Quantitative Aptitude PYQs

Shortcut Trick

Apply Componendo and Dividendo rule directly:

⇒ [√(3+x)] / [√(3−x)] = (2+1) / (2−1)

⇒ [√(3+x)] / [√(3−x)] = 3

Squaring both sides:

⇒ (3+x) / (3−x) = 9 ⇒ 3 + x = 27 − 9x ⇒ 10x = 24 ⇒ x = 12/5

∴ The correct answer is 12/5.

Alternate Method

Given:

[√(3+x) + √(3−x)] / [√(3+x) − √(3−x)] = 2

Method Used: Cross-multiplication

⇒ √(3+x) + √(3−x) = 2 × [√(3+x) − √(3−x)]

⇒ √(3+x) + √(3−x) = 2√(3+x) − 2√(3−x)

⇒ √(3−x) + 2√(3−x) = 2√(3+x) − √(3+x)

⇒ 3√(3−x) = √(3+x)

Squaring both sides:

⇒ [3√(3−x)]² = [√(3+x)]²

⇒ 9 × (3 − x) = 3 + x

⇒ 27 − 9x = 3 + x

⇒ 27 − 3 = x + 9x

⇒ 24 = 10x

⇒ x = 24 / 10 = 12/5

∴ The correct answer is 12/5.

Additional Information

Componendo and Dividendo

If a/b = c/d, then by this rule, (a+b) / (a−b) = (c+d) / (c−d). It quickly reduces complex fractions involving sums and differences of the same terms.

Squaring Equations

When dealing with expressions inside a square root (√), squaring both sides is the primary method to eliminate the root. Always verify roots back into the original equation to identify any extraneous solutions.

Cross Multiplication

For an equation of the form A/B = C/D, cross multiplying yields A × D = B × C, converting fractions into a simple linear expression.

Shortcut Trick

Assume the initial selling price Z = 100.

Since initial profit is 25%, Cost Price (CP) = 100 ÷ 1.25 = 80.

The new Marked Price (MP) during the sale = 1.6 × 100 = 160.

New Selling Price (SP) after 20% discount = 160 × (1 − 0.20) = 128.

Profit % = [(128 − 80) ÷ 80] × 100 = 60%.

∴ The correct answer is 60%.

Alternate Method

Given:

Initial Selling Price (SP₁) = ₹Z

Initial Profit Margin = 25%

New Marked Price (MP) = ₹1.6Z

Discount during festive sale = 20%

Formula Used:

CP = SP ÷ (1 + Profit%)

New SP = MP × (1 − Discount%)

Profit% = [(New SP − CP) ÷ CP] × 100

Markup

−20%

Profit = 60%

CP = 0.8Z

MP = 1.6Z

SP = 1.28Z

Profit & Loss Flow

Calculations:

⇒ CP = Z ÷ 1.25 = 0.8Z

⇒ New SP = 1.6Z × (1 − 0.20)

⇒ New SP = 1.6Z × 0.8 = 1.28Z

⇒ Profit Earned = New SP − CP

⇒ Profit Earned = 1.28Z − 0.8Z = 0.48Z

⇒ Percentage Profit = (0.48Z ÷ 0.8Z) × 100

⇒ Percentage Profit = 0.6 × 100 = 60%

∴ The correct answer is 60%.

Additional Information

Marked Price and Discount

Discount is always calculated on the Marked Price (MP) of an article, not its Cost Price. Final Selling Price = MP × (1 − Discount%).

Profit and Loss Percentage

Profit or Loss percentages are traditionally evaluated over the Cost Price (CP). Profit% = [(SP − CP) ÷ CP] × 100.

Successive Discounts

If two successive discounts of a% and b% are applied on a product, the effective equivalent single discount is given by: [a + b − (a × b) ÷ 100]%.

Shortcut Trick

Time = 2 years = 24 months = 3 cycles of 8 months.

Rate per 8 months = 15% × (8 ÷ 12) = 10%.

Effective compound interest rate for 10% over 3 cycles = 33.1%.

CI = 33.1% of ₹12,500 = (33.1 ÷ 100) × 12500

CI = 331 × 12.5 = ₹4137.50

∴ The correct answer is ₹4137.50.

Alternate Method

Given:

Principal (P) = ₹12,500

Rate of interest (R) = 15% p.a.

Time (T) = 2 years

Compounding frequency = 8-monthly

0 M 8 M 16 M 24 M

P = ₹12500 A = ?

