Search Previous Year Questions

Shortcut Trick

Apply the algebraic identity (a + 1/a)² = a² + (1/a)² + 2.

Substitute a = √2 ⇒ a² = 2 and (1/a)² = 1/2.

Expression = 2 + 0.5 + 2 = 4.5.

∴ The correct answer is 4.5.

Alternate Method

Given: Diagonal length of a unit square = √2 and its reciprocal = 1/√2.

Formula Used: (a + b)² = a² + 2ab + b²

⇒ Let a = √2 and b = 1/√2

⇒ (√2 + 1/√2)² = (√2)² + 2 × (√2) × (1/√2) + (1/√2)²

⇒ (√2)² = 2

⇒ 2 × (√2) × (1/√2) = 2 × 1 = 2

⇒ (1/√2)² = 1/2 = 0.5

⇒ Total Value = 2 + 2 + 0.5 = 4.5

∴ The correct answer is 4.5.

Additional Information

Standard Algebraic Identity

The square of a sum is given by (a + b)² = a² + 2ab + b², while the square of a difference is (a − b)² = a² − 2ab + b².

Diagonal of a Square

For a square with side length 's', the diagonal 'd' is always s√2. For a unit square (s=1), the diagonal is exactly √2 units.

Rationalization

The reciprocal of √2, which is 1/√2, can be rationalized by multiplying the numerator and denominator by √2 to get √2/2.

Shortcut Trick

Equate the packing capacity of both groups: 5P + 2H = 3 × (1P + 1H)

Solve for the variables: 5P + 2H = 3P + 3H

Simplify the equation: 2P = 1H

The ratio of P:H = 1:2

∴ The correct answer is 1:2.

Alternate Method

Given:

Let the capacity of one packer be P and one helper be H.

Total capacity of 5 packers and 2 helpers = 5P + 2H

Total capacity of 1 packer and 1 helper = 1P + 1H

According to the question:

⇒ (5P + 2H) = 3 × (P + H)

⇒ 5P + 2H = 3P + 3H

⇒ 5P − 3P = 3H − 2H

⇒ 2P = 1H

⇒ P/H = 1/2

⇒ P : H = 1 : 2

∴ The correct answer is 1:2.

Additional Information

Work-Efficiency Relation

Total work is directly proportional to efficiency when time is constant. Work = Efficiency × Time.

Ratio of Efficiency

If the efficiency of A is x and B is y, the ratio of work done in the same time is x:y. Conversely, the ratio of time taken to complete the same work is 1/x : 1/y.

Man-Days Formula

(M₁ × D₁ × E₁) / W₁ = (M₂ × D₂ × E₂) / W₂, where M is number of workers, D is days, E is efficiency, and W is work.

Shortcut Trick

The Arc Length is a fraction of the total Circumference based on the Central Angle.

For a central angle of 60°, the arc length is exactly 1/6^th of the full circumference (since 60°/360° = 1/6).

Arc Length = (1 ÷ 6) × 2 × π × r = (1 ÷ 6) × 2 × 3.14159 × 12 = 4 × 3.14159

Arc Length ≈ 12.566 m

∴ The correct answer is 12.57 m.

Alternate Method

Given:

⇒ Radius (r) of the circular stage = 12 m

⇒ Central Angle (θ) subtended by the arc = 60°

Formula Used:

⇒ Length of an arc = (θ ÷ 360) × 2πr

Calculations:

⇒ Arc Length = (60 ÷ 360) × 2 × (22 ÷ 7) × 12

⇒ Arc Length = (1 ÷ 6) × 2 × 3.14159 × 12

⇒ Arc Length = (1 ÷ 6) × 24 × 3.14159

⇒ Arc Length = 4 × 3.14159

⇒ Arc Length = 12.56637... m

⇒ Rounding to two decimal places, Arc Length ≈ 12.57 m

∴ The correct answer is 12.57 m.

Additional Information

Area of a Sector

The region bounded by two radii and an arc is a sector. Area = (θ ÷ 360) × πr².

Perimeter of a Sector

The total boundary length of a sector. Perimeter = Arc Length + 2 × Radius.

Radian Measure

If the angle θ is in radians, the arc length is simply calculated as s = r × θ.

Shortcut Trick

For a right circular cone where radius (r) is 7, the Total Surface Area (TSA) calculation simplifies as the 7 in the denominator of π (22/7) cancels out.

Shortcut: TSA = 22 × (l + r)

Given r = 7 cm and l = 25 cm:

TSA = 22 × (25 + 7) = 22 × 32 = 704 cm².

∴ The correct answer is 704 cm².

