Quantitative Aptitude PYQs

Shortcut Trick

Since it is an internal transfer, the total number of employees remains constant.

Initial Ratio A : B = 5 : 7 (Total 12 units) | Final Ratio = 1 : 1 = 6 : 6 (Total 12 units).

Shift from Dept B to Dept A = 7 − 6 = 1 unit (or 6 − 5 = 1 unit).

Given, 1 unit = 28 employees.

Original Dept A = 5 × 28 = 140 | Original Dept B = 7 × 28 = 196.

∴ The correct answer is 140 in A, 196 in B.

Alternate Method

Given:

Original ratio of employees in A and B = 5 : 7

Number of employees moved from B to A = 28

New ratio after transfer = 1 : 1

Calculations:

Let the original number of employees in Department A be 5x and in Department B be 7x.

⇒ New count in A = 5x + 28

⇒ New count in B = 7x − 28

⇒ (5x + 28) / (7x − 28) = 1 / 1

⇒ 5x + 28 = 7x − 28

⇒ 7x − 5x = 28 + 28

⇒ 2x = 56

⇒ x = 28

⇒ Original Dept A = 5 × 28 = 140

⇒ Original Dept B = 7 × 28 = 196

∴ The correct answer is 140 in A, 196 in B.

Additional Information

Constant Sum Property

When items are shifted between two groups, the total sum of items across both groups remains unchanged (Initial Sum = Final Sum).

Ratio Balancing

To solve ratio changes involving constant sums, equate the sum of ratio parts by multiplying with suitable constants.

Cross-Multiplication Method

For problems with additions/subtractions in ratios: (a × d) − (b × c) logic can be used to find the value of one ratio unit directly.

Shortcut Trick

The Simple Interest (SI) is calculated as a constant percentage of the Principal amount for every time period.

By definition, Simple Interest = Principal × Rate (%) × Time.

Converting the percentage Rate into a fraction gives: SI = (Principal × Rate × Time) ÷ 100.

∴ The correct answer is Simple Interest = (Principal × Rate × Time) / 100.

Alternate Method

Given: Principal (P), Rate of Interest per annum (R), and Time in years (T).

Formula Used: Simple Interest (SI) = (P × R × T) / 100.

⇒ Simple Interest is defined as the cost of borrowing money, where the interest is calculated only on the original Principal amount.

⇒ Interest for one year = P × (R / 100)

⇒ Interest for T years = (P × R / 100) × T

⇒ Simple Interest (SI) = (Principal × Rate × Time) / 100

∴ The correct answer is Simple Interest = (Principal × Rate × Time) / 100.

Additional Information

Total Amount (A)

The final value including both Principal and Interest is calculated as A = P + SI, which can also be written as A = P(1 + (R × T)/100).

Time Unit Consistency

The Time (T) must be in the same unit as the Rate (R). If the rate is per annum, time must be in years. (Divide months by 12 or days by 365 if necessary).

Simple vs. Compound Interest

Unlike Simple Interest, Compound Interest is calculated on the principal plus any accumulated interest from previous periods.

Shortcut Trick

The markup percentage is +20% and the discount percentage is (2 ÷ 30) × 100 = 20/3%.

Using the successive percentage formula: Net Change = Markup − Discount − [(Markup × Discount) ÷ 100]

Net Gain % = 20 − 20/3 − [20 × (20/3) ÷ 100] = 20 − 20/3 − 4/3

Net Gain % = 20 − 24/3 = 20 − 8 = 12%.

∴ The correct answer is 12% gain.

Alternate Method

Given:

The markup on the article is 20% above the cost price.

A discount of ₹2 is offered on a purchase of ₹30.

Formula Used:

Marked Price (MP) = CP × [1 + (Markup% ÷ 100)]

Selling Price (SP) = MP − Discount

Profit % = [(SP − CP) ÷ CP] × 100

⇒ Let the cost price (CP) of the article be ₹100.

⇒ The marked price (MP) is 20% above CP: MP = 100 + 20% of 100 = ₹120.

