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Shortcut Trick

When an object is melted and recast, its total volume remains completely conserved.

Volume of Original Sphere = 8 × Volume of One Cone

4/3 × π × R³ = 8 × (1/3 × π × r² × h)

R³ = 2 × 3² × 4 = 72

R = ³√72 ≈ 4.16 cm

∴ The correct answer is 4.16 cm.

Alternate Method

Given:

Radius of each cone (r) = 3 cm

Height of each cone (h) = 4 cm

Number of cones (n) = 8

Formula Used:

Volume of a sphere = 4/3 × π × R³

Volume of a cone = 1/3 × π × r² × h

R = ? Original Sphere

Melted & Recast

h=4 r=3 8 Identical Cones

Calculations:

⇒ Volume of the original sphere = n × Volume of one cone

⇒ 4/3 × π × R³ = 8 × [1/3 × π × (3)² × 4]

⇒ 4 × R³ = 8 × 9 × 4

⇒ 4 × R³ = 288

⇒ R³ = 288 ÷ 4

⇒ R³ = 72

⇒ R = ³√72

⇒ We know that 4³ = 64 and 5³ = 125.

⇒ Since 72 is slightly greater than 64, its cube root will be slightly greater than 4.

⇒ From the given options, R ≈ 4.16 cm.

∴ The correct answer is 4.16 cm.

Additional Information

Conservation of Volume

When a three-dimensional object is melted and reshaped into another form, the total volume of the material always remains conserved. Only the surface area changes.

Volume of a Sphere

The total space occupied by a sphere of radius R is calculated using the formula V = 4/3 × π × R³.

Volume of a Cone

A cone with base radius r and vertical height h has a volume equal to V = 1/3 × π × r² × h, which is exactly one-third of the volume of a cylinder having the exact same base and height.

Shortcut Trick

A regular hexagon has 6 equal sides and 6 equal interior angles of 120°.

By joining the opposite vertices, the hexagon is symmetrically divided by its diagonals passing through the center.

The full central angle of 360° is split equally into 6 parts of 60° each.

Because the sides of the hexagon are equal and the angles are all 60°, exactly 6 perfectly equilateral triangles are formed.

∴ The correct answer is 6.

Alternate Method

O 1 2 3 4 5 6

Given:

A polygon which is a regular hexagon.

Formula Used:

Sum of interior angles of an n-sided polygon = (n − 2) × 180°

Calculations:

⇒ For a regular hexagon, the number of sides n = 6.

⇒ Each interior angle of the hexagon = [(6 − 2) × 180°] ÷ 6 = 720° ÷ 6 = 120°.

⇒ The 3 main diagonals connecting opposite vertices intersect perfectly at the geometric center (Point O).

⇒ These diagonals bisect the interior angles of the hexagon, giving base angles of 120° ÷ 2 = 60° for each formed triangle.

⇒ At the center O, the full 360° angle is divided equally into 6 parts, so the central angle for each triangle = 360° ÷ 6 = 60°.

⇒ Since all three angles of each triangle are exactly 60°, they are all equilateral triangles.

⇒ As shown in the figure above, counting them around the center results in exactly 6 equilateral triangles.

∴ The correct answer is 6.

Additional Information

Area of a Regular Hexagon

Since it is composed of 6 equilateral triangles of side length a, its total Area = 6 × (√3 ÷ 4) × a² = (3√3 ÷ 2) × a².

Perimeter of a Regular Hexagon

The sum of all six equal outer sides of length a is simply given by Perimeter = 6a.

Radius of the Circumcircle

For any regular hexagon, the side length a is exactly equal to the radius of its circumscribed circle (R = a), due to the internal equilateral triangles.

Shortcut Trick

Ratio of inner radius (fountain) to outer radius (total) = 7 : (7 + 3.5) = 7 : 10.5 = 2 : 3.

Ratio of fountain area to total area = (2)² : (3)² = 4 : 9.

Area of grass bed in ratio units = Total area − Fountain area = 9 − 4 = 5.

Ratio of grass bed area to fountain area = 5 : 4.

∴ The correct answer is 5:4.

Alternate Method

r = 7 m

w = 3.5 m

R = 10.5 m

Fountain Grass Bed

Given:

Radius of the circular fountain (r) = 7 m

Width of the grass bed (w) = 3.5 m

Formula Used:

Area of a circle = πr²

Area of a circular ring = π(R² − r²), where R is the outer radius and r is the inner radius.

