Search Previous Year Questions

Given:

9² + (√4)² − 7√16 − 7

Formula used:

Square: a² = a × a

Square Root identity: (√a)² = a

Calculations:

To simplify the expression, solve each term separately:

9² + (√4)² − 7√16 − 7

⇒ 81 + 4 − (7 × 4) − 7

⇒ 81 + 4 − 28 − 7

⇒ 85 − 35

⇒ 50

∴ The correct answer is option (1).

Shortcut Trick

Cost Price (CP) of 1 notebook = 1000 ÷ 50 = ₹20.

Profit percentage = 20%.

Selling Price (SP) = CP × (100 + Profit)% = 20 × 120% = 20 × 1.2 = ₹24.

∴ The correct answer is ₹24.

Alternate Method

Given: Total quantity = 50 notebooks, Total CP = ₹1000, Profit = 20%.

Formula Used: CP per unit = Total CP ÷ Quantity; SP = CP + (Profit% of CP).

⇒ Cost Price of 1 notebook = 1000 ÷ 50

⇒ CP = ₹20

⇒ Profit amount = 20% of 20 = (20 ÷ 100) × 20

⇒ Profit = 0.2 × 20 = ₹4

⇒ Selling Price = CP + Profit = 20 + 4

⇒ SP = ₹24

∴ The correct answer is ₹24.

Additional Information

Profit and Loss Formulas

Profit % = (Profit ÷ CP) × 100. Loss % = (Loss ÷ CP) × 100. Always calculate percentage on the Cost Price unless stated otherwise.

Selling Price Calculation

SP = CP × (100 + Gain%) ÷ 100 when there is a profit, and SP = CP × (100 − Loss%) ÷ 100 when there is a loss.

Marked Price and Discount

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

Shortcut Trick

Ratio of Cost Price (CP) to Marked Price (MP) = (100 − Discount%) : (100 + Gain%)

⇒ CP : MP = (100 − 10) : (100 + 8) = 90 : 108 = 5 : 6

If no discount is allowed, Selling Price (SP) = Marked Price (MP) = 6 units.

⇒ Profit = SP − CP = 6 − 5 = 1 unit.

⇒ Gain percentage = (Profit / CP) × 100 = (1 / 5) × 100 = 20%.

∴ The correct answer is 20%.

Alternate Method

Given: Marked Price of refrigerator = ₹4,884, Discount = 10%, Gain = 8%.

Formula Used: CP = [MP × (100 − D%)] / (100 + G%)

⇒ Cost Price (CP) = [4884 × (100 − 10)] / (100 + 8)

⇒ CP = (4884 × 90) / 108

⇒ CP = 4884 × (5 / 6) = ₹4,070

⇒ If no discount is allowed, new Selling Price (SP) = Marked Price = ₹4,884

⇒ Profit = SP − CP = 4884 − 4070 = ₹814

⇒ Gain percentage = (Profit / CP) × 100

⇒ Gain percentage = (814 / 4070) × 100 = 20%

∴ The correct answer is 20%.

Additional Information

Relationship between CP and MP

The direct formula connecting Cost Price and Marked Price is CP/MP = (100 − D%)/(100 + G%). This eliminates the need to calculate SP.

Marked Price (MP)

Also known as the list price or tag price. Discounts are always calculated on the Marked Price.

Successive Discounts

If two discounts d₁ and d₂ are given, the effective discount is d₁ + d₂ − (d₁ × d₂)/100.

Shortcut Trick

Sum 5 years ago = 50 − (2 × 5) = 40 years.

Ratio 5 years ago (Father : Son) = 3 : 1.

Total units = 3 + 1 = 4 units ⇒ 4 units = 40 years ⇒ 1 unit = 10 years.

Age Difference = 3 − 1 = 2 units = 2 × 10 = 20 years.

Age difference always remains constant over time.

∴ The correct answer is 20 years.

Alternate Method

Given:

Present sum of Father and Son = 50 years.

Age ratio 5 years ago = 3 : 1.

Let the Son's age 5 years ago be x years.

⇒ Father's age 5 years ago = 3x years.

⇒ Present Age of Father = 3x + 5

⇒ Present Age of Son = x + 5

⇒ (3x + 5) + (x + 5) = 50

⇒ 4x + 10 = 50

⇒ 4x = 40 ⇒ x = 10

⇒ Father's present age = 3(10) + 5 = 35 years.

⇒ Son's present age = 10 + 5 = 15 years.

⇒ Age difference = 35 − 15 = 20 years.

∴ The correct answer is 20 years.

Additional Information

Constant Age Difference

The numerical difference between the ages of two individuals never changes with time, regardless of how many years pass or go back.

