Quantitative Aptitude PYQs

Shortcut Trick

Let the total registered voters be 100x.

Votes cast = 80% of 100x = 80x, Valid votes = 95% of 80x = 76x.

Winner gets 75% of valid votes ⇒ Loser gets 25% of valid votes.

Winning Margin = (75% − 25%) = 50% of valid votes.

Margin = 50% of 76x = 38x = 1805 ⇒ x = 1805 ÷ 38 = 47.5

∴ The correct answer is 4750.

Alternate Method

Given:

Percentage of registered voters who cast their vote = 80%

Percentage of invalid votes = 5% of cast votes

Winner gets 75% of valid votes and wins by 1805 votes.

Registered: 100x

Cast: 80x

Not Cast: 20x

Valid: 76x

Invalid: 4x

Win: 57x

Lose: 19x

Margin = 57x − 19x = 38x
38x = 1805
x = 47.5
∴ 100x = 4750

Calculations:

Let the total number of registered voters be V.

⇒ Total cast votes = 80% of V = 0.80 V

⇒ Valid votes = (100% − 5%) of cast votes = 95% of 0.80 V

⇒ Valid votes = 0.95 × 0.80 V = 0.76 V

⇒ Winning candidate's vote share = 75% of valid votes.

⇒ Losing candidate's vote share = (100% − 75%) = 25% of valid votes.

⇒ Winning margin = (75% − 25%) = 50% of valid votes.

⇒ Margin = 0.50 × 0.76 V = 0.38 V

⇒ 0.38 V = 1805

⇒ V = 1805 ÷ 0.38 = 180500 ÷ 38 = 4750

∴ The correct answer is 4750.

Additional Information

Election Voting Tree

Elections follow a structured percentage split: Registered Voters → Cast Votes → Valid Votes → Candidates. Percentage losses (not cast, invalid) must be subtracted successively at each step.

Successive Percentages

Instead of calculating intermediate values individually, successive percentage chains can be written in a single equation. Final Value = Base × (1 − Loss₁%) × (1 − Loss₂%).

Winning Margin

When there are exactly two candidates, if the winner gets x% of the valid votes, the loser naturally gets (100 − x)%. The winning margin in terms of percentage is always (2x − 100)% of valid votes.

Shortcut Trick

Total Capacity = LCM(16, 25, 12) = 1200 units

Combined efficiency = (1200 ÷ 16) + (1200 ÷ 25) + (1200 ÷ 12) = 75 + 48 + 100 = 223 units/hour

Time taken together = Total Capacity ÷ Combined efficiency = 1200 ÷ 223 = 5 ⁸⁵/₂₂₃ hours

∴ The correct answer is 5 ⁸⁵/₂₂₃ hours.

Alternate Method

Given:

Time taken by Pipe A = 16 hours

Time taken by Pipe B = 25 hours

Time taken by Pipe C = 12 hours

Formula Used:

Total Work = LCM of individual times

Efficiency = Total Work ÷ Time

Time Taken = Total Work ÷ Total Efficiency

Calculation:

⇒ Total Efficiency of A, B, and C = 75 + 48 + 100 = 223 units/hour

⇒ Total time taken together = Total Work ÷ Total Efficiency

⇒ Time taken = 1200 ÷ 223

⇒ Time taken = 5 ⁸⁵/₂₂₃ hours

∴ The correct answer is 5 ⁸⁵/₂₂₃ hours.

Additional Information

Inlet vs Outlet Pipes

Inlet pipes add water to the tank (positive efficiency), while outlet pipes drain water from the tank (negative efficiency). The net efficiency is the algebraic sum of all pipes in operation.

Work Equivalency Principle

If a pipe takes x hours to fill a tank, its 1 hour work is 1/x. When multiple pipes operate together, their combined 1 hour work is computed as 1/T = 1/x + 1/y + 1/z.

Shortcut Trick

The mean proportional of two numbers a and b is directly given by the square root of their product: √(a × b).

⇒ Mean proportional = √(25 × 16)

⇒ Mean proportional = 5 × 4

⇒ Mean proportional = 20

∴ The correct answer is 20.

