Search Previous Year Questions

Given:

$\sqrt{\dfrac{169^2 \times 2.56^2 \times 0.25^2}{13^4 \times 1.6^4 \times 0.5^2}}$ = ?

Formula used:

Perfect Squares: 169 = 13², 2.56 = 1.6², and 0.25 = 0.5²

Laws of Indices: (a^m)ⁿ = a^mn and a^m ÷ aⁿ = a^m−n

Calculations:

Substitute the perfect square values into the expression:

⇒ $\sqrt{\dfrac{(13^2)^2 \times (1.6^2)^2 \times (0.5^2)^2}{13^4 \times 1.6^4 \times 0.5^2}}$

⇒ $\sqrt{\dfrac{13^4 \times 1.6^4 \times 0.5^4}{13^4 \times 1.6^4 \times 0.5^2}}$

Cancel common terms 13⁴ and 1.6⁴ from the numerator and denominator:

⇒ √0.5^{(4 − 2)}

⇒ √0.5²

⇒ 0.5

∴ The correct answer is option (1).

Shortcut Trick

Difference in percentage = 40% − 25% = 15%

Given that 15% of the number corresponds to 42.

Therefore, 1% of the number = 42 ÷ 15 = 2.8

The total number (100%) = 2.8 × 100 = 280.

∴ The correct answer is 280.

Alternate Method

Given: Let the number be x. The value of 40% of x is 42 more than 25% of x.

Formula Used: Percentage Value = (Percentage ÷ 100) × Number

⇒ (40 ÷ 100) × x − (25 ÷ 100) × x = 42

⇒ 0.40x − 0.25x = 42

⇒ 0.15x = 42

⇒ x = 42 ÷ 0.15

⇒ x = (42 × 100) ÷ 15

⇒ x = 280

∴ The correct answer is 280.

Additional Information

Basic Percentage Formula

To find p% of a number (N), use the relation: Value = (p × N) ÷ 100.

Percentage Difference Principle

When comparing two percentages of the same base, the difference in their percentage values is directly proportional to the difference in their absolute values.

Fraction Equivalents

Using fractions can speed up calculations: 25% = 1/4, 40% = 2/5, 50% = 1/2, and 75% = 3/4.

Shortcut Trick

The required number must be of the form: LCM(11, 7, 14) × k + 2.

LCM of 11, 7, and 14 = 154.

Largest 4-digit multiple of 154: 154 × 64 = 9856.

Required number = 9856 + 2 = 9858.

∴ The correct answer is 9858.

Alternate Method

Given: Divisors = 11, 7, 14; Remainder = 2 in each case; Target = Largest 4-digit number.

Step 1: Calculate the LCM of the given divisors.

⇒ Prime factorization: 11 = 11¹, 7 = 7¹, 14 = 2 × 7 ⇒ LCM(11, 7, 14) = 11 × 2 × 7 = 154

Step 2: Find the largest 4-digit multiple of 154.

⇒ Largest 4-digit number = 9999 ⇒ 9999 ÷ 154 → Quotient = 64, Remainder = 143 ⇒ Largest multiple = 9999 − 143 = 9856

Step 3: Add the common remainder to the multiple.

⇒ Required number = 9856 + 2 = 9858

∴ The correct answer is 9858.

Additional Information

General Remainder Theorem

For a number that leaves the same remainder 'r' when divided by a, b, and c, the general form is LCM(a, b, c) × k + r.

Divisibility Rule of 7

Double the last digit and subtract it from the remaining number; if the result is divisible by 7, the original number is also divisible by 7.

Divisibility Rule of 11

A number is divisible by 11 if the difference between the sum of digits at odd places and the sum of digits at even places is 0 or a multiple of 11.

Given:

Amount of sugar = 0.75 cups

Amount of flour = 1 1/4 cups

Formula used:

Total amount = Amount of sugar + Amount of flour

Calculations:

First, convert the mixed fraction representing the amount of flour into a decimal:

1 1/4 = 1 + (1 ÷ 4)

⇒ 1 + 0.25 = 1.25 cups

Now, add the quantity of sugar and flour to find the total ingredients used in the recipe:

Total = 0.75 + 1.25

⇒ Total = 2.0 cups

∴ The correct answer is option (4).

