RRB Technician Grade III Official Paper (Held On: 06 March, 2026 Shift 2) (Previous Year Paper)

Shortcut Trick

The total difference in percentage = Gain % − (Loss %) = 19% − (−6%) = 25%.

This 25% change is equal to the increase in Selling Price of ₹960.

25% of Cost Price = ₹960

⇒ 100% of Cost Price = 960 × 4 = ₹3840.

∴ The correct answer is ₹3840.

Alternate Method

Given:

Initial Loss = 6%

Final Gain = 19%

Increase in Selling Price (SP) = ₹960

Formula Used:

SP = CP × (100 ± Profit or Loss %)/100

Let the Cost Price (CP) be ₹x.

⇒ Initial Selling Price (SP₁) = x × (100 − 6)/100 = 0.94x

⇒ New Selling Price (SP₂) = x × (100 + 19)/100 = 1.19x

⇒ According to the question: SP₂ − SP₁ = 960

⇒ 1.19x − 0.94x = 960

⇒ 0.25x = 960

⇒ x = 960 ÷ 0.25

⇒ x = 960 × 4 = 3840

∴ The correct answer is ₹3840.

Additional Information

Profit and Loss Percentage

Profit % = (Profit ÷ CP) × 100 and Loss % = (Loss ÷ CP) × 100. Both are always calculated on the Cost Price unless stated otherwise.

Selling Price Calculation

When Profit % is P, SP = CP × (100 + P)/100. When Loss % is L, SP = CP × (100 − L)/100.

Marked Price and Discount

Discount is always calculated on the Marked Price (MP). SP = MP − Discount. Discount % = (Discount ÷ MP) × 100.

Shortcut Trick

Total Work = LCM(10, 15, 20) = 60 units.

Efficiencies: A = 6, B = 4, C = 3; Combined (A+B+C) = 13 units/hr.

Work done by A and B in 2 hr = (6 + 4) × 2 = 20 units.

Remaining work = 60 − 20 = 40 units.

Time for remaining work (t) = 40 ÷ 13 ≈ 3 hr

∴ The correct answer is 3 hr.

Alternate Method

Given: Time taken by A = 10 hr, B = 15 hr, and C = 20 hr.

Formula: Efficiency = Total Work ÷ Time Taken

⇒ Efficiency of A and B working together = 6 + 4 = 10 units/hr

⇒ Work done by A and B in the first 2 hr = 10 × 2 = 20 units

⇒ Remaining work to be completed = 60 − 20 = 40 units

⇒ Combined efficiency of A, B, and C = 6 + 4 + 3 = 13 units/hr

⇒ Time taken to finish remaining work (t) = 40 ÷ 13 ≈ 3.08 hr

⇒ So, the value of t = 3 hr

∴ The correct answer is 3 hr.

Additional Information

Total Work using LCM

Taking the Least Common Multiple (LCM) of individual time periods is the most efficient way to assign a value to total work.

Efficiency Relation

Efficiency is inversely proportional to the time taken to complete the work (E ∝ 1/T).

Combined Work Formula

If multiple people work together, their individual efficiencies are added to find the total rate of work per unit time.

Shortcut Trick

Total Area printed = Number of revolutions × Curved Surface Area (CSA)

4158 = 350 × 2 × (22/7) × r × 2.1

4158 = 44 × 350 × 0.3 × r

4158 = 4620 × r ⇒ r = 4158 ÷ 4620 = 0.9 m

r = 0.9 × 100 cm = 90 cm

∴ The correct answer is 90 cm.

Alternate Method

Given:

Length of the cylindrical drum (h) = 2.1 m

Number of revolutions (n) = 350

Total printed area = 4158 m²

Formula Used:

Curved Surface Area of Cylinder = 2 × π × r × h

Total Area = n × Curved Surface Area

Calculations:

⇒ Total Area = n × (2 × π × r × h)

⇒ 4158 = 350 × 2 × (22/7) × r × 2.1

⇒ 4158 = 350 × 2 × 22 × r × 0.3

⇒ 4158 = 4620 × r

⇒ r = 4158 ÷ 4620

⇒ r = 0.9 m

⇒ r = 0.9 × 100 cm

⇒ r = 90 cm

∴ The correct answer is 90 cm.

Additional Information

Volume of a Cylinder

The capacity of a cylindrical container is given by the formula V = πr²h.

Total Surface Area of a Cylinder

It includes the curved area and the areas of the two circular bases: TSA = 2πr(r + h).

Unit Conversion

Always ensure units are consistent. 1 meter (m) = 100 centimeters (cm) and 1 m² = 10,000 cm².

Shortcut Trick

Test the standard trigonometric values for θ within the range 0° ≤ θ ≤ 90°.

If we put θ = 90°, then cos 90° = 0 and sin 90° = 1.

LHS: 10(0)² + 7(1)² = 0 + 7 = 7. Since LHS = RHS, θ = 90° is correct.

Value required: sin 90° + cos 90° = 1 + 0 = 1.

∴ The correct answer is 1.

Alternate Method

Given: 10 cos² θ + 7 sin² θ = 7 and 0° ≤ θ ≤ 90°

Formula Used: sin² θ + cos² θ = 1

⇒ 10 cos² θ + 7 sin² θ = 7

⇒ 3 cos² θ + 7 cos² θ + 7 sin² θ = 7

⇒ 3 cos² θ + 7(cos² θ + sin² θ) = 7

⇒ 3 cos² θ + 7(1) = 7

⇒ 3 cos² θ = 7 − 7

⇒ 3 cos² θ = 0

⇒ cos θ = 0

⇒ Since 0° ≤ θ ≤ 90°, θ = 90°

⇒ Now, sin θ + cos θ = sin 90° + cos 90°

⇒ 1 + 0 = 1

∴ The correct answer is 1.

Additional Information

Trigonometric Identities

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

Specific Values for Sine and Cosine

sin 0° = 0, sin 90° = 1 | cos 0° = 1, cos 90° = 0.

Quadratic Form in Trig

Equations of the form a cos²θ + b sin²θ = c can be solved by converting all terms to either sin²θ or cos²θ using basic identities.

Shortcut Trick

Value of one-third of the number = Final Result − Increase

⇒ 51 − 37 = 14

Original number = 14 × 3 = 42

Sum of the digits = 4 + 2 = 6

∴ The correct answer is 6.

Alternate Method

Given:

Let the original number be x.

According to the question, one-third of x increased by 37 equals 51.

Calculations:

⇒ (x/3) + 37 = 51

⇒ x/3 = 51 − 37

⇒ x/3 = 14

⇒ x = 14 × 3

⇒ x = 42

Now, calculate the sum of the digits of 42:

⇒ Sum = 4 + 2 = 6

∴ The correct answer is 6.

Additional Information

Linear Equation in One Variable

An equation of the form ax + b = c, where x is the unknown variable and a, b, c are constants.

Digit Sum Concept

The digit sum is the sum of all individual digits of a number. For example, for 123, the sum is 1 + 2 + 3 = 6.

Divisibility by 3 and 9

A number is divisible by 3 (or 9) if the sum of its digits is divisible by 3 (or 9).

Shortcut Trick

Side of square = √9 = 3 cm ⇒ Perimeter = 4 × 3 = 12 cm.

Side of regular hexagon = 12 ÷ 6 = 2 cm.

Area of hexagon = 6 × (Area of equilateral triangle with side 2 cm).

Area = 6 × (√3 ÷ 4 × 2²) = 6 × √3 = 6√3 cm².

∴ The correct answer is 6√3 cm².

Alternate Method

Given:

Area of square = 9 cm²

Perimeter of square = Perimeter of regular hexagon

Formula Used:

1. Area of square = a² (where a = side)
2. Perimeter of square = 4a
3. Perimeter of regular hexagon = 6s (where s = side)
4. Area of regular hexagon = (3√3 ÷ 2) × s²

Calculations:

⇒ a² = 9 cm²

⇒ a = √9 = 3 cm

⇒ Perimeter of square = 4 × 3 = 12 cm

⇒ Perimeter of hexagon = 6s = 12 cm

⇒ s = 12 ÷ 6 = 2 cm

⇒ Area of hexagon = (3√3 ÷ 2) × 2²

⇒ Area of hexagon = (3√3 ÷ 2) × 4

⇒ Area of hexagon = 3√3 × 2 = 6√3 cm²

∴ The correct answer is 6√3 cm².

Additional Information

Regular Hexagon Area

A regular hexagon is composed of 6 equilateral triangles. Area = 6 × (√3/4 × s²) = (3√3/2)s².

Internal Angles

Each internal angle of a regular hexagon is 120° and the sum of all internal angles is 720°.

Circumradius and Inradius

For a regular hexagon of side s, the circumradius (R) = s and the inradius (r) = (√3/2)s.

Shortcut Trick

Difference in percentage = 84% − 45% = 39%

Total difference in marks = 62 (pass) + 35.5 (fail) = 97.5 marks

Since 39% corresponds to 97.5, then 1% = 97.5 ÷ 39 = 2.5

Maximum Marks (100%) = 2.5 × 100 = 250

∴ The correct answer is 250.

Alternate Method

Given:

Case 1: Score = 84%, Result = Passed by 62 marks

Case 2: Score = 45%, Result = Failed by 35.5 marks

Formula Used:

Passing Marks = (Score% × Max Marks) − Pass Margin

Passing Marks = (Score% × Max Marks) + Fail Margin

Calculations:

⇒ Let Maximum Marks be x.

⇒ Passing Marks = 0.84x − 62

⇒ Passing Marks = 0.45x + 35.5

⇒ 0.84x − 62 = 0.45x + 35.5

⇒ 0.84x − 0.45x = 35.5 + 62

⇒ 0.39x = 97.5

⇒ x = 97.5 ÷ 0.39

⇒ x = 250

∴ The correct answer is 250.

Additional Information

Passing Percentage

The minimum score required to pass. Passing % = (Passing Marks ÷ Maximum Marks) × 100.

Percentage to Absolute Value

To find the total value when a percentage difference is known: Total = (Difference in values ÷ Difference in %) × 100.

Passing Marks Relation

Obtained Marks + Failing Marks = Passing Marks = Obtained Marks − Passing Margin.

Shortcut Trick

Sum of ages of husband and wife at marriage = 28 × 2 = 56 years.

Sum of ages 2 years later (when daughter was born) = 56 + (2 × 2) + 0 = 60 years.

Present total age of 3 family members = 24 × 3 = 72 years.

Increase in total age since daughter's birth = 72 − 60 = 12 years.

Years passed since daughter's birth = 12 ÷ 3 = 4 years.

Total years since marriage = 2 (before birth) + 4 (after birth) = 6 years.

∴ The correct answer is 6 years.

Alternate Method

Given:

Average age of husband and wife at marriage = 28 years

Time after marriage when daughter was born = 2 years

Present average age of 3 members = 24 years

Formula Used:

Total Age = Average Age × Number of persons

Calculations:

⇒ Sum of ages of husband (H) and wife (W) at marriage = 28 × 2 = 56

⇒ Let they got married 'x' years ago.

⇒ Present age of H + W = 56 + 2x

⇒ Present age of daughter (born 2 years after marriage) = x − 2

⇒ Present total sum = (56 + 2x) + (x − 2)

⇒ 24 × 3 = 54 + 3x

⇒ 72 = 54 + 3x

⇒ 3x = 72 − 54

⇒ 3x = 18

⇒ x = 6

∴ The correct answer is 6 years.

Additional Information

Average Age Concept

If the current average age of n people is A, then after k years, the average age will be A + k.

Sum of Ages

Total sum of ages increases by (Number of people × Number of years) over a period of time.

Birth in Average Problems

When a child is born, the number of members in the family increases, which immediately affects the denominator in the average calculation.