⇒ The discount is ₹2 for a purchase of ₹30. This means the discount rate = 2 ÷ 30 = 1/15.

⇒ Total discount on the MP of ₹120 = 120 × (1 ÷ 15) = ₹8.

⇒ The selling price (SP) = MP − Discount = 120 − 8 = ₹112.

⇒ Gain = SP − CP = 112 − 100 = ₹12.

⇒ Gain percentage = (12 ÷ 100) × 100 = 12%.

∴ The correct answer is 12% gain.

Additional Information

Relationship between CP and MP

The ratio of MP to CP can be expressed as MP/CP = (100 + Profit%) ÷ (100 − Discount%).

Successive Discount Formula

If two successive discounts of a% and b% are given, the equivalent single discount is [a + b − (ab/100)]%.

Markup Percentage

Markup % = [(MP − CP) ÷ CP] × 100. It represents how much the retailer has increased the price above the original cost.

Shortcut Trick

The smallest number which leaves a remainder r when divided by a, b, and c is given by: LCM(a, b, c) + r

⇒ Required number = LCM(9, 12, 15) + 3

⇒ Required number = 180 + 3 = 183

∴ The correct answer is 183.

Alternate Method

Given:

Divisors = 9, 12, and 15

Common remainder in each case = 3

Formula Used:

Required Number = LCM(divisors) × k + Remainder

For the smallest possible number, we take k = 1.

Calculations:

Find the prime factorization of each divisor to calculate the LCM:

⇒ 9 = 3²

⇒ 12 = 2² × 3

⇒ 15 = 3 × 5

The Least Common Multiple (LCM) is the product of the highest powers of all prime factors involved:

⇒ LCM(9, 12, 15) = 2² × 3² × 5

⇒ LCM(9, 12, 15) = 4 × 9 × 5

⇒ LCM(9, 12, 15) = 180

Now, add the common remainder to obtain the required smallest number:

⇒ Required number = LCM(9, 12, 15) + 3

⇒ Required number = 180 + 3 = 183

∴ The correct answer is 183.

Additional Information

General Form of Such Numbers

Any number that leaves a remainder r when divided by divisors a, b, and c is written in the form: (LCM(a, b, c) × k) + r, where k ≥ 1 is an integer.

LCM with Different Remainders (Constant Difference)

If a number leaves remainders p, q, and r when divided by a, b, and c respectively, such that (a − p) = (b − q) = (c − r) = k (constant difference), then the smallest such number is: LCM(a, b, c) − k.

HCF with Remainders

To find the greatest number that divides x, y, and z leaving remainders p, q, and r respectively, calculate: HCF(x − p, y − q, z − r).

Shortcut Trick

Mode is the value that appears most frequently in a given data set.

Frequency of 46 → 3 times (46, 46, 46).

Frequency of 41, 43, and 45 → 2 times each.

Frequency of all other values (40, 42, 44, 48, 50, 51, 52, 53, 55) → 1 time each.

Since 46 has the highest frequency of 3, it is the mode.

∴ The correct answer is 46.

Alternate Method

Given data: 43, 40, 41, 53, 50, 41, 44, 45, 43, 46, 48, 46, 51, 55, 52, 42, 46, 45

Formula: Mode = Observation with the maximum frequency.

To find the Mode, we construct a frequency distribution table:

⇒ 40: 1 time

⇒ 41: 2 times

⇒ 42: 1 time

⇒ 43: 2 times

⇒ 44: 1 time

⇒ 45: 2 times

⇒ 46: 3 times

⇒ 48: 1 time

⇒ 50, 51, 52, 53, 55: 1 time each

⇒ The observation with the maximum frequency is 46.

∴ The correct answer is 46.

Additional Information

Empirical Relation

For a moderately skewed distribution, the relationship is: Mode = 3 × Median − 2 × Mean.

Median

The Median is the middle value when the data is arranged in ascending or descending order.

Bimodal Distribution

If a data set contains two values that share the highest frequency, the distribution is called bimodal.