Shortcut Trick

The age difference between two people remains constant over time (8 years).

7 years ago, the ratio of Ramu's age to Keeran's age was 3 : 1.

Difference in ratio units = 3 − 1 = 2 units → 8 years.

1 ratio unit = 8 ÷ 2 = 4 years.

Ramu's age 7 years ago = 3 units × 4 = 12 years.

Ramu's present age = 12 + 7 = 19 years.

∴ The correct answer is 19 years.

Alternate Method

Given:

Difference between Ramu and Keeran = 8 years

Condition: 7 years ago, Ramu was three times as old as Keeran.

Formula Used:

Present Age = Age (n years ago) + n; Age difference is always constant.

Calculations:

⇒ Let Keeran's present age be K years.

⇒ Ramu's present age = (K + 8) years.

⇒ 7 years ago, Keeran's age = (K − 7) and Ramu's age = (K + 8 − 7) = (K + 1).

⇒ According to the question: (K + 1) = 3 × (K − 7)

⇒ K + 1 = 3K − 21

⇒ 3K − K = 21 + 1

⇒ 2K = 22

⇒ K = 11 years

⇒ Ramu's present age = 11 + 8 = 19 years.

∴ The correct answer is 19 years.

Additional Information

Constant Age Difference

The numerical difference between the ages of two individuals never changes, regardless of how many years pass or how many years ago the comparison was made.

Ratio Method for Ages

When a ratio is given for a specific time, calculate the difference in ratio units and equate it to the actual constant age difference to find the value of one unit.

Past and Future age Calculation

If the present age is A, then the age 'n' years ago was (A − n) and the age after 'n' years will be (A + n).

Shortcut Trick

Ratio of investments A : B = 12,000 : 8,000 = 3 : 2.

Total units = 3 + 2 = 5 units.

5 units = ₹10,000 → 1 unit = ₹2,000.

A's share = 3 units = 3 × ₹2,000 = ₹6,000.

∴ The correct answer is ₹6,000.

Alternate Method

Given:

Investment of Partner A = ₹12,000

Investment of Partner B = ₹8,000

Total profit = ₹10,000

Formula Used:

In a partnership, if the time period is constant, the profit is distributed in the ratio of the capitals invested: Profit Ratio = Capital Ratio.

⇒ Profit ratio (A : B) = 12,000 : 8,000

⇒ Profit ratio (A : B) = 3 : 2

⇒ Sum of ratio parts = 3 + 2 = 5 units

⇒ 5 units = ₹10,000

⇒ A's share of profit = (3 ÷ 5) × 10,000 = ₹6,000

∴ The correct answer is ₹6,000.

Additional Information

Basic Partnership

When investments are made for the same duration, the profit share is directly proportional to the capital: P₁ : P₂ = C₁ : C₂.

Compound Partnership

When investments are for different durations, profit share depends on both capital and time: P₁ : P₂ = (C₁ × T₁) : (C₂ × T₂).

Working vs. Sleeping Partner

A working partner manages the business and may receive a salary or fixed percentage of profit before the remaining profit is divided.

Shortcut Trick

Check if a + b + c = 0 for the terms 23, −19, and −4.

Since 23 + (−19) + (−4) = 0, apply the Algebraic Identity: a³ + b³ + c³ = 3abc.

Calculation → 3 × 23 × (−19) × (−4) = 3 × 23 × 76.

Product → 69 × 76 = 5244.

∴ The correct answer is 5244.

Alternate Method

Given: a = 23, b = −19, c = −4

Formula Used: a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca)

Step-by-step Calculation:

⇒ First, check the sum: 23 − 19 − 4 = 0.

⇒ When the sum of three numbers is zero, the sum of their cubes equals three times their product.

⇒ Result = 3 × (23) × (−19) × (−4)

⇒ Result = 3 × 23 × 76

⇒ Result = 69 × 76

⇒ Result = 5244.

∴ The correct answer is 5244.

