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Shortcut Trick

Using the tangent ratio for both points A and B directly:

tan(x) = Height ÷ Distance A = 25√3 ÷ 25 = √3 ⇒ x = 60°

tan(y) = Height ÷ Distance B = 25√3 ÷ 75 = 1/√3 ⇒ y = 30°

Sum of angles = x + y = 60° + 30° = 90°.

∴ The correct answer is 90°.

Alternate Method

x°

y°

T F A B

25√3 m 25 m

75 m

Given:

Height of the vertical pole, TF = 25√3 m

Distance of point A from the base, FA = 25 m

Distance of point B from the base, FB = 75 m

Formula Used:

In a right-angled triangle, tan(θ) = Perpendicular ÷ Base

Calculations:

In right-angled triangle TFA:

⇒ tan(x°) = TF ÷ FA

⇒ tan(x°) = 25√3 ÷ 25

⇒ tan(x°) = √3

⇒ x° = 60°   (since tan 60° = √3)

In right-angled triangle TFB:

⇒ tan(y°) = TF ÷ FB

⇒ tan(y°) = 25√3 ÷ 75

⇒ tan(y°) = √3 ÷ 3 = 1 ÷ √3

⇒ y° = 30°   (since tan 30° = 1/√3)

We need to find the value of x° + y°:

⇒ x° + y° = 60° + 30° = 90°

∴ The correct answer is 90°.

Additional Information

Angle of Elevation

The angle formed by the line of sight with the horizontal plane when the object is placed above the horizontal level of the observer.

Complementary Angles in Heights & Distances

If the angles of elevation of a tower from two points at distances a and b from its base are complementary (sum is 90°), then the height of the tower is given by h = √(a × b).

Standard Trigonometric Values

Important values to remember: tan(30°) = 1/√3, tan(45°) = 1, tan(60°) = √3. These are extremely useful for quick mental calculations in geometry.

Shortcut Trick

Two identical solid hemispheres perfectly combine to form exactly one full solid sphere of the same radius.

Therefore, the volume and dimensions match perfectly without any relative change in the radius.

Radius of the new sphere = Radius of the given hemisphere = 21 cm.

∴ The correct answer is 21 cm.

Alternate Method

Given:

Radius of each hemisphere (R) = 21 cm

Number of identical hemispheres melted = 2

R=21 Hemisphere 1 Vol: V

R=21 Hemisphere 2 Vol: V

→

r=? New Sphere Vol: 2V

Formula Used:

Volume of a hemisphere = (2/3) × π × R³

Volume of a full sphere = (4/3) × π × r³

Calculations:

⇒ Total volume of the two melted hemispheres = 2 × [(2/3) × π × (21)³]

⇒ Total volume = (4/3) × π × (21)³

⇒ Let the radius of the newly cast sphere be r.

⇒ Volume of the new sphere = Total volume of the two hemispheres

⇒ (4/3) × π × r³ = (4/3) × π × (21)³

⇒ Canceling the common factor (4/3) × π from both sides:

⇒ r³ = 21³

⇒ r = 21 cm

∴ The correct answer is 21 cm.

Additional Information

Recasting of 3D Objects

When one or more solid shapes are melted and recast into a completely new shape, the total volume of the material strictly remains constant throughout the transformation.

Surface Area of a Hemisphere

Curved Surface Area (CSA) = 2 × π × R² | Total Surface Area (TSA) = 3 × π × R².

Surface Area of a Sphere

Total Surface Area (TSA) = 4 × π × R². A full sphere has only one continuous curved surface, so its CSA and TSA are exactly the same.

Shortcut Trick

Ratio of corresponding heights of the small cut cone to the large original cone = 10 / 30 = 1 / 3

For similar figures, the ratio of their volumes is the cube of the ratio of their heights.

Ratio of their volumes = (1 / 3)³ = 1 / 27

Volume of frustum = Volume of large cone − Volume of small cone = 27 − 1 = 26

Required ratio = Frustum Volume : Total Cone Volume = 26 : 27

∴ The correct answer is 26:27.

Alternate Method

Given:

Height of the original total cone (H) = 30 cm

A cut is made parallel to the base, 10 cm from the vertex.

