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Shortcut Trick

The hypotenuse of the right triangle is the diameter of the inscribed circle, so AC = 10 cm.

By the right triangle altitude theorem: AB² = AD × AC

⇒ 6² = AD × 10

⇒ 36 = 10 × AD

⇒ AD = 3.6 cm

∴ The correct answer is 3.6 cm.

Alternate Method

Given:

⇒ Diameter (Hypotenuse AC) = 10 cm

⇒ Leg AB = 6 cm

O A C B D

6 cm

AD = ?

AC = 10 cm

Formula Used:

⇒ Angle in a semicircle is always a right angle, so ∠ABC = 90°.

⇒ In a right-angled triangle, the altitude drawn to the hypotenuse creates similar triangles: ΔADB ∼ ΔABC.

Calculations:

⇒ From the similarity of ΔADB and ΔABC: AB / AC = AD / AB

⇒ By cross-multiplying: AB² = AD × AC

⇒ Substituting the given values: 6² = AD × 10

⇒ 36 = 10 × AD

⇒ AD = 36 ÷ 10

⇒ AD = 3.6 cm

∴ The correct answer is 3.6 cm.

Additional Information

Right Triangle Similarity Theorem

When an altitude is drawn to the hypotenuse of a right triangle, it divides the triangle into two smaller triangles that are similar to the original triangle and to each other.

Geometric Mean Theorem

The length of the altitude drawn to the hypotenuse is the geometric mean of the lengths of the two segments of the hypotenuse: BD² = AD × CD.

Angle in a Semicircle

Any angle subtended by a diameter at the circumference of a circle is always exactly a right angle (90°).

Shortcut Trick When two circles intersect, they overlap each other, meaning the distance between their centers is less than the sum of their radii. Any internal (transverse) common tangent must pass between their centers, which is impossible without intersecting the overlapping area. Thus, no internal common tangents can exist for intersecting circles. Only external (direct) tangents can be drawn to touch the outer boundaries of both circles simultaneously. Consequently, exactly 2 direct common tangents can be drawn. ∴ The correct answer is Only two common tangents exist.

Alternate Method Given: Two circles intersect at two distinct points. Concept Used: The possible number of common tangents depends entirely on the distance between the circle centers (d) and their radii (r1 and r2).

C₁ C₂ Direct Common Tangent 1 Direct Common Tangent 2

Calculations: ⇒ For intersecting circles, the distance between their centers follows the condition: |r1 − r2| < d < r1 + r2. ⇒ Because d < r1 + r2, the Transverse (internal) Common Tangents are restricted as they would cut through the overlapping region of the circles instead of behaving as tangents. ⇒ Only Direct (external) Common Tangents can be drawn, touching both circles on their outer boundaries without crossing the central intersecting zone. ⇒ Therefore, exactly 2 Direct Common Tangents are geometrically possible. ∴ The correct answer is Only two common tangents exist.

Additional Information Number of Common Tangents Rule If d > r1 + r2 → 4 tangents. If d = r1 + r2 → 3 tangents.If d = |r1 − r2| → 1 tangent. If d < |r1 − r2| → 0 tangents.

Length of Direct Common Tangent (DCT) The formula to find the length of a direct tangent when circles are separated by distance d is: DCT = √[d2 − (r1 − r2)2]

Length of Transverse Common Tangent (TCT) The formula to find the length of an internal cross tangent when circles are non-intersecting is: TCT = √[d2 − (r1 + r2)2]

Shortcut Trick

When two circles with radii r₁ and r₂ touch externally, the distance between their centers is exactly d = r₁ + r₂.

By applying this to the standard Direct Common Tangent formula, it reduces to a direct standard result.

The length of the direct common tangent is the geometric mean of their diameters.

Direct Formula: T = √(4 × r₁ × r₂)

This simplifies directly to T = 2√(r₁r₂).

∴ The correct answer is 2√(r₁r₂).

Alternate Method

r₁ r₂ T (Tangent)

C₁ C₂

r₁ − r₂ T r₁ + r₂

Given:

Radius of the larger circle = r₁

Radius of the smaller circle = r₂

Since the circles touch each other externally, the distance between their centers (d) = r₁ + r₂.

Length of the direct common tangent = T

Formula Used:

In the right-angled triangle formed by drawing a horizontal line from C₂ to the radius of C₁:

⇒ Base² + Perpendicular² = Hypotenuse²

Calculations:

⇒ T² + (r₁ − r₂)² = (r₁ + r₂)²

⇒ T² = (r₁ + r₂)² − (r₁ − r₂)²

Applying the standard algebraic identity (a + b)² − (a − b)² = 4ab:

⇒ T² = 4 × r₁ × r₂

⇒ T = √(4r₁r₂)

⇒ T = 2√(r₁r₂)

∴ The correct answer is 2√(r₁r₂).

