Quantitative Aptitude PYQs

Shortcut Trick

The effective compound interest rate for 10% per annum over 5 years can be directly derived from the multiplying factor 1.1

We know that 1.1⁵ = 1.61051, meaning the total amount becomes 161.051% of the principal.

Amount = 1.61051 × 4500

⇒ Amount = 7247.295

⇒ Rounding to one decimal place, we get 7247.3

∴ The correct answer is ₹7247.3.

Alternate Method

Given:

Principal (P) = ₹4500

Rate of Interest (R) = 10% per annum

Time (n) = 5 years

Formula Used:

Amount (A) = P × (1 + R/100)ⁿ

Year 0 Year 1 Year 2 Year 3 Year 4 Year 5

4500 4950 5445 5989.5 6588.5 7247.3

Each step × 1.1 (+10%)

Calculations:

⇒ A = 4500 × (1 + 10/100)⁵

⇒ A = 4500 × (11/10)⁵

⇒ A = 4500 × 1.61051

⇒ A = 7247.295

⇒ Amount ≈ ₹7247.3

∴ The correct answer is ₹7247.3.

Additional Information

Effective Compound Interest Rates

For a 10% interest rate, the effective CI is 21% for 2 years, 33.1% for 3 years, and 46.41% for 4 years. Remembering these saves significant calculation time in exams.

Difference Between CI and SI

For 2 years, the difference between Compound Interest and Simple Interest on the same principal is computed as Diff = P × (R/100)².

Rule of 72

The approximate time required to double your principal amount at a given compound interest rate can be found directly using 72 ÷ R.

Shortcut Trick

A fraction in its simplest form p/q is a terminating decimal only if the prime factorization of its denominator (q) contains exclusively 2, 5, or both.

1/3 → Denominator is 3, hence non-terminating.

4/5 → Denominator is 5, hence terminating.

√3 and π are standard irrational numbers (non-terminating and non-repeating).

∴ The correct answer is 4/5.

Alternate Method

Given:

The numbers provided in the options are 1/3, 4/5, √3, and π.

Method Used:

Convert each value to its decimal equivalent to identify which one is both rational and terminating.

Calculations:

⇒ Option 1: 1/3 = 0.3333... (This is a rational number but a non-terminating repeating decimal)

⇒ Option 2: 4/5 = 0.8 (This decimal stops at 8, so it is a rational and terminating decimal)

⇒ Option 3: √3 ≈ 1.73205... (The square root of a prime number is irrational, non-terminating, and non-repeating)

⇒ Option 4: π ≈ 3.14159... (Pi is an irrational number, non-terminating, and non-repeating)

∴ The correct answer is 4/5.

Additional Information

Rational Numbers

Numbers that can be expressed in the form p/q, where p and q are integers and q is not equal to 0. They either terminate or repeat in their decimal form.

Irrational Numbers

Numbers that cannot be written as a simple fraction. Their decimal expansions are non-terminating and non-repeating (e.g., √2, √5, π).

Terminating Decimals

Decimals that have an end. A fraction p/q is terminating if the prime factors of q are limited strictly to 2 and/or 5.

Shortcut Trick

Average Speed = Total Distance ÷ Total Time

Total Distance = 6 km + 8 km = 14 km

Total Time = (6 ÷ 5) + (8 ÷ 6) = 1.2 + 1.333 = 2.533 hours

Average Speed = 14 ÷ 2.533 = 5.526 km/h ≈ 5.53 km/h

∴ The correct answer is 5.53 km/h.

Alternate Method

Given:

Distance 1 (D₁) = 6 km, Speed 1 (S₁) = 5 km/h

Distance 2 (D₂) = 8 km, Speed 2 (S₂) = 6 km/h

Formula Used:

Average Speed = Total Distance ÷ Total Time

Time = Distance ÷ Speed

D1 = 6 km S1 = 5 km/h

D2 = 8 km S2 = 6 km/h

Total Distance = 14 km

Total Time = T1 + T2 = 1.2 h + 1.33 h = 2.53 h Average Speed = 14 ÷ 2.53 ≈ 5.53 km/h

Calculations:

⇒ Time for the first part = 6 ÷ 5 = 1.2 hours = 6/5 hours

⇒ Time for the second part = 8 ÷ 6 = 4/3 hours

⇒ Total Time = 6/5 + 4/3 = (18 + 20) ÷ 15 = 38/15 hours

⇒ Total Distance = 6 + 8 = 14 km

⇒ Average Speed = 14 ÷ (38/15)

⇒ Average Speed = (14 × 15) ÷ 38 = 210 ÷ 38

⇒ Average Speed = 105 ÷ 19 = 5.526 km/h

⇒ Rounding to two decimal places, we get 5.53 km/h.