Height of the smaller cone formed at the top (h) = 10 cm

Formula Used:

When a cone is cut parallel to its base, the smaller cone formed is similar to the original cone. The ratio of their volumes is equal to the cube of the ratio of their heights:

V₁ / V₂ = (h₁ / h₂)³

H = 30 cm

h = 10 cm

Small Cone Frustum Volume Ratio = Frustum : Total Cone

Calculations:

⇒ Volume of small cone (V_small) / Volume of original cone (V_large) = (10 / 30)³

⇒ V_small / V_large = (1 / 3)³

⇒ V_small / V_large = 1 / 27

⇒ Let the volume of the small cone be 1 unit and the large cone be 27 units.

⇒ Volume of the Frustum = V_large − V_small

⇒ Volume of the Frustum = 27 − 1 = 26 units

⇒ Ratio of the volume of the frustum to the entire cone = 26 : 27

∴ The correct answer is 26:27.

Additional Information

Similarity of Solid Figures

If two 3D shapes are similar with a scale factor of a:b, then the ratio of their corresponding surface areas is a² : b², and the ratio of their volumes is a³ : b³.

Volume of a Frustum

The volume of a frustum of a right circular cone is given by V = (1/3) × π × h × (R² + r² + R × r), where R and r are radii of the two parallel bases and h is the height of the frustum.

Slant Height of Frustum

The slant height of a frustum is computed as l = √(h² + (R − r)²), where h is the height and R, r are the radii of the larger and smaller circular bases respectively.

Shortcut Trick

Ratio of the number of sides of an octagon to a square = 8 : 4 = 2 : 1.

Ratio of their side lengths = 1 : 2 (since the octagon side is half of the square side).

Perimeter ratio = (Number of sides ratio) × (Side length ratio) = 2 × (1/2) = 1 : 1.

Since the perimeter ratio is exactly 1 : 1, the perimeters of both polygons are equal.

∴ The correct answer is 20 cm.

Alternate Method

Given:

Perimeter of the regular quadrilateral (square) = 20 cm

Side of the regular octagon = 1/2 × Side of the square

Formula Used:

Perimeter of a regular polygon = Number of sides × Length of one side

Perimeter = 20 cm Square (n=4) Side = 5 cm

Perimeter = 20 cm Octagon (n=8) Side = 2.5 cm

Side ÷ 2

Calculations:

⇒ Perimeter of the square = 4 × Side of square

⇒ 4 × Side of square = 20

⇒ Side of square = 5 cm

⇒ Side of regular octagon = 5 ÷ 2 = 2.5 cm

⇒ Perimeter of regular octagon = 8 × Side of octagon

⇒ Perimeter = 8 × 2.5 = 20 cm

∴ The correct answer is 20 cm.

Additional Information

Interior Angles of a Regular Polygon

Each interior angle = [(n − 2) × 180°] ÷ n, where n is the total number of sides.

Number of Diagonals

The total number of diagonals that can be drawn in an n-sided polygon = n(n − 3) ÷ 2.

Exterior Angles of a Polygon

The sum of all exterior angles of any convex polygon is always 360°. Each exterior angle of a regular polygon = 360° ÷ n.

Shortcut Trick

Total height of the column = Sum of the 4 sections in AP = 5 + 7 + 9 + 11 = 32 cm.

Base area of the regular hexagon = 6 × (√3 ÷ 4) × (Side)² = 6 × (√3 ÷ 4) × 100 = 150√3 cm².

Total Volume = Base Area × Total Height = 150√3 × 32 = 4800√3 cm³.

∴ The correct answer is 4800√3 cm³.

Alternate Method

Given:

Side length of the regular hexagon base, a = 10 cm

Heights of the 4 sections form an Arithmetic Progression (AP): First height = 5 cm, Common difference = 2 cm.

Formula Used:

Area of a regular hexagon = 3√3 ÷ 2 × a²

Volume of a prism = Area of base × Total height

a = 10 cm Hexagon Base

11 cm 9 cm 7 cm 5 cm

Total = 32 cm Column Heights

Calculations:

⇒ Total height (H) = Sum of heights of all 4 sections

⇒ H = 5 + 7 + 9 + 11 = 32 cm

⇒ Area of hexagonal base = 3√3 ÷ 2 × (10)²

⇒ Area of hexagonal base = 3√3 ÷ 2 × 100 = 150√3 cm²

⇒ Total Volume = Area of base × Total height

⇒ Volume = 150√3 × 32 = 4800√3 cm³

∴ The correct answer is 4800√3 cm³.