+10% +10% +10%

n = 3 cycles

Formula Used:

Amount (A) = P × (1 + r ÷ 100)ⁿ

Compound Interest (CI) = A − P

Calculations:

⇒ Total time in months = 2 × 12 = 24 months

⇒ Number of 8-month cycles (n) = 24 ÷ 8 = 3

⇒ Rate of interest for 8 months (r) = 15% × (8 ÷ 12) = 10%

⇒ Amount (A) = 12500 × (1 + 10 ÷ 100)³

⇒ A = 12500 × (11 ÷ 10) × (11 ÷ 10) × (11 ÷ 10)

⇒ A = 12500 × (1331 ÷ 1000)

⇒ A = 12.5 × 1331 = ₹16637.50

⇒ CI = A − P

⇒ CI = 16637.50 − 12500 = ₹4137.50

∴ The correct answer is ₹4137.50.

Additional Information

Half-Yearly Compounding

When interest is compounded half-yearly, the rate is halved (R ÷ 2) and the time period is doubled (2T). Amount = P(1 + R ÷ 200)^2T.

Quarterly Compounding

For quarterly compounding, the rate becomes one-fourth (R ÷ 4) and time period becomes four times (4T). Amount = P(1 + R ÷ 400)^4T.

Effective Rate of Interest for 3 Cycles

If the rate of interest per cycle is r%, the effective CI rate for 3 cycles is given by 3r + (3r² ÷ 100) + (r³ ÷ 10000) %.

Simple Interest vs Compound Interest

Difference between CI and SI for 2 years = P(R ÷ 100)². For 3 years = P(R ÷ 100)² × (3 + R ÷ 100).

Shortcut Trick

Equating remaining work directly using the MDH rule: M₁ × D₁ × H₁ = M₂ × D₂ × H₂

Remaining work for 15 men would take 18 days (24 − 6) at 8 hours/day.

For the new group of 20 men (15 + 5) working at 6 hours/day, the required days is D₂.

15 × 18 × 8 = 20 × D₂ × 6 ⇒ 2160 = 120 × D₂ ⇒ D₂ = 18

∴ The correct answer is 18 days.

Alternate Method

Given:

Initial group = 15 men, Total scheduled days = 24, Hours/day = 8

Work completed before changes = 6 days

New group = 15 + 5 = 20 men, New hours/day = 6

Formula Used:

Total Work = Men × Days × Hours/day

Calculations:

⇒ Let 1 man's work in 1 hour be considered as 1 unit.

⇒ Total Work to be done = 15 × 24 × 8 = 2880 units

⇒ Work completed in the first 6 days = 15 × 6 × 8 = 720 units

⇒ Remaining Work = 2880 − 720 = 2160 units

⇒ Efficiency of the new group = 20 men × 6 hours/day = 120 units/day

⇒ Time required for remaining work (D₂) = Remaining Work ÷ Efficiency

⇒ D₂ = 2160 ÷ 120 = 18 days

∴ The correct answer is 18 days.

Additional Information

MDH Concept (Chain Rule)

When multiple groups are working on portions of the same work, the relationship is given by: (M₁ × D₁ × H₁) ÷ W₁ = (M₂ × D₂ × H₂) ÷ W₂. This formula is highly useful for equating work capacities directly.

Total Work Rule

Total work is always evaluated as the product of the number of workers, the number of days they work, and the hours they put in per day. Work size is directly proportional to all three components.

Inverse Proportions in Work

For a constant amount of work, the total men employed is inversely proportional to the days required. As the men or daily working hours increase, the required days decrease proportionally.

Shortcut Trick

Relative speed (same direction) = 500 ÷ 10 = 50 meters/minute.

Relative speed (opposite direction) = 500 ÷ 2 = 250 meters/minute.

Speed of thief = (Relative speed in opposite direction − Relative speed in same direction) ÷ 2

Speed of thief = (250 − 50) ÷ 2 = 100 meters/minute.

∴ The correct answer is 100 meters/minute.

Alternate Method

   Case 1: Same Direction (Time = 10 min)  Gap = 500 m  Police   Thief  Speed = V<sub>p</sub> − V<sub>t</sub>  Case 2: Opposite Direction (Time = 2 min)  Gap = 500 m  Police   Thief  Speed = V<sub>p</sub> + V<sub>t</sub>

Given:

Initial distance between thief and police officer = 500 m

Time to catch in the same direction = 10 minutes

Time to meet in the opposite direction = 2 minutes

Formula Used:

Relative Speed (same direction) = V_p − V_t = Distance ÷ Time

Relative Speed (opposite direction) = V_p + V_t = Distance ÷ Time

Calculations:

⇒ Let the speed of the police officer be V_p and the speed of the thief be V_t.

