Quantitative Aptitude PYQs

Shortcut Trick

Solve the denominator directly: 24² + 6² − 144 = 576 + 36 − 144 = 468.

Convert the decimal numerator to an integer with a negative power of 10: 0.01404 = 1404 × 10⁻⁵.

Divide the converted numerator by the denominator: 1404 ÷ 468 = 3.

Combine with the power of 10 to get the final result: 3 × 10⁻⁵.

∴ The correct answer is 3 × 10⁻⁵.

Alternate Method

Given:

Expression = 0.01404 ÷ (24² + 6² − 144)

Calculations:

⇒ First, simplify the terms inside the parentheses (denominator).

⇒ 24² = 576

⇒ 6² = 36

⇒ Denominator = 576 + 36 − 144

⇒ Denominator = 612 − 144 = 468

⇒ Substitute the simplified denominator back into the expression.

⇒ Expression = 0.01404 ÷ 468

⇒ To make the division easier, write 0.01404 as an integer multiplied by a base of 10.

⇒ 0.01404 = 1404 × 10⁻⁵

⇒ Expression = (1404 × 10⁻⁵) ÷ 468

⇒ Expression = (1404 ÷ 468) × 10⁻⁵

⇒ Expression = 3 × 10⁻⁵

∴ The correct answer is 3 × 10⁻⁵.

Additional Information

BODMAS Rule

Simplification problems strictly follow the BODMAS order: Brackets, Orders (powers/roots), Division/Multiplication, and Addition/Subtraction.

Decimal to Exponent Conversion

A decimal number can be seamlessly converted to an integer by multiplying it with a negative power of 10 that perfectly matches the number of decimal places shifted (e.g., 0.01404 = 1404 × 10⁻⁵).

Algebraic Identity Application

The denominator pattern in the expression resembles the partial factor (a² + b² − ab), which inherently stems from the sum of cubes formula: a³ + b³ = (a + b)(a² − ab + b²).

Shortcut Trick

If a sum becomes n times in t years at compound interest, it becomes n^m times in m × t years.

Here, the sum becomes 2 times in 4 years.

We need to find the time for the sum to become 8 times itself.

8 can be written as 2³ (so m = 3).

Time required = 3 × 4 = 12 years.

∴ The correct answer is 12 years.

Alternate Method

Given:

A sum invested at compound interest doubles itself in 4 years.

Formula Used:

A = P(1 + R/100)^T

0 yrs P

4 yrs 2P

8 yrs 4P

12 yrs 8P

×2

×2

×2

Calculations:

⇒ 2P = P(1 + R/100)⁴

⇒ 2 = (1 + R/100)⁴

Let the sum become 8 times in T years.

⇒ 8P = P(1 + R/100)^T

⇒ 8 = (1 + R/100)^T

We know that 8 can be written as 2³:

⇒ 2³ = (1 + R/100)^T

Substitute the value of 2 from the first equation:

⇒ ((1 + R/100)⁴)³ = (1 + R/100)^T

⇒ (1 + R/100)¹² = (1 + R/100)^T

Comparing the exponents on both sides:

⇒ T = 12

∴ The correct answer is 12 years.

Additional Information

Simple vs Compound Interest

Simple Interest grows by adding a constant amount each year, whereas Compound Interest grows by multiplying a constant factor each year.

Compound Interest Formula

Amount A = P(1 + R/100)ⁿ and CI = A − P, where P is the principal sum, R is the rate of interest, and n is the time period.

Rule of 72

The Rule of 72 is a quick mental math shortcut to estimate the number of years required to double your money at a given annual compound interest rate (Years ≈ 72 ÷ Rate).

Shortcut Trick

Every Whole number is automatically an Integer, a Rational number, and a Real number.

The set of Whole numbers is {0, 1, 2, 3, ...}.

We need a non-zero number, which eliminates 0.

Among the given options, only 1 is a non-zero whole number.

∴ The correct answer is 1.

Alternate Method

Given:

Sets to check: Whole, Integer, Rational, and Real numbers.

Condition: The number must be non-zero.

Definitions:

⇒ Whole numbers = {0, 1, 2, 3, ...}

⇒ Integers = {..., −2, −1, 0, 1, 2, ...}

⇒ Rational numbers = Numbers of the form p/q (where q ≠ 0).