Additional Information

General Direct Common Tangent (DCT)

DCT = √[d² − (r₁ − r₂)²], where d is the straight line distance between the centers of the two circles.

Transverse Common Tangent (TCT)

TCT = √[d² − (r₁ + r₂)²]. The transverse tangent diagonally crosses the line joining the centers.

Maximum Common Tangents

When two circles touch each other externally, exactly 3 common tangents can be drawn (2 direct common tangents and 1 transverse common tangent touching at the intersection point).

Internally Touching Circles

If two circles touch each other internally, the distance between their centers is d = |r₁ − r₂| and only 1 common tangent can be drawn.

Shortcut Trick

The angle at the center is twice the angle at the circumference → 2 × 30° = 60°.

The triangle formed by the center and the chord is an isosceles triangle with a 60° vertex, making it an equilateral triangle.

Radius of the circle = Chord length = 12 cm.

The distance from the center to the chord is simply the altitude of this equilateral triangle.

Distance = (√3 ÷ 2) × Side = (√3 ÷ 2) × 12 = 6√3 cm.

∴ The correct answer is 6√3 cm.

Alternate Method

Given:

Length of the chord (AB) = 12 cm

Angle subtended at the circumference (∠ACB) = 30°

Formula Used:

Angle at the center = 2 × Angle at the circumference

Pythagoras Theorem: Hypotenuse² = Perpendicular² + Base²

60° 30°

O A B C M

AB = 12 cm d = ?

Calculation:

⇒ Angle subtended at the center (∠AOB) = 2 × 30° = 60°

⇒ In ▵AOB, OA = OB (radii of the same circle), making it an isosceles triangle.

⇒ Base angles ∠OAB = ∠OBA = (180° − 60°) ÷ 2 = 60°

⇒ Since all internal angles are exactly 60°, ▵AOB is an equilateral triangle.

⇒ Radius (OA) = AB = 12 cm

⇒ Let OM be the perpendicular drawn from the center O to the chord AB.

⇒ The perpendicular from the center bisects the chord perfectly, so AM = AB ÷ 2 = 12 ÷ 2 = 6 cm.

⇒ In right-angled ▵OMA, applying Pythagoras Theorem:

⇒ OA² = OM² + AM²

⇒ 12² = OM² + 6²

⇒ 144 = OM² + 36

⇒ OM² = 144 − 36 = 108

⇒ OM = √108 = √(36 × 3) = 6√3 cm

∴ The correct answer is 6√3 cm.

Additional Information

Angle Subtended by an Arc

The angle subtended by an arc at the center is always double the angle subtended by it at any remaining part of the circle's circumference.

Perpendicular from Center to Chord

A perpendicular drawn from the center of a circle to any chord will perfectly bisect the chord into two equal halves.

Equilateral Triangle Altitude

In an equilateral triangle where all sides are equal, the height or altitude can be calculated directly using the standard formula h = (√3 ÷ 2) × side.

Shortcut Trick

Area of segment = Area of sector − Area of triangle

Area = [(θ ÷ 360°) × πr²] − [(1 ÷ 2) × r² × sin(θ)]

Substitute r = 8 cm and θ = 60°:

Area = [(60 ÷ 360) × π × 8²] − [(1 ÷ 2) × 8² × sin(60°)]

Area = (64π ÷ 6) − (32 × √3 ÷ 2)

Area = 32π/3 − 16√3 cm²

∴ The correct answer is 32π/3 − 16√3 cm².

Alternate Method

O A B 8 cm 8 cm 60°

Segment Area = Sector OAB − Triangle OAB

Given:

Radius of the circle, r = 8 cm

Angle subtended at the center by the chord, θ = 60°

Formula Used:

Area of Segment = Area of Sector − Area of Triangle

Area of Sector = (θ ÷ 360°) × πr²

Since the two sides bounding the angle 60° are the radii of the same circle, they are equal in length, forming an isosceles triangle. With the included angle being 60°, it becomes an equilateral triangle.

Area of Equilateral Triangle = (√3 ÷ 4) × a² (where a = r)

Calculations:

⇒ Area of Sector = (60° ÷ 360°) × π × 8²

⇒ Area of Sector = (1 ÷ 6) × 64π = 32π/3 cm²

⇒ Area of Triangle = (√3 ÷ 4) × 8²

⇒ Area of Triangle = (√3 ÷ 4) × 64 = 16√3 cm²

⇒ Area of Segment = Area of Sector − Area of Triangle

⇒ Area of Segment = 32π/3 − 16√3 cm²

∴ The correct answer is 32π/3 − 16√3 cm².