Additional Information

Volume of a Prism

The volume of any uniform prism is simply the cross-sectional (base) area multiplied by its perpendicular length or height. Formula: V = Area of base × Height.

Properties of a Regular Hexagon

A regular hexagon is composed of 6 congruent equilateral triangles. The area is calculated as Area = 3√3 ÷ 2 × a², and its perimeter is P = 6a.

Arithmetic Progression (AP)

A numerical sequence where the difference between consecutive terms is constant. The sum of the first n terms can be calculated using S_n = n/2 × [2a + (n−1)d].

Shortcut Trick

The coordinates of the midpoint are the simple average of the given x and y coordinates.

x-coordinate = (3 + 7) ÷ 2 = 5

y-coordinate = (4 + 8) ÷ 2 = 6

∴ The correct answer is (5, 6).

Alternate Method

Given:

First point (x₁, y₁) = (3, 4)

Second point (x₂, y₂) = (7, 8)

Formula Used:

Midpoint Formula: M(x, y) = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)

A (3, 4) B (7, 8) M (x, y)

Calculation:

⇒ x-coordinate = (3 + 7) ÷ 2 = 10 ÷ 2 = 5

⇒ y-coordinate = (4 + 8) ÷ 2 = 12 ÷ 2 = 6

⇒ The midpoint is (5, 6).

∴ The correct answer is (5, 6).

Additional Information

Distance Formula

The distance d between two points (x₁, y₁) and (x₂, y₂) is given by d = √((x₂ − x₁)² + (y₂ − y₁)²).

Section Formula

A point P(x, y) dividing a line segment joining two points internally in the ratio m:n has coordinates x = (mx₂ + nx₁) ÷ (m + n) and y = (my₂ + ny₁) ÷ (m + n).

Centroid of a Triangle

The centroid coordinates of a triangle with vertices (x₁, y₁), (x₂, y₂) and (x₃, y₃) are given by ((x₁ + x₂ + x₃) ÷ 3, (y₁ + y₂ + y₃) ÷ 3).

Shortcut Trick

For any polygon with n sides, the direct formula for the number of diagonals is n(n − 3) ÷ 2.

For a hexagon, n = 6.

Number of diagonals = 6 × (6 − 3) ÷ 2 = 6 × 3 ÷ 2 = 9

∴ The correct answer is 9.

Alternate Method

Given:

A hexagon (number of sides, n = 6).

Formula Used:

Number of diagonals in a polygon of n sides = n(n − 3) ÷ 2

A B C D E F

Calculations:

⇒ Number of diagonals = 6 × (6 − 3) ÷ 2

⇒ Number of diagonals = 6 × 3 ÷ 2

⇒ Number of diagonals = 18 ÷ 2 = 9

∴ The correct answer is 9.

Additional Information

Sum of Interior Angles

The sum of all interior angles of a polygon with n sides is given by (n − 2) × 180°.

Sum of Exterior Angles

The sum of the exterior angles of any convex polygon is always 360°, regardless of the number of sides.

Interior Angle of a Regular Polygon

Each interior angle of a regular polygon with n sides can be calculated as [(n − 2) × 180°] ÷ n.

Shortcut Trick

The centroid G divides the median AD internally in the ratio 2:1.

Using the inverse section formula: 3G = 2D + A ⇒ 2D = 3G − A

⇒ D_x = (3 × 4 − 2) ÷ 2 = 10 ÷ 2 = 5

⇒ D_y = (3 × 5 − 3) ÷ 2 = 12 ÷ 2 = 6

∴ The correct answer is (5, 6).

Alternate Method

Given:

Centroid G = (4, 5)

Vertex A = (2, 3)

D is the midpoint of BC, meaning AD is the median of triangle ABC.

A(2, 3) G(4, 5) D(x, y) B C 2 1

Formula Used:

The centroid divides the median in the ratio 2:1 internally from the vertex to the midpoint of the opposite side.

Section formula: P(x, y) = [(m × x₂ + n × x₁) ÷ (m + n), (m × y₂ + n × y₁) ÷ (m + n)]

Calculations:

⇒ Let the coordinates of D be (x, y).

⇒ Here, ratio m:n = 2:1, (x₁, y₁) = (2, 3) and (x₂, y₂) = (x, y).

