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

Given:

Perimeter of the rectangle = 120 cm

Ratio of length to breadth = 7 : 8

Area of the square is 4 cm² more than the area of the rectangle.

Formula used:

Perimeter of rectangle = 2 × (Length + Breadth)

Area of rectangle = Length × Breadth

Area of square = Side²

Calculations:

Let the length = 7x and the breadth = 8x (from the ratio 7:8).

Perimeter = 2 × (Length + Breadth) = 2 × (7x + 8x) = 120

⇒ 2 × 15x = 120

⇒ 30x = 120

⇒ x = 4

Length = 7x = 7 × 4 = 28 cm

Breadth = 8x = 8 × 4 = 32 cm

Area of rectangle = Length × Breadth = 28 × 32 = 896 cm²

Let the side of the square be 's'.

Area of square = s²

We are given that the area of the square is 4 cm² more than the area of the rectangle:

s² = 896 + 4

s² = 900

⇒ s = √900

⇒ s = 30 cm

∴ The side of the square is 30 cm.

Given:

The dimensions of the rectangular floor are:

Length (l) = 5.25 m

Width (w) = 5.10 m

Formula used:

To find the minimum number of square tiles, we calculate the side of the largest square tile that can exactly divide both the length and width of the floor using the HCF (Highest Common Factor).

Number of tiles = Area of floor / Area of one tile

Area of floor = l × w

Area of one tile = (HCF)²

Calculation:

l = 5.25 m = 525 cm

w = 5.10 m = 510 cm

HCF of 525 and 510:

Prime factorization of 525 = 3 × 5² × 7

Prime factorization of 510 = 2 × 3 × 5 × 17

Common factors = 3 × 5 = 15 ⇒ HCF = 15 cm

Area of floor = 525 × 510 = 267750 cm²

Area of one tile = 15 × 15 = 225 cm²

Number of tiles = 267750 / 225

⇒ Number of tiles = 1190

∴ The correct answer is option (1).

Given:

Side of square = diameter of smaller circle = d₁

Diagonal of square = diameter of larger circle = d₂

Formula used:

Diagonal of square = \( \sqrt{2} \times \text{side} \)

Area of circle = \( \pi r^2 = \pi \left(\frac{d}{2}\right)^2 = \frac{\pi d^2}{4} \)

Calculation:

Let side of square = d₁ ⇒ diagonal = d₂ = \( \sqrt{2} \times d₁ \)

Area of smaller circle = \( \frac{\pi d₁^2}{4} \)

Area of larger circle = \( \frac{\pi d₂^2}{4} = \frac{\pi (\sqrt{2} \times d₁)^2}{4} = \frac{\pi \times 2d₁^2}{4} \)

Ratio = \( \frac{\pi d₁^2/4}{2\pi d₁^2/4} = \frac{1}{2} \)

∴ Ratio of areas = 1 : 2

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:

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:

The side of the triangle is 6 cm, 8 cm, 10 cm.

Formula used:

Area of triangle = 1/2 × base × height

Area of square = side²

Calculation:

Let the side of the square be ‘a’

Area of  Δ ABC = Area of  Δ ADE + Area of Δ EFC + Area of square

⇒ 1/2 × 6 × 8 = 1/2 × a × (8 – a) + 1/2 × (6 – a) × a + a²

⇒ 24 = 7a – a² + a²

⇒ a = 24/7

Area of square = side² = a²

⇒ (24/7)²

⇒ 576/49

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.