Search Previous Year Questions

Shortcut Trick

Use the algebraic identity: If a + b + c = 0, then a³ + b³ + c³ = 3abc.

Here, a = 20, b = −9, and c = −11.

Sum = 20 + (−9) + (−11) = 20 − 20 = 0.

∴ 20³ + (−9)³ + (−11)³ = 3 × 20 × (−9) × (−11) = 60 × 99 = 5940.

∴ The correct answer is 5940.

Alternate Method

Given: 20³ + (−9)³ + (−11)³

Formula Used: Direct calculation of cubes.

⇒ 20³ = 20 × 20 × 20 = 8000

⇒ (−9)³ = (−9) × (−9) × (−9) = −729

⇒ (−11)³ = (−11) × (−11) × (−11) = −1331

⇒ 8000 + (−729) + (−1331)

⇒ 8000 − (729 + 1331)

⇒ 8000 − 2060

⇒ 5940

∴ The correct answer is 5940.

Additional Information

Special Algebraic Identity

The conditional identity states that if a + b + c = 0, then the sum of their cubes is exactly 3abc.

General Cube Formula

a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca). This formula explains why the sum equals 3abc when the first factor is zero.

Negative Base Property

The cube of a negative number is always negative: (−x)³ = −(x³).

Shortcut Trick

Directly substitute the perfect squares: √225 = 15 and √484 = 22.

Find variable x: x = 15 + 22 = 37.

Solve target expression: 37² − (15² + 32²).

Simplify: 1369 − (225 + 1024) = 1369 − 1249 = 120.

∴ The correct answer is 120.

Alternate Method

Given:

x = √225 + √484

Expression to solve: x² − [(√225)² + (32)²]

Step-by-step Calculation:

⇒ First, determine the value of x by calculating square roots:

⇒ √225 = 15

⇒ √484 = 22

⇒ x = 15 + 22 = 37

⇒ Now, substitute x into the expression:

⇒ (37)² − [(15)² + (32)²]

⇒ 1369 − [225 + 1024]

⇒ 1369 − 1249

⇒ 120

∴ The correct answer is 120.

Additional Information

Perfect Square Roots

Memorizing squares up to 40 helps in simplification; for example, 15² = 225, 22² = 484, 32² = 1024, and 37² = 1369.

BODMAS Rule

Always prioritize Brackets, then Orders (powers/roots), followed by Division, Multiplication, Addition, and lastly Subtraction.

Algebraic Identity Shortcut

If x = a + b, then x² = a² + b² + 2ab. In this problem, substituting a = √225 and b = √484 can also simplify the calculation steps.

Shortcut Trick

Net successive discounts are 17% and 10% on the Marked Price.

Final Selling Price = Marked Price × (100 − d₁)% × (100 − d₂)%

Final Selling Price = 2000 × 83% × 90%

Final Selling Price = 2000 × 0.83 × 0.9 = 1494

∴ The correct answer is ₹1494.

Alternate Method

Given:

Marked Price (MP) = ₹2,000

First Discount = 17%

Additional Discount (Credit Card) = 10%

Calculations:

⇒ Price after 17% discount = 2000 − (17/100 × 2000)

⇒ Price after 17% discount = 2000 − 340 = ₹1,660

⇒ Additional 10% discount on ₹1,660 = 10/100 × 1660 = ₹166

⇒ Final amount paid by Pooja = 1660 − 166 = ₹1,494

∴ The correct answer is ₹1494.

Additional Information

Effective Successive Discount

For two successive discounts of a% and b%, the single equivalent discount is (a + b − ab/100)%.

Discount and Selling Price

Selling Price (SP) = Marked Price (MP) − Discount. Also, Discount % = (Discount ÷ MP) × 100.

Marked Price vs Cost Price

The relation between CP and MP is given by MP ÷ CP = (100 + Profit%) ÷ (100 − Discount%).

Shortcut Trick

The age difference between two people always remains constant (30 years).

In 12 years, the ratio of Father : Son = 3 : 1.

Difference in ratio parts = 3 − 1 = 2 units ⇒ 30 years.

1 unit = 15 years (This is the son's age after 12 years).

Present age of son = 15 − 12 = 3 years.

∴ The correct answer is 3 years.

Alternate Method

Given: Current age difference = 30 years. Multiplier after 12 years = 3.

Formula Used: Age after n years = Present age + n.

⇒ Let the Present age of the son be S.

