Search Previous Year Questions

Shortcut Trick

For a series of a repeated digit 'd' in the form d + dd + ddd + ... up to n terms (where n ≤ 9):

Sum = d × 123...n

Here, the digit d = 4 and there are 5 terms.

Sum = 4 × 12345 = 49380

∴ The correct answer is 49380.

Alternate Method

Given:

Expression = 4 + 44 + 444 + 4444 + 44444

Calculation:

Factoring out 4 from the expression:

⇒ 4 × (1 + 11 + 111 + 1111 + 11111)

Adding the numbers inside the bracket step-by-step:

⇒ 1 + 11 = 12

⇒ 12 + 111 = 123

⇒ 123 + 1111 = 1234

⇒ 1234 + 11111 = 12345

Now, multiply the sum by 4:

⇒ 4 × 12345

Breaking it down:

⇒ 4 × (10000 + 2000 + 300 + 40 + 5)

⇒ 40000 + 8000 + 1200 + 160 + 20 = 49380

∴ The correct answer is 49380.

Additional Information

Sum of Repeated 1s

The sum of a series 1 + 11 + 111 + ... up to n terms (for n ≤ 9) is always a sequential number 123...n.

Sum of Repeated 9s

For a series like 9 + 99 + 999 + ... up to n terms, rewrite each term as (10 − 1) + (100 − 1) + (1000 − 1) ... to simplify the calculation to (10 + 100 + 1000 + ...) − n.

Column Addition Shortcut for Repeated Digits

To add repeated digit terms vertically, count the digits in each column. The units column has n terms, tens has n−1 terms, and so on. Multiply the digit by the count and sequentially carry over the extra value to the next place.

Shortcut Trick

Convert all mixed fractions directly into decimals for faster calculation.

1 ²/₅ = 1 + 0.4 = 1.4

1 ³/₄ = 1 + 0.75 = 1.75

⇒ (1.4 + 2.6) − 1.75

⇒ 4.0 − 1.75 = 2.25

∴ The correct answer is 2.25.

Alternate Method

Given:

(1 ²/₅ + 2.6) − 1 ³/₄

Method Used:

Convert mixed fractions and decimals into improper fractions and simplify using BODMAS.

Calculations:

⇒ 1 ²/₅ = ^{(1 × 5 + 2)}/₅ = ⁷/₅

⇒ 2.6 = ²⁶/₁₀ = ¹³/₅

⇒ 1 ³/₄ = ^{(1 × 4 + 3)}/₄ = ⁷/₄

⇒ Expression = (⁷/₅ + ¹³/₅) − ⁷/₄

⇒ Expression = (²⁰/₅) − ⁷/₄

⇒ Expression = 4 − ⁷/₄

⇒ Expression = ^{(16 − 7)}/₄

⇒ Expression = ⁹/₄

⇒ Expression = 2.25

∴ The correct answer is 2.25.

Additional Information

BODMAS Rule

Simplification problems always follow the BODMAS order: Brackets, Orders, Division, Multiplication, Addition, and Subtraction.

Fraction to Decimal Conversion

Converting mixed fractions directly to decimals (e.g., ²/₅ = 0.4 and ³/₄ = 0.75) avoids LCM calculations and saves significant time in exams.

Mixed to Improper Fraction

A mixed fraction A ^B/_C can be converted to an improper fraction using the standard formula (A × C + B) ÷ C.

Shortcut Trick

Rewrite 3 as √9 and 2 as √4 to see the continuous pattern.

Rationalizing 1/(√a − √b) gives √a + √b when a − b = 1.

The given expression evaluates to: (√9 + √8) − (√8 + √7) + (√7 + √6) − (√6 + √5) + (√5 + √4)

All intermediate terms cancel out, leaving only the first and last parts: √9 + √4 = 3 + 2 = 5

∴ The correct answer is 5.

Alternate Method

Given:

1/(3 − √8) − 1/(√8 − √7) + 1/(√7 − √6) − 1/(√6 − √5) + 1/(√5 − 2)

Formula Used:

(a − b)(a + b) = a² − b²

Calculations:

⇒ Rationalize the first term:

⇒ 1/(3 − √8) = (3 + √8) / (3² − (√8)²) = (3 + √8) / (9 − 8) = 3 + √8

⇒ Rationalize the second term:

⇒ 1/(√8 − √7) = (√8 + √7) / ((√8)² − (√7)²) = (√8 + √7) / (8 − 7) = √8 + √7

⇒ Rationalize the third term:

⇒ 1/(√7 − √6) = (√7 + √6) / (7 − 6) = √7 + √6

⇒ Rationalize the fourth term:

⇒ 1/(√6 − √5) = (√6 + √5) / (6 − 5) = √6 + √5

⇒ Rationalize the fifth term:

⇒ 1/(√5 − 2) = (√5 + 2) / ((√5)² − 2²) = (√5 + 2) / (5 − 4) = √5 + 2

⇒ Substitute all these simplified values back into the original expression:

⇒ (3 + √8) − (√8 + √7) + (√7 + √6) − (√6 + √5) + (√5 + 2)

⇒ Expand by applying the minus signs:

⇒ 3 + √8 − √8 − √7 + √7 + √6 − √6 − √5 + √5 + 2

⇒ Cancel out the opposite matching pairs:

⇒ 3 + 0 + 0 + 0 + 0 + 2 = 5

∴ The correct answer is 5.

