Two Figures MCQ Quiz - Objective Question with Answer for Two Figures - Download Free PDF
Last updated on Jun 9, 2025
Latest Two Figures MCQ Objective Questions
Two Figures Question 1:
The combined perimeter of the top and bottom circular faces of a right circular cylinder is 176 cm. The volume of the cylinder is given as 3528π cm³. If the height of the cylinder is three-fourths the length of a side of a square, what is the area of the square (in cm²)?
Answer (Detailed Solution Below)
Two Figures Question 1 Detailed Solution
Calculation
ATQ, 2 ×(2πr) = 176
Or, 2πr = 88
Or, πr = 44
So, r = 14 cm
Volume of the cylinder = [22/7] r2h = 3528π
So, 196h = 3528
So, h = 18
Side of the square = 18 × [4 / 3] = 24 cm
Required area = 24 × 24 = 576 cm
Two Figures Question 2:
A solid iron cube of edge 24 cm is melted and recast into a rectangular sheet of thickness 2 mm. If the length (l) and breadth (b) of the sheet are in the ratio 6:5, then l + b (in cm) is
Answer (Detailed Solution Below)
Two Figures Question 2 Detailed Solution
Given:
Edge of solid iron cube = 24 cm
Thickness of rectangular sheet = 2 mm = 0.2 cm
Ratio of length (l) to breadth (b) = 6:5
Formula Used:
Volume of cube = Volume of rectangular sheet
Volume of cube = Edge3
Volume of rectangular sheet = Length × Breadth × Thickness
l / b = 6 / 5
Calculation:
Volume of cube = Edge3
⇒ Volume = 243
⇒ Volume = 13824 cm3
Volume of rectangular sheet = l × b × 0.2
⇒ 13824 = l × b × 0.2
⇒ l × b = 13824 / 0.2
⇒ l × b = 69120
l / b = 6 / 5
⇒ l = 6k, b = 5k
⇒ l × b = 6k × 5k
⇒ 69120 = 30k2
⇒ k2 = 69120 / 30
⇒ k2 = 2304
⇒ k = √2304
⇒ k = 48
l = 6k = 6 × 48 = 288 cm
b = 5k = 5 × 48 = 240 cm
l + b = 288 + 240
⇒ l + b = 528 cm
The correct answer is option 2 (528 cm).
Two Figures Question 3:
A solid metallic right circular cylinder of radius 3 inches and height 8 inches is melted and recast into identical hemispheres whose base is equal to the base of the cylinder. Then the number of hemispheres thus formed is
Answer (Detailed Solution Below)
Two Figures Question 3 Detailed Solution
Given:
Radius of cylinder = 3 inches.
Height of cylinder = 8 inches.
Formula Used:
Volume of cylinder = πr2h
Volume of hemisphere = (2/3)πr3
Calculation:
Volume of cylinder = π × 32 × 8
⇒ Volume of cylinder = π × 9 × 8
⇒ Volume of cylinder = 72π cubic inches
Volume of hemisphere = (2/3)π × 33
⇒ Volume of hemisphere = (2/3)π × 27
⇒ Volume of hemisphere = 18π cubic inches
Number of hemispheres = Volume of cylinder / Volume of hemisphere
⇒ Number of hemispheres = 72π / 18π
⇒ Number of hemispheres = 4
The number of hemispheres formed is 4.
Two Figures Question 4:
From a wooden cube of edge 14 cm, a right circular cone of maximum volume is carved out. If the volume of the removed portion of the cube is V, then 3 V (in cu.cm) is
Answer (Detailed Solution Below)
Two Figures Question 4 Detailed Solution
Given:
Edge of the wooden cube = 14 cm.
Formula Used:
Volume of the cube = Edge3
Volume of the cone = (1/3) × π × r2 × h
Maximum volume of the cone when it is carved from a cube: r = h/2, where h = edge of the cube.
Volume of the removed portion = Volume of the cube - Volume of the cone.
Calculation:
Volume of the cube = Edge3
⇒ Volume of the cube = 143
⇒ Volume of the cube = 2744 cm3
Radius (r) of the cone = h / 2 = 14 / 2 = 7 cm
Height (h) of the cone = 14 cm
Volume of the cone = (1/3) × π × r2 × h
⇒ Volume of the cone = (1/3) × 22/7 × 72 × 14
⇒ Volume of the cone = (1/3) × 22 × 49 × 2
⇒ Volume of the cone = (1/3) × 2156
⇒ Volume of the cone = 718.67 cm3
Volume of the removed portion (V) = Volume of the cube - Volume of the cone
⇒ V = 2744 - 718.67
⇒ V = 2025.33 cm3
3V = 3 × 2025.33
⇒ 3V = 6076 cm3
The value of 3V is 6076 cm3.