⇒ In the same direction: V_p − V_t = 500 ÷ 10

⇒ V_p − V_t = 50 meters/minute    --- (Equation 1)

⇒ In opposite directions: V_p + V_t = 500 ÷ 2

⇒ V_p + V_t = 250 meters/minute    --- (Equation 2)

⇒ Subtracting Equation 1 from Equation 2:

⇒ (V_p + V_t) − (V_p − V_t) = 250 − 50

⇒ 2V_t = 200

⇒ V_t = 100 meters/minute

∴ The correct answer is 100 meters/minute.

Additional Information

Relative Speed (Same Direction)

When two bodies move in the same direction at speeds V₁ and V₂ (where V₁ > V₂), their relative speed is calculated as the difference between their speeds: V₁ − V₂.

Relative Speed (Opposite Direction)

When two bodies move towards each other at speeds V₁ and V₂, their relative speed is calculated as the sum of their individual speeds: V₁ + V₂.

Time to Catch / Meet

The time taken for two moving bodies to meet or catch each other is directly given by the formula: Time = Initial Gap ÷ Relative Speed.

Shortcut Trick

Check the two extreme cases by taking examples of a rational number.

Case 1: If rational number = 0, then 0 × √3 = 0 (Rational).

Case 2: If rational number = 2, then 2 × √3 = 2√3 (Irrational).

Since the product changes based on the value of the rational number being zero or non-zero.

∴ The correct answer is Sometimes Rational, Sometimes Irrational.

Alternate Method

Given:

Multiplication of a Rational Number and an Irrational Number.

Concept Used:

Rational number = any number that can be expressed as p/q where q ≠ 0.

Irrational number = a number that cannot be written as a simple fraction (like √2, π).

Calculations:

⇒ Let Rational Number = R and Irrational Number = I.

⇒ Case 1: Let R = 0 (since 0/1 = 0, it is a valid rational number) and I = √5.

⇒ Product = R × I = 0 × √5 = 0.

⇒ Here, the result 0 is a Rational Number.

⇒ Case 2: Let R = 4 (a non-zero rational number) and I = √2.

⇒ Product = R × I = 4 × √2 = 4√2.

⇒ Here, the result 4√2 is an Irrational Number.

⇒ Therefore, the product of a rational and an irrational number can be rational or irrational depending purely on the value of the rational number.

∴ The correct answer is Sometimes Rational, Sometimes Irrational.

Additional Information

Sum and Difference Rule

The sum or difference of a rational number and an irrational number is always an irrational number (e.g., 2 + √3 is always irrational).

Product of Two Irrational Numbers

The product of two irrational numbers can be either rational or irrational. For example, √2 × √2 = 2 (Rational), but √2 × √3 = √6 (Irrational).

Non-Zero Rational Rule

If a rational number is strictly non-zero, its product with any irrational number will always result in an irrational number.

Shortcut Trick

When multiple spheres are melted into a single solid sphere, the sum of their cubes directly gives the cube of the new radius.

Using the standard cubic identity for radii 3, 4, and 5:

R³ = r₁³ + r₂³ + r₃³

R³ = 3³ + 4³ + 5³

R³ = 27 + 64 + 125 = 216 ⇒ R = 6

∴ The correct answer is 6 cm.

Alternate Method

3 cm +

4 cm +

5 cm →

R = ? Total Volume = Volume of New Sphere

Given:

Radius of first sphere (r₁) = 3 cm

Radius of second sphere (r₂) = 4 cm

Radius of third sphere (r₃) = 5 cm

Formula Used:

Volume of a sphere = (4/3)×π×r³

When metallic shapes are melted down and reformed, the overall volume is perfectly conserved.

Calculation:

⇒ Volume of new sphere = Volume of 1^st + Volume of 2^nd + Volume of 3^rd

⇒ (4/3)×π×R³ = (4/3)×π×r₁³ + (4/3)×π×r₂³ + (4/3)×π×r₃³

⇒ (4/3)×π×R³ = (4/3)×π×(3³ + 4³ + 5³)

⇒ Canceling (4/3)×π from both sides:

⇒ R³ = 3³ + 4³ + 5³

⇒ R³ = 27 + 64 + 125

⇒ R³ = 216

⇒ R³ = 6³

⇒ R = 6 cm

∴ The correct answer is 6 cm.

