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Shortcut Trick

Since the formula for total surface area involves π (22/7), the answer must be a multiple of 11.

Check divisibility by 11 for the options: (Sum of digits at odd places) − (Sum of digits at even places)

For 9152: (9 + 5) − (1 + 2) = 14 − 3 = 11 (Divisible by 11).

Other options: 9352 (9), 9052 (12), 9252 (10) are not divisible by 11.

∴ The correct answer is 9152 cm².

Alternate Method

Given: Capacity (Volume) = 55440 cm³, Base Radius (r) = 14 cm.

Formula Used: Volume of cylinder = πr²h | Total Surface Area (TSA) = 2πr(h + r)

⇒ Volume = πr²h

⇒ 55440 = (22/7) × 14 × 14 × h

⇒ 55440 = 22 × 2 × 14 × h

⇒ 55440 = 616 × h

⇒ h = 55440 ÷ 616 = 90 cm

⇒ TSA = 2πr(h + r)

⇒ TSA = 2 × (22/7) × 14 × (90 + 14)

⇒ TSA = 2 × 22 × 2 × 104

⇒ TSA = 88 × 104 = 9152 cm²

∴ The correct answer is 9152 cm².

Additional Information

Curved Surface Area (CSA) of Cylinder

The area of the side surface excluding the top and base circles is given by CSA = 2πrh.

Hollow Cylinder Volume

If R is the outer radius and r is the inner radius, Volume = π(R² − r²)h.

Total Surface Area of Hollow Cylinder

TSA = 2πRh + 2πrh + 2π(R² − r²) = 2π(R + r)(h + R − r).

Shortcut Trick

Use the property: HCF × LCM = Product of two numbers.

Other number = (HCF × LCM) ÷ First number

Other number = (7 × 84) ÷ 21

Other number = 588 ÷ 21 = 28

∴ The correct answer is 28.

Alternate Method

Given:

HCF of two numbers = 7

LCM of two numbers = 84

One number = 21

Formula Used:

The product of HCF and LCM of two numbers is equal to the product of the numbers themselves.

Number₁ × Number₂ = HCF × LCM

Calculations:

⇒ 21 × Number₂ = 7 × 84

⇒ Number₂ = (7 × 84) ÷ 21

⇒ Number₂ = 588 ÷ 21

⇒ Number₂ = 28

∴ The correct answer is 28.

Additional Information

Relationship between HCF and LCM

For any two numbers, the HCF is always a factor of the LCM. In this case, 7 is a factor of 84 (84 ÷ 7 = 12).

HCF of Coprime Numbers

If the HCF of two numbers is 1, the numbers are called coprime numbers, and their LCM is simply their product.

HCF and LCM for Fractions

HCF of fractions = HCF of Numerators ÷ LCM of Denominators | LCM of fractions = LCM of Numerators ÷ HCF of Denominators.

Shortcut Trick

Relative Speed in the same direction = 72 − 36 = 36 km/h.

Convert to m/s: 36 × 5/18 = 10 m/s.

Total distance (L₁ + L₂) = Relative Speed × Time = 10 × 27 = 270 m.

Length of faster train = 270 − 180 = 90 m.

∴ The correct answer is 90 m.

Alternate Method

Given:

Speed of faster train (S₁) = 72 km/h

Speed of slower train (S₂) = 36 km/h

Length of slower train (L₂) = 180 m

Time taken to overtake (t) = 27 seconds

Formula Used:

When two objects move in the same direction:

Relative Speed = S₁ − S₂

Total Distance = Length₁ + Length₂

Distance = Speed × Time

Calculations:

⇒ Relative Speed = 72 − 36 = 36 km/h

⇒ Speed in m/s = 36 × (5/18) = 10 m/s

⇒ Total Distance = L₁ + L₂

⇒ Total Distance = 10 × 27 = 270 m

⇒ L₁ + 180 = 270

⇒ L₁ = 270 − 180

⇒ L₁ = 90 m

∴ The correct answer is 90 m.

Additional Information

Relative Speed Concept

For the same direction, Relative Speed = |v₁ − v₂|. For opposite directions, Relative Speed = v₁ + v₂.

Speed Unit Conversion

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

Train Crossing Distance

When a train crosses another train or a bridge/platform, the total distance is the sum of both lengths.