Alternate Method

Given:

First number (a) = 25

Second number (b) = 16

Formula Used:

If x is the mean proportional between a and b, they form a continuous proportion: a : x :: x : b

25
16
Area = 400
Rectangle (a × b)

Equivalent to

x
x
Area = x²
Square (x²)

Calculations:

⇒ Let the mean proportional be x.

⇒ 25 : x = x : 16

⇒ 25 ÷ x = x ÷ 16

⇒ x² = 25 × 16

⇒ x² = 400

⇒ x = √400

⇒ x = 20

∴ The correct answer is 20.

Additional Information

Third Proportional

If a : b :: b : c, then c is called the third proportional to a and b. It is calculated using the formula: c = b² ÷ a.

Fourth Proportional

If a : b :: c : d, then d is called the fourth proportional to a, b, and c. It is calculated using the formula: d = (b × c) ÷ a.

Duplicate & Triplicate Ratio

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

Shortcut Trick

Trade Discount = List Price − Buying Price

Discount = 3600 − 2800 = 800

Discount % = (Discount ÷ List Price) × 100

Discount % = (800 ÷ 3600) × 100 = 200 ÷ 9 ≈ 22.2%

∴ The correct answer is 22.2%.

Alternate Method

Given:

List Price (MP) of the fan = ₹ 3,600

Buying Price (SP) of the fan = ₹ 2,800

Formula Used:

Discount = MP − SP

Discount % = (Discount ÷ MP) × 100

MP
₹ 3600

SP
₹ 2800

Discount = ₹ 800

Discount % = (800 ÷ 3600) × 100 = 22.2%

Calculations:

⇒ Discount = 3600 − 2800 = ₹ 800

⇒ Discount % = (800 ÷ 3600) × 100

⇒ Discount % = 80000 ÷ 3600

⇒ Discount % = 200 ÷ 9

⇒ Discount % = 22.222... %

⇒ Correct to one decimal place, Discount % ≈ 22.2%

∴ The correct answer is 22.2%.

Additional Information

Marked Price (MP)

The price on the label of an article is known as the marked price. Discount is always calculated on the marked price.

Successive Discount

If two successive discounts of a% and b% are given, then the single equivalent discount = [a + b − (a × b) ÷ 100]%.

Selling Price (SP) Formula

SP = MP × (100 − Discount%) ÷ 100.

Shortcut Trick

When Salaj's speed is doubled, his time taken becomes exactly half.

Total time difference for Salaj = 4 hours (more than Kevin) + 9 hours (less than Kevin) = 13 hours.

Since doubling the speed reduces the time by half, this 13 hours reduction is exactly half of his normal time.

Therefore, Salaj's normal time = 13 × 2 = 26 hours.

Salaj's normal speed = Distance ÷ Time = 182 ÷ 26 = 7 km/hr.

∴ The correct answer is 7 km/hr.

Alternate Method

Given:

Total distance = 182 km

Let Salaj's normal time be T_S and Kevin's time be T_K.

Formula Used:

Speed = Distance ÷ Time

Distance = 182 km

Salaj (Normal)
Time = T

Kevin
Time = T − 4

Salaj (Doubled Speed)
Time = T/2 = T − 13

Calculations:

⇒ In the normal case, Salaj takes 4 hours more than Kevin.

⇒ T_S − T_K = 4 → T_K = T_S − 4 ... (Equation i)

⇒ When speed is doubled, Salaj's new time becomes T_S / 2

⇒ In this case, Salaj takes 9 hours less than Kevin: T_K − (T_S / 2) = 9

⇒ Substituting the value of T_K from Equation (i):

⇒ (T_S − 4) − (T_S / 2) = 9

⇒ T_S / 2 = 9 + 4

⇒ T_S / 2 = 13

⇒ T_S = 26 hours

⇒ Salaj's normal speed = Distance ÷ Time

⇒ Salaj's normal speed = 182 ÷ 26 = 7 km/hr

∴ The correct answer is 7 km/hr.

Additional Information

Inverse Relation of Speed and Time

For a constant distance, speed is inversely proportional to time (Speed ∝ 1 / Time). If speed is multiplied by a factor 'n', the time taken is divided by 'n'.

Relative Speed

When two bodies move in the same direction, their relative speed is the difference of their individual speeds |S₁ − S₂|. In opposite directions, it is S₁ + S₂.