Additional Information

Melting and Recasting Conservation

When solid objects are melted and recast into a new shape, their total volume remains perfectly conserved, though the total surface area will change.

Volume of a Sphere

The volume of a sphere of radius r is given by V = (4/3)×π×r³.

Surface Area of a Sphere

The entire surface of a sphere is curved, and its total surface area is derived as A = 4×π×r².

Shortcut Trick

Mileage is the distance covered per unit of fuel.

Mileage = Distance ÷ Fuel Volume = 18.4 ÷ 2.5

Multiply numerator and denominator by 4 for a quick decimal calculation:

Mileage = (18.4 × 4) ÷ (2.5 × 4) = 73.6 ÷ 10

Mileage = 7.36 km/liter

∴ The correct answer is 7.36 km/liter.

Alternate Method

Given:

Total distance covered = 18.4 km

Quantity of petrol used = 2 ½ liters = 2.5 liters

Formula Used:

Mileage = Total Distance ÷ Total Fuel Consumed

Total Distance = 18.4 km

1 Liter 7.36 km

1 Liter 7.36 km

0.5 L 3.68 km

Total Fuel = 2.5 Liters

Calculations:

⇒ Mileage = 18.4 ÷ 2.5

To remove the decimal from the denominator, convert 2.5 to a fraction:

⇒ Mileage = 18.4 ÷ (5 ÷ 2)

⇒ Mileage = (18.4 × 2) ÷ 5

⇒ Mileage = 36.8 ÷ 5

⇒ Mileage = 7.36 km/liter

∴ The correct answer is 7.36 km/liter.

Additional Information

Mileage Calculation

Mileage represents the efficiency of a vehicle, calculated as the ratio of distance traveled to the fuel consumed. Formula: Mileage = Distance ÷ Fuel Volume.

Decimal Division Shortcut

To simplify division by 2.5, multiply both the numerator and denominator by 4. This converts the denominator to 10, allowing you to easily find the result by shifting the decimal point.

Unitary Method Application

If 2.5 liters give 18.4 km, then 1 liter gives (18.4 ÷ 2.5) km. This direct proportion is fundamental in solving rate and efficiency problems.

Shortcut Trick

Total profit ratio = 3 : 2 : 2 ⇒ Total = 7 parts

Strategist's share = (2 / 7) × Total Profit

Strategist's share = (2 / 7) × 1,40,000

Strategist's share = 2 × 20,000 = Rs. 40,000

∴ The correct answer is Rs. 40,000.

Alternate Method

Given:

Ratio of profit sharing (Investor : Strategist : Supplier) = 3 : 2 : 2

Total profit earned in the first year = Rs. 1,40,000

Profit Sharing Ratio

Total Profit = Rs. 1,40,000 (7 parts)

Investor 3 parts

Strategist 2 parts

Supplier 2 parts

Rs. 40,000

Calculations:

⇒ Total ratio parts = 3 + 2 + 2 = 7 parts

⇒ Value of 7 parts = Rs. 1,40,000

⇒ Value of 1 part = 1,40,000 ÷ 7 = Rs. 20,000

⇒ The second partner (Strategist) receives 2 parts.

⇒ Strategist's share = 2 × 20,000 = Rs. 40,000

∴ The correct answer is Rs. 40,000.

Additional Information

Simple Partnership

When partners invest capital or resources for the same time period, the generated profit or loss is distributed in the ratio of their respective investments.

Compound Partnership

When partners invest different amounts for different durations of time, the profit sharing ratio is proportional to the product of capital and time: Profit Ratio = (C₁ × T₁) : (C₂ × T₂).

Active vs Sleeping Partner

An active (working) partner usually receives a direct salary or a fixed percentage from the total profit as compensation for managing the business, before the remaining profit is distributed among all partners.

Shortcut Trick

Total ratio parts of the ribbon = 2 + 3 + 5 = 10 parts.

The longest piece corresponds to the largest ratio value, which is 5 parts.

Given length of the 5 parts = 20 cm ⇒ 1 part = 4 cm.

Total length of the ribbon = 10 parts = 10 × 4 = 40 cm.

∴ The correct answer is 40 cm.

Alternate Method

Given:

Ratio of the three cut pieces of ribbon = 2 : 3 : 5

Length of the longest piece = 20 cm

2x

3x

5x = 20 cm

8 cm

12 cm

20 cm

Total Length = 10x = 40 cm

Calculations:

⇒ Let the lengths of the three pieces of ribbon be 2x, 3x, and 5x.

