RRB NTPC (UG) CBT-I Official Paper (Held On 16 Jun, 2026 Shift 2) (Previous Year Paper)

Shortcut Trick

Remaining food is sufficient for 360 soldiers for 40 days (50 − 10).

Total soldiers after arrival = 360 + 120 = 480 soldiers.

Using M₁D₁ = M₂D₂ logic on remaining stock: 360 × 40 = 480 × Days.

Days = 14400 ÷ 480 = 30 days.

∴ The correct answer is 30 days.

Alternate Method

Given:

Initial soldiers (M₁) = 360

Initial provision time (D₁) = 50 days

Elapsed time = 10 days

Newly joined soldiers = 120

Calculations:

⇒ Total provision units = 360 × 50 = 18,000 units

⇒ Provision consumed in 10 days = 360 × 10 = 3,600 units

⇒ Remaining provision = 18,000 − 3,600 = 14,400 units

⇒ Total soldiers now = 360 + 120 = 480

⇒ Number of days for remaining provisions = Remaining stock ÷ Total soldiers

⇒ Number of days = 14,400 ÷ 480

⇒ Number of days = 30 days

∴ The correct answer is 30 days.

Additional Information

Chain Rule (Work and Time)

The amount of work or provisions is directly proportional to the number of people and the number of days: M₁×D₁×H₁ = M₂×D₂×H₂.

Inverse Variation

For a constant amount of food, the number of days the food lasts is inversely proportional to the number of soldiers. If men increase, days decrease.

Effective Man-Days

The total quantity of provisions is often measured in man-days, which is the product of the number of men and the number of days they can be fed.

Shortcut Trick

The Mean Proportional between two numbers a and b is calculated as √(a × b).

Given a = 0.0009 and b = 0.25.

Mean proportional = √(0.0009 × 0.25) = 0.03 × 0.5 = 0.015.

∴ The correct answer is 0.015.

Alternate Method

Given: First number (a) = 0.0009 and Second number (b) = 0.25.

In a mean proportion a : x = x : b, where x is the mean proportional value.

Calculation:

⇒ x² = a × b

⇒ x² = 0.0009 × 0.25

⇒ x = √(0.0009 × 0.25)

⇒ x = √(9/10000 × 25/100)

⇒ x = √(225/1000000)

⇒ x = 15/1000

⇒ x = 0.015

∴ The correct answer is 0.015.

Additional Information

Third Proportional

For two numbers a and b, the third proportional c is such that a : b = b : c, calculated as c = b²/a.

Fourth Proportional

For three numbers a, b, and c, the fourth proportional d is such that a : b = c : d, calculated as d = (b × c)/a.

Continued Proportion

A sequence of numbers a, b, c, d... is in continued proportion if a/b = b/c = c/d and so on.

Shortcut Trick

Average Speed = (D₁ + D₂) ÷ (D₁/S₁ + D₂/S₂)

Average Speed = (60 + 90) ÷ (60/45 + 90/60)

Average Speed = 150 ÷ (4/3 + 3/2) = 150 ÷ (17/6)

Average Speed = 900 ÷ 17 ≈ 52.94 km/hr

∴ The correct answer is 52.94 km/hr.

Alternate Method

Given:

First distance (d₁) = 60 km | Speed (s₁) = 45 km/hr

Second distance (d₂) = 90 km | Speed (s₂) = 60 km/hr

Formula Used:

Average Speed = Total Distance ÷ Total Time

Time = Distance ÷ Speed

Calculations:

⇒ Time taken for the first part (t₁) = 60 ÷ 45 = 4/3 hours

⇒ Time taken for the second part (t₂) = 90 ÷ 60 = 3/2 hours

⇒ Total Time = t₁ + t₂ = 4/3 + 3/2

⇒ Total Time = (8 + 9) ÷ 6 = 17/6 hours

⇒ Total Distance = 60 + 90 = 150 km

⇒ Average Speed = 150 ÷ (17/6) = (150 × 6) ÷ 17

⇒ Average Speed = 900 ÷ 17 = 52.9411... km/hr

∴ The correct answer is 52.94 km/hr.

Additional Information

Average Speed for Equal Distances

If a body travels two equal distances at speeds x and y, the average speed is given by (2 × x × y) ÷ (x + y).

Average Speed for Equal Time Intervals

If a body travels at speeds x and y for equal time intervals, the average speed is simply the arithmetic mean: (x + y) ÷ 2.

Relative Speed Concept

For objects moving in the same direction, relative speed = S₁ − S₂. For opposite directions, relative speed = S₁ + S₂.

Shortcut Trick

The Shortcut formula to find the Difference when Sum and Product are given:

Difference² = Sum² − 4 × Product

⇒ Difference² = 36² − 4 × 180

⇒ Difference² = 1296 − 720 = 576

⇒ Difference = √576 = 24

∴ The correct answer is 24.

Alternate Method

Given:

Sum of two Positive Numbers (x + y) = 36

Product of the numbers (xy) = 180

Formula Used: (x − y)² = (x + y)² − 4xy

⇒ (x − y)² = (36)² − 4 × 180

⇒ (x − y)² = 1296 − 720

⇒ (x − y)² = 576

⇒ x − y = √576

⇒ x − y = 24

∴ The correct answer is 24.

Additional Information

Algebraic Identities

(x + y)² = x² + y² + 2xy and (x − y)² = x² + y² − 2xy. Subtracting these gives (x + y)² − (x − y)² = 4xy.

Finding Individual Numbers

If Sum (S) and Difference (D) are known, the larger number is (S + D) ÷ 2 and the smaller number is (S − D) ÷ 2.

Sum of Squares

The sum of squares of two numbers can be found using x² + y² = (x + y)² − 2xy.

Shortcut Trick

Check the ratio of the sides: 75 : 21 : 72 ÷ 3 = 25 : 7 : 24.

The numbers 7, 24, and 25 form a Pythagorean triplet, which means the triangle is right-angled.

In a right-angled triangle, the altitude corresponding to one leg is simply the length of the other leg.

The sides forming the right angle are 21 cm and 72 cm.

∴ The correct answer is 72 cm.

Alternate Method

Given: The sides of the triangle are 75 cm, 21 cm, and 72 cm.

Formula Used: Area of a triangle = (1 ÷ 2) × base × altitude

⇒ First, identify the type of triangle using the Pythagoras Theorem: 21² + 72² = 441 + 5184 = 5625.

⇒ Since 75² = 5625, the sides satisfy a² + b² = c², making it a right-angled triangle.

⇒ In this triangle, the longest side (75 cm) is the hypotenuse, and the other two sides (21 cm and 72 cm) are the legs (base and perpendicular).

⇒ Area = (1 ÷ 2) × product of legs = (1 ÷ 2) × 21 × 72.

⇒ Also, Area = (1 ÷ 2) × base (21 cm) × altitude (h).

⇒ (1 ÷ 2) × 21 × 72 = (1 ÷ 2) × 21 × h.

⇒ h = 72 cm.

∴ The correct answer is 72 cm.

Additional Information

Pythagorean Triplets

Sets of three positive integers a, b, and c, such that a² + b² = c². Common triplets include (3, 4, 5), (5, 12, 13), and (7, 24, 25).

Altitude of a Right Triangle

The two legs of a right-angled triangle act as altitudes to each other. The altitude to the hypotenuse can be found using: (Leg₁ × Leg₂) ÷ Hypotenuse.

Heron's Formula

For any triangle with sides a, b, and c, Area = √(s × (s − a) × (s − b) × (s − c)), where semi-perimeter s = (a + b + c) ÷ 2.

Shortcut Trick

Age of new student = New Average + (Initial count of students × Increase in average)

⇒ Increase in Average Age = 18 − 17 = 1 year

⇒ Age = 18 + (7 × 1) = 25 years

∴ The correct answer is 25 years.

Alternate Method

Given:

Initial number of students = 7

Initial average age = 17 years

New average age = 18 years

Formula Used:

Total Age = Average Age × Number of Students

Calculations:

⇒ Total age of 7 students = 7 × 17 = 119 years

⇒ Total number of students after 1 joins = 8

⇒ Total age of 8 students = 8 × 18 = 144 years

⇒ Age of new student = Total age of 8 students − Total age of 7 students

⇒ Age = 144 − 119

⇒ Age = 25 years

∴ The correct answer is 25 years.

Additional Information

Basic Average Formula

The Average of a set of numbers is the sum of the values divided by the total number of values: Average = (Sum of observations) ÷ (Total number of observations).

Effect of Adding a Constant

If a constant value 'k' is added to each observation in a data set, the original average also increases by exactly 'k'.

Weighted Average

When two groups with different averages are combined, the combined average is (n₁x₁ + n₂x₂) ÷ (n₁ + n₂), where n is the count and x is the average of each group.

Shortcut Trick

The LCM of 12, 15, and 20 is 60.

The smallest 5-digit number is 10000.

Dividing 10000 by 60 leaves a remainder of 40.

Smallest 5-digit multiple = 10000 + (60 − 40) = 10020.

∴ The correct answer is 10020.

Alternate Method

Given: The numbers are 12, 15, and 20. We need to find the smallest 5-digit number divisible by all three.

Formula Used: Required Number = Smallest 5-digit number + (LCM − Remainder)

⇒ Prime factorization of divisors:

⇒ 12 = 2 × 2 × 3 = 2² × 3

⇒ 15 = 3 × 5

⇒ 20 = 2 × 2 × 5 = 2² × 5

⇒ LCM = 2² × 3 × 5 = 4 × 3 × 5 = 60

⇒ Smallest 5-digit number = 10000

⇒ Dividing 10000 by 60: 10000 = (166 × 60) + 40

⇒ Remainder = 40

⇒ Number to be added = 60 − 40 = 20

⇒ Required number = 10000 + 20 = 10020

∴ The correct answer is 10020.

Additional Information

LCM (Least Common Multiple)

The smallest positive integer that is perfectly divisible by a given set of numbers. It is found by taking the highest power of all prime factors involved.

Divisibility by 60

A number is divisible by 60 if it satisfies the divisibility rules of 3 (sum of digits), 4 (last two digits), and 5 (ends in 0 or 5) simultaneously.

Smallest N-digit Divisible Number

To find the smallest N-digit number divisible by x, use: 10^N−1 + (x − Remainder(10^N−1 ÷ x)).

Shortcut Trick

Ratio of volumes of two cylinders: V₁/V₂ = (r₁/r₂)² × (h₁/h₂)

V₁ = 317900 × (3/5)² × (21/14)

V₁ = 317900 × (9/25) × (3/2) = 317900 × (27/50)

V₁ = 6358 × 27 = 171666 cm³

∴ The correct answer is 171666 cm³.

Alternate Method

Given:

Height of first cylinder (h₁) = 21 cm

Volume of second cylinder (V₂) = 317900 cm³

Height of second cylinder (h₂) = 14 cm

Radius of first cylinder (r₁) = 3/5 × radius of second cylinder (r₂)

Formula Used: Volume of cylinder = πr²h

Calculations:

⇒ Volume of second cylinder (V₂) = π × r₂² × h₂

⇒ 317900 = (22/7) × r₂² × 14

⇒ 317900 = 22 × r₂² × 2

⇒ 317900 = 44 × r₂²

⇒ r₂² = 317900 ÷ 44 = 7225

⇒ r₂ = √7225 = 85 cm

⇒ Radius of first cylinder (r₁) = 3/5 × 85 = 3 × 17 = 51 cm

⇒ Volume of first cylinder (V₁) = π × r₁² × h₁

⇒ V₁ = (22/7) × (51)² × 21

⇒ V₁ = 22 × 2601 × 3

⇒ V₁ = 66 × 2601 = 171666 cm³

∴ The correct answer is 171666 cm³.

Additional Information

Curved Surface Area of Cylinder

The area of the curved surface is given by the formula CSA = 2πrh.

Total Surface Area of Cylinder

The sum of the areas of the curved surface and the two circular bases: TSA = 2πr(h + r).

Volume and Ratio Property

If the radius and height of two cylinders are in the ratio r₁:r₂ and h₁:h₂, then the ratio of their volumes is (r₁/r₂)² × (h₁/h₂).