Additional Information

Area of a Sector

The region bounded by two radii and an arc is called a sector. Its area is calculated as (θ ÷ 360°) × πr².

Length of an Arc

The distance along the curved line making up the arc is given by the formula (θ ÷ 360°) × 2πr.

Area of Triangle in a Circle

For a triangle formed by two circle radii and a connecting chord, the general area can be found using (1 ÷ 2) × r² × sin(θ).

Perimeter of a Sector

The perimeter of a circular sector is the sum of its arc length and the two boundary radii: Length of Arc + 2r.

Shortcut Trick

For a homogeneous ratio expression, directly substitute the ratio values.

Put p = 4 and q = 1 in the expression (5p + 3q) : (5p − 3q).

⇒ (5 × 4 + 3 × 1) : (5 × 4 − 3 × 1)

⇒ (20 + 3) : (20 − 3)

⇒ 23 : 17

∴ The correct answer is 23 : 17.

Alternate Method

Given:

p : q = 4 : 1

p = 4k

q = k

5(4k) + 3(k) : 5(4k) − 3(k) 23k : 17k

Calculations:

⇒ Let p = 4k and q = k, where k is a common multiple.

⇒ We need to find the value of the expression: (5p + 3q) : (5p − 3q)

⇒ Substitute the values of p and q into the expression:

⇒ [5(4k) + 3(k)] : [5(4k) − 3(k)]

⇒ (20k + 3k) : (20k − 3k)

⇒ 23k : 17k

⇒ Canceling the common term k from both sides, we get 23 : 17.

∴ The correct answer is 23 : 17.

Additional Information

Homogeneous Expression

An expression where the sum of the powers of variables in each term is identical. In ratio forms like (ax + by)/(cx + dy), substituting equivalent proportional values directly always gives the correct simplified ratio.

Componendo and Dividendo

A fundamental proportion rule stating that if a : b = c : d, then (a + b) : (a − b) = (c + d) : (c − d). This is extremely useful for quickly simplifying complex rational equations.

Proportion Rule

When two ratios are equivalent, they form a proportion. For any relation a : b = c : d, the product of the extremes equals the product of the means (a × d = b × c).

Shortcut Trick

Simplify the surds first: √18 = 3√2 and √8 = 2√2.

Substitute into the expression: (3√2 + √2)² − (2√2 + √2)²

This simplifies to: (4√2)² − (3√2)²

Calculate the squares directly: 32 − 18 = 14.

∴ The correct answer is 14.

Alternate Method

Given:

(√18 + √2)² − (√8 + √2)²

Formula Used:

(a + b)² = a² + b² + 2ab

Calculations:

⇒ Expand the first term: (√18 + √2)² = 18 + 2 + 2√(18 × 2)

⇒ 20 + 2√36 = 20 + 2(6) = 32

⇒ Expand the second term: (√8 + √2)² = 8 + 2 + 2√(8 × 2)

⇒ 10 + 2√16 = 10 + 2(4) = 18

⇒ Subtract the two results: 32 − 18

⇒ 14

∴ The correct answer is 14.

Additional Information

Properties of Surds

√(a × b) = √a × √b, which helps in simplifying large square roots, e.g., √18 = √(9 × 2) = 3√2.

Algebraic Identities

Using standard formulas like (a + b)² = a² + b² + 2ab and a² − b² = (a − b)(a + b) simplifies complex algebraic and numerical calculations significantly.

Simplification Rules

According to BODMAS, always solve the expressions inside the brackets first (when possible) before evaluating powers and applying subtraction.

Shortcut Trick

First bracket: LCM of 6 and 4 is 12 → (10 + 9)/12 = 19/12

Second bracket: LCM of 8 and 3 is 24 → (21 − 8)/24 = 13/24

Divide the fractions: (19/12) × (24/13) = 19 × 2 / 13 = 38/13

∴ The correct answer is 38/13.

Alternate Method

Given:

Expression: (5/6 + 3/4) ÷ (7/8 − 1/3)

Concept Used:

Apply the BODMAS rule to solve the values inside the brackets first, then perform the division of fractions.

Calculations:

⇒ Step 1: Solve the first bracket (5/6 + 3/4)

⇒ LCM of denominators 6 and 4 is 12

⇒ (5 × 2)/(6 × 2) + (3 × 3)/(4 × 3)

⇒ 10/12 + 9/12 = 19/12

⇒ Step 2: Solve the second bracket (7/8 − 1/3)

⇒ LCM of denominators 8 and 3 is 24

⇒ (7 × 3)/(8 × 3) − (1 × 8)/(3 × 8)

⇒ 21/24 − 8/24 = 13/24

⇒ Step 3: Divide the results

⇒ (19/12) ÷ (13/24)

⇒ (19/12) × (24/13)

⇒ (19 × 24) / (12 × 13)

⇒ (19 × 2) / 13

⇒ 38/13

∴ The correct answer is 38/13.

