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Shortcut Trick

Curved Surface Area of a cone = πrl

Given: r = 3.5 cm, l = 10 cm, π = 22/7

Area = (22/7) × 3.5 × 10

Area = 22 × 0.5 × 10 = 110 cm²

∴ The correct answer is 110 cm².

Alternate Method

Given:

Radius of the cone (r) = 3.5 cm

Slant height of the cone (l) = 10 cm

Use π = 22/7

Formula Used:

Curved Surface Area (CSA) of a cone = π × r × l

Calculations:

⇒ CSA = (22/7) × 3.5 × 10

⇒ CSA = (22/7) × (7/2) × 10

⇒ CSA = 11 × 10

⇒ CSA = 110 cm²

∴ The correct answer is 110 cm².

Additional Information

Total Surface Area of a Cone

The total surface area includes the curved surface and the base area: TSA = πrl + πr² = πr(l + r).

Volume of a Cone

The volume of a cone is one-third the volume of a cylinder with the same base and height: V = (1/3)πr²h.

Relation between Slant Height and Vertical Height

The relationship is derived from Pythagoras theorem: l² = r² + h², where h is the vertical height.

Shortcut Trick

When the chord length of a circle is equal to its radius, it forms an equilateral triangle with the center.

In an equilateral triangle, all internal angles are equal to 60°.

∴ The central angle subtended by the chord = 60°.

∴ The correct answer is 60°.

Alternate Method

Given: Chord length = Radius (r)

Formula Used: For a triangle with sides a, b, and c, if a = b = c, then the triangle is equilateral and each angle is 60°.

Let O be the center of the circle and AB be the chord.

⇒ OA and OB are the radii of the circle.

⇒ OA = OB = r

⇒ Given: Chord AB = r

⇒ In ΔOAB, OA = OB = AB = r

⇒ Since all three sides are equal, ΔOAB is an equilateral triangle.

⇒ In an equilateral triangle, each interior angle is 60°.

⇒ ∠AOB = 60°

∴ The correct answer is 60°.

Additional Information

Chord Length Formula

The length of a chord subtending an angle θ at the center is given by L = 2r sin(θ/2). If L = r, then 1/2 = sin(θ/2), which gives θ/2 = 30° or θ = 60°.

Angle at Circumference

The angle subtended by a chord at any point on the major arc is half the angle subtended at the center. For this chord, the angle at the circumference would be 30°.

Angle in a Semi-circle

A diameter is a special chord that subtends an angle of 180° at the center and 90° at any point on the circle.

Shortcut Trick

For any point situated outside a circle, exactly two tangents can be drawn to that circle.

The number of tangents depends on the position of the point relative to the circle boundary.

∴ The correct answer is 2.

Alternate Method

Given: A point P located in the exterior of a circle with center O.

Property: From an external point, two and only two tangents can be drawn to a circle.

⇒ Let the external point be P and the circle center be O.

⇒ Lines PA and PB touch the circle at exactly one point each (A and B).

⇒ These lines are defined as tangents.

⇒ Any other line from P will either intersect the circle at two points (forming a secant) or not touch the circle at all.

∴ The correct answer is 2.

Additional Information

Tangents from Different Points

If a point is on the circle, exactly 1 tangent can be drawn. If a point is inside the circle, 0 tangents can be drawn.

Length of Tangents

The lengths of the two tangents drawn from an external point to a circle are equal (PA = PB).

Radius-Tangent Relationship

The radius is always perpendicular to the tangent at the point of contact (OA ⊥ PA and OB ⊥ PB).

Angle Property

The center of the circle lies on the angle bisector of the angle between the two tangents (∠APO = ∠BPO).

Shortcut Trick

Profit Ratio = (Capital × Time)

X : Y : Z = (5 × 12) : (7 × 12) : (14/3 × 7)

X : Y : Z = 180 : 252 : 98 = 90 : 126 : 49

49 units = ₹30,000 ⇒ Total (265 units) = (30,000 ÷ 49) × 265 ≈ ₹162245

∴ The correct answer is ₹162245.

Alternate Method

Given:

Ratio of capitals = 5x : 7x : (14x/3)

Investment time for X = 12 months, Y = 12 months, Z = 7 months.

Z's profit share = ₹30,000.

Formula Used:

Profit Ratio = (Capital₁ × Time₁) : (Capital₂ × Time₂) : (Capital₃ × Time₃)

Calculations:

⇒ Ratio of profits (X : Y : Z) = (5 × 12) : (7 × 12) : (14/3 × 7)

⇒ X : Y : Z = 60 : 84 : 98/3

⇒ Multiply by 3 to eliminate fraction: (60 × 3) : (84 × 3) : 98

⇒ Profit Ratio = 180 : 252 : 98

⇒ Simplify by 2: 90 : 126 : 49

⇒ Z's share (49 units) = ₹30,000

⇒ Value of 1 unit = 30,000 ÷ 49

⇒ Total profit units = 90 + 126 + 49 = 265 units

⇒ Total profit = 265 × (30,000 ÷ 49)

⇒ Total profit = 7,950,000 ÷ 49 ≈ ₹162244.89

⇒ Rounded to nearest integer = ₹162245

∴ The correct answer is ₹162245.