Sum of Ages Rule

If the sum of ages of 'n' people is S at present, then 't' years ago, the sum was S − (n × t).

Ratio of Ages

If the ratio of ages is a : b and the difference is D, then the ages are [a/(a−b)] × D and [b/(a−b)] × D respectively.

Shortcut Trick

Difference in Time Period = 5 years − 3 years = 2 years.

Effective Interest Percentage difference = 2 years × 4% = 8%.

Given that 8% of the Principal corresponds to ₹784.

Required Simple Interest for 8 years = 8 years × 4% = 32%.

By unitary method: 32% = (784 ÷ 8) × 32 = 98 × 32 = 3136.

∴ The correct answer is ₹3136.

Alternate Method

Given:

Rate of Interest (R) = 4% per annum

Difference in Simple Interest = ₹784

Time Durations: T₁ = 5 years, T₂ = 3 years, T₃ = 8 years

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

⇒ Let the Principal amount be P.

⇒ SI for 5 years − SI for 3 years = 784

⇒ (P × 4 × 5 ÷ 100) − (P × 4 × 3 ÷ 100) = 784

⇒ 20P ÷ 100 − 12P ÷ 100 = 784

⇒ 8P ÷ 100 = 784

⇒ P = (784 × 100) ÷ 8

⇒ P = ₹9800

⇒ SI for 8 years = (9800 × 4 × 8) ÷ 100

⇒ SI = 98 × 32

⇒ SI = ₹3136

∴ The correct answer is ₹3136.

Additional Information

Simple Interest Properties

For a fixed Principal and Rate, Simple Interest is directly proportional to Time. SI₁ ÷ T₁ = SI₂ ÷ T₂.

Amount Formula

The total Amount (A) is the sum of Principal (P) and Simple Interest (SI): A = P + (P × R × T ÷ 100) = P(1 + RT ÷ 100).

Rate Change Impact

If the Rate increases by r%, the additional interest earned for T years is (P × r × T) ÷ 100.

Shortcut Trick

When a sphere is melted and recast into a cone with the same radius (R = r):

Volume relationship: 4 × R = h

⇒ 4 × R = 84 cm ⇒ R = 21 cm

⇒ Volume of sphere = (4/3) × (22/7) × 21 × 21 × 21 = 88 × 441 = 38808 cm³.

∴ The correct answer is 38808 cm³.

Alternate Method

Given:

Radius of Sphere (R) = Radius of Cone (r)

Height of Cone (h) = 84 cm

Formula Used:

Volume of Sphere = (4/3) π R³

Volume of Cone = (1/3) π r² h

Calculations:

Since the sphere is melted and recast into a cone:

⇒ Volume of Sphere = Volume of Cone

⇒ (4/3) × π × R³ = (1/3) × π × r² × h

⇒ (4/3) × R³ = (1/3) × R² × 84 (Since R = r)

⇒ 4 × R = 84

⇒ R = 84 ÷ 4 = 21 cm

Now, calculate the volume of the original sphere:

⇒ Volume = (4/3) × (22/7) × 21 × 21 × 21

⇒ Volume = (4 × 22 × 21 × 21 × 21) ÷ 21

⇒ Volume = 88 × 441

⇒ Volume = 38808 cm³

∴ The correct answer is 38808 cm³.

Additional Information

Conservation of Volume

When a solid is melted and reshaped into another form, the total amount of material or volume remains constant (V₁ = V₂).

Total Surface Area of Sphere

The total surface area (TSA) of a sphere is given by the formula 4πR².

Slant Height of a Cone

The slant height (l) of a cone can be calculated using the Pythagorean theorem: l = √(r² + h²).

Shortcut Trick

Mass = Density × Volume

Weight (kg) = [4/3 × 22/7 × (13³ − 12³) × 39] ÷ 1000

⇒ Weight = [4/3 × 22/7 × 469 × 39] ÷ 1000

⇒ Weight = (4 × 22 × 67 × 13) ÷ 1000

⇒ Weight = 76648 ÷ 1000 = 76.648 kg

∴ The correct answer is 76.648 kg.

Alternate Method

Given:

Internal radius (r) = 12 cm

External radius (R) = 13 cm

Density of metal = 39 g/cm³

Formula Used:

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

Mass = Density × Volume

Calculations:

⇒ Volume = 4/3 × 22/7 × (13³ − 12³)

⇒ Volume = 4/3 × 22/7 × (2197 − 1728)

⇒ Volume = 4/3 × 22/7 × 469

⇒ Volume = 4/3 × 22 × 67 = 5896/3 cm³

⇒ Mass = 39 × (5896/3)

⇒ Mass = 13 × 5896 = 76648 g

⇒ Weight in kg = 76648 ÷ 1000

∴ The correct answer is 76.648 kg.