Additional Information

Special Algebraic Identity

The general identity is a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca). If (a + b + c) = 0, the entire right-hand side becomes zero.

Alternative Form

The term (a² + b² + c² − ab − bc − ca) can also be written as 1/2[(a − b)² + (b − c)² + (c − a)²].

Product Calculation Hint

Multiplying 69 × 76 can be simplified as (70 − 1) × 76 = 5320 − 76 = 5244.

Shortcut Trick

Assume Total Work = LCM(12, 30) = 60 units.

Efficiency of A = 5 units/day and Efficiency of B = 2 units/day.

Work done in 4 days by both = (5 + 2) × 4 = 28 units.

Remaining work = 60 − 28 = 32 units.

Time taken by B = 32 ÷ 2 = 16 days.

∴ The correct answer is 16 days.

Alternate Method

Given: Time taken by A = 12 days, Time taken by B = 30 days, Working together time = 4 days.

Formula: Work = Efficiency × Time

⇒ Total work to be done = LCM of 12 and 30 = 60 units.

⇒ Daily work (Efficiency) of A = 60 ÷ 12 = 5 units/day.

⇒ Daily work (Efficiency) of B = 60 ÷ 30 = 2 units/day.

⇒ Work completed by A and B together in 4 days = (5 + 2) × 4 = 7 × 4 = 28 units.

⇒ Remaining work = 60 − 28 = 32 units.

⇒ Time taken by B to complete the remaining work = 32 ÷ 2 = 16 days.

∴ The correct answer is 16 days.

Additional Information

Total Work Concept

The Total Work is usually taken as the LCM of the individual days to avoid working with complex fractions.

Efficiency and Time

Efficiency is the amount of work done in a single unit of time. It is inversely proportional to the Time taken to complete the task.

Work and Wages

When workers are paid for a job, the wages are distributed in the ratio of the work done by each person, which is the product of Efficiency × Time.

Given:

Mathematical expression = 0.5 × 0.125 ÷ 8⁻²

Formula used:

Using laws of indices: a⁻ⁿ = 1 / aⁿ

Calculation:

Convert the decimal values into standard fractions:

⇒ 0.5 = 1/2

⇒ 0.125 = 1/8

Substitute these values into the given expression:

⇒ (1/2) × (1/8) ÷ (1 / 8²)

⇒ (1/16) ÷ (1/64)

Applying the division of fractions rule:

⇒ (1/16) × 64

⇒ 4

∴ The correct answer is option (3).

Shortcut Trick

Use the ratio CP : MP = (100 − Discount%) : (100 + Profit%)

Initial Ratio: CP : MP = (100 − 25) : (100 + 10) = 75 : 110

New Selling Price (SP) at 10% discount = 110 − 11 = 99 units

New Profit % = [(99 − 75) ÷ 75] × 100 = 32%

∴ The correct answer is 32%.

Alternate Method

Given:

Initial Discount = 25% | Initial Profit = 10% | New Discount = 10%

Formula Used:

1. SP = CP × (1 + P/100) | 2. SP = MP × (1 − D/100) | 3. CP/MP = (100 − D)/(100 + P)

Calculations:

⇒ Let Cost Price (CP) = 100 − 25 = 75 units

⇒ Marked Price (MP) = 100 + 10 = 110 units

⇒ New Discount = 10% of MP = 0.10 × 110 = 11 units

⇒ New Selling Price (SP) = MP − Discount = 110 − 11 = 99 units

⇒ Profit = SP − CP = 99 − 75 = 24 units

⇒ Profit % = (Profit ÷ CP) × 100

⇒ Profit % = (24 ÷ 75) × 100 = 32%

∴ The correct answer is 32%.

Additional Information

Relationship between CP and MP

The ratio MP ÷ CP = (100 + P%) ÷ (100 − D%) is a direct result of equate SP in both profit and discount scenarios.

Markup Percentage

It is the percentage by which the Cost Price is increased to set the Marked Price. Formula: Markup % = [(MP − CP) ÷ CP] × 100.

Successive Discounts

If two successive discounts of a% and b% are given, the Effective Discount = (a + b − ab/100)%.