Additional Information

Rationalization of Surds

To remove a square root surd from the denominator, multiply both the numerator and denominator by its conjugate. The conjugate of (√a − √b) is (√a + √b).

Difference of Squares Formula

The standard algebraic identity (x − y)(x + y) = x² − y² is widely applied during rationalization to eliminate square root symbols entirely from denominators.

Continuous Series Simplification

When solving a continuous sequence of surd fractions, locating patterns like differences of 1 will allow quick cancellation of intermediate terms, drastically reducing calculation time.

Shortcut Trick

Total Football players = 25 (Both) + 45 (Only Football) = 70

Total Basketball players = 25 (Both) + 35 (Only Basketball) = 60

Ratio of Football to Basketball players = 70 ÷ 60 = 7 : 6

∴ The correct answer is 7:6.

Alternate Method

Given:

Students playing both Football and Basketball = 25

Students playing Football but not Basketball = 45

Students playing Basketball but not Football = 35

Formula Used:

Total players of a sport = Players of only that sport + Players of both sports

Calculation:

⇒ Number of students who play Football = 45 + 25 = 70

⇒ Number of students who play Basketball = 35 + 25 = 60

⇒ Required Ratio = Total Football players : Total Basketball players

⇒ Required Ratio = 70 : 60

⇒ Required Ratio = 7 : 6

∴ The correct answer is 7:6.

Additional Information

Venn Diagram Fundamentals

In a two-set Venn diagram, the total number of elements in set A is calculated by n(A) = n(Only A) + n(A and B).

Union of Two Sets

The total number of elements in at least one set is found using the formula: n(A or B) = n(A) + n(B) − n(A and B).

Ratio & Proportion

A ratio compares two quantities of the same kind. The ratio a:b is in its simplest form when a and b have no common factor other than 1.

Shortcut Trick

Work with ratios in 10-thousands to simplify the calculation (1 unit = ₹10,000).

Tina's equivalent capital = 10 × 6 + 8 × 6 = 60 + 48 = 108

Meera's equivalent capital = 8 × 6 + 12 × 6 = 48 + 72 = 120

Profit Ratio = 108 : 120 = 9 : 10

∴ The correct answer is 9:10.

Alternate Method

Given:

Tina's initial investment = ₹1,00,000

Meera's initial investment = ₹80,000

After 6 months, Tina withdrew ₹20,000 and Meera added ₹40,000.

₹100k × 6m

₹80k × 6m Tina

₹80k × 6m

₹120k × 6m Meera

Total = 1080k Total = 1200k Final Ratio = 1080 : 1200 = 9 : 10

Formula Used:

Profit Ratio = (C₁ × T₁ + C₂ × T₂) : (C₃ × T₃ + C₄ × T₄)

Calculations:

⇒ For the first 6 months, Tina's investment = ₹1,00,000

⇒ For the next 6 months, Tina's investment = ₹1,00,000 − ₹20,000 = ₹80,000

⇒ Total equivalent capital for Tina = (1,00,000 × 6) + (80,000 × 6) = 6,00,000 + 4,80,000 = ₹10,80,000

⇒ For the first 6 months, Meera's investment = ₹80,000

⇒ For the next 6 months, Meera's investment = ₹80,000 + ₹40,000 = ₹1,20,000

⇒ Total equivalent capital for Meera = (80,000 × 6) + (1,20,000 × 6) = 4,80,000 + 7,20,000 = ₹12,00,000

⇒ Profit sharing ratio = Tina's equivalent capital : Meera's equivalent capital

⇒ Profit sharing ratio = 10,80,000 : 12,00,000

⇒ Profit sharing ratio = 108 : 120 = 9 : 10

∴ The correct answer is 9:10.

Additional Information

Partnership Basic Rule

When investments are made for different time periods, the profit is distributed in the ratio of the product of capital and time. P₁ : P₂ = (C₁ × T₁) : (C₂ × T₂).

Compound Partnership

If a partner's capital changes during the year, calculate the equivalent capital by multiplying each unique capital amount by the respective number of months it remained invested, then sum them up.

Active vs Sleeping Partner

An active partner takes part in running the business and typically gets a salary or a fixed percentage of the total profits before the remaining profit is divided based on their capital ratios.