⇒ Real numbers = All rational and irrational numbers.

Checking the Options:

⇒ Option 1: 1 is non-zero. It is a Whole number, Integer, Rational (1/1), and Real. (Satisfies all)

⇒ Option 2: 0 is a Whole number, but the question asks for a non-zero number. (Rejected)

⇒ Option 3: −1 is an Integer but NOT a Whole number. (Rejected)

⇒ Option 4: √2 is a Real number but NOT a Whole, Integer, or Rational number. (Rejected)

∴ The correct answer is 1.

Additional Information

Natural Numbers

Counting numbers starting from 1 are called Natural numbers (N = {1, 2, 3, 4, ...}). They do not include 0, fractions, or decimals.

Irrational Numbers

Numbers that cannot be written as a simple fraction p/q. Their decimal expansions are non-terminating and non-repeating. Examples: √2, π, e.

Prime Numbers

Natural numbers greater than 1 that have exactly two distinct positive divisors: 1 and themselves. Examples: 2, 3, 5, 7, 11.

Shortcut Trick

To directly find the speed in km/h, calculate distance over time and multiply by the conversion factor (18 ÷ 5).

Speed = (300 ÷ 24) × (18 ÷ 5)

Speed = 12.5 × 3.6 = 45 km/h

∴ The correct answer is 45 km/h.

Alternate Method

Given:

Distance covered by the athlete = 300 m

Time taken = 24 seconds

Formula Used:

Speed = Distance ÷ Time

Speed in km/h = Speed in m/s × (18 ÷ 5)

Start Finish Distance = 300 m  A   Time = 24 s Speed = ? km/h

Calculations:

⇒ Speed in m/s = 300 ÷ 24

⇒ Speed = 12.5 m/s

⇒ Speed in km/h = 12.5 × (18 ÷ 5)

⇒ Speed in km/h = 2.5 × 18

⇒ Speed = 45 km/h

∴ The correct answer is 45 km/h.

Additional Information

Speed Conversion Factors

To convert speed from m/s to km/h, always multiply by 18 ÷ 5. Conversely, to convert from km/h to m/s, multiply by 5 ÷ 18.

Average Speed

When an object travels equal distances at two different speeds S₁ and S₂, the average speed for the entire journey is given by (2 × S₁ × S₂) ÷ (S₁ + S₂).

Relative Speed

If two bodies move in the same direction at speeds S₁ and S₂, their relative speed is |S₁ − S₂|. If they move in opposite directions, the relative speed is S₁ + S₂.

Shortcut Trick

According to the given condition, directly equate the volume and surface area of the sphere:

⇒ (4/3)πr³ = 4πr²

⇒ r/3 = 1 (Canceling 4πr² from both sides)

⇒ r = 3

∴ The correct answer is 3 units.

Alternate Method

Given:

Volume of the sphere = Surface area of the sphere

Formula Used:

Volume of a sphere = (4/3)πr³

Surface Area of a sphere = 4πr²

 O r Volume = Surface Area

Calculation:

According to the given condition:

⇒ (4/3) × π × r³ = 4 × π × r²

⇒ (4/3) × r³ = 4 × r²

⇒ (1/3) × r = 1 (Since radius r ≠ 0, we can divide both sides by 4r²)

⇒ r = 3 × 1

⇒ r = 3 units

∴ The correct answer is 3 units.

Additional Information

Volume of a Hemisphere

When a sphere is divided into two equal parts, each part is called a hemisphere. Volume = (2/3)πr³

Curved Surface Area (CSA) of a Hemisphere

The surface area of the curved surface only, excluding the flat circular top. CSA = 2πr²

Total Surface Area (TSA) of a Hemisphere

Includes the curved surface area plus the area of the flat circular base (πr²). TSA = 3πr²

Shortcut Trick

Convert time directly to hours: 48 minutes = 48 ÷ 60 = 4/5 hours.

Speed = Distance ÷ Time

Speed = 7.2 ÷ (4/5) = 7.2 × 5/4 = 9 km/h.

∴ The correct answer is 9 km/h.

Alternate Method

Start End Distance = 7.2 km Time = 48 minutes Speed = ? km/h

Given:

Distance = 7.2 km

Time = 48 minutes

Formula Used:

Speed = Distance ÷ Time

Calculations:

⇒ First, convert time from minutes to hours to match the required speed unit (km/h).

