Two Figures MCQ Quiz - Objective Question with Answer for Two Figures - Download Free PDF

Given:

Painting rate = Rs.75/m²

Total painting cost = Rs.6825

Area of square wall = 196 m²

Length of rectangular wall = side of square wall

Formula used:

Area = Total Cost / Rate

Area of rectangle = Length × Breadth

Side of square = √Area

Calculations:

Area of rectangular wall = 6825 / 75 = 91 m²

Side of square wall = √196 = 14 m ⇒ Length of rectangular wall = 14 m

Now, 14 × Breadth = 91

⇒ Breadth = 91 / 14 = 6.5 m

∴ The breadth of the rectangular wall is 6.5 metres.

Calculation:

Let the sides of the rectangular field be 5x and 4x. The area of the rectangle is:

Area = 5x × 4x = 500

⇒ 20x² = 500

⇒ x² = 25

⇒ x = 5.

Thus, the length = 5x = 25 m, and the breadth = 4x = 20 m.

The perimeter of the rectangular field = 2 × (25 + 20) = 90 m.

The sides of the triangular field are in the ratio 3 : 2 : 4. Let the sides be 3y, 2y, and 4y.

The perimeter of the triangle is: 3y + 2y + 4y = 9y.

Since the perimeter is 90 m, we have: 9y = 90 ⇒ y = 10.

The longest side of the triangle is: 4y = 4 × 10 = 40 m.

∴ The longer side of the triangular field is 40 meters.

Given:

Triangle sides: a = 28 cm, b = 45 cm, c = 53 cm

π = 3.14

Formula used:

Area of triangle = √(s(s - a)(s - b)(s - c)), where s = semi-perimeter = (a + b + c)/2

Area of circle = π × r², where r = inradius = Area of triangle / s

Area excluding circle = Area of triangle - Area of circle

Calculations:

s = (28 + 45 + 53)/2

⇒ s = 63 cm

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

⇒ Area of triangle = √(63 × (63 - 28) × (63 - 45) × (63 - 53))

⇒ Area of triangle = √(63 × 35 × 18 × 10)

⇒ Area of triangle = √396900

⇒ Area of triangle = 630 cm²

Inradius (r) = Area of triangle / s

⇒ r = 630 / 63

⇒ r = 10 cm

Area of circle = π × r²

⇒ Area of circle = 3.14 × 10²

⇒ Area of circle = 314 cm²

Area excluding circle = Area of triangle - Area of circle

⇒ Area excluding circle = 630 - 314

⇒ Area excluding circle = 316 cm²

∴ The correct answer is option (3).

Given:

The sides of the triangle are 21 cm, 35 cm and 28 cm.

Concept used:

Semi-permieter of a triangle, S = (A + B + C)/2

Area of a triangle

⇒ \(\sqrt {S (S - A) (S - B) (S - C)}\)

Where, A,B,C are the length of three sides of the triangle.

The measure of the inradius of a triangle

⇒ \({Area\ of\ the\ triangle} \over {Semi\ perimeter\ of\ the\ triangle}\)

Calculation:

Semi-permieter of the triangular park

⇒ (21 + 35 + 28)/2

⇒ 84/2 = 42 m

Area of the triangular park

⇒ \(\sqrt {42 (42 - 21) (42 - 35) (42- 28)}\)

⇒ \(\sqrt {42 × 21 × 7 × 14}\)

⇒ \(\sqrt { 7 × 3 × 2 × 7 × 3 × 7 × 7 × 2}\)

⇒ 7 × 7 × 3 × 2

⇒ 294 cm²

Now, the inradius of the triangle

⇒ 294/42 = 7 cm

∴ The measure of the inradius is 7 cm.

Alternate Method

Concept used:

Pythagoras Theorem states that "In a right-angled triangle. the square of the hypotenuse side is equal to the sum of squares of the other two sides."

Area of △ = 1/2 × Base × Height

Calculation:

Here, 352 = 282 + 212

So, the given triangle is a right-angled triangle.
 
 

Area of △

⇒ 1/2 × 21 × 28

⇒ 294 cm² 

Semi-perimeter of the triangular park

⇒ (21 + 35 + 28)/2

⇒ 84/2 = 42 cm

So, the inradius of the triangle

⇒ (Area of △)/Semi-perimeter

⇒ 294/72 = 7 cm

∴ The measure of the inradius is 7 cm.

Given:

Length of the rectangular garden = 32 m

Breadth of the rectangular garden = 8 m

Area of square garden = Area of rectangular garden

Formula Used:

Area of rectangle = Length × Breadth

Area of square = Side²

Perimeter of square = 4 × Side

Calculation:

Area of rectangular garden = 32 × 8

Area of square garden = 256 m²

Let the side of the square garden be 'a'.

a² = 256

⇒ a = √256

⇒ a = 16 m

Perimeter of square garden = 4 × a

⇒ Perimeter = 4 × 16

⇒ Perimeter = 64 m

The perimeter of the square garden is 64 m.