Shortcut Trick

A's management fee = 20% of ₹60,000 = ₹12,000

Remaining profit to be shared = 60,000 − 12,000 = ₹48,000

Profit sharing ratio of A and B = 20,000 : 30,000 = 2 : 3

A's profit share = (2/5) × 48,000 = ₹19,200

A's total share = 12,000 + 19,200 = ₹31,200

∴ The correct answer is ₹31,200.

Alternate Method

Given:

A's investment = ₹20,000

B's investment = ₹30,000

Total profit = ₹60,000

A's management commission = 20% of total profit

Total Profit: ₹60,000

A's Mgmt: ₹12,000

Remaining: ₹48,000

A's Share: ₹19,200

B's: ₹28,800

A's Total: ₹31,200

Calculations:

⇒ Profit distribution ratio = Ratio of investments (since time period is equal)

⇒ Ratio of A : B = 20,000 : 30,000 = 2 : 3

⇒ A's management fee = (20 / 100) × 60,000 = ₹12,000

⇒ Remaining profit = 60,000 − 12,000 = ₹48,000

⇒ A's share from remaining profit = 48,000 × [2 ÷ (2 + 3)]

⇒ A's share = 48,000 × (2 / 5) = ₹19,200

⇒ Total share of A = Management fee + Profit share

⇒ Total share of A = 12,000 + 19,200 = ₹31,200

∴ The correct answer is ₹31,200.

Additional Information

Active vs Sleeping Partner

An active partner manages the business and receives a salary or a fixed percentage of the profit before the remainder is split. A sleeping partner only invests capital and takes a share of the remaining profit based on investment and time.

Partnership Profit Formula

Profit is directly proportional to the product of capital invested and the time period for which it is invested. Profit Ratio = (C₁ × T₁) : (C₂ × T₂).

Shortcut Trick

Net increase in average = 48 − 44 = 4 runs

Total extra runs scored above the old average = 132 − 44 = 88 runs

Total number of innings played (new) = Total extra runs ÷ Net increase = 88 ÷ 4 = 22

Earlier innings played = 22 − 1 = 21

∴ The correct answer is 21 innings.

Alternate Method

Given:

Initial average = 44

Score in the next inning = 132

New average = 48

Formula Used:

Average = Total Runs ÷ Number of Innings

Avg = 44x inningsDeficit = 4 × xAvg = 48Next inningScore = 132Surplus = 84New Avg = 48132 runs

Calculation:

Let the number of innings played earlier be x.

⇒ Total runs scored in earlier innings = 44 × x = 44x

⇒ Total runs after the next inning = 44x + 132

⇒ New average = (44x + 132) ÷ (x + 1)

⇒ 48 = (44x + 132) ÷ (x + 1)

⇒ 48 × (x + 1) = 44x + 132

⇒ 48x + 48 = 44x + 132

⇒ 48x − 44x = 132 − 48

⇒ 4x = 84

⇒ x = 21

∴ The correct answer is 21 innings.

Additional Information

Batting Average

Batting average is calculated as the total number of runs scored divided by the total number of times the batsman was dismissed. Formula: Total Runs = Average × Innings.

Bowling Average

Bowling average is the number of runs conceded per wicket taken. Formula: Bowling Average = Total Runs Conceded ÷ Total Wickets Taken.

Deviation Method for Averages

When a new observation is added, the new average changes based on the deviation of the new value from the old average. Change in Average = (New Value − Old Average) ÷ New Total Number of Observations.

Shortcut Trick

Net deviation of the first 10 numbers from the overall average 70 = 5 × (60 − 70) + 5 × (75 − 70) = −50 + 25 = −25.

To balance the overall average, the sum of the last 3 numbers must be = 3 × 70 − (−25) = 235.

Let the 13^th number be x. Then 11^th = x − 6, and 12^th = x + 4.

Sum = (x − 6) + (x + 4) + x = 3x − 2 = 235 ⇒ x = 79.

The 11^th number = 73, and the 12^th number = 83. Their average = (73 + 83) ÷ 2 = 78.

∴ The correct answer is 78.

Alternate Method

Given:

Average of 13 numbers = 70

Average of first 5 numbers = 60

Average of next 5 numbers = 60 + (25% of 60) = 75

11^th number = 13^th number − 6

12^th number = 13^th number + 4

1^st 5 Nums Avg: 60 Sum: 300

Next 5 Nums Avg: 75 Sum: 375

11^th, 12^th, 13^th Avg: ? Sum: 235

Total Sum of 13 Nums = 910 (Avg = 70)

Formula Used:

Sum of observations = Average × Number of observations

Calculations:

⇒ Total sum of 13 numbers = 13 × 70 = 910

⇒ Sum of first 5 numbers = 5 × 60 = 300

⇒ Sum of next 5 numbers = 5 × 75 = 375

⇒ Sum of the remaining 3 numbers (11^th, 12^th, 13^th) = 910 − (300 + 375) = 235

⇒ Let the 13^th number be x.

