Mensuration MCQ Quiz - Objective Question with Answer for Mensuration - Download Free PDF

Calculation

First, side of square:

Side = 64/4 = 16 m

Length of rectangle = 16 + 4 = 20 

Area of rectangle = Length × Breadth = 20 × 12 = 240

Given:

Ratio of volumes of two cones = 3 : 2

Ratio of radii of the two cones = 3 : 4

Formula:

The volume of a cone is given by the formula:

V = (1/3)πr²h

Solution:

Let's assume the radii of the two cones are 3x and 4x, where x is a common factor.

So, the ratio of their radii is 3x : 4x.

The volume of the first cone (V1) can be expressed as:

V1 = (1/3)π(3x)²h1

V1 = (1/3)π9x²h1

The volume of the second cone (V2) can be expressed as:

V2 = (1/3)π(4x)2h1

V2 = (1/3)π16x2h1

Given that the ratio of volumes of the two cones is 3 : 2, we have:
V1/V2 = 3/2

9x​2h1/16x​2h2 = 3/2

h1 / h2 = 48x²/18x​2

h1/h2 = 8/3

Therefore, the ratio of the heights of the two cones is 8 : 3.

Concept used:

If the number of sides of a regular polygon be n, then

The interior angle = \( \frac{(n - 2) \times 180^\circ }{n}\)

Calculation:

Let the required difference be x.

No. of sides of a regular polygon = 10

According to the question

\(\Rightarrow \frac{{\left( {n - 2} \right) \times 180^\circ }}{n} - \frac{{360^\circ }}{n} = x\)

\(\Rightarrow \frac{{\left( {10 - 2} \right) \times 180^\circ }}{{10}} - \frac{{360^\circ }}{{10}} = x\)

\(\Rightarrow \frac{{8 \times 180^\circ }}{{10}} - 36^\circ = x\)

⇒ 144 - 36° = x

Given:

Area of circular field = 6 × area of triangular field

Formula used:

Area of circular field = πr², where r is a radius of circle.

Area of triangle = √s (s - a) (s - b) (s - c),

where s = (a + b + c)/2

a, b and c are sides of triangle respectively.

Calculation:

a = 35 m, b = 53 m and c = 66 m

s = (35 + 53 + 66)/2

⇒ s = 77

Area of triangle = √s(s - a) (s - b) (s - c)

⇒ √77 (77 - 35) (77 - 53) (77 - 66)

⇒ √77 × 42 × 24 × 11

⇒ √7 × 11 × 2 × 3 × 7 × 2 × 3 × 2 × 2 × 11

⇒ 11 × 7 × 2 × 2 × 3

⇒ 924 m²

Now, 6 × Area of triangle = Area of circle

⇒ 924 × 6 = πr²

⇒ (924 × 6 × 7)/22 = r²

⇒ 1764 = r²

⇒ 42 = r

∴ Radius of circle is 42 m.

Given:

Number of revolutions = 5000

Distance covered = 11 km = 1100000 cm

Formula used:

Circumference of circle = 2πr

Calculation:

Circumference of circle = 1100000/5000

⇒ 220 cm

According to the question,

⇒ 2πr = 220 cm

⇒ 2 × (22/7) × r = 220 cm

⇒ r/7 = 5 cm

⇒ r = 35 cm

Diameter = 2r

⇒ 2 × 35

⇒ 70 cm

∴ The diameter of circle is 70 cm

Given:

Diameter of semicircle = 14√2 cm

Radius = 14√2/2 = 7√2 cm

Total no. of chords = 6

Concept:

Since the chords are equal in length, they will subtend equal angles at the centre. Calculate the area of one sector and subtract the area of the isosceles triangle formed by a chord and radius, then multiply the result by 6 to get the desired result.

Formula used:

Area of sector = (θ/360°) × πr²

Area of triangle = 1/2 × a × b × Sin θ

Calculation:

The angle subtended by each chord = 180°/no. of chord

⇒ 180°/6

⇒ 30°

Area of sector AOB = (30°/360°) × (22/7) × 7√2 × 7√2

⇒ (1/12) × 22 × 7 × 2

⇒ (77/3) cm²

Area of triangle AOB = 1/2 × a × b × Sin θ

⇒ 1/2 × 7√2 × 7√2 × Sin 30°

⇒ 1/2 × 7√2 × 7√2 × 1/2

⇒ 49/2 cm²

∴ Area of shaded region = 6 × (Area of sector AOB - Area of triangle AOB)

⇒ 6 × [(77/3) – (49/2)]

⇒ 6 × [(154 – 147)/6]

⇒ 7 cm²

∴ Area of shaded region is 7 cm²

Formula used

Area = length × breath

Calculation

The garden EFGH is shown in the figure. Where EF = 220 meters & EH = 70 meters.

The width of the path is 4 meters.

Now the area of the path leaving the four colored corners

= [2 × (220 × 4)] + [2 × (70 × 4)]

= (1760 + 560) square meter

= 2320 square meters

Now, the area of 4 square colored corners:

4 × (4 × 4)

{∵ Side of each square = 4 meter}

= 64 square meter

The total area of the path = the area of the path leaving the four colored corners + square colored corners

⇒ Total area of the path = 2320 + 64 = 2384 square meter

∴ Option 4 is the correct answer.