Average Speed

When equal distances are covered at speeds a and b, Average Speed = (2 × a × b) ÷ (a + b). For unequal distances, it is Total Distance ÷ Total Time.

Shortcut Trick

Profit ratio of Anubhav to Gaurav = (7155 + 1431)/2 : (7155 − 1431)/2 = 4293 : 2862 = 3 : 2

Capital ratio = (Profit ratio) ÷ (Time ratio) = (3/2) : (2/5) = 15 : 4

Difference in capital ratio = 15 − 4 = 11 parts

11 parts = ₹ 55,000 ⇒ 1 part = ₹ 5,000

Anubhav's capital = 15 parts = 15 × 5,000 = ₹ 75,000

∴ The correct answer is ₹ 75,000.

Alternate Method

Given:

Total profit = ₹ 7,155

Difference in profit = ₹ 1,431

Time of Anubhav = 2 months, Time of Gaurav = 5 months

Formula Used:

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

Anubhav
Gaurav

Capital: x + 55K
Time: 2 months
Profit Ratio: 3

Capital: x
Time: 5 months
Profit Ratio: 2

VS

Capital Ratio = 15 : 4

Calculations:

⇒ Anubhav's profit + Gaurav's profit = ₹ 7,155

⇒ Anubhav's profit − Gaurav's profit = ₹ 1,431

⇒ 2 × Anubhav's profit = 8586 ⇒ Anubhav's profit = ₹ 4,293

⇒ Gaurav's profit = 7155 − 4293 = ₹ 2,862

⇒ Ratio of profit = 4293 : 2862 = 3 : 2

Let Gaurav's capital be x, so Anubhav's capital is x + 55000.

⇒ [(x + 55000) × 2] / (x × 5) = 3 / 2

⇒ 4 × (x + 55000) = 15x

⇒ 4x + 220000 = 15x

⇒ 11x = 220000

⇒ x = 20000

⇒ Capital contributed by Anubhav = 20000 + 55000 = ₹ 75,000

∴ The correct answer is ₹ 75,000.

Additional Information

Profit Sharing in Partnership

When investments are made for different time periods, the profit is distributed in the ratio of the product of capital and time for each partner. Formula: P₁ : P₂ = (C₁ × T₁) : (C₂ × T₂).

Working vs Sleeping Partner

A working partner manages the business and usually receives a specific percentage of the profit or a fixed salary before the remaining profit is divided based on capital ratios.

Finding Capital from Profit

If the profit ratio and time ratio are known, the capital ratio can be found by dividing the respective profit parts by the time parts: C₁ : C₂ = (P₁ / T₁) : (P₂ / T₂).

Shortcut Trick

Convert decimals to their simplest fractional forms: 2.7 = 27/10 and 0.096 = 96/1000 = 12/125.

LCM of fractions = LCM of numerators ÷ HCF of denominators.

⇒ LCM(27, 12) = 108

⇒ HCF(10, 125) = 5

⇒ Result = 108 ÷ 5 = 21.6

∴ The correct answer is 21.6.

Alternate Method

Given:

Numbers: 2.7 and 0.096

Method Used:

Convert decimals to integers by multiplying by a power of 10, find their LCM, and divide the result by the same power of 10.

Calculations:

⇒ The maximum number of decimal places is 3 (in 0.096). So, multiply both numbers by 1000 to convert them to integers.

⇒ 2.7 × 1000 = 2700

⇒ 0.096 × 1000 = 96

⇒ Prime factorization of 2700 = 2² × 3³ × 5²

⇒ Prime factorization of 96 = 2⁵ × 3¹

⇒ LCM(2700, 96) = 2⁵ × 3³ × 5²

⇒ LCM(2700, 96) = 32 × 27 × 25 = 21600

⇒ To find the LCM of the original decimal numbers, divide the result by 1000:

⇒ LCM = 21600 ÷ 1000 = 21.6

∴ The correct answer is 21.6.

Additional Information

LCM of Fractions

Fractions must be reduced to their simplest form first. LCM = (LCM of numerators) ÷ (HCF of denominators).

HCF of Fractions

Fractions must be reduced to their simplest form first. HCF = (HCF of numerators) ÷ (LCM of denominators).

LCM of Decimals

Convert decimals into integers by multiplying them with the highest power of 10 present in any decimal, find the LCM, and then divide the result by the same power of 10.