Additional Information

BODMAS Rule

It dictates the order of operations: Brackets, Orders (powers/roots), Division, Multiplication, Addition, and Subtraction. Expressions inside brackets are always evaluated first.

Dividing Fractions

To divide by a fraction, you simply multiply by its reciprocal: (a/b) ÷ (c/d) = (a/b) × (d/c).

Lowest Common Multiple (LCM)

When adding or subtracting fractions with different denominators, you must find a common denominator by taking the LCM of the original denominators.

Shortcut Trick

Profit ratio = (Capital_A × Time_A) : (Capital_B × Time_B) : (Capital_C × Time_C)

Profit ratio = (24 × 12) : (36 × 8) : (40 × 6)

Profit ratio = 288 : 288 : 240 = 6 : 6 : 5

B's share = (6 ÷ 17) × 50000

B's share = 300000 ÷ 17 ≈ 17647.06

∴ The correct answer is ₹17,647.06.

Alternate Method

Given:

A's investment = ₹24,000 for 12 months

B's investment = ₹36,000 for 8 months

C's investment = ₹40,000 for 6 months

Total profit = ₹50,000

Formula Used:

Ratio of Profit = (Investment × Time Period)

A B C

24k × 12m = 288k Ratio: 6

36k × 8m = 288k Ratio: 6

40k × 6m = 240k Ratio: 5

Total Profit Ratio = 6 : 6 : 5 (Total 17 parts)

Calculations:

⇒ Profit ratio of A : B : C = (24000 × 12) : (36000 × 8) : (40000 × 6)

⇒ Profit ratio = 288000 : 288000 : 240000

⇒ Dividing by 1000: Profit ratio = 288 : 288 : 240

⇒ Dividing by 48: Profit ratio = 6 : 6 : 5

⇒ Total ratio parts = 6 + 6 + 5 = 17 parts

⇒ B's share of profit = (6 ÷ 17) × 50000

⇒ B's share = 300000 ÷ 17

⇒ B's share = 17647.0588...

⇒ B's share ≈ ₹17,647.06

∴ The correct answer is ₹17,647.06.

Additional Information

Compound Partnership

When investments are made for different time periods, profit is shared based on the product of capital and time. Profit Ratio = (C₁ × T₁) : (C₂ × T₂).

Simple Partnership

When all partners invest for the exact same time period, the profit is shared directly in the ratio of their invested capital amounts.

Working vs Sleeping Partner

A working partner manages the business and receives a fixed salary or commission from the total profit before the remaining profit is divided among all partners (including the working partner) based on investments.

Shortcut Trick

Total age increase for the 9 committee members = 9 × 3 = 27 years.

Sum of the ages of the two replaced men = 42 + 44 = 86 years.

Total age of the two incoming women = Age of replaced men + Total age increase.

Total age of the two women = 86 + 27 = 113 years.

Average age of the two women = 113 ÷ 2 = 56.5 years.

∴ The correct answer is 56.5 years.

Alternate Method

Given:

Number of members in the committee = 9

Increase in the average age = 3 years

Ages of the replaced men = 42 years and 44 years

Formula Used:

Total Sum = Average × Number of observations

New Sum = Old Sum − Sum of outgoing + Sum of incoming

Outgoing: 42 + 44 = 86

Increase: 9 × 3 = 27

Incoming 2 Women: 86 + 27 = 113

Average = 113 ÷ 2 = 56.5 years

Calculations:

⇒ Let the initial average age of the committee be A years.

⇒ Initial total age of the 9 members = 9A

⇒ After replacing the two men with two women, the new average becomes (A + 3) years.

⇒ New total age = 9 × (A + 3) = 9A + 27

⇒ Also, New total age = Initial total age − (42 + 44) + Sum of ages of the two women

⇒ 9A + 27 = 9A − 86 + Sum of ages of the two women

⇒ Sum of ages of the two women = 27 + 86

⇒ Sum of ages of the two women = 113 years

⇒ Average age of the two women = 113 ÷ 2 = 56.5 years

∴ The correct answer is 56.5 years.

Additional Information

Replacement Method in Averages

When individuals are replaced and the overall average changes, the value of the incoming individuals can be found using: New Value = Replaced Value + (Increase in Average × Total Members).

Total Sum Logic

The sum of elements in a group is strictly linked to their average by the equation: Sum = Average × Count. Any deviation in the sum proportionally affects the average.

Combined Group Average

When merging two groups of sizes n₁ and n₂ having individual averages A₁ and A₂, the resulting average is represented as (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).