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polygon, the sum of all interior angles is given by (n − 2) × 180°ree;.

 

Cyclic Quadrilateral Property

If all vertices of a quadrilateral lie on a circle, the sum of opposite angles is 180°ree;.

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Shortcut Trick

In a standard 30°−60°−90° triangle, the side ratio is 1 : √3 : 2.

The tower height (side opposite to 60°) corresponds to √3 units = 18 m.

The horizontal distance (side opposite to 30°) corresponds to 1 unit.

1 unit = 18 ÷ √3 = (18 × √3) ÷ 3 = 6√3 m.

∴ The correct answer is 6√3 m.

Alternate Method

Given:

Height of the vertical tower (AB) = 18 m

Angle of elevation (θ) = 60°

Horizontal distance between person and tower = BC

Formula Used:

tan θ = Opposite Side / Adjacent Side

Calculations:

⇒ tan 60° = AB / BC

⇒ √3 = 18 / BC

⇒ BC = 18 / √3

⇒ BC = (18 × √3) / (√3 × √3)

⇒ BC = 18√3 / 3

⇒ BC = 6√3 m

∴ The correct answer is 6√3 m.

Additional Information

Trigonometric Values

Common values include tan 30° = 1/√3, tan 45° = 1, and tan 60° = √3.

Angle of Elevation

The angle between the horizontal plane and the line of sight when an observer looks at an object above their eye level.

Rationalization

To simplify a fraction with a radical denominator, multiply the numerator and denominator by the radical: a/√b = (a√b)/b.

Given:

(2 + 3/2)² − [4 − (6/3)²] ÷ 1/2

Formula used:

The BODMAS rule is used to solve the expression: Brackets, Orders (powers), Division, Multiplication, Addition, and Subtraction.

Calculation:

⇒ (7/2)² − [4 − 2²] ÷ 1/2

⇒ 49/4 − [4 − 4] ÷ 1/2

⇒ 49/4 − 0 ÷ 1/2

⇒ 49/4 − 0

⇒ 49/4

∴ The correct answer is option (3).

Shortcut Trick

Numbers are in Ratio 9 : 7 : 5, which are co-prime to each other.

Shortcut formula: LCM = (Product of ratio parts) × HCF

2205 = (9 × 7 × 5) × HCF

2205 = 315 × HCF

HCF = 2205 ÷ 315 = 7

∴ The correct answer is 7.

Alternate Method

Given:

Ratio of three numbers = 9 : 7 : 5

LCM (Least Common Multiple) = 2205

Formula Used:

If numbers are ax, bx, and cx where a, b, and c are co-prime, then HCF = x and LCM = (a × b × c) × x.

Calculations:

⇒ Let the three numbers be 9x, 7x, and 5x.

⇒ Since 9, 7, and 5 have no common factors other than 1, they are co-prime.

⇒ HCF of (9x, 7x, 5x) = x

⇒ LCM of (9x, 7x, 5x) = 9 × 7 × 5 × x = 315x

⇒ 315x = 2205

⇒ x = 2205 ÷ 315

⇒ x = 7

⇒ HCF = 7

∴ The correct answer is 7.

Additional Information

Product of Two Numbers

For any two numbers A and B, A × B = HCF(A, B) × LCM(A, B).

Co-prime Numbers

Numbers that have only 1 as their common factor are called co-prime; their HCF is always 1.

LCM and HCF Relationship

The HCF of a set of numbers is always a factor of their LCM (LCM ÷ HCF is always an integer).

The solution is based on the relationship between Cost Price (CP), Selling Price (SP), and Profit Percentage.

Initial Case:

Let the Original Cost Price (CP₁) of the article be 100x.

Given the man sells it at a profit of 15%:

Original Selling Price (SP₁) = 100x + (15% of 100x) = 115x

New Scenario:

If the article was bought at 25% less cost price and sold for ₹294 less than the original selling price, the details change as follows:

In this new scenario, the Profit made is 30%.

We can express the New Selling Price in terms of the New Cost Price and Profit:

SP₂ = CP₂ + 30% of CP₂

SP₂ = 75x + (30/100) × 75x

SP₂ = 75x + 22.5x = 97.5x

Equating the New Selling Price values:

Now, we equate the two different expressions for SP₂ to find the value of x:

115x − 294 = 97.5x

115x − 97.5x = 294

17.5x = 294

x = 294 / 17.5

x = 16.8

Calculating the Original Cost Price:

The Original Cost Price (CP₁) was assumed to be 100x.

