✦ NCF 2023 ALIGNED • CBSE CLASS X ✦
Chapter 12: Surface Areas and Volumes
Assertion-Reason Question Bank
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Assertion (A)
If the surface area of a sphere is \(616\text{ cm}^2\), its diameter is \(14\text{ cm}\).
Reason (R)
Surface area \(4\pi r^2 = 616 \implies 4 \times \frac{22}{7} \times r^2 = 616 \implies r^2 = 49 \implies r = 7\text{ cm} \implies d = 14\text{ cm}\).
Correct Option: (A)
\(r = 7\text{ cm} \implies d = 14\text{ cm}\). Reason explains Assertion.
\(r = 7\text{ cm} \implies d = 14\text{ cm}\). Reason explains Assertion.
Correct Answer: (A)
\(r = 7\text{ cm} \implies d = 14\text{ cm}\). Reason explains Assertion.
\(r = 7\text{ cm} \implies d = 14\text{ cm}\). Reason explains Assertion.
Assertion (A)
Ratio of volumes of two spheres is \(8:27\), then ratio of their surface areas is \(4:9\).
Reason (R)
Ratio of volumes \(V_1/V_2 = r_1^3/r_2^3 = 8/27 \implies r_1/r_2 = 2/3\). Ratio of surface areas \(= r_1^2/r_2^2 = (2/3)^2 = 4/9\).
Correct Option: (B)
Both A and R are true area/volume scaling facts.
Both A and R are true area/volume scaling facts.
Correct Answer: (B)
Both A and R are true.
Both A and R are true.
Assertion (A)
Slant height of cone with \(r=3\text{ cm}, h=4\text{ cm}\) is \(7\text{ cm}\).
Reason (R)
Slant height \(l = \sqrt{r^2 + h^2} = \sqrt{3^2 + 4^2} = 5\text{ cm}\).
Correct Option: (D)
Assertion (A) is false (it is \(5\text{ cm}\)). Reason (R) is true.
Assertion (A) is false (it is \(5\text{ cm}\)). Reason (R) is true.
Correct Answer: (D)
Assertion false. Reason (R) is true.
Assertion false. Reason (R) is true.
Assertion (A)
When a cone is cut by a plane parallel to base, the top portion is called frustum.
Reason (R)
The bottom portion containing the base is called frustum, while top portion is a smaller cone.
Correct Option: (D)
Assertion (A) is false (bottom portion is frustum). Reason (R) is true.
Assertion (A) is false (bottom portion is frustum). Reason (R) is true.
Correct Answer: (D)
Assertion false. Reason (R) is true.
Assertion false. Reason (R) is true.
Assertion (A)
Volume of a sphere of radius \(r\) is \(\frac{4}{3}\pi r^3\).
Reason (R)
Surface area of a sphere of radius \(r\) is \(4\pi r^2\).
Correct Option: (B)
Both A and R are true basic formulas for a sphere.
Both A and R are true basic formulas for a sphere.
Correct Answer: (B)
Both A and R are true.
Both A and R are true.
Assertion (A)
If two cubes each of volume \(64\text{ cm}^3\) are joined end to end, the surface area of the resulting cuboid is \(160\text{ cm}^2\).
Reason (R)
Side of cube \(a = \sqrt[3]{64} = 4\text{ cm}\). Resulting cuboid has length \(l = 8\text{ cm}\), breadth \(b = 4\text{ cm}\), height \(h = 4\text{ cm}\). Surface area \(= 2(lb + bh + hl) = 2(32 + 16 + 32) = 2(80) = 160\text{ cm}^2\).
Correct Option: (A)
Cuboid dimensions \(8, 4, 4\). \(SA = 2(32+16+32) = 160\text{ cm}^2\). Reason proves Assertion.
Cuboid dimensions \(8, 4, 4\). \(SA = 2(32+16+32) = 160\text{ cm}^2\). Reason proves Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
Total surface area of a solid cylinder of radius \(r\) and height \(h\) is \(2\pi r(h + r)\).
Reason (R)
TSA = CSA + \(2 \times\) Base Area \(= 2\pi r h + 2\pi r^2 = 2\pi r(h + r)\).
Correct Option: (B)
Both A and R are true cylinder formulas.
Both A and R are true cylinder formulas.
Correct Answer: (B)
Both A and R are true.
Both A and R are true.