Product of Two Numbers

For any two numbers a and b: LCM(a, b) × HCF(a, b) = a × b.

Shortcut Trick

Second Number = (H.C.F. × L.C.M.) ÷ First Number

⇒ Second Number = (2 × 492) ÷ 82

⇒ Second Number = 984 ÷ 82 = 12

∴ The correct answer is 12.

Alternate Method

Given:

H.C.F. of two numbers = 2

L.C.M. of two numbers = 492

First Number = 82

Formula Used:

First Number × Second Number = H.C.F. × L.C.M.

Calculations:

⇒ 82 × Second Number = 2 × 492

⇒ 82 × Second Number = 984

⇒ Second Number = 984 ÷ 82

⇒ Second Number = 12

∴ The correct answer is 12.

Additional Information

Co-prime Numbers

Two numbers are said to be co-prime if their H.C.F. is 1. The L.C.M. of two co-prime numbers is always their product.

H.C.F. and L.C.M. of Fractions

H.C.F. of fractions = (H.C.F. of Numerators) ÷ (L.C.M. of Denominators).

L.C.M. of fractions = (L.C.M. of Numerators) ÷ (H.C.F. of Denominators).

Relation between H.C.F. and L.C.M.

For any two natural numbers A and B, their H.C.F. always completely divides their L.C.M.

Shortcut Trick

Simplify the fraction inside the bracket: 4 ÷ 2 = 2.

Expand both brackets directly: 4x² − 42x + 30 − 4x² − 20x + 60.

Cancel 4x² and −4x².

Combine the remaining terms: (−42x − 20x) + (30 + 60) = −62x + 90.

∴ The correct answer is −62x + 90.

Alternate Method

Given:

Expression: 2((4/2)x² − 21x + 15) − 4(x² + 5x − 15)

Calculation:

⇒ Simplify the fraction inside the first bracket: 4 ÷ 2 = 2

⇒ Expression becomes: 2(2x² − 21x + 15) − 4(x² + 5x − 15)

⇒ Multiply terms inside the first bracket by 2:

⇒ 4x² − 42x + 30

⇒ Multiply terms inside the second bracket by −4:

⇒ −4x² − 20x + 60

⇒ Now, combine both expanded parts:

⇒ (4x² − 4x²) + (−42x − 20x) + (30 + 60)

⇒ 0 − 62x + 90

⇒ −62x + 90

∴ The correct answer is −62x + 90.

Additional Information

BODMAS Rule

Follow the exact order: Brackets, Orders (powers/roots), Division, Multiplication, Addition, Subtraction to solve expressions systematically.

Algebraic Like Terms

Only terms with the exact same variable and power can be added or subtracted together (e.g., 4x² and −4x²).

Distributive Property

When multiplying a number outside a bracket, multiply it by every term inside: a(b + c) = ab + ac.

Shortcut Trick

Let the Actual Selling Price (SP) = 100 units.

Selling at 2/5 means SP = 40 units, which results in a 24% loss (i.e., 76% of CP).

Since 40 units → 76% of CP ⇒ 1 unit → 1.9% of CP.

If sold at 88% of Actual SP, the New SP = 88 units.

Value of 88 units = 88 × 1.9% of CP = 167.2% of CP.

Profit % = 167.2% − 100% = 67.2%.

∴ The correct answer is 67.2%.

Alternate Method

Given:

Selling an article at 2/5 of its actual SP causes a 24% loss.

The new selling price is 88% of the actual SP.

Formula Used:

SP = CP × (100 − Loss%) ÷ 100

Profit% = [(New SP − CP) ÷ CP] × 100

× 2/5

× 0.88

−24% Loss

+67.2% Profit

2/5 of SP
76% of CP

CP
100%

88% of SP
167.2% of CP

Actual SP
190% of CP

Calculations:

⇒ Let the actual Selling Price (SP) be 100x and Cost Price be CP.

⇒ First Selling Price = 100x × (2 ÷ 5) = 40x

⇒ Since loss is 24%, the selling price is equivalent to 76% of CP.