⇒ Time = 48 ÷ 60 hours

⇒ Time = 0.8 hours

⇒ Speed = 7.2 ÷ 0.8

⇒ Speed = 72 ÷ 8

⇒ Speed = 9 km/h

∴ The correct answer is 9 km/h.

Additional Information

Conversion of Speed Units

To convert speed from km/h to m/s, multiply the speed by 5/18. To convert from m/s to km/h, multiply by 18/5.

Average Speed

Average Speed is calculated as the total distance traveled divided by the total time taken (Total Distance ÷ Total Time).

Relative Speed

When two bodies move in the same direction, their relative speed is the difference of their speeds (S₁ − S₂). When moving in opposite directions, it is the sum of their speeds (S₁ + S₂).

Shortcut Trick

Profit ratio is directly proportional to (Capital × Time).

Ratio of X : Y : Z = (50 × 12) : (100 × 12) : (150 × 6)

Ratio = 600 : 1200 : 900 = 2 : 4 : 3

Total profit = 2 + 4 + 3 = 9 parts = Rs. 90,000 ⇒ 1 part = Rs. 10,000

Difference in earnings of Z and X = 3 − 2 = 1 part.

∴ The correct answer is Rs. 10,000.

Alternate Method

Given:

Investment by X = Rs. 50,000

Investment by Y = Rs. 1,00,000

Investment by Z = Rs. 1,50,000

Total Profit at year-end = Rs. 90,000

Formula Used:

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

X 50k 12 mths Ratio: 2 Y 100k 12 mths Ratio: 4 Z 150k 6 mths Ratio: 3 Diff (Z − X) = 3 − 2 = 1 part

Calculations:

⇒ Time period for X = 12 months

⇒ Time period for Y = 12 months

⇒ Z joined after 6 months, so Time period for Z = 12 − 6 = 6 months

⇒ Profit Ratio (X : Y : Z) = (50,000 × 12) : (1,00,000 × 12) : (1,50,000 × 6)

⇒ Ratio = 6,00,000 : 12,00,000 : 9,00,000

⇒ Dividing each term by 3,00,000 ⇒ Ratio = 2 : 4 : 3

⇒ Total profit parts = 2 + 4 + 3 = 9 parts

⇒ 9 parts = Rs. 90,000

⇒ 1 part = 90,000 ÷ 9 = Rs. 10,000

⇒ Profit share of X = 2 × 10,000 = Rs. 20,000

⇒ Profit share of Z = 3 × 10,000 = Rs. 30,000

⇒ Difference (Z − X) = 30,000 − 20,000 = Rs. 10,000

∴ The correct answer is Rs. 10,000.

Additional Information

Compound Partnership

When partners invest different capital amounts for different time durations, the profit is divided in the ratio of the product of their respective capital and time period: (C₁ × T₁ : C₂ × T₂).

Joining Mid-Way

If a partner joins the business later in the year, their effective time period is calculated by subtracting their joining time from the total business duration. Here, Z joined after 6 months, so the effective active time was 12 − 6 = 6 months.

Active vs Sleeping Partner

A partner who physically manages the business is called an active or working partner. They often receive a fixed salary or commission from the net profit before the remaining amount is distributed among all partners based on their capital ratios.

Shortcut Trick

Total parts in ratio = 2 + 3 + 4 = 9 parts.

Value of 9 parts = ₹9000 ⇒ 1 part = ₹1000.

Difference in ratio (Highest − Lowest) = 4 − 2 = 2 parts.

Difference amount = 2 × 1000 = ₹2000.

∴ The correct answer is ₹2000.

Alternate Method

Given:

Total prize amount = ₹9000

Ratio of shares of three friends = 2 : 3 : 4

2 Parts 3 Parts 4 Parts Total Prize = ₹9000 (9 Parts) Lowest Highest Difference = 4 − 2 = 2 Parts

Calculations:

Let the shares of the three friends be 2x, 3x, and 4x respectively.

⇒ Total sum = 2x + 3x + 4x = 9x

⇒ We are given that the total sum is ₹9000.