∴ The correct answer is 5.53 km/h.

Additional Information

Average Speed (Equal Distances)

If a person covers two equal consecutive distances at speeds x and y respectively, the average speed for the entire journey is given by 2xy ÷ (x + y).

Average Speed (Equal Time Intervals)

If a person travels at varying speeds x and y for equal time periods, the average speed over the total duration is simply the arithmetic mean: (x + y) ÷ 2.

Relative Speed

For two bodies moving in the same direction at speeds x and y (where x > y), relative speed = x − y. When moving in opposite directions towards each other, relative speed = x + y.

Shortcut Trick

If the radius or linear dimension of a 3D figure is multiplied by a factor of n, its volume gets multiplied by n³.

Here, the radius is doubled (n = 2), so the new volume becomes 2³ = 8 times the original volume.

Increase in volume = 8 − 1 = 7 times.

Percentage increase = 7 × 100% = 700%.

(Note: The given initial volume 144π cm³ is not needed to find the percentage increase.)

∴ The correct answer is 700%.

Alternate Method

Given:

Original volume of the sphere (V₁) = 144π cm³

Formula Used:

Volume of a sphere = (4/3)πr³

r Original Sphere V₁ = 144π

Radius × 2

2r New Sphere V₂ = 8 × 144π

Calculations:

⇒ Let the original radius of the sphere = r cm.

⇒ When the radius is doubled, the new radius = 2r cm.

⇒ Since Volume is directly proportional to the cube of the radius (r³), the ratio of the new volume to the original volume is (2r ÷ r)³ = 2³ = 8.

⇒ New volume (V₂) = 8 × Original Volume = 8 × 144π = 1152π cm³

⇒ Increase in volume = 1152π − 144π = 1008π cm³

⇒ Percentage increase = (Increase ÷ Original Volume) × 100

⇒ Percentage increase = (1008π ÷ 144π) × 100 = 7 × 100 = 700%

∴ The correct answer is 700%.

Additional Information

Area Scaling Rule

If the linear dimension of a 2D or 3D shape is multiplied by n, its surface area is multiplied by n². The percentage change is given by (n² − 1) × 100%.

Volume Scaling Rule

If the linear dimensions of any 3D shape are multiplied by n, its volume is multiplied by n³. The percentage change is given by (n³ − 1) × 100%.

Sphere vs Hemisphere

The volume of a hemisphere is exactly half the volume of a full sphere: V = (2/3)πr³, and its total surface area is 3πr².

Shortcut Trick

Convert the decimal values into fractions for faster multiplication:

Length = 12.75 = 51/4 m

Width = 6.2 = 31/5 m

Area = (51/4) × (31/5) = 1581/20

⇒ Area = 158.1 ÷ 2 = 79.05 m²

∴ The correct answer is 79.05 m².

Alternate Method

Length = 12.75 m Width = 6.2 m Area = ?

Given:

Length (L) = 12.75 m

Width (W) = 6.2 m

Formula Used:

Area of a rectangle = Length × Width

Calculation:

⇒ Area = 12.75 × 6.2

⇒ Area = 12.75 × (6 + 0.2)

⇒ Area = (12.75 × 6) + (12.75 × 0.2)

⇒ Area = 76.5 + 2.55

⇒ Area = 79.05 m²

∴ The correct answer is 79.05 m².

Additional Information

Perimeter of a Rectangle

The total distance around the outside of a rectangle. Formula: P = 2 × (Length + Width)

Diagonal of a Rectangle

The straight line segment connecting opposite corners. Formula: D = √(Length² + Width²)

Area of a Square

A square is a special type of rectangle where all four sides are equal. Formula: Area = Side²

Shortcut Trick

Since the amount of milk and servings are directly proportional:

⇒ Multiplying factor = Total milk ÷ Milk for 5 servings

⇒ Multiplying factor = 3.75 ÷ 0.625 = 6

⇒ Total servings = 5 × 6 = 30

∴ The correct answer is 30 servings.