Alternate Method

Given:

Radius (r) of the cone = 7 cm

Slant height (l) of the cone = 25 cm

Formula Used:

Total Surface Area (TSA) = Curved Surface Area (CSA) + Base Area

TSA = πrl + πr² = πr(l + r)

Calculations:

⇒ Base Area = πr² = (22/7) × 7 × 7 = 154 cm²

⇒ Curved Surface Area (CSA) = πrl = (22/7) × 7 × 25 = 550 cm²

⇒ Total Surface Area (TSA) = 154 + 550

⇒ TSA = 704 cm²

∴ The correct answer is 704 cm².

Additional Information

Curved Surface Area (CSA)

The area of the slanted side of the cone is given by CSA = πrl.

Volume of a Cone

The capacity or space inside a cone is calculated as Volume = (1/3)πr²h, where h is the vertical height.

Slant Height Relation

In a right circular cone, the relation between height (h), radius (r), and slant height (l) is defined by l² = h² + r².

Total Surface Area of Cylinder

For comparison, the TSA of a cylinder is given by 2πr(h + r), which includes two circular bases and the curved wall.

Shortcut Trick

An increased by 100% means the value becomes double (2 times).

The volume of a cylinder is proportional to radius² × height.

If both radius and height are doubled: New Volume Factor = (2)² × 2 = 4 × 2 = 8.

∴ The correct answer is 8x.

Alternate Method

Given:

Initial radius = r

Initial height = h

Increase in radius and height = 100%

Formula Used:

Volume of Cylinder (V) = πr²h

Calculations:

⇒ Let original volume V₁ = πr²h

⇒ New radius (R) = r + (100/100) × r = 2r

⇒ New height (H) = h + (100/100) × h = 2h

⇒ New volume V₂ = π × R² × H

⇒ V₂ = π × (2r)² × (2h)

⇒ V₂ = π × (4r²) × (2h)

⇒ V₂ = 8 × (πr²h) = 8V₁

∴ The correct answer is 8x.

Additional Information

Cylinder Surface Area

The Curved Surface Area is 2πrh and the Total Surface Area is 2πr(r + h).

Net Percentage Change

For a product of three terms (like r × r × h), the net percentage change can be found using the successive change formula: a + b + (ab/100).

Ratio of Volumes

If radius ratio is r₁:r₂ and height ratio is h₁:h₂, then the volume ratio is (r₁²h₁) : (r₂²h₂).

Shortcut Trick

Volume of the bubble layer = 4/3 × π × (R³ − r³)

Difference of cubes = 7³ − 6.8³ = 343 − 314.432 = 28.568

Approximate calculation: 4.187 × 28.568 ≈ 119.59

∴ The correct answer is 119.59 cm².

Alternate Method

Given:

Outer radius (R) = 7 cm

Inner radius (r) = 6.8 cm

Formula Used:

Volume (V) of a Spherical Shell = 4/3 × π × (R³ − r³)

Calculations:

⇒ Volume = 4/3 × 3.14 × (7³ − 6.8³)

⇒ Volume = 1.333 × 3.14 × (343 − 314.432)

⇒ Volume = 4.1866 × 28.568

⇒ Volume = 119.599 ≈ 119.59 cm²

∴ The correct answer is 119.59 cm².

Additional Information

Volume of a Sphere

The total space occupied by a solid sphere is given by V = 4/3 × π × r³.

Surface Area of a Sphere

The total outer area of a sphere is given by A = 4 × π × r².

Thickness of a Shell

The thickness of a spherical bubble or shell is the difference between the outer and inner radii: t = R − r.

Shortcut Trick

Radius (r) = 3 cm

Volume of Hemisphere = (2/3) × π × r³

Substitute r = 3 and π = 3.14: Volume = (2/3) × 3.14 × 27

Simplifying: 18 × 3.14 = 56.52 cm³

∴ The correct answer is 56.52 cm³.

Alternate Method

Given:

Radius (r) of the Hemisphere = 3 cm

Formula Used:

Volume of Hemisphere = (2/3) × π × r³

⇒ Radius (r) = 3 cm

⇒ Volume = (2/3) × 3.14 × (3)³

⇒ Volume = (2/3) × 3.14 × 27

⇒ Volume = 2 × 3.14 × 9

⇒ Volume = 18 × 3.14

⇒ Volume = 56.52 cm³

∴ The correct answer is 56.52 cm³.

Additional Information

Volume of a Sphere

The volume of a complete sphere is double that of a hemisphere and is calculated as (4/3) × π × r³.

Surface Area of Hemisphere

The Curved Surface Area (CSA) is 2 × π × r² and the Total Surface Area (TSA) is 3 × π × r².

Relation between Volume and Radius

The volume of a 3D curved object like a hemisphere is proportional to the cube of its radius.