Two Figures Question 5:
The part above the cone filled with ice-cream looks as a hemisphere surmounted on its top. If the height and the radius of the cone are 13 cm and 4 cm respectively, then the volume (in cu.cm) of the ice-cream is
Answer (Detailed Solution Below)
Two Figures Question 5 Detailed Solution
Given:
Height of cone = 13 cm
Radius of cone = 4 cm
Radius of hemisphere = 4 cm
Formula Used:
Volume of cone = (1/3) × π × r2 × h
Volume of hemisphere = (2/3) × π × r3
Total volume = Volume of cone + Volume of hemisphere
Calculation:
Volume of cone = (1/3) × π × 42 × 13
⇒ Volume of cone = (1/3) × π × 16 × 13
⇒ Volume of cone = (208/3)π
Volume of hemisphere = (2/3) × π × 43
⇒ Volume of hemisphere = (2/3) × π × 64
⇒ Volume of hemisphere = (128/3)π
Total volume = (208/3)π + (128/3)π
⇒ Total volume = (336/3)π
⇒ Total volume = 112π
Using π ≈ 3.14:
Total volume = 112 × 3.14
⇒ Total volume = 351.68 cm3
The volume of the ice-cream is approximately 352 cm3.
Top Two Figures MCQ Objective Questions
A sphere of radius 42 cm is melted and recast into a cylindrical wire of radius 21 cm. Find the length of the wire.
Answer (Detailed Solution Below)
Two Figures Question 6 Detailed Solution
Download Solution PDFGiven:
Radius of Sphere = 42 cm
Radius of wire = 21 cm
Formula:
Volume of cylinder = πr2h
Volume of sphere = [4/3]πr3
Calculation:
Let length of the wire be x, then
According to the question
π × 21 × 21 × x = [4/3] × π × 42 × 42 × 42 [As volume will remain constant]
⇒ x = (4 × 42 × 42 × 42)/(21 × 21 × 3)
⇒ x = 224 cm
∴ The length of the wire is 224 cm
A sphere of radius 12 cm is melted and re-casted into a right circular cone of height 12 cm. The radius of the cone is.
Answer (Detailed Solution Below)
Two Figures Question 7 Detailed Solution
Download Solution PDFGiven:
Radius of sphere = 12 cm
Height of cone = 12 cm
Formula:
Volume of cone = (1/3) × πr2h
Volume of sphere = (4/3) × πr3
Calculation:
Let radius of the cone be r cm
According to the question
(1/3) × π × r2 × 12 = (4/3) × π × 12 × 12 × 12
⇒ r2 = 12 × 12 × 4
⇒ r = 12 × 2
∴ r = 24 cmTwo cubes each of volume 729 cm3 are joined end to end. The total surface area of the resulting cuboid is:
Answer (Detailed Solution Below)
Two Figures Question 8 Detailed Solution
Download Solution PDFFORMULA USED:
Total surface area of cuboid = 2lb + 2bh + 2hl or 2(lb + bh + hl)
Here l, b and h are length, breadth and height.
Volume of cube = a3
CALCULATION:
a3 = 729
⇒ a = 9 cm.
Length of cuboid = 9 + 9 = 18 cm.
Breadth = 9 cm
Height = 9 cm
Total surface area of cuboid = 2 (18 × 9 + 9 × 9 + 9 × 18) = 810 cm2
∴ Total surface area of the cuboid is 810 cm2
A solid sphere of radius 3 cm is melted to form a right circular cone such that the height of the cone is half the radius of the cone. Find the radius of the cone.
Answer (Detailed Solution Below)
Two Figures Question 9 Detailed Solution
Download Solution PDFGIVEN:
Radius of sphere = 3 cm
Height of the cone = Half of the radius of the cone
CONCEPT:
Volume of Sphere = Volume of Cone
FORMULA USED:
Volume of Sphere = 4/3 × πR3
Volume of Cone = 1/3 × πr2h
CALCULATION:
Suppose the height and radius of the cone are ‘h’ and ‘r’ respectively.
∴ h = r/2
Now,
Applying the formula:
\(\frac{4}{3} \times \pi \times 3 \times 3 \times 3 = \frac{1}{3} \times \pi \times r \times r \times h\)
Put h = r/2
⇒ \(\frac{4}{3} \times 3 \times 3 \times 3 = \frac{1}{3} \times r \times r \times \frac{r}{2}\)
⇒ r3 = 216
⇒ r = 6
∴ Radius of the cone = 6 cm.
A glass cylinder with diameter 20 cm has water to a height of 9 cm. A metal cube of 8 cm edge is immersed in it completely. Calculate the height (correct to I decimal place) by which the water will rise in the cylinder (by taking π = 3.142).
Answer (Detailed Solution Below)
Two Figures Question 10 Detailed Solution
Download Solution PDFGiven:
A glass cylinder with diameter 20 cm has water to a height of 9 cm. A metal cube of 8 cm edge is immersed in it completely.