⇒ 9x = 9000

⇒ x = 9000 ÷ 9 = 1000

⇒ The highest share = 4x = 4 × 1000 = 4000

⇒ The lowest share = 2x = 2 × 1000 = 2000

⇒ Difference = 4000 − 2000 = 2000

∴ The correct answer is ₹2000.

Additional Information

Compound Ratio

The ratio of the product of the antecedents to that of the consequents of two or more given ratios is called the compound ratio. If a:b and c:d are two ratios, their compound ratio is (a × c):(b × d).

Duplicate & Triplicate Ratios

The duplicate ratio of a:b is a²:b². The triplicate ratio of a:b is a³:b³.

Proportion Rule

An equality of two ratios is called a proportion. If a:b = c:d, then a, b, c, and d are in proportion. The fundamental rule is: Product of extremes = Product of means (a × d = b × c).

Shortcut Trick

Let the base and height of the triangle be 5x and 4x respectively.

Area = (1 ÷ 2) × Base × Height = (1 ÷ 2) × 5x × 4x = 10x².

Given Area = 160 cm² ⇒ 10x² = 160 ⇒ x² = 16 ⇒ x = 4.

Base = 5 × 4 = 20 cm, and Height = 4 × 4 = 16 cm.

∴ The correct answer is 20 cm and 16 cm.

Alternate Method

Given:

Ratio of base to height = 5 : 4

Area of the triangular flag = 160 cm²

Formula Used:

Area of a Triangle = (1 ÷ 2) × Base × Height

Base = 5x Height = 4x Area = 160 sq cm Base = 20 cm, Height = 16 cm

Calculations:

⇒ Let the base of the triangle be 5x.

⇒ Let the height of the triangle be 4x.

⇒ Area = (1 ÷ 2) × (5x) × (4x)

⇒ 160 = 10x²

⇒ x² = 160 ÷ 10

⇒ x² = 16

⇒ x = √16 = 4

⇒ Base = 5 × 4 = 20 cm

⇒ Height = 4 × 4 = 16 cm

∴ The correct answer is 20 cm and 16 cm.

Additional Information

Area of an Equilateral Triangle

For a triangle with all three sides equal to a, Area = (√3 ÷ 4) × a².

Heron's Formula

For a triangle with side lengths a, b, and c, Area = √[s(s − a)(s − b)(s − c)], where s = (a + b + c) ÷ 2.

Right-Angled Triangle

Area = (1 ÷ 2) × Base × Perpendicular. The sides satisfy the Pythagorean theorem: Hypotenuse² = Base² + Perpendicular².

Shortcut Trick

Fractional equivalent of 8% = 8/100 = 2/25

Multiplying factor for 2 years = (1 + 2/25)² = (27/25)² = 729/625

Here, 729 units represent the current population and 625 units represent the past population.

⇒ 729 units = 58320 ⇒ 1 unit = 80

⇒ Past Population = 625 × 80 = 50000

∴ The correct answer is 50000.

Alternate Method

Given:

Current Population = 58,320

Annual increase rate (R) = 8%

Time (T) = 2 years

Formula Used:

Current Population = P × (1 + R/100)^T

Past (P) Year 1 58,320        + 8% + 8% Population Growth over 2 Years P × 1.08 × 1.08 = 58,320

Calculations:

Let the population 2 years ago be P.

⇒ 58320 = P × (1 + 8/100)²

⇒ 58320 = P × (108/100)²

⇒ 58320 = P × (27/25)²

⇒ 58320 = P × (729/625)

⇒ P = 58320 × (625/729)

⇒ P = 80 × 625

⇒ P = 50000

∴ The correct answer is 50000.

Additional Information

Population Decrease (Depreciation)

If the population decreases by R% annually, the population after T years is P × (1 − R/100)^T.

Successive Percentage Change

If a value undergoes two successive percentage changes of A% and B%, the net percentage change is given by: Net Change = A + B + (A × B)/100.

Population After n Years vs n Years Ago

Population after n years = P × (1 + R/100)ⁿ. Population n years ago = P ÷ (1 + R/100)ⁿ.

Shortcut Trick

Overall profit percentage = (100 ÷ 5000) × 100 = +2%

By Alligation, ratio of profit to loss investment = (2 − (−5)) : (10 − 2) = 7 : 8

Total investment parts = 7 + 8 = 15 parts

Investment in profit item = (7 ÷ 15) × 5000 ⇒ 35000 ÷ 15 = 2333.33

∴ The correct answer is Rs. 2,333.33.

