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Shortcut Trick

In melting and recasting, the total volume remains constant.

Volume of Hemisphere = n × Volume of Cone

(2/3)πr³ = n × (1/3)πr²h

By canceling (1/3), π, and r²: 2r = n × h

n = 2r / h

∴ The correct answer is 2r/h.

Alternate Method

Given:

Radius of hemisphere = r

Base radius of cone = r

Height of cone = h

Number of identical cones = n

Formula Used:

Volume of Hemisphere = (2/3)πr³

Volume of Cone = (1/3)πr²h

Calculations:

⇒ Total volume of melted hemisphere = Total volume of n cones formed

⇒ (2/3) × π × r³ = n × (1/3) × π × r² × h

⇒ 2 × r³ / 3 = n × r² × h / 3

⇒ 2 × r³ = n × r² × h

⇒ 2 × r = n × h

⇒ n = 2r / h

∴ The correct answer is 2r/h.

Additional Information

Volume of Sphere

The volume of a full sphere is exactly double that of a hemisphere: V = (4/3)πr³.

Surface Area of Hemisphere

The curved surface area (CSA) is 2πr² and the total surface area (TSA) is 3πr².

Volume of Cylinder

A cylinder with the same radius and height as a cone has three times the volume: V = πr²h.

Shortcut Trick

Given height (h) is 4 times the radius (r) ⇒ h = 4r.

Total Surface Area (TSA) formula becomes: 2πr(h + r) = 2πr(4r + r) = 2πr(5r) = 10πr².

For r = 7 cm, the base area (πr²) is always 154 cm².

TSA = 10 × 154 = 1540 cm².

∴ The correct answer is 1540 cm².

Alternate Method

Given:

Base radius (r) = 7 cm

Height (h) = 4 × radius = 4 × 7 = 28 cm

Formula Used:

Total Surface Area of Cylinder = 2πr(h + r)

Calculations:

⇒ Total Surface Area = 2 × (22/7) × 7 × (28 + 7)

⇒ Total Surface Area = 2 × 22 × 35

⇒ Total Surface Area = 44 × 35

⇒ Total Surface Area = 1540 cm²

∴ The correct answer is 1540 cm².

Additional Information

Curved Surface Area (CSA)

The area of the side surface excluding the top and bottom bases is given by 2πrh.

Volume of a Cylinder

The space occupied by the cylinder is calculated using the formula V = πr²h.

Base Area

Since the base of a cylinder is a circle, its area is πr². A cylinder has two such bases.

Shortcut Trick

Ratio of radius (r) to height (h) = 7 : 12. Let r = 7x and h = 12x.

Volume = 1/3 × π × r² × h ⇒ 77 = 1/3 × 22/7 × 49x² × 12x.

77 = 616x³ ⇒ x³ = 77/616 = 1/8.

x = √³(1/8) = 0.5. Hence, Radius (r) = 7 × 0.5 = 3.5 cm.

∴ The correct answer is 3.5 cm.

Alternate Method

Given:

• Ratio of radius (r) and height (h) = 7 : 12

• Volume of the cone (V) = 77 cm³

• Value of π = 22/7

Formula Used:

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

Calculations:

⇒ Let the radius (r) be 7x and the height (h) be 12x.

⇒ Volume (V) = (1/3) × (22/7) × (7x)² × (12x) = 77

⇒ (1/3) × (22/7) × 49x² × 12x = 77

⇒ 22 × 7 × 4 × x³ = 77

⇒ 616x³ = 77

⇒ x³ = 77 ÷ 616

⇒ x³ = 1/8

⇒ x = 1/2 = 0.5

⇒ Radius (r) = 7 × 0.5 = 3.5 cm.

∴ The correct answer is 3.5 cm.

Additional Information

Slant Height of a Cone

The slant height (l) is the distance from the apex to any point on the base circumference: l = √(r² + h²).

Curved Surface Area (CSA)

The area of the curved surface of a cone is given by the formula: CSA = π × r × l.

Total Surface Area (TSA)

The total area including the circular base: TSA = π × r × (l + r).

Shortcut Trick

Given: sin x = 4/5 and cos x = 3/5.

Expression: sin[2(x + π/8)] = sin(2x + π/4) = (sin 2x + cos 2x) / √2.

sin 2x = 2 × (4/5) × (3/5) = 24/25.

cos 2x = cos² x − sin² x = (3/5)² − (4/5)² = −7/25.

Value = (24/25 − 7/25) / √2 = 17 / (25√2).

∴ The correct answer is 17 / (25√2).