Shortcut Trick

Subtract the extra cost of tables from the total to equalize all items to chair prices.

Extra cost of 5 tables = 5 × 400 = ₹2000.

Remaining total for 12 items (5 + 7) = 7040 − 2000 = ₹5040.

Price of 1 chair = 5040 ÷ 12 = ₹420.

Price of 1 table = 420 + 400 = ₹820.

∴ The correct answer is ₹820.

Alternate Method

Given: Total cost of 5 tables and 7 chairs = ₹7040. A table costs ₹400 more than a chair.

Let the cost of one chair = ₹x

Then, the cost of one table = ₹(x + 400)

⇒ 5 × (x + 400) + 7 × x = 7040

⇒ 5x + 2000 + 7x = 7040

⇒ 12x + 2000 = 7040

⇒ 12x = 7040 − 2000

⇒ 12x = 5040

⇒ x = 5040 ÷ 12

⇒ x = 420

Cost of one table = x + 400 = 420 + 400 = 820.

∴ The correct answer is ₹820.

Additional Information

Linear Equations in One Variable

A linear equation is an algebraic equation in which each term has an exponent of 1. It is solved by isolating the variable on one side.

Unitary Method

This method involves finding the value of a single unit (like one chair) and then calculating the required value for other units.

Substitution Method

When one variable (Table) is expressed in terms of another (Chair), we use substitution to simplify the equation into a single variable.

Shortcut Trick

Since the Median is 28.5, the Median Class is 20−30.

Total frequency N = 52 + x ⇒ N/2 = 26 + 0.5x.

Cumulative frequency before median class cf = 5 + x and frequency f = 20.

Applying the simplified relation: 28.5 − 20 = [(26 + 0.5x − 5 − x) ÷ 20] × 10.

8.5 = (21 − 0.5x) ÷ 2 ⇒ 17 = 21 − 0.5x ⇒ 0.5x = 4 ⇒ x = 8.

∴ The correct answer is 8.

Alternate Method

Given: Median = 28.5, Class Intervals and Frequencies as provided.

Formula Used: Median = L + [((N/2) − cf) ÷ f] × h

Calculations:

⇒ Total frequency N = 5 + x + 20 + 15 + 7 + 5 = 52 + x

⇒ Median 28.5 lies in class 20−30, so L = 20, f = 20, h = 10

⇒ Cumulative frequency of preceding class cf = 5 + x

⇒ 28.5 = 20 + [((52 + x) ÷ 2 − (5 + x)) ÷ 20] × 10

⇒ 8.5 = [((52 + x) − 2(5 + x)) ÷ 2] × (1 ÷ 2)

⇒ 8.5 = (52 + x − 10 − 2x) ÷ 4

⇒ 8.5 × 4 = 42 − x

⇒ 34 = 42 − x

⇒ x = 42 − 34 = 8

∴ The correct answer is 8.

Additional Information

Median for Grouped Data

It is the value that divides the data into two equal halves. Formula: L + [((N/2) − cf) ÷ f] × h, where cf is the cumulative frequency of the class preceding the median class.

Empirical Formula

The relationship between Mean, Median, and Mode is: 3 × Median = Mode + 2 × Mean.

Mode for Grouped Data

The value with the highest frequency. Formula: L + [(f₁ − f₀) ÷ (2f₁ − f₀ − f₂)] × h.

Shortcut Trick

Effective Compound Interest (CI) rate for 2 years at 10% = 10 + 10 + (10×10)/100 = 21%.

Effective Simple Interest (SI) rate for 2 years at 10% = 10 × 2 = 20%.

Ratio of CI to SI for 2 years = 21 : 20.

Given 21 units = ₹136.50 ⇒ 1 unit = 136.50 ÷ 21 = ₹6.50.

SI (20 units) = 20 × 6.50 = ₹130.

∴ The correct answer is ₹130.