Alternate Method

Given:

Total Investment = Rs. 5000

Profit on 1^st item = 10%

Loss on 2^nd item = 5%

Net profit = Rs. 100

Formula Used:

Net Profit = (Profit on 1^st item) − (Loss on 2^nd item)

 Profit Item Loss Item  +10%  −5%  +2%  7  8 Ratio = 7 : 8

Calculations:

Let the investment in the profit-making item be x.

⇒ Investment in the loss-making item = 5000 − x

⇒ Profit from 1^st item = 10% of x = 0.10x

⇒ Loss from 2^nd item = 5% of (5000 − x) = 0.05 × (5000 − x)

⇒ Net profit = 0.10x − 0.05 × (5000 − x)

⇒ 100 = 0.10x − 250 + 0.05x

⇒ 100 + 250 = 0.15x

⇒ 350 = 0.15x

⇒ x = 350 ÷ 0.15

⇒ x = 35000 ÷ 15

⇒ x = 7000 ÷ 3 ≈ 2333.33

∴ The correct answer is Rs. 2,333.33.

Additional Information

Alligation Method

Used to find the ratio in which two components at given prices/percentages are mixed to produce a mixture at a target value. The components are subtracted diagonally with respect to the overall mixture average.

Overall Profit/Loss Percentage

Net Profit % = (Net Profit ÷ Total CP) × 100. It combines individual profits and losses across parts of an investment.

Profit and Loss Basics

Profit occurs when Selling Price > Cost Price. Loss occurs when Cost Price > Selling Price. Both values are fundamentally calculated on the initial Cost Price or Investment.

Shortcut Trick

For questions where groups work as 'OR', Time = Given Days ÷ [(M_and ÷ M_or) + (W_and ÷ W_or)]

Time = 70 ÷ [(5 ÷ 5) + (12 ÷ 12)]

Time = 70 ÷ (1 + 1) = 70 ÷ 2

Time = 35 days

∴ The correct answer is 35 Days.

Alternate Method

Given:

Time taken by a group of 5 men = 70 days

Time taken by a group of 12 women = 70 days

Formula Used:

Efficiency = Total Work ÷ Time Taken

Calculations:

⇒ Let the total work be the LCM of given days (70, 70) = 70 units.

⇒ Efficiency of 5 men = 1 unit/day

⇒ Efficiency of 12 women = 1 unit/day

⇒ Combined efficiency of (5 men + 12 women) = 1 + 1 = 2 units/day.

⇒ Time taken by 5 men and 12 women together = Total Work ÷ Combined Efficiency

⇒ Required Time = 70 ÷ 2 = 35 days.

∴ The correct answer is 35 Days.

Additional Information

OR - AND Concept

When a task is done by 'x' men OR 'y' women in D days, the time taken by 'a' men AND 'b' women together directly uses the formula: D ÷ [(a ÷ x) + (b ÷ y)].

Total Work Equivalence

Total Work = Men × Days. Equating different groups (M₁ × D₁ = M₂ × D₂) helps in finding individual efficiency ratios.

Efficiency and Time Relationship

Time taken to complete a task is inversely proportional to the working efficiency. Increasing efficiency reduces the time proportionately.

Shortcut Trick

Using the unitary method, we can first find the number of bottles filled in 1 minute.

Bottles filled per minute = 120 ÷ 6 = 20 bottles.

Bottles filled in 14 minutes = 20 × 14 = 280 bottles.

∴ The correct answer is 280 milk bottles.

Alternate Method

Given:

Initial work (W₁) = 120 bottles

Initial time (T₁) = 6 minutes

New time (T₂) = 14 minutes

Formula Used:

Using the chain rule for direct proportion: W₁ / T₁ = W₂ / T₂

6 minutes × 20 (Rate) 120 bottles 14 minutes × 20 (Rate) 280 bottles

Calculations:

⇒ 120 / 6 = W₂ / 14

⇒ 20 = W₂ / 14

⇒ W₂ = 20 × 14

⇒ W₂ = 280

∴ The correct answer is 280 milk bottles.

