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Shortcut Trick

Volume of Cone = (1/3) × π × r² × h

⇒ 1232 = (1/3) × (22/7) × r² × 24

⇒ 1232 = 8 × (22/7) × r²

⇒ 154 = (22/7) × r² ⇒ r² = 7 × 7 = 49

⇒ r = 7 cm

∴ The correct answer is 7 cm.

Alternate Method

Given:

Volume (V) of the right circular cone = 1232 cm³

Height (h) of the cone = 24 cm

Formula Used:

Volume of a right circular cone = (1/3) × π × r² × h

Calculations:

⇒ 1232 = (1/3) × (22/7) × r² × 24

⇒ 1232 = (22/7) × r² × 8

⇒ r² = (1232 × 7) / (22 × 8)

⇒ r² = (154 × 7) / 22

⇒ r² = 7 × 7

⇒ r² = 49

⇒ r = √49 = 7 cm

∴ The correct answer is 7 cm.

Additional Information

Slant Height (l)

The relationship between slant height (l), height (h), and radius (r) is l = √(h² + r²).

Curved Surface Area (CSA)

The Curved Surface Area of a cone is given by the formula π × r × l.

Total Surface Area (TSA)

The Total Surface Area of a cone is π × r × (l + r), which includes the base area πr².

Shortcut Trick

Direct Formula for Volume of a Pyramid = (1÷3) × Base Area × Vertical Height

Base Area (Square) = side² = 10 × 10 = 100 cm²

Volume = (1÷3) × 100 × 12 = 100 × 4 = 400 cm³

∴ The correct answer is 400 cm³.

Alternate Method

Given:

Base side (a) = 10 cm

Vertical height (h) = 12 cm

Step 1: Calculate the Area of the Square Base.

⇒ Base Area = side × side

⇒ Base Area = 10 cm × 10 cm = 100 cm²

Step 2: Calculate the Volume of the Pyramid using the standard formula.

⇒ Volume = (1÷3) × Base Area × Height

⇒ Volume = (1÷3) × 100 × 12

⇒ Volume = 100 × 4

⇒ Volume = 400 cm³

∴ The correct answer is 400 cm³.

Additional Information

Slant Height of a Square Pyramid

The slant height (l) is the distance from the apex to the midpoint of a base edge: l = √(h² + (a÷2)²).

Lateral Surface Area (LSA)

The LSA of a square pyramid is the sum of areas of its four triangular faces: LSA = 2 × a × l (where l is slant height).

Total Surface Area (TSA)

The TSA includes the base area and the lateral area: TSA = a² + 2 × a × l.

General Pyramid Volume

For any pyramid (triangular, hexagonal, etc.), the volume is always (1÷3) × Area of the Base × Vertical Height.

Shortcut Trick

The Incenter is the point where the internal angle bisectors of a triangle intersect.

By geometric definition, the internal bisectors of any triangle (Acute, Right, or Obtuse) must stay within the boundaries of the triangle.

Since all three bisectors are internal, their point of concurrency must lie inside the triangle.

Thus, Assertion (A) is true, and Reason (R) provides the fundamental logic for it.

∴ The correct answer is Both (A) and (R) are true, and (R) is the correct explanation of (A).

Alternate Method

Given: A triangle with its angle bisectors and incenter.

Property: The Incenter (I) is equidistant from all sides of the triangle, serving as the center of the Incircle.

⇒ Assertion (A): The incenter is the center of the largest circle that can fit inside the triangle.

⇒ Reason (R): This point is generated by the intersection of internal angle bisectors.

⇒ Since internal bisectors begin at the vertices and extend into the interior, they must meet inside.

⇒ Both statements are conceptually accurate and linked.

∴ The correct answer is Both (A) and (R) are true, and (R) is the correct explanation of (A).

Additional Information

Incenter and Inradius

The radius of the incircle (r) is calculated as Area (A) ÷ Semi-perimeter (s).

Points Always Inside

The Centroid and the Incenter are the only two major centers that always lie inside any triangle.

Points That Can Be Outside

The Circumcenter and Orthocenter lie outside in an obtuse-angled triangle and on the boundary in a right-angled triangle.

Equilateral Triangle Special Case

In an equilateral triangle, the Centroid, Incenter, Orthocenter, and Circumcenter all coincide at the same point.

Shortcut Trick

Net profit for distribution = 80% of 48,000 = ₹38,400.

Modified profit sharing ratio P : Q : R = 6 : 7 : (9 ÷ 2) = 6 : 7 : 4.5.

Total ratio parts = 6 + 7 + 4.5 = 17.5.

Q share = (7 ÷ 17.5) × 38,400 = (2 ÷ 5) × 38,400 = ₹15,360.

∴ The correct answer is ₹15,360.