Alternate Method

Given:

Rate (R) = 10% per annum

Time (T) = 2 years

Compound Interest (CI) = ₹136.50

Formula Used:

CI = P[(1 + R/100)^T − 1]

SI = (P × R × T) ÷ 100

Calculations:

⇒ 136.50 = P[(1 + 10/100)² − 1]

⇒ 136.50 = P[(1.1)² − 1]

⇒ 136.50 = P[1.21 − 1]

⇒ 136.50 = 0.21 × P

⇒ P = 136.50 ÷ 0.21 = ₹650

Now, calculate Simple Interest:

⇒ SI = (650 × 10 × 2) ÷ 100

⇒ SI = 13000 ÷ 100 = ₹130

∴ The correct answer is ₹130.

Additional Information

Difference between CI and SI for 2 years

The difference (D) between Compound Interest and Simple Interest for 2 years is given by D = P(R/100)².

Successive Percentage Formula

For a two-year period, the effective compound interest rate can be calculated using the successive change formula: x + y + (xy/100).

CI and SI for the First Year

When interest is compounded annually, the Simple Interest and Compound Interest for the first year are always equal.

Shortcut Trick

To find the lowest price, calculate the product of the remaining price factors (100% − discount%).

Shop I: 0.13 × 0.63 = 0.0819

Shop II: 0.19 × 0.63 = 0.1197

Shop III: 0.43 × 0.46 = 0.1978

Shop IV: 0.09 × 0.75 × 0.41 = 0.0277

The shop with the minimum factor (Shop IV) will have the lowest price.

∴ The correct answer is Shop IV.

Alternate Method

Given: Marked Price (MP) is the same for all shops. Successive discounts are provided for four shops (I, II, III, and IV).

Formula Used: Selling Price (SP) = MP × (1 − d₁/100) × (1 − d₂/100) × ... where d₁, d₂ are successive discount percentages.

⇒ Let the Marked Price (MP) be 100 units.

⇒ Shop I SP = 100 × (100 − 87)/100 × (100 − 37)/100 = 100 × 0.13 × 0.63 = 8.19 units.

⇒ Shop II SP = 100 × (100 − 81)/100 × (100 − 37)/100 = 100 × 0.19 × 0.63 = 11.97 units.

⇒ Shop III SP = 100 × (100 − 57)/100 × (100 − 54)/100 = 100 × 0.43 × 0.46 = 19.78 units.

⇒ Shop IV SP = 100 × (100 − 91)/100 × (100 − 25)/100 × (100 − 59)/100 = 100 × 0.09 × 0.75 × 0.41 = 2.7675 units.

⇒ Comparing all Selling Prices: 2.7675 < 8.19 < 11.97 < 19.78.

∴ The correct answer is Shop IV.

Additional Information

Single Equivalent Discount

For two successive discounts a% and b%, the effective discount is (a + b − ab/100)%.

Discount and Selling Price

Selling Price = Marked Price × (100 − Discount%)/100. A higher total discount results in a lower selling price.

Successive Multiplication

When multiple discounts are applied, the order of discounts does not change the final selling price.

Given:

4^x − 4^x−1 = 48

Formula used:

a^m−n = a^m / aⁿ

Calculation:

4^x − (4^x / 4¹) = 48

⇒ 4^x (1 − 1/4) = 48

⇒ 4^x (3/4) = 48

⇒ 4^x = (48 × 4) / 3

⇒ 4^x = 16 × 4

⇒ 4^x = 64

⇒ 4^x = 4³

⇒ x = 3

Now, substitute x = 3 into the expression (2x − 1) / (2x + 3):

⇒ (2 × 3 − 1) / (2 × 3 + 3)

⇒ (6 − 1) / (6 + 3)

⇒ 5/9

∴ The correct answer is option (2).

Shortcut Trick

In any regular polygon, Interior angle + Exterior angle = 180^o.

Given: Interior angle − Exterior angle = 100^o.

2 × Exterior angle = (Sum − Difference) = 180^o − 100^o = 80^o.

⇒ Exterior angle = 40^o.

Number of sides (n) = 360^o ÷ Exterior angle = 360^o ÷ 40^o = 9.

∴ The correct answer is 9.