Shortcut Trick

Radius of the hemisphere (r) = Side ÷ 2 = 12 ÷ 2 = 6 cm.

Unused Space = Volume of Cube − Volume of Hemisphere

Volume = 12³ − (2/3) × 3.1416 × 6³

Volume = 1728 − 452.39 = 1275.61 cm³.

∴ The correct answer is 1275.61 cm³.

Alternate Method

Given:

Side of the cube (a) = 12 cm

The diameter of the largest hemisphere that can be placed on the base is equal to the side of the cube.

⇒ Radius of hemisphere (r) = 12 ÷ 2 = 6 cm

Formula Used:

⇒ Volume of Cube = a³

⇒ Volume of Hemisphere = (2/3)πr³

Calculations:

⇒ Volume of Cube = 12 × 12 × 12 = 1728 cm³

⇒ Volume of Hemisphere = (2/3) × (22/7) × 6 × 6 × 6

⇒ Volume of Hemisphere = (2/3) × 3.14159 × 216

⇒ Volume of Hemisphere = 2 × 3.14159 × 72

⇒ Volume of Hemisphere ≈ 452.39 cm³

⇒ Unused Space = 1728 − 452.39

⇒ Unused Space = 1275.61 cm³

∴ The correct answer is 1275.61 cm³.

Additional Information

Total Surface Area of Hemisphere

The total surface area of a solid hemisphere includes the curved surface and the flat base area: TSA = 3πr².

Volume of a Sphere

A hemisphere is exactly half of a sphere. The volume of a full sphere is given by V = (4/3)πr³.

Diagonal of a Cube

The longest rod that can be placed inside a cube is its space diagonal, calculated as d = a√3.

Shortcut Trick

Ratio of heights of the original pyramid to the smaller top pyramid = 18 : 9 = 2 : 1.

Ratio of their volumes = 2³ : 1³ = 8 : 1.

The frustum volume ratio = Total volume − Top volume = 8 − 1 = 7 units.

Total Volume = (1/3) × 16² × 18 = 1536 cm³.

Volume of Frustum = (7/8) × 1536 = 1344 cm³.

∴ The correct answer is 1344 cm².

Alternate Method

Given: Base side (a₁) = 16 cm, Total height (H) = 18 cm, Slice height (h) = 9 cm (halfway).

Formula: Volume of Frustum = (h/3) × [A₁ + A₂ + √(A₁ × A₂)]

⇒ Area of base (A₁) = 16 × 16 = 256 cm²

⇒ By similar triangles, top side (a₂) = 16 × (9/18) = 8 cm

⇒ Area of top (A₂) = 8 × 8 = 64 cm²

⇒ Volume = (9/3) × [256 + 64 + √(256 × 64)]

⇒ Volume = 3 × [256 + 64 + (16 × 8)]

⇒ Volume = 3 × [256 + 64 + 128]

⇒ Volume = 3 × 448 = 1344 cm³

∴ The correct answer is 1344 cm².

Additional Information

Volume of a Right Pyramid

The volume of any pyramid is given by V = (1/3) × Base Area × Height.

Slant Height of a Frustum

The slant height (l) of a square frustum is l = √[h² + ((a₁ − a₂)/2)²], where h is the vertical height.

Similar Solids Property

If the linear dimensions of two similar solids are in ratio x : y, then their volumes are in ratio x³ : y³.

Shortcut Trick

A chord subtending 90° at the center of a circle forms a right-angled isosceles triangle with the radii.

In such a triangle, the Radius (r) is calculated as: r = Chord ÷ √2.

Radius = 12 ÷ 1.4142 ≈ 8.485 m.

∴ The correct answer is 8.49 m.

Alternate Method

Given:

Length of the support beam (chord AB) = 12 m

Angle subtended at the center (θ) = 90°

Formula Used:

In a right-angled triangle AOB, by Pythagoras Theorem:

OA² + OB² = AB²

Calculations:

⇒ r² + r² = 12²

⇒ 2r² = 144

⇒ r² = 144 ÷ 2

⇒ r² = 72

⇒ r = √72

⇒ r = 6√2

⇒ r = 6 × 1.4142

⇒ r = 8.4852... m

∴ The correct answer is 8.49 m.

Additional Information

General Chord Length Formula

The length of a chord subtending an angle θ at the center of a circle of radius r is given by L = 2r sin(θ/2).

Right-Angled Isosceles Triangle

In a triangle with angles 45°-45°-90°, the sides are in the ratio 1 : 1 : √2. Here, radii are the legs and chord is the hypotenuse.

Area of the Triangle

The area of the triangle formed by the radii and the chord is (1/2) × r² sin(θ). For θ = 90°, Area = 0.5 × r².