Shortcut Trick

Find the percentage difference: 45% − 20% = 25%.

The question states that 25% of the number is 36.

To get the total number (100%), multiply by 4 (since 25% × 4 = 100%).

Result = 36 × 4 = 144.

∴ The correct answer is 144.

Alternate Method

Given:

First percentage = 45%

Second percentage = 20%

Difference in values = 36

Calculations:

⇒ Let the number be x.

⇒ According to the problem: 45% of x − 20% of x = 36

⇒ (45 − 20)% of x = 36

⇒ 25% of x = 36

⇒ (25 ÷ 100) × x = 36

⇒ (1 ÷ 4) × x = 36

⇒ x = 36 × 4

⇒ x = 144

∴ The correct answer is 144.

Additional Information

Percentage Formula

The value of P% of a number N is calculated as (P × N) ÷ 100.

Finding the Base Number

If x% of a number is V, the original number is (V ÷ x) × 100.

Common Fraction Equivalents

Knowing common fractions like 20% = 1/5, 25% = 1/4, and 50% = 1/2 helps in faster calculation during competitive exams.

Shortcut Trick

Tap B (fills in 9 hr) and Tap C (empties in 9 hr) have equal rates and opposite effects.

Their net Efficiency is zero, meaning only Tap A effectively fills the cistern.

Tap A takes 4 hr to fill the complete cistern.

Time to fill one-fourth of the cistern = 4 hr × 1/4 = 1 hr.

∴ The correct answer is 1 hr.

Alternate Method

Given:

Time taken by Tap A = 4 hr

Time taken by Tap B = 9 hr

Time taken by Tap C (Emptying) = 9 hr

Formula Used:

Efficiency = Total Work ÷ Time

Calculations:

⇒ Total Efficiency of all taps = 9 + 4 − 4 = 9 units/hr.

⇒ Target to fill one-fourth of the cistern = 36 ÷ 4 = 9 units.

⇒ Time required = 9 units ÷ 9 units/hr = 1 hr.

∴ The correct answer is 1 hr.

Additional Information

Inlet and Outlet Pipes

An inlet pipe fills the tank (positive efficiency), while an outlet pipe empties the tank (negative efficiency).

Net Efficiency

When multiple pipes work together, the net filling or emptying rate is the algebraic sum of their individual efficiencies.

LCM Method

The capacity of the tank is usually assumed to be the Least Common Multiple (LCM) of the individual time periods given in the problem.

Shortcut Trick

Price after 40% discount = 60% of 1200 = ₹720.

Final Sale Price = ₹520.

Additional Discount value = 720 − 520 = ₹200.

Additional Discount % = (200 ÷ 720) × 100 = (5 ÷ 18) × 100 = 250/9% = 27(7)/(9)%.

∴ The correct answer is 27(7)/(9)%.

Alternate Method

Given:

Marked Price (MP) = ₹1,200

First Discount = 40%

Final Sale Price (SP) = ₹520

Formula:

Price after discount = MP × (100 − Discount%) ÷ 100

Calculations:

⇒ Price after 40% discount = 1200 × (1 − 0.40)

⇒ Price after 40% discount = 1200 × 0.60 = ₹720

⇒ Total Additional Discount needed = 720 − 520 = ₹200

⇒ Additional Discount % = (Additional Discount ÷ Mid Price) × 100

⇒ Additional Discount % = (200 ÷ 720) × 100

⇒ Additional Discount % = (5 ÷ 18) × 100 = 500/18%

⇒ Additional Discount % = 250/9% = 27 7/9%

∴ The correct answer is 27(7)/(9)%.

Additional Information

Successive Discounts

When two discounts a% and b% are given, the equivalent single discount is calculated as: [a + b − (ab ÷ 100)]%.

Marked Price (MP)

The price printed on the product. Discounts are always calculated on the Marked Price unless stated otherwise.

Selling Price (SP)

The actual price at which an item is sold after all discounts. SP = MP − Discount.

Given:

$\sqrt{\sqrt{\sqrt{256} + \sqrt{144} + (2\sqrt{2})^2} + \sqrt{841} \times (\sqrt{2})^2}$

Formula used:

√x² = x

(a√b)² = a² × b

Calculations:

⇒ $\sqrt{\sqrt{16 + 12 + 8} + 29 \times 2}$

⇒ $\sqrt{\sqrt{36} + 58}$

⇒ $\sqrt{6 + 58}$

⇒ $\sqrt{64}$

⇒ 8

Now, checking the options:

A. (√2)² = 2

B. (2√2)² = 4 × 2 = 8

C. (4√2)² = 16 × 2 = 32

D. (3√2)² = 9 × 2 = 18

∴ The correct answer is option (2).

Shortcut Trick

Effective rate of Compound Interest for 2 years at 2% = 2 + 2 + (2 × 2)/100 = 4.04%.

Given Compound Interest (4.04%) = ₹404.

⇒ Principal (y) = (404 ÷ 4.04) × 100 = ₹10,000.

Total Amount = Principal + Interest = 10,000 + 404 = ₹10,404.

∴ The correct answer is ₹10,404.

Alternate Method

Given: Interest (CI) = ₹404, Rate (r) = 2% per annum, Time (n) = 2 years.

Formula Used: Amount (A) = P(1 + r/100)ⁿ and CI = A − P

⇒ CI = y[(1 + 2/100)² − 1]

⇒ 404 = y[(51/50)² − 1]

⇒ 404 = y[2601/2500 − 1]

⇒ 404 = y[101/2500]

⇒ y = (404 × 2500) ÷ 101

⇒ y = 4 × 2500 = 10,000

⇒ Total Amount (A) = Principal (y) + Interest

⇒ Total Amount (A) = 10,000 + 404 = 10,404

∴ The correct answer is ₹10,404.

Additional Information

Effective Percentage Rate

For two successive changes of a% and b%, the effective change is (a + b + ab/100)%.

CI vs SI Difference

The difference between Compound Interest and Simple Interest for 2 years is P(r/100)².

Amount Property

In Compound Interest, the Total Amount of the previous year becomes the Principal for the next year.

Shortcut Trick

A natural number is a positive integer starting from 1 (1, 2, 3, ...).

Evaluate the expression 2x + 1 sequentially:

For x = 1, 2, 3 → 2x + 1 = 3, 5, 7 (all are prime numbers)

For x = 4 → 2(4) + 1 = 9 (composite number, not prime)

∴ The correct answer is 4.

Alternate Method

Given:

An algebraic expression: 2x + 1

We need to find the smallest natural number x such that 2x + 1 is not a prime number.

Definition and Method:

A natural number belongs to the set {1, 2, 3, 4, ...}.

A prime number is a positive integer greater than 1 that has no positive divisors other than 1 and itself.

We substitute sequential natural numbers for x into the expression and check for primality.

Calculations:

⇒ For x = 1:

⇒ 2(1) + 1 = 3 (Prime number)

⇒ For x = 2:

⇒ 2(2) + 1 = 5 (Prime number)

⇒ For x = 3:

⇒ 2(3) + 1 = 7 (Prime number)

⇒ For x = 4:

⇒ 2(4) + 1 = 9 (Composite number, since 9 = 3 × 3)

Since 9 is divisible by 3, it is not a prime number. Hence, the smallest natural number satisfying the given condition is x = 4.

∴ The correct answer is 4.

Additional Information

Prime Numbers

A prime number is a positive integer greater than 1 that has exactly two distinct positive divisors: 1 and itself. Examples include 2, 3, 5, 7, 11, etc.

Composite Numbers

A composite number is a positive integer greater than 1 that has more than two distinct positive divisors. For example, 4, 6, 8, 9, 10 are composite numbers.

Natural Numbers

Natural numbers are the set of positive integers starting from 1, i.e., {1, 2, 3, 4, ...}. Note that 0 is not considered a natural number in this context.

Primality of 2x + 1

For any integer x, the expression 2x + 1 always represents an odd number. Thus, we are looking for the smallest odd composite number greater than 1, which is 9.

Shortcut Trick

When the sum (S) and difference (D) of two ages are known:

Son's age (Smaller value) = (Sum − Difference) ÷ 2

Son's age = (56 − 24) ÷ 2

Son's age = 32 ÷ 2 = 16 years

∴ The correct answer is 16 years.

Alternate Method

Given:

⇒ Sum of Father and Son's ages = 56 years

⇒ Difference between Father and Son's ages = 24 years

⇒ Let the age of the son be x years.

⇒ According to the question, the father is 24 years older than the son.

⇒ Father's age = (x + 24) years

⇒ Sum of their ages: x + (x + 24) = 56

⇒ 2x + 24 = 56

⇒ 2x = 56 − 24

⇒ 2x = 32

⇒ x = 32 ÷ 2

⇒ x = 16

∴ The correct answer is 16 years.

Additional Information

Linear Equations in Ages

Age problems are typically solved by forming a linear equation where the unknown age is represented by a variable like 'x'.

Constant Age Difference

The difference between the ages of two individuals never changes over time. If the father is 24 years older now, he will always be 24 years older.

Sum and Difference Rule

If (A + B) = S and (A − B) = D, then the larger value A = (S + D) ÷ 2 and the smaller value B = (S − D) ÷ 2.

Shortcut Trick

Error in total sum = 37 − 22 = 15 marks.

Given average marks increase = 5/4 = 1.25.

Number of students = Total Increase in Sum ÷ Increase in Average.

⇒ 15 ÷ 1.25 = (15 × 4) ÷ 5 = 12.

∴ The correct answer is 12.

Alternate Method

Given:

Incorrect entry = 37

Correct entry = 22

Average increase = 5/4

Formula used:

Average = Total Sum of Marks ÷ Number of Students

Total Increase in Sum = Number of Students × Increase in Average

Calculations:

⇒ Net effect on total sum = 37 − 22 = 15.

⇒ Let the total number of students be n.

⇒ (New Sum − Old Sum) ÷ n = 5/4

⇒ 15 ÷ n = 5/4

⇒ 5 × n = 15 × 4

⇒ 5 × n = 60

⇒ n = 60 ÷ 5 = 12.

∴ The correct answer is 12.

Additional Information

Average Property (Constant Change)

If every value in a data set is increased or decreased by a constant k, the average also increases or decreases by k.

Relationship between Sum and Average

The change in total sum is always equal to the number of terms × the change in average (i.e., ΔSum = n × ΔAvg).

Correction of Errors

Correct Average = [Incorrect Sum + (Correct Value − Incorrect Value)] ÷ Total Number of Items.

Shortcut Trick

Ratio of numbers = 19 : 7 : 13

Since 19, 7, and 13 are all prime numbers, they are co-prime to each other.

For co-prime ratios, LCM = (Product of Ratios) × HCF

8645 = (19 × 7 × 13) × HCF ⇒ 8645 = 1729 × HCF

HCF = 8645 ÷ 1729 = 5

∴ The correct answer is 5.

Alternate Method

Given:

Ratio of three numbers = 19 : 7 : 13

Least Common Multiple (LCM) = 8645

Formula Used:

Let the HCF be x. Then the numbers are ax, bx, and cx.

LCM of (ax, bx, cx) = x × LCM(a, b, c)

Calculations:

⇒ Let the three numbers be 19x, 7x, and 13x, where x is the HCF.

⇒ Since 19, 7, and 13 are prime numbers, their LCM is the product of these numbers and their common factor x.

⇒ LCM = 19 × 7 × 13 × x

⇒ 8645 = 1729 × x

⇒ x = 8645 ÷ 1729

⇒ x = 5

Therefore, the Highest Common Factor (HCF) of the numbers is 5, which corresponds to Option 4.

∴ The correct answer is 5.

Additional Information

Relationship between HCF and LCM

For any two numbers a and b, the product of the numbers is equal to the product of their HCF and LCM: a × b = HCF(a, b) × LCM(a, b).

HCF of Co-prime Numbers

The HCF of two or more co-prime numbers (numbers that have no common factor other than 1) is always 1.

LCM and HCF of Fractions

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

Shortcut Trick

Use Option Verification to identify the number satisfying both conditions.

If the larger number is 9, then the smaller number is 9 − 8 = 1.

Check the second condition: (2 × 9) − (5 × 1) = 18 − 5 = 13.

