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Shortcut Trick

Height of lower part (h) = Height of upper part (d) ÷ [tan(higher angle) ÷ tan(lower angle) − 1]

Given d = 1.6 m, higher angle = 60°, lower angle = 45°

h = 1.6 ÷ (tan 60° ÷ tan 45° − 1) = 1.6 ÷ (√3 ÷ 1 − 1) = 1.6 ÷ (√3 − 1)

Rationalizing the denominator: h = 1.6 × (√3 + 1) ÷ [(√3 − 1)(√3 + 1)]

h = 1.6(√3 + 1) ÷ (3 − 1) = 0.8(√3 + 1) m

∴ The correct answer is 0.8(√3 + 1) m.

Alternate Method

Given: Height of the statue (AB) = 1.6 m.

Let the height of the pedestal (BC) be h and the horizontal distance to observation point D (CD) be x.

45° 60°

A B C D

x

h

1.6 m

⇒ In right-angled triangle BCD, tan 45° = BC ÷ CD

⇒ 1 = h ÷ x

⇒ x = h

⇒ In right-angled triangle ACD, tan 60° = AC ÷ CD

⇒ √3 = (AB + BC) ÷ x

⇒ √3 = (1.6 + h) ÷ h

⇒ h√3 = 1.6 + h

⇒ h(√3 − 1) = 1.6

⇒ h = 1.6 ÷ (√3 − 1)

⇒ Rationalizing the denominator: h = 1.6(√3 + 1) ÷ [(√3 − 1)(√3 + 1)]

⇒ h = 1.6(√3 + 1) ÷ (3 − 1) = 0.8(√3 + 1) m

∴ The correct answer is 0.8(√3 + 1) m.

Additional Information

Trigonometric Values

Standard trigonometric ratios frequently used in heights and distances: tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.

Rationalization of Surds

To simplify an expression of the form a / (√b − c), multiply both the numerator and the denominator by its conjugate (√b + c) to eliminate the square root from the denominator.

Angle of Elevation

It is the angle formed by the line of sight with the horizontal when an object is viewed from below. A higher angle of elevation corresponds to a greater vertical height for objects observed from the same horizontal distance.

Shortcut Trick

Direct formula for the volume of a hollow hemisphere: V = ²/₃ × π × (R³ − r³)

Substitute R = 14 cm, r = 7 cm, and π = 3.14:

V = ²/₃ × 3.14 × (14³ − 7³)

V = ²/₃ × 3.14 × (2744 − 343) = ²/₃ × 7539.14

V ≈ 5026.09 cm³

∴ The correct answer is 5026.09 cm³.

Alternate Method

Given:

Outer radius of the hemispherical shell (R) = 14 cm

Inner radius of the hemispherical shell (r) = 7 cm

Formula Used:

Volume of the metal used = Volume of outer hemisphere − Volume of inner hemisphere

Volume = ²/₃ × π × (R³ − r³)

r = 7 cm

R = 14 cm

Hollow Hemispherical Shell

Calculation:

⇒ Volume = ²/₃ × π × (14³ − 7³)

⇒ Cube values: 14³ = 2744 and 7³ = 343

⇒ Difference of cubes = 2744 − 343 = 2401

⇒ Volume = ²/₃ × 3.14 × 2401   (Taking π ≈ 3.14 for approximation)

⇒ Volume = ²/₃ × 7539.14

⇒ Volume = 5026.0933... cm³

⇒ Volume ≈ 5026.09 cm³

∴ The correct answer is 5026.09 cm³.

Additional Information

Total Surface Area of a Hollow Hemisphere

The total surface area includes the outer curved surface, inner curved surface, and the flat circular ring at the base. Formula: TSA = 2πR² + 2πr² + π(R² − r²) = π(3R² + r²).

Volume of a Solid Hemisphere

For a solid hemisphere of radius R, the volume is exactly half the volume of a full sphere. Formula: V = ²/₃ × πR³.

Approximation of Pi (π)

For standard calculations, π is usually taken as ²²/₇. However, in options involving specific decimal values, taking π ≈ 3.14 or 3.1416 often yields the required exact approximation.

Shortcut Trick

For similar cones, the volume is proportional to the cube of the height: V ∝ H³

Volume of smaller cone = 1 unit ⇒ Volume of larger cone = 1 + 7 = 8 units.

Ratio of their heights (smaller : larger) = ³√1 : ³√8 = 1 : 2

Height of frustum = Height of larger cone − Height of smaller cone = 2 − 1 = 1 unit.

Ratio of heights of frustum and smaller cone = 1 : 1

∴ The correct answer is 1:1.