Additional Information

Partnership Profit Sharing Rule

Profits in a partnership are always distributed in the ratio of the product of the capital invested and the duration of the investment.

Active vs. Sleeping Partner

An active partner manages the business and may receive a salary/commission before the remaining profit is shared, while a sleeping partner only provides capital.

Total Profit Calculation

Total Profit = (Sum of all Ratio Units) × (Value of One Unit). One unit value is found by dividing a known partner's share by their ratio part.

Shortcut Trick

Identify the remaining fraction of milk in each step: (60 − 12) ÷ 60 = 48/60 = 4/5.

Since the process is performed 3 times in total, the milk left = Initial Quantity × (Fraction)³.

Calculation: 60 × (4/5)³ = 60 × (64/125) = 12 × 64/25 = 768 ÷ 25.

Final Value: 30.72 liters.

∴ The correct answer is 30.72 liters.

Alternate Method

Given:

Initial quantity of milk (V) = 60 liters

Quantity removed and replaced (x) = 12 liters

Total number of operations (n) = 1 (initial) + 2 (repeated) = 3 operations

Formula Used:

Remaining Quantity = V × (1 − x/V)ⁿ

Calculations:

⇒ Quantity of milk left = 60 × (1 − 12/60)³

⇒ Quantity of milk left = 60 × (1 − 1/5)³

⇒ Quantity of milk left = 60 × (4/5)³

⇒ Quantity of milk left = 60 × 64/125

⇒ Quantity of milk left = 3840 ÷ 125

⇒ Quantity of milk left = 30.72 liters

∴ The correct answer is 30.72 liters.

Additional Information

Successive Replacement Formula

When a volume x is replaced n times from total volume V, the remaining amount of original liquid is V(1 − x/V)ⁿ.

Concentration Ratio

The ratio of Final Milk to Total Mixture is given by (1 − x/V)ⁿ, as the total volume remains constant throughout.

Mixture Alligation Rule

This problem can also be viewed as repeated dilution where each step reduces the concentration by a factor of (V−x)/V.

Shortcut Trick

Apply the Alligation Rule directly to the costs of the two types of sugar.

Ratio of Type A to Type B = (Price of Type B − Mean Price) : (Mean Price − Price of Type A)

Ratio = (30 − 28) : (28 − 25)

Ratio = 2 : 3

∴ The correct answer is 2 : 3.

Alternate Method

Given:

Cost of Type A sugar (C₁) = ₹25/kg

Cost of Type B sugar (C₂) = ₹30/kg

Mean Price of the mixture (C_m) = ₹28/kg

Calculations:

⇒ Let the quantity of Type A be q₁ and Type B be q₂.

⇒ According to the Rule of Alligation:

⇒ q₁ / q₂ = (C₂ − C_m) / (C_m − C₁)

⇒ q₁ / q₂ = (30 − 28) / (28 − 25)

⇒ q₁ / q₂ = 2 / 3

⇒ Ratio = 2 : 3

∴ The correct answer is 2 : 3.

Additional Information

Alligation Rule

It is a modified form of weighted average used to find the ratio in which two ingredients at given prices must be mixed to produce a mixture of a desired price.

Mean Price Formula

The cost price of a unit quantity of the mixture is called the Mean Price. Formula: C_m = (q₁c₁ + q₂c₂) ÷ (q₁ + q₂).

Selling Price and Profit

If the mixture is sold at a profit, always convert the selling price to cost price first: CP = [SP × 100] ÷ [100 + Profit %] before applying alligation.

Shortcut Trick

To compare a fraction and a decimal, convert the fraction to a decimal by making the denominator 100.

Multiply both numerator and denominator of 17/20 by 5.

⇒ (17 × 5) ÷ (20 × 5) = 85 ÷ 100 = 0.85.

Since both values are 0.85, they are equal.

∴ The correct answer is Both are equal.

Alternate Method

Given: First value = 17/20, Second value = 0.85

Method: Convert the decimal value into a fraction and simplify.

⇒ 0.85 = 85/100

⇒ Divide both numerator and denominator by their HCF (5):

⇒ 85 ÷ 5 = 17

⇒ 100 ÷ 5 = 20

⇒ Simplified fraction = 17/20

⇒ Comparing 17/20 and 17/20, both are equal.

∴ The correct answer is Both are equal.

Additional Information

Fraction to Decimal Conversion

Divide the numerator by the denominator. For example, 1/4 = 1 ÷ 4 = 0.25.

Decimal to Fraction Conversion

Write the number as a fraction with a denominator of 10, 100, etc., based on decimal places, then simplify.