The result matches the condition given in the problem.

∴ The correct answer is 9.

Alternate Method

Given:

Difference between numbers = 8

Result of (2 × Larger − 5 × Smaller) = 13

Formula Used:

Method of Linear Equations using Substitution.

Calculations:

⇒ Let the larger number be x and the smaller number be y.

⇒ From the first condition: x − y = 8 → y = x − 8

⇒ From the second condition: 2x − 5y = 13

⇒ Substitute the value of y in the second equation:

⇒ 2x − 5(x − 8) = 13

⇒ 2x − 5x + 40 = 13

⇒ −3x = 13 − 40

⇒ −3x = −27

⇒ x = −27 ÷ (−3) = 9

∴ The correct answer is 9.

Additional Information

Linear Equations in One Variable

Equations where the highest power of the variable is 1, generally expressed as ax + b = 0.

Substitution Method

A technique to solve simultaneous equations by expressing one variable in terms of another and substituting it into the other equation.

Verification Technique

Fastest method for objective exams where you plug option values into the given constraints to see which one satisfies all conditions.

Shortcut Trick

Equate the total work done in both cases: (15M + 19W) × 8 days = (9M + 4W) × 14 days.

Divide both sides by 2: (15M + 19W) × 4 = (9M + 4W) × 7.

Expand and group terms: 60M + 76W = 63M + 28W ⇒ 3M = 48W.

Thus, 1 man is equivalent to 16 women.

∴ The correct answer is 16.

Alternate Method

Given:

Case 1: (15 men + 19 women) can do the work in 8 days.

Case 2: (9 men + 4 women) can do the same work in 14 days.

Formula Used:

Total Work = Efficiency × Time

Calculations:

⇒ 15M + 19W = 7 ... (i)

⇒ 9M + 4W = 4 ... (ii)

⇒ To find the ratio 1M/1W, use cross-multiplication or elimination.

⇒ From (ii), 4W = 4 − 9M.

⇒ Substitute into (i) by multiplying (i) by 4: 60M + 19(4W) = 28.

⇒ 60M + 19(4 − 9M) = 28.

⇒ 60M + 76 − 171M = 28.

⇒ 111M = 48 ⇒ M = 48/111 = 16/37.

⇒ Substituting M in 4W = 4 − 9M ⇒ 4W = 4 − 9(16/37) = (148 − 144)/37 = 4/37.

⇒ W = 1/37.

⇒ Ratio M/W = (16/37) ÷ (1/37) = 16.

⇒ 1 man = 16 women.

∴ The correct answer is 16.

Additional Information

Work-Efficiency Relation

Efficiency (E) is inversely proportional to Time (T). For the same work, E₁ × T₁ = E₂ × T₂.

Man-Day Concept

Total work is often measured in man-days, calculated as (Number of persons × Number of days taken).

Composite Efficiency

When different types of workers (men, women, children) work together, their individual efficiencies are summed to find the group's total efficiency.

Shortcut Trick

Net Profit Ratio = (True Weight ÷ Faulty Weight) × (Markup Factor) × (Discount Factor)

Net Ratio = (1000 ÷ 960) × (120 ÷ 100) × (90 ÷ 100)

Net Ratio = (25/24) × (6/5) × (9/10) = 45/40 = 9/8

Profit % = [(9 − 8) ÷ 8] × 100 = 1/8 × 100 = 12.5%

∴ The correct answer is 12.5% profit.

Alternate Method

Given:

Cost Price (CP) per kg = ₹48

Markup percentage = 20%

Discount percentage = 10%

Faulty weight used = 960 g instead of 1000 g

Calculations:

⇒ Marked Price (MP) for 1000 g = 48 + (20 ÷ 100) × 48 = 48 + 9.6 = ₹57.6

⇒ Selling Price (SP) for 1000 g = 57.6 − (10 ÷ 100) × 57.6 = 57.6 − 5.76 = ₹51.84

⇒ The shopkeeper delivers only 960 g, so his actual cost is for 960 g.

⇒ Actual CP for 960 g = (48 ÷ 1000) × 960 = 0.048 × 960 = ₹46.08

⇒ Total Profit = SP for 1000 g (received from customer) − Actual CP for 960 g

⇒ Total Profit = 51.84 − 46.08 = ₹5.76

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

⇒ Profit % = (5.76 ÷ 46.08) × 100 = 1/8 × 100 = 12.5%

∴ The correct answer is 12.5% profit.

Additional Information

Dishonest Dealer Formula

Profit % = [ (True Weight − False Weight) ÷ False Weight ] × 100 when selling at cost price.

Successive Discounts/Markups

Effective change of two percentages x and y is given by the formula: [ x + y + (xy ÷ 100) ].

General Profit Relation

Selling Price (SP) = CP × (100 + Profit%) ÷ 100 = MP × (100 − Discount%) ÷ 100.

Shortcut Trick

Rate = 5% = 1/20 → Principal : Amount ratio for 1 year = 20 : 21.

For 3 years, Ratio = (20)³ : (21)³ = 8000 : 9261.

Given 8000 units = ₹4,000 ⇒ 1 unit = ₹0.5.

Total Amount = 9261 × 0.5 = ₹4,630.50.

∴ The correct answer is ₹4,630.50.

Alternate Method

Given:

Principal (P) = ₹4,000

Rate (R) = 5% per annum

Time (n) = 3 years

Formula Used:

Amount (A) = P(1 + R/100)ⁿ

Calculations:

⇒ A = 4000 × (1 + 5/100)³

⇒ A = 4000 × (1 + 1/20)³

⇒ A = 4000 × (21/20)³

⇒ A = 4000 × (9261 / 8000)

⇒ A = 9261 / 2

⇒ A = ₹4,630.50

∴ The correct answer is ₹4,630.50.

Additional Information

Compound Interest (CI) Formula

CI = P[(1 + R/100)ⁿ − 1], where P is Principal, R is Rate, and n is Time.

Difference between CI and SI for 3 Years

Difference = P(R/100)² × (3 + R/100). This formula helps find the gap between simple and compound interest quickly.

Successive Percentage Method

The effective rate for 5% over 3 years is calculated as [5 + 5 + (5×5)/100] = 10.25% for 2 years, then combining 10.25% and 5% to get 15.7625% for 3 years.

Shortcut Trick

Use the Value Substitution Method by assuming A = 90°.

At A = 90°, the values are sin 90° = 1 and cos 90° = 0.

Substitute these into the expression: (1²) / (1 − 0) + (1²) / (1 + 0).

⇒ 1 / 1 + 1 / 1 = 2.

∴ The correct answer is 2.

Alternate Method

Given: (sin² A) / (1 − cos A) + (sin² A) / (1 + cos A)

Formula Used: sin² A + cos² A = 1 ⇒ sin² A = 1 − cos² A

⇒ Factor out sin² A: sin² A [ 1 / (1 − cos A) + 1 / (1 + cos A) ]

⇒ Take the LCM inside the bracket: sin² A [ (1 + cos A + 1 − cos A) / ((1 − cos A)(1 + cos A)) ]

⇒ Simplify the numerator and apply (a−b)(a+b) = a²−b²: sin² A [ 2 / (1 − cos² A) ]

⇒ Substitute 1 − cos² A = sin² A: sin² A [ 2 / sin² A ]

⇒ 2

∴ The correct answer is 2.

Additional Information

Fundamental Trigonometric Identities

The three primary identities are sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ.

Algebraic Manipulations

Trigonometric expressions often utilize a² − b² = (a − b)(a + b), such as 1 − cos² A = (1 − cos A)(1 + cos A).

Substitution Method

In objective exams, substituting standard angles like 0°, 30°, 45°, 60°, or 90° can solve complex identities rapidly, provided the expression remains defined.

Given:

{(3³ + 5²) × 2³ − 4³} ÷ (7 − 3)

Formula used:

BODMAS rule (Brackets, Orders/Exponents, Division, Multiplication, Addition, Subtraction)

Calculations:

⇒ {(27 + 25) × 8 − 64} ÷ (7 − 3)

⇒ {52 × 8 − 64} ÷ 4

⇒ {416 − 64} ÷ 4

⇒ 352 ÷ 4

⇒ 88

∴ The correct answer is option (3).

Shortcut Trick

The Buy 5, Get 2 Free offer implies that out of every 7 items, 5 are paid and 2 are free.

Effective Price = [Paid Items ÷ Total Items] × Price per Item

Effective Price = [5 ÷ (5 + 2)] × 300 = (5 ÷ 7) × 300

Effective Price = 1500 ÷ 7 = ₹214.285...

∴ The correct answer is ₹214.29.

Alternate Method

Given:

Price of each item (MRP) = ₹300

Scheme = Buy 5, Get 2 Free

Total items needed = 21 items

⇒ Total items in one complete offer set = 5 + 2 = 7 items

⇒ Number of such sets required to get 21 items = 21 ÷ 7 = 3 sets

⇒ Total items paid for by the customer = 3 sets × 5 items/set = 15 items

⇒ Total cost paid = 15 items × ₹300 = ₹4500

⇒ Total items received = 21 items

⇒ Effective cost per item = Total cost paid ÷ Total items received

⇒ Effective cost = 4500 ÷ 21 = 1500 ÷ 7

⇒ Effective cost = 214.2857... ≈ ₹214.29

Therefore, the correct option is 2.

∴ The correct answer is ₹214.29.

Additional Information

Discount Percentage Formula

In a 'Buy X, Get Y Free' offer, the Discount % = [Y ÷ (X + Y)] × 100.

Effective Price with Discount

The Effective Cost Price can also be calculated as: Marked Price × (1 − Discount Rate). Here, Discount Rate = 2/7.

Successive Discounts

If an additional cash discount is given on top of a 'Buy X, Get Y' offer, the net discount is calculated using the successive percentage formula: d_net = d₁ + d₂ − (d₁ × d₂)/100.

Shortcut Trick

Ratio of investments of A : B : C = 1960 : 1660 : 1130 = 196 : 166 : 113.

Difference in profit shares of A and C = 196 − 113 = 83 units.

Given, B's share = 166 units = ₹968.

Since 83 is exactly half of 166, the required difference = 968 ÷ 2 = ₹484.

∴ The correct answer is ₹484.

Alternate Method

Given:

Investment of A = ₹1,960

Investment of B = ₹1,660

Investment of C = ₹1,130

Profit share of B = ₹968

Formula Used:

In a partnership, Profit Ratio = Investment Ratio × Time Period (where Time is same for all).

Calculations:

⇒ Ratio of profits of A : B : C = 1960 : 1660 : 1130

⇒ Ratio = 196 : 166 : 113

⇒ Let the profits be 196x, 166x, and 113x respectively.

⇒ 166x = 968

⇒ x = 968 ÷ 166 = 484 ÷ 83

⇒ Difference in profit earned by A and C = 196x − 113x = 83x

⇒ Difference = 83 × (484 ÷ 83)

⇒ Difference = ₹484

∴ The correct answer is ₹484.

Additional Information

Partnership Profit Sharing

Profit is distributed among partners in the ratio of (Investment₁ × Time₁) : (Investment₂ × Time₂). If the time period is equal, profit depends solely on the investment ratio.

Simple vs. Compound Partnership

In a simple partnership, all partners invest for the same duration. In a compound partnership, the duration of investments varies for different partners.

Active and Sleeping Partners

An active partner manages the business and may receive a fixed salary or a percentage of the profit before the remaining profit is distributed according to investment ratios.

Shortcut Trick

Relative Speed (Same Direction) = 54 − 36 = 18 km/h

Convert to m/s: 18 × (5/18) = 5 m/s

Total Distance = Sum of lengths = 90 + 75 = 165 m

Time taken = Total Distance ÷ Relative Speed = 165 ÷ 5 = 33 sec

∴ The correct answer is 33 sec.

Alternate Method

Given:

Length of Train A (L₁) = 90 m | Speed of Train A (V₁) = 54 km/h

Length of Train B (L₂) = 75 m | Speed of Train B (V₂) = 36 km/h

Formula Used:

Relative Speed (Same Direction) = V₁ − V₂

Time = (L₁ + L₂) ÷ Relative Speed

Calculations:

⇒ Total distance to be covered = 90 + 75 = 165 m

⇒ Relative speed of trains = (54 − 36) km/h = 18 km/h

⇒ Converting km/h to m/s: 18 × (5/18) m/s = 5 m/s

⇒ Time taken to overtake = Total distance ÷ Relative speed

⇒ Time = 165 ÷ 5 = 33 seconds

Therefore, Option (2) is the correct answer.