Alternate Method

Given:

Milk for 5 servings = 0.625 liters

Total milk available = 3.75 liters

Concept Used:

Unitary Method: Find the requirement for 1 serving and multiply or divide accordingly.

  Proportional Mapping  0.625 Liters  3.75 Liters  × 6  5 Servings  30 Servings  × 6

Calculations:

⇒ Milk required for 1 serving = 0.625 ÷ 5 = 0.125 liters

⇒ Number of servings possible = Total milk ÷ Milk per serving

⇒ Number of servings possible = 3.75 ÷ 0.125

⇒ Multiply the numerator and denominator by 1000 to eliminate decimals.

⇒ Number of servings possible = 3750 ÷ 125 = 30

∴ The correct answer is 30 servings.

Additional Information

Unitary Method

It is a technique for solving a problem by first finding the value of a single unit, and then calculating the desired quantity by multiplying the single unit value accordingly.

Direct Proportion

When two variables scale by the same exact factor, they are in direct proportion. If x increases, y also increases linearly. Example: x₁ ÷ y₁ = x₂ ÷ y₂.

Inverse Proportion

When an increase in one variable leads to a proportional decrease in another variable (like more workers requiring fewer days). Example: x₁ × y₁ = x₂ × y₂.

Shortcut Trick

D's Profit : E's Profit = (33000 − 15000) : 15000 = 18000 : 15000 = 6 : 5.

The extra 1 unit of profit ratio is due to D's extra capital investment.

Effective extra capital of D = 10000 × 8 months = 80000.

Effective capital ratio = (12x + 80000) / 12x = 6 / 5 ⇒ 1 + (80000 / 12x) = 6 / 5.

80000 / 12x = 1 / 5 ⇒ 12x = 400000 ⇒ x = 33333.33.

∴ The correct answer is Rs. 33,333.33.

Alternate Method

Given:

Initial investment for both D and E = Rs. x

After 4 months, D added Rs. 10000.

Total profit at the end of 1 year = Rs. 33,000

E's share of profit = Rs. 15,000

Formula Used:

Profit Ratio = (Capital of A × Time of A) : (Capital of B × Time of B)

D's Capital E's Capital x × 4 (x + 10000) × 8 x × 12 Profit Ratio (D : E) = 18000 : 15000 = 6 : 5

Calculations:

⇒ D's profit = Total Profit − E's Profit

⇒ D's profit = 33000 − 15000 = Rs. 18,000

⇒ Profit Ratio of D to E = 18000 : 15000 = 6 : 5

⇒ Effective capital of D for 12 months = (x × 4) + ((x + 10000) × 8)

⇒ D's effective capital = 4x + 8x + 80000 = 12x + 80000

⇒ Effective capital of E for 12 months = x × 12 = 12x

⇒ Ratio of effective capitals = Profit Ratio

⇒ (12x + 80000) / 12x = 6 / 5

⇒ 5 × (12x + 80000) = 6 × 12x

⇒ 60x + 400000 = 72x

⇒ 72x − 60x = 400000

⇒ 12x = 400000

⇒ x = 400000 / 12

⇒ x = 33333.33

∴ The correct answer is Rs. 33,333.33.

Additional Information

Simple Partnership

When all partners invest their capitals for the exact same duration, the profit or loss generated is distributed purely in the ratio of their original investments.

Compound Partnership

When capitals are invested for varying time periods, the equivalent capital is computed as Capital × Time. The final profit distribution follows the ratio of these calculated equivalent capitals.

Working vs Sleeping Partner

A working partner actively manages the operations and typically receives a salary or fixed percentage of the profit, while a sleeping partner only supplies capital. The residual profit is divided according to effective investment ratios.

Shortcut Trick

For homogeneous ratio expressions, we can directly substitute the given ratio values into the equation.

Given x:y = 2:5, directly assume x = 2 and y = 5.

Substitute these into the expression (2x + 3y) : (x + y).

Numerator = 2(2) + 3(5) = 4 + 15 = 19

Denominator = 2 + 5 = 7

∴ The correct answer is 19:7.

Alternate Method

Given:

x:y = 2:5

Formula Used:

If x:y = a:b, then x = ak and y = bk (where k is a common constant multiplier).