Additional Information

Direct Proportion

When two quantities increase or decrease together, they are in direct proportion. The ratio of their values remains constant: X₁ / Y₁ = X₂ / Y₂.

Inverse Proportion

When one quantity increases while the other decreases, they are in inverse proportion. Their product remains constant: X₁ × Y₁ = X₂ × Y₂.

Unitary Method

A technique to solve proportion problems by first finding the value of a single unit, and then multiplying that rate to find the required proportional value.

Shortcut Trick

Average weight of the two removed boxes = (21 + 10) ÷ 2 = 15.5 kg.

If the average of the removed items is exactly equal to the original average of the group, the new average remains unchanged.

Since the removed boxes have the same average as the whole group, there is 0 net deviation.

∴ The correct answer is 15.5 kg.

Alternate Method

Given:

Average weight of 10 boxes = 15.5 kg

Weight of the heaviest box = 21 kg

Weight of the lightest box = 10 kg

Formula Used:

Sum of observations = Average × Number of observations

Total 10 boxes = 155 kg (15.5 kg/box)  10 kg  Remaining 8 boxes (124 kg)  21 kg    New Average = 124 ÷ 8 = 15.5 kg

Calculations:

⇒ Total weight of 10 boxes = 10 × 15.5 = 155 kg

⇒ Total weight of the heaviest and lightest boxes = 21 + 10 = 31 kg

⇒ Weight of the remaining 8 boxes = 155 − 31 = 124 kg

⇒ Average weight of the remaining boxes = 124 ÷ 8

⇒ Average weight = 15.5 kg

∴ The correct answer is 15.5 kg.

Additional Information

Net Deviation Method

If the average of a sub-group removed from a total group is perfectly equal to the original average, the new average remains unchanged.

Combined Average

When combining two separate groups, the combined average is represented by: (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Replacement in Average

New value = Old value + (Number of items × Change in average). A positive change means the incoming item is heavier.

Shortcut Trick

Initial acid percentage in the mixture = (2 / 5) × 100 = 40%

Fraction of original mixture remaining = (150 − 30) / 150 = 120 / 150 = 4 / 5

Since the removed part is replaced by pure water, the acid concentration reduces proportionally.

Final acid percentage = Initial % × Fraction remaining = 40% × (4 / 5) = 32%

∴ The correct answer is 32%.

Alternate Method

Given:

Total initial volume of mixture = 150 litres

Ratio of acid to water = 2 : 3

Amount of mixture replaced with water = 30 litres

      150 Litres Ratio 2:3 Acid: 60L Water: 90L  − 30L Mixture  + 30L Water  150 Litres Acid: 48L Water: 102L Acid = 32%

Calculations:

⇒ Quantity of acid initially = [2 / (2 + 3)] × 150 = (2 / 5) × 150 = 60 litres

⇒ When 30 litres of mixture is removed, the acid removed proportionally = (2 / 5) × 30 = 12 litres

⇒ Remaining acid in the mixture = 60 − 12 = 48 litres

⇒ Total volume after replacing with 30 litres of water remains 150 litres

⇒ Percentage of acid in final mixture = (Remaining Acid / Total Volume) × 100

⇒ Percentage of acid = (48 / 150) × 100 = 32%

∴ The correct answer is 32%.

Additional Information

Successive Replacement Formula

If a vessel contains V litres of pure liquid, and R litres are repeatedly removed and replaced with another liquid n times, the remaining pure liquid = V × (1 − R/V)ⁿ.

Alligation Rule

Ratio of quantities = (Cheaper quantity / Dearer quantity) = (Price of dearer − Mean price) / (Mean price − Price of cheaper). This easily computes mixture proportions.

Concentration in Mixtures

When two mixtures of volumes V₁ and V₂ with concentrations C₁ and C₂ are mixed, the final concentration is (C₁V₁ + C₂V₂) ÷ (V₁ + V₂).

Shortcut Trick

Convert all heights to cm: 0.8 m = 80 cm and 1.2 m = 120 cm.

Volume of top part = Length × Width × Height = 40 × 30 × 80 = 96,000 cm³ = 96 L.

Volume of bottom part = Base Area × Height = 60 × 120 = 7,200 cm³ = 7.2 L.

Total Volume = 96 L + 7.2 L = 103.2 L.