Alternate Method

Given: cosec x = 5/4 and sec x = 5/3.

Formula Used: sin(A + B) = sin A cos B + cos A sin B; sin 2x = 2 sin x cos x; cos 2x = cos² x − sin² x.

⇒ sin x = 1/cosec x = 4/5

⇒ cos x = 1/sec x = 3/5

⇒ sin 2x = 2 × (4/5) × (3/5) = 24/25

⇒ cos 2x = (3/5)² − (4/5)² = 9/25 − 16/25 = −7/25

⇒ sin[2(x + π/8)] = sin(2x + π/4)

⇒ sin(2x + π/4) = sin 2x cos(π/4) + cos 2x sin(π/4)

⇒ sin(2x + π/4) = (24/25) × (1/√2) + (−7/25) × (1/√2)

⇒ sin(2x + π/4) = (24 − 7) / (25√2) = 17 / (25√2)

∴ The correct answer is 17 / (25√2).

Additional Information

Double Angle Identities

sin 2θ = 2 sin θ cos θ and cos 2θ = cos² θ − sin² θ = 2 cos² θ − 1 = 1 − 2 sin² θ.

Compound Angle Formulas

sin(A ± B) = sin A cos B ± cos A sin B and cos(A ± B) = cos A cos B ∓ sin A sin B.

Trigonometric Values of π/4

At π/4 (45°), the values are sin(π/4) = 1/√2 and cos(π/4) = 1/√2.

Shortcut Trick

A quadrantal angle is an angle in standard position whose terminal side lies on any coordinate axis.

In radians, quadrantal angles are always integral multiples of π/2 (e.g., 0, π/2, π, 3π/2, 2π etc.).

Checking the options: π/6, π/3, and π/4 are not multiples of π/2.

Only π/2 satisfies this condition as it lies on the positive y-axis.

∴ The correct answer is π/2.

Alternate Method

Given: Angles in radians π/6, π/3, π/4, and π/2.

Formula: An angle θ is quadrantal if θ = n × 90° or n × (π/2) radians, where n is an integer.

Step-by-step Calculation:

⇒ Convert π/2 to degrees: (π/2) × (180/π) = 90° (Lies on positive Y-axis).

⇒ Convert π/3 to degrees: (π/3) × (180/π) = 60° (Lies in Quadrant I).

⇒ Convert π/4 to degrees: (π/4) × (180/π) = 45° (Lies in Quadrant I).

⇒ Convert π/6 to degrees: (π/6) × (180/π) = 30° (Lies in Quadrant I).

⇒ Since only 90° falls on an axis, π/2 is the quadrantal angle.

∴ The correct answer is π/2.

Additional Information

Quadrantal Angles

These are angles whose terminal side coincides with the x-axis or y-axis. Examples include 0, π/2, π, 3π/2, 2π, and their negative counterparts like −π/2.

Trigonometric Values of Quadrantal Angles

For π/2 (90°), sin(π/2) = 1, cos(π/2) = 0, and tan(π/2) is undefined because the x-coordinate on the unit circle is 0.

General Form

Any angle θ that can be written as θ = n × (π/2) for any integer n is a quadrantal angle.

Shortcut Trick Sum of two complementary angles is always 90°. Larger angle = (Sum + Difference) ÷ 2 = (90° + 52°) ÷ 2 = 142° ÷ 2 = 71°. Smaller angle = (Sum − Difference) ÷ 2 = (90° − 52°) ÷ 2 = 38° ÷ 2 = 19°. ∴ The correct answer is 19°, 71°.

Alternate Method Given: Difference between angles = 52° Sum of complementary angles = 90°

Let the two angles be x and y (where x > y). ⇒ x + y = 90° --- (Equation 1) ⇒ x − y = 52° --- (Equation 2) Adding Equation 1 and Equation 2: ⇒ (x + y) + (x − y) = 90° + 52° ⇒ 2x = 142° ⇒ x = 142° ÷ 2 = 71° Substitute x in Equation 1: ⇒ 71° + y = 90° ⇒ y = 90° − 71° = 19° ∴ The correct answer is 19°, 71°.

Additional Information Complementary Angles Two angles are called complementary if their sum is exactly 90°. Supplementary Angles Two angles are called supplementary if their sum is exactly 180°. Conjugate Angles Two angles are called conjugate or explementary if their sum is 360°. Linear Pair A pair of adjacent angles formed by intersecting lines that are supplementary.

Shortcut Trick

The standard algebraic identity for the sum of two cubes is a³ + b³ = (a + b)(a² − ab + b²).