∴ The correct answer is 33 sec.

Additional Information

Relative Speed (Opposite Direction)

When two objects move towards each other, their relative speed is the sum of their individual speeds: V_rel = V₁ + V₂.

Speed Conversion Unit

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

Train Overtaking/Crossing a Point

If a train crosses a stationary pole or a person, the distance covered is equal to the length of the train only.

Shortcut Trick

Net monthly savings = 17,000 − (4,000 + 2,500) = ₹10,500.

Since the savings of the birthday month is spent entirely, effective saving occurs for 11 months.

Total annual savings = 10,500 × 11 = ₹1,15,500.

∴ The correct answer is ₹1,15,500.

Alternate Method

Given:

Monthly salary of Chetan = ₹17,000

Monthly house rent = ₹4,000

Monthly bills = ₹2,500

Formula Used:

Monthly savings = Monthly salary − Total expenditure

Annual savings = Monthly savings × Number of months saved

Calculations:

⇒ Total monthly expenditure = 4,000 + 2,500 = ₹6,500

⇒ Monthly savings = 17,000 − 6,500 = ₹10,500

⇒ In the birthday month, savings = ₹0 (as he spent the complete amount for celebration)

⇒ Total months in a year = 12

⇒ Effective months for savings = 12 − 1 = 11 months

⇒ Total savings in a year = 10,500 × 11

⇒ Total savings = ₹1,15,500

∴ The correct answer is ₹1,15,500.

Additional Information

Income and Savings Relationship

Income is always the sum of expenditure and savings (Income = Expenditure + Savings). Any increase in expenditure without a change in income directly reduces savings.

Annual Calculation Adjustments

When calculating annual values from monthly data, verify if any specific months have deviations or zero values due to special events like a birthday celebration.

Savings Percentage

The savings percentage can be calculated as (Monthly Savings ÷ Monthly Salary) × 100. In this case: (10,500 ÷ 17,000) × 100 ≈ 61.76%.

Shortcut Trick

Ratio of Cost Price (CP) to Selling Price (SP) = 1000 : (20 × 60)

CP : SP = 1000 : 1200 = 5 : 6

Profit = 6 − 5 = 1 unit

Profit % = (1 ÷ 5) × 100 = 20%

∴ The correct answer is 20%.

Alternate Method

Given:

Quantity of rice = 20 kg

Cost Price (CP) of 20 kg rice = ₹1000

Selling Price (SP) of 1 kg rice = ₹60

Formula Used:

Profit = Total SP − Total CP

Profit % = (Profit ÷ CP) × 100

Calculations:

⇒ Total Selling Price (SP) = 20 × 60 = ₹1200

⇒ Profit = 1200 − 1000 = ₹200

⇒ Profit % = (200 ÷ 1000) × 100

⇒ Profit % = 20%

∴ The correct answer is 20%.

Additional Information

Loss Percentage

When CP > SP, a loss occurs. Loss % = (Loss ÷ CP) × 100, where Loss = CP − SP.

Marked Price & Discount

Discount is a reduction offered on the Marked Price (MP). SP = MP − Discount.

Profit Margin on SP

Sometimes profit is calculated on SP for business analysis. Profit % on SP = (Profit ÷ SP) × 100.

Shortcut Trick

Volume of shell = 4/3 × π × (R³ − r³) = 4/3 × 22/7 × (4³ − 1³)

⇒ Volume = 4/3 × 22/7 × 63 = 4 × 22 × 3 = 264 cm³

Mass = Density × Volume = 10 × 264 = 2640 g

Weight in kg = 2640 ÷ 1000 = 2.64 kg

∴ The correct answer is 2.64 kg.

Alternate Method

Given:

Metal density (ρ) = 10 g/cm³

Internal radius (r) = 1 cm

External radius (R) = 4 cm

Formula Used:

Volume of hollow sphere = 4/3 × π × (R³ − r³)

Mass = Density × Volume

Calculations:

⇒ Volume = 4/3 × 22/7 × (4³ − 1³)

⇒ Volume = 4/3 × 22/7 × (64 − 1)

⇒ Volume = 4/3 × 22/7 × 63

⇒ Volume = 4 × 22 × 3 = 264 cm³

⇒ Mass = 10 × 264 = 2640 g

⇒ Weight (in kg) = 2640 ÷ 1000 = 2.64 kg

∴ The correct answer is 2.64 kg.

Additional Information

Volume of a Solid Sphere

The volume of a complete solid sphere is given by V = 4/3 × π × r³.

Surface Area of a Sphere

The total surface area of a sphere is 4 × π × r², and for a hollow shell, the total surface area is 4πR² + 4πr².

Unit Conversion

Always ensure units are consistent; 1 kg = 1000 g and 1 m³ = 10⁶ cm³.

Shortcut Trick

The Ratio of Rice to Water is 2 : 3.

Current Rice quantity = 6 cups, which is 3 times the original amount (6 ÷ 2 = 3).

To maintain the same proportion, multiply the Water quantity by the same factor: 3 × 3 = 9 cups.

∴ The correct answer is 9 cups.

Alternate Method

Given:

Initial Rice (R₁) = 2 cups

Initial Water (W₁) = 3 cups

New Rice (R₂) = 6 cups

Formula Used:

Direct Proportion: R₁ / W₁ = R₂ / W₂

Calculations:

⇒ 2 / 3 = 6 / W₂

⇒ 2 × W₂ = 6 × 3

⇒ 2 × W₂ = 18

⇒ W₂ = 18 ÷ 2

⇒ W₂ = 9

∴ The correct answer is 9 cups.

Additional Information

Direct Proportion

Two quantities are in direct proportion if an increase in one leads to a corresponding increase in the other such that their ratio (x/y = k) remains constant.

Unitary Method

To solve using the unitary method, first find the amount of water needed for 1 cup of rice (3 ÷ 2 = 1.5) and then multiply by the desired amount (1.5 × 6 = 9).

Scaling Factor

The scaling factor is the constant multiplier used to increase or decrease both parts of a ratio simultaneously while maintaining the original relationship.

The correct answer is Random Access Memory.

Key Points

• RAM stands for Random Access Memory, which serves as the primary memory or main memory of a computer system.
• It is called Random Access because any memory cell or storage location can be accessed directly by the CPU in the same amount of time, regardless of its physical position, unlike sequential access devices like magnetic tapes.
• The RAM stores the operating system, application programs, and data currently in use so that they can be quickly reached by the processor.
• It is volatile memory, meaning that all the information stored in RAM is lost as soon as the power supply is switched off or the system is rebooted.
• RAM is a Read/Write memory, which allows the computer to both read existing data and write new data into it at extremely high speeds.

 Additional Information

• Types of RAM:
• SRAM (Static Random Access Memory): It uses transistors to store data and does not need to be refreshed periodically. It is faster and more expensive than DRAM and is primarily used as Cache Memory.
• DRAM (Dynamic Random Access Memory): It stores data using capacitors and transistors. Since capacitors leak charge, DRAM must be refreshed thousands of times per second. It is used as the main system memory.
• SDRAM (Synchronous DRAM): This type synchronizes its speed with the system clock, allowing the CPU to process instructions more efficiently.
• Comparison with ROM:
• Unlike RAM, ROM (Read Only Memory) is non-volatile, meaning it retains its data even after power is lost. It typically contains the BIOS (Basic Input/Output System) necessary for the computer to start up.
• Capacity and Speed:
• Modern RAM capacity is measured in Gigabytes (GB), and its speed is often measured in Megahertz (MHz) or Megatransfers per second (MT/s). Higher capacity and speed allow for better multitasking performance.

The correct answer is Article 133.

Key Points

• Article 133 of the Constitution of India defines the appellate jurisdiction of the Supreme Court in appeals from High Courts specifically regarding civil matters.
• According to this provision, an appeal lies to the Supreme Court from any judgment, decree, or final order in a civil proceeding of a High Court if the High Court certifies under Article 134A that the case involves a substantial question of law of general importance.
• The High Court must also be of the opinion that the said question needs to be decided by the Supreme Court to grant the certificate.
• Prior to the 30^th Constitutional Amendment Act, 1972, civil appeals were allowed if the amount or value of the subject matter of the dispute was not less than 20,000 rupees; however, this monetary requirement was abolished, and the focus shifted entirely to the substantial question of law.
• Under clause (2) of Article 133, any party appealing to the Supreme Court may also argue that a substantial question of law as to the interpretation of the Constitution has been wrongly decided.

 Additional Information

• Article 131:
• Specifies the Original Jurisdiction of the Supreme Court.
• It empowers the court to hear disputes between the Government of India and one or more States, or between two or more States, to the exclusion of any other court.
• Article 132:
• Deals with the appellate jurisdiction of the Supreme Court in appeals from High Courts in certain cases involving constitutional matters.
• It applies if the High Court certifies that the case involves a substantial question of law regarding the interpretation of the Constitution.
• Article 134:
• Delineates the appellate jurisdiction of the Supreme Court in criminal matters.
• It provides for an appeal if a High Court has, on appeal, reversed an order of acquittal and sentenced an accused to death, or has withdrawn a trial to itself and awarded a death sentence.

The correct answer is Palaeolithic Age and Neolithic Age.

Key Points

• The Mesolithic Age, often referred to as the Middle Stone Age, represents a crucial cultural and technological bridge between the Palaeolithic Age (Old Stone Age) and the Neolithic Age (New Stone Age).
• This period is termed transitional because it reflects the shift from the hunter-gatherer lifestyle of the Pleistocene epoch to the food-producing, sedentary societies of the Holocene epoch.
• Geographically and climatically, it coincided with the end of the last Ice Age around 10,000 BCE, leading to warmer climates, increased rainfall, and the expansion of forests and grasslands.
• Technologically, it is defined by the production of microliths, which are very small, geometric stone tools made of chert, chalcedony, or agate, typically measuring 1 to 5 cm in length.
• The economy of the Mesolithic people remained based on hunting, fishing, and gathering, but it also saw the earliest evidence of animal domestication, particularly the dog, which paved the way for the agricultural revolution.
• Prominent Mesolithic sites in India include Bagor in Rajasthan, Adamgarh in Madhya Pradesh, and Langhnaj in Gujarat, providing evidence of both microlithic industries and early burials.

 Additional Information

• Palaeolithic Age:
• The earliest period of human history, characterized by the use of crude, unpolished stone tools like hand-axes and cleavers.
• Humans were entirely nomadic and relied on hunting large animals and gathering wild plants during the Ice Age.
• Neolithic Age:
• Marked by the Neolithic Revolution, where humans began settled agriculture and pottery making.
• Tools became polished and grounded for higher efficiency in farming and construction.
• Chalcolithic Age:
• Known as the Copper-Stone Age, it marked the first use of metals alongside stone tools.
• It serves as the transition between the Neolithic and the Bronze Age.
• Iron Age:
• Characterized by the widespread smelting of iron ore, leading to the creation of stronger tools and weapons.
• In the Indian context, it is often associated with the Painted Grey Ware (PGW) culture and the Vedic period.

The correct answer is Mumbai.

Key Points

• Mumbai is home to the world's second-largest collection of Art Deco buildings, a title it holds globally after Miami, USA. This architectural movement flourished in the city between the 1930s and 1950s, symbolizing the city's transition toward modernity.
• In 2018, the Victorian Gothic and Art Deco Ensembles of Mumbai were collectively inscribed as a UNESCO World Heritage Site. This recognition covers a total of 94 structures, primarily concentrated in the Oval Maidan and Marine Drive precincts.
• The Oval Maidan serves as a unique architectural boundary; its western side is flanked by a seamless row of Art Deco apartment blocks, while the eastern side features 19^th-century Victorian Gothic public buildings, creating a grand urban ensemble.
• The style in Mumbai is often referred to as Indo-Deco, as it blended the sleek, streamlined aesthetic of international Art Deco with Indian motifs and imagery, such as tropical flora and mythological figures.
• As of October 2025, these heritage structures continue to define the city's skyline, with iconic landmarks like Eros Cinema and Regal Cinema serving as prime examples of the movement's cinematic and maritime influences. Therefore, the correct option is (4).