Shortcut Trick

Simplify the first ratio: 8 : 12 = 2 : 3.

Since the ratios are in proportion, 2 units correspond to 20.

⇒ 1 unit = 20 ÷ 2 = 10.

⇒ The value of x corresponds to 3 units: 3 × 10 = 30.

∴ The correct answer is 30.

Alternate Method

Given: The terms are in proportion, 8 : 12 :: 20 : x.

Formula Used: Product of Extremes = Product of Means.

⇒ According to the property of proportion, (8) × (x) = (12) × (20)

⇒ 8x = 240

⇒ x = 240 ÷ 8

⇒ x = 30

∴ The correct answer is 30.

Additional Information

Fourth Proportional

If four quantities a, b, c, and d are in proportion (a:b = c:d), then d is called the fourth proportional and is calculated as d = (b × c) ÷ a.

Third Proportional

When a, b, and c are in continued proportion (a:b = b:c), then c is the third proportional where c = b² ÷ a.

Mean Proportional

The mean proportional between two numbers a and b is √(a × b).

Given:

Find the value of sec²(60°) − 1.

Formula Used:

secθ = 1 / cosθ

cos60° = 1 / 2

Calculation:

⇒ sec60° = 1 ÷ (1 / 2)

⇒ sec60° = 2

⇒ sec²(60°) − 1 = 2² − 1

⇒ = 4 − 1

⇒ = 3

∴ The value of sec²(60°) − 1 is 3.

Shortcut Trick

Apply BODMAS: Solve innermost brackets, then of, then division/multiplication from left to right.

Solve curly bracket: { ((18 ÷ 12) ÷ 3) + (16 ÷ 2) } → { (1.5 ÷ 3) + 8 } → 8.5

Solve first part: (18 ÷ 9 × 3) → 2 × 3 → 6

Combine with of: 6 of 8.5 = 6 × 8.5 = 51

∴ The correct answer is 51.

Alternate Method

Given Expression: [ (18 ÷ 9 × 3) of { ((18 ÷ (6 × 2)) ÷ 3) + (16 ÷ (5 + 3 − 6)) } ]

Method: Step-by-step BODMAS (Brackets, Of, Division, Multiplication, Addition, Subtraction) simplification.

⇒ [ (2 × 3) of { ((18 ÷ 12) ÷ 3) + (16 ÷ 2) } ]

⇒ [ 6 of { (1.5 ÷ 3) + 8 } ]

⇒ [ 6 of { 0.5 + 8 } ]

⇒ [ 6 of 8.5 ]

⇒ 6 × 8.5 = 51

∴ The correct answer is 51.

Additional Information

BODMAS Rule

It defines the order of operations: Brackets (), Of (powers/multiplication), Division and Multiplication (left to right), Addition and Subtraction (left to right).

Hierarchy of Brackets

Expressions within parentheses () are solved first, followed by curly braces {}, and then square brackets [].

The 'Of' Operator

The of operator represents multiplication but must be performed before any standard division or multiplication outside of brackets.

Shortcut Trick

Identify the number that repeats most frequently in the given series.

Frequency of 50 = 3 times

Frequency of 52 = 4 times

Other numbers repeat 1 or 2 times each.

Since 52 has the maximum frequency, it is the Mode.

∴ The correct answer is 52.

Alternate Method

Given: Data set = 52, 44, 47, 50, 45, 40, 43, 53, 53, 45, 52, 50, 52, 52, 44, 47, 54, 50

Formula Used: Mode is the value that appears most often in a set of data values.

Tabulating frequencies of each observation:

⇒ Observation 40: 1 time

⇒ Observation 43: 1 time

⇒ Observation 44: 2 times (44, 44)

⇒ Observation 45: 2 times (45, 45)

⇒ Observation 47: 2 times (47, 47)

⇒ Observation 50: 3 times (50, 50, 50)

⇒ Observation 52: 4 times (52, 52, 52, 52)

⇒ Observation 53: 2 times (53, 53)

⇒ Observation 54: 1 time

⇒ Maximum frequency is 4, which corresponds to the value 52.

∴ The correct answer is 52.

Additional Information

Empirical Formula

Relationship between Mean, Median and Mode: Mode = 3 × Median − 2 × Mean.

Types of Data Distribution

A data set is Unimodal if it has one mode, Bimodal if it has two modes, and Multimodal if it has more than two modes.