Calculations:

⇒ Outer radius of the grass bed (R) = r + w

⇒ R = 7 + 3.5 = 10.5 m

⇒ Area of the fountain = π × (7)² = 49π m²

⇒ Area of the grass bed = π × (R² − r²)

⇒ Area of the grass bed = π × (10.5² − 7²)

⇒ Area of the grass bed = π × (110.25 − 49) = 61.25π m²

⇒ Required Ratio = Area of grass bed ÷ Area of fountain

⇒ Required Ratio = 61.25π ÷ 49π

⇒ Required Ratio = 61.25 ÷ 49 = 6125 ÷ 4900

⇒ Simplifying the fraction, Required Ratio = 5 ÷ 4

∴ The correct answer is 5:4.

Additional Information

Area of a Circular Path

When a path of uniform width 'w' runs around a circular field of radius 'r', the area of the path can be directly calculated using the formula π × w × (2r + w).

Perimeter of a Circle

The total boundary length of a circular region is known as its circumference, which is calculated as 2πr or πd, where d is the diameter.

Ratio of Areas of Circles

The ratio of the areas of any two circles is always equal to the square of the ratio of their corresponding radii or diameters.

Shortcut Trick

For any uniform geometric prism, the volume is calculated by simply multiplying its base area by its height (or length).

Volume = Base Area × Height

Volume = 20 × 6 = 120 m³

∴ The correct answer is 120 m³.

Alternate Method

Given:

Area of the triangular base = 20 m²

Height of the tent (length of the prism) = 6 m

Formula Used:

Volume of a Prism = Base Area × Height

Base Area = 20 sq. m

Height = 6 m

Volume = Base Area × Height

Calculation:

⇒ Volume of the tent = Base Area × Height

⇒ Volume of the tent = 20 × 6

⇒ Volume of the tent = 120 m³

∴ The correct answer is 120 m³.

Additional Information

Volume of a Prism

For any prism (triangular, rectangular, etc.), the volume is calculated by multiplying the area of its base by its perpendicular height or length: V = Base Area × h.

Surface Area of a Prism

The lateral surface area of a prism is the perimeter of the base multiplied by the height: LSA = Base Perimeter × h. The total surface area is TSA = LSA + 2 × Base Area.

Volume of a Pyramid

Unlike a prism, the volume of a pyramid is one-third the product of its base area and height: V = (1 ÷ 3) × Base Area × h.

Shortcut Trick

Check the given options to find the line that has a slope of −1 and satisfies the point (2, 3).

The standard slope-intercept form is y = mx + c, where m is the slope.

Option 1: y = −x + 5 → Here, slope m = −1.

Substitute x = 2 into this equation: y = −2 + 5 = 3. This directly matches our given point (2, 3).

∴ The correct answer is y = −x + 5.

Alternate Method

Given:

Slope of the line (m) = −1

The line passes through the point (x₁, y₁) = (2, 3)

Formula Used:

The point-slope form of a linear equation is given by:

y − y₁ = m(x − x₁)

(2, 3) 5 5 2 3 0 X-axis Y-axis

y = −x + 5 Slope (m) = −1

Calculations:

⇒ Substitute the given point and slope into the formula:

⇒ y − 3 = −1(x − 2)

⇒ y − 3 = −x + 2

⇒ y = −x + 2 + 3

⇒ y = −x + 5

∴ The correct answer is y = −x + 5.

Additional Information

Slope-Intercept Form

The linear equation of a straight line can also be written directly as y = mx + c, where m is the slope and c is the y-intercept (the coordinate where the line crosses the y-axis).

Slope Formula

The slope m between any two valid points (x₁, y₁) and (x₂, y₂) on a line is calculated as m = (y₂ − y₁) ÷ (x₂ − x₁).

Parallel and Perpendicular Lines

Parallel lines always share the exact same slope value (m₁ = m₂). Conversely, two lines are perpendicular if their slopes are negative reciprocals of each other (m₁ × m₂ = −1).

Shortcut Trick

Express each term as a multiple of the simplest surd form.

√45 = √(9 × 5) = 3√5

√20 = √(4 × 5) = 2√5

Now, substitute the simplified terms back into the expression:

3√5 + 2√5 − √5 = (3 + 2 − 1)√5 = 4√5

∴ The correct answer is 4√5.