Additional Information

Volume of a Solid Sphere

The total space occupied by a solid sphere is given by the formula V = 4/3 πr³.

Surface Area of a Sphere

The total outer surface area of a sphere is calculated as A = 4πr².

Density-Mass-Volume Triangle

Mass = Density × Volume | Density = Mass ÷ Volume | Volume = Mass ÷ Density.

Unit Conversion

To convert grams to kilograms, divide the value by 1000 (1 kg = 1000 g).

Shortcut Trick

Equalize decimal places for all numbers: 5.00, 3.20, and 0.03.

Remove the decimal to treat them as integers: 500, 320, and 3.

Calculate LCM(500, 320, 3) = 24000.

Adjust the decimal back by dividing by 100: 24000 ÷ 100 = 240.

∴ The correct answer is 240.

Alternate Method

Given: Numbers are 5, 3.2, and 0.03.

Formula: LCM of Fractions = LCM of Numerators ÷ HCF of Denominators.

⇒ Convert decimals to simplest fractions:

⇒ 5 = 5/1

⇒ 3.2 = 32/10 = 16/5

⇒ 0.03 = 3/100

⇒ LCM of Numerators (5, 16, 3):

⇒ Prime factors: 5 = 5¹, 16 = 2⁴, 3 = 3¹

⇒ LCM = 2⁴ × 5¹ × 3¹ = 16 × 5 × 3 = 240

⇒ HCF of Denominators (1, 5, 100) = 1

⇒ Required LCM = 240 ÷ 1 = 240

∴ The correct answer is 240.

Additional Information

LCM of Fractions

LCM = LCM of numerators ÷ HCF of denominators. Ensure fractions are in their simplest form before applying the formula.

HCF of Fractions

HCF = HCF of numerators ÷ LCM of denominators. This is the inverse logic used for finding the LCM of fractions.

Product Rule for Two Numbers

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).

Shortcut Trick

Ratio of Initial Revenue to Final Revenue = 2,50,000 : 2,12,500

Divide both by 12,500 ⇒ 20 : 17

Decrease in units = 20 − 17 = 3 units

Percentage decrease = (3 ÷ 20) × 100 = 15%

∴ The correct answer is 15%.

Alternate Method

Given:

Initial Revenue = ₹2,50,000

Final Revenue = ₹2,12,500

Formula Used:

Percentage Decrease = [(Initial Value − Final Value) ÷ Initial Value] × 100

Calculations:

⇒ Amount of decrease = 2,50,000 − 2,12,500

⇒ Amount of decrease = ₹37,500

⇒ Percentage decrease = (37,500 ÷ 2,50,000) × 100

⇒ Percentage decrease = 3,75,00,00 ÷ 2,50,000

⇒ Percentage decrease = 375 ÷ 25

⇒ Percentage decrease = 15%

∴ The correct answer is 15%.

Additional Information

Percentage Increase

It is calculated as [(New Value − Old Value) ÷ Old Value] × 100 when the value grows over time.

Base Change Rule

In percentage calculations, the denominator is always the original (initial) value unless specified otherwise.

Fraction to Percentage Conversion

Multiplying a fraction by 100 converts it into a percentage. For example, 3/20 × 100 = 15%.

Shortcut Trick

Under the Buy 5, Get 2 Free offer, a customer pays for 5 markers but receives a total of 7 markers.

Total amount paid = 5 × ₹43 = ₹215

Effective price per marker = ₹215 ÷ 7 ≈ ₹30.71

∴ The correct answer is ₹30.71.

Alternate Method

Given:

Cost of one marker = ₹43

Offer = Buy 5, Get 2 Free

Formula Used:

Effective price per marker = Total cost paid ÷ Total markers received

Calculations:

⇒ Total markers received = 5 + 2 = 7 markers

⇒ Total price paid = 5 × ₹43 = ₹215

⇒ Effective price per marker = ₹215 ÷ 7

⇒ Effective price per marker = ₹30.7142...

⇒ Effective price per marker ≈ ₹30.71

∴ The correct answer is ₹30.71.

Additional Information

Discount % in 'Buy X Get Y Free' Offer

The discount percentage is calculated as: Discount % = [Y ÷ (X + Y)] × 100. For 'Buy 5, Get 2 Free', the equivalent discount is [2 ÷ 7] × 100 ≈ 28.57%.

Effective Price Formula

Effective Price = Original Price × (1 − Discount Rate) = Original Price × [X ÷ (X + Y)].

Successive Discounts Rule

If two successive discounts of a% and b% are offered on an item, the equivalent single discount is given by: (a + b − ab/100)%.