Shortcut Trick

Profit Ratio = Capital × Time

Ratio of (Anusha : Minal) = (55,000 × 12) : (Capital_M × 5) = 8 : 3

⇒ (11,000 × 12) : (Capital_M) = 8 : 3

⇒ 1,32,000 / Capital_M = 8 / 3

⇒ Capital_M = (1,32,000 × 3) / 8 = 16,500 × 3 = 49,500

∴ The correct answer is ₹49,500.

Alternate Method

Given:

Anusha's Capital (C_A) = ₹55,000

Anusha's Time (T_A) = 12 months

Minal's Time (T_M) = 12 − 7 = 5 months

Profit Ratio (P_A : P_M) = 8 : 3

Formula Used:

Ratio of Profits = (Capital₁ × Time₁) / (Capital₂ × Time₂)

Calculations:

⇒ Let Minal's capital be ₹x.

⇒ P_A / P_M = (C_A × T_A) / (C_M × T_M)

⇒ 8 / 3 = (55,000 × 12) / (x × 5)

⇒ 8 / 3 = 6,60,000 / 5x

⇒ 8 × 5x = 6,60,000 × 3

⇒ 40x = 19,80,000

⇒ x = 19,80,000 / 40

⇒ x = 49,500

∴ The correct answer is ₹49,500.

Additional Information

Partnership Basic Rule

Profit is shared in the ratio of the product of Investment and Time Period: P1 : P2 = C1 × T1 : C2 × T2.

Compound Partnership

When partners invest different amounts for different time periods, it is called a compound partnership.

Working Partner

A partner who manages the business is a working partner and often receives a salary or fixed percentage of profit before the remaining profit is divided.

Shortcut Trick

Net Efficiency of both pipes = 9 (Filling) − 4 (Emptying) = 5 units/min.

Total Capacity (LCM of 20, 45) = 180 units.

Work to fill one-fourth of the tank = 180 ÷ 4 = 45 units.

Time taken = 45 ÷ 5 = 9 min.

∴ The correct answer is 9 min.

Alternate Method

Given:

Time taken by Filling Pipe = 20 min

Time taken by Emptying Pipe = 45 min

Formula Used:

Total Work = Efficiency × Time

Calculations:

⇒ Total capacity of the tank = LCM(20, 45) = 180 units.

⇒ Net efficiency when both pipes work together = 9 − 4 = 5 units/min.

⇒ Required part of the tank to be filled = 1/4 × 180 = 45 units.

⇒ Time taken = Required Capacity ÷ Net Efficiency

⇒ Time taken = 45 ÷ 5

⇒ Time taken = 9 min.

∴ The correct answer is 9 min.

Additional Information

Pipes & Cisterns Basics

Filling pipes are considered as positive work (+), while emptying pipes (leaks) are considered as negative work (−).

Net Work Done

If pipe A fills in 'x' hours and pipe B empties in 'y' hours, net work in 1 hour = (1/x − 1/y).

Time and Efficiency Relationship

Efficiency is inversely proportional to the time taken to complete the work (Efficiency = 1/Time).

Shortcut Trick

Total divisors for any prime power pⁿ is (n + 1).

Option 1 & 3: p³ and q³ ⇒ 3 + 1 = 4 divisors.

Option 2: p × q ⇒ (1 + 1) × (1 + 1) = 4 divisors.

Option 4: p² ⇒ 2 + 1 = 3 divisors.

∴ The correct answer is p² for prime p.

Alternate Method

Given: Four expressions where p and q are prime numbers.

Formula Used: If a number is written in its prime factored form N = a^m × bⁿ, the total number of positive divisors is (m + 1) × (n + 1).

⇒ For p³: The divisors are 1, p, p², p³. Total = 4.

⇒ For p × q: The divisors are 1, p, q, pq. Total = 4.

⇒ For q³: Similar to p³, the divisors are 1, q, q², q³. Total = 4.

⇒ For p²: The divisors are 1, p, p². Total = 3.

⇒ Therefore, p² has 3 divisors, which is not equal to 4.

∴ The correct answer is p² for prime p.

Additional Information

Divisor Rule for Powers

If N = p^a, where p is prime, the number of factors is always (a + 1). For example, 2² = 4 has (2 + 1) = 3 factors (1, 2, 4).

Perfect Squares Property

Every perfect square has an odd number of distinct factors, while non-perfect squares always have an even number of factors.

Prime Product Property

The product of two distinct prime numbers (p × q) always results in exactly 4 divisors: {1, p, q, pq}.

Shortcut Trick

The mode is the value that appears most frequently in a data set.

Frequency of 55 → 4 times (at positions 1, 4, 11, 17)

Frequency of 44 → 3 times (at positions 5, 9, 13)

Frequency of 52, 45, and 54 → 2 times each

Frequency of 48, 50, 49, 43, and 47 → 1 time each

Highest frequency is 4, which belongs to the number 55.

∴ The correct answer is 55.

Alternate Method

Given data set: 55, 48, 50, 55, 44, 52, 45, 52, 44, 49, 55, 54, 44, 54, 43, 45, 55, 47

Formula: Mode = Observation with the maximum frequency.

Frequency Distribution Value: 55 Freq: 4 (MAX) Value: 44 Freq: 3 Value: 52/45 Freq: 2 Mode = 55 (Highest occurring value)

⇒ Counting the occurrences of each unique number in the list:

⇒ 43: 1 time | 44: 3 times | 45: 2 times | 47: 1 time | 48: 1 time

⇒ 49: 1 time | 50: 1 time | 52: 2 times | 54: 2 times | 55: 4 times

⇒ The value 55 occurs 4 times, which is more than any other value.

⇒ Therefore, Mode = 55.

∴ The correct answer is 55.

Additional Information

Empirical Formula

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

Types of Data Distributions

Unimodal has one mode, Bimodal has two modes, and Multimodal has more than two modes. If no value repeats, there is no mode.

Central Tendency Measures

Mean is the average, Median is the middle value after sorting, and Mode is the most frequent value.

Shortcut Trick

Total interest percentage for 2 years = Rate × Time = 5% × 2 = 10%.

Given that 10% of Principal = ₹720.

To find the sum of money (100%), we multiply by 10.

Principal = 720 × 10 = ₹7200.

∴ The correct answer is ₹7200.

Alternate Method

Given:

Simple Interest (SI) = ₹720

Time (T) = 2 years

Rate (R) = 5% per annum

Formula Used:

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

Calculations:

⇒ 720 = (P × 5 × 2) ÷ 100

⇒ 720 = (P × 10) ÷ 100

⇒ 720 = P ÷ 10

⇒ P = 720 × 10

⇒ P = ₹7200

∴ The correct answer is ₹7200.

Additional Information

Amount in Simple Interest

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

Rate of Interest Calculation

If the sum, interest, and time are known, the Rate (R) can be calculated as R = (SI × 100) ÷ (P × T).

Constant Interest Property

In Simple Interest, the interest earned remains the same every year because it is always calculated on the original Principal.

Shortcut Trick

Total interest percentage for 9 months at 9% per annum = 9 × (9 ÷ 12) = 6.75%.

Simple Interest = 6.75% of ₹28,260.

⇒ 28260 × 0.0675 = 1907.55.

∴ The correct answer is ₹1907.55.

Alternate Method

Given:

Principal (P) = ₹28,260

Rate of interest (R) = 9% per annum

Time period (T) = 9 months

Formula Used:

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

Calculations:

⇒ T in years = 9 ÷ 12 = 3/4 years

⇒ SI = (28260 × 9 × 3/4) ÷ 100

⇒ SI = (28260 × 27) ÷ 400

⇒ SI = 763020 ÷ 400

⇒ SI = 1907.55

∴ The correct answer is ₹1907.55.

Additional Information

Total Amount Formula

The total amount (A) to be returned after the time period is the sum of the Principal and the Interest: A = P + SI.

Time Conversion for Interest

When time is given in months, divide by 12 to convert to years. When given in days, divide by 365 (standard year) or 366 (leap year).

Effective Rate of Interest

Effective Rate = R × T. In this case, 9% × 0.75 = 6.75% is the total interest percentage applied to the principal.

Shortcut Trick

SP of Article X = 1300 + 734 = 2034.

Effective selling price factor for X = (1 − 0.84) × (1 − 0.25) = 0.16 × 0.75 = 0.12 of MP.

Selling price for Y = (1 − 0.87) = 0.13 of MP.

Since 12 units = 2034, then 13 units = (2034 ÷ 12) × 13 = 169.5 × 13 = 2203.5.

Profit on Article Y = 2203.5 − 1300 = 903.5.

Profit % = 903.5 ÷ 13 = 69.5%.

∴ The correct answer is 69.5%.

Alternate Method

Given:

Cost Price (CP) of Article X and Article Y = ₹1300 each.

Successive discounts on X = 84% and 25%.

Profit on X = ₹734.

Discount on Y = 87%.

Formula Used:

1. Selling Price (SP) = CP + Profit
2. SP = MP × (1 − d₁/100) × (1 − d₂/100)
3. Profit % = (Profit / CP) × 100

Calculations:

⇒ SP of Article X = 1300 + 734 = ₹2034.

⇒ Let the Marked Price (MP) be M.

⇒ M × (1 − 84/100) × (1 − 25/100) = 2034

⇒ M × 0.16 × 0.75 = 2034

⇒ M × 0.12 = 2034

⇒ M = 2034 / 0.12 = ₹16950.

⇒ SP of Article Y = MP × (1 − 87/100) = 16950 × 0.13

⇒ SP of Article Y = ₹2203.5.

⇒ Profit on Article Y = 2203.5 − 1300 = ₹903.5.

⇒ Profit % = (903.5 / 1300) × 100 = 69.5%.

∴ The correct answer is 69.5%.

Additional Information

Successive Discounts Formula

When two successive discounts of a% and b% are given, the single equivalent discount is calculated as (a + b − ab/100)%.

Markup and Discount Relation

Profit % can be related to markup and discount as: (100 + P)/(100 − D) = MP/CP, where P is profit % and D is discount %.

Successive Percentage Change

For any quantity changing by x% and then y% successively, the net percentage change is given by (x + y + xy/100)%.

Shortcut Trick

Given Ratio = 4 : 5 and HCF = 16.

Trick: LCM = HCF × Product of the Ratio terms.

LCM = 16 × 4 × 5 = 320.

∴ The correct answer is 320.

Alternate Method

Given:

Ratio of two numbers = 4 : 5

Highest Common Factor (HCF) = 16

Formula Used:

If the ratio of two numbers is a : b and their HCF is x, then the numbers are ax and bx.

LCM of these numbers = x × a × b (where a and b are coprime).

Let the numbers be 4k and 5k.

Since the HCF is the common factor k, we have:

⇒ k = 16

⇒ First number = 4 × 16 = 64

⇒ Second number = 5 × 16 = 80

⇒ LCM(64, 80) = 16 × 4 × 5

⇒ LCM = 320

∴ The correct answer is 320.

Additional Information

Product of Two Numbers Rule

For any two positive integers, the product of the numbers is equal to the product of their HCF and LCM: n₁ × n₂ = HCF × LCM.

HCF as a Factor of LCM

The Highest Common Factor of any set of numbers is always a factor of their Least Common Multiple.

Coprime Numbers Property

If two numbers are coprime (HCF = 1), then their LCM is simply the product of the numbers themselves.

Shortcut Trick

For any algebraic expression where xⁿ + 1/xⁿ = k, the value of the squared power term x²ⁿ + 1/x²ⁿ is given by k² − 2.

Here, the power is being doubled from 2 to 4, so n = 2 and the given value k = 16.

Applying the trick: x⁴ + 1/x⁴ = 16² − 2.