CP₁ = 100 × 16.8 = 1,680

The cost price of the article is ₹1,680.

The correct answer is option 4

Shortcut Trick

For divisibility by 2, the units digit Q must be an even number (0, 2, 4, 6, or 8).

For divisibility by 3, the sum of digits (3 + P + 2 + Q) = 5 + P + Q must be a multiple of 3.

The smallest multiple of 3 greater than or equal to 5 is 6.

Setting 5 + P + Q = 6 ⇒ P + Q = 1.

With Q = 0 (even) and P = 1, both conditions are satisfied, so the smallest value is 1.

∴ The correct answer is 1.

Alternate Method

Given: A four-digit number 3P2Q is divisible by both 2 and 3.

Formula Used:

1. Divisibility by 2: Last digit must be even (0, 2, 4, 6, 8).
2. Divisibility by 3: Sum of all digits must be divisible by 3.

Calculations:

⇒ For the number to be divisible by 2, Q ∈ {0, 2, 4, 6, 8}.

⇒ For the number to be divisible by 3, Sum = 3 + P + 2 + Q = 5 + P + Q must be a multiple of 3.

⇒ Possible multiples of 3 are 6, 9, 12, 15, ...

⇒ To minimize P + Q, we test the smallest possible sum: 5 + P + Q = 6.

⇒ P + Q = 6 − 5 = 1.

⇒ Now, check if there exist digits P and Q such that P + Q = 1 and Q is even.

⇒ If Q = 0 (even) and P = 1, then P + Q = 1.

⇒ The resulting number 3120 is divisible by both 2 and 3.

∴ The correct answer is 1.

Additional Information

Divisibility by 6

A number is divisible by 6 if it satisfies the divisibility rules for both 2 (even units digit) and 3 (sum of digits divisible by 3).

Divisibility by 4

A number is divisible by 4 if the number formed by its last two digits is divisible by 4 (e.g., in 3P2Q, the number 2Q must be divisible by 4).

Divisibility by 9

A number is divisible by 9 if the sum of all its digits is divisible by 9.

Shortcut Trick

Compare the Effective Discount for 5 toys in each scheme:

Scheme A: 24 + 33 − (24 × 33 ÷ 100) = 49.08%

Scheme B: Successive 39%, 37%, 23% → Obviously higher than 50% (approx 70.4%)

Scheme C: (2 × 24% + 3 × 12%) ÷ 5 = (48 + 36) ÷ 5 = 16.8%

Scheme D: 2 ÷ (5 + 2) × 100 ≈ 28.57%

Comparing all, 16.8% is the minimum discount.

∴ The correct answer is C.

Alternate Method

Assume the Marked Price (MP) of each toy = ₹100. For 5 toys, total MP = ₹500.

Scheme A (Successive 24% and 33%):

⇒ Net Discount % = 24 + 33 − (24 × 33)/100 = 57 − 7.92 = 49.08%

Scheme B (Successive 39%, 37% and 23%):

⇒ After 39% and 37%: 39 + 37 − (39 × 37)/100 = 76 − 14.43 = 61.57%

⇒ Net Discount % = 61.57 + 23 − (61.57 × 23)/100 = 84.57 − 14.16 = 70.41%

Scheme C (24% on first 2, 12% on rest):

⇒ Total Discount = (2 × 24) + (3 × 12) = 48 + 36 = ₹84

⇒ Effective Discount % = (84 ÷ 500) × 100 = 16.8%

Scheme D (2 toys free on 5 toys bought):

⇒ Total items = 5 + 2 = 7 toys

⇒ Discount % = (Free items ÷ Total items) × 100 = (2 ÷ 7) × 100 ≈ 28.57%

The Least Beneficial scheme for the customer is the one with the minimum discount.

∴ The correct answer is C.

Additional Information

Successive Discount Formula

For two successive discounts a% and b%, the single equivalent discount is [a + b − (a × b ÷ 100)]%.

Buy X Get Y Free

The effective discount percentage for such schemes is calculated as [Y ÷ (X + Y)] × 100.

Customer vs Shopkeeper Benefit

A customer seeks the maximum discount (Most Beneficial), while the shopkeeper seeks the minimum discount (Least Beneficial to customer).

Shortcut Trick

Use the direct algebraic identity: (Difference)² = (Sum)² − 4 × (Product)

Given Sum = 32 and Product = 220.

