CBSE Class 10 Mathematics Surface Areas and Volumes MCQs Set H

Question. If the radius of the sphere is increased by 100%, the volume of the corresponding sphere is increased by
(a) 200%
(b) 500%
(c) 700%
(d) 800%
Answer: C
When the radius is increased by 100%, the corresponding volume becomes 800% and thus increase is 700%.

Question. A sphere is melted and half of the melted liquid is used to form 11 identical cubes, whereas the remaining half is used to form 7 identical smaller spheres. The ratio of the side of the cube to the radius of the new small sphere is
(a) \((\frac{4}{3})^{1/3}\)
(b) \((\frac{8}{3})^{1/3}\)
(c) \((3)^{1/3}\)
(d) 2
Answer: B
As per the given conditions,
\(11a^3 = 7 \times \frac{4}{3} \times \pi \times r^3\)
\(\frac{a}{r} = (\frac{8}{3})^{1/3}\)

Question. The base radii of a cone and a cylinder are equal. If their curved surface areas are also equal, then the ratio of the slant height of the cone to the height of the cylinder is
(a) 2 : 1
(b) 1 : 2
(c) 1 : 3
(d) 3 : 1
Answer: A
\(\pi rl = 2\pi rh\)
\(\frac{l}{h} = \frac{2}{1}\)

Question. A slab of ice 8 inches in length, 11 inches in breadth, and 2 inches thick was melted and resolidified in the form of a rod of 8 inches diameter. The length of such a rod, in inches, is nearest to
(a) 3
(b) 3.5
(c) 4
(d) 4.5
Answer: B
Volume of the given ice cuboid = \(8 \times 11 \times 2 = 176\)
Let the length of the required rod be \(l\).
\(\pi \frac{8^2}{4} l = 176\)
\(l = 3.5\) inches

Question. If the perimeter of one face of a cube is 20 cm, then its surface area is
(a) \(120 \text{ cm}^2\)
(b) \(150 \text{ cm}^2\)
(c) \(125 \text{ cm}^2\)
(d) \(400 \text{ cm}^2\)
Answer: B
Edge of cube = \(\frac{20}{4} \text{ cm} = 5 \text{ cm}\)
Surface area = \(6 \times 5^2 \text{ cm}^2 = 150 \text{ cm}^2\)

Question. Ratio of lateral surface areas of two cylinders with equal height is
(a) 1 : 2
(b) \(H : h\)
(c) \(R : r\)
(d) None of these
Answer: C
\(2\pi Rh : 2\pi rh = R : r\)

Question. Ratio of volumes of two cylinders with equal height is
(a) \(H : h\)
(b) \(R : r\)
(c) \(R^2 : r^2\)
(d) None of these
Answer: C
\(\pi R^2 h : \pi r^2 h = R^2 : r^2\)

Question. Ratio of volumes of two cones with same radii is
(a) \(h_1 : h_2\)
(b) \(s_1 : s_2\)
(c) \(r_1 : r_2\)
(d) None of these
Answer: A
\(\frac{1}{3}\pi r_1^2 h_1 : \frac{1}{3}\pi r_2^2 h_2\)
\(\frac{1}{3}\pi r_1^2 h_1 : \frac{1}{3}\pi r_1^2 h_2\) (\(r_1 = r_2\))
\(h_1 : h_2\)

Question. The diameter of hollow cone is equal to the diameter of a spherical ball. If the ball is placed at the base of the cone, what portion of the ball will be outside the cone?
(a) 50%
(b) less than 50%
(c) more then 50%
(d) 100%
Answer: C

Question. If a solid of one shape is converted to another, then the volume of the new solid.
(a) remains same
(b) increases
(c) decreases
(d) can’t say
Answer: A

FILL IN THE BLANK

Question. The volume and surface area of a sphere are numerically equal, then the radius of sphere is .......... units.
Answer: 3

Question. In a right circular cone, the cross-section made by a plane parallel to the base is a ..........
Answer: Circle

Question. Volume of the frustum of cone is ..........
Answer: \( \frac{1}{3}\pi h(r_1^2 + r_2^2 + r_1 r_2) \)

Question. Total curved surface area of the frustum is ..........
Answer: \( \pi(r_1 + r_2)l + \pi r_1^2 + \pi r_2^2 \)

Question. The TSA, CSA stand for .......... and .......... respectively.
Answer: Total surface area, Curved surface area.