Visual Scale Representation x = 2k y = 5k 2x + 3y = 2(2k) + 3(5k) = 19k x + y = 2k + 5k = 7k 19 : 7

Calculations:

⇒ Let x = 2k and y = 5k

⇒ Expression = (2x + 3y) : (x + y)

⇒ Substitute the assumed values of x and y into the expression:

⇒ Expression = (2 × 2k + 3 × 5k) : (2k + 5k)

⇒ Expression = (4k + 15k) : (7k)

⇒ Expression = 19k : 7k

⇒ The common factor 'k' cancels out from both sides.

⇒ Ratio = 19 : 7

∴ The correct answer is 19:7.

Additional Information

Duplicate & Triplicate Ratios

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

Componendo and Dividendo Rule

If a/b = c/d, then according to this algebraic identity, (a + b) ÷ (a − b) = (c + d) ÷ (c − d).

Proportions Rule

When four quantities are in proportion (a:b = c:d), the product of the extremes always equals the product of the means (a × d = b × c).

Shortcut Trick

We can solve this directly by verifying the given options.

Let's test Option 1: 10.

Square of the number = 10² = 100.

Twice the number = 2 × 10 = 20.

Difference = 100 − 20 = 80. (This matches the given condition perfectly)

∴ The correct answer is 10.

Alternate Method

Given:

If you square a number and then subtract twice the number, the result is 80.

Calculation:

Let the unknown number be x.

⇒ x² − 2x = 80

⇒ x² − 2x − 80 = 0

⇒ x² − 10x + 8x − 80 = 0

⇒ x(x − 10) + 8(x − 10) = 0

⇒ (x − 10)(x + 8) = 0

⇒ x = 10 or x = −8

Since 10 is available in our given options, the required number is 10.

∴ The correct answer is 10.

Additional Information

Standard Form of a Quadratic Equation

Any equation of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0, is a quadratic equation.

Sum and Product of Roots

For a quadratic equation ax² + bx + c = 0, the sum of its roots is −b ÷ a and the product of its roots is c ÷ a.

Nature of Roots (Discriminant)

The nature of roots is determined by the discriminant D = b² − 4ac. If D > 0, the roots are real and distinct. If D = 0, roots are real and equal. If D < 0, roots are imaginary.

Shortcut Trick

Using the Shoelace method: write coordinates vertically and cross-multiply.

Area = 1/2 × |(x₁y₂ + x₂y₃ + x₃y₁) − (y₁x₂ + y₂x₃ + y₃x₁)|

⇒ Area = 1/2 × |(2×1 + 7×6 + 4×3) − (3×7 + 1×4 + 6×2)|

⇒ Area = 1/2 × |(2 + 42 + 12) − (21 + 4 + 12)|

⇒ Area = 1/2 × |56 − 37| = 1/2 × 19 = 9.5

∴ The correct answer is 9.5 square units.

Alternate Method

Given:

Vertices of the triangle are A(2, 3), B(7, 1), and C(4, 6).

A(2, 3)

B(7, 1)

C(4, 6)

Area of Triangle = ?

Formula Used:

Area of a triangle = 1/2 × |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|

Calculation:

⇒ Let (x₁, y₁) = (2, 3)

⇒ Let (x₂, y₂) = (7, 1)

⇒ Let (x₃, y₃) = (4, 6)

⇒ Area = 1/2 × |2(1 − 6) + 7(6 − 3) + 4(3 − 1)|

⇒ Area = 1/2 × |2(−5) + 7(3) + 4(2)|

⇒ Area = 1/2 × |−10 + 21 + 8|

⇒ Area = 1/2 × |19|

⇒ Area = 9.5

∴ The correct answer is 9.5 square units.

Additional Information

Centroid of a Triangle

The centroid is the point where the three medians of a triangle intersect. Its coordinates can be found using the formula: ((x₁+x₂+x₃)÷3, (y₁+y₂+y₃)÷3).

Collinear Points Condition

Three points are considered collinear if they lie exactly on the same straight line. In coordinate geometry, the area of the triangle formed by three collinear points will always be zero.

Area via Base and Height

When the length of the base and its corresponding perpendicular height (altitude) are known, the area can be easily calculated using: Area = 1/2 × base × height.