Alternate Method

Given:

Difference between interior angle (I) and exterior angle (E) of a regular polygon is 100^o.

⇒ I − E = 100^o --- (i)

Formula Used:

1. I + E = 180^o (Linear pair property)
2. Each Exterior angle (E) = 360^o ÷ n, where n is the number of sides.

Calculations:

⇒ We know that I + E = 180^o --- (ii)

⇒ Adding equation (i) and (ii):

⇒ (I − E) + (I + E) = 100^o + 180^o

⇒ 2I = 280^o

⇒ I = 140^o

⇒ Now, substitute I in (ii):

⇒ 140^o + E = 180^o

⇒ E = 180^o − 140^o = 40^o

⇒ Using the formula for number of sides:

⇒ n = 360^o ÷ E

⇒ n = 360^o ÷ 40^o

⇒ n = 9

∴ The correct answer is 9.

Additional Information

Sum of Interior Angles

The sum of all interior angles of a polygon with n sides is given by (n − 2) × 180^o.

Sum of Exterior Angles

The sum of the exterior angles of any convex polygon, taking one at each vertex, is always 360^o.

Regular Polygon Properties

In a regular polygon, all interior angles are equal and all exterior angles are equal. Each interior angle = [(n − 2) × 180^o] ÷ n.

Shortcut Trick

Ratio of original Cost Price (CP) to Selling Price (SP) is 100 : 121.

New Profit is 13.6%, so the new ratio is 100 : 113.6.

Using the Cross-Product Method with changes (+₹50 for CP and −₹69 for SP):

(100 × 113.6 − 121 × 100) units = (100 × −69 − 113.6 × 50)

−740 units = −6900 − 5680 = −12580 ⇒ 1 unit = ₹17.

∴ The correct answer is ₹1,700.

Alternate Method

Given:

Original Profit = 21%

Increase in CP = ₹50

Decrease in SP = ₹69

New Profit = 13.6%

Formula Used:

SP = CP × (1 + Profit%)

⇒ Let original Cost Price (CP) = x

⇒ Original Selling Price (SP) = x + 21% of x = 1.21x

⇒ New CP = x + 50 and New SP = 1.21x − 69

⇒ According to the question: 1.136 × (x + 50) = 1.21x − 69

⇒ 1.136x + 56.8 = 1.21x − 69

⇒ 1.21x − 1.136x = 56.8 + 69

⇒ 0.074x = 125.8

⇒ x = 125.8 ÷ 0.074 = 1700

∴ The correct answer is ₹1,700.

Additional Information

Profit Percentage

Profit % = (Selling Price − Cost Price) ÷ Cost Price × 100.

Relationship between CP and SP

If an article is sold at a profit of P%, then SP = CP × (100 + P) ÷ 100.

Cross-Product Method

Used when initial ratios and absolute changes are known to find the original values directly.

Shortcut Trick

Test the options to find the smallest positive number.

For 1: Numbers become (3 − 1) = 2 and (5 − 1) = 4.

Third proportional = b² ÷ a = 4² ÷ 2 = 16 ÷ 2 = 8.

The condition is satisfied as 8 is the required third proportional.

∴ The correct answer is 1.

Alternate Method

Given: Initial numbers are 3 and 5. The third proportional to be obtained is 8.

Formula Used: If c is the third proportional to a and b, then a : b = b : c, which means b² = a × c.

Calculations:

⇒ Let x be the smallest positive number to be subtracted.

⇒ First number (a) = 3 − x

⇒ Second number (b) = 5 − x

⇒ Third proportional (c) = 8

⇒ According to the property: (5 − x)² = (3 − x) × 8

⇒ 25 − 10x + x² = 24 − 8x

⇒ x² − 10x + 8x + 25 − 24 = 0

⇒ x² − 2x + 1 = 0

⇒ (x − 1)² = 0

⇒ x − 1 = 0

⇒ x = 1

∴ The correct answer is 1.

Additional Information

Mean Proportional

If b is the mean proportional between a and c, then b = √(a × c) or b² = ac.