Alternate Method

Given:

Ratio of the volume of the frustum to the volume of the smaller cone = 7 : 1

Formula Used:

Ratio of volumes of similar geometric solids = (Ratio of their corresponding heights)³

h1

h2

Vol = 1 Vol = 7

Height Ratio = h2 : h1 = 1 : 1

Calculations:

⇒ Let the volume of the smaller cone be V₁ = x

⇒ Volume of the frustum = 7x

⇒ Total volume of the original larger cone V₂ = x + 7x = 8x

⇒ Since the cutting plane is parallel to the base, the smaller cone and the larger cone are similar geometric solids.

⇒ V₁ / V₂ = (h₁ / H)³

⇒ x / 8x = (h₁ / H)³

⇒ 1 / 8 = (h₁ / H)³

⇒ h₁ / H = 1 / 2

⇒ H = 2h₁

⇒ Height of the frustum h₂ = H − h₁

⇒ h₂ = 2h₁ − h₁ = h₁

⇒ Ratio of heights of the frustum and the smaller cone = h₂ : h₁ = h₁ : h₁ = 1 : 1

∴ The correct answer is 1:1.

Additional Information

Similar Solids Property

When a solid is cut by a plane parallel to its base, the smaller shape formed at the top is mathematically similar to the original solid. The ratio of their volumes strictly equals the cube of the ratio of their corresponding linear dimensions (height, radius, or slant height).

Frustum of a Cone

The volume of a frustum of a right circular cone is given by V = πh(r² + R² + rR) ÷ 3, where r and R are the radii of the top and bottom circular bases and h is the vertical height of the frustum.

Curved Surface Area of a Frustum

The lateral or curved surface area is given by CSA = πl(R + r), where l is the slant height of the frustum calculated as l = √(h² + (R − r)²).

Shortcut Trick

For any 2D geometrical figure, the area is directly proportional to the square of its side length.

If the side length is scaled by a factor of n, the area scales by a factor of n².

Given the side is doubled, n = 2.

New Area = 2² = 4 times the original area.

∴ The correct answer is Quadrupled.

Alternate Method

Given:

The side of a regular hexagon is doubled.

Formula Used:

Area of a regular hexagon = (3√3 ÷ 2) × (side)²

a Area = A

2a Area = 4A

Side × 2

Calculations:

⇒ Let the original side of the regular hexagon be a.

⇒ Original Area = (3√3 ÷ 2) × a²

⇒ New side = 2a

⇒ New Area = (3√3 ÷ 2) × (2a)²

⇒ New Area = (3√3 ÷ 2) × 4a²

⇒ New Area = 4 × [(3√3 ÷ 2) × a²]

⇒ New Area = 4 × Original Area

∴ The correct answer is Quadrupled.

Additional Information

Area of a Regular Hexagon

A regular hexagon is made up of 6 equilateral triangles. Hence, Area = 6 × (√3 ÷ 4) × a² = (3√3 ÷ 2) × a².

Scaling of 2D Figures

When the corresponding sides of any 2D shape are scaled by a factor k, the perimeter scales by k, and the area scales by k².

Scaling of 3D Figures

When the corresponding edges of a 3D object are scaled by a factor k, its surface area scales by k², and its volume scales by k³.

Shortcut Trick

Total height of the container = Sum of the arithmetic progression (AP)

⇒ Total height (H) = (n ÷ 2) × (a + l) = (5 ÷ 2) × (2 + 10) = 30 cm

Total Volume = π × r² × H

⇒ Volume = 3.14 × 3² × 30 = 847.8 cm³

∴ The correct answer is 847.8 cm³.

Alternate Method

r = 3 cm

h₅ = 10 cm h₄ = 8 cm h₃ = 6 cm h₂ = 4 cm h₁ = 2 cm

H = 30 cm

Total Volume = π × r² × H

Given:

⇒ Base radius (r) = 3 cm

⇒ Number of sections (n) = 5

⇒ Height of bottom section (a) = 2 cm

⇒ Height of topmost section (l) = 10 cm

Formula Used:

⇒ Sum of an AP = (n ÷ 2) × (a + l)

⇒ Volume of a cylinder = π × r² × H

Calculations:

⇒ Total height (H) of the container is the sum of the heights of 5 sections in an AP.

⇒ H = (5 ÷ 2) × (2 + 10)

⇒ H = 2.5 × 12 = 30 cm

⇒ Total volume = 3.14 × (3)² × 30

⇒ Total volume = 3.14 × 9 × 30

⇒ Total volume = 3.14 × 270 = 847.8 cm³

∴ The correct answer is 847.8 cm³.

Additional Information

Arithmetic Progression (AP)

A sequence of numbers where the difference between consecutive terms is constant. The sum of the first n terms can be calculated directly using S = (n ÷ 2) × (a + l), where a is the first term and l is the last term.