⇒ The longest piece clearly corresponds to the 5x part.

⇒ 5x = 20

⇒ x = 20 ÷ 5

⇒ x = 4 cm

⇒ Total length of the ribbon = Sum of all three pieces

⇒ Total length = 2x + 3x + 5x

⇒ Total length = 10x

⇒ Total length = 10 × 4 = 40 cm

∴ The correct answer is 40 cm.

Additional Information

Dividing a Quantity in a Given Ratio

If a total quantity T is distributed among different segments in the ratio a : b : c, the parts are determined as [a / (a + b + c)] × T, [b / (a + b + c)] × T, and [c / (a + b + c)] × T respectively.

Equivalent Ratios

Multiplying or dividing each term of a ratio by the same non-zero constant does not alter the absolute value of the ratio (i.e., a : b is structurally equal to k×a : k×b).

Proportional Equality

When two individual ratios equate to each other, they are described as being in proportion. Example: a / b = c / d implies that ad = bc.

Shortcut Trick

The side ratio 3:4:5 indicates a right-angled triangle.

Total perimeter ratio = 3 + 4 + 5 = 12 units.

12 units = 60 m ⇒ 1 unit = 5 m.

Base = 3 × 5 = 15 m and Height = 4 × 5 = 20 m.

Area = 1/2 × Base × Height = 1/2 × 15 × 20 = 150 m².

∴ The correct answer is 150 m².

Alternate Method

Given:

Side ratio of the triangular plot = 3:4:5

Perimeter of the plot = 60 m

Formula Used:

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

Base = 20 m Height = 15 m Hypotenuse = 25 m Right-Angled Triangle (3:4:5 ratio)

Calculations:

⇒ Let the sides of the triangle be 3x, 4x, and 5x.

⇒ Perimeter = Sum of all sides = 3x + 4x + 5x = 60 m

⇒ 12x = 60 m ⇒ x = 5 m

⇒ The lengths of the sides are: 3 × 5 = 15 m, 4 × 5 = 20 m, and 5 × 5 = 25 m.

⇒ Since 15² + 20² = 225 + 400 = 625 = 25², the triangle is a right-angled triangle.

⇒ The sides forming the right angle represent the base and height (15 m and 20 m).

⇒ Area = 1/2 × 15 × 20

⇒ Area = 150 m²

∴ The correct answer is 150 m².

Additional Information

Pythagorean Triples

Any set of positive integers satisfying a² + b² = c² is a Pythagorean triple. Common examples include (3, 4, 5), (5, 12, 13), and (8, 15, 17).

Heron's Formula

For any triangle with sides a, b, and c, the area is √[s(s − a)(s − b)(s − c)], where s = (a + b + c) ÷ 2 is the semi-perimeter.

Inradius of a Right Triangle

The radius of the incircle (r) for a right-angled triangle is given by r = (a + b − c) ÷ 2, where c is the hypotenuse.

Shortcut Trick

Marks in Mathematics = Physics marks × Multiplying factor for 20% increase

Maths marks = 72 × (1 + 20 ÷ 100)

Maths marks = 72 × 1.2 = 86.4

∴ The correct answer is 86.4 marks.

Alternate Method

Given:

Marks scored in Physics = 72

Marks in Mathematics = 20% more than in Physics

Physics: 72

Maths: ?

• 20%

Maths = 72 + (20% of 72)

Formula Used:

New Value = Old Value × (100 + Percentage Increase) ÷ 100

Calculations:

⇒ Marks in Mathematics = 72 × (100 + 20) ÷ 100

⇒ Marks in Mathematics = 72 × 120 ÷ 100

⇒ Marks in Mathematics = 72 × 1.2

⇒ Marks in Mathematics = 86.4

∴ The correct answer is 86.4 marks.

Additional Information

Percentage Change Formula

Percentage change = (New Value − Old Value) ÷ Old Value × 100.

Multiplying Factor Concept

For an x% increase, the direct multiplying factor is (1 + x ÷ 100). For an x% decrease, it is (1 − x ÷ 100).

Successive Percentage

When a value is changed by a% and then b%, the effective combined percentage change is given by the formula: a + b + (a × b) ÷ 100.

Shortcut Trick

Equating efficiencies since the days taken are equal: 6W + 4M = 4W + 5M ⇒ 1M = 2W.