∴ The correct answer is 103.2 L.

Alternate Method

Given:

Base dimensions of the top rectangular prism = 40 cm × 30 cm

Height of the top rectangular prism = 0.8 m = 80 cm

Base area of the bottom triangular prism = 60 cm²

Height of the bottom triangular prism = 1.2 m = 120 cm

Formula Used:

Volume of a rectangular prism = Length × Width × Height

Volume of any prism = Base Area × Height

1 Liter (L) = 1,000 cm³

Top Part (Rectangular Prism) Base = 40 × 30 cm, Height = 80 cm Volume = 96,000 cu cm = 96 L Bottom Part (Triangular Prism) Base Area = 60 sq cm, Height = 120 cm Volume = 7,200 cu cm = 7.2 L Total Volume = 96 L + 7.2 L = 103.2 L

Calculations:

⇒ Volume of the top rectangular part = 40 × 30 × 80

⇒ Volume of the top rectangular part = 96,000 cm³

⇒ Convert top part to liters: 96,000 ÷ 1000 = 96 L

⇒ Volume of the bottom triangular part = 60 × 120

⇒ Volume of the bottom triangular part = 7,200 cm³

⇒ Convert bottom part to liters: 7,200 ÷ 1000 = 7.2 L

⇒ Total capacity of the water tank = 96 L + 7.2 L = 103.2 L

∴ The correct answer is 103.2 L.

Additional Information

Volume of a Prism

The volume of any prism is calculated by the product of its uniform base area and its height (V = Base Area × Height), regardless of the geometric shape of its base.

Important Volume Conversions

1 m³ = 1,000 Liters and 1 Liter = 1,000 cm³. By relation, 1 m³ equals 1,000,000 cm³.

Composite Solids

The total volume of a composite figure is simply the direct sum of the volumes of its individual 3D components without any geometric loss.

Shortcut Trick

The standard trigonometric identity is cosec²A = 1 + cot²A.

Substitute cotA = x + (1/x) directly into the formula.

cosec²A = 1 + [x + (1/x)]²

cosec²A = 1 + x² + (1/x²) + 2

cosec²A = x² + (1/x²) + 3

∴ The correct answer is x² + (1/x²) + 3.

Alternate Method

Given:

cotA = x + (1/x)

Formula Used:

cosec²A − cot²A = 1

(a + b)² = a² + b² + 2ab

Calculations:

⇒ cosec²A = 1 + cot²A

⇒ cosec²A = 1 + [x + (1/x)]²

⇒ cosec²A = 1 + x² + (1/x)² + 2 × x × (1/x)

⇒ cosec²A = 1 + x² + (1/x²) + 2

⇒ cosec²A = x² + (1/x²) + 3

∴ The correct answer is x² + (1/x²) + 3.

Additional Information

Pythagorean Trigonometric Identities

• sin²θ + cos²θ = 1

• sec²θ − tan²θ = 1

• cosec²θ − cot²θ = 1

Reciprocal Trigonometric Identities

• sinθ = 1 ÷ cosecθ

• cosθ = 1 ÷ secθ

• tanθ = 1 ÷ cotθ

Standard Algebraic Identity

• (a + b)² = a² + b² + 2ab

Shortcut Trick

The slope of a straight line passing through any two given points is calculated directly by the ratio of the change in y to the change in x.

Slope = Change in y ÷ Change in x

Slope = (−7 − 5) ÷ (4 − (−2))

Slope = −12 ÷ 6 = −2

∴ The correct answer is −2.

Alternate Method

Given:

Coordinates of the first point: (x₁, y₁) = (−2, 5)

Coordinates of the second point: (x₂, y₂) = (4, −7)

Formula Used:

Slope (m) = (y₂ − y₁) ÷ (x₂ − x₁)

X Y O Δx = 6 Δy = −12 (−2, 5) (4, −7) m = −12 ÷ 6 = −2

Calculations:

⇒ m = (−7 − 5) ÷ (4 − (−2))

⇒ m = −12 ÷ (4 + 2)

⇒ m = −12 ÷ 6

⇒ m = −2

∴ The correct answer is −2.

Additional Information

Equation of a Line (Two-Point Form)

The equation of a straight line passing through (x₁, y₁) and (x₂, y₂) is given by y − y₁ = m(x − x₁), where m is the slope.