Assertion (A)
If a solid cone is melted and recast into a solid cylinder of same base radius, height of cylinder is \(1/3\) of height of cone.
Reason (R)
Volume of cone \(= \frac{1}{3}\pi r^2 h_{\text{cone}}\) and Volume of cylinder \(= \pi r^2 h_{\text{cyl}}\). Equating gives \(h_{\text{cyl}} = \frac{1}{3}h_{\text{cone}}\).
Correct Option: (B)
Both A and R are true volume conservation proofs.
Both A and R are true volume conservation proofs.
Correct Answer: (B)
Both A and R are true.
Both A and R are true.
Assertion (A)
Curved surface area of a cone of base radius \(r\) and slant height \(l\) is \(\pi r l\).
Reason (R)
Total surface area of cone is \(\pi r(l + r)\).
Correct Option: (B)
Both A and R are true cone surface area formulas.
Both A and R are true cone surface area formulas.
Correct Answer: (B)
Both A and R are true.
Both A and R are true.
Assertion (A)
Volume of a cone of height \(h\) and radius \(r\) is \(\frac{1}{3}\pi r^2 h\).
Reason (R)
Volume of a cone is \(3\) times volume of cylinder of same base radius and height.
Correct Option: (C)
Assertion (A) is true. Reason (R) is false because volume of cone is \(1/3\) (one-third) of cylinder, not \(3\) times.
Assertion (A) is true. Reason (R) is false because volume of cone is \(1/3\) (one-third) of cylinder, not \(3\) times.
Correct Answer: (C)
Assertion (A) is true. Reason (R) is false.
Assertion (A) is true. Reason (R) is false.
Assertion (A)
The slant height of a right circular cone of height \(h\) and radius \(r\) is \(l = \sqrt{r^2 + h^2}\).
Reason (R)
The vertical height, radius, and slant height form a right-angled triangle, so by Pythagoras theorem \(l^2 = r^2 + h^2 \implies l = \sqrt{r^2 + h^2}\).
Correct Option: (A)
Pythagoras theorem on right triangle with legs \(r, h\) and hypotenuse \(l\). Reason explains Assertion.
Pythagoras theorem on right triangle with legs \(r, h\) and hypotenuse \(l\). Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
If radius of a cylinder is \(r\) and height \(h\), its total surface area is \(2\pi r h\).
Reason (R)
Curved surface area is \(2\pi r h\), whereas total surface area is \(2\pi r(h + r)\).
Correct Option: (D)
Assertion (A) is false (\(2\pi rh\) is CSA, not TSA). Reason (R) is true.
Assertion (A) is false (\(2\pi rh\) is CSA, not TSA). Reason (R) is true.
Correct Answer: (D)
Assertion false. Reason (R) is true.
Assertion false. Reason (R) is true.
Assertion (A)
Volume of a cuboid of dimensions \(a, b, c\) is \(2(ab + bc + ca)\).
Reason (R)
Volume of cuboid is \(a \times b \times c = abc\), while \(2(ab + bc + ca)\) is its total surface area.
Correct Option: (D)
Assertion (A) is false (\(2(ab+bc+ca)\) is surface area). Reason (R) is true.
Assertion (A) is false (\(2(ab+bc+ca)\) is surface area). Reason (R) is true.
Correct Answer: (D)
Assertion false. Reason (R) is true.
Assertion false. Reason (R) is true.
Assertion (A)
The maximum length of a rod that can be placed in a room of dimensions \(10\text{ m} \times 10\text{ m} \times 5\text{ m}\) is \(15\text{ m}\).
Reason (R)
The maximum length corresponds to the space diagonal of cuboid \(= \sqrt{l^2 + b^2 + h^2} = \sqrt{10^2 + 10^2 + 5^2} = \sqrt{100 + 100 + 25} = \sqrt{225} = 15\text{ m}\).
Correct Option: (A)
Diagonal formula \(\sqrt{100+100+25} = 15\text{ m}\). Reason explains Assertion.
Diagonal formula \(\sqrt{100+100+25} = 15\text{ m}\). Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
When a solid is remoulded into another shape, its volume remains unchanged.
Reason (R)
When a solid is remoulded, its total surface area always remains unchanged.
Correct Option: (C)
Assertion (A) is true (conservation of volume). Reason (R) is false because total surface area changes when shape changes.
Assertion (A) is true (conservation of volume). Reason (R) is false because total surface area changes when shape changes.