⇒ 256 − 2 = 254.

∴ The correct answer is 254.

Alternate Method

Given: x² + 1/x² = 16

Formula Used: (a + b)² = a² + b² + 2ab (Algebraic Identity for Squaring)

⇒ To find the value of x⁴ + 1/x⁴, we square the given equation on both sides:

⇒ (x² + 1/x²)² = (16)²

⇒ (x²)² + (1/x²)² + 2 × (x²) × (1/x²) = 256

⇒ x⁴ + 1/x⁴ + 2 = 256

⇒ x⁴ + 1/x⁴ = 256 − 2

⇒ x⁴ + 1/x⁴ = 254

∴ The correct answer is 254.

Additional Information

Successive Squaring

If x + 1/x = a, then x² + 1/x² = a² − 2. To find x⁴ + 1/x⁴, we apply the same rule to the new result: (a² − 2)² − 2.

Reciprocal Cube Identity

If x + 1/x = a, then the value of x³ + 1/x³ is calculated as a³ − 3a.

Negative Reciprocal Relation

If x − 1/x = a, then x² + 1/x² = a² + 2. Note the sign change in the constant term when the original expression is negative.

Shortcut Trick

Profit Ratio (X : Y) = (Capital × Time)_X : (Capital × Time)_Y

Profit Ratio = (12,000 × 12) : (18,000 × 6) = 144,000 : 108,000 = 4 : 3

Total Ratio parts = 4 + 3 = 7 units → ₹25,200

1 unit = ₹3,600

X's share = 4 × ₹3,600 = ₹14,400

∴ The correct answer is ₹14,400.

Alternate Method

Given:

Investment of X = ₹12,000 for 12 months

Investment of Y = ₹18,000 for 6 months

Total Profit = ₹25,200

Formula Used:

Profit Ratio = (Investment₁ × Time₁) : (Investment₂ × Time₂)

Calculations:

⇒ Profit Ratio (X : Y) = (12,000 × 12) : (18,000 × 6)

⇒ Profit Ratio (X : Y) = 1,44,000 : 1,08,000

⇒ Simplifying by 36,000: Ratio = 4 : 3

⇒ Total profit units = 4 + 3 = 7 units

⇒ 7 units = ₹25,200

⇒ 1 unit = ₹25,200 ÷ 7 = ₹3,600

⇒ X's share = 4 units = 4 × ₹3,600 = ₹14,400

∴ The correct answer is ₹14,400.

Additional Information

Compound Partnership

When partners invest different amounts for different periods of time, the profit is shared in the ratio of the product of their capital and time period (Capital × Time).

Ratio of Capitals

If the profit ratio and time ratio are given, the capital ratio can be found as (Profit₁/Time₁) : (Profit₂/Time₂).

Working vs Sleeping Partner

A working partner manages the business and often receives a fixed percentage of total profit as salary before the remaining profit is distributed among all partners.

Shortcut Trick

The area of a triangle is fundamentally defined by half the product of its base and corresponding altitude.

Formula: Area = 1/2 × base × altitude.

In the given options, statement (ii) specifies the fraction as 1/3, which is used for calculating the volume of pyramids or cones, not the area of a 2D triangle.

Other properties (i), (iii), and (iv) are standard geometric theorems for all triangles.

∴ The correct answer is (ii).

Alternate Method

Given: Four statements regarding the properties of a triangle.

Formula Used: Angle Sum Property, Area Formula, Perimeter Definition, and Exterior Angle Theorem.

Verification of properties:

⇒ Property (i): Sum of internal angles = ∠A + ∠B + ∠C = 180°. This is always true for any Euclidean triangle.

⇒ Property (ii): Mathematical Area of a triangle = 1/2 × base × altitude. The given statement (ii) uses 1/3, making it incorrect.

⇒ Property (iii): Perimeter = Length(AB) + Length(BC) + Length(CA). This is the standard definition of perimeter for any polygon.

⇒ Property (iv): According to the Exterior Angle Theorem, the exterior angle at any vertex equals the sum of the two opposite interior angles.

⇒ Therefore, statement (ii) is not a property of a triangle.

∴ The correct answer is (ii).

Additional Information

Heron's Formula

For a triangle with sides a, b, and c, the area is √[s(s−a)(s−b)(s−c)], where s is the semi-perimeter (a+b+c)/2.

Triangle Inequality Theorem

The sum of the lengths of any two sides of a triangle must always be strictly greater than the length of the third side.

Pythagoras Theorem

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (a² + b² = c²).

Shortcut Trick

वृद्धि राशि ज्ञात करने के लिए सीधे प्रतिशत निकालें:

300 का 5% = 300 × 0.05

⇒ 3 × 5 = 15

∴ The correct answer is ₹15.

Alternate Method

दिया गया है:

पुस्तक की मूल कीमत = ₹300

कीमत में वृद्धि प्रतिशत = 5%

सूत्र: वृद्धि राशि = मूल कीमत × (वृद्धि प्रतिशत ÷ 100)

⇒ वृद्धि राशि = 300 × (5 ÷ 100)

⇒ वृद्धि राशि = 300 × 0.05

⇒ वृद्धि राशि = 15

अतः, नई कीमत मूल कीमत से ₹15 अधिक होगी।

∴ The correct answer is ₹15.

Additional Information

प्रतिशत वृद्धि (Percentage Increase)

जब किसी वस्तु के मूल्य में प्रतिशत वृद्धि होती है, तो बढ़ी हुई राशि (मूल मूल्य × दर) ÷ 100 होती है।

नया मूल्य (New Price)

वृद्धि के बाद नया मूल्य = मूल मूल्य + वृद्धि राशि। इस प्रश्न में नया मूल्य 300 + 15 = ₹315 होगा।

प्रतिशत कमी (Percentage Decrease)

यदि कीमत में 5% की कमी होती, तो कमी की राशि ₹15 होती और नया मूल्य ₹285 होता।

Shortcut Trick

If x = (√a + √b)/(√a − √b) and y is its reciprocal, then x + y = 2(a + b)/(a − b) and xy = 1.

Here, a = 5 and b = 4 (since 2 = √4), so x + y = 2(5 + 4) / (5 − 4) = 18.

Using the algebraic identity, x² + y² = (x + y)² − 2xy.

x² + y² = 18² − 2(1) = 324 − 2 = 322.

∴ The correct answer is 322.

Alternate Method

Given: x = (√5 + 2)/(√5 − 2) and y = (√5 − 2)/(√5 + 2)

Formula Used: x² + y² = (x + y)² − 2xy

⇒ Rationalizing x: x = [(√5 + 2) × (√5 + 2)] / [(√5 − 2) × (√5 + 2)]

⇒ x = (√5 + 2)² / (5 − 4) = 5 + 4 + 4√5 = 9 + 4√5

⇒ Rationalizing y: y = [(√5 − 2) × (√5 − 2)] / [(√5 + 2) × (√5 − 2)]

⇒ y = (√5 − 2)² / (5 − 4) = 5 + 4 − 4√5 = 9 − 4√5

⇒ x + y = (9 + 4√5) + (9 − 4√5) = 18

⇒ xy = (9 + 4√5)(9 − 4√5) = 81 − 16(5) = 81 − 80 = 1

⇒ x² + y² = (18)² − 2(1)

⇒ x² + y² = 324 − 2 = 322

∴ The correct answer is 322.

Additional Information

Sum of Squares Identity

The expression a² + b² can be written as (a + b)² − 2ab or (a − b)² + 2ab depending on the given values.

Conjugate Surds

The conjugate of (√a + b) is (√a − b). Their product (√a + b)(√a − b) = a − b² results in a rational number.

Reciprocal Sum Property

If x = √a + √b and y = √a − √b where a − b = 1, then x and y are reciprocals of each other, meaning xy = 1.

Shortcut Trick

Relative Speed = (54 + 72) × 5/18 = 35 m/s.

Total Length (P + Q) = 35 × 30 = 1050 m.

Length of Train P = (3/5) × 1050 = 630 m.

Total Distance covered by P in 90 sec = 15 × 90 = 1350 m.

Length of Bridge = 1350 − 630 = 720 m.

∴ The correct answer is 720 m.

Alternate Method

Given:

Ratio of lengths of Train P and Train Q = 3 : 2

Speed of Train P (S_P) = 54 km/h | Speed of Train Q (S_Q) = 72 km/h

Time to cross each other = 30 sec | Time for P to cross bridge = 90 sec

Formula Used:

Distance = Speed × Time | Relative Speed (Opposite) = S₁ + S₂

Calculations:

⇒ Let Length of Train P = 3x and Train Q = 2x

⇒ S_P in m/s = 54 × (5/18) = 15 m/s

⇒ S_Q in m/s = 72 × (5/18) = 20 m/s

⇒ Relative Speed = 15 + 20 = 35 m/s

⇒ (3x + 2x) = 35 × 30

⇒ 5x = 1050 ⇒ x = 210

⇒ Length of Train P = 3 × 210 = 630 m

⇒ When P crosses bridge of length L:

⇒ 630 + L = 15 × 90

⇒ 630 + L = 1350

⇒ L = 1350 − 630 = 720 m

∴ The correct answer is 720 m.

Additional Information

Relative Speed Concept

When two objects move in opposite directions, their relative speed is S₁ + S₂. In the same direction, it is |S₁ − S₂|.

Unit Conversion

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

Train Crossing Objects

Total distance = Train length + Object length (if object has length like a bridge, platform, or tunnel). For a pole or man, distance = Train length only.

Shortcut Trick

Total years added to the sum of ages in 14 years = 14 × 2 = 28 years.

Given: Future Sum = 2 × Present Sum ⇒ Present Sum + 28 = 2 × Present Sum.

⇒ Present Sum (A + B) = 28 years.

Age difference (A − B) = 8 years.

A's present age = (Sum + Difference) ÷ 2 = (28 + 8) ÷ 2 = 18 years.

∴ The correct answer is 18.

Alternate Method

Given:

Difference between A's age and B's age = 8 years.

Condition: Age sum 14 years later = 2 × (Age sum now).

Let B's present age be x years.

Then A's present age = (x + 8) years.

According to the question:

⇒ [(x + 8) + 14] + [x + 14] = 2 × [(x + 8) + x]

⇒ (x + 22) + (x + 14) = 2 × (2x + 8)

⇒ 2x + 36 = 4x + 16

⇒ 4x − 2x = 36 − 16

⇒ 2x = 20

⇒ x = 10

A's present age = x + 8 = 10 + 8 = 18 years.

∴ The correct answer is 18.

Additional Information

Constant Age Difference

The numerical difference between the ages of two individuals remains the same throughout their lives, regardless of how many years pass.

Sum and Difference Rule

If the Sum (S) and Difference (D) of two ages are known, then Elder age = (S + D) ÷ 2 and Younger age = (S − D) ÷ 2.

Ratio Method in Ages

When ages are given in ratios at different time periods, ensure the ratio difference between the two individuals is made equal to solve for the multiplier.

Shortcut Trick

Solve the inner brackets first: 3 + 2 × 3 = 9 and 40 − 12 = 28.

Simplify the first major term: (36 + 4 × 9) ÷ 9 ⇒ 72 ÷ 9 = 8.

Simplify the second major term: 28 + (16 − 6) ⇒ 28 + 10 = 38.

Combine the results: 8 + 38 = 46.

∴ The correct answer is 46.

Alternate Method

Given: [ ( 36 + 4 × ( 3 + 2 × ( 5 − 2 ) ) ) ÷ ( 6 + 3 ) ] + [ ( 40 − ( 6 + 3 × 2 ) ) + ( 8 × 2 − 6 ) ]

According to the BODMAS rule, we solve in the order: Brackets, Of, Division, Multiplication, Addition, and Subtraction.

