u) = 2 u + 16 u = 18 u.
• The unit u stands for the unified atomic mass unit, which is the standard unit for expressing masses at the atomic scale.

 Additional Information

• Unified Atomic Mass Unit (u):
• It is defined as a mass exactly equal to one-twelfth (1/12th) of the mass of one Carbon-12 atom.
• This unit replaced the older amu (atomic mass unit) to provide a more precise international standard.
• Molar Mass Comparison:
• While molecular mass is 18 u (mass of one molecule), the molar mass of water is 18 g/mol.
• This represents the mass of one mole of water molecules, which equals 6.022 × 10²³ molecules (Avogadro number).
• Heavy Water (D₂O):
• If Hydrogen is replaced by its isotope Deuterium (which has an atomic mass of 2 u), the resulting substance is called Heavy Water.
• The molecular mass of D₂O is (2 × 2 u) + 16 u = 20 u, which is frequently used as a moderator in nuclear reactors.

The correct answer is Ligament connects bone to bone; tendon connects muscle to bone.

Key Points

• Ligaments are specialized connective tissues that attach bone to bone. They play a critical role in stabilizing joints and preventing bones from shifting out of place during physical activity. Hence, the description of ligaments connecting bones is correct.
• Tendons are tough, high-tensile strength fibrous bands that anchor skeletal muscles to bones. Their primary function is to transmit the mechanical force generated by muscle contraction to the bone to initiate movement. Hence, the description of tendons connecting muscle to bone is correct.
• While both are dense connective tissues, they differ fundamentally in their cellular composition and collagen arrangement to suit their specific mechanical roles in the human musculoskeletal system. Hence, the differentiation provided in the first option is correct.

 Additional Information

• Ligament Properties:
• They contain a higher proportion of elastin fibers compared to tendons, which allows for a certain degree of elasticity and flexibility within the joint capsule.
• A common injury involving the overstretching or tearing of a ligament is known as a sprain.
• Tendon Properties:
• They are composed of densely packed, parallel collagen type I fibers, providing the tensile strength needed to withstand heavy loads during locomotion.
• An injury or overextension of a tendon or the muscle it attaches to is medically termed a strain.
• Unlike ligaments, tendons are relatively inelastic, which ensures that the force of a muscle contraction is efficiently transferred to the bone without energy loss through stretching.

The correct answer is Sunlight.

Key Points

• Methane (CH₄) is a saturated hydrocarbon belonging to the alkane family, characterized by single covalent bonds between carbon and hydrogen atoms.
• Alkanes are generally inert at room temperature and do not react with reagents like chlorine in the dark; however, they undergo substitution reactions under specific conditions.
• The reaction between methane and chlorine is a photochemical reaction that requires the presence of Sunlight or Ultraviolet (UV) radiation to proceed.
• Light energy provides the necessary activation energy to initiate homolytic fission of the chlorine-chlorine bond, where the shared pair of electrons is split equally between the two atoms.
• This fission creates highly reactive chlorine free radicals, which are essential intermediates that attack the stable methane molecules to begin the substitution process.

 Additional Information

• Free Radical Substitution Mechanism:
• Initiation: The UV light photons break the Cl-Cl bond: $Cl_{2} \xrightarrow{h\nu} 2Cl^{\bullet}$.
• Propagation: A chlorine radical reacts with methane to form hydrogen chloride (HCl) and a methyl radical (CH₃^•). This methyl radical then reacts with another chlorine molecule to produce chloromethane (CH₃Cl).
• Termination: The reaction ends when two radicals combine to form a stable molecule, such as dichloromethane, ethane, or more chloromethane.
• Successive Substitution:
• If chlorine is provided in excess, the reaction continues through a chain reaction, successively replacing all hydrogen atoms to form dichloromethane (CH₂Cl₂), trichloromethane (CHCl₃ or chloroform), and finally tetrachloromethane (CCl₄ or carbon tetrachloride).
• Reactivity of Alkanes: Alkanes are often termed paraffins (from Latin 'parum affinis' meaning 'little affinity') because they exhibit low chemical reactivity toward most laboratory reagents due to the high stability of C-H and C-C sigma bonds.

The correct answer is It ensures optimal water supply at critical growth stages..