⇒ Then, Father's present age = S + 30.

⇒ After 12 years, Father's age = (S + 30) + 12 = S + 42.

⇒ After 12 years, Son's age = S + 12.

⇒ According to the question: S + 42 = 3 × (S + 12)

⇒ S + 42 = 3S + 36

⇒ 42 − 36 = 3S − S

⇒ 6 = 2S

⇒ S = 3

∴ The correct answer is 3 years.

Additional Information

Constant Age Difference

The numerical difference between the ages of two individuals never changes, regardless of how many years pass.

Ratio Method in Ages

When ages are given in ratios at different times, adjust the ratio difference to match the actual age difference to find values quickly.

Average Age Calculation

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

Shortcut Trick

Total work is constant, so use the formula: M₁ × D₁ = M₂ × D₂.

9 workers × 12 days = 6 workers × D₂

108 = 6 × D₂

D₂ = 108 ÷ 6 = 18 days.

∴ The correct answer is 18 days.

Alternate Method

Given:

Initial number of workers (M₁) = 9

Initial time taken (D₁) = 12 days

New number of workers (M₂) = 6

Formula Used:

Total Work = Number of Workers × Number of Days

Calculations:

⇒ Total work done by 9 workers in 12 days = 9 × 12 = 108 man-days

⇒ Since the wall is the same, 6 workers must complete the same 108 man-days of work.

⇒ Days taken by 6 workers = Total Work ÷ M₂

⇒ Days = 108 ÷ 6

⇒ Days = 18 days

∴ The correct answer is 18 days.

Additional Information

Inverse Variation in Work

When the rate of work is constant, the number of workers is inversely proportional to the time taken. If workers decrease, time increases proportionally.

The Chain Rule (M-D-H Formula)

For different work quantities or hours per day: (M₁×D₁×H₁)/W₁ = (M₂×D₂×H₂)/W₂, where H is hours and W is units of work.

Efficiency Concept

Efficiency × Time = Total Work. Efficiency is the work done per unit of time (e.g., 1 day or 1 hour).

Shortcut Trick

Convert all terms to the same base (19/10) to compare the exponents directly.

LHS total power = 7 + (−6) + 10 = 11.

RHS power = −(6m + 18) = −6m − 18.

Equating powers: 11 = −6m − 18 ⇒ 6m = −29.

∴ The correct answer is −29/6.

Alternate Method

Given:

((19)/(10))⁷ × ((10)/(19))⁶ × ((19)/(10))¹⁰ = ((10)/(19))^{6m + 18}

Formula Used:

Product Law: a^x × a^y = a^x+y

Reciprocal Rule: (a/b)^x = (b/a)^−x

Calculations:

⇒ ((19)/(10))⁷ × ((19)/(10))⁻⁶ × ((19)/(10))¹⁰ = ((10)/(19))^{6m + 18} ⇒ ((19)/(10))^{(7 − 6 + 10)} = ((19)/(10))^{−(6m + 18)} ⇒ ((19)/(10))¹¹ = ((19)/(10))^{−6m − 18}

Since the bases are equal, we can equate the exponents:

⇒ 11 = −6m − 18 ⇒ 6m = −18 − 11 ⇒ 6m = −29 ⇒ m = −29/6

∴ The correct answer is −29/6.

Additional Information

Product Rule of Exponents

When multiplying terms with the same base, the powers are added: a^m × aⁿ = a^m+n.

Negative Exponent Rule

A negative power indicates the reciprocal of the base with a positive power: (a/b)⁻ⁿ = (b/a)ⁿ.

Property of Equality

In an equation where the bases are the same (and not 0, 1, or −1), the powers must be equal: a^x = a^y ⇒ x = y.

Shortcut Trick

The mean proportional x between two numbers m and n is calculated using x = √(m × n).

Substitute m = (a + b) and n = (a − b).

x = √((a + b) × (a − b))

Using the identity (a + b)(a − b) = a² − b², we get x = √(a² − b²).

∴ The correct answer is √(a² − b²).

Alternate Method

Given: The terms are (a + b) and (a − b), and their mean proportional is x.

Formula Used: If three quantities A, B, and C are in continued proportion, then A : B = B : C, which means B² = A × C. Here, B is the mean proportional.

⇒ (a + b) / x = x / (a − b)

⇒ x² = (a + b) × (a − b)

⇒ We know that (x + y)(x − y) = x² − y²

⇒ x² = a² − b²

⇒ x = √(a² − b²)

∴ The correct answer is √(a² − b²).