Converting the first group to women: 6W + 4(2W) = 14 Women.

Total work = 14 Women × 10 days = 140 units.

Time for 8 women = 140 ÷ 8 = 17.5 days.

∴ The correct answer is 17.5 days.

Alternate Method

Given:

6 women and 4 men complete the work in 10 days.

4 women and 5 men complete the work in 10 days.

Formula Used:

Total Work = Efficiency × Days

Calculations:

⇒ 6W + 4M = 1 ... (i)

⇒ 4W + 5M = 1 ... (ii)

⇒ Equating (i) and (ii): 6W + 4M = 4W + 5M

⇒ 5M − 4M = 6W − 4W

⇒ 1M = 2W

⇒ Let the efficiency of 1 woman (W) = 1 unit/day and 1 man (M) = 2 units/day.

⇒ Total work based on the first group = (6 × 1 + 4 × 2) × 10 = (6 + 8) × 10 = 140 units.

⇒ Efficiency of 8 women = 8 × 1 = 8 units/day.

⇒ Time required for 8 women = Total Work ÷ Efficiency of 8 women

⇒ Time required = 140 ÷ 8 = 17.5 days.

∴ The correct answer is 17.5 days.

Additional Information

Work Equivalence (MDH Rule)

If M₁ persons can do W₁ work in D₁ days, and M₂ persons can do W₂ work in D₂ days, then (M₁ × D₁) ÷ W₁ = (M₂ × D₂) ÷ W₂.

Total Work Concept

Total work is considered constant and is generally found using the LCM of the days taken by different persons or groups.

Efficiency & Time Relation

Efficiency is inversely proportional to time. A person with higher efficiency takes less time to complete the same amount of work.

Shortcut Trick

For any set of numbers in an Arithmetic Progression (like consecutive even numbers), the average is equal to the average of the first and last terms.

Average = (Smallest + Largest) ÷ 2

⇒ Sum of Smallest and Largest = 2 × Average

⇒ Sum = 2 × 41 = 82

∴ The correct answer is 82.

Alternate Method

Given:

Number of consecutive even numbers = 8

Average of these 8 numbers = 41

Formula Used:

Average = Sum of all observations ÷ Total number of observations

Avg = 41

34 Min

36

38

40

42

44

46

48 Max

Sum = 82

Calculations:

Let the 8 consecutive even numbers be x, x + 2, x + 4, x + 6, x + 8, x + 10, x + 12, and x + 14.

⇒ Average = (x + x + 2 + x + 4 + x + 6 + x + 8 + x + 10 + x + 12 + x + 14) ÷ 8

⇒ 41 = (8x + 56) ÷ 8

⇒ 41 = x + 7

⇒ x = 41 − 7 = 34

⇒ Smallest number = x = 34

⇒ Largest number = x + 14 = 34 + 14 = 48

⇒ Sum of smallest and largest numbers = 34 + 48 = 82

∴ The correct answer is 82.

Additional Information

Average of an Arithmetic Progression

For any sequence in an Arithmetic Progression (like consecutive integers, even/odd numbers), the average is exactly equal to the middle term, or the average of the first and last terms: Avg = (First + Last) ÷ 2.

Sum of 'n' Consecutive Even Numbers

The sum of the first 'n' consecutive even numbers starting from 2 is given by the formula: Sum = n × (n + 1).

Average of 'n' Consecutive Even Numbers

The average of the first 'n' consecutive even numbers starting from 2 is strictly equal to (n + 1).

Shortcut Trick

The heights form an AP with 4 layers: 4 cm, 8 cm, 12 cm, and 16 cm.

Total Height = 4 + 8 + 12 + 16 = 40 cm.

Base Area = 5 × 5 = 25 cm².

Total Volume = Base Area × Total Height = 25 × 40 = 1000 cm³.

∴ The correct answer is 1000 cm³.

Alternate Method

Given:

Side of the square base = 5 cm

First layer height (a) = 4 cm

Fourth layer height (a + 3d) = 16 cm

Formula Used:

Volume of Prism = Base Area × Total Height

Sum of AP = n/2 × (first term + last term)

Total H = 40 cm

16 cm

12 cm

8 cm

4 cm

Square Base (5 × 5)

Calculations:

⇒ Total height (H) is the sum of the 4 arithmetic progression layers.