 Additional Information

• Distinct Features of Mumbai Art Deco:
• Nautical Themes: Influenced by Mumbai's status as a port city, many buildings feature porthole windows, curved balconies, and railing designs that mimic ocean liners.
• Tropical Motifs: Buildings often incorporate bas-reliefs depicting palm trees, sunbursts, and local agricultural scenes, grounding the global style in the Indian context.
• UNESCO Significance:
• The ensemble represents the modernization of Mumbai through two distinct phases of urban planning: the Victorian era for public administration and the Art Deco era for residential and commercial expansion.
• The Backbay Reclamation project provided the necessary land for this 20^th-century architectural explosion, turning the waterfront into a modernist corridor.

The correct answer is January 12^th.

Key Points

• National Youth Day, also known as Rashtriya Yuva Diwas, is observed every year on January 12 to celebrate the birth anniversary of Swami Vivekananda, one of India's most influential spiritual leaders.
• The Government of India officially declared this day in 1984, and the first nationwide celebration took place on January 12, 1985, to inspire the youth of the country.
• The primary objective of the day is to propagate the philosophy and ideals of Swami Vivekananda among the Indian youth, encouraging them to contribute to the nation's progress with energy and character.
• Swami Vivekananda was born on January 12, 1863, in Kolkata; his teachings focus on self-reliance, spirituality, and the potential of the youth to transform society.

 Additional Information

• Swami Vivekananda and his Legacy:
• He was the chief disciple of the mystic Ramakrishna Paramahansa and established the Ramakrishna Mission in 1897 to provide social service and spiritual education.
• He gained global recognition after his historic speech at the Parliament of the World's Religions in Chicago in 1893, where he introduced Vedanta and Yoga to the West.
• His works and speeches, such as 'Karma Yoga' and 'Raja Yoga', remain fundamental texts for modern spiritual and philosophical study.
• Other Notable Observances:
• March 20^th: This day is celebrated globally as World Sparrow Day to raise awareness about the conservation of sparrows and as the International Day of Happiness.
• April 30^th: Observed as Ayushman Bharat Diwas in India, highlighting the goals of the Ayushman Bharat scheme to provide health insurance to the underprivileged.

The correct answer is It is an equatorial low-pressure belt where trade winds meet..

Key Points

• The Inter Tropical Convergence Zone (ITCZ) is a permanent low-pressure belt encircling the Earth near the Equator, where the solar insolation is maximum throughout the year.
• It is the region where the Northeast trade winds from the Northern Hemisphere and the Southeast trade winds from the Southern Hemisphere converge, forcing the air to rise into the atmosphere.
• As the converged air mass rises due to convection, it cools adiabatically, leading to the formation of towering cumulonimbus clouds and frequent, intense thunderstorms.
• The ITCZ is not stationary; it migrates North or South depending on the apparent movement of the Sun. For instance, during the Northern summer, it moves over the Indian subcontinent, establishing the Monsoon Trough.
• Due to the vertical movement of air rather than horizontal flow, this zone is often characterized by very light or calm winds, traditionally known to sailors as the Doldrums.

 Additional Information

• Climatic Significance
• Hadley Cell: The ITCZ represents the ascending branch of the Hadley Cell, a major component of the Earth’s atmospheric circulation system.
• Monsoon Connection: The shift of the ITCZ to about 20°N−25°N latitude over India in July is a primary driver for the Southwest Monsoon.
• Precipitation: This belt is responsible for the high annual rainfall found in tropical rainforest regions like the Amazon Basin and the Congo Basin.
• Distinction from other zones
• High-Pressure Belts: Unlike the ITCZ, the Subtropical Highs (Horse Latitudes) and Polar Highs are regions of descending air and high atmospheric pressure.
• Westerlies: These are prevailing winds in the mid-latitudes (between 30° and 60°), which flow towards the poles, unlike the trade winds that converge at the ITCZ.
• Frontal Systems: Cold fronts and warm fronts are features of the Extra-tropical Cyclones in temperate regions, whereas the ITCZ is a thermal, tropical phenomenon.

The correct answer is Joule.

Key Points

• Heat generated in an electrical resistor is a form of thermal energy produced by the collision of electrons with the atoms of the resistive material.
• The International System of Units (SI) identifies the Joule (J) as the standard unit for all forms of energy, work, and heat.
• The relationship between electricity and heat is defined by Joule's Law of Heating, which states that the heat produced (H) in a conductor is equal to the product of the square of the current (I), the resistance (R), and the time (t) for which the current flows.
• The mathematical expression for this law is H = I²Rt, where the resulting energy is measured in Joules.
• One Joule is defined as the work done or energy transferred when a force of one Newton moves an object through a distance of one meter in the direction of the force.

 Additional Information

• Coulomb:
• It is the SI unit of electric charge.
• One Coulomb represents the amount of charge transported by a constant current of one Ampere in one second.
• Watt:
• It is the SI unit of power, which defines the rate at which energy is consumed or converted.
• One Watt is equivalent to one Joule per second (1 W = 1 J/s). While heat is energy (Joules), the rate at which a resistor generates heat is measured in Watts.
• Volt:
• It is the SI unit of electric potential, potential difference, and electromotive force (EMF).
• One Volt is defined as the potential difference across a conductor when a current of one Ampere dissipates one Watt of power.

The correct answer is 1956.

Key Points

• The Industrial Policy Resolution (IPR) of 1956 was adopted by the Indian Parliament in April 1956, replacing the previous 1948 resolution to align with the goal of a socialist pattern of society.
• This resolution is famously regarded as the Economic Constitution of India because it provided the basic framework for the country's industrial strategy for several decades.
• The policy categorized industries into three distinct schedules: Schedule A (17 industries under exclusive state responsibility), Schedule B (12 industries progressively state-owned but supplemented by the private sector), and Schedule C (remaining industries left to the private sector).
• It aimed to accelerate the rate of economic growth and speed up industrialization, specifically focusing on the development of heavy industries and machine-making sectors.
• The IPR 1956 emphasized the Public Sector taking the lead role, ensuring that the government controlled the commanding heights of the economy to prevent the concentration of wealth.
• It provided the theoretical and legal basis for the Second Five-Year Plan, which was built upon the Mahalanobis Model of development.

 Additional Information

• Second Five-Year Plan (1956–1961):
• Formulated by Prasanta Chandra Mahalanobis, it prioritized heavy industrialization over agriculture.
• Focus was placed on iron and steel plants, chemical industries, and mining to build a self-reliant economy.
• Regional Balanced Development:
• The IPR 1956 used industrial licensing as a tool to promote regional equality by offering tax concessions and lower electricity tariffs to industries set up in backward regions.
• Small Scale Industries (SSI):
• In 1955, the Karve Committee (Village and Small-scale Industries Committee) highlighted the potential of small-scale industries for rural development.
• The 1956 resolution protected these industries by reserving products and providing subsidies to generate mass employment.
• Industrial Development and Regulation Act (IDRA), 1951:
• This legislative act provided the government with the necessary powers to regulate and license new and existing industrial units according to the priorities of the 1956 resolution.

The correct answer is 1,027 million.

Key Points

• The Census of India 2001 was the 14^th Census of the country since 1872 and the 6^th Census conducted after Independence.
• The reference moment for the 2001 Census was 00:00 hours of 1^st March 2001, which served as the official date for calculating the nationwide population count.
• As of this reference date, the total population of India was recorded at 1,027,015,247, which is approximately 1,027 million or 1.027 billion.
• This census officially confirmed that India had crossed the one billion mark, making it the second country in the world after China to reach this demographic threshold.
• The population comprised 532.1 million males and 494.8 million females, reflecting the vast human resource and demographic diversity of the nation at the start of the 21^st century.

 Additional Information

• Population Growth and Density:
• The decadal growth rate from 1991 to 2001 was 21.54%, showing a slight decline compared to the previous decade.
• The population density in 2001 was 324 persons per square kilometer, increasing significantly from 267 in 1991.
• The sex ratio was recorded as 933 females per 1,000 males, showing a marginal improvement from 927 in the 1991 Census.
• Literacy and States:
• The literacy rate in 2001 stood at 64.84%, with Kerala maintaining the highest literacy and Uttar Pradesh being the most populous state.
• The subsequent 2011 Census recorded the population at 1,210 million, continuing the trend of substantial demographic expansion.

The correct answer is INS Rana and INS Jyoti.

Key Points

• SLINEX-25 (Sri Lanka-India Naval Exercise 2025) is the 12^th edition of the bilateral maritime exercise conducted between the Indian Navy and the Sri Lanka Navy in August 2025.
• The exercise was structured into two distinct phases: the Harbour Phase held at Colombo and the Sea Phase conducted in the Laccadive Sea off the western coast of Sri Lanka.
• The Indian Navy deployment for this iteration featured INS Rana, a Rajput-class guided-missile destroyer, and INS Jyoti, a fleet replenishment tanker, representing a balanced mix of combat power and logistical support.
• The primary objective of SLINEX-25 was to improve interoperability, strengthen mutual understanding, and exchange best practices in multi-faceted maritime operations including surface firing and tactical maneuvers.
• This exercise aligns with India's SAGAR (Security and Growth for All in the Region) policy and the Neighborhood First initiative, fostering regional maritime security and cooperation in the Indian Ocean Region (IOR).

 Additional Information

• Participating Vessels and Specifics:
• INS Rana (D52): It is the fourth ship of the Rajput-class destroyers. Commissioned in 1982, it is equipped with BrahMos supersonic cruise missiles, Styx anti-ship missiles, and advanced anti-submarine warfare (ASW) capabilities.
• INS Jyoti (A58): A Komandarm Fedko-class replenishment oiler that provides essential logistics like fuel, ammunition, and provisions to the fleet, extending the blue-water reach of the Indian Navy.
• SLNS Sayurala: The Sri Lanka Navy typically participates with its Advanced Offshore Patrol Vessels (AOPV), such as SLNS Sayurala, to conduct joint visit, board, search, and seizure (VBSS) drills.
• Technical Concepts in Exercises:
• Sea Phase Operations: Focuses on complex drills such as asymmetric threat simulations, cross-deck helicopter operations, and coordinated maritime surveillance.
• Harbour Phase Activities: Includes professional interactions, subject matter expert exchanges (SMEE), and sports fixtures to build camaraderie between personnel.
• Other Notable Ships (Incorrect Options):
• INS Vikrant: India's first indigenous aircraft carrier (IAC-1), which serves as the flagship of the Western or Eastern fleets.
• INS Talwar: The lead ship of the Talwar-class stealth frigates, known for its stealth hull design and multi-role versatility.
• INS Kolkata: A Project 15A destroyer featuring the MF-STAR AESA radar and Barak-8 long-range surface-to-air missiles.

The correct answer is Sleep keeps data in RAM; Hibernate saves it to disk.

Key Points

• Sleep Mode is a low-power state that allows a Windows computer to resume full-power operation quickly. It achieves this by maintaining the power supply to the Random Access Memory (RAM), where all open applications, files, and the System State are stored. Because RAM is Volatile Memory, it requires a continuous trickle of electricity to retain data.
• Hibernate Mode is designed specifically to save power over longer durations. Instead of keeping data in the RAM, the Operating System takes a snapshot of the current session and writes it to the Hard Disk Drive (HDD) or Solid State Drive (SSD). Once the data is safely written to the Non-volatile Storage, the computer completely powers down, consuming zero electricity.
• The primary technical difference involves the Storage Medium and Data Persistence. In Sleep, if power is lost (e.g., a battery dies or a plug is pulled), any unsaved data in the RAM is permanently lost. In Hibernate, the data remains safe on the Disk even without power, and the session is restored when the user restarts the machine.
• In terms of Performance, waking from Sleep is significantly faster (often 1−3 seconds) because the CPU simply resumes processing from the existing RAM state. Resuming from Hibernate takes longer (15−60 seconds) because the Windows Boot Manager must read the saved session file from the Disk and reload it into the RAM.