Alternate Method

Given:

Expression = √45 + √20 − √5

Method Used:

Factorize the numbers inside the square roots to extract the perfect square factors.

Calculations:

⇒ Expression = √(3 × 3 × 5) + √(2 × 2 × 5) − √5

⇒ Expression = 3√5 + 2√5 − 1√5

⇒ Taking √5 common from all the terms

⇒ Expression = (3 + 2 − 1)√5

⇒ Expression = 4√5

∴ The correct answer is 4√5.

Additional Information

Rules of Surds

√(a × b) = √a × √b. This property is strictly used to simplify square roots by factoring out perfect squares.

Addition and Subtraction of Surds

Only like surds (surds having the exact same irrational part) can be directly added or subtracted. For example: x√a + y√a = (x + y)√a.

BODMAS Rule

Always follow the correct order of operations: Brackets, Orders (powers/roots), Division, Multiplication, Addition, and Subtraction for successful simplification.

Shortcut Trick

The maximum number of distinct pairs that can be formed from 3 non-overlapping, non-touching circles is 3.

For each distinct pair of separate circles, a maximum of 4 common tangents (2 direct and 2 transverse) can be drawn.

To maximize the total number of distinct tangents, we assume no single line acts as a tangent to all three circles simultaneously.

Maximum distinct common tangents = 3 pairs × 4 tangents/pair = 12.

∴ The correct answer is 12 tangents.

Alternate Method

Given:

Three non-overlapping, non-touching circles are drawn in a plane.

Formula Used:

Maximum common tangents for 2 disjoint circles = 4

Number of pairs from n items = n(n − 1) ÷ 2

C1C2C34 distinct common tangents shown between C1 and C2Total for all 3 pairs = 3 × 4 = 12 distinct tangents

Calculations:

⇒ Number of given circles = 3

⇒ Number of possible distinct pairs of circles = 3 × 2 ÷ 2 = 3 pairs

⇒ Let the 3 circles be C₁, C₂, and C₃. The respective pairs are (C₁, C₂), (C₂, C₃), and (C₃, C₁).

⇒ For each pair of distinctly separated circles, we can draw exactly 4 common tangents (2 Direct Common Tangents + 2 Transverse Common Tangents).

⇒ To achieve the maximum total distinct tangents, we place the circles such that no tangent is shared between all three circles simultaneously.

⇒ Maximum distinct common tangents = 3 pairs × 4 tangents/pair = 12.

∴ The correct answer is 12 tangents.

Additional Information

Direct Common Tangent (DCT)

It is a common tangent that does not intersect the line segment joining the centers of the two circles. The formula for the length of a DCT is √(d² − (r₁ − r₂)²).

Transverse Common Tangent (TCT)

It is a common tangent that intersects the line segment joining the centers of the two circles. The formula for the length of a TCT is √(d² − (r₁ + r₂)²).

Number of Tangents Based on Circle Positions

Two completely separated circles have exactly 4 common tangents. Externally touching circles have 3. Intersecting circles have 2. Internally touching circles have 1. If one circle is completely inside the other, there are 0 common tangents.

Shortcut Trick

By Intersecting Chords Theorem, the product of the segments of intersecting chords are equal.

⇒ (2/5 × x) × (3/5 × x) = (5/7 × y) × (2/7 × y)

⇒ 6x² / 25 = 10y² / 49

⇒ x² / y² = (10 × 25) / (6 × 49)

∴ The correct answer is x²/y² = 125/147.

Alternate Method

Given:

AE : EB = 2 : 3 and AB = x

CE : ED = 5 : 2 and CD = y

Formula Used:

Intersecting Chords Theorem: If two chords AB and CD intersect at E, then AE × EB = CE × ED.

A B C D E

2x/5 3x/5 5y/7 2y/7

Calculation:

⇒ Let AE = 2k and EB = 3k.

⇒ Total length AB = 2k + 3k = 5k.

⇒ Given AB = x, so 5k = x ⇒ k = x/5.

⇒ Thus, AE = 2x/5 and EB = 3x/5.

⇒ Similarly, let CE = 5m and ED = 2m.

⇒ Total length CD = 5m + 2m = 7m.

⇒ Given CD = y, so 7m = y ⇒ m = y/7.