Alternate Method

Given:

Investment ratio of P, Q, and R = 6 : 7 : 9

Total Profit = ₹48,000

Reinvestment = 20%

Condition: R is a silent partner, so he gets only 50% of his investment-based share.

Formula Used:

Profit Share = (Individual Ratio ÷ Total Ratio) × Distributed Profit

Calculations:

⇒ Amount to be distributed = 48,000 − (20% of 48,000)

⇒ Amount to be distributed = 48,000 − 9,600 = ₹38,400

⇒ Effective Profit Ratio P : Q : R = 6 : 7 : (9 ÷ 2)

⇒ P : Q : R = 6 : 7 : 4.5 = 12 : 14 : 9

⇒ Sum of ratios = 12 + 14 + 9 = 35

⇒ Q share = (14 ÷ 35) × 38,400

⇒ Q share = (2 ÷ 5) × 38,400

⇒ Q share = 2 × 7,680 = ₹15,360

∴ The correct answer is ₹15,360.

Additional Information

Partnership Profit Sharing

Profit is divided based on the product of Capital and Time. Ratio = (C₁ × T₁) : (C₂ × T₂).

Active vs Silent Partner

An active partner manages the business, while a silent (sleeping) partner only provides capital and may receive a different share as per the agreement.

Reserve Fund / Reinvestment

If a portion of the profit is reinvested or kept as a reserve, it must be subtracted from the Gross Profit before calculating individual shares.

Shortcut Trick

Apply the Alligation Rule to find the required ratio of the two fertilizers.

Higher concentration (40%) and Lower concentration (25%) are mixed to get 30%.

Ratio = (Mean − Lower) : (Higher − Mean)

Ratio = (30 − 25) : (40 − 30) = 5 : 10 = 1 : 2

∴ The correct answer is 1 : 2.

Alternate Method

Given:

Nitrogen percentage in Fertilizer 1 = 40%

Nitrogen percentage in Fertilizer 2 = 25%

Desired Mean Mixture concentration = 30%

Formula Used:

Quantity of Cheaper / Quantity of Dearer = (CP of Dearer − Mean Price) / (Mean Price − CP of Cheaper)

Calculations:

⇒ Let the quantity of Fertilizer 1 be Q₁ and Fertilizer 2 be Q₂.

⇒ By Rule of Alligation:

⇒ Q₁ : Q₂ = (30 − 25) : (40 − 30)

⇒ Q₁ : Q₂ = 5 : 10

⇒ Q₁ : Q₂ = 1 : 2

∴ The correct answer is 1 : 2.

Additional Information

Weighted Average Formula

The mean concentration (C_m) can also be calculated as (n₁c₁ + n₂c₂) ÷ (n₁ + n₂), where n is the quantity and c is the concentration.

Mixture Replacement Rule

If a quantity x is removed from a container of volume V and replaced with water n times, the final concentration is Initial × (1 − x/V)ⁿ.

Alligation for Three Components

For three ingredients, pair them (Cheaper with Mean and Dearer with Mean) and sum the respective ratio parts.

Shortcut Trick

Identify the decimal part: 0.625 is a known multiple of 0.125 (1/8).

Since 0.125 = 1/8, then 0.625 = 5 × 0.125 = 5/8.

The total value is 3 + 5/8.

Convert mixed fraction to improper: (3 × 8 + 5) / 8 = 29/8.

∴ The correct answer is 29/8.

Alternate Method

Given: Current level = 3.625 times the base level.

To convert a decimal to a fraction, divide by the power of 10 corresponding to the decimal places.

⇒ 3.625 = 3625/1000

⇒ Dividing numerator and denominator by 5: 725/200

⇒ Dividing numerator and denominator by 5: 145/40

⇒ Dividing numerator and denominator by 5: 29/8

∴ The correct answer is 29/8.

Additional Information

Conversion of Decimal to Fraction

Count the number of digits after the decimal point and place the number over 10 raised to that power (e.g., 0.abc = abc/1000).

Standard Fraction to Decimal Table (Base 8)

1/8 = 0.125 | 3/8 = 0.375 | 5/8 = 0.625 | 7/8 = 0.875. Remembering these helps in fast calculation.

Improper Fraction

An improper fraction has a numerator larger than the denominator. It can be written as a mixed number: 29/8 = 3 5/8.

Shortcut Trick

Recognize the Pythagorean triplet (3, 4, 5) where tan θ = Perpendicular / Base = 3/4.

Here, Perpendicular (P) = 3 and Base (B) = 4, so the Hypotenuse (H) must be 5.

The ratio for sin θ is defined as Perpendicular / Hypotenuse.

By direct substitution: sin θ = 3/5.

∴ The correct answer is 3/5.