⇒ 40x = 76% of CP ⇒ CP = 40x ÷ 0.76 = 1000x ÷ 19

⇒ New Selling Price = 88% of 100x = 88x

⇒ Profit = New SP − CP = 88x − (1000x ÷ 19) = (1672x − 1000x) ÷ 19 = 672x ÷ 19

⇒ Profit % = [(672x ÷ 19) ÷ (1000x ÷ 19)] × 100

⇒ Profit % = (672 ÷ 1000) × 100 = 67.2%

∴ The correct answer is 67.2%.

Additional Information

Successive Percentage Change

When a value is changed by a% and then by b%, the net percentage change is given by (a + b + ab ÷ 100)%.

Marked Price and Discount

Discount is always calculated on the Marked Price (MP). Selling Price = MP × (100 − Discount%) ÷ 100.

Dishonest Dealer Concept

If a dealer claims to sell at Cost Price but uses false weight, Profit% = (True Weight − False Weight) ÷ False Weight × 100.

Shortcut Trick

When the distance is constant, speed is inversely proportional to time.

Ratio of speeds (Ishaan : Meghna) = 77 : 33 = 7 : 3

Ratio of time taken (Ishaan : Meghna) = 3 : 7

Ishaan's time = 3 units = 12 hours ⇒ 1 unit = 4 hours

Meghna's time = 7 units = 7 × 4 = 28 hours.

∴ The correct answer is 28 hours.

Alternate Method

Given:

Speed of Ishaan = 77 km/hr

Speed of Meghna = 33 km/hr

Time taken by Ishaan = 12 hours

Formula Used:

Distance = Speed × Time

Delhi
Kanpur
Distance (D) = Constant

Ishaan
77 km/hr | 12 hr

Meghna
33 km/hr | T hr

Calculations:

⇒ Distance covered by Ishaan = 77 × 12

⇒ Distance = 924 km

⇒ Since both travel the same distance, distance covered by Meghna = 924 km

⇒ Time taken by Meghna = Distance ÷ Speed of Meghna

⇒ Time taken by Meghna = 924 ÷ 33

⇒ Time taken by Meghna = 28 hours

∴ The correct answer is 28 hours.

Additional Information

Unit Conversion

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

Average Speed

When traveling two equal distances at different speeds S₁ and S₂, the Average Speed is given by (2 × S₁ × S₂) ÷ (S₁ + S₂).

Relative Speed

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

Shortcut Trick

Simplified investment ratio of X, Y and Z = 43 ∶ 30 ∶ 12

Total parts = 43 + 30 + 12 = 85 parts

Value of 85 parts = ₹ 4590 → 1 part = 4590 ÷ 85 = ₹ 54

Difference in shares of Y and Z = 30 − 12 = 18 parts

Required difference = 18 × 54 = ₹ 972

∴ The correct answer is ₹ 972.

Alternate Method

Given:

Investment ratio of X, Y and Z = 86 ∶ 60 ∶ 24

Total profit earned at the end of the year = ₹ 4590

43 parts
X

30 parts
Y

12 parts
Z
Total Profit = 85 parts = ₹ 4590

Calculations:

Since the time period of investment is not mentioned, it is assumed to be the same for all three partners. Therefore, the profit will be distributed in the exact ratio of their investments.

⇒ Profit Ratio of X, Y and Z = 86 ∶ 60 ∶ 24

⇒ Simplified Profit Ratio = 43 ∶ 30 ∶ 12

⇒ Total sum of ratios = 43 + 30 + 12 = 85 parts

⇒ 85 parts = ₹ 4590

⇒ 1 part = 4590 ÷ 85 = ₹ 54

Now, we need to find the difference between the profit shares of Y and Z.

⇒ Difference in parts = 30 parts (Y's share) − 12 parts (Z's share) = 18 parts

⇒ Difference in amount = 18 × 54

⇒ Difference in amount = ₹ 972

∴ The correct answer is ₹ 972.

Additional Information

Simple Partnership

When partners invest their capital for the identical time period, the profits or losses are distributed precisely in the ratio of their invested capitals. Formula: P₁ ∶ P₂ = C₁ ∶ C₂

Compound Partnership

When partners invest for varying time intervals, the profit share is determined by the product of the capital invested and the time period. Formula: P₁ ∶ P₂ = (C₁ × T₁) ∶ (C₂ × T₂)

Working vs Sleeping Partner

A working partner actively runs the business and usually takes a fixed salary or commission from the profit before the remainder is split. A sleeping partner merely invests capital without taking part in management operations.