Volume of a Cylinder

The amount of space contained within a cylindrical object. The formula is V = π × r² × h, where r is the base radius and h is the height.

Curved Surface Area (CSA)

The total area of the curved surface excluding the flat circular ends. It is calculated using the formula CSA = 2 × π × r × h.

Shortcut Trick

For any linear equation in the standard form Ax + By + C = 0 or Ax + By = C,

The slope (m) is directly given by the formula: m = −A ÷ B

Here, the equation is 7x + 4y = 8, which gives A = 7 and B = 4.

Slope = −7 ÷ 4 = −7/4

∴ The correct answer is −7/4.

Alternate Method

Given:

The linear equation is 7x + 4y = 8.

Run: +4 Rise: −7

(0, 2)

(4, −5)

(8/7, 0)

Line: 7x + 4y = 8

Formula Used:

The slope-intercept form of a line is y = mx + c, where m represents the slope and c represents the y-intercept.

Calculations:

⇒ The given equation is 7x + 4y = 8

⇒ Rearrange the equation to isolate y on one side:

⇒ 4y = −7x + 8

⇒ Divide the entire equation by 4:

⇒ y = (−7 ÷ 4)x + (8 ÷ 4)

⇒ y = (−7/4)x + 2

⇒ Comparing this directly with y = mx + c, we get:

⇒ Slope (m) = −7/4

∴ The correct answer is −7/4.

Additional Information

Slope of a Line

The slope m defines the steepness and direction of a line. For two points (x₁, y₁) and (x₂, y₂), the formula is m = (y₂ − y₁) ÷ (x₂ − x₁).

Parallel Lines

Two non-vertical lines are parallel if and only if their slopes are exactly equal (m₁ = m₂).

Perpendicular Lines

Two straight lines are perpendicular if the product of their slopes is −1 (m₁ × m₂ = −1).

Intercept Form

The equation x/a + y/b = 1 represents a line with x-intercept a and y-intercept b. Its slope is given by −b/a.

Shortcut Trick

A fundamental property of any convex polygon is that a full trip around its perimeter involves exactly one complete rotation.

Therefore, the total sum of all exterior angles, taking one at each vertex, is equivalent to one full circle.

One complete circle rotation is always equal to 360°.

∴ The correct answer is 360°.

Alternate Method

Given:

Any arbitrary polygon with n sides.

Formula Used:

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

At any vertex, Interior Angle + Exterior Angle = 180° (Linear Pair)

e1

e2

e3

e4

e5

Sum of exterior angles = 360°

Calculations:

⇒ Total sum of all interior and exterior angles for n vertices = n × 180°

⇒ Sum of exterior angles = Total sum − Sum of interior angles

⇒ Sum of exterior angles = n × 180° − (n − 2) × 180°

⇒ Sum of exterior angles = 180° × (n − n + 2)

⇒ Sum of exterior angles = 180° × 2 = 360°

⇒ The sum evaluates to 360° independent of the number of sides.

∴ The correct answer is 360°.

Additional Information

Interior Angles of a Polygon

The sum of interior angles of an n-sided polygon is defined as (n − 2) × 180°. For a regular polygon, each interior angle is equal to ((n − 2) × 180°) ÷ n.

Exterior Angles of a Regular Polygon

Since the total sum of exterior angles is always fixed at 360°, each exterior angle of an n-sided regular polygon is given exactly by 360° ÷ n.

Number of Diagonals

The total number of unique diagonals that can be drawn within an n-sided polygon is calculated using the formula n(n − 3) ÷ 2.

Shortcut Trick

The centroid (G) of any triangle connects to its three vertices to form three smaller triangles of equal area.

These three triangles are ΔGAB, ΔGBC, and ΔGCA.

Area of ΔGBC = 1/3 × Area of ΔABC

Area of ΔGBC = 1/3 × 60 = 20 cm²

∴ The correct answer is 20 cm².

Alternate Method

Given:

⇒ Area of ΔABC = 60 cm²

⇒ AD, BE, and CF are the medians intersecting at the centroid G.

A B C D E F G

10 cm² 10 cm²

Area of ΔGBC = 20 cm²

Property of Medians:

⇒ The three medians of a triangle divide it into 6 smaller triangles of exactly equal area.

⇒ The 6 equal triangles are: ΔAFG, ΔBFG, ΔBDG, ΔCDG, ΔCEG, and ΔAEG.

Calculations:

⇒ Area of each smaller triangle = Area of ΔABC ÷ 6

⇒ Area of each smaller triangle = 60 ÷ 6 = 10 cm²

⇒ ΔGBC is formed by combining two of these smaller triangles: ΔBDG and ΔCDG.