⇒ [ ( 36 + 4 × ( 3 + 2 × 3 ) ) ÷ 9 ] + [ ( 40 − ( 6 + 6 ) ) + ( 16 − 6 ) ]

⇒ [ ( 36 + 4 × 9 ) ÷ 9 ] + [ ( 40 − 12 ) + 10 ]

⇒ [ ( 36 + 36 ) ÷ 9 ] + [ 28 + 10 ]

⇒ [ 72 ÷ 9 ] + 38

⇒ 8 + 38 = 46

∴ The correct answer is 46.

Additional Information

BODMAS / VBODMAS Rule

It defines the sequence of operations: Vinculum (Bar), Brackets, Of (multiplication), Division, Multiplication, Addition, and Subtraction.

Order of Brackets

Always simplify parentheses ( ) first, then curly braces { }, and finally square brackets [ ].

Arithmetic Precedence

In expressions with only addition and subtraction or only multiplication and division, solve from left to right.

Shortcut Trick

Convert all terms to simple fractions: (5/6 + 2/6) ÷ (4/10 + 3/10) × 1/2 + 5/6

Solve brackets: (7/6) ÷ (7/10) × 1/2 + 5/6

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

Final addition: 5/6 + 5/6 = 10/6 = 5/3

∴ The correct answer is 5/3.

Alternate Method

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

Method Used: BODMAS Rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction)

⇒ Step 1 (Solve first bracket): (5/6 + 2/6) = 7/6

⇒ Step 2 (Solve second bracket): (2/5 + 3/10) = (4/10 + 3/10) = 7/10

⇒ Step 3 (Substitute values): (7/6) ÷ (7/10) × 0.5 + 5/6

⇒ Step 4 (Perform division): 7/6 × 10/7 = 10/6

⇒ Step 5 (Perform multiplication): 10/6 × 1/2 = 5/6

⇒ Step 6 (Final addition): 5/6 + 5/6 = 10/6

⇒ Simplify: 5/3

∴ The correct answer is 5/3.

Additional Information

BODMAS Rule

This rule defines the order of operations: Brackets (), Of (multiplication), Division (÷), Multiplication (×), Addition (+), and Subtraction (−).

Fraction Division

To divide two fractions, multiply the first fraction by the reciprocal of the second fraction: a/b ÷ c/d = a/b × d/c.

Decimal to Fraction Conversion

Convert decimals to fractions by placing the number over a power of 10. For example, 0.3 = 3/10 and 0.5 = 5/10 = 1/2.

1. Since all persons face the centre: right = counter-clockwise (CCW), left = clockwise (CW). Fix O at pos(1). L sits fourth to the right of O (CCW): O(1) → 2 → 3 → 4 → L(5).

2. L sits third to the left of C. This means C sits third to the right of L (CCW): L(5) → 6 → 1 → C(2). So, C is at pos(2).

3. B is an immediate neighbor of C and N. Since O is at pos(1), B must be at pos(3) to be next to C, and N must be at pos(4) so that B is also next to N.

4. The last person M fills the remaining pos(6). This fits the condition: one person (L) sits between M and N from the left of M (CW): M(6) → L(5) → N(4).

<br>

To find the number of people between L and B when counted from the left of B (clockwise direction): Starting from B(3) going CW: C(2), O(1), M(6), and then we reach L(5). The people sitting between them are C, O, and M, which is a total of three people.

 

Hence, the correct answer is 'Option 1'.

The logic followed here is the gap between the letters based on their positional values in the English alphabetical series.

Logic: 1st Letter + 5 = 2nd Letter and 2nd Letter + 6 = 3rd Letter.

Option 1) TYE,

Option 2) MRX,

Option 3) PUA

Option 4) GKP

This letter-cluster follows a +4, +5 pattern and does not belong to the group.

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

Hence, the correct answer is 'Option 4'.

The logic followed here is based on the English alphabetical series where each letter of the subsequent term is obtained by subtracting 1 from the positional value of the corresponding letter in the previous term.

The next term in the series is HLB.

Hence, the correct answer is "Option 1".

The logic followed here is to compare the position of each letter in the word COMBINED with its position when the letters are arranged in English alphabetical order.

Original Word:

 

Arranging letters in English Alphabetical Order:

 

Comparison of positions:

• Position 1: C → B (Changed)
• Position 2: O → C (Changed)
• Position 3: M → D (Changed)
• Position 4: B → E (Changed)
• Position 5: I → I (Unchanged)
• Position 6: N → M (Changed)
• Position 7: E → N (Changed)
• Position 8: D → O (Changed)

Only the letter I remains in the same position as in the original word.

Therefore, the position of One letter remains unchanged.

Hence, the correct answer is "Option 4".

Logic followed here is.

The code for a specific word is determined by comparing the given sentences to identify the unique common word and its corresponding common code.

Analysis of the given sentences:

1. 'door closed now' is coded as 'vt xg uk'

2. 'school is closed' is coded as 'cw uk ne'

Therefore, 'closed' is coded as 'uk'.

Hence, the correct answer is "Option 1".

Logic followed here is:

Logic: The difference between consecutive terms in the series increases by 1 at each step, representing a sequence of consecutive natural numbers starting from 8.

 

Therefore, the missing number that should come in place of the question mark is 94.

Hence, the correct answer is "Option 2".

The arrangement of seven people A, B, C, P, S, T, and U sitting in a straight line facing North is determined as follows:

1) Only three people are seated to the left of S.

2) Only A is seated to the right of B.

3) Only three people are seated between B and C.

4) P is seated at some place to the left of T, but at some place to the right of U.

 

Final Arrangement:

Calculation of people between U and T:

U is at the 1st position and T is at the 5th position.

The people seated between them are C (2nd), P (3rd), and S (4th).

Total number of people = 3.

Therefore, there are three people seated between U and T.

Hence, the correct answer is "Option 1".

The logic followed in this pattern is based on a consistent shift in the alphabetical positions of the letters.

For 'KSD' related to 'HPA':

For 'FXM' related to 'CUJ':

Applying the same logic to 'LIQ':

Thus, LIQ is related to IFN by the same pattern.

Hence, the correct answer is 'Option 3'.

The logic followed here is

Logic: First letter of the second cluster = First letter of the first cluster + 5; Second letter of the second cluster = Second letter of the first cluster − 3.

Option 1)  QK - VH,

Option 2) AE - FB,

Option 3) WH - BD

This pair does not follow the rule.

Option 4) GQ - LN,

Therefore, the letter-cluster pair that is different from the others is WH - BD.

Hence, the correct answer is 'Option 3'.

The logic followed here is based on the positional values of letters in the English alphabetical series.

The series consists of three letter positions in each term. Let's analyze the pattern for each position independently:

Therefore, the term that should come in place of the question mark is FTA.

Hence, the correct answer is "Option 1".

Given:

KL 36 : MN 41 →

QR 91 : ST 96 →

EF 64 : GH 69 →

Hence, EF 64 relates to GH 69 following the same pattern.

Hence, the correct answer is 'Option 4'.

Given:

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

 

On checking the expression: F # L − R @ O

1. F # L - 'F # L' means 'F is the father of L'.

2. L − R - 'L − R' means 'L is the brother of R'.

3. R @ O - 'R @ O' means 'R is the daughter of O'.

Therefore, F is the father of L and R, and O is the mother of R. Thus, F is the husband of O.

Hence, the correct answer is 'Option 3'.

The logic followed here is:

Position of Mr. Kell from the top = 84^th

Position of Mr. Kell from the bottom = 123^rd

To find the Total number of students in the class, we use the formula:

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

= (84 + 123) − 1

= 207 − 1

= 206

Hence, the correct answer is 'Option 1'.

Given:

Based on the relationship description, let us draw a family tree.

1. Ritik is the brother of Kajal - Ritik (Male) and Kajal (Female) are siblings.

2. Kajal is the wife of Abhishek - Kajal and Abhishek (Male) are a married couple.

3. Abhishek is the father of Megha - Abhishek is the parent of Megha (Female).

4. Megha is the sister of Priti - Megha and Priti are siblings.

Since Kajal is the mother of Priti and Ritik is the brother of Kajal, Ritik is the Mother's brother of Priti.

Hence, the correct answer is 'Option 2'.

Given Word:

DANGEROUS

Arrange the letters in alphabetical order:

<br>

Required letters:

• 4th from the left = G
• 4th from the right = O

Letters between G and O in the English alphabet are:

H, I, J, K, L, M, N

Total letters between them = 7.

Hence, the correct answer is "Option 2".

The logic followed here is:

Logic: Second Number = First Number + 7; Third Number = (Second Number × 2) + 4

For the first set (5, 12, 28)

⇒ 5 + 7 = 12.

⇒ (12 × 2) + 4 = 24 + 4 = 28.

For the second set (4, 11, 26)

⇒ 4 + 7 = 11.

⇒ (11 × 2) + 4 = 22 + 4 = 26.

Applying the same logic to the set in Option 4

(6, 13, 30)

⇒ 6 + 7 = 13.

⇒ (13 × 2) + 4 = 26 + 4 = 30.

Hence, the correct answer is 'Option 4'.

Given Series:

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

Required pattern:

Even Digit → Odd Digit → Odd Digit

Checking the series:

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

Odd digits satisfying the condition:

→ 5 in (8 5 1)

→ 3 in (4 3 5)

→ 9 in (6 9 7)

→ 1 in (4 1 3)

Therefore, there are 5 such odd digits.

Hence, the correct answer is "Option 4 — More than three".

The logic followed in this pattern is a combination of specific alphabetical shifts for each letter position.

For 'WIND' related to 'VHOE':

For 'FIRE' related to 'EHSF':

Applying the same logic to 'GOLD':

Thus, GOLD is related to FNME.

Hence, the correct answer is 'Option 1'.

Logic followed here is the substitution of letters with their assigned mathematical operators and then solving the expression using the BODMAS rule.

 

Given Expression:

1 B 3 D 10 A 5 C 2 = ?

After replacing the letters with mathematical signs, we get:

= 1 × 3 − 10 ÷ 5 + 2

= 3 - 2 + 2

= 3

Therefore, the value that will come in place of the question mark is 3.

Hence, the correct answer is "Option 3".

The logic followed here is:

Total number of people in the row = 50

Position of Cara from the right end = 19^th

Position of Gita relative to Cara = 19^th to the left

Position of Gita from the right end = Position of Cara from the right end + 19

= 19 + 19

= 38^th

Position of Gita from the left end = (Total number of people − Position of Gita from the right end) + 1

= (50 − 38) + 1

= 12 + 1

= 13^th

Hence, the correct answer is 'Option 3'.

Logic followed here is based on the common word comparison method.

Given:

‘conflict of interest’ is coded as ‘gu ky bz’

‘end this conflict’ is coded as ‘ky tg cn’

The code for conflict is ky.

Hence, the correct answer is "Option 1".

Given:

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

 

On checking: Z × C ge L o M @ N

1. Z × C - 'Z × C' means 'Z is the daughter of C'.

2. C ge L - 'C ge L' means 'C is the wife of L'.

3. L o M - 'L o M' means 'L is the father of M'.

4. M @ N - 'M @ N' means 'M is the brother of N'.

Therefore, Z and M are siblings of N, and all are children of L and C. Since Z is female, she is the sister of N.

Hence, the correct answer is 'Option 3'.

Sign Interchanges

Equation Transformation

The original equation is: 6 ÷ 2 + 24 × 6 − 4 →

By substituting the signs according to the given logic:

New Equation: 

= 6 × 2 − 24 ÷ 6 + 4

= 6 × 2 - 4 + 4

= 12 - 4 + 4

= 12

Therefore, the value that will come in the place of ? is 12.