Alternate Method

Given: tan θ = 3/4

Formula Used: tan θ = Perpendicular (P) / Base (B) and sin θ = Perpendicular (P) / Hypotenuse (H)

Calculations:

⇒ Let Perpendicular (P) = 3k and Base (B) = 4k

⇒ According to Pythagoras Theorem: H² = P² + B²

⇒ H² = (3k)² + (4k)²

⇒ H² = 9k² + 16k² = 25k²

⇒ H = √(25k²) = 5k

⇒ sin θ = P / H = 3k / 5k

⇒ sin θ = 3/5

∴ The correct answer is 3/5.

Additional Information

Trigonometric Identities

The fundamental identity relates sine and cosine as: sin²θ + cos²θ = 1.

Common Pythagorean Triplets

Standard sets of integers that satisfy a² + b² = c² include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25).

Ratio Relations

tan θ can also be expressed as sin θ / cos θ. Since tan θ = 3/4 and sin θ = 3/5, then cos θ must be 4/5.

Shortcut Trick

Use the direct relationship between arc length, radius, and angle in radians.

Formula: Radius = Arc Length ÷ Angle (in radians)

Radius = 44 ÷ 4 = 11 cm

∴ The correct answer is 11 cm.

Alternate Method

Given:

Length of the arc (s) = 44 cm

Central angle (θ) = 4 radians

Formula Used:

The arc length (s) of a circle is given by: s = r × θ

Where 'r' is the radius and 'θ' is the angle in radians.

Calculations:

⇒ s = r × θ

⇒ 44 = r × 4

⇒ r = 44 ÷ 4

⇒ r = 11 cm

∴ The correct answer is 11 cm.

Additional Information

Area of Sector

The area of a circular sector is given by A = (1/2) × r² × θ, where θ is in radians.

Radian to Degree Conversion

To convert an angle from radians to degrees, multiply by (180/π). 1 radian ≈ 57.3°.

Arc Length in Degrees

If the angle is given in degrees, the arc length formula is s = (θ/360) × 2πr.

Shortcut Trick

Use the Trigonometric Identity for Complementary Angles: cot(90° − θ) = tan θ.

Given that tan θ = 1.

Directly substituting the value, we get cot(90° − θ) = 1.

∴ The correct answer is 1.

Alternate Method

Given: tan θ = 1

Formula Used: tan θ = Opposite / Adjacent and cot(90° − θ) = tan θ

⇒ tan θ = 1

⇒ Since tan 45° = 1, we have θ = 45°

⇒ Now, find cot(90° − θ)

⇒ cot(90° − 45°) = cot 45°

⇒ cot 45° = 1

∴ The correct answer is 1.

Additional Information

Complementary Angle Identities

Trigonometric ratios of complementary angles follow: sin(90°−θ) = cosθ, sec(90°−θ) = cosecθ, and tan(90°−θ) = cotθ.

Reciprocal Identities

The tangent and cotangent functions are reciprocals: cot θ = 1 / tan θ.

Standard Trigonometric Values at 45°

For an angle of 45°, both tan 45° and cot 45° are equal to 1, while sin 45° and cos 45° are both 1/√2.

Shortcut Trick

Initial total parts = 5 (Milk) + 3 (Water) = 8 parts.

Total solution = 56 L ⇒ 8 parts = 56 L ⇒ 1 part = 7 L.

Since Water remains constant (3 parts), compare Milk parts.

Milk increase = 7 parts − 5 parts = 2 parts.

Quantity of milk to be added = 2 parts × 7 L = 14 L.

∴ The correct answer is 14 L.

Alternate Method

Given: Total solution = 56 L, Initial Ratio (M:W) = 5:3, Final Ratio (M:W) = 7:3.

Formula Used: Ratio = Quantity of Milk ÷ Quantity of Water.

⇒ Quantity of milk = (5 / (5+3)) × 56 = (5/8) × 56 = 35 L

⇒ Quantity of water = 56 − 35 = 21 L

⇒ Let the quantity of milk added be x litres.

⇒ New quantity of milk = (35 + x) L

⇒ (35 + x) / 21 = 7 / 3

⇒ 3 × (35 + x) = 7 × 21

⇒ 105 + 3x = 147

⇒ 3x = 147 − 105 = 42

⇒ x = 42 ÷ 3 = 14 L

∴ The correct answer is 14 L.

Additional Information

Ratio Balancing

When one component in a mixture is added or removed, the other component remains constant. Scale the ratios so that the constant component has the same number of units in both ratios.

Alligation Rule

Used to find the proportion in which two ingredients are mixed to produce a mixture of a desired price or concentration: (Cheaper quantity) ÷ (Dearer quantity) = (d − m) ÷ (m − c).

Concentration of Milk

Concentration % = (Quantity of Milk ÷ Total Volume) × 100. In this case, initial concentration was 62.5% and final concentration is 70%.