Given:

The width of the path around a square field = 4.5 m

The area of the path = 105.75 m2

Formula used:

The perimeter of a square = 4 × Side

The area of a square = (Side)²

Calculation:

Let, each side of the field = x

Then, each side with the path = x + 4.5 + 4.5 = x + 9

So, (x + 9)² - x² = 105.75

⇒ x2 + 18x + 81 - x2 = 105.75

⇒ 18x + 81 = 105.75

⇒ 18x = 105.75 - 81 = 24.75

⇒ x = 24.75/18 = 11/8

∴ Each side of the square field = 11/8 m

The perimterer = 4 × (11/8) = 11/2 m

So, the total cost of fencing = (11/2) × 100 = Rs. 550

∴ The cost of fencing of the field is Rs. 550

Shortcut TrickIn such types of questions, 

Area of path outside the Square is, 

⇒ (2a + 2w)2w = 105.75

here, a is a side of a square and w is width of a square

⇒ (2a + 9)9 = 105.75

⇒ 2a + 9 = 11.75

⇒ 2a = 2.75

Perimeter of a square = 4a

⇒ 2 × 2a = 2 × 2.75 = 5.50

costing of fencing = 5.50 × 100 = 550

∴ The cost of fencing of the field is Rs. 550

Given:

The sides of an equilateral triangle are increased by 34%.

Formula used:

Effective increment % = Inc.% + Inc.% + (Inc.²/100) 

Calculation:

Effective increment = 34 + 34 + {(34 × 34)/100}

⇒ 68 + 11.56 = 79.56%

∴ The correct answer is 79.56%.

Given : 

Length of an arc of a circle is 4.5π.

Area of ​​the sector circumscribed by it is 27π cm2.

Formula Used : 

Area of sector = θ/360 × πr²

Length of arc = θ/360 × 2πr

Calculation : 

According to question,

⇒ 4.5π = θ/360 × 2πr 

⇒ 4.5 = θ/360 × 2r   -----------------(1)

⇒ 27π = θ/360 × πr2 

⇒ 27 = θ/360 × r2       ---------------(2)

Doing equation (1) ÷ (2)

⇒ 4.5/27 = 2r/πr²

⇒ 4.5/27 = 2/r

⇒ r = (27 × 2)/4.5

⇒ Diameter = 2r = 24

∴ The correct answer is 24.

Given:

The side of the square = 22 cm

Formula used:

The perimeter of the square = 4 × a    (Where a = Side of the square)

The circumference of the circle = 2 × π × r     (Where r = The radius of the circle)

Calculation:

Let us assume the radius of the circle be r

⇒ The perimeter of the square = 4 × 22 = 88 cm

⇒ The circumference of the circle = 2 × π ×  r

⇒ 88 = 2 × (22/7) × r

⇒ \(r = {{88\ \times\ 7 }\over {22\ \times \ 2}}\)

⇒ r = 14 cm

∴ The required result will be 14 cm.

Given:

The radius of a solid hemisphere is 21 cm.

The ratio of the cylinder's curved surface area to its Total surface area is 2/5.

Formula used:

The curved surface area of the cylinder = 2πRh

The total surface area of cylinder = 2πR(R + h)

The volume of the cylinder = πR²h

The volume of the solid hemisphere = 2/3πr³ 

(where r is the radius of a solid hemisphere and R is the radius of a cylinder)

Calculations:

According to the question,

CSA/TSA = 2/5

⇒ [2πRh]/[2πR(R + h)] = 2/5

⇒ h/(R + h) = 2/5

⇒ 5h = 2R + 2h

⇒ h = (2/3)R .......(1)

The cylinder's volume and the volume of a solid hemisphere are equal.

⇒ πR²h = (2/3)πr³

⇒ R2 × (2/3)R = (2/3) × (21)³

⇒ R³ = (21)3

⇒ R = 21 cm

∴ The radius (in cm) of its base is 21 cm.

The surface area of three faces of a cuboid sharing a vertex are 20 m², 32 m² and 40 m²,

⇒ L × B = 20 sq. Mt

⇒ B × H = 32 sq. Mt

⇒ L × H = 40 sq. Mt

⇒ L × B × B × H × L × H = 20 × 32 × 40

⇒ L²B²H² = 25600

⇒ LBH = 160

Given:

Each side of the cube = 8 cm

The rectangular container has a length of 16 cm, breadth of 8 cm, and height of 15 cm

Formula used:

The volume of cube = (Edge)3

The volume of a cuboid = Length × Breadth × Height

Calculation:

The volume of cube = The volume of the rectangular container with a length of 16 cm, breadth of 8 cm, and height of the water level rise

Let, the height of the water level will rise = x cm

So, 8³ = 16 × 8 × x

⇒ 512 = 128 × x

⇒ x = 512/128 = 4

∴ The rise of water level (in cm) is 4 cm

Given:

Sum of length,, breadth and height of a cuboid = 21 cm

Length of the diagonal(d) = 13 cm

Formula used:

d² = l² + b² + h²

T.S.A of cuboid = 2(lb + hb +lh)

Calculation:

⇒ l² + b² + h² = 13² = 169

According to question,

⇒ (l + b + h)² = 441

⇒ l² + b² + h² + 2(lb + hb +lh) = 441

⇒ 2(lb + hb +lh) = 441 - 169 = 272

∴ The answer is 272 cm² .