⇒ Area of ΔGBC = Area of ΔBDG + Area of ΔCDG

⇒ Area of ΔGBC = 10 + 10 = 20 cm²

∴ The correct answer is 20 cm².

Additional Information

Centroid Ratio Rule

The centroid divides each median into segments with a 2:1 ratio, with the longer segment being on the side of the vertex. For example, AG : GD = 2 : 1.

Apollonius Theorem

It relates the length of a median to the lengths of its adjacent sides: AB² + AC² = 2 × (AD² + BD²), where AD is the median to side BC.

Incenter vs Centroid

While the centroid is the intersection of medians, the incenter is the intersection of the interior angle bisectors and is exactly equidistant from all three sides of the triangle.

Orthocenter vs Circumcenter

The orthocenter is the intersection of the triangle's altitudes, whereas the circumcenter is the intersection of the perpendicular bisectors of its sides and is equidistant from all vertices.

Shortcut Trick

By Componendo and Dividendo rule: if (a + b) / (a − b) = c / d, then a / b = (c + d) / (c − d).

Given a = √(5+x), b = √(5-x), and c/d = 3/1.

Apply the rule directly: √(5+x) / √(5-x) = (3 + 1) / (3 − 1) = 4 / 2 = 2.

Squaring both sides: (5 + x) / (5 − x) = 4

5 + x = 20 − 4x ⇒ 5x = 15 ⇒ x = 3.

∴ The correct answer is 3.

Alternate Method

Given:

[√(5+x) + √(5-x)] ÷ [√(5+x) − √(5-x)] = 3

Calculation:

⇒ By cross-multiplying, we get:

⇒ √(5+x) + √(5-x) = 3 × [√(5+x) − √(5-x)]

⇒ √(5+x) + √(5-x) = 3√(5+x) − 3√(5-x)

⇒ Grouping similar terms on each side:

⇒ √(5-x) + 3√(5-x) = 3√(5+x) − √(5+x)

⇒ 4√(5-x) = 2√(5+x)

⇒ Dividing both sides by 2:

⇒ 2√(5-x) = √(5+x)

⇒ Squaring both sides to eliminate the square root:

⇒ [2√(5-x)]² = [√(5+x)]²

⇒ 4 × (5 − x) = 5 + x

⇒ 20 − 4x = 5 + x

⇒ 20 − 5 = x + 4x

⇒ 15 = 5x

⇒ x = 3

∴ The correct answer is 3.

Additional Information

Componendo and Dividendo Rule

If a/b = c/d, then according to this rule: (a+b)/(a−b) = (c+d)/(c−d). It helps to quickly simplify fractions involving sum and difference of the same terms.

Rationalization

It is a process to eliminate surds or square roots from the denominator by multiplying the numerator and denominator by the conjugate of the denominator.

Linear Equations

A mathematical equation in which the highest power of the variable is 1. Solving involves isolating the variable on one side by balancing the operations.

Shortcut Trick

By drawing a line parallel to the direct common tangent, a right-angled triangle is formed where hypotenuse = distance between centers, base = DCT, and perpendicular = R − r.

Given the hypotenuse = 10 cm and base = 8 cm.

Using the standard Pythagorean triplet (6, 8, 10), the perpendicular must be 6 cm.

So, R − r = 6 cm. Since R : r = 2 : 1, we know R = 2r.

Thus, 2r − r = 6 cm ⇒ r = 6 cm.

∴ The correct answer is 6 cm.

Alternate Method

d = 10 cm

DCT = 8 cm

R r

C1 C2

Given:

Ratio of radii R : r = 2 : 1

Distance between centers (d) = 10 cm

Direct Common Tangent (DCT) = 8 cm

Formula Used:

DCT = √(d² − (R − r)²)

Calculations:

⇒ Let the radius of the smaller circle be r and the larger circle be 2r.

⇒ 8 = √(10² − (2r − r)²)

⇒ 8 = √(100 − r²)

⇒ Squaring both sides to remove the square root:

⇒ 64 = 100 − r²

⇒ r² = 100 − 64

⇒ r² = 36

⇒ r = 6 cm

∴ The correct answer is 6 cm.

Additional Information

Direct Common Tangent (DCT)

It is a tangent that touches two circles without crossing the line joining their centers. The formula to calculate its length is √(d² − (R − r)²).

Transverse Common Tangent (TCT)

It is a tangent that crosses the line connecting the centers of the two circles. Its length is calculated as √(d² − (R + r)²).

Externally Touching Circles

When two circles touch each other externally, the distance between their centers is exactly R + r. In this case, the length of the Direct Common Tangent simplifies to 2√(Rr).