⇒ Thus, CE = 5y/7 and ED = 2y/7.

⇒ Applying the Intersecting Chords Theorem:

⇒ AE × EB = CE × ED

⇒ (2x/5) × (3x/5) = (5y/7) × (2y/7)

⇒ 6x² / 25 = 10y² / 49

⇒ x² / y² = (10 / 49) × (25 / 6)

⇒ x² / y² = 250 / 294

⇒ x² / y² = 125 / 147

∴ The correct answer is x²/y² = 125/147.

Additional Information

Intersecting Chords Theorem

When two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other. Formula: AE × EB = CE × ED

Secant-Secant Theorem

If two secant lines intersect outside a circle, the product of the outer segment and the whole secant is equal for both lines. Formula: PA × PB = PC × PD

Tangent-Secant Theorem

If a tangent segment and a secant segment are drawn to a circle from an exterior point, the square of the tangent length equals the product of the secant and its outer segment. Formula: PT² = PA × PB

Shortcut Trick

Area of minor segment = Area of a Quadrant − Area of the Right-Angled Triangle

⇒ Area = (πr² ÷ 4) − (r² ÷ 2)

⇒ Area = (π × 100 ÷ 4) − (100 ÷ 2)

⇒ Area = 25π − 50

∴ The correct answer is (25π−50) sq. cm.

Alternate Method

OAB10 cm10 cm90°Minor Segment

Given:

Radius of the circle (r) = 10 cm

Angle subtended at the center (θ) = 90°

Formula Used:

Area of minor segment = Area of sector − Area of triangle

Area of sector = (θ ÷ 360°) × πr²

Area of a right-angled triangle = (1/2) × base × height

Calculations:

⇒ Area of sector OAB = (90° ÷ 360°) × π × (10)²

⇒ Area of sector OAB = (1 ÷ 4) × 100π = 25π sq. cm

⇒ Area of triangle OAB = (1 ÷ 2) × 10 × 10

⇒ Area of triangle OAB = 50 sq. cm

⇒ Area of minor segment = (25π − 50) sq. cm

∴ The correct answer is (25π−50) sq. cm.

Additional Information

Area of a Major Segment

The Area of the Major Segment is calculated as Area of the Circle − Area of the Minor Segment.

Length of an Arc

The length of an arc subtending angle θ at the center is determined by (θ ÷ 360°) × 2πr.

Area of Triangle in a Circle

For any chord subtending angle θ at the center, the area of the triangle formed with the center is (1/2)r²sin(θ).

Shortcut Trick

Observe the relation between the numerator and denominator terms.

Each term in the denominator is exactly 5 times the corresponding term in the numerator.

Denominator = (5 × 0.05)³ + (5 × 0.01)³ = 5³ × (0.05³ + 0.01³)

Expression = 1 ÷ 5³ = 1 ÷ 125 = 0.008

∴ The correct answer is 0.008.

Alternate Method

Given:

Expression: (0.05³ + 0.01³) ÷ (0.25³ + 0.05³)

Calculations:

⇒ Numerator = 0.05³ + 0.01³

⇒ Denominator = 0.25³ + 0.05³

⇒ Rewrite the denominator by extracting 5 as a common multiple:

⇒ Denominator = (5 × 0.05)³ + (5 × 0.01)³

⇒ Denominator = 5³ × 0.05³ + 5³ × 0.01³

⇒ Denominator = 5³ × (0.05³ + 0.01³)

⇒ Substitute this back into the expression:

⇒ (0.05³ + 0.01³) ÷ [125 × (0.05³ + 0.01³)]

⇒ Cancel out the common terms from the numerator and denominator:

⇒ Expression = 1 ÷ 125

⇒ 1 ÷ 125 = 0.008

∴ The correct answer is 0.008.

Additional Information

BODMAS Rule

BODMAS stands for Brackets, Orders, Division, Multiplication, Addition, and Subtraction, dictating the correct sequence for evaluating mathematical expressions.

Laws of Exponents

When calculating the power of a product, distribute the exponent to each individual factor: (a × b)ⁿ = aⁿ × bⁿ.

Decimal Conversion Trick

To convert 1 ÷ 125 to a decimal quickly, multiply both the numerator and denominator by 8: (1 × 8) ÷ (125 × 8) = 8 ÷ 1000 = 0.008.