⇒ 11^th number = x − 6

⇒ 12^th number = x + 4

⇒ Sum of these three numbers = (x − 6) + (x + 4) + x = 3x − 2

⇒ 3x − 2 = 235

⇒ 3x = 237

⇒ x = 79

⇒ 11^th number = 79 − 6 = 73

⇒ 12^th number = 79 + 4 = 83

⇒ Average of 11^th and 12^th numbers = (73 + 83) ÷ 2 = 156 ÷ 2 = 78

∴ The correct answer is 78.

Additional Information

Weighted Average

When different groups have different averages, the combined average is calculated as (n₁A₁ + n₂A₂ + ...) ÷ (n₁ + n₂ + ...).

Net Deviation Method

The sum of deviations of all observations from their arithmetic mean is always zero. This property is highly useful for finding missing observations in a data set without calculating heavy sum values.

Successive Percentage Increase

If a base value is increased by x%, the new value is obtained by multiplying the base by (1 + x/100). For example, a 25% increase on a base of 60 makes it 60 × 1.25 = 75.

Shortcut Trick

Let the total eligible voters be V.

V × 80% × 95% × 60% = 11400

V × (4/5) × (19/20) × (3/5) = 11400

V × (228/500) = 11400 ⇒ V = (11400 × 500) ÷ 228 = 25000

∴ The correct answer is 25000.

Alternate Method

Given:

Votes cast = 80% of total eligible voters

Invalid votes = 5% of cast votes

Winner's votes = 60% of valid votes = 11400

Total Eligible Voters = 100x

Votes Cast = 80x

Valid Votes = 76x

Winner's Votes = 45.6x = 11400

80% cast

95% valid

60% of valid

Calculations:

⇒ Let the total eligible voters be 100x.

⇒ Votes cast = 80% of 100x = 80x

⇒ Valid votes = 80x − (5% of 80x) = 80x − 4x = 76x

⇒ Winner's votes = 60% of 76x = 45.6x

⇒ We are given that Winner's votes = 11400

⇒ 45.6x = 11400

⇒ x = 11400 ÷ 45.6 = 250

⇒ Total eligible voters = 100x = 100 × 250 = 25000

∴ The correct answer is 25000.

Additional Information

Election Percentage Concept

Always maintain the chain: Total Voters → Votes Cast → Valid Votes → Winner's Votes. Subtract invalid votes from cast votes, not from total eligible voters.

Successive Percentage

When a value is subjected to multiple percentage changes sequentially, Final Value = Initial Value × (1 ± p₁/100) × (1 ± p₂/100).

Base Change in Percentage

The base for the winner's 60% share is 'Valid Votes' and not 'Total Voters'. Always correctly identify the base "of what" the percentage is calculated.

Shortcut Trick

Effective rate for the 1^st year = 18%

Effective rate for the next 1/2 year = 18% ÷ 2 = 9%

Total effective Compound Interest % = 18 + 9 + (18 × 9) / 100 = 28.62%

According to the question, 28.62% of Principal = ₹2862

Principal = (2862 ÷ 28.62) × 100 = 10000

∴ The correct answer is ₹10,000.

Alternate Method

Given:

Compound Interest (CI) = ₹2862

Rate of Interest (R) = 18% p.a.

Time (T) = 1.5 years

Formula Used:

For time = n + f years (where n is the complete year and f is the fraction), compounded yearly:

Amount (A) = P × (1 + R/100)ⁿ × (1 + fR/100)

Compound Interest (CI) = A − P

100% (P)

+18% (1 Year)

118%

+9% (1/2 Year)

128.62%

Total CI = 28.62% = ₹2862

Calculations:

⇒ Amount (A) = P × (1 + 18/100) × (1 + (1/2 × 18)/100)

⇒ A = P × (118/100) × (109/100)

⇒ A = P × 1.18 × 1.09

⇒ A = 1.2862 P

⇒ CI = A − P = 1.2862 P − P = 0.2862 P

⇒ 0.2862 P = 2862

⇒ P = 2862 ÷ 0.2862

⇒ P = 10000

∴ The correct answer is ₹10,000.

Additional Information

Fractional Year Compounding

When time is given in fractions (e.g., n full years and f fraction of a year), interest for the fractional year is applied as simple interest on the compounded amount: A = P × (1 + R/100)ⁿ × (1 + fR/100).

Effective Interest Rate

For two successive periods with different rates R₁ and R₂, the net effective rate is given by the formula: R₁ + R₂ + (R₁ × R₂) / 100.

Half-Yearly & Quarterly Compounding

If compounded half-yearly, the rate becomes R/2 and the time periods become 2T. If quarterly, the rate is R/4 and the time is 4T.