Additional Information

Third Proportional

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

Fourth Proportional

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

Continued Proportion

Quantities a, b, c, d are in continued proportion if a/b = b/c = c/d.

Shortcut Trick

Effective successive discount of 20% and 15% = 20 + 15 − (20 × 15)/100 = 32%.

Difference between 35% and 32% = 3%.

Given, 3% of Marked Price = ₹70.

Total MP (100%) = (70 ÷ 3) × 100 = 7000 ÷ 3 ≈ ₹2,333.

∴ The correct answer is ₹2,333.

Alternate Method

Given:

Single discount percentage = 35%

Two successive discount percentages = 20% and 15%

Difference between the two discount amounts = ₹70

Formula Used:

Equivalent single discount for two successive discounts a% and b% = [a + b − (ab/100)]%

Discount Amount = (Discount% / 100) × Marked Price

Calculations:

⇒ Let the Marked Price be M.

⇒ Equivalent of successive discounts = 20 + 15 − (20 × 15)/100 = 35 − 3 = 32%

⇒ First discount amount = 0.35M

⇒ Second discount amount = 0.32M

⇒ Difference = 0.35M − 0.32M = 0.03M

⇒ 0.03M = 70

⇒ M = 70 / 0.03 = 7000 / 3

⇒ M = 2333.333...

⇒ Rounded to nearest integer, M = 2,333.

∴ The correct answer is ₹2,333.

Additional Information

Successive Discount Formula

For three successive discounts a%, b%, and c%, first find equivalent of a and b as x%, then find equivalent of x and c. Effective = x + c − (xc/100).

Relationship between MP and CP

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

Selling Price Calculation

SP = MP × (1 − d₁/100) × (1 − d₂/100) for multiple discounts.

Shortcut Trick

Net percentage change = 10 − 5 − (10 × 5 ÷ 100) = 4.5% increase.

Final enrolment in June = 100% + 4.5% = 104.5%.

Given, 104.5% = 1045 students.

March enrolment (100%) = (1045 ÷ 104.5) × 100 = 1000.

∴ The correct answer is 1000.

Alternate Method

Given:

Enrolment in June = 1045

Increase in April = 10%

Decrease in May/June = 5%

Let the number of students in March be x.

⇒ Students in April = x + (10 ÷ 100) × x = 1.1x

⇒ Students in June = 1.1x − (5 ÷ 100) × 1.1x

⇒ 1045 = 1.1x × 0.95

⇒ 1045 = 1.045x

⇒ x = 1045 ÷ 1.045

⇒ x = 1000

∴ The correct answer is 1000.

Additional Information

Successive Percentage Change

If a value is changed by a% and then by b%, the net percentage change is given by the formula: [a + b + (ab/100)]%.

Multiplying Factor Method

An increase of r% is equivalent to multiplying by (1 + r/100), and a decrease of r% is equivalent to multiplying by (1 − r/100).

Base Value Property

The original value (March in this case) is always considered the 100% base for initial calculations.

Shortcut Trick

Assume an Assumed Mean (A) = 60.

Deviations from 60: −48, −35, −23, −12, −4, +1, +13, +25, +32, +44.

Sum of deviations (Σd) = −7.

Average deviation = Σd ÷ n = −7 ÷ 10 = −0.7.

Actual Mean = Assumed Mean + Average Deviation = 60 − 0.7 = 59.3.

∴ The correct answer is 59.3.

Alternate Method

Given:

Dataset = {12, 25, 37, 48, 56, 61, 73, 85, 92, 104}

Formula Used:

Arithmetic Mean = (Sum of all observations) ÷ (Total number of observations)

Calculations:

⇒ Sum = 12 + 25 + 37 + 48 + 56 + 61 + 73 + 85 + 92 + 104

⇒ Sum = 593

⇒ Total count (n) = 10

⇒ Arithmetic Mean = 593 ÷ 10

⇒ Arithmetic Mean = 59.3

∴ The correct answer is 59.3.

Additional Information

Median

The middle value of a sorted dataset. For even n, it is the average of the two middle terms: (56 + 61) ÷ 2 = 58.5.

Range

The difference between the maximum and minimum values: 104 − 12 = 92.

Relation between Mean, Median, and Mode

Empirical formula: Mode = 3 × Median − 2 × Mean.