 Additional Information

• Hybrid Sleep
• This feature, primarily found on Desktop PCs, combines Sleep and Hibernate. It writes the session to the Disk but also keeps it in RAM.
• Its primary Key Function is to provide the speed of Sleep while offering the Data Redundancy of Hibernate in case of a sudden power outage.
• hiberfil.sys File
• This is a hidden System File located in the root folder of the drive where the Operating System is installed.
• The size of hiberfil.sys is approximately equal to the amount of Physical RAM installed on the system, as it must accommodate the entire Memory Dump during hibernation.
• Fast Startup
• Introduced in Windows 8 and continuing through Windows 10 and Windows 11, this feature uses Hibernation technology to save the Kernel Session to the disk during a standard shutdown.
• This allows the Operating System to boot much faster by skipping the full hardware initialization and loading the saved Kernel State directly from the Disk.

The correct answer is Software that manages hardware and provides platform for application software.

Key Points

• System software is defined as a type of computer program that is designed to run a computer's hardware and application programs.
• It acts as the primary interface between the physical components of the computer (the hardware) and the user-facing software, ensuring that the machine functions efficiently and reliably.
• The most prominent example of system software is the Operating System (OS), such as Windows, macOS, or Linux, which manages the Central Processing Unit (CPU), Main Memory (RAM), and Input/Output (I/O) devices.
• It provides a necessary runtime environment or platform; without system software, application software like games or web browsers would have no way to communicate instructions to the hardware.
• Beyond the OS, system software includes device drivers, which are specialized programs that allow the OS to interact with specific hardware like Graphic Cards, Printers, and Network Adapters.
• It also encompasses firmware like BIOS (Basic Input/Output System) or UEFI, which initializes the hardware during the booting process before the operating system takes control.
• System software typically runs in the background and is responsible for resource allocation, file management, and memory management tasks that are transparent to the user.

 Additional Information

• Application Software: Unlike system software, application software is designed to perform specific tasks for the end-user.
• Document Creation: Tools like Microsoft Word, LibreOffice Writer, or Notepad are productivity applications used for text manipulation.
• Multimedia Players: Software such as VLC Media Player or Windows Media Player is used specifically for decoding and playing audio and video file formats.
• Web Browsers: Applications like Google Chrome, Mozilla Firefox, and Microsoft Edge are used to access and navigate the World Wide Web.
• Utility Software: Often categorized under system software, these programs help analyze, configure, and maintain the computer system, such as Antivirus software, Disk Cleanup tools, and Compression utilities like WinRAR.
• Programming Language Translators: These include Compilers, Interpreters, and Assemblers that convert high-level programming code into Machine Code that the hardware can execute.

The correct answer is Gender Inequality Index.

Key Points

• The Gender Inequality Index (GII) is a composite metric introduced by the United Nations Development Programme (UNDP) in the 2010 Human Development Report to measure the loss in potential human development due to inequality between female and male achievements.
• It evaluates gender disparity across three fundamental dimensions: Reproductive Health, Empowerment, and Labour Market Participation.
• The Reproductive Health dimension is measured by two indicators: the Maternal Mortality Ratio (MMR) and the Adolescent Birth Rate (ABR).
• The Empowerment dimension is assessed by the share of parliamentary seats held by each gender and the attainment of at least secondary education by both women and men.
• The Labour Market dimension is represented by the labour force participation rate for both female and male populations aged 15 years and older.
• The GII value ranges from 0 to 1, where 0 signifies that women and men fare equally (perfect equality) and 1 signifies that one gender fares as poorly as possible in all measured dimensions (perfect inequality).

 Additional Information

• Comparison with Other Indices:
• Human Development Index (HDI): A summary measure of average achievement in key dimensions of human development, including a long and healthy life, knowledge, and a decent standard of living.
• Gender Development Index (GDI): Introduced in 2014, it measures gender gaps in human development by calculating the ratio of female to male HDI values.
• Multidimensional Poverty Index (MPI): Captures the various deprivations experienced by people in their daily lives, such as poor health, lack of education, and low living standards.
• Global Context:
• The GII helps policy makers understand the opportunity cost of gender inequality and provides a mechanism to monitor progress toward Sustainable Development Goal 5 (SDG 5), which aims for Gender Equality.

The correct answer is 20^th Edition.

Key Points

• The Global Risks Report is a flagship annual publication released by the World Economic Forum (WEF) ahead of its high-profile Annual Meeting in Davos, Switzerland.
• The 20^th Edition of the report was launched in January 2025, marking two full decades of the WEF tracking and analyzing the global risk landscape.
• The 2025 report is based on the results of the Global Risks Perception Survey (GRPS), which captures the insights of more than 1,500 experts across various sectors, including business, government, and civil society.
• The 20^th Edition focuses on the theme of a 'polycrisis,' where interconnected risks such as environmental degradation, technological disruption, and geopolitical instability compound one another.
• The report serves as a critical tool for policy-makers and global leaders to understand the transition from short-term volatility (2-year horizon) to long-term structural shifts (10-year horizon).

 Additional Information

• World Economic Forum (WEF):
• It was established in 1971 by Klaus Schwab as a not-for-profit foundation and is headquartered in Cologny, Geneva, Switzerland.
• The forum is committed to improving the state of the world through public-private cooperation.
• Historical Timeline of the Report:
• The 1^st Edition of the Global Risks Report was published in 2006, which initially focused on economic imbalances and the threat of a global pandemic.
• The 19^th Edition, released in 2024, prominently highlighted misinformation and disinformation as the top short-term global risks.
• The 18^th Edition, released in 2023, focused heavily on the cost-of-living crisis following the post-pandemic inflationary pressures.
• Risk Classification:
• The WEF classifies risks into five categories: Economic, Environmental, Geopolitical, Societal, and Technological.
• In recent editions, Environmental risks like biodiversity loss and climate action failure have consistently ranked as the most severe long-term threats.

The correct answer is 1909.

Key Points

• The Morley-Minto Reforms, officially known as the Indian Councils Act 1909, were enacted by the British Parliament to introduce limited self-governance and increase Indian participation in the legislative process.
• The reforms were named after Lord John Morley, the then Secretary of State for India, and Lord Minto, who served as the Viceroy of India from 1905 to 1910.
• A defining and historically significant feature of the Act was the introduction of separate electorates for Muslims, a system where Muslim representatives were elected exclusively by Muslim voters.
• The Act significantly increased the size of the legislative councils at both the Central and Provincial levels; for instance, the membership of the Central Legislative Council was expanded from 16 to 60.
• It empowered members of the council to ask supplementary questions and move resolutions on the budget, although they could not vote on the entire budget.
• The reforms allowed for the first time the inclusion of an Indian in the Viceroy’s Executive Council, with Satyendra Prasad Sinha being appointed as the Law Member.

 Additional Information

• Background and Evolution of Reforms:
• The reforms were preceded by the Simla Deputation of 1906, where a group of Muslim leaders led by the Aga Khan demanded separate representation to safeguard their political interests.
• Due to the institutionalization of communal representation, Lord Minto is formally recognized in Indian history as the Father of Communal Electorate.
• The political trajectory established in 1909 was followed by the Montagu-Chelmsford Reforms in 1919, which introduced Dyarchy and a bicameral legislature at the center.
• Later, the Government of India Act 1935 further expanded the constitutional framework by proposing an All-India Federation and granting Provincial Autonomy, marking a major step toward the eventual independence of India.
• While the 1909 Act aimed to appease moderates in the Indian National Congress, it was later criticized for sowing the seeds of partition through religious segregation in politics.

The correct answer is (i) True, (ii) True.

Key Points

• A Microphone is a specialized transducer designed to convert sound waves (acoustic energy) into electrical signals (analog or digital) for computer processing; since its hardware architecture is strictly unidirectional for data entry and lacks the components to convert electrical signals back into audible sound waves, it is classified solely as an input device. Hence, statement (i) is correct.
• The Central Processing Unit (CPU) operates at extremely high speeds, with clock rates typically measured in Gigahertz (GHz), which allows it to process billions of operations per second, while external peripherals like printers are limited by mechanical constraints and the relatively slower data transfer rates of I/O interfaces (like USB or Wi-Fi). Hence, statement (ii) is correct.

 Additional Information

• Input Devices:
• These hardware components allow users to feed data into a computer system.
• Key examples include Keyboards (text entry), Optical Scanners (image digitization), and Joysticks (control signals).
• The primary function is to translate human-understandable data into binary format.
• Output Devices:
• These devices receive data from the computer and convert it into a form that is perceptible to humans.
• Common examples include Monitors (visual), Printers (hard copy), and Speakers (audio).
• While a microphone captures sound, a Speaker or Headset is required to output it.
• Speed Gap Management:
• Because the CPU is significantly faster than I/O devices, computers use Buffering and Spooling techniques.
• A Buffer is a temporary storage area in RAM that holds data while it is being transferred between two devices with different speeds.
• Print Spooling allows the CPU to offload a print job to the Hard Drive or Memory so the user can continue working while the slow printer catches up.

The correct answer is Article 51 'A'.

Key Points

• The Fundamental Duties of Indian citizens are contained in Article 51A, which is the only article under Part IV-A of the Indian Constitution.
• These duties were not part of the original Constitution; they were incorporated by the 42^nd Constitutional Amendment Act, 1976, based on the recommendations of the Sardar Swaran Singh Committee.
• The concept of Fundamental Duties was borrowed from the Constitution of the erstwhile USSR (now Russia).
• Originally, the amendment introduced 10 Fundamental Duties. The 11^th Fundamental Duty, regarding providing education opportunities to children between 6 and 14 years, was added by the 86^th Constitutional Amendment Act, 2002.
• Fundamental Duties are non-justiciable in nature, meaning they cannot be directly enforced by the courts, but they serve as a guide for citizens to maintain discipline and commitment to the nation.

 Additional Information

• Article 52:
• This article provides that there shall be a President of India, who is the head of the Indian State and the first citizen of India.
• Article 50:
• Contained within the Directive Principles of State Policy (DPSP), it directs the State to take steps to separate the judiciary from the executive in the public services of the State.
• Article 51:
• This article mandates the State to promote international peace and security, maintain just and honorable relations between nations, and encourage settlement of international disputes by arbitration.

The correct answer is Indian princes.

Key Points

• The Second Round Table Conference took place in London from September 7, 1931, to December 1, 1931, following the Gandhi-Irwin Pact.
• During this conference, Mahatma Gandhi claimed that the Indian National Congress was the sole representative of the entire Indian population, including those living in the Princely States.
• The Indian princes, who represented over 560 semi-autonomous states, strongly contested this claim by asserting that the Congress possessed no legal or political authority over their territories.
• The princes maintained that their relationship was exclusively with the British Crown through the doctrine of Paramountcy and individual treaties, which exempted them from the jurisdiction of any political party in British India.
• This dispute over sovereignty was a major reason for the failure of the conference to reach a consensus on the All-India Federation, as the rulers feared the Congress would interfere with their internal autocratic administration.

 Additional Information

• Second Round Table Conference:
• It was the only conference attended by the Indian National Congress after the suspension of the Civil Disobedience Movement.
• Other key stakeholders included B.R. Ambedkar, who demanded separate electorates for the Depressed Classes, and representatives of the Muslim League.
• Zamindars:
• This group consisted of influential landlords who held significant socio-economic power in British India but did not possess the sovereign territorial status that the princes did.
• Provincial Governors:
• These were British officials appointed to head the executive branch in British Indian provinces like Bengal and Bombay, operating under the Government of India Act.

The correct answer is Amoebiasis.

Key Points

• Amoebiasis, also referred to as amoebic dysentery, is a gastrointestinal infection caused by the protozoan parasite Entamoeba histolytica.
• The parasite primarily inhabits the large intestine of humans, where it can cause ulceration of the intestinal lining.
• Clinical symptoms of the disease include abdominal pain, constipation, cramps, and the passage of stools characterized by excess mucous and blood clots.
• The transmission of the disease is facilitated by houseflies (Musca domestica), which serve as mechanical carriers by transferring the parasite from the feces of infected individuals to food and water supplies.
• Infection occurs through the fecal-oral route, specifically via the ingestion of contaminated food and drinking water containing the cysts of the parasite.

