Search Previous Year Questions

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.