Formula used:
Volume of cylinder = Πr2h
Volume of cube = a3
Calculation:
Diameter of cylinder = 20 cm
⇒ Radius of cylinder = 10 cm
Now, Volume of water displaced (because of which water will rise in cylinder) = Volume of cube
∴ πr2h = a3
⇒ 3.142 × 10 × 10 × h = 83
⇒ 3142 × 1/10 × h = 512
⇒ h = 5120/3142
⇒ h = 1.62 cm ~ 1.6 cm
The height of a solid cylinder is 30 cm and the diameter of its base is 10 cm. Two identical conical holes each of radius 5 cm and height 12 cm are drilled out. What is the surface area (in cm2) of the remaining solid?
Answer (Detailed Solution Below)
Two Figures Question 11 Detailed Solution
Download Solution PDFGiven:
Height of cylinder = 30 cm
Radius of cylinder = 5 cm
Height of cone = 12 cm
Radius of cone = 5 cm
Formula used:
The surface area of cylinder = 2πrh
The surface area of cone = πrl
l2 = h2 + r2
Where,
l = slant height of the cone
h = height
r = radius
Calculation:
l2 = h2 + r2
⇒ l2 = 122 + 52
⇒ l2 = 144 + 25
⇒ l = 13 cm
The surface area of the remaining figure = surface area of cylinder + 2 × Curved surface area of the cone
⇒ 2πrh + 2πrl
⇒ 2πr(h + l)
⇒ 2π × 5(30 + 13)
⇒ 430π
∴ The surface area of the remaining solid is 430π.
Additional Information
When the cones are drilled out then the volume of a cylinder is decreasing. But the surface area will increase. Surface area means the area which we can touch. When the cones drilled out then we can touch both outer and the inner surface. So we have to add both surface areas.
A few lead spheres of diameter 6 cm are dropped into a cylindrical beaker containing some water such that they are fully submerged. If the diameter of the beaker is 9 cm and the water level has risen by 32 cm, find the number of lead spheres dropped into the beaker.
Answer (Detailed Solution Below)
Two Figures Question 12 Detailed Solution
Download Solution PDFGiven:
The radius of the sphere is 3 cm
Radius & height of cylinder are 4.5 cm & 32cm
Formula used:
The volume of sphere = 4/3πr³
The volume of cylinder = πr²h
Calculations:
The volume of sphere inserted = volume of water risen
Radius of sphere = 6/2= 3 cm
Radius of cylinder = 9/2 = 4.5
So,
(4/3)π × (3)³ × Number of sphere = π (4.5) × (4.5) × 32
⇒ Number of sphere = 18
∴ The correct choice is option 2.
A solid metallic cube of side 20cm is melted and recast into a cuboid of length 40 cm and breadth 40 cm. What is the length (in cm) of the body diagonal of the cuboid?
Answer (Detailed Solution Below)
Two Figures Question 13 Detailed Solution
Download Solution PDFGiven:
Cube of side 20 cm is recast into a cuboid of length and breadth 40cm respectively.
Formulas used:
Volume of cube = (side)3
Volume of cuboid = l × b × h
Diagonal of cuboid = √ l2 + b2 + h2
Calculation:
Volume of cube = Volume of cuboid
⇒ 20 × 20 × 20 = 40 × 40 × h
⇒ h = (20 × 20 × 20) ÷ (40 × 40)
⇒ h = 5 cm
Diagonal of the cuboid = √ 402 + 402 + 52
⇒ √ 1600 + 1600 + 25 = √ 3225
⇒ 5√129 cm
∴ The diagonal of the cuboid = 5√129 cm
Find the volume of the largest sphere that can be carved out of a cube of side 21 cm.
Answer (Detailed Solution Below)
Two Figures Question 14 Detailed Solution
Download Solution PDFGiven:
Side of cube = 21 cm
Formulas used:
Volume of sphere = 4/3πr3
Calculation:
The largest sphere that can be carved out of a cube of side 21 cm will have the diameter equal to 21 cm.
Radius of sphere = 21/2 cm
Volume of sphere = 4/3 × 22/7 × 21/2 × 21/2 × 21/2
⇒ 11 × 21 × 21
⇒ 4851 cm3
∴ The required result is 4851 cm3.
From a solid right circular cylinder with height 9 cm and base radius 5 cm, a right circular cone of the same height and same base is removed. The volume (in cm3) of the remaining solid is
Answer (Detailed Solution Below)
Two Figures Question 15 Detailed Solution
Download Solution PDFGiven :
Radius of right circular cone and cylinder(r) = 5 cm
Height of right circular cone and cylinder(h) = 9 cm
Formula used :
Volume of right circular cylinder(V1) = πr2h
The volume of the right circular cone (V2) = \(\dfrac{1}{3}\)πr2h
Calculation :
Volume of remaining solid = V1 - V2
⇒ \(π× 5^2× 9 - \dfrac{1}{3}× π× 5^2× 9\)
⇒ π × 52 × 9 (1 - \(\dfrac{1}{3}\))
⇒ \(\dfrac{2}{3}× π× 5^2× 9\)
⇒ 150π
∴ The answer is 150π .