Median Calculation

Median is the middle term of sorted data. For even n, Median = [(n/2)^th + (n/2 + 1)^th]/2 term. For odd n, Median = [(n+1)/2]^th term.

Shortcut Trick

Quarterly Rate = 12% ÷ 4 = 3% per quarter.

Time = 6 months = 2 quarters.

Effective Percentage for 2 periods = 3 + 3 + (3 × 3)/100 = 6.09%.

Total Amount = 24000 × (100 + 6.09)% = 24000 × 1.0609 = 25461.60.

∴ The correct answer is ₹25,461.60.

Alternate Method

Given:

Principal (P) = ₹24,000

Annual Rate (R) = 12%

Time (T) = 6 months

Compounding = Quarterly (4 times a year)

Formula Used:

Amount (A) = P(1 + r/100)ⁿ

Where r = Quarterly Rate and n = Number of Quarters

Calculations:

⇒ r = 12% ÷ 4 = 3%

⇒ n = 6 months ÷ 3 months = 2

⇒ A = 24000 × (1 + 3/100)²

⇒ A = 24000 × (1.03)²

⇒ A = 24000 × 1.0609

⇒ A = 25461.60

∴ The correct answer is ₹25,461.60.

Additional Information

Quarterly Compounding Rule

When interest is compounded quarterly, the annual rate is divided by 4 and the time in years is multiplied by 4.

Half-Yearly Compounding

For half-yearly compounding, divide the annual rate by 2 and multiply the time in years by 2. Formula: A = P(1 + R/200)^2T.

Effective Annual Rate (EAR)

EAR is the actual interest rate earned due to compounding. For quarterly compounding, EAR = [(1 + R/400)⁴ − 1] × 100.

Shortcut Trick

Total Work = LCM(12, 6) = 12 units.

Efficiencies: A = 12 ÷ 12 = 1, B = 12 ÷ 6 = 2.

Combined work in 2 days = (1 + 2) × 2 = 6 units.

Remaining work = 12 − 6 = 6 units.

Time taken by A alone = 6 ÷ 1 = 6 days.

∴ The correct answer is 6 days.

Alternate Method

Given: Time taken by A = 12 days, Time taken by B = 6 days. Working period together = 2 days.

Formula Used: Efficiency = Total Work ÷ Time; Work Done = Efficiency × Time.

⇒ Total Efficiency of (A + B) = 1 + 2 = 3 units/day

⇒ Work completed by both in 2 days = 3 × 2 = 6 units

⇒ Remaining Work = Total Work − Completed Work = 12 − 6 = 6 units

⇒ Days taken by A to complete remaining work = 6 ÷ 1 = 6 days

∴ The correct answer is 6 days.

Additional Information

Total Work using LCM

Taking the Least Common Multiple (LCM) of individual days as the total work simplifies calculation by providing integer efficiencies.

Work and Wages

When people work together, wages are distributed in the ratio of the total work done by each individual, which is (Efficiency × Days worked).

Alternate Day Work

In alternate day problems, calculate the work done in one full cycle (e.g., 2 days) and then divide the total work by the cycle efficiency.

Shortcut Trick

Express B's marks as a ratio of A's marks: 50 : 80 = 5 : 8.

Since 1/8 = 12.5%, then 5/8 = 5 × 12.5%.

∴ 5/8 = 62.5%.

∴ The correct answer is 62.5%.

Alternate Method

Given:

Marks obtained by A = 80

Marks obtained by B = 50

Maximum Marks = 120 (Not required for this specific comparison)

Formula Used:

Required Percentage = (Marks of B ÷ Marks of A) × 100

⇒ Percentage of A's marks obtained by B = (50 ÷ 80) × 100

⇒ Required Percentage = (5 ÷ 8) × 100

⇒ Required Percentage = 0.625 × 100

⇒ Required Percentage = 62.5%

∴ The correct answer is 62.5%.

Additional Information

Base Value in Percentage

In any percentage comparison, the quantity following the word 'of' or 'than' is always treated as the Base (denominator).

Percentage Difference

To find by what percent A is more than B: [(A − B) ÷ B] × 100. Here, B becomes the base.

Standard Fraction Conversions

Memorizing 1/8 = 12.5%, 1/4 = 25%, and 3/8 = 37.5% helps in solving quantitative aptitude questions faster without manual division.