Hence, the correct answer is "Option 4".

Given Series:

(Left) D U A L J X E I H R S B W P K T C V M F Y (Right)

Required pattern:

Consonant → Vowel → Vowel

Checking the series:

(Left) D U A L J X E I H R S B W P K T C V M F Y (Right)

Vowels satisfying the condition:

→ U in (D U A)

→ E in (X E I)

Therefore, there are 2 such vowels.

Hence, the correct answer is "Option 3".

Given:

1. Ankur starts from Point A and drives 11 km towards east. 2. He then takes a right turn, drives 7 km. 3. Then he turns right and drives 14 km. 4. He then takes a right turn and drives 12 km. 5. He takes a final right turn, drives 3 km and stops at Point P.

Following the movements described in the question, we can track Ankur path step-by-step from Point A to Point P.

Therefore, he should drive 5 km to the south to reach Point A.

Hence, the correct answer is "Option 3".

The correct answer is Graphite.

Key Points

• Graphite is a unique allotrope of carbon that functions as a conductor despite the general insulating nature of non-metals.
• The crystal structure of graphite consists of hexagonal layers where each carbon atom is sp² hybridized and covalently bonded to three other carbon atoms.
• Because carbon has four valence electrons, only three are utilized in forming strong sigma bonds within the planar sheets.
• The fourth valence electron is located in an unhybridized p-orbital perpendicular to the plane of the carbon atoms.
• These p-orbitals overlap laterally to form a delocalized pi-electron system that extends across the entire layer of the lattice.
• The presence of these mobile, delocalized electrons allows graphite to conduct electricity efficiently along the direction of its layers when a potential difference is applied.
• In contrast, the van der Waals forces between the layers are weak, contributing to graphite's lubricating properties but not its conductivity.

 Additional Information

• Sulphur:
• Exists predominantly as S₈ molecules with a puckered ring structure.
• It is an insulator because all its valence electrons are locked in covalent bonds, leaving no free electrons for conduction.
• Phosphorus:
• Found in various allotropes like white phosphorus (P₄) and red phosphorus (polymeric).
• These structures consist of discrete molecules or chains where electrons are localized, making it a poor conductor.
• Iodine:
• A halogen that forms diatomic molecules (I₂) arranged in a molecular crystal lattice.
• The lack of delocalized electrons or ions in the solid state prevents the flow of electric current.
• Diamond:
• Another carbon allotrope with sp³ hybridization forming a three-dimensional tetrahedral structure.
• All valence electrons are involved in strong C-C bonds, resulting in it being an excellent electrical insulator.

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

Key Points

• In a distance-time graph, the slope (gradient) is defined as the change in distance divided by the change in time (Δd/Δt), which represents the speed of the object. Acceleration is the rate of change of velocity over time and is not directly represented by the slope of a distance-time graph. Hence, statement (i) is incorrect.
• For a body in uniform motion, the graph is a straight line. A lesser slope (a flatter line) indicates that the body covers less distance in a given time interval, meaning the magnitude of its speed is lower. Hence, statement (ii) is correct.
• The area under a graph is the integral of the y-axis variable with respect to the x-axis variable. For a distance-time graph, this would be the product of distance and time (unit: meter·seconds), which does not correspond to speed (unit: meters per second) or any standard kinematic quantity. Hence, statement (iii) is incorrect.

 Additional Information

• Uniform Motion Characteristics:
• An object is in uniform motion if it travels in a straight line and covers equal distances in equal intervals of time.
• In this state, the velocity is constant, and the resultant acceleration is zero.
• The distance-time graph is always a linear plot with a constant gradient.
• Speed-Time (or Velocity-Time) Graphs:
• In a velocity-time graph, the slope represent the acceleration of the body.
• The area under the velocity-time graph represents the displacement or total distance traveled by the body.
• For uniform motion, the velocity-time graph is a horizontal line parallel to the time axis, indicating a slope of zero (zero acceleration).

The correct answer is − 2000 N.

Key Points

• According to Newton's Second Law of Motion, the force (F) applied to an object is directly proportional to the rate of change of its linear momentum, commonly expressed as F = m × a, where 'm' is mass and 'a' is acceleration.
• The acceleration (a) of the car is calculated using the first equation of motion or the definition of acceleration: a = (v − u) / t. Here, u is the initial velocity, v is the final velocity, and t is the time interval.
• Given parameters: Mass (m) = 1000 kg, Initial Velocity (u) = 20 m/s, Final Velocity (v) = 0 m/s (as the car comes to rest), and Time (t) = 10 s.
• Calculating acceleration: a = (0 − 20) / 10 = −2 m/s². The negative sign indicates retardation or deceleration due to braking.
• Applying the force formula: F = 1000 kg × (−2 m/s²) = −2000 N. The negative sign signifies that the braking force is acting in the direction opposite to the car's motion.

 Additional Information

• Kinematics and Dynamics:
• Newton (N): The SI unit of force. One Newton is the force required to accelerate a mass of 1 kg at a rate of 1 m/s².
• Retardation: It is defined as negative acceleration. In this case, the brakes provide a resistive force that reduces the kinetic energy of the vehicle.
• Impulse: The change in momentum (m × Δv) is also known as impulse, which is equal to the product of force and the time interval (F × t).
• Direction of Force: In physics, the sign of the force indicates vector direction relative to the chosen coordinate system; a negative result confirms it is a resistive force.

The correct answer is Power.

Key Points

• The Power of a lens represents its capacity to bend light rays. It is defined as the tangent of the angle by which it converges or diverges a beam of light falling at a unit distant from the optical center.
• It is mathematically defined as the reciprocal of the focal length (f) of the lens. The formula is expressed as P = 1/f, where the focal length must be in meters.
• The SI unit of optical power is the Dioptre (represented by D). One Dioptre is equal to 1 m⁻¹.
• A convex lens has a positive focal length, and therefore its power is positive. Conversely, a concave lens has a negative focal length, making its power negative.
• A lens with a shorter focal length has a greater ability to bend light rays, meaning it has higher power compared to a lens with a longer focal length.

 Additional Information

• Magnification:
• It is the ratio of the height of the image to the height of the object.
• It indicates how many times the image is enlarged or diminished relative to the object size.
• Aperture:
• It refers to the effective diameter of the light-transmitting area of the lens.
• The intensity of the image formed depends on the size of the aperture, as it controls the amount of light entering the system.
• Image Distance:
• It is the distance between the optical center of the lens and the position of the image formed, denoted by v.
• According to the Lens Formula, 1/f = 1/v − 1/u, where u is the object distance.

The correct answer is They contain their own DNA and ribosomes.

Key Points

• Mitochondria are often referred to as the powerhouse of the cell because they are the primary sites for ATP (Adenosine Triphosphate) synthesis through aerobic respiration.
• They are classified as semi-autonomous organelles because they possess their own genetic material, known as mitochondrial DNA (mtDNA), and their own ribosomes.
• The mtDNA is typically a small, circular molecule that encodes for several essential proteins, tRNAs, and rRNAs required for the organelle's function.
• Mitochondria contain 70S ribosomes, which are structurally similar to those found in prokaryotic organisms, allowing them to perform protein synthesis within the mitochondrial matrix independently of the cell nucleus.
• While most cellular proteins are encoded by nuclear DNA and imported from the cytoplasm, these organelles produce about 13 critical polypeptides involved in the oxidative phosphorylation system.

 Additional Information

• Endosymbiotic Theory:
• The presence of independent DNA and ribosomes supports the theory that mitochondria evolved from an ancient endosymbiosis between a primitive eukaryote and an aerobic bacterium.
• Structural Components:
• Double Membrane: Mitochondria consist of an inner membrane and an outer membrane. The inner membrane is folded into cristae to maximize the surface area for energy production.
• Porous Outer Membrane: This membrane contains special proteins called porins, which permit the free diffusion of molecules smaller than 5,000 Daltons.
• ATP Production:
• While mitochondria produce ATP, this is a metabolic function and does not directly provide the genetic blueprint or machinery for protein manufacturing.

The correct answer is 1 and 2 only.

Key Points

• A precipitation reaction occurs when two aqueous solutions containing soluble salts are mixed together, resulting in a double displacement reaction that yields an insoluble ionic solid. Hence, statement 1 is correct.
• The insoluble solid that emerges from the liquid mixture is scientifically referred to as a precipitate, and its formation indicates that the product has exceeded its solubility limit in the given solvent. Hence, statement 2 is correct.
• While some acid-base reactions might produce a precipitate, the vast majority of precipitation reactions involve salts; conversely, acid-base reactions are fundamentally neutralization reactions that primarily produce water and a soluble salt. Hence, statement 3 is incorrect.

 Additional Information

• Common Precipitation Examples:
• Silver Chloride (AgCl): When Silver Nitrate (AgNO₃) reacts with Sodium Chloride (NaCl), a white precipitate of AgCl is formed.
• Barium Sulphate (BaSO₄): The reaction between Barium Chloride (BaCl₂) and Sodium Sulphate (Na₂SO₄) produces a white insoluble solid.
• Lead Iodide (PbI₂): Mixing Lead Nitrate and Potassium Iodide results in a striking yellow precipitate.
• Industrial and Laboratory Uses:
• Qualitative Analysis: Used to detect the presence of specific cations or anions in an unknown solution based on the color and solubility of the precipitate.
• Water Treatment: Used to remove contaminants like magnesium, calcium, and heavy metals from hard water or industrial wastewater.
• Pigment Production: Many industrial dyes and pigments are manufactured through controlled precipitation processes.

The correct answer is Sodium chloride (NaCl).

Key Points

• Sodium chloride (NaCl) is classified as a neutral salt because it is the product of a neutralization reaction between a strong acid (Hydrochloric acid, HCl) and a strong base (Sodium hydroxide, NaOH).
• When dissolved in water, NaCl dissociates completely into its constituent ions, Na⁺ and Cl⁻. These ions are the conjugate species of a strong base and a strong acid, respectively, and therefore have negligible tendency to react with water.
• Since neither the cation nor the anion undergoes salt hydrolysis, the concentration of hydrogen ions (H⁺) and hydroxyl ions (OH⁻) remains equal to that of pure water.
• Consequently, the pH value of the resulting aqueous solution is approximately 7.0 at 25°C, characterizing the solution as neutral.

 Additional Information

• Potassium hydroxide (KOH):
• It is not a salt but a strong base. It dissociates to release a high concentration of OH⁻ ions, resulting in a strongly alkaline solution with a pH typically greater than 13.
• Ammonium chloride (NH₄Cl):
• This is an acidic salt formed from a strong acid (HCl) and a weak base (Ammonium hydroxide). The NH₄⁺ ion undergoes cationic hydrolysis, releasing H⁺ ions into the solution and lowering the pH below 7.
• Sodium acetate (CH₃COONa):
• This is a basic salt derived from a weak acid (Acetic acid) and a strong base (NaOH). The acetate ion (CH₃COO⁻) undergoes anionic hydrolysis, producing OH⁻ ions and making the solution alkaline (pH > 7).

The correct answer is Three.

Key Points

• Among the five substances buried by the gardener, Banana peels, Newspaper, and Cotton cloth are the three that are classified as biodegradable.
• Banana peels are organic waste products from plants that consist of biological tissue, which microorganisms like bacteria and fungi can decompose into humus within approximately 2 to 5 weeks.
• Newspaper is primarily made from wood pulp, which contains cellulose; this organic compound is easily broken down by soil enzymes and moisture, typically taking 10 to 30 days to disappear.
• Cotton cloth is a natural textile derived from the Gossypium plant; because it is composed of natural fibers, it is susceptible to biological decay and generally decomposes within 2 to 5 months in a garden environment.