Shortcut Trick

Working backwards from SP to MP:

Total CP = SP × 100 ÷ (100 + Gain%) = 4800 × 100 ÷ 125 = ₹3840

Initial CP (before repairs) = Total CP − Repairs = 3840 − 400 = ₹3440

MP = Initial CP × 100 ÷ (100 − Discount%) = 3440 × 100 ÷ 80 = ₹4300

∴ The correct answer is ₹4300.

Alternate Method

Given:

Discount = 20%

Repairs = ₹400

Selling Price (SP) = ₹4800

Gain = 25%

Formula Used:

Total CP = Initial CP + Overhead charges

SP = Total CP × (100 + Gain%) ÷ 100

Initial CP = MP × (100 − Discount%) ÷ 100

MP ₹4300

−20%

Initial CP ₹3440

+₹400

Total CP ₹3840

+25%

SP ₹4800

Reverse Calculation to find Marked Price

Calculation:

⇒ Let Marked Price be x.

⇒ Initial CP = x − 20% of x = 0.8x

⇒ Total CP = Initial CP + Repairs = 0.8x + 400

⇒ SP = Total CP × 1.25

⇒ 4800 = (0.8x + 400) × 1.25

⇒ 0.8x + 400 = 4800 ÷ 1.25

⇒ 0.8x + 400 = 3840

⇒ 0.8x = 3840 − 400 = 3440

⇒ x = 3440 ÷ 0.8

⇒ x = 4300

∴ The correct answer is ₹4300.

Additional Information

Marked Price and Discount

Marked Price (MP) is the list price or printed price. Discount is always calculated on the MP. Formula: SP = MP × (100 − Discount%) ÷ 100.

Overhead Charges

Extra expenses like repairs or transportation are added to the initial buying price to get the Total Cost Price (CP). Profit or loss is always calculated on the Total CP.

Successive Discounts

If two discounts of a% and b% are offered consecutively, the equivalent discount % = a + b − (a × b) ÷ 100.

Shortcut Trick

Since only water is added, the amount of milk remains constant.

Initial Milk Quantity = Final Milk Quantity

80 × 60% = (80 + 20) × New Milk %

48 = 100 × New Milk %

New Milk % = 48%

∴ The correct answer is 48%.

Alternate Method

Given:

Initial quantity of the mixture = 80 liters

Percentage of milk in the mixture = 60%

Quantity of water added = 20 liters

Formula Used:

Quantity of Milk = (Percentage of Milk ÷ 100) × Total Mixture

New Percentage = (Quantity of Milk ÷ New Total Mixture) × 100

Initial

80 L 60% Milk

Added

20 L Water

=

Final

100 L ? % Milk

Result: 48%

Calculations:

⇒ Quantity of milk in the initial mixture = (60 ÷ 100) × 80

⇒ Quantity of milk = 48 liters

⇒ New total quantity of the mixture = 80 + 20 = 100 liters

⇒ Since only water is added, the quantity of milk remains unchanged at 48 liters.

⇒ New percentage of milk = (48 ÷ 100) × 100

⇒ New percentage of milk = 48%

∴ The correct answer is 48%.

Additional Information

Constant Quantity Principle

When one component is added to or evaporated from a mixture, the total quantity of the other component remains unchanged. By equating its quantity before and after the change, you can easily solve the problem.

Alligation Rule

Alligation is used to quickly find the ratio in which two ingredients at given concentrations must be mixed to produce a desired final concentration. The ratio is found crosswise: (Cheaper − Mean) : (Mean − Dearer).

Replacement Formula

If an initial volume 'V' contains pure liquid, and 'x' volume is removed and replaced by water 'n' times successively, the final amount of pure liquid is given by V × (1 − x/V)ⁿ.

Shortcut Trick

According to the chain rule, M₁ × D₁ = M₂ × D₂

15 × 20 = 20 × D₂

D₂ = 15

∴ The correct answer is 15 days.

Alternate Method

Given:

Initial number of boys (M₁) = 15

Initial days required (D₁) = 20

New number of boys (M₂) = 20

Formula Used:

M₁ × D₁ = M₂ × D₂

Calculation:

⇒ Total work = M₁ × D₁

⇒ Total work = 15 × 20 = 300 boy-days

⇒ Now, the same work is done by 20 boys in D₂ days.

⇒ 300 = 20 × D₂

⇒ D₂ = 300 ÷ 20

⇒ D₂ = 15

∴ The correct answer is 15 days.