Question. A shuttle cock used for playing badminton has the shape of the combination of .......... of cone and hemisphere.
Answer: Frustum

Question. .......... is measured in square units.
Answer: Area

Question. In the gilli-danda game, the shape of a gilli is a combination of two cones and ..........
Answer: Cylinder

Question. The volume of a cube with diagonal d is ..........
Answer: \( \frac{d^3}{3\sqrt{3}} \) cu units.

Question. .......... is measured in cubic units.
Answer: Volume

Question. A cube is a special type of ..........
Answer: Cuboid

Question. The total surface area of a solid hemisphere having radius r is ..........
Answer: \( 3\pi r^2 \)

TRUE/FALSE

DIRECTION : Read the following statements and write your answer as true or false.

Question. Two identical solid cubes of side ‘a’ are joined end to end. Then the total surface area of the resulting cuboid is \( 12a^2 \).
Answer: False

Question. If the base area and the volume of a cone are numerically equal, then its height is 3 units.
Answer: True

Question. A circle is revolved about any of its diameters, a hollow sphere is generated.
Answer: True

Question. If the curved surface of a right circular cylinder is \( 1760 \text{ cm}^2 \) and its radius is 21 cm, then its height is \( \frac{80}{3} \text{ cm} \).
Answer: False

Question. If a right circular cone and a cylinder have equal circles as their base and have equal heights, then the ratio of their volume is 2 : 3.
Answer: False

Question. The curved surface area of a frustum of a cone is \( \pi(r_1 + r_2)l \), where \( l = \sqrt{h^2 + (r_1 - r_2)^2} \), \( r_1 \) and \( r_2 \) are the radii of the two ends of the frustum and h is the vertical height.
Answer: False

Question. Volume of cone is \( \frac{1}{3}\pi r^2 h \).
Answer: True

Question. All faces of a cuboid must be rectangular.
Answer: False

Question. If the total surface area of a cube is \( \frac{50}{3} m^2 \), then its side is (5/3) m.
Answer: True

Question. The volume of cylinder is \( \pi r^3 h \).
Answer: False

Question. Surface area of a square pyramid is \( S = s^2 + 2sl \).
Answer: True

Question. If we double the radius of a hemisphere, its surface area will also be doubled.
Answer: False

MATCHING QUESTIONS

DIRECTION : Each question contains statements given in two columns which have to be matched. Statements (A, B, C, D) in Column-I have to be matched with statements (p, q, r, s) in Column-II.

Question. From a solid cylinder of height 2.4 cm and diameter 1.4 cm, a conical cavity of the same height and some diameter is hollowed out then match the column.
Column-I
(A) Area of bottom of cylinder
(B) Outer curved surface area
(C) Curved area of conical cavity
(D) Total surface area
Column-II
(p) 10.56
(q) 1.54
(r) 5.5
(s) 17.6
Answer: (A) - q, (B) - p, (C) - r, (D) - s

Question. Match the Following:
Column I
(A) Solid
(B) Area
(C) Volume
(D) Cube
(E) Cuboid
(F) Cylinder
(G) Cone
(H) Sphere
(I) Frustum of a cone
Column II
(p) a set of points in the space which are at equal distances from a fixed point.
(q) a mathematical term used for a rigid three-dimensional shape.
(r) quantitative measure of a plane or curved surface.
(s) a solid whose faces are rectangles.
(t) a solid whose faces are all congruent squares.
(u) a solid with a circular base tapering to a point.
(v) a solid whose cross-sections are all circles of the same radii.
(w) a solid which is obtained by removing the upper portion of the cone by a plane parallel to its base.
(x) amount of space occupied by a solid.
Answer: (A) - q, (B) - r, (C) - x, (D) - t, (E) - s, (F) - v, (G) - u, (H) - p, (I) - w

Question. Match the following:
Column-I
(A) Solids
(B) Road rollers
(C) Ice-cream cone
(D) Volleyball
Column-II
(p) Right circular cone
(q) Sphere
(r) Cylinder
(s) Cuboid
(t) Cube
Answer: (A) - (p, q, r, s, t), (B) - r, (C) - p, (D) - q

ASSERTION AND REASON

DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.