Correct Answer: (C)
Assertion (A) is true. Reason (R) is false.
Assertion (A) is true. Reason (R) is false.
Assertion (A)
Volume of a cylinder of radius \(r\) and height \(h\) is \(\pi r^2 h\).
Reason (R)
Curved surface area of a cylinder of radius \(r\) and height \(h\) is \(2\pi r h\).
Correct Option: (B)
Both A and R are true formulas for a cylinder.
Both A and R are true formulas for a cylinder.
Correct Answer: (B)
Both A and R are true.
Both A and R are true.
Assertion (A)
If radius of a sphere is increased by \(10\%\), its volume increases by \(33.1\%\).
Reason (R)
Volume of sphere is directly proportional to square of radius.
Correct Option: (C)
Assertion (A) is true (\(V' = (1.1)^3 V = 1.331 V \implies 33.1\%\) increase). Reason (R) is false because volume is proportional to CUBE of radius (\(r^3\)), not square.
Assertion (A) is true (\(V' = (1.1)^3 V = 1.331 V \implies 33.1\%\) increase). Reason (R) is false because volume is proportional to CUBE of radius (\(r^3\)), not square.
Correct Answer: (C)
Assertion (A) is true. Reason (R) is false.
Assertion (A) is true. Reason (R) is false.
Assertion (A)
A wooden toy is in the form of a cone surmounted on a hemisphere of same radius \(r\). Total surface area of toy is \(\pi r(l + 2r)\).
Reason (R)
Total surface area of combined solid = Curved surface area of cone + Curved surface area of hemisphere \(= \pi r l + 2\pi r^2 = \pi r(l + 2r)\).
Correct Option: (A)
Combined CSA \(= \pi r l + 2\pi r^2 = \pi r(l+2r)\). Reason explains Assertion.
Combined CSA \(= \pi r l + 2\pi r^2 = \pi r(l+2r)\). Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
Total surface area of a hemisphere of radius \(r\) is \(2\pi r^2\).
Reason (R)
Curved surface area of hemisphere is \(2\pi r^2\) and total surface area is \(3\pi r^2\).
Correct Option: (D)
Assertion (A) is false (\(2\pi r^2\) is CSA, TSA is \(3\pi r^2\)). Reason (R) is true.
Assertion (A) is false (\(2\pi r^2\) is CSA, TSA is \(3\pi r^2\)). Reason (R) is true.
Correct Answer: (D)
Assertion false. Reason (R) is true.
Assertion false. Reason (R) is true.
Assertion (A)
Volume of a sphere of radius \(3\text{ cm}\) is \(36\text{ cm}^3\).
Reason (R)
Volume \(V = \frac{4}{3}\pi (3)^3 = 36\pi\text{ cm}^3\).
Correct Option: (D)
Assertion (A) is false (it is \(36\pi\text{ cm}^3\), missing \(\pi\)). Reason (R) is true.
Assertion (A) is false (it is \(36\pi\text{ cm}^3\), missing \(\pi\)). Reason (R) is true.
Correct Answer: (D)
Assertion false. Reason (R) is true.
Assertion false. Reason (R) is true.
Assertion (A)
If radius of base of cylinder is \(r\) and height is \(h = 2r\), its total surface area is \(6\pi r^2\).
Reason (R)
TSA of cylinder with \(h = 2r\) is \(3\pi r^2\).
Correct Option: (C)
Assertion (A) is true (\(2\pi r(2r + r) = 6\pi r^2\)). Reason (R) is false because it is \(6\pi r^2\), not \(3\pi r^2\).
Assertion (A) is true (\(2\pi r(2r + r) = 6\pi r^2\)). Reason (R) is false because it is \(6\pi r^2\), not \(3\pi r^2\).
Correct Answer: (C)
Assertion (A) is true. Reason (R) is false.
Assertion (A) is true. Reason (R) is false.
Assertion (A)
When a solid metallic sphere of radius \(R\) is melted and recast into smaller spheres of radius \(r\), the number of smaller spheres is \(\frac{R^3}{r^3}\).
Reason (R)
Number of spheres \(= \frac{\text{Volume of large sphere}}{\text{Volume of small sphere}} = \frac{\frac{4}{3}\pi R^3}{\frac{4}{3}\pi r^3} = \frac{R^3}{r^3}\).
Correct Option: (A)
Ratio of volumes gives number of spheres. Reason explains Assertion.