Additional Information

Chain Rule (MDH Concept)

If M₁ persons can do W₁ work in D₁ days working H₁ hours, and M₂ persons do W₂ work in D₂ days working H₂ hours, then (M₁ × D₁ × H₁) ÷ W₁ = (M₂ × D₂ × H₂) ÷ W₂.

Total Work Concept

Total work is considered constant. It is typically the product of the number of workers and the time taken to complete the job.

Work and Efficiency

Efficiency is inversely proportional to the time taken. If the number of workers increases, the time required to complete the same work decreases proportionally.

Shortcut Trick

Using the basic Man-Days concept for Time and Work:

Total Work = M₁ × D₁ = 20 × 45 = 900 man-days.

Work completed by 60 men in 6 days = 60 × 6 = 360 man-days.

Percentage of work done = (Work completed ÷ Total Work) × 100.

Percentage = (360 ÷ 900) × 100 = 40%.

∴ The correct answer is 40%.

Alternate Method

Given:

Initial workforce = 20 men

Time taken by 20 men to finish the work = 45 days

New workforce = 60 men

Days worked by new workforce = 6 days

Formula Used:

M₁ × D₁ = M₂ × D₂

First, we find out how many days 60 men would take to complete the entire work.

⇒ 20 × 45 = 60 × D₂

⇒ D₂ = 900 ÷ 60 = 15 days

Now, we can form an LCM efficiency table treating the two groups as entities:

Calculations:

⇒ Efficiency of 60 men = 3 units/day

⇒ Work completed by 60 men in 6 days = 3 units/day × 6 days = 18 units

⇒ Required Percentage = (Completed Work ÷ Total Work) × 100

⇒ Required Percentage = (18 ÷ 45) × 100

⇒ Required Percentage = (2 ÷ 5) × 100

⇒ Required Percentage = 0.4 × 100 = 40%

∴ The correct answer is 40%.

Additional Information

MDH Rule (Work Equivalence)

If M₁ men can do W₁ work in D₁ days working H₁ hours per day, and M₂ men can do W₂ work in D₂ days working H₂ hours per day, then: (M₁ × D₁ × H₁) ÷ W₁ = (M₂ × D₂ × H₂) ÷ W₂.

Total Work Concept

Total work is considered to be the product of the total number of workers and the number of days required to complete the task. Total Work = Men × Days.

Efficiency

Efficiency is the amount of work done by a person (or group) in a single unit of time. It is inversely proportional to the time taken to complete a given task when the total work remains constant.

Shortcut Trick

Given L = 2W. Base perimeter = 2 × (2W + W) = 6W = 30 ⇒ W = 5 cm, L = 10 cm.

Base Area = L × W = 10 × 5 = 50 cm².

Heights form an AP for 4 layers: 15 cm to 9 cm.

Total Height = Sum of AP = (4 ÷ 2) × (15 + 9) = 2 × 24 = 48 cm.

Total Volume = Base Area × Total Height = 50 × 48 = 2400 cm³.

∴ The correct answer is 2400 cm³.

Alternate Method

Given:

⇒ Base perimeter = 30 cm

⇒ Length (L) = 2 × Width (W)

⇒ Heights of 4 layers are in AP with 1^st term = 15 cm and 4^th term = 9 cm

Formula Used:

⇒ Perimeter of rectangle = 2 × (L + W)

⇒ Area of rectangle = L × W

⇒ Sum of AP = n ÷ 2 × (First term + Last term)

⇒ Total Volume = Base Area × Total Height

 Total Height 48 cm  Prism Layers  Layer 4: 9 cm  Layer 3: 11 cm  Layer 2: 13 cm  Layer 1: 15 cm  Base of Prism L = 10 cm  W = 5 cm Base Area 50 cm² Perimeter = 30 cm

Calculations:

⇒ 2 × (2W + W) = 30

⇒ 6W = 30 ⇒ W = 5 cm

⇒ L = 2 × 5 = 10 cm

⇒ Base Area = 10 × 5 = 50 cm²

⇒ The heights of the 4 layers form an AP where n = 4, first term a = 15, last term l = 9.

⇒ Total Height = 4 ÷ 2 × (15 + 9) = 2 × 24 = 48 cm

⇒ Total Volume = Base Area × Total Height

⇒ Total Volume = 50 × 48 = 2400 cm³

∴ The correct answer is 2400 cm³.