Question. Assertion : The radii of two cones are in the ratio 2 : 3 and their volumes in the ratio 1 : 3. Then the ratio of their heights is 3 : 2.
Reason : Volume of the cone \( = \frac{1}{3}\pi r^2 h \).
(a) Choice (a)
(b) Choice (b)
(c) Choice (c)
(d) Choice (d)
Answer: D
Assertion (A) is false but reason (R) is true. We have, ratio of volume \( = \frac{\frac{1}{3}\pi \times (2x)^2 \times h_1}{\frac{1}{3}\pi \times (3x)^2 \times h_2} = \frac{1}{3} \implies \frac{4}{9} \times \frac{h_1}{h_2} = \frac{1}{3} \implies \frac{h_1}{h_2} = \frac{3}{4} \). \( h_1 : h_2 = 3 : 4 \).

Question. Assertion : If a ball is in the shape of a sphere has a surface area of \( 221.76 \text{ cm}^2 \), then its diameter is 8.4 cm.
Reason : If the radius of the sphere be r , then surface area, \( S = 4\pi r^2 \), i.e. \( r = \frac{1}{2}\sqrt{\frac{S}{\pi}} \).
(a) Choice (a)
(b) Choice (b)
(c) Choice (c)
(d) Choice (d)
Answer: A
Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

Question. Assertion : The number of coins 1.75 cm in diameter and 2 mm thick is formed from a melted cuboid \( 10 \text{ cm} \times 5.5 \text{ cm} \times 3.5 \text{ cm} \) is 400.
Reason : Volume of a cylinder \( = \pi r^2 h \) cubic units and area of cuboid \( = (l \times b \times h) \) cubic units.
(a) Choice (a)
(b) Choice (b)
(c) Choice (c)
(d) Choice (d)
Answer: A
Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A). Number of coins \( = \frac{\text{volume of cuboid}}{\text{volume of one coin}} = \frac{10 \times 5.5 \times 3.5}{\pi \times (\frac{1.75}{2}) \times (\frac{1.75}{2}) \times 0.2} = \frac{10 \times 5.5 \times 3.5}{\frac{22}{7} \times \frac{1.75}{2} \times \frac{1.75}{2} \times 0.2} = 400 \)

Question. Assertion : No. of spherical balls that can be made out of a solid cube of lead whose edge is 44 cm, each ball being 4 cm. in diameter, is 2541.
Reason : Number of balls \( = \frac{\text{volume of lead}}{\text{Volume of one ball}} \).
(a) Choice (a)
(b) Choice (b)
(c) Choice (c)
(d) Choice (d)
Answer: C
Assertion (A) is true but reason (R) is false.

Question. Assertion : If the volumes of two spheres are in the ratio 27 : 8. Then their surface areas are in the ratio 3 : 2.
Reason : Volume of the sphere \( = \frac{4}{3}\pi r^3 \) and its surface area \( = 4\pi r^2 \).
(a) Choice (a)
(b) Choice (b)
(c) Choice (c)
(d) Choice (d)
Answer: D
Assertion (A) is false but reason (R) is true. We have, \( \frac{\frac{4}{3}\pi R^3}{\frac{4}{3}\pi r^3} = \frac{27}{8} \implies \frac{R^3}{r^3} = \frac{27}{8} \implies \frac{R}{r} = \frac{3}{2} \). Ratio of surface area \( = \frac{4\pi R^2}{4\pi r^2} = \frac{R^2}{r^2} = (\frac{3}{2})^2 = \frac{9}{4} \).