Shortcut Trick

For divisibility by 24, a number must be divisible by its co-prime factors 3 and 8.

Check divisibility by 8 (last 3 digits): 904 ÷ 8 = 113. Only 6830904 is divisible by 8.

Check divisibility by 3 (sum of digits): 6 + 8 + 3 + 0 + 9 + 0 + 4 = 30. Divisible by 3.

Hence, 6830904 is divisible by both 3 and 8.

∴ The correct answer is 6830904.

Alternate Method

Given:

Numbers to check: 7991987, 7340390, 6830904, 7132978

Concept Used:

If a number is exactly divisible by a composite number, it must be divisible by all its co-prime factors.

The co-prime factors of 24 are 3 and 8. So, the number must be completely divisible by both 3 and 8.

Calculations:

⇒ Divisibility by 8 check for Option 1: 7991987 is an odd number, so it is not divisible by 8.

⇒ Divisibility by 8 check for Option 2: 7340390 ends in 390. Since 390 ÷ 8 = 48.75, it is not divisible by 8.

⇒ Divisibility by 8 check for Option 4: 7132978 ends in 978. Since 978 ÷ 8 = 122.25, it is not divisible by 8.

⇒ Divisibility by 8 check for Option 3: 6830904 ends in 904. Since 904 ÷ 8 = 113, it is divisible by 8.

⇒ Divisibility by 3 check for Option 3: Sum of the digits = 6 + 8 + 3 + 0 + 9 + 0 + 4 = 30.

⇒ Since 30 ÷ 3 = 10, the sum of the digits is fully divisible by 3.

⇒ Therefore, 6830904 is divisible by both 3 and 8, making it divisible by 24.

∴ The correct answer is 6830904.

Additional Information

Divisibility Rule of 3

A number is divisible by 3 if the total sum of all its digits is a multiple of 3.

Divisibility Rule of 8

A number is divisible by 8 if the number formed by its last three digits is a multiple of 8.

Co-Prime Factors Check

To check if a number is divisible by a larger composite number (like 36 or 72), check its divisibility by the co-prime factors of that number (e.g., for 36, check 4 and 9; for 72, check 8 and 9).

Co-Prime Numbers

Two numbers are considered co-prime if their Highest Common Factor (HCF) is exactly 1.

Shortcut Trick

Given Discount = 15%

⇒ Selling Price = 100% − 15% = 85% of Marked Price

⇒ 85% = ₹ 3,400

⇒ 100% = (3400 ÷ 85) × 100 = ₹ 4,000

∴ The correct answer is ₹ 4,000.

Alternate Method

Given:

Selling Price (SP) = ₹ 3,400

Discount = 15%

Formula Used:

Marked Price (MP) = [SP × 100] ÷ [100 − Discount%]

SP = ₹ 3,400 (85%)

15%

Discount

MP = 100%

Calculations:

⇒ MP = (3400 × 100) ÷ (100 − 15)

⇒ MP = (3400 × 100) ÷ 85

⇒ MP = 40 × 100

⇒ MP = ₹ 4,000

∴ The correct answer is ₹ 4,000.

Additional Information

Marked Price & Discount

Discount is an allowance given to the buyer and is always calculated on the Marked Price (MP) of an article, not on the Cost Price (CP).

Successive Discounts

When two consecutive discounts of a% and b% are given, the equivalent single discount is given by: [a + b − (a × b)÷100]%.

Relationship Between CP, MP, Profit% and Discount%

A highly useful direct ratio relation is: MP ÷ CP = (100 + Profit%) ÷ (100 − Discount%).

Shortcut Trick

Given sinθ = 1/2, the angle θ must be 30°.

For θ = 30°, the value of cosec 30° = 2 and cot 30° = √3.

Substitute these values directly into the given expression:

⇒ 13 × (2)² − 3 × (√3)² − 15

⇒ 52 − 9 − 15 = 28

∴ The correct answer is 28.