Fourth Proportional

If d is the fourth proportional to a, b, and c, then a : b = c : d, which gives d = (b × c) ÷ a.

Continued Proportion

Numbers a, b, c, d are in continued proportion if a/b = b/c = c/d.

Shortcut Trick

Total Work (LCM of 6, 34, 36) = 612 units.

Efficiencies: Eduardo = 102, Abraham = 18, Xavier = 17.

Adjusted Work = Total Work − (Eduardo's 2-day work) + (Abraham's work if he stayed for the last 3 days).

Adjusted Work = 612 − (102 × 2) + (18 × 3) = 612 − 204 + 54 = 462 units.

Total time = Adjusted Work ÷ (Combined Efficiency of Abraham and Xavier) = 462 ÷ (18 + 17) = 13.2 days.

∴ The correct answer is 13.2 days.

Alternate Method

Given: Time taken by Eduardo = 6 days, Abraham = 34 days, and Xavier = 36 days.

Formula Used: Total Work = LCM of individual time; Efficiency = Total Work ÷ Time.

⇒ Let the total number of days taken to finish the work be T.

⇒ Work done by Eduardo (for 2 days) = 102 × 2 = 204 units.

⇒ Work done by Abraham (for T − 3 days) = 18 × (T − 3) units.

⇒ Work done by Xavier (for T days) = 17 × T units.

⇒ Total Work = (Work by Eduardo) + (Work by Abraham) + (Work by Xavier)

⇒ 612 = 204 + 18(T − 3) + 17T

⇒ 612 = 204 + 18T − 54 + 17T

⇒ 612 = 150 + 35T

⇒ 35T = 612 − 150 ⇒ 35T = 462

⇒ T = 462 ÷ 35 = 13.2 days.

∴ The correct answer is 13.2 days.

Additional Information

Time and Efficiency Relation

Efficiency is inversely proportional to the time taken to complete a task. Work = Efficiency × Time.

Leaving Before Completion

If a person leaves 'n' days before completion, adding their 'n' days' potential work to the total work allows you to treat them as having worked until the end.

LCM Method for Total Work

Assuming Total Work as the LCM of given time durations helps in calculating efficiencies as whole numbers, simplifying arithmetic steps.

Shortcut Trick

Time taken: Patna Express = 9h, Delhi Express = 11h. Let Distance = 99 units (LCM).

Speeds: Patna Express = 11 units/h, Delhi Express = 9 units/h.

Distance covered by Patna Express before 2:00 PM (4 hours) = 11 × 4 = 44 units.

Meeting time from 2:00 PM = Remaining distance ÷ Relative speed = (99 − 44) ÷ (11 + 9) = 55 ÷ 20 = 2 hours 45 min.

∴ The correct answer is 04:45PM.

Alternate Method

Given:

Patna Express (Delhi to Patna): 10:00 AM to 07:00 PM = 9 hours.

Delhi Express (Patna to Delhi): 02:00 PM to 01:00 AM = 11 hours.

Formula Used:

Time = Distance ÷ Relative Speed

⇒ Let total distance = 99 km.

⇒ Speed of Patna Express = 99 ÷ 9 = 11 km/h.

⇒ Speed of Delhi Express = 99 ÷ 11 = 9 km/h.

⇒ Patna Express distance from 10:00 AM to 02:00 PM = 11 × 4 = 44 km.

⇒ Distance between trains at 02:00 PM = 99 − 44 = 55 km.

⇒ Relative Speed = 11 + 9 = 20 km/h.

⇒ Time taken to meet = 55 ÷ 20 = 2.75 hours.

⇒ 2.75 hours = 2 hours + (0.75 × 60) min = 2 hours 45 min.

⇒ Meeting time = 02:00 PM + 2 hours 45 min = 04:45 PM.

∴ The correct answer is 04:45PM.

Additional Information

Relative Speed in Opposite Directions

When two objects move toward each other, their relative speed is the sum of their individual speeds: V_rel = S₁ + S₂.