Comparing Fractions

To compare different fractions, use cross-multiplication (a/b vs c/d → compare a × d and b × c) or find a common denominator.

Shortcut Trick

Profit ratio = (Investment × Time) ratio

Ravi : Sita = (15,000 × 9) : (10,000 × 6) = 135 : 60 = 9 : 4

Total Ratio parts = 9 + 4 = 13 units

Ravi profit share = (9 ÷ 13) × 18,000 ≈ ₹12461.54

∴ The correct answer is ₹12461.54.

Alternate Method

Given:

Investment of Ravi (C_R) = ₹15,000 | Investment of Sita (C_S) = ₹10,000

Time period for Ravi (T_R) = 9 months | Time period for Sita (T_S) = 6 months

Total Business Profit = ₹18,000

Formula Used: Profit Share Ratio = (C₁ × T₁) : (C₂ × T₂)

Calculations:

⇒ Profit Ratio (Ravi : Sita) = (15000 × 9) : (10000 × 6)

⇒ Profit Ratio = 135000 : 60000

⇒ Profit Ratio = 135 : 60

⇒ Profit Ratio = 9 : 4 (Dividing by 15)

⇒ Ravi share = [9 ÷ (9 + 4)] × 18000

⇒ Ravi share = (9 ÷ 13) × 18000

⇒ Ravi share = 162000 ÷ 13

⇒ Ravi share = ₹12461.538...

∴ The correct answer is ₹12461.54.

Additional Information

Compound Partnership

When partners invest capital for different time periods, the profit is shared in the ratio of the product of their capital and time duration.

Active vs. Sleeping Partner

An active partner manages the business and may receive a fixed salary or commission from the profit before the remaining profit is shared among all partners.

Ratio of Capitals

If the profit ratio and time ratio are given, the capital ratio can be found as: C₁ : C₂ = (P₁ ÷ T₁) : (P₂ ÷ T₂).

Shortcut Trick

Net Profit % = Markup % − Discount % − [(Markup % × Discount %) ÷ 100]

Net Profit % = 60 − 25 − [(60 × 25) ÷ 100]

Net Profit % = 35 − (1500 ÷ 100) = 35 − 15 = 20%

Final Selling Price = 120% of 5,000 = ₹6,000

∴ The correct answer is 20%.

Alternate Method

Given:

Cost Price (CP) of the jacket = ₹5,000

Markup Percentage = 60%

Discount Percentage = 25%

Calculations:

⇒ Marked Price (MP) = CP + (60% of CP)

⇒ MP = 5,000 + (0.60 × 5,000) = 5,000 + 3,000 = ₹8,000

⇒ Selling Price (SP) = MP − (25% of MP)

⇒ SP = 8,000 − (0.25 × 8,000) = 8,000 − 2,000 = ₹6,000

⇒ Total Profit = SP − CP = 6,000 − 5,000 = ₹1,000

⇒ Profit % = (Total Profit ÷ CP) × 100

⇒ Profit % = (1,000 ÷ 5,000) × 100 = 20%

∴ The correct answer is 20%.

Additional Information

Relation between CP and MP

The ratio of MP to CP is given by MP ÷ CP = (100 + Profit %) ÷ (100 − Discount %).

Markup Percentage Formula

Markup % = [(MP − CP) ÷ CP] × 100. It represents how much more the price is set above the cost price.

Successive Percentage Change

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

Shortcut Trick

For expression of form (√x + √y) / (√x − √y), the result is (x + y + 2√xy) / (x − y).

Here, x = 3 and y = 2 ⇒ Result = (3 + 2 + 2√6) / (3 − 2) = 5 + 2√6.

Comparing with a + b√6: a = 5 and b = 2.

Therefore, a + b = 5 + 2 = 7.

∴ The correct answer is 7.

Alternate Method

Given: (√3 + √2) / (√3 − √2) = a + b√6

Formula Used: (x − y)(x + y) = x² − y² and (x + y)² = x² + y² + 2xy

Calculations:

⇒ [(√3 + √2) × (√3 + √2)] / [(√3 − √2) × (√3 + √2)] = a + b√6

⇒ [(√3)² + (√2)² + 2(√3)(√2)] / [(√3)² − (√2)²] = a + b√6

⇒ [3 + 2 + 2√6] / [3 − 2] = a + b√6

⇒ 5 + 2√6 = a + b√6

⇒ On comparing both sides, a = 5 and b = 2

⇒ a + b = 5 + 2 = 7

∴ The correct answer is 7.

Additional Information

Rationalization of Denominator

It is the process of converting a fraction with an irrational denominator into an equivalent fraction with a rational denominator by multiplying both numerator and denominator by a suitable factor.

Conjugate Surds

For a surd expression (√x + √y), its conjugate is (√x − √y). Their product always results in a rational number: (√x + √y)(√x − √y) = x − y.

Comparison Principle

If p + q√n = x + y√n, where p, q, x, y are rational numbers and √n is irrational, then p = x and q = y.