 Additional Information

• Giardiasis:
• Caused by the flagellated protozoan Giardia lamblia.
• It affects the small intestine and typically leads to diarrhea, fatigue, and greasy stools that tend to float.
• Cholera:
• An acute diarrheal infection caused by the bacterium Vibrio cholerae.
• It is characterized by severe watery diarrhea, often described as rice-water stools, leading to rapid dehydration.
• Ascariasis:
• A helminthic disease caused by the common roundworm Ascaris lumbricoides.
• Symptoms include internal bleeding, muscular pain, fever, anemia, and potential blockage of the intestinal passage.

The correct answer is Electromagnetic Energy.

Key Points

• Light is fundamentally composed of photons, which are discrete packets or quanta of energy that lack rest mass but carry momentum.
• The energy of these photons is a manifestation of electromagnetic radiation, which occurs due to the synchronization of oscillating electric and magnetic fields.
• According to quantum mechanics, the energy (E) of a photon is determined by the Planck-Einstein relation: E = hf, where h is the Planck constant (approximately 6.626 × 10⁻³⁴ J·s) and f is the frequency.
• Electromagnetic energy covers the entire electromagnetic spectrum, including visible light, infrared radiation, and ultraviolet rays, all of which travel at the speed of light (approximately 3 × 10⁸ m/s) in a vacuum.
• Since light behaves as both a wave and a particle (wave-particle duality), its energy is transferred via electromagnetic waves that do not require a physical medium for conduction.

 Additional Information

• Nuclear Energy
• Refers to the binding energy held within the atomic nucleus, typically harnessed via nuclear fission or nuclear fusion reactions.
• Gravitational Potential Energy
• A form of stored energy related to an object mass and its altitude or height within a gravitational field.
• Mechanical Energy
• Defined as the sum of kinetic energy and potential energy in a system, pertaining to macroscopic objects in motion or specific physical configurations.

The correct answer is Spiti River.

Key Points

• The Pin River is recognized as the most significant right-bank tributary of the Spiti River, located within the Lahaul and Spiti district of Himachal Pradesh.
• It originates from the base of the Pin-Parvati Pass, which acts as a high-altitude trekking corridor connecting the Parvati Valley in Kullu to the Spiti Valley.
• The river flows through the Pin Valley National Park, a high-altitude cold desert ecosystem famous for the conservation of the Snow Leopard and the Siberian Ibex.
• The confluence of the Pin River and the Spiti River occurs near Dankar (specifically near the Dhankar Monastery), where it adds substantial discharge to the main Spiti channel.

 Additional Information

• Spiti River:
• It is a major mountain torrent that originates from the Kunzum Range of the Himalayas.
• The river system ultimately merges with the Sutlej River at Khab in the Kinnaur district.
• Chandra and Bhaga Rivers:
• The Chandra River originates near the Bara-lacha la pass, while the Bhaga River flows from the opposite side of the same pass.
• These two rivers meet at Tandi in Lahaul to form the Chenab River, which is a principal tributary of the Indus.
• Tsarab River:
• It is a key tributary of the Zanskar River, primarily flowing through the Zanskar region of Ladakh.
• It contributes to the Indus River basin and is known for its rugged, high-altitude path.

The correct answer is 15% of the strength of the Legislative Assembly.

Key Points

• Article 164(1A) of the Constitution of India explicitly states that the total number of Ministers, including the Chief Minister, in the Council of Ministers in a State shall not exceed 15% of the total number of members of the Legislative Assembly of that State.
• This specific provision was inserted into the Constitution through the 91^st Constitutional Amendment Act, 2003 to prevent the formation of jumbo cabinets and to reduce the financial burden on the state exchequer.
• The article also provides a mandatory minimum limit, specifying that the number of Ministers, including the Chief Minister, in a State shall not be less than twelve.
• The primary objective of this percentage cap is to ensure administrative efficiency and political stability by discouraging the practice of offering ministerial berths solely to appease various political factions or defectors.

 Additional Information

• 91^st Constitutional Amendment Act, 2003:
• This amendment also introduced a similar ceiling for the Union Council of Ministers under Article 75(1A), limiting their number to 15% of the total strength of the Lok Sabha.
• It strengthened the Anti-Defection Law (Tenth Schedule) by removing the provision that protected a split of one-third of the members of a political party.
• Article 164 Overview:
• Article 164(1): States that the Chief Minister shall be appointed by the Governor and other Ministers shall be appointed by the Governor on the advice of the Chief Minister.
• Article 164(2): Mandates that the Council of Ministers shall be collectively responsible to the Legislative Assembly of the State.

The correct answer is Pakhawaj.

Key Points

• The Pakhawaj is a traditional barrel-shaped, two-headed percussion instrument that serves as the primary accompaniment for the Dhrupad genre of Hindustani classical music.
• Musicologists and historians widely regard the Pakhawaj as the direct ancestor of the Tabla due to structural similarities in the drum heads, specifically the application of the syahi (black permanent paste) used to modulate pitch and resonance.
• According to popular musical lore, the Tabla was created in the 13^th century by the Sufi musician and poet Amir Khusrau, who reportedly divided a Pakhawaj into two separate pieces to produce a more versatile and portable sound.
• While the Pakhawaj is played with the full open palm and fingers, the Tabla evolved to accommodate more intricate finger-tapping techniques, though the right drum (Dayan) of the Tabla remains almost identical to the right head of the Pakhawaj.

 Additional Information

• Dhol
• It is a large double-sided barrel drum used primarily in regional folk music across North India and Pakistan, most notably in Bhangra performances.
• Unlike classical drums, it is played using wooden sticks (the dagga and tilli) and lacks the specialized syahi coating required for melodic precision.
• Thavil
• A high-pitched, barrel-shaped percussion instrument from South India, traditionally used in Carnatic music and temple festivals.
• It is typically paired with the Nadaswaram and is played with a stick on one side and fingers protected by plaster caps on the other.
• Kanjira
• The Kanjira is a frame drum from the tambourine family, used as a secondary percussion instrument in South Indian classical concerts.
• Its frame is traditionally made from jackfruit wood, and the drumhead was historically made from monitor lizard skin, giving it a unique, high-pitched snapping sound.

The correct answer is Brazil.

Key Points

• In October 2025, Brazil formally signed the Multilateral Convention (MLI) to implement measures related to the Base Erosion and Profit Shifting (BEPS) project, specifically targeting the Subject to Tax Rule (STTR) and Pillar Two components.
• The BEPS initiative, led by the OECD/G20 Inclusive Framework, aims to prevent multinational enterprises from exploiting gaps and mismatches in tax rules to avoid paying tax in jurisdictions where economic activities are performed.
• By signing the convention, Brazil has committed to aligning its domestic tax legislation with international standards, facilitating its ongoing process for OECD accession.
• The Multilateral Instrument (MLI) allows countries to modify their existing bilateral tax treaties simultaneously to include anti-abuse provisions without the need for lengthy individual renegotiations.
• This move represents a significant shift for Brazil, which historically maintained a unique transfer pricing system, but has now transitioned toward the OECD Transfer Pricing Guidelines to promote tax transparency and certainty.

 Additional Information

• BEPS 2.0 Framework:
• Pillar One: Focuses on the reallocation of taxing rights over the largest and most profitable multinational enterprises to the jurisdictions where consumers are located.
• Pillar Two: Establishes a Global Minimum Tax rate of 15% to ensure that large corporations pay a fair share of tax regardless of where they operate.
• Subject to Tax Rule (STTR): A treaty-based rule that allows developing countries to "tax back" certain intra-group payments where these are subject to low nominal tax rates in the recipient country.
• OECD Role:
• The Organisation for Economic Co-operation and Development serves as a global forum for developing international standards on taxation, environment, and governance.
• The Inclusive Framework on BEPS currently brings together over 140 jurisdictions to collaborate on the implementation of the BEPS package.

The correct answer is IT and Services.

Key Points

• The Economic Reforms of 1991 in India, characterized by Liberalisation, Privatisation, and Globalisation (LPG), created a conducive environment for the Information Technology (IT) and Services sectors to flourish.
• India became a global hub for outsourcing because of its ability to provide high-quality technical services at a fraction of the cost compared to developed nations, leveraging its vast pool of English-speaking and skilled professionals.
• Major Multinational Corporations (MNCs) set up their Global Delivery Centres in India, leading to the rapid growth of companies like TCS, Infosys, and Wipro.
• Technological advancements in telecommunications and the internet bridged the geographical gap, allowing the tertiary sector to contribute nearly 55 percent to India's Gross Domestic Product (GDP) post-globalisation.

 Additional Information

• IT-Enabled Services (ITeS):
• This includes Business Process Outsourcing (BPO) and Knowledge Process Outsourcing (KPO), providing 24/7 customer support, payroll management, and data analytics to global clients.
• Cities such as Bengaluru (the Silicon Valley of India), Hyderabad, and Gurugram became epicenters for this growth.
• Impact on Other Sectors:
• Handicrafts: Conversely, this sector faced a decline as global competition introduced cheap, mass-produced machine goods, making it difficult for traditional artisans to compete.
• Foreign Direct Investment (FDI): The services sector consistently receives the highest share of FDI inflows in India, emphasizing its importance in the globalized economy.

The correct answer is Rajmahal Hills.

Key Points

• The Malda Gap, frequently referred to as the Garo-Rajmahal Gap, is a major geological depression that separates the main Indian Peninsular Plateau from its northeastern extension, the Meghalaya Plateau.
• This gap originated during the Tertiary period due to down-faulting of the crust, creating a tectonic rift that was subsequently filled with alluvial sediments carried by the Ganga and Brahmaputra river systems.
• The Rajmahal Hills, situated primarily in Jharkhand, represent the northeastern edge of the Chota Nagpur Plateau and serve as the physical western boundary of the Malda Gap.
• While the gap itself facilitates the merging of the Ganga Plains and the Brahmaputra Plains, the Rajmahal Hills act as the topographical feature that demarcates the transition from the stable peninsular block to the sedimentary depression. Hence, Rajmahal Hills is the correct feature.

 Additional Information

• Meghalaya Plateau:
• It consists of the Garo, Khasi, and Jaintia Hills and is geologically part of the Deccan Plateau.
• It serves as the eastern boundary of the Malda Gap, rather than the feature separating the gap from the plains.
• Aravalli Range:
• Stretching from Gujarat to Delhi, these are among the oldest fold mountains in the world.
• They are located in the western part of India and act as a divide between the Thar Desert and the Ganga Basin.
• Bundelkhand Plateau:
• Located in central India across Uttar Pradesh and Madhya Pradesh, it is characterized by Archaean gneiss and granite rocks.
• It lies to the south of the Yamuna River and is far removed from the Malda Gap region.

The correct answer is It is a collection of hymns composed by various sages over generations.

Key Points

• The Rigveda is the oldest of the four Vedas and represents the earliest form of Indo-Aryan civilization literature, dating back approximately between 1500 BCE and 1000 BCE.
• It is a massive collection of 1,028 hymns (known as Suktas) that are organized into 10 Mandalas (books) of varying lengths and ages.
• Authorship is attributed to numerous Rishis (seer-poets) rather than a single individual; these sages belonged to different lineages or clans who composed these verses over several centuries.
• The Family Books (Mandalas 2 through 7) form the core and oldest part of the text, with each book dedicated to hymns composed by a specific family of sages like Ghritsamada, Vishvamitra, Atri, and Vashistha.
• The transmission of the Rigveda was strictly oral for many centuries, categorized as Shruti (that which is heard), ensuring the preservation of precise phonetics and accents across generations before it was ever committed to writing.