Dividing the identity (a + b)(a² − ab + b²) by the factor (a + b) directly yields the remaining factor.

Result = (a² − ab + b²).

∴ The correct answer is a² − ab + b².

Alternate Method

Given: (a³ + b³) ÷ (a + b)

Formula Used: a³ + b³ = (a + b)(a² − ab + b²)

⇒ Expression = (a³ + b³) / (a + b)

⇒ Expression = [(a + b) × (a² − ab + b²)] / (a + b)

⇒ Expression = a² − ab + b²

∴ The correct answer is a² − ab + b².

Additional Information

Difference of Two Cubes

The expression for the difference of cubes is a³ − b³ = (a − b)(a² + ab + b²).

Square of Binomials

Note the middle term difference: (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b².

Difference of Two Squares

The identity for the difference of squares is a² − b² = (a − b)(a + b).

Shortcut Trick

Break each term into perfect square factors: √50 = √(25 × 2) and √8 = √(4 × 2).

Simplify to 5√2 and 2√2 respectively.

Sum the coefficients of the like surds: 5 + 2 = 7.

∴ The correct answer is 7√2.

Alternate Method

Given Expression: √50 + √8

Method: Simplification of radical expressions

⇒ √(25 × 2) + √(4 × 2)

⇒ (5 × √2) + (2 × √2)

⇒ (5 + 2) × √2

⇒ 7√2

∴ The correct answer is 7√2.

Additional Information

Like Surds

Surds that have the same irrational factor (radicand) in their simplest form are called like surds and can be added or subtracted.

Simplifying Radicals

A surd is simplified by extracting the greatest perfect square factor from the radicand using the property √(a²b) = a√b.

Product Rule for Surds

For any two positive numbers a and b, the rule √a × √b = √(ab) is used to combine or split square roots.

Shortcut Trick The ratio of perimeters of two similar triangles is always equal to the ratio of their corresponding sides. Given Side Ratio = 5 : 7 (Smaller : Larger) 7 units (Larger Perimeter) → 35 cm 1 unit → 5 cm 5 units (Smaller Perimeter) → 5 × 5 = 25 cm ∴ The correct answer is 25 cm. Alternate Method Given: Ratio of sides of triangular logo = 5 : 7 Perimeter of the larger logo = 35 cm Formula Used: Perimeter of Smaller Logo ÷ Perimeter of Larger Logo = Smaller Side ÷ Larger Side

Calculations: ⇒ Let the perimeter of the smaller logo be P. ⇒ P ÷ 35 = 5 ÷ 7 ⇒ P = (5 ÷ 7) × 35 ⇒ P = 5 × 5 ⇒ P = 25 cm ∴ The correct answer is 25 cm. Additional Information Properties of Similar Figures When two shapes are similar, the ratio of any linear measurements (like medians, altitudes, or angle bisectors) is equal to the side ratio. Area Ratio Theorem The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides: Area1 ÷ Area2 = (s1 ÷ s2)2. Scale Factor (k) The scale factor represents the ratio of new dimension to original dimension. If k < 1, the figure is reduced; if k > 1, the figure is enlarged.

Shortcut Trick

For a circumscribed circle (circle outside), the diameter of the circle is equal to the diagonal of the square.

Diagonal of square = Side × √2 = 14 × 1.414 ≈ 19.8 cm.

Radius = Diagonal ÷ 2 = 19.8 ÷ 2 = 9.9 cm.

Rounding to the nearest option, we get 10 cm.

∴ The correct answer is ≈ 10 cm.

Alternate Method

Given:

Side of the square (a) = 14 cm

Formula Used:

Diagonal of a square = a√2

Radius of circumcircle (R) = Diagonal ÷ 2

Calculations:

⇒ Diagonal of the square = 14 × √2 cm

⇒ Diameter of the circumscribed circle = Diagonal = 14√2 cm

⇒ Radius of the circle (R) = (14√2) ÷ 2

⇒ R = 7√2 cm

⇒ Taking √2 ≈ 1.414

⇒ R ≈ 7 × 1.414

⇒ R ≈ 9.898 cm

⇒ R ≈ 10 cm (approx.)

∴ The correct answer is ≈ 10 cm.

Additional Information

Incircle of a Square

When a circle is inscribed inside a square, its diameter is equal to the side of the square. Radius (r) = Side ÷ 2.

Area Ratio

The ratio of the area of the circumcircle to the area of the incircle of the same square is always 2 : 1.

Diagonal Property

The diagonal of a square also acts as the angle bisector and measures Side × √2.