Alternate Method

Given:

sinθ = 1/2

Formula Used:

cosecθ = 1 / sinθ

cot²θ = cosec²θ − 1

Calculation:

⇒ cosecθ = 1 / (1/2) = 2

⇒ cosec²θ = (2)² = 4

⇒ cot²θ = 4 − 1 = 3

Now, substitute the squared values into the expression: 13 × cosec²θ − 3 × cot²θ − 15

⇒ 13 × 4 − 3 × 3 − 15

⇒ 52 − 9 − 15

⇒ 52 − 24

⇒ 28

∴ The correct answer is 28.

Additional Information

Basic Trigonometric Identities

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

Reciprocal Trigonometric Ratios

The reciprocal trigonometric ratios are: cosecθ = 1 / sinθ, secθ = 1 / cosθ, and cotθ = 1 / tanθ.

Values for Common Angles

For 30°, the standard values are sin 30° = 1/2, cos 30° = √3/2, and tan 30° = 1/√3.

Shortcut Trick

Total work = LCM(13, 4) = 52 units, so Naman's efficiency = 4 units/day and Omar's efficiency = 13 units/day.

Work done by both in 2 days = (4 + 13) × 2 = 34 units.

Remaining work to be done = 52 − 34 = 18 units.

Time taken by Naman to finish the remaining work = 18 ÷ 4 = 4.5 days.

∴ The correct answer is 4.5 days.

Alternate Method

Given:

⇒ Time taken by Naman to complete the work = 13 days

⇒ Time taken by Omar to complete the work = 4 days

⇒ They worked together for 2 days before Omar left.

Calculations:

⇒ Combined efficiency of Naman and Omar = 4 + 13 = 17 units/day

⇒ Work completed by both working together in 2 days = 17 × 2 = 34 units

⇒ Remaining work = 52 − 34 = 18 units

⇒ Time taken by Naman to complete the remaining work = Remaining Work ÷ Naman's efficiency

⇒ Time = 18 ÷ 4 = 4.5 days

∴ The correct answer is 4.5 days.

Additional Information

Pipes and Cisterns

This concept is identical to Time & Work logic, but an inlet pipe has positive efficiency while an outlet pipe (leak) has negative efficiency.

Work and Wages

Wages are always distributed in the ratio of the total work done by each person. If they work for the same number of days, wages are distributed in the ratio of their efficiencies.

Alternate Days Work

When individuals work on alternate days, calculate the total work done in one complete cycle (e.g., 2 days) and scale it up to reach close to the total work before calculating the remainder.

Shortcut Trick

By applying the laws of exponents directly from left to right for the same base:

⇒ a^m ÷ aⁿ × a^p = a^{m − n + p}

⇒ 83^{−6 − 16 + (−8)}

⇒ 83⁻³⁰

∴ The correct answer is 83⁻³⁰.

Alternate Method

Given:

Expression: 83⁻⁶ ÷ 83¹⁶ × 83⁻⁸

Formula Used:

Law of Quotient: a^m ÷ aⁿ = a^{m − n}

Law of Product: a^m × aⁿ = a^{m + n}

Calculation:

According to the BODMAS rule, division and multiplication have the same precedence, so we evaluate the expression from left to right.

⇒ (83⁻⁶ ÷ 83¹⁶) × 83⁻⁸

Applying the Law of Quotient for the first part:

⇒ 83^{−6 − 16} × 83⁻⁸

⇒ 83⁻²² × 83⁻⁸

Now, applying the Law of Product:

⇒ 83^{−22 + (−8)}

⇒ 83⁻³⁰

∴ The correct answer is 83⁻³⁰.

Additional Information

Law of Power of a Power

When a power is raised to another power, you multiply the exponents: (a^m)ⁿ = a^{m × n}

Negative Exponent Rule

A negative exponent denotes the reciprocal of the base raised to the positive exponent: a⁻ⁿ = 1 ÷ aⁿ

Zero Exponent Rule

Any non-zero base raised to the power of zero is always equal to 1: a⁰ = 1

Order of Operations (BODMAS)

Operations of equal priority like Division and Multiplication are always solved strictly from left to right unless brackets dictate otherwise.

Shortcut Trick

Let the assumed average be 50 kg.

Deviations of the weights from 50 = (40 − 50), (49 − 50), (56 − 50), (73 − 50), and (38 − 50)

Net deviation = −10 − 1 + 6 + 23 − 12 = +6

Average = Assumed average + (Net deviation ÷ 5)

Average = 50 + (6 ÷ 5) = 50 + 1.2 = 51.2 kg

∴ The correct answer is 51.2 kg.