 Additional Information

• Biodegradable substances are materials that can be broken down into simpler, non-toxic substances by the action of living organisms like decomposers.
• Natural materials like paper, wood, animal waste, and plant remains are essential for maintaining the nutrient cycle in the soil.
• The process of biodegradation requires specific environmental conditions including moisture, oxygen, and the presence of aerobic or anaerobic bacteria.
• Non-biodegradable substances are materials that cannot be decomposed by biological processes and persist in the environment for long periods.
• Plastic bottles are made from synthetic polymers like Polyethylene Terephthalate (PET), which lack the chemical bonds that bacteria can recognize and break, lasting for up to 450 years.
• Metal spoons, often made of stainless steel or aluminum, are inorganic and do not undergo biological decay, though they may eventually undergo corrosion or oxidation over centuries.

The correct answer is Electrons are embedded in a positively charged sphere.

Key Points

• J.J. Thomson proposed his model of the atom in 1898, shortly after he discovered the electron, the first subatomic particle.
• According to this model, an atom is envisioned as a positively charged sphere where negatively charged electrons are embedded into it, similar to seeds in a fruit.
• This model is widely known as the Plum Pudding Model or the Watermelon Model due to the specific arrangement of its components.
• In the Watermelon analogy, the red edible part represents the positive charge distributed throughout the atom, while the black seeds represent the electrons.
• Thomson hypothesized that the magnitude of the positive and negative charges is equal, which ensures that the atom remains electrically neutral.

 Additional Information

• Rutherford Atomic Model:
• Unlike Thomson's static model, Ernest Rutherford proposed in 1911 that electrons revolve around a nucleus in circular paths.
• This model was based on the Alpha-particle scattering experiment, which proved the existence of a dense, central nucleus.
• Bohr Atomic Model:
• Proposed by Niels Bohr in 1913, it refined the arrangement by stating that electrons move in discrete orbits with fixed energy levels.
• Shortcomings of Thomson Model:
• It could not explain the stability of the atom or the results of later experiments showing that most of an atom is empty space.
• It failed to account for the scattering of alpha particles through large angles, which necessitated the concept of a central positive core.

The correct answer is Regulate movement of substances across the cell.

Key Points

• The plasma membrane, also referred to as the cell membrane, is a biological membrane that separates the interior of all cells from the outside environment.
• It is selectively permeable, which is its most vital characteristic, allowing it to control the influx and efflux of nutrients, ions, and metabolic waste products.
• The structure is primarily composed of a phospholipid bilayer with embedded integral proteins and peripheral proteins that act as channels or pumps.
• It maintains homeostasis within the cell by facilitating transport mechanisms such as simple diffusion, facilitated diffusion, and active transport using ATP.
• The Fluid Mosaic Model describes the membrane as a dynamic structure where proteins float in or on the fluid lipid bilayer, enabling cell signaling and recognition.

 Additional Information

• ATP Synthesis:
• In eukaryotic cells, the primary site for ATP (Adenosine Triphosphate) production is the mitochondria via the process of oxidative phosphorylation.
• Cell Wall:
• Provides rigid support and structural integrity; it is found in plants (cellulose), fungi (chitin), and prokaryotes (peptidoglycan) but is absent in animal cells.
• Nucleus:
• Acts as the genetic control center of the cell, housing the chromosomes and DNA (Deoxyribonucleic Acid), protecting them from the metabolic activities of the cytoplasm.

The correct answer is Transporting food from leaves.

Key Points

• Xylem is a complex conducting tissue in vascular plants that primarily facilitates the upward movement of water and dissolved minerals from the roots.
• It consists of four specialized components: tracheids, vessel elements, xylem parenchyma, and xylem fibers.
• Tracheids and vessels are the primary water-conducting elements that form long, hollow tubes to ensure vertical transport of fluids throughout the plant.
• Xylem fibers are highly lignified, dead cells with thick walls that function to provide mechanical support and structural rigidity to the plant body.
• Xylem parenchyma is the only living component of the xylem tissue, specifically designed for storing food materials such as starch and assisting in lateral water conduction.
• The process of transporting organic solutes and food from the leaves to other parts of the plant is called translocation, which is the exclusive function of the phloem tissue.

 Additional Information

• Phloem Tissue:
• Sieve Tube Elements: These are long, tube-like structures that lack a nucleus at maturity and are responsible for the bidirectional transport of sucrose.
• Companion Cells: These specialized cells are closely associated with sieve tubes and maintain the pressure gradient required for nutrient movement.
• Phloem Parenchyma: Responsible for storing food and secretion of substances like resins and mucilage.
• Direction of Flow: While xylem flow is unidirectional (root to leaf), phloem flow is multidirectional, moving nutrients from source to sink.
• Vascular Bundle:
• The combination of xylem and phloem constitutes the vascular bundle, which serves as the circulatory system for higher plants.
• The cambium layer, found in dicotyledonous plants between xylem and phloem, is responsible for secondary growth and thickening of the stem.

The correct answer is It is the sum of protons and neutrons in an atom.

Key Points

• The mass number (represented by the symbol A) of an atom is defined as the total number of protons and neutrons present in its nucleus. Since these two particles reside in the nucleus, they are collectively known as nucleons. Hence, statement 1 is correct.
• In many light elements, the mass number is approximately twice the atomic number (e.g., Helium has 2 protons and 2 neutrons, A=4), but this is not a universal law. As elements become heavier, the proportion of neutrons increases to provide nuclear stability, making the mass number significantly more than twice the atomic number. Hence, statement 2 is incorrect.
• The number of electrons in a neutral atom is equal to the atomic number (Z), which is the number of protons. The mass number focuses on the dense particles within the nucleus, and electrons have negligible mass compared to nucleons. Hence, statement 3 is incorrect.
• While the sum of protons and electrons equals the total charge components of a neutral atom, it does not define the mass number. The mass number is strictly the count of heavy subatomic particles in the nucleus. Hence, statement 4 is incorrect.

 Additional Information

• Atomic Notation:
• In standard chemical notation, an element is written as ^A_ZX, where A is the mass number, Z is the atomic number, and X is the chemical symbol.
• The number of neutrons (n) can be calculated by the formula: n = A − Z.
• Isotopes:
• Isotopes are atoms of the same element that have the same atomic number (protons) but different mass numbers due to a different number of neutrons.
• For example, Carbon-12 (6 protons, 6 neutrons) and Carbon-14 (6 protons, 8 neutrons) are isotopes used in radiocarbon dating.
• Atomic Mass vs. Mass Number:
• The mass number is always a whole number representing a count of particles.
• The atomic mass is the weighted average of the masses of all naturally occurring isotopes of that element, which is why it often appears as a decimal on the Periodic Table.

The correct answer is Methyl chloride (CH₃Cl).

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When methane reacts with chlorine in the presence of ultraviolet (UV) light, a free-radical substitution reaction occurs, and the first product formed is methyl chloride (chloromethane), CH₃Cl.

Key Points

• The first substitution product of methane and chlorine is methyl chloride (CH₃Cl).
• The reaction takes place in the presence of UV light or sunlight, which initiates free-radical formation.
• One hydrogen atom of methane is replaced by one chlorine atom during the first substitution step.
• Further chlorination can produce dichloromethane (CH₂Cl₂), chloroform (CHCl₃), and carbon tetrachloride (CCl₄).

Additional Information

• Substitution Reaction
• A reaction in which one atom or group in a molecule is replaced by another atom or group.
• Alkanes typically undergo free-radical substitution reactions with halogens.
• Methyl Chloride (CH₃Cl)
• Also known as chloromethane.
• It is the first product formed during the chlorination of methane.
• Used in the production of silicones and other organic compounds.
• Dichloromethane (CH₂Cl₂)
• Formed during further chlorination of methyl chloride.
• Commonly used as an industrial solvent.
• Carbon Tetrachloride (CCl₄)
• The final product when all four hydrogen atoms of methane are replaced by chlorine atoms.
• Historically used in fire extinguishers and cleaning agents.
• Free Radical Mechanism
• Initiation: Cl₂ → 2Cl· (under UV light)
• Propagation: CH₄ + Cl· → CH₃· + HCl
• Propagation: CH₃· + Cl₂ → CH₃Cl + Cl·

The correct answer is Sand settles at the bottom, while salt dissolves in water.

Key Points

• Salt (Sodium Chloride, NaCl) is an ionic compound that is highly soluble in water; its ions dissociate and are surrounded by water molecules through a process called hydration, forming a homogeneous solution.
• Sand, primarily composed of Silicon Dioxide (SiO₂), is insoluble in water because the strong covalent bonds within its crystal lattice cannot be overcome by the dipole-dipole interactions of water.
• The combination of sand and water forms a suspension, a type of heterogeneous mixture where large particles are visible and do not chemically bond with the solvent.
• Because sand has a higher density (approximately 2.65 g/cm³) than water (1.0 g/cm³), the particles undergo sedimentation and sink to the bottom of the beaker due to gravity once stirring stops.

 Additional Information

• Mixture Classification:
• Homogeneous Mixture: A mixture with a uniform composition, such as the brine solution (saltwater) where the solute is evenly distributed.
• Heterogeneous Mixture: A mixture where components remain physically separate, like the sand-water suspension.
• Separation Techniques:
• Filtration: This method uses a porous barrier to separate the insoluble sand from the liquid phase based on particle size.
• Decantation: This involves carefully pouring out the supernatant liquid (the salt solution) after the sand has completely settled.
• Evaporation: This process recovers the dissolved salt by heating the solution until the water vaporizes, leaving the salt crystals behind.
• Solubility Limit: At room temperature (25°C), the solubility of salt in water is approximately 36 grams per 100 ml.

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

Key Points

• When white light is incident on a glass prism, it undergoes dispersion, splitting into its constituent colors (VIBGYOR) because each color travels at a different speed in the glass medium.
• Statement (i) explains that red light has the longest wavelength in the visible spectrum; since the angle of deviation is inversely proportional to wavelength, red bends the least. Hence, statement (i) is correct.
• Statement (ii) pertains to the speed of light, which is calculated as $v = \frac{c}{n}$. Because violet light experiences the highest refractive index, it travels at the slowest speed within the prism. Hence, statement (ii) is correct.
• Statement (iii) suggests that green has a higher refractive index than violet. In reality, according to Cauchy's formula, the refractive index decreases as wavelength increases; since violet has a shorter wavelength than green, violet has a higher refractive index. Hence, statement (iii) is incorrect.
• Statement (iv) claims the refractive index of red is more than blue. However, red light has a significantly longer wavelength than blue light, resulting in a lower refractive index for red. Hence, statement (iv) is incorrect.

 Additional Information

• Refractive Index and Wavelength Relationship:
• The refractive index (n) of a material is not constant but varies with the frequency of light, a phenomenon known as chromatic dispersion.
• Violet Light: It has the shortest wavelength (approximately 400 nm) and highest frequency, leading to the maximum deviation from the original path.
• Red Light: It has the longest wavelength (approximately 700 nm) and lowest frequency, leading to the minimum deviation.
• Visible Spectrum Order: The sequence from longest to shortest wavelength is Red, Orange, Yellow, Green, Blue, Indigo, and Violet.
• Speed in Vacuum: All colors of light travel at the same speed of approximately $3 \times 10^{8}$ m/s in a vacuum, but their speeds differ when they enter a dense medium like glass.

The correct answer is Uniform circular motion.

Key Points

• Uniform circular motion is defined as the motion of an object traveling at a constant speed along a circular path.
• In this scenario, while the cyclist maintains a steady speed, the direction of motion changes continuously at every point along the circular track, meaning the velocity is constantly changing.
• Because the velocity is changing due to the direction, the cyclist undergoes centripetal acceleration, which is always directed toward the center of the circle.
• The magnitude of velocity (speed) remains unchanged, which distinguishes it from non-uniform circular motion where the speed would fluctuate.