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:

Diameter of each lead shot = 8.4 cm

Dimension of the rectangular solid = 88 × 63 × 42 (cm)

Concept used:

1. Volume of a sphere = \(\frac {4\pi × (Radius)^3}{3}\)

2. Volume of a cuboid = Length × Breadth × Height

3. The collective volume of all lead shots obtained should be equal to the volume of the rectangular solid.

4. Diameter = Radius × 2

Calculation:

Let N number of shots can be obtained.

Radius of each lead shot = 8.4/2 = 4.2 cm

​According to the concept,

N × \(\frac {4\pi × (4.2)^3}{3}\) = 88 × 63 × 42

⇒ N × \(\frac {4 × 22 × (42)^2}{3 × 7 × 1000}\) = 88 × 63

⇒ N = 750

∴ 750 lead shots can be obtained.

Given:

A rhombus has one of its diagonal 65% of the other.

A square is drawn using the longer diagonal as a side.

Concept used:

Area of rhombus = ½(diagonals product)

Area of square = side × side

Calculations:

Let diagonal(larger) of rhombus be 100 cm

Let the diagonal(smaller) diagonal be 65 cm (65% of larger diagonal)

Area of Rhombus = ½(100 × 65) = 3250

Side of square = 100 cm (equal to larger diagonal)

Area of square = (100 × 100) = 10000

Ratio,

⇒ Rhombus : Square = 3250 : 10000

⇒ 13 : 40

∴ The correct choice is option 3.

Given:

The volume of a cuboid is twice that of a cube.

Height = 16 cm

Breadth = 8 cm

Length = 8 cm

Formula Used:

The volume of the cuboid =  Length × Breadth × Height

Volume of cube= (edge)³

Calculation:

The volume of the cuboid = 8 × 8 × 16 

= 1024

The volume of a cuboid = 2 × volume of a cube.

The volume of a cuboid = 2 × (edge)3

(edge)3 = 1024/2 = 512 m

edge = 8 m

The total surface area of the cube = 6 × 64

= 384 m 2

Given:

Cone diameter = 14 cm, Height = 24 cm

Cube of side = 14 cm

Formula used:

Curved surface area of cone = π × Radius × Slant height 

Surface area of cube = 6 × side2 

Slant height = √(height2 + radius2)

Area of circle = π × radius2

Calculation:

Slant height = √(height² + radius²) = √(24² + 7²) = 25

Curved surface area of cone = π × Radius × Slant height = (22/7) × 7 × 25 = 550 cm²

Surface area of cube = 6 × side² = 6 × (14)² = 1176 cm²

But some area covered by cone's base = π × radius2

⇒ (22/7) × 7² = 154 cm²

⇒ Total surface area = 550 + 1176 - 154 = 1,572 cm²

Let the length of side of a cube and square be a and b units respectively

Now,

⇒ Sum of length of edges of a cube = (1/8) × perimeter of square

⇒ 12a = (1/8) × 4b

⇒ 24a = b

Also,

⇒ Volume of cube = area of square

⇒ a³ = b²

⇒ a³ = (24a)²

Given:

The radius of the circle = 21 cm

The ratio of base and height of the right-angled triangle formed = 3 : 4

Formulae Used:

In a right-angled triangle,

(Hypotenuse)2 = (Base)2 + (Height)2

The perimeter of the circle = 2πr, where r is the radius of the circle

Calculation:

Let the base and height of the given right angle triangle be 3x and 4x.

⇒ Hypotenuse = √{(3x)2 + (4x)2} = 5x

Radius of circle = r = 21 cm

According to the question,

The perimeter of circle = Perimeter of right angle triangle

⇒ 2πr = 3x + 4x + 5x

⇒ 2 × (22/7) × 21 = 12x

⇒ x = 11

Given:

Side of the square = 11 cm

Formula:

Perimeter of the square = 4a

Circumference of the circle = 2πr

Area of the circle = πr²

Calculation:

According to the question,

Circumference of the circle = Perimeter of the square

⇒ 2πr = 4a

⇒ 2πr = 4 × 11

⇒ 2 × (22 / 7) × r = 44

⇒ r = 7 cm

Area of the circle =  πr2

⇒ (22 / 7) × 7 × 7 = 154 cm²

∴ Area of circle is 154 cm².

Given:

Radius of semi-circle = r cm

Formula used:

Pythagoras theorem

H² = P² + B²

Area of square = side2

Calculation:

Let the side of the square be ‘a’ cm

The maximum size square exists with side ‘r’ cm.

⇒ H² = P² + B²

⇒ r² = a² + (a/2)²

⇒ r² = a² + a²/4

⇒ r² = 5a²/4

⇒ a = 2r/√5

Given:

Length of a rectangular metal sheet = 24 cm

The breadth of a rectangular sheet = 18 cm

Formula used:

The volume of cuboid = lbh

Where, l = length, b = breadth and h = height

Calculation:

As shown in the above figures,

Length of box = (24 – 2x)

Breadth of box = (18 – 2x)

Height of box = x

The volume of the box = lbh

⇒ (24 – 2x)(18 – 2x)(x) = 640

From Option (3): If x = 4

⇒ 16 × 10 × 4 = 640

∴ The correct value of x is 4.