⇒ H = 4/2 × (4 + 16) = 2 × 20 = 40 cm

⇒ Area of the square base = 5 × 5 = 25 cm²

⇒ Total Volume = Base Area × Total Height

⇒ Total Volume = 25 × 40 = 1000 cm³

∴ The correct answer is 1000 cm³.

Additional Information

Arithmetic Progression (AP)

An arithmetic progression is a sequence where the differences between every two consecutive terms are identical. The sum of an AP is evaluated as S_n = n/2 × (a + l), where a is the first term and l is the last term.

Volume of a General Prism

The volume of any uniform cross-section solid is strictly the area of its base multiplied by its vertical height. Formula: V = A_base × h.

Square Prism

A square prism is a specific type of cuboid where the top and bottom faces are perfect squares. Its total surface area is 2a² + 4ah, where a is the base side and h is the total height.

Shortcut Trick

Remaining solution after removing 12 litres = 60 − 12 = 48 litres.

Quantity of Sugar in 48 litres = 48 × (5/6) = 40 litres.

Quantity of Water in 48 litres = 48 × (1/6) = 8 litres.

After adding 12 litres of water, new water quantity = 8 + 12 = 20 litres.

New ratio of Sugar to Water = 40 : 20 = 2 : 1.

∴ The correct answer is 2:1.

Alternate Method

Given:

Total volume of solution = 60 litres

Initial ratio of Sugar to Water = 5 : 1

Volume of mixture replaced with water = 12 litres

Formula Used:

Quantity of component = (Ratio of component ÷ Total Ratio parts) × Total Quantity

Replacement ProcessSugar: 40 LWater: 8 LRemaining 48L+Water: 12 LAdded 12L=Sugar: 40 LWater: 20 LFinal 60L

⇒ Amount of mixture remaining after removal = 60 − 12 = 48 litres

⇒ The ratio of Sugar to Water in the remaining mixture remains exactly 5 : 1

⇒ Total ratio parts = 5 + 1 = 6 parts

⇒ Amount of Sugar in remaining mixture = (5 ÷ 6) × 48 = 40 litres

⇒ Amount of Water in remaining mixture = (1 ÷ 6) × 48 = 8 litres

⇒ Now, 12 litres of pure water is added to the mixture.

⇒ Total Water in the new mixture = 8 + 12 = 20 litres

⇒ Total Sugar in the new mixture remains 40 litres

⇒ New ratio of Sugar to Water = 40 : 20 = 2 : 1

∴ The correct answer is 2:1.

Additional Information

Replacement Principle

When a part of a uniformly mixed solution is removed, the concentration or ratio of the components in the remaining mixture stays exactly the same as the original mixture.

Rule of Alligation

Used to find the ratio in which two ingredients at given concentrations/prices must be mixed to produce a mixture of a desired concentration: Ratio = (C₂ − C_m) ÷ (C_m − C₁).

Successive Replacement Formula

If an initial volume V of pure liquid is repeatedly replaced by x volume of water 'n' times, the final volume of pure liquid left in the container = V × (1 − x/V)ⁿ.

Shortcut Trick

Given cot A = Base ÷ Perpendicular = 7 ÷ 24.

The lengths 7, 24, and 25 form a standard Pythagorean triplet.

Therefore, the Hypotenuse directly evaluates to 25.

cosec A = Hypotenuse ÷ Perpendicular = 25 ÷ 24.

∴ The correct answer is 25/24.

Alternate Method

ABase = 7Perp = 24Hyp = 25

Given:

cot A = 7 / 24

Formula Used:

cot A = Base / Perpendicular

cosec A = Hypotenuse / Perpendicular

Hypotenuse² = Base² + Perpendicular²

Calculations:

⇒ Base = 7 and Perpendicular = 24

⇒ Hypotenuse = √(Base² + Perpendicular²)

⇒ Hypotenuse = √(7² + 24²)

⇒ Hypotenuse = √(49 + 576)

⇒ Hypotenuse = √625 = 25

⇒ cosec A = 25 / 24

∴ The correct answer is 25/24.

Additional Information

Cosecant and Cotangent Identity

The difference between the squares of cosecant and cotangent is exactly 1: cosec²θ − cot²θ = 1.

Pythagorean Triplets

A set of three positive integers that satisfy the Pythagoras theorem (a² + b² = c²) is called a Pythagorean triplet. Common triplets are (3, 4, 5), (5, 12, 13), (7, 24, 25), and (8, 15, 17).

Other Major Identities

Sine & Cosine: sin²θ + cos²θ = 1 | Secant & Tangent: sec²θ − tan²θ = 1.