⇒ (Difference)² = 32² − 4 × 220

⇒ (Difference)² = 1024 − 880 = 144

⇒ Difference = √144 = 12

∴ The correct answer is 12.

Alternate Method

Let the two positive numbers be x and y.

Given:

1. x + y = 32
2. x × y = 220

Calculations:

⇒ (x + y)² = 32²

⇒ x² + y² + 2xy = 1024

⇒ x² + y² + 2(220) = 1024

⇒ x² + y² = 1024 − 440 = 584

⇒ (x − y)² = x² + y² − 2xy

⇒ (x − y)² = 584 − 440 = 144

⇒ x − y = √144 = 12

∴ The correct answer is 12.

Additional Information

Algebraic Identity for Sum and Difference

The relation between sum, difference, and product is given by the identity: (a + b)² − (a − b)² = 4ab.

Quadratic Equation Roots

If the sum (S) and product (P) of two numbers are known, the numbers are roots of the equation: t² − St + P = 0.

Finding Individual Values

Once the sum (S) and difference (D) are known, the larger number is (S + D) ÷ 2 and the smaller number is (S − D) ÷ 2.

Shortcut Trick

When a value is first increased by x% and then decreased by x%, the net change is always a decrease given by the formula: (x/10)²%.

Here, x = 20.

Net percentage change = (20 / 10)² = 2² = 4% decrease.

Note: The original price (₹500) does not affect the net percentage change.

∴ The correct answer is 4% decrease.

Alternate Method

Given: Original price = ₹500, first increase = 20%, subsequent decrease = 20%.

Formula: New Value = Original Value × (100 + change%)/100

⇒ Original Price = ₹500

⇒ Price after 20% increase = 500 + (20/100) × 500 = 500 + 100 = ₹600

⇒ Price after 20% decrease = 600 − (20/100) × 600 = 600 − 120 = ₹480

⇒ Net change in price = 500 − 480 = ₹20

⇒ Net percentage change = (Net Change / Original Price) × 100

⇒ Net percentage change = (20 / 500) × 100 = 4%

⇒ Since the final price (₹480) is less than the original price (₹500), it is a decrease.

∴ The correct answer is 4% decrease.

Additional Information

Successive Percentage Change Formula

For two successive changes of a% and b%, the effective change is [a + b + (ab/100)]%. Use positive for increase and negative for decrease.

Order of Changes

The final net percentage change remains the same regardless of the order of the percentage increases or decreases (i.e., a 20% increase followed by a 20% decrease is the same as a 20% decrease followed by a 20% increase).

Constant Percentage Rule

If a quantity is increased by x% and then the resulting quantity is decreased by x%, the overall result is always a net decrease of (x/10)²% or x²/100%.

Shortcut Trick

Loss of 12% ⇒ SP is 88% of Cost Price (CP).

Desired Profit of 14% ⇒ New SP should be 114% of CP.

Using the ratio: (1786.40 ÷ 88) × 114

⇒ 20.3 × 114 = ₹2,314.20.

∴ The correct answer is ₹2,314.20.

Alternate Method

Given:

Initial Selling Price (SP₁) = ₹1,786.40

Initial Loss = 12%

Required Profit = 14%

Formula Used:

Cost Price (CP) = SP × [100 ÷ (100 − Loss%)]

New Selling Price (SP₂) = CP × [(100 + Profit%) ÷ 100]

Calculations:

⇒ CP = 1786.40 × [100 ÷ (100 − 12)]

⇒ CP = 1786.40 × (100 ÷ 88)

⇒ CP = 20.3 × 100 = ₹2,030

⇒ New SP = 2030 × [(100 + 14) ÷ 100]

⇒ New SP = 2030 × (114 ÷ 100)

⇒ New SP = 20.3 × 114

⇒ New SP = ₹2,314.20

∴ The correct answer is ₹2,314.20.

Additional Information

Relationship between CP and SP

Profit % = (Profit ÷ CP) × 100, where Profit = SP − CP. Similarly, Loss % = (Loss ÷ CP) × 100.

Direct SP Calculation

When CP is constant, SP₁ ÷ (100 ± Gain/Loss₁) = SP₂ ÷ (100 ± Gain/Loss₂).

Successive Changes

If an article is sold through multiple hands at a profit/loss of x%, y%, and z%, then Final SP = CP × (1 ± x/100) × (1 ± y/100) × (1 ± z/100).