 Additional Information

• Linear Motion:
• Also referred to as rectilinear motion, it occurs when an object moves in a straight line.
• In this type of motion, the direction remains constant unless the object reverses its path.
• Oscillatory Motion:
• This is the to-and-fro or periodic motion of an object about a fixed mean position.
• Classic examples include the movement of a simple pendulum or a tuning fork.
• Non-uniform Motion:
• An object is said to be in non-uniform motion if its speed changes as it moves, covering unequal distances in equal intervals of time.
• If the cyclist were accelerating or braking on the track, the motion would be classified as non-uniform circular motion.

The correct answer is Less than 1 nm.

Key Points

• A homogeneous mixture is a type of mixture where the components are distributed with uniform composition and properties throughout the entire mass, commonly known as a True Solution.
• In a True Solution, the size of the solute particles is extremely small, typically measuring less than 1 nanometer (1 nm = 10⁻⁹ meters) in diameter.
• Because these particles are at the molecular or ionic level, they cannot be seen even with the aid of a powerful microscope.
• Due to their minute size, they do not scatter a beam of light passing through the medium, meaning they do not exhibit the Tyndall effect.
• These particles pass easily through filter paper and parchment membranes, making them impossible to separate from the solvent using standard filtration techniques.
• The mixture is highly stable, and the solute particles do not settle down when left undisturbed because of their constant kinetic motion and negligible mass.

 Additional Information

• Colloids:
• These are heterogeneous mixtures where the particle size ranges between 1 nm and 1000 nm.
• They are large enough to scatter a beam of light (Tyndall effect) but small enough to remain suspended indefinitely without settling.
• Suspensions:
• These are heterogeneous mixtures containing large particles, typically larger than 1000 nm (1 μm).
• The particles are visible to the naked eye, do not pass through filter paper, and eventually settle at the bottom due to gravity.
• Units of Measurement:
• 1 Nanometer (nm) is equal to 10⁻⁹ meters.
• 1 Micrometer (μm) is equal to 10⁻⁶ meters or 1000 nm.

The correct answer is Many short, branched parts.

Key Points

• The neuron or nerve cell is the structural and functional unit of the nervous system, specialized for the reception and transmission of electrical impulses.
• A typical neuron consists of three distinct regions: the cell body (Cyton), dendrites, and a single axon.
• The axon is a long, slender, and single cytoplasmic extension that originates from the axon hillock; its primary role is to carry nerve impulses away from the cell body toward a synapse or effector organ.
• Dendrites are the multiple, short, and highly branched protoplasmic processes that project from the cell body.
• The primary function of dendrites is to receive afferent signals (stimuli) from other neurons or sensory receptors and conduct them toward the soma or cell body.
• The branching nature of dendrites increases the surface area significantly, allowing a single neuron to receive thousands of synaptic inputs simultaneously.

 Additional Information

• Cell body with nucleus:
• Commonly known as the Soma or Cyton, it contains the nucleus and Nissl’s granules (groups of ribosomes and rough endoplasmic reticulum).
• It serves as the metabolic center of the neuron where protein synthesis and cellular maintenance occur.
• Protective connective tissue:
• In the peripheral nervous system, nerves are bundled and protected by layers such as the epineurium, perineurium, and endoneurium.
• The myelin sheath, produced by Schwann cells, provides electrical insulation around the axon.
• Rapid stimulus transmission:
• This refers to the functional efficiency of the axon, especially those that are myelinated.
• The Nodes of Ranvier allow for saltatory conduction, which significantly speeds up the transmission of action potentials.

The correct answer is Voltmeter.

Key Points

• In electrical circuit diagrams, a voltmeter is represented by a circle with the uppercase letter 'V' inscribed inside it.
• The primary function of this instrument is to measure the potential difference (voltage) between two specific points in an electric circuit, with the standard unit being the Volt (V).
• To ensure measurement accuracy, a voltmeter is always connected in parallel with the component whose voltage is being measured, as voltage remains constant across parallel branches.
• A characteristic feature of a voltmeter is its high internal resistance; this prevents it from drawing significant current from the main circuit. In theoretical physics, an ideal voltmeter is considered to have infinite resistance.
• The sensitivity of the device can be modified using a high-value resistor, known as a multiplier, connected in series with the internal galvanometer mechanism.

 Additional Information

• Variable Resistance (Rheostat):
• Represented by a zigzag line with an arrow passing through it or a sliding contact arrow pointing toward it.
• It is used to vary the current in a circuit by manually changing the resistance value without interrupting the circuit flow.
• Vacuum Diode:
• A legacy electronic device symbolized by a circle containing a cathode and an anode.
• It functions as a rectifier, allowing electric current to flow in only one direction, and was the precursor to the modern semiconductor p-n junction diode.
• Voltage Regulator:
• Often represented by a functional block or a Zener diode symbol (a triangle with a 'Z' shaped bar).
• Its core purpose is to maintain a constant output voltage level despite fluctuations in input voltage or varying load conditions.

The correct answer is If the forefinger points in the direction of magnetic field and the middle finger in the direction of current, the thumb shows the direction of motion or force on the conductor.

Key Points

• Fleming’s Left-Hand Rule is a mnemonic used to determine the direction of force acting on a current-carrying conductor when placed within an external magnetic field. Hence, statement 2 is correct.
• The rule assigns the Thumb to the direction of Force (or motion), the Forefinger to the Magnetic Field, and the Middle Finger to the Electric Current, provided they are held mutually perpendicular. Hence, statement 1 is incorrect.
• This principle is the fundamental basis for the operation of Electric Motors, where electrical energy is converted into mechanical motion, and Loudspeakers. Hence, statement 3 is incorrect.
• Correct orientation requires the Forefinger to represent the field and the Middle Finger to represent the current to correctly identify the Thumb as the force vector. Hence, statement 4 is incorrect.

 Additional Information

• Physical Components:
• Magnetic Field (B): Represents the flux lines moving from the North Pole to the South Pole.
• Electric Current (I): Represents the flow of conventional current (from positive to negative).
• Force (F): Also known as the Lorentz Force, it is calculated by the formula F = BIL sin(θ), where θ is the angle between the conductor and the field.
• Comparison with Right-Hand Rule:
• While the Left-Hand Rule is applied to Motors (input current produces motion), Fleming’s Right-Hand Rule is applied to Generators (input motion produces induced current).
• In the right-hand version, the Thumb represents motion, the Forefinger field, and the Middle Finger represents the direction of the Induced Current.
• Applications:
• Used in designing Galvanometers, DC Motors, and electromagnetic actuators.
• Critical for understanding how charged particles (like electrons or protons) deflect when entering a magnetic field in Particle Accelerators.

The correct answer is Biological magnification.

Key Points

• Biological magnification, also termed biomagnification or bioamplification, refers to the progressive increase in the concentration of toxic substances at each successive trophic level of a food chain.
• This phenomenon occurs with chemicals that are non-biodegradable and lipophilic (fat-soluble), meaning they cannot be broken down by metabolic processes or excreted easily, causing them to accumulate in adipose tissues.
• As energy is transferred through the food web, predators must consume large quantities of prey from lower levels; consequently, they ingest all the accumulated toxins stored in those organisms, leading to a massive increase in chemical concentration.
• A scientific landmark for this process was the study of DDT (Dichlorodiphenyltrichloroethane), which showed that while water contained only 0.000003 parts per million (ppm), fish-eating birds at the top of the chain had concentrations as high as 25 ppm.
• High levels of these toxins in apex predators can result in reproductive failure, immune system suppression, and in the case of birds, the thinning of eggshells, which significantly impacts biodiversity.

 Additional Information

• Photosynthesis:
• A biological process used by photoautotrophs, such as green plants and cyanobacteria, to convert light energy into chemical energy.
• It utilizes chlorophyll to transform carbon dioxide and water into glucose, releasing oxygen as a vital byproduct for aerobic life.
• Biodegradation:
• The natural breakdown of organic matter by microorganisms like bacteria and fungi into simpler, non-toxic substances like CO2 and water.
• It is a crucial component of the nutrient cycle, though many modern synthetic pesticides are specifically designed to resist this process.
• Eutrophication:
• An environmental process where water bodies receive excess nutrients, particularly nitrates and phosphates, often from agricultural runoff.
• This leads to algal blooms that block sunlight and deplete dissolved oxygen, creating hypoxic conditions or dead zones where aquatic life cannot survive.

The correct answer is Broilers are raised for meat; layers for eggs.

Key Points

• Broilers are poultry birds specifically bred and reared for meat production. They are known for their fast growth rate and high muscle development, typically reaching market weight within 6 to 8 weeks.
• Layers are chickens raised primarily for commercial egg production. These birds reach sexual maturity and start laying eggs at approximately 18 to 20 weeks of age.
• The nutritional requirements differ significantly; broilers need a diet rich in protein and vitamins for rapid growth, while layers require high amounts of calcium and minerals to produce strong eggshells.
• Environmental management for layers involves strictly controlled lighting cycles (photoperiodism) to stimulate the pituitary gland for consistent egg laying, whereas broilers require space and conditions optimized for feed conversion efficiency.

 Additional Information

• Broiler Characteristics:
• Common breeds include the Cornish Cross and Plymouth Rock.
• They exhibit a high feed conversion ratio (FCR), meaning they efficiently turn feed into body mass.
• Management focuses on floor space, ventilation, and hygiene to prevent diseases like Coccidiosis.
• Layer Characteristics:
• The White Leghorn is one of the most popular breeds used for white egg production.
• Layers are kept for a longer duration, often producing eggs for 52 to 72 weeks before their productivity declines.
• Egg quality is maintained through a diet balanced in phosphorus and Vitamin D₃.

The logic followed here is:

Logic: Mode of asexual reproduction in specific organisms.

Here,

Binary Fission : Bacteria → Binary Fission is a method of asexual reproduction where a parent cell divides into two identical daughter cells, which is the primary mode of reproduction in Bacteria.

Similarly,

Fragmentation : Algae → Fragmentation is a type of asexual reproduction where an organism breaks into several fragments, each of which matures into a new individual. This process is commonly observed in Algae (such as Spirogyra).

Hence, the correct answer is 'Option 2'.

The correct answer is Only 1 and 3.

Key Points

• The atomic size refers to the distance between the center of the nucleus and the outermost shell of an atom. Most atoms have a size or diameter that falls within the range of 0.1 nm to 0.5 nm (1 to 5 Angstroms). Hence, statement 1 is correct.
• Across a period in the modern periodic table, the atomic radius actually decreases from left to right. This occurs because the atomic number increases, leading to a higher nuclear charge (more protons) that pulls the valence electrons closer to the nucleus, despite them being in the same energy level. Hence, statement 2 is incorrect.
• The hydrogen atom is the smallest neutral atom. Its atomic radius is approximately 37 pm (picometers) to 53 pm (Bohr radius). In terms of order of magnitude, 1 Angstrom is equal to 10⁻¹⁰ m, which represents the standard scale for the radius of a hydrogen atom. Hence, statement 3 is correct.

 Additional Information

• Group Trends:
• In a group, the atomic size increases from top to bottom.
• This is due to the addition of new electron shells at each successive element, which increases the distance between the nucleus and the outermost electrons, overriding the increase in nuclear charge.
• Measurement Units:
• Nanometer (nm): 1 nm = 10⁻⁹ m.
• Angstrom (Å): 1 Å = 10⁻¹⁰ m or 0.1 nm.
• Picometer (pm): 1 pm = 10⁻¹² m.
• Effective Nuclear Charge (Z_eff):
• This is the net positive charge experienced by an electron in a multi-electron atom. It is the primary driver for the contraction of atomic size across a period.