Key Points

• Irrigation is the artificial application of water to the soil for assisting in the growth of agricultural crops. Plants contain nearly 90% water, and regular supply is essential for maintaining cell turgidity and photosynthesis.
• Controlled irrigation ensures that water is available at critical growth stages, such as crown root initiation (CRI), tillering, flowering, and grain filling. For example, in Wheat, the CRI stage (usually 21 days after sowing) is the most sensitive to moisture stress.
• Water serves as a solvent for mineral nutrients and fertilizers, facilitating their uptake by the root system and transport to other plant parts through the xylem.
• Regular supply maintains the soil temperature and protects the crops from both frost and hot air currents, ensuring a stable microclimate for plant development.
• Controlled application prevents waterlogging, a condition that causes anaerobic environments in the soil, leading to root rot and the accumulation of toxic substances like ethanol.

 Additional Information

• Irrigation Methods and Efficiency:
• Drip Irrigation: Also known as trickle irrigation, it provides water directly to the root zone drop by drop, achieving a Water Use Efficiency (WUE) of over 90%.
• Sprinkler Irrigation: Useful for sandy soils and uneven land, it mimics natural rainfall and prevents excessive percolation losses.
• Surface Irrigation: Includes furrow, basin, and border irrigation; while widely used, it often leads to higher evaporation and runoff losses.
• Soil Salinity:
• Incorrect irrigation practices, especially in arid regions, can lead to capillary action drawing salts to the surface, causing soil salinization which degrades land fertility.
• Water Productivity:
• In India, the Pradhan Mantri Krishi Sinchayee Yojana (PMKSY) focuses on Per Drop More Crop to promote efficient water management techniques.

The correct answer is Skeletal muscle.

Key Points

• Skeletal muscle is the specific type of muscle tissue that is primarily responsible for the locomotion and voluntary movement of the human body and its various limbs.
• These muscles are physically attached to the skeleton by tough connective tissues called tendons, allowing them to pull on bones to create movement at joints.
• They are classified as voluntary muscles because they are under the direct conscious control of the somatic nervous system.
• Under a microscope, skeletal muscle cells exhibit a striated (striped) appearance due to the highly organized arrangement of actin and myosin filaments within sarcomeres.
• Beyond movement, these muscles play a critical role in maintaining posture and generating metabolic heat to maintain body temperature.

 Additional Information

• Cardiac muscle:
• Located exclusively in the myocardium (the wall of the heart).
• Responsible for the involuntary, rhythmic contraction required to pump blood throughout the circulatory system.
• Smooth muscle:
• Found in the walls of visceral organs such as the esophagus, stomach, intestines, and blood vessels.
• Controls involuntary actions such as peristalsis and the regulation of blood pressure via vasoconstriction.
• Epithelial tissue:
• Serves as a protective barrier covering the body surface and lining internal cavities and ducts.
• Main functions include secretion, absorption, and filtration, but it does not possess contractile properties for movement.

The correct answer is 1 × 10⁻⁷ M.

Key Points

• Pure water undergoes a process known as self-ionization or auto-ionization, where a tiny fraction of water molecules act as both an acid and a base to produce ions.
• The chemical equilibrium for this reversible reaction is represented as: H₂O + H₂O ↔ H₃O⁺ + OH⁻, or more simply as H₂O ↔ H⁺ + OH⁻.
• The equilibrium constant for this dissociation is called the ionic product of water, symbolized as K_w.
• At a standard temperature of 25°C (or 298 K), the value of K_w is experimentally determined to be exactly 1.0 × 10⁻¹⁴ mol²/L².
• In pure water, every single dissociation event produces one hydrogen ion (H⁺) and one hydroxide ion (OH⁻), ensuring their concentrations remain perfectly equal: [H⁺] = [OH⁻].
• By substituting this equality into the expression K_w = [H⁺] × [OH⁻], we obtain the equation 1.0 × 10⁻¹⁴ = [H⁺]².
• Taking the square root of both sides leads to the concentration of [H⁺] = 1.0 × 10⁻⁷ moles per liter (M).

 Additional Information

• pH Scale and Neutrality:
• The pH is the negative base-10 logarithm of the hydrogen ion concentration: pH = −log[H⁺].
• For pure water at 25°C, the pH is −log(10⁻⁷) = 7, defining the neutral point on the scale.
• Temperature Dependence:
• The auto-ionization of water is an endothermic process. According to Le Chatelier's Principle, increasing the temperature shifts the equilibrium to the right.
• At temperatures higher than 25°C, K_w increases, resulting in a higher concentration of H⁺ ions and a neutral pH lower than 7.
• Molarity (M):
• Molarity is a unit of concentration expressing the number of moles of solute per liter of solution.
• Even in highly acidic or basic solutions, the product of [H⁺] and [OH⁻] always equals K_w at that specific temperature.