Additional Information

Volume of a General Prism

The volume of any uniform prism is calculated as the product of its base area and total vertical height: V = Base Area × Height.

Arithmetic Progression (AP) Sum Formula

When the first term (a) and last term (l) of an AP are known, the sum of n terms is derived using the formula: S_n = n ÷ 2 × (a + l).

Composite Solids Concept

For a solid structure generated by stacking individual uniformly based layers on top of each other, the total aggregate volume is essentially the continuous base area multiplied by the net sum of all vertical layer heights.

Shortcut Trick

Using the rule of alligation directly for the given prices:

Ratio = (Mean Price − Price of Tea 2) : (Price of Tea 1 − Mean Price)

Ratio = (270 − 240) : (300 − 270)

Ratio = 30 : 30 = 1 : 1

∴ The correct answer is 1:1.

Alternate Method

Given:

Price of Type 1 tea = ₹300/kg

Price of Type 2 tea = ₹240/kg

Since the mixture is sold without profit or loss, Cost Price (Mean Price) = Selling Price = ₹270/kg

Method Used:

We use the Alligation Cross Method to find the required ratio.

Type 1 ₹300/kg Type 2 ₹240/kg Mean ₹270/kg 30 270 − 240 30 300 − 270 Ratio = 30 : 30 = 1 : 1

Calculations:

Let the quantities of Type 1 and Type 2 tea be x and y respectively.

⇒ Total Cost Price = 300 × x + 240 × y

⇒ Total Selling Price = 270 × (x + y)

Equating Cost Price and Selling Price:

⇒ 300x + 240y = 270x + 270y

⇒ 300x − 270x = 270y − 240y

⇒ 30x = 30y

⇒ x / y = 30 / 30

⇒ x : y = 1 : 1

∴ The correct answer is 1:1.

Additional Information

Rule of Alligation

It is a method used to find the ratio in which two or more ingredients at given prices must be mixed to produce a mixture of a desired price. Formula: Quantity of Cheaper ÷ Quantity of Dearer = (CP of Dearer − Mean Price) ÷ (Mean Price − CP of Cheaper).

Mean Price

The cost price of a unit quantity of the mixture is called the mean price. Always ensure that the Mean Price represents Cost Price before applying alligation.

No Profit No Loss Condition

When a mixture is sold without any profit or loss (0% profit), the Selling Price (SP) of the mixture is exactly equal to its Cost Price (CP).

Shortcut Trick

Net change in marks = Correct Marks − Wrong Marks = 62 − 82 = −20

Change in average = Net change ÷ Total students = −20 ÷ 30 = −0.67

Correct Average = Old Average + Change in average

Correct Average = 62 − 0.67 = 61.33

∴ The correct answer is 61.33 marks.

Alternate Method

Given:

Total students (n) = 30

Wrong average = 62

Correct marks = 62

Wrong marks entered = 82

Formula Used:

Correct Sum = (Wrong Average × n) − Wrong Value + Correct Value

Correct Average = Correct Sum ÷ n

   Old Average 62   Correction −20 ÷ 30 −0.67   New Avg 61.33 Average Deviation Shift

Calculations:

⇒ Total sum before correction = 30 × 62 = 1860

⇒ Correct sum = 1860 − 82 + 62

⇒ Correct sum = 1860 − 20 = 1840

⇒ Correct average = 1840 ÷ 30

⇒ Correct average = 61.333...

∴ The correct answer is 61.33 marks.

Additional Information

1. Deviation Method for Errors

When an item is misread, New Average = Old Average + (Correct Value − Incorrect Value) ÷ Total Items.

2. Combined Average

If two groups of sizes n₁ and n₂ have averages A₁ and A₂, then Combined Average = (n₁ × A₁ + n₂ × A₂) ÷ (n₁ + n₂).

3. Inclusion/Exclusion Principle

If a new item is added and the average increases, New Item Value = Old Average + Total Items × Increase in Average.

Shortcut Trick

The expression sin²A + cos²A + 2sinAcosA is the exact algebraic expansion of (sinA + cosA)².

Substitute the given value sinA + cosA = √2 / 2 directly into the perfect square.

⇒ Result = (√2 / 2)²

⇒ Result = 2 / 4 = 1/2

∴ The correct answer is 1/2.