Standard Meeting Time Logic

To find the meeting time, first calculate the distance covered by the early-starting train to bring both trains to the same starting time, then divide the remaining distance by the relative speed.

Average Speed for Same Distance

If a journey is covered at speed S₁ and returned at speed S₂, the Average Speed is (2 × S₁ × S₂) ÷ (S₁ + S₂).

Shortcut Trick

Average Speed = (Total Distance) ÷ (Total Time)

Total Distance = 6 + 18 = 24 km

Total Time = (6 ÷ 6) + (18 ÷ 9) = 1 + 2 = 3 hours

Average Speed = 24 ÷ 3 = 8 km/h

∴ The correct answer is 8 km/h.

Alternate Method

Given:

Distance covered while walking (d₁) = 6 km

Speed while walking (s₁) = 6 km/h

Distance covered while running (d₂) = 18 km

Speed while running (s₂) = 9 km/h

Formula Used:

Average Speed = Total Distance ÷ Total Time

Time = Distance ÷ Speed

⇒ Time taken for walking (t₁) = 6 ÷ 6 = 1 hour

⇒ Time taken for running (t₂) = 18 ÷ 9 = 2 hours

⇒ Total Distance = 6 + 18 = 24 km

⇒ Total Time = 1 + 2 = 3 hours

⇒ Average Speed = 24 ÷ 3 = 8 km/h

∴ The correct answer is 8 km/h.

Additional Information

Average Speed (Equal Distances)

If a person travels equal distances at speeds u and v, the average speed is given by the harmonic mean: 2uv ÷ (u + v).

Unit Conversion

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

Relative Speed

When two objects move in the same direction, relative speed = |u − v|. When they move in opposite directions, relative speed = u + v.

Shortcut Trick

Diameter of the sphere = 28 cm → Radius (r) = 14 cm.

Area of one circle with r = 14 cm is πr² = 616 cm².

Total Surface Area of a sphere = 4 × πr² = 4 × 616 = 2464 cm².

∴ The correct answer is 2464 cm².

Alternate Method

Given:

Diameter of the spherical tank (d) = 28 cm

Value of π = 22/7

Formula Used:

Radius of sphere (r) = d ÷ 2

Total Surface Area of sphere = 4 × π × r²

Calculations:

⇒ Radius (r) = 28 ÷ 2 = 14 cm

⇒ Total Surface Area = 4 × (22/7) × 14 × 14

⇒ Total Surface Area = 4 × 22 × 2 × 14

⇒ Total Surface Area = 88 × 28

⇒ Total Surface Area = 2464 cm²

∴ The correct answer is 2464 cm².

Additional Information

Volume of a Sphere

The total space enclosed inside a sphere is given by the formula: Volume = (4/3) × π × r³.

Curved Surface Area of a Hemisphere

For a hemisphere of radius r, the curved surface area is exactly half of the sphere's surface area: CSA = 2 × π × r².

Total Surface Area of a Hemisphere

For a solid hemisphere, the total surface area includes the curved surface and the flat circular base: TSA = 3 × π × r².

Shortcut Trick

Use the direct trigonometric identity: sec²θ = 1 + tan²θ.

Substitute tan θ = 2 into the formula: sec²θ = 1 + (2)² = 1 + 4 = 5.

Taking square root on both sides: sec θ = √5.

Since θ is in the first quadrant, sec θ must be positive.

Thus, the value matches Option (3).

∴ The correct answer is √5.

Alternate Method

Given: tan θ = 2 and θ is in the first quadrant.

Formula Used: In a right-angled triangle, tan θ = Perpendicular / Base and sec θ = Hypotenuse / Base. By Pythagoras Theorem: H² = P² + B².

⇒ tan θ = 2 / 1 → Perpendicular (P) = 2 and Base (B) = 1

⇒ Hypotenuse (H) = √(P² + B²)

⇒ H = √(2² + 1²) = √(4 + 1) = √5

⇒ sec θ = Hypotenuse / Base = √5 / 1 = √5

Since the angle is in the first quadrant, all trigonometric ratios are positive, confirming the result matches Option (3).