Shortcut Trick

Since both equations are equal to y, equate them to find x directly.

3x − 4 = −x + 2 ⇒ 4x = 6 ⇒ x = 1.5

Substitute x = 1.5 in y = −x + 2 ⇒ y = 0.5

∴ The correct answer is (1.5, 0.5).

Alternate Method

Given:

Equation 1: y = 3x − 4

Equation 2: y = −x + 2

Method Used:

Substitution method for solving simultaneous linear equations.

Calculations:

⇒ Substitute the value of y from Equation 2 into Equation 1:

⇒ −x + 2 = 3x − 4

⇒ Move all x terms to one side and constants to the other:

⇒ 2 + 4 = 3x + x

⇒ 6 = 4x

⇒ x = 6 ÷ 4

⇒ x = 1.5

⇒ Now, substitute x = 1.5 back into Equation 2:

⇒ y = −(1.5) + 2

⇒ y = 0.5

∴ The correct answer is (1.5, 0.5).

Additional Information

System of Linear Equations

A system of linear equations consists of two or more linear equations involving the same variables. The solution is the coordinate point (x, y) where the lines intersect.

Types of Solutions

1. Unique Solution: Intersecting lines with different slopes (m₁ ≠ m₂).
2. No Solution: Parallel lines with the same slope but different y-intercepts.
3. Infinite Solutions: Identical overlapping lines with the same slope and y-intercept.

Methods of Solving

Linear systems can be solved algebraically using substitution or elimination methods, or visually by graphing both equations on a coordinate plane.

Shortcut Trick

According to the basic geometric properties of parallel lines intersected by a transversal:

Alternate interior angles are always equal to each other.

Since one alternate interior angle is given as 115°, the angle opposite to it on the alternate side must also be identical.

∴ The correct answer is 115°.

Alternate Method

Given:

⇒ Two parallel lines are cut by a transversal line.

⇒ One alternate interior angle measures 115°.

115° ? = 115°

Line 1 Line 2 Transversal

Concept Used:

⇒ Alternate Interior Angles Theorem: When a transversal intersects two strictly parallel lines, the pairs of alternate interior angles are congruent (equal).

Calculation:

⇒ Let the given interior angle be ∠A = 115°.

⇒ Let the opposite alternate interior angle be ∠B.

⇒ By the theorem, we know that ∠A = ∠B

⇒ Substituting the value gives ∠B = 115°.

∴ The correct answer is 115°.

Additional Information

Alternate Exterior Angles

These are pairs of angles that lie outside the parallel lines and on opposite sides of the transversal. They are always equal in measure.

Corresponding Angles

Pairs of corresponding angles occupy the same relative position at each intersection where the straight line crosses the parallel lines. They are always equal in measure.

Consecutive Interior Angles

Also known as co-interior angles, these lie on the same side of the transversal and between the parallel lines. Their sum is always 180° (they are supplementary).

Shortcut Trick

Since the triangles are similar, their sides are proportional.

Scale factor from ΔABC to ΔXYZ = XY ÷ AB = 8 ÷ 4 = 2

Side YZ corresponds to side BC.

Therefore, YZ = BC × Scale factor = 6 × 2 = 12

∴ The correct answer is 12 cm.

Alternate Method

Given:

ΔABC ~ ΔXYZ

AB = 4 cm

BC = 6 cm

XY = 8 cm

Formula Used:

If two triangles are similar, the ratio of their corresponding sides is equal.

⇒ AB / XY = BC / YZ = AC / XZ

A B C 4 cm 6 cm

X Y Z 8 cm YZ = ?

Calculations:

⇒ AB / XY = BC / YZ

⇒ 4 / 8 = 6 / YZ

⇒ 1 / 2 = 6 / YZ

⇒ YZ = 6 × 2

⇒ YZ = 12

∴ The correct answer is 12 cm.

Additional Information

Properties of Similar Triangles

If two triangles are similar, their corresponding angles are equal and the ratio of their corresponding sides is constant (AB/XY = BC/YZ = AC/XZ).

Area Theorem for Similar Triangles

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides: Area(ΔABC)/Area(ΔXYZ) = (AB/XY)².

Perimeter of Similar Triangles

The ratio of the perimeters of two similar triangles is identical to the ratio of their corresponding sides.

Criteria for Similarity

Triangles can be proven similar using AA (Angle-Angle), SAS (Side-Angle-Side proportion), or SSS (Side-Side-Side proportion) criteria.