Alternate Method

Given:

sinA + cosA = √2 / 2

Formula Used:

(a + b)² = a² + b² + 2ab

Calculations:

⇒ We need to evaluate the given trigonometric expression: sin²A + cos²A + 2sinAcosA

⇒ By comparing the expression with the algebraic identity, we substitute a = sinA and b = cosA.

⇒ sin²A + cos²A + 2sinAcosA = (sinA + cosA)²

⇒ Now, substitute the given value of (sinA + cosA):

⇒ (sinA + cosA)² = (√2 / 2)²

⇒ (√2 / 2)² = 2 / 4

⇒ 2 / 4 = 1/2

∴ The correct answer is 1/2.

Additional Information

Algebraic Expansion Identity

(a + b)² = a² + b² + 2ab. This standard algebraic formula is used frequently to simplify both trigonometric and polynomial equations.

Pythagorean Trigonometric Identity

sin²A + cos²A = 1 is the fundamental Pythagorean identity. Using this rule, the expression can also be simplified and written as 1 + 2sinAcosA.

Double Angle Identity

2sinAcosA = sin(2A). Substituting this into the simplified expression gives the alternative form 1 + sin(2A).

Shortcut Trick

The point where a line crosses the x-axis always has a y-coordinate of 0.

Substitute y = 0 into the given equation: y = 5x − 10

⇒ 0 = 5x − 10

⇒ 5x = 10 ⇒ x = 2

∴ The correct answer is (2, 0).

Alternate Method

Given:

Equation of the line: y = 5x − 10

Concept Used:

The x-axis is the horizontal line representing the equation y = 0.

Any point lying perfectly on the x-axis will always have coordinates in the form (x, 0).

  x y   (2, 0)  (0, -10)  O y = 5x − 10

Calculations:

⇒ To find the x-intercept, put y = 0 into the given linear equation.

⇒ 0 = 5x − 10

⇒ 5x = 10

⇒ x = 10 ÷ 5

⇒ x = 2

⇒ Thus, the x-coordinate is 2 and the y-coordinate is 0.

⇒ The point of intersection on the x-axis is (2, 0).

∴ The correct answer is (2, 0).

Additional Information

x-intercept

The x-intercept is the point where a graph crosses the x-axis. At this point, the y-value is always exactly 0. Substitute y = 0 to directly solve for the x-coordinate.

y-intercept

The y-intercept is the point where a graph crosses the y-axis. At this point, the x-value is always exactly 0. Substitute x = 0 to directly solve for the y-coordinate.

Slope-Intercept Form

The standard equation of a straight line is commonly written as y = mx + c, where m represents the slope of the line and c is the point of the y-intercept.

Equation of the Axes

The mathematical equation representing the entire x-axis is y = 0, and the equation representing the entire y-axis is x = 0.

Shortcut Trick

Total sum of 8 interior angles = (8 − 2) × 180° = 1080°.

Measure of the 8^th interior angle = 1080° − 1000° = 80°.

Exterior angle = 180° − Interior angle.

8^th exterior angle = 180° − 80° = 100°.

∴ The correct answer is 100°.

Alternate Method

Given:

Number of sides of the polygon (n) = 8

Sum of 7 interior angles = 1000°

Formula Used:

Sum of all interior angles = (n − 2) × 180°

Interior angle + Exterior angle = 180°

80° 100° Interior Exterior

Interior + Exterior = 180°

Calculation:

⇒ Sum of all 8 interior angles = (8 − 2) × 180°

⇒ Sum of all 8 interior angles = 6 × 180° = 1080°

⇒ 8^th interior angle = Total sum − Sum of 7 given angles

⇒ 8^th interior angle = 1080° − 1000° = 80°

⇒ 8^th exterior angle = 180° − 80°

⇒ 8^th exterior angle = 100°

∴ The correct answer is 100°.

Additional Information

Sum of Exterior Angles

The sum of the exterior angles of any convex polygon is always 360°, regardless of the number of sides.

Regular Polygon Angles

For a regular polygon with n sides, each interior angle = ((n − 2) × 180°) ÷ n, and each exterior angle = 360° ÷ n.

Number of Diagonals

The total number of unique diagonals in an n-sided polygon is given by the formula: n(n − 3) ÷ 2.