Surface Areas and Volumes – Assertion and Reason Questions

Q 1 :

Assertion (A): Two cubes each of edge length 10 cm are joined together. The total surface area of newly formed cuboid is 1200 ${cm}^{2}$ .

Reason (R): Area of each surface of a cube of side 10 cm is 100 ${cm}^{2}$ .

Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A)
Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of Assertion (A)
Assertion (A) is true but reason(R) is false.
Assertion (A) is false but reason(R) is true.

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(4) Assertion (A) is false but reason(R) is true

Q 2 :

Assertion (A): Length of diagonal of a cube of side 7 cm is $7 \sqrt{3} c m .$

Reason (R): Length of diagonal of a cube of edge $x u n i t = \frac{x}{\sqrt{3}} u n i t s .$

Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A)
Assertion (A) is true but Reason (R) is false
Assertion (A) is false but Reason (R) is true.

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(3)

We have side of a cube, a=7 cm.

$∴ Length of the diagonal of a cube = \sqrt{3} a = \sqrt{3} \times 7 = 7 \sqrt{3} cm ∴ Assertion (A) is true but Reason (R) is false . ∴ Option (c) is correct .$

Q 3 :

Assertion (A): Total surface area of a hemisphere of radius 2 cm is $4 π {cm}^{2}$

Reason (R): Total surface area of a hemisphere of radius $r = 3 π r^{2} .$

Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
Assertion (A) is true but Reason (R) is false.
Assertion (A) is false but Reason (R) is true

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(4)

We have,
Radius (r) of the hemisphere = 2 cm

$∴ Total surface area of the hemisphere = 3 π r^{2} = 3 π (2)^{2} = 12 π {cm}^{2}$

$∴ Assertion (A) is false but Reason (R) is true .$

Q 4 :

Assertion (A): A solid hemisphere of radius 4 cm is to be painted. Then the total cost of painting at the rate of $R s 2 p e r c m ² i s R s 96 π .$

Reason (R): Total surface area of a hemisphere $= 3 π r^{2} .$

Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
Assertion (A) is true but Reason (R) is false
Assertion (A) is false but Reason (R) is true.

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(1)

Radius of solid hemisphere, r = 4 cm

$∴ Total surface area of solid hemisphere = 3 π r^{2} = 3 π (4)^{2} = 48 π {cm}^{2} T o t a l \cos t o f p a i n t i n g = R s 48 π \times 2 = ₹ 96 π Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A) .$

Q 5 :

Assertion (A): If the areas of three adjacent faces of a cuboid are x,y,z respectively, then the volume of the cuboid is $\sqrt{x y z} .$

Reason (R): Volume of a cuboid whose edges are $l, b, h u n i t s i s l b h$ cube units.

Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
Assertion (A) is true but Reason (R) is false
Assertion (A) is false but Reason (R) is true

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(1)

The edges