Parallel and Perpendicular Lines

Two lines are considered parallel if their slopes are equal (m₁ = m₂) and perpendicular if the product of their slopes is −1 (m₁ × m₂ = −1).

Distance Formula

The straight-line distance between two points (x₁, y₁) and (x₂, y₂) is d = √[(x₂ − x₁)² + (y₂ − y₁)²].

Shortcut Trick

The points A(3, 0), B(0, 4) and Origin O(0, 0) form a right-angled triangle.

The minimum distance from the origin to the line segment AB is simply the altitude drawn to the hypotenuse.

Altitude (h) = (Base × Perpendicular) ÷ Hypotenuse

Here, Base = 3, Perpendicular = 4, Hypotenuse = √(3² + 4²) = 5.

h = (3 × 4) ÷ 5 = 12 ÷ 5 = 2.4

∴ The correct answer is 2.4 units.

Alternate Method

Given:

Origin O = (0, 0), Point A = (3, 0) and Point B = (0, 4)

Formula Used:

Area of right-angled triangle = 1/2 × base × height

Hypotenuse = √(base² + height²)

         O(0, 0) A(3, 0) B(0, 4)  3 4 AB = 5  h  h = (3 × 4) ÷ 5 = 2.4 units

Calculations:

⇒ OA = 3 units and OB = 4 units

⇒ In ΔOAB, Hypotenuse AB = √(OA² + OB²)

⇒ AB = √(3² + 4²) = √(9 + 16) = √25 = 5 units

⇒ The minimum distance from origin O to line AB is the perpendicular altitude (h) from O to AB.

⇒ Area of ΔOAB = 1/2 × base × height = 1/2 × OA × OB

⇒ Area of ΔOAB = 1/2 × 3 × 4 = 6 sq. units

⇒ Using AB as the base, Area of ΔOAB = 1/2 × AB × h

⇒ 1/2 × 5 × h = 6

⇒ 5 × h = 12

⇒ h = 12 ÷ 5 = 2.4 units

∴ The correct answer is 2.4 units.

Additional Information

Perpendicular Distance from Origin

The perpendicular distance from the origin (0, 0) to any straight line given by the equation ax + by + c = 0 is determined using the formula d = |c| ÷ √(a² + b²).

Altitude to Hypotenuse

In any right-angled triangle with base 'a', perpendicular 'b' and hypotenuse 'c', the length of the altitude drawn to the hypotenuse is h = (a × b) ÷ c.

Line Intercept Form

The standard equation of a straight line making an x-intercept 'a' and a y-intercept 'b' on the coordinate axes is x/a + y/b = 1.

Shortcut Trick

Identify the position of the given equal parts in △LMN and △OPQ:

1. ∠L = ∠O (Angle)
2. LM = OP (Included Side)
3. ∠M = ∠P (Angle)

Since the equal side is positioned exactly between the two equal angles, the triangles are congruent by the ASA (Angle-Side-Angle) rule.

∴ The correct answer is ASA.

Alternate Method

Given:

⇒ In △LMN and △OPQ, ∠L = ∠O, LM = OP, and ∠M = ∠P

L M N O P Q

Step-by-step Proof:

⇒ ∠L = ∠O (Given equal angles)

⇒ LM = OP (Given equal included sides)

⇒ ∠M = ∠P (Given equal angles)

⇒ When two angles and the included side of one triangle are equal to the corresponding two angles and included side of another triangle, the two triangles are defined as congruent.

⇒ Therefore, by the Angle-Side-Angle (ASA) property, △LMN ≅ △OPQ.

∴ The correct answer is ASA.

Additional Information

SAS (Side-Angle-Side) Congruence

Two triangles are congruent if two sides and the included angle of one are equal to the corresponding sides and included angle of the other.

SSS (Side-Side-Side) Congruence

Two triangles are congruent if all three sides of one triangle are respectively equal to the three sides of the other.

AAS (Angle-Angle-Side) Congruence

Two triangles are congruent if any two angles and a non-included side of one are equal to the corresponding parts of the other.

RHS (Right Angle-Hypotenuse-Side) Congruence

Two right-angled triangles are congruent if the hypotenuse and one side of one triangle are equal to the hypotenuse and corresponding side of the other.