Ratio of volumes gives number of spheres. Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
Total surface area of a solid hemisphere of radius \(7\text{ cm}\) is \(462\text{ cm}^2\).
Reason (R)
Total surface area of hemisphere formula is \(2\pi r^2\).
Correct Option: (C)
Assertion (A) is true (\(3 \times (22/7) \times 49 = 462\text{ cm}^2\)). Reason (R) is false because TSA formula is \(3\pi r^2\), whereas \(2\pi r^2\) is CSA.
Assertion (A) is true (\(3 \times (22/7) \times 49 = 462\text{ cm}^2\)). Reason (R) is false because TSA formula is \(3\pi r^2\), whereas \(2\pi r^2\) is CSA.
Correct Answer: (C)
Assertion (A) is true. Reason (R) is false.
Assertion (A) is true. Reason (R) is false.
Assertion (A)
If radius of base of a cylinder is doubled and height is halved, its volume is doubled.
Reason (R)
New volume \(V' = \pi (2r)^2 (h/2) = 2 \pi r^2 h = 2V\).
Correct Option: (B)
Both A and R are true algebraic proofs.
Both A and R are true algebraic proofs.
Correct Answer: (B)
Both A and R are true.
Both A and R are true.
Assertion (A)
Surface area of a sphere of radius \(r\) is \(\frac{4}{3}\pi r^2\).
Reason (R)
Surface area of a sphere of radius \(r\) is \(4\pi r^2\).
Correct Option: (D)
Assertion (A) is false (has wrong coefficient \(4/3\)). Reason (R) is true.
Assertion (A) is false (has wrong coefficient \(4/3\)). Reason (R) is true.
Correct Answer: (D)
Assertion false. Reason (R) is true.
Assertion false. Reason (R) is true.
Assertion (A)
If height of a cone is \(24\text{ cm}\) and radius is \(7\text{ cm}\), its slant height is \(25\text{ cm}\).
Reason (R)
Slant height formula is \(l = r + h\).
Correct Option: (C)
Assertion (A) is true (\(l = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = 25\text{ cm}\)). Reason (R) is false because \(l = \sqrt{r^2+h^2}\), not \(r+h\).
Assertion (A) is true (\(l = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = 25\text{ cm}\)). Reason (R) is false because \(l = \sqrt{r^2+h^2}\), not \(r+h\).
Correct Answer: (C)
Assertion (A) is true. Reason (R) is false.
Assertion (A) is true. Reason (R) is false.
Assertion (A)
If side of a cube is doubled, its volume becomes \(4\) times.
Reason (R)
New volume \(V' = (2a)^3 = 8a^3 = 8V\), so volume becomes \(8\) times.
Correct Option: (D)
Assertion (A) is false (it becomes \(8\) times). Reason (R) is true.
Assertion (A) is false (it becomes \(8\) times). Reason (R) is true.
Correct Answer: (D)
Assertion false. Reason (R) is true.
Assertion false. Reason (R) is true.
Assertion (A)
Total surface area of a solid hemisphere of radius \(r\) is \(3\pi r^2\).
Reason (R)
Total surface area = Curved surface area + Base area \(= 2\pi r^2 + \pi r^2 = 3\pi r^2\).
Correct Option: (A)
\(2\pi r^2 + \pi r^2 = 3\pi r^2\). Reason proves Assertion.
\(2\pi r^2 + \pi r^2 = 3\pi r^2\). Reason proves Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
If edge of a cube is \(a\), its total surface area is \(6a^2\).
Reason (R)
A cube has \(4\) square faces.
Correct Option: (C)
Assertion (A) is true (\(6a^2\)). Reason (R) is false because a cube has \(6\) faces, not \(4\).
Assertion (A) is true (\(6a^2\)). Reason (R) is false because a cube has \(6\) faces, not \(4\).
Correct Answer: (C)
Assertion (A) is true. Reason (R) is false.
Assertion (A) is true. Reason (R) is false.
Assertion (A)
A hollow sphere of internal and external radii \(r\) and \(R\) has volume \(\frac{4}{3}\pi (R^3 - r^3)\).
Reason (R)
Volume of hollow sphere = External volume - Internal volume.
Correct Option: (B)
Both A and R are true hollow sphere facts.
Both A and R are true hollow sphere facts.
Correct Answer: (B)
Both A and R are true.
Both A and R are true.