⇒ x-coordinate of G = 4 = (2 × x + 1 × 2) ÷ (2 + 1)

⇒ 4 × 3 = 2x + 2

⇒ 12 − 2 = 2x

⇒ x = 10 ÷ 2 = 5

⇒ y-coordinate of G = 5 = (2 × y + 1 × 3) ÷ (2 + 1)

⇒ 5 × 3 = 2y + 3

⇒ 15 − 3 = 2y

⇒ y = 12 ÷ 2 = 6

∴ The correct answer is (5, 6).

Additional Information

Centroid of a Triangle

The centroid is the point of intersection of the three medians of the triangle. It divides each median in the ratio 2:1, with the larger part towards the vertex.

Coordinates of Centroid

If the vertices of a triangle are (x₁, y₁), (x₂, y₂), and (x₃, y₃), the centroid G is given by ((x₁ + x₂ + x₃) ÷ 3, (y₁ + y₂ + y₃) ÷ 3).

Midpoint Formula

If a line segment joins two points (x₁, y₁) and (x₂, y₂), its midpoint M is given by ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2).

Shortcut Trick

For equations involving algebraic square roots, substituting the given options is the fastest method.

Let us check Option 2: x = 6.

LHS = √(6 + 3) + √(6 − 2)

LHS = √9 + √4 = 3 + 2 = 5

Since LHS = RHS, x = 6 is the correct value.

∴ The correct answer is 6.

Alternate Method

Given:

√(x + 3) + √(x − 2) = 5

Method:

Let a = √(x + 3) and b = √(x − 2).

⇒ a + b = 5 . . . . (Equation 1)

Now, finding the difference of their squares:

⇒ a² − b² = (x + 3) − (x − 2)

⇒ a² − b² = x + 3 − x + 2 = 5

Using the algebraic identity a² − b² = (a − b)(a + b):

⇒ (a − b) × (a + b) = 5

⇒ (a − b) × 5 = 5

⇒ a − b = 1 . . . . (Equation 2)

Adding Equation 1 and Equation 2:

⇒ (a + b) + (a − b) = 5 + 1

⇒ 2a = 6

⇒ a = 3

Substituting the value of a back into our initial assumption:

⇒ √(x + 3) = 3

Squaring both sides:

⇒ x + 3 = 9

⇒ x = 9 − 3

⇒ x = 6

∴ The correct answer is 6.

Additional Information

Algebraic Identities

The difference of squares formula a² − b² = (a − b)(a + b) is extremely useful to eliminate surds when the difference of the terms under the root is a constant.

Extraneous Roots

When solving equations by squaring both sides, extraneous roots (false solutions) can be introduced. Always substitute the final answer back into the original equation to verify.

Surds and Indices

A surd is an unresolved radical, such as √2 or √3. Two surds of the form √a + √b and √a − √b are called conjugate surds, and their product is a rational number.

Shortcut Trick

For a Transverse Common Tangent, the distance (d), TCT length, and sum of radii (R + r) form a right-angled triangle.

By Pythagoras theorem: (R + r)² + TCT² = d²

Given TCT = 9 cm and d = 15 cm, this perfectly forms a 9-12-15 right triangle triplet.

Therefore, R + r = 12 cm.

Since R = 8 cm, r = 12 − 8 = 4 cm.

∴ The correct answer is 4 cm.

Alternate Method

Given:

Length of Transverse Common Tangent (TCT) = 9 cm

Distance between centers (d) = 15 cm

Radius of larger circle (R) = 8 cm

Let the radius of the smaller circle be r.

8 cm r TCT = 9 cm d = 15 cm

Formula Used:

Length of TCT = √(d² − (R + r)²)

Calculations:

⇒ 9 = √(15² − (8 + r)²)

⇒ 9² = 15² − (8 + r)²

⇒ 81 = 225 − (8 + r)²

⇒ (8 + r)² = 225 − 81

⇒ (8 + r)² = 144

⇒ 8 + r = 12

⇒ r = 12 − 8 = 4 cm

∴ The correct answer is 4 cm.

Additional Information

Direct Common Tangent (DCT)

A tangent that touches two circles and does not cross the line segment joining their centers. The length formula is DCT = √(d² − (R − r)²).

Transverse Common Tangent (TCT)

A tangent that crosses the line joining the centers of the two circles. Its length is always less than the length of the Direct Common Tangent for the same set of circles.

Number of Common Tangents

When two circles do not intersect and are strictly outside each other, there can be exactly 4 common tangents drawn between them (2 direct and 2 transverse).