The correct answer is Balanced forces.

Key Points

• Newton’s First Law of Motion, often referred to as the Law of Inertia, states that an object will persist in its state of rest or uniform motion in a straight line unless acted upon by an external unbalanced force.
• The law fundamentally explains the behavior of objects under the influence of balanced forces, which occurs when the vector sum of all external forces acting on a body is exactly zero.
• When forces are balanced, there is no change in the magnitude or direction of the object’s velocity, meaning the acceleration remains zero.
• This principle was first published by Sir Isaac Newton in 1687 within his monumental work, Philosophiæ Naturalis Principia Mathematica, establishing the foundation for Classical Mechanics.
• The law serves as a qualitative definition of force as the external agent capable of changing a body's state of inertia.

 Additional Information

• Inertia:
• It is the inherent property of matter that resists changes in its state of motion or rest.
• The mass of an object is the quantitative measure of its inertia; objects with higher mass require greater force to change their velocity.
• Common examples include Inertia of Rest (dust falling from a carpet when beaten) and Inertia of Motion (passengers falling forward when a moving bus brakes suddenly).
• Net Force and Acceleration:
• A non-zero net force refers to unbalanced forces, which cause an object to accelerate as per Newton’s Second Law.
• Continuous acceleration implies a constant application of an unbalanced force, which contradicts the equilibrium state described in the First Law.
• The change in momentum is defined as the product of mass and the change in velocity, which is directly proportional to the impressed force.

The correct answer is Speedometers.

Key Points

• Archimedes' Principle states that when a body is immersed fully or partially in a fluid, it experiences an upward buoyant force that is equal to the weight of the fluid displaced by it.
• Speedometers are instruments designed to measure and display the instantaneous speed of a vehicle. They typically function using Eddy currents, magnetic sensors, or GPS technology and do not rely on fluid displacement or buoyancy.
• Lactometers are used to determine the purity of milk by measuring its density. Since the instrument floats at different levels based on the fluid's weight and displacement, it is a direct application of Archimedes' Principle.
• Hydrometers are devices used to measure the specific gravity or relative density of liquids. They operate by observing the depth to which a calibrated glass tube sinks in a liquid, which is governed by the upward thrust of the displaced fluid.
• The Designing of submarines and ships heavily relies on this principle to ensure stability and buoyancy. Submarines specifically utilize ballast tanks to alter their weight by taking in or discharging water, allowing them to submerge or surface by manipulating the displaced volume.

 Additional Information

• Buoyant Force:
• It is the upward force exerted by a fluid on an object.
• The magnitude of this force is calculated as the product of the density of the fluid, the acceleration due to gravity, and the volume of the displaced fluid.
• Specific Gravity:
• It is a dimensionless quantity representing the ratio of the density of a substance to the density of pure water at 4 degrees Celsius.
• Instruments like Hydrometers use this ratio to identify the composition of various liquids like battery acid or antifreeze.
• Eddy Current Mechanism:
• Found in traditional speedometers, where a rotating magnet creates a magnetic field that induces currents in a non-ferrous metal cup, creating torque without physical contact.

The correct answer is CuSO4 and AgNO3 only.

Key Points

• Chemical reactions between a metal and a salt solution depend on the Reactivity Series, where a more reactive metal displaces a less reactive metal from its compound.
• Zinc (Zn) is positioned higher than Copper (Cu) in the reactivity series. When Zinc metal is added to CuSO₄, a displacement reaction occurs, forming Zinc Sulphate and solid Copper.
• Zinc is significantly more reactive than Silver (Ag). Consequently, Zinc reacts with AgNO₃ to displace Silver, resulting in the formation of Zinc Nitrate and metallic Silver deposits.
• Magnesium (Mg) is located above Zinc in the reactivity series, making it more reactive. Therefore, Zinc cannot displace Magnesium from the MgSO₄ solution, and no reaction occurs.
• Adding Zinc to a ZnSO₄ solution results in no observable chemical change because a metal does not react with its own salt solution to produce a new substance.

 Additional Information

• Reactivity Gradient: The metals involved follow the reactivity order: Magnesium (Mg) > Zinc (Zn) > Copper (Cu) > Silver (Ag).
• Magnesium: An alkaline earth metal that is highly electropositive.
• Zinc: A transition metal often used in galvanization and as a reducing agent in chemical labs.
• Copper and Silver: These are noble-like metals found lower in the series, making them easily displaced by Zinc.
• Visual Indicators:
• The Blue vitriol (CuSO₄) solution loses its blue color as Cu²⁺ ions are reduced to Copper metal.
• Displacement of Silver from Silver Nitrate typically results in a lustrous grey or silver precipitate on the surface of the Zinc strip.
• Standard Reduction Potential: Zinc has a standard reduction potential of −0.76 V, which is lower than Copper (+0.34 V) and Silver (+0.80 V), allowing it to act as the anode or reducing agent in these specific redox pairs.

The correct answer is To increase groundwater by storing rainwater.

Key Points

• A check-dam is a small, often temporary, barrier constructed across a drainage ditch, swale, or channel to lower the velocity of flow.
• The primary objective of these structures is to promote groundwater recharge. By slowing down the surface runoff, the dam creates a small pool that allows water to percolate through the soil layers into the underground aquifers.
• In arid and semi-arid regions, check-dams serve as a vital tool for water harvesting, ensuring that rainwater is not lost as flash floods but is instead stored beneath the surface for future irrigation needs.
• These structures significantly reduce soil erosion by trapping sediments and silt, which would otherwise be carried away by fast-moving water, thus preserving the fertility of the land.
• By maintaining a higher water table, check-dams ensure that nearby borewells and open wells remain functional even during the dry summer months.

 Additional Information

• Watershed Management:
• Check-dams are a core component of integrated watershed management, which aims to conserve every drop of rain within a specific geographical area.
• They can be constructed using various materials such as boulders, gabions (wire cages filled with rocks), logs, or reinforced concrete.
• Environmental Impact:
• Unlike large-scale dams, check-dams have a minimal ecological footprint and do not involve large-scale displacement of local populations.
• They improve soil moisture content, which directly benefits the growth of natural vegetation and crops in the immediate vicinity.
• Siltation control provided by these dams prevents the clogging of larger downstream reservoirs and riverbeds.

The correct answer is Double.

Key Points

• The Weight of an object is defined as the gravitational force with which a celestial body, such as the Earth, attracts the object toward its center.
• Mathematically, weight is expressed by the formula W = m × g, where m represents the mass of the object and g represents the acceleration due to gravity.
• This equation indicates that weight is directly proportional to the mass of the object when the gravitational acceleration remains constant.
• If the mass is doubled (2m) and g is kept constant, the new weight (W_new) becomes (2m) × g, which is equivalent to 2W.
• Consequently, any linear change in mass results in an identical linear change in weight, provided the gravitational field strength does not fluctuate.

 Additional Information

• Mass:
• It is a scalar quantity that measures the inertia or the total quantity of matter within a body.
• The SI unit of mass is the kilogram (kg), and it remains constant regardless of the object's location in the universe.
• Weight:
• It is a vector quantity because it has a specific direction, which is always towards the center of the planet.
• The SI unit of weight is the Newton (N).
• Unlike mass, weight varies depending on the value of g; for instance, an object's weight on the Moon is approximately 1/6th of its weight on Earth due to the Moon's weaker gravity.
• Acceleration due to Gravity (g):
• On the surface of the Earth, its standard value is taken as 9.8 m/s².
• It reaches its maximum at the poles and its minimum at the equator, and becomes zero at the center of the Earth.

The correct answer is C₃H₈.

• Alkanes have the general molecular formula C_nH_2n+2. The molecular mass of the alkane can be calculated using the atomic masses of carbon (12 u) and hydrogen (1 u).

Key Points

• For propane (C₃H₈), molecular mass = (3 × 12) + (8 × 1) = 36 + 8 = 44 u.
• Therefore, the alkane having molecular mass 44 is C₃H₈ (Propane).
• Propane is the third member of the alkane homologous series.
• Alkanes are saturated hydrocarbons containing only single covalent bonds between carbon atoms.

Additional Information

• Methane (CH₄)
• First member of the alkane series.
• Molecular mass = 12 + 4 = 16 u.
• Ethane (C₂H₆)
• Second member of the alkane series.
• Molecular mass = (2 × 12) + (6 × 1) = 30 u.
• Propane (C₃H₈)
• Third member of the alkane series.
• Molecular mass = 44 u.
• Commonly used as LPG fuel.
• Butane (C₄H₁₀)
• Fourth member of the alkane series.
• Molecular mass = (4 × 12) + (10 × 1) = 58 u.
• Alkanes
• General formula: C_nH_2n+2.
• Known as saturated hydrocarbons because they contain only single bonds.
• Major constituents of natural gas and petroleum.

The correct answer is Locate underwater objects.

Key Points

• SONAR is an acronym for Sound Navigation and Ranging. It is a sophisticated device that utilizes ultrasonic waves to determine the distance, direction, and speed of underwater objects such as submarines, icebergs, or sunken ships.
• The system typically comprises a transmitter and a detector mounted on the hull of a vessel. The transmitter emits high-frequency ultrasound waves that travel through the water medium.
• When these waves encounter an object or the seabed, they are reflected back as echoes. These reflected waves are captured by the detector, which converts the acoustic energy into electrical signals for processing.
• The distance to the object is calculated using the formula 2d = v × t, where 'd' is the distance, 'v' is the speed of sound in water, and 't' is the total time interval between the transmission and reception. This specific technique is known as echo-ranging.

 Additional Information

• Ultrasound refers to sound waves with frequencies above 20,000 Hz, which are beyond the audible range of human hearing.
• Different scientific instruments serve the other functions mentioned:
• Barometer: A scientific instrument primarily used to measure atmospheric pressure to forecast weather changes.
• Ammeter: An electrical instrument used to measure the electric current flowing through a circuit in Amperes.
• Pyrometer: A type of remote-sensing thermometer used to measure exceptionally high temperatures of surfaces, often in industrial furnaces.
• SONAR is also used by marine animals like bats and dolphins (biological sonar) for echolocation to navigate and find prey in dark or murky environments.

The correct answer is 20√10 m/s.

Key Points

• The block is in a state of free fall, which occurs when an object is dropped from a height and moves solely under the influence of gravity.
• The initial velocity (u) is 0 m/s because the object starts from a stationary position at the top of the 200 m building.
• The acceleration due to gravity (g) is constant and given as 10 m/s², acting downwards towards the ground.
• To find the final velocity (v), we utilize the third equation of motion: $v^2 = u^2 + 2gh$.
• By substituting the known values, the equation becomes: $v^2 = 0^2 + 2 × 10 × 200$.
• Simplifying the expression leads to $v^2 = 4000$.
• The final velocity (v) is obtained by taking the square root of the result: $v = \sqrt{4000} = \sqrt{400 × 10} = 20\sqrt{10}$ m/s.

 Additional Information

• Equations of Motion:
• These equations are valid only for objects moving with uniform acceleration.
• First Equation: $v = u + at$ relates velocity and time.
• Second Equation: $s = ut + \frac{1}{2}at^2$ relates displacement and time.
• Conservation of Energy:
• The Potential Energy (PE) at the top ($mgh$) is converted into Kinetic Energy (KE) at the ground ($\frac{1}{2}mv^2$).
• Equating PE and KE shows that the velocity is independent of mass, as the variable m cancels out from both sides.
• Free Fall Characteristics:
• In a vacuum, a 20 kg block and a lighter object would hit the ground with the same velocity if dropped from the same height.
• The displacement (s) in vertical motion is simply the height (h) of the fall.