 Additional Information

• Structure and Composition:
• The Purusha Sukta, found in the 10^th Mandala, is a famous hymn that describes the spiritual unity of the universe and the origin of the Varna system.
• The Gayatri Mantra, one of the most sacred verses in Hinduism, is located in the 3^rd Mandala and is attributed to Sage Vishvamitra.
• While predominantly male, the text includes hymns composed by nearly 27 female sages (Rishikas) such as Lopamudra, Ghosha, and Apala, highlighting the inclusive nature of early Vedic spiritual discourse.
• Context of Other Options:
• Valmiki is traditionally honored as the Adi Kavi and the author of the epic Ramayana, which belongs to the Itihasa category of literature.
• Sage Vyasa, or Veda Vyasa, is historically credited with the compilation and classification of the Vedas into four distinct collections (Rig, Sama, Yajur, and Atharva) rather than being the sole author of the content.
• The Mahabharata era is associated with the Later Vedic Period or the transition to the Epic Period, whereas the Rigveda was largely finalized during the Early Vedic Period.

The correct answer is 1.42degreeC.

Key Points

• The World Meteorological Organization (WMO) analysis of global temperature data for the period of January to August 2025 revealed that the global average temperature was approximately 1.42degreeC above the 1850−1900 pre-industrial baseline.
• This measurement is derived from a synthesis of six major international datasets, including NASA GISS, NOAA NCEI, and the Copernicus Climate Change Service (C3S), which utilize millions of observations from ships, buoys, and land-based weather stations.
• The figure of 1.42degreeC highlights a continuing trend where global temperatures are dangerously close to the 1.5degreeC threshold defined by the Paris Agreement as a critical limit for avoiding the most severe impacts of climate change.
• The warming observed during the first eight months of 2025 was driven by the unprecedented accumulation of greenhouse gases in the atmosphere, alongside the transition from a strong El Niño event to neutral or La Niña conditions.

 Additional Information

• World Meteorological Organization (WMO):
• It is a specialized agency of the United Nations established in 1950 to coordinate international atmospheric science and meteorology.
• Headquartered in Geneva, Switzerland, it issues the annual State of the Global Climate report, which serves as a benchmark for international climate policy.
• Pre-industrial Baseline:
• The period 1850−1900 is used as a reference because it is the earliest period for which high-quality, global-scale instrumental temperature records are available.
• Warming relative to this period is the primary metric used by the Intergovernmental Panel on Climate Change (IPCC) to assess progress toward global climate goals.
• Atmospheric Drivers:
• The concentration of Carbon Dioxide (CO₂) reached record levels in 2025, continuing to trap heat and increase the thermal energy of the oceans.
• Other key gases like Methane (CH₄) and Nitrous Oxide (N₂O) also contribute significantly to the total warming anomaly.
• Global Implications:
• A warming level of 1.42degreeC is associated with increased frequency of extreme weather events, including marine heatwaves, intense tropical cyclones, and prolonged droughts.

The correct answer is Dr. Anil Nayak.

Key Points

• Dr. Anil Nayak (officially Dr. Anilkumar J. Nayak) officially assumed charge as the National President of the Indian Medical Association (IMA) for the 2025-2026 term in December 2025.
• The installation took place during NATCON 2025, the 100^th national conference of the IMA, held in Ahmedabad, Gujarat, where the swearing-in ceremony was graced by Union Home Minister Shri Amit Shah and Gujarat CM Shri Bhupendra Patel.
• A distinguished Orthopedic Surgeon from Mehsana, Gujarat, Dr. Nayak has an extensive leadership record, having previously served as the National Honorary Secretary General of the IMA and President of the IMA Gujarat State Branch.
• He succeeded Dr. Dilip Bhanushali, who concluded his term as the National President for the 2024-2025 period.

 Additional Information

• Indian Medical Association (IMA):
• Established in 1928 and headquartered in New Delhi, it is the largest voluntary organization of Allopathic doctors in India.
• It plays a critical role in healthcare policy, medical ethics, and safeguarding the interests of over 3.5 lakh doctor members.
• Dr. Dilip Bhanushali:
• Held the office of IMA National President for the year 2024-2025.
• He is a noted physician from Telangana who represented India at several global medical summits during his tenure.
• Dr. Ketan Desai:
• A former National President of the IMA and former President of the Medical Council of India (MCI) and the World Medical Association (WMA).
• Currently acts as the Chief Patron of the association.
• Dr. Amit Chakrabarty:
• A prominent urologist and former President of the Association of American Physicians of Indian Origin (AAPI), though not the IMA National President for 2025.

The correct answer is Promoting monotheistic worship and rejecting ritual sacrifices entirely..

Key Points

• The Brahmo Samaj, established by Raja Ram Mohan Roy in 1828, was founded on the principle of Monotheism, which emphasized the worship of one single, formless God (the Brahman). This was a direct challenge to the prevalent polytheistic practices and deity worship in 19^th-century India.
• The movement sought to reform Hinduism by returning to the original philosophical roots found in the Upanishads and Vedanta, arguing that these ancient texts supported a rational, ethical, and monotheistic framework rather than the complex ritualism that had developed over centuries.
• A central tenet of the Samaj was the total rejection of idol worship and ritual sacrifices. The members believed that God should be reached through meditation, prayer, and the reading of scriptures, rather than through external physical offerings or the mediation of a priestly class.
• By advocating for the removal of meaningless ceremonies, the Brahmo Samaj aimed to create a simplified and purified form of religion that was accessible to all, regardless of caste or social status, thereby promoting social equality and humanism.

 Additional Information

• Raja Ram Mohan Roy:
• Known as the Father of Indian Renaissance, he founded the Atmiya Sabha in 1815 before establishing the Brahmo Samaj.
• He played a pivotal role in the abolition of Sati in 1829 through the efforts of Lord William Bentinck.
• Evolution of the Samaj:
• After Roy, the leadership passed to Debendranath Tagore, who introduced the Brahmo Covenant.
• In 1866, the movement split into the Brahmo Samaj of India led by Keshab Chandra Sen and the Adi Brahmo Samaj led by Tagore.
• Vedic Polytheism vs. Reformism:
• While some later movements like the Arya Samaj emphasized Back to the Vedas, the Brahmo Samaj was distinct in its heavy reliance on the rationalist and universalist interpretation of the Upanishads.

The correct answer is 44^th Amendment.

Key Points

• The 44^th Constitutional Amendment Act was enacted in 1978 by the Janata Party government primarily to undo several changes made by the 42^nd Amendment and to provide safeguards against future misuse of emergency provisions.
• Before this amendment, the President could declare a National Emergency on the oral advice of the Prime Minister, as witnessed during the 1975 Emergency when Indira Gandhi advised the President without consulting her Cabinet.
• The act amended Article 352 to stipulate that the President can only issue a Proclamation of Emergency after receiving a written recommendation from the Union Cabinet (the Prime Minister and other ministers of Cabinet rank).
• It replaced the vague term 'internal disturbance' with 'armed rebellion' as a ground for declaring an emergency, ensuring it cannot be invoked for simple political unrest or protests.
• The amendment also mandated that the proclamation must be approved by both Houses of Parliament within one month (reduced from two months) by a special majority.
• Therefore, the 44^th Amendment is the legislation that made written Cabinet advice mandatory for imposing an emergency in India.

 Additional Information

• 42^nd Amendment Act (1976):
• Known as the Mini-Constitution, it was passed during the Emergency to centralize power.
• It added Fundamental Duties via Article 51A and inserted Socialist, Secular, and Integrity into the Preamble.
• 24^th Amendment Act (1971):
• Enacted to counter the Golaknath case judgment, it affirmed Parliament's power to amend any part of the Constitution, including Fundamental Rights.
• It made it mandatory for the President to give assent to Constitution Amendment Bills.
• 26^th Amendment Act (1971):
• This act formally abolished the Privy Purses and privileges of the former rulers of Princely States in India.
• It omitted Articles 291 and 362 to ensure equality of status among all citizens.

The correct answer is Krishna.

Key Points

• The Nagarjuna Sagar Dam is recognized as one of the largest and tallest masonry dams in the world, serving as a pivotal multipurpose river valley project in India.
• It is constructed across the Krishna River, which is the fourth-largest river in India in terms of water inflow and basin area.
• The dam is geographically located on the border between Nalgonda district in Telangana and Palnadu district (formerly part of Guntur) in Andhra Pradesh.
• The foundation stone for this massive project was laid by the first Prime Minister of India, Pandit Jawaharlal Nehru, on December 10, 1955, and the dam was completed in 1967.
• The project is named after Acharya Nagarjuna, a renowned Buddhist philosopher and scholar of the 2^nd century AD, who established a center of learning at the site now submerged under the reservoir, known as Nagarjunakonda.
• The structure stands approximately 124 meters high and features 26 radial floodgates, facilitating irrigation for over 10 lakh acres of agricultural land.

 Additional Information

• Godavari River:
• Originates at Trimbakeshwar in Nashik, Maharashtra, and is the longest river in Peninsular India.
• It is often called the Dakshin Ganga and hosts major dams like the Jayakwadi and Sriram Sagar.
• Tungabhadra River:
• A major right-bank tributary of the Krishna River, formed by the confluence of the Tunga and Bhadra rivers at Koodli in Karnataka.
• The Tungabhadra Dam near Hospet is a significant multipurpose project on this river.
• Bhima River:
• Originates from the Bhimashankar hills in the Western Ghats of Maharashtra.
• It is the most important left-bank tributary of the Krishna River, flowing through Maharashtra, Karnataka, and Telangana.

Concept:

Corporate Average Fuel Efficiency (CAFE) norms are regulatory standards designed to reduce the carbon footprint of the automotive industry. These norms aim to lower the average fuel consumption (and thus CO₂ emissions) of a vehicle manufacturer's entire fleet. In India, these standards are notified by the Ministry of Power through the Bureau of Energy Efficiency (BEE).

The implementation of CAFE norms in India is divided into specific phases, each with progressively stricter targets for fuel efficiency and emission levels.

Given

Target Phase: CAFE-3 (Phase 3)

Calculation

To determine the implementation date for CAFE-3, we look at the timeline established for CAFE norms in India:

• Phase 1 (CAFE-1): This phase was implemented from April 2017 to March 2022.
• Phase 2 (CAFE-2): This phase is currently in effect, having started in April 2022, and is scheduled to conclude in March 2027.
• Phase 3 (CAFE-3): Following the completion of Phase 2, the third phase of these efficiency norms is scheduled to begin on April 1, 2027.
• Phase 4 (CAFE-4): This subsequent phase is proposed to start after five years of Phase 3, beginning in April 2032.

The transition to CAFE-3 in April 2027 will require automobile manufacturers to further enhance engine technology, promote hybrid vehicles, and increase the share of Electric Vehicles (EVs) in their total sales to meet the stricter CO₂ emission targets.

Final Answer: April 2027

The correct answer is Article 72.

Key Points

• Article 72 of the Indian Constitution empowers the President of India to grant pardons, reprieves, respites, or remissions of punishment, or to suspend, remit, or commute the sentence of any person convicted of any offense.
• This power extends to all cases where the punishment is by a Court Martial, cases involving an offense against a Union Law, and all cases where the sentence is a death sentence.
• A Pardon completely absolves the offender from all sentences, punishments, and disqualifications, effectively placing them in the same position as if they had never committed the crime.
• Commutation denotes the substitution of one form of punishment for a lighter form; for instance, a death sentence may be commuted to rigorous imprisonment.
• Remission implies reducing the duration of the sentence without changing its nature, such as reducing a sentence of two years of rigorous imprisonment to one year.
• Respite involves awarding a lesser sentence in place of one originally imposed due to some special fact, such as the physical disability of a convict or the pregnancy of a woman offender.
• Reprieve signifies a temporary stay of the execution of a sentence, especially a death sentence, to allow the convict time to seek a pardon or commutation from the President.

 Additional Information

• Article 53:
• Declares that the executive power of the Union is vested in the President and shall be exercised by him either directly or through officers subordinate to him in accordance with the Constitution.
• It also establishes the President as the Supreme Commander of the Defence Forces of the Union.
• Article 74:
• Provides for a Council of Ministers with the Prime Minister at the head to aid and advise the President, who shall, in the exercise of his functions, act in accordance with such advice.
• Article 75:
• Deals with other provisions as to Ministers, including their appointment by the President on the advice of the Prime Minister and their collective responsibility to the Lok Sabha.
• Article 161:
• While the President acts under Article 72, the Governor of a State possesses similar powers under Article 161 for offenses against State Laws, though the Governor cannot pardon a death sentence or a Court Martial sentence.