Alternate Method

Given:

Weights of 5 family members = 40 kg, 49 kg, 56 kg, 73 kg, and 38 kg.

Formula Used:

Average = Sum of all observations ÷ Total number of observations

40
49
56
73
38

Avg = 51.2

Weights of 5 Family Members (in kg)

Calculations:

⇒ Sum of all weights = 40 + 49 + 56 + 73 + 38

⇒ Sum of all weights = 256 kg

⇒ Total members = 5

⇒ Average = 256 ÷ 5

⇒ Average = 51.2 kg

∴ The correct answer is 51.2 kg.

Additional Information

Deviation Method for Averages

It simplifies large calculations by picking an assumed average 'A'. The actual average is derived using: Actual Average = A + (Sum of deviations ÷ Total items).

Combined Average

If group 1 has n₁ items with an average of A₁ and group 2 has n₂ items with an average of A₂, then the Combined Average = (n₁ × A₁ + n₂ × A₂) ÷ (n₁ + n₂).

Operations on Averages

If every observation in a dataset is increased, decreased, multiplied, or divided by a constant value 'k', the new average changes identically by applying the same operation with 'k'.

Shortcut Trick

The sum of an interior and an exterior angle of a regular polygon is always 180°.

Ratio of Exterior : Interior = 1 : 8 ⇒ Total ratio parts = 1 + 8 = 9.

Exterior angle = (1 ÷ 9) × 180° = 20°.

Number of sides (n) = 360° ÷ Exterior angle = 360° ÷ 20° = 18.

∴ The correct answer is 18.

Alternate Method

Given:

Ratio of an exterior angle to an interior angle = 1 : 8

Formula Used:

Interior angle + Exterior angle = 180°

Number of sides (n) = 360° ÷ Exterior angle

Interior = 8x

Exterior = 1x

Vertex

Straight line = 180°

Calculations:

⇒ Let the exterior angle be x and the interior angle be 8x.

⇒ We know that angles on a straight line add up to 180°.

⇒ x + 8x = 180°

⇒ 9x = 180°

⇒ x = 20° (Exterior angle)

⇒ The sum of all exterior angles of any regular polygon is exactly 360°.

⇒ Number of sides (n) = 360° ÷ x

⇒ n = 360° ÷ 20°

⇒ n = 18

∴ The correct answer is 18.

Additional Information

Sum of Interior Angles

The sum of all interior angles of a polygon with n sides is given by the formula: Sum = (n − 2) × 180°.

Sum of Exterior Angles

The sum of exterior angles of any convex polygon is always 360°, regardless of the number of sides. This property directly helps find the number of sides.

Number of Diagonals

The total number of diagonals that can be drawn in an n-sided polygon is given by the formula: D = n(n − 3) ÷ 2.

Shortcut Trick

Total interest rate for 12 months = 3% × 12 = 36%

Simple Interest = 36% of Principal

SI = 36% of 1200

SI = 36 × 12 = 432

∴ The correct answer is ₹ 432.

Alternate Method

Given:

Principal (P) = ₹ 1,200

Rate of Interest (R) = 3% per month

Time (T) = 12 months

Time = 0
P = ₹ 1200

Time = 12 months
SI = ₹ 432

Rate = 3% per month

Total Interest = 12 × 3% = 36%
36% of 1200 = 432

Formula Used:

Simple Interest (SI) = (P × R × T) ÷ 100

Note:

Since the rate is given per month and time is in months, we use T = 12 directly.*

Calculations:

⇒ SI = (1200 × 3 × 12) ÷ 100

⇒ SI = 12 × 36

⇒ SI = 432

∴ The correct answer is ₹ 432.

Additional Information

Compound Interest Formula

When interest is compounded annually, Amount = P(1 + R/100)^T. The Compound Interest is calculated as Amount − P.

Difference between SI and CI for 2 Years

For a given principal P and rate R%, the difference between CI and SI for exactly 2 years is given by the formula D = P(R/100)².

Rule of 72

A quick formula to estimate the number of years required to double the invested money at a given annual interest rate (R). Time to double ≈ 72 ÷ R years.