∴ The correct answer is √5.

Additional Information

Pythagorean Identities

There are three fundamental identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ.

Quadrant Rules (ASTC)

In the First Quadrant (0 to 90°), all trigonometric ratios are positive. In the Second, only sin and cosec are positive.

Reciprocal Relations

Trigonometric functions are related by reciprocals: sec θ = 1/cos θ, cosec θ = 1/sin θ, and cot θ = 1/tan θ.

Shortcut Trick

For a constant Rupee amount, Quantity₁ × SP%₁ = Quantity₂ × SP%₂

Here, Quantity₁ = 5, Loss = 34% (SP%₁ = 66%), Gain = 10% (SP%₂ = 110%)

⇒ 5 × 66 = x × 110

⇒ 330 = 110 × x

⇒ x = 3

∴ The correct answer is 3.

Alternate Method

Given:

Selling Price (SP₁) of 5 lemons = ₹1

Loss = 34% | Target Gain = 10%

Formula Used:

Cost Price (CP) = SP ÷ (1 − Loss%) | New SP = CP × (1 + Gain%)

Calculations:

⇒ SP of 1 lemon = ₹1 ÷ 5 = ₹0.20

⇒ Since this is at 34% loss, 66% of CP = ₹0.20

⇒ CP of 1 lemon = 0.20 ÷ 0.66 = ₹20/66 = ₹10/33

⇒ To gain 10%, New SP = 110% of (₹10/33)

⇒ New SP per lemon = (110 ÷ 100) × (10 ÷ 33) = (11 ÷ 10) × (10 ÷ 33) = 1 ÷ 3

⇒ New SP of 1 lemon = ₹1/3

⇒ Number of lemons sold for ₹1 = 1 ÷ (1/3) = 3

∴ The correct answer is 3.

Additional Information

Inverse Variation in Profit and Loss

When the total amount received remains the same, the quantity and the selling price percentage (100 ± P/L%) are inversely proportional. n₁ × %SP₁ = n₂ × %SP₂.

Successive Percentage Change

If a value is changed by a% and then by b%, the net change is given by the formula [a + b + (a × b) ÷ 100]%.

Relation between CP and SP

Cost Price is always considered as the base (100%). Profit increases the percentage above 100, while Loss decreases it below 100.

Shortcut Trick

For a hollow cylinder, Total Surface Area = 2π(R + r)(h + R − r).

Given: Outer radius (R) = 10 cm, Inner radius (r) = 8 cm, Height (h) = 200 cm.

TSA = 2 × (22/7) × (10 + 8) × (200 + 10 − 8)

TSA = 2 × (22/7) × 18 × 202 = 159984 ÷ 7 ≈ 22854.85 cm².

∴ The correct answer is 22854.85 cm².

Alternate Method

Given:

⇒ Outer radius (R) = 10 cm

⇒ Inner radius (r) = 8 cm

⇒ Length (h) = 2 m = 2 × 100 = 200 cm

Formula Used:

⇒ Total Surface Area (TSA) = Outer CSA + Inner CSA + 2 × Area of base rings

⇒ TSA = 2πRh + 2πrh + 2π(R² − r²)

Calculations:

⇒ TSA = 2 × (22/7) × [ (10 × 200) + (8 × 200) + (10² − 8²) ]

⇒ TSA = (44/7) × [ 2000 + 1600 + (100 − 64) ]

⇒ TSA = (44/7) × [ 3600 + 36 ]

⇒ TSA = (44 × 3636) ÷ 7

⇒ TSA = 159984 ÷ 7

⇒ TSA ≈ 22854.857 cm²

∴ The correct answer is 22854.85 cm².

Additional Information

Volume of Hollow Cylinder

The volume of the material in a hollow cylinder is given by V = π × (R² − r²) × h.

Curved Surface Area (CSA)

The total curved surface area is the sum of the inner and outer surface areas: CSA = 2πh(R + r).

Cross-sectional Area of the Ring

The area of the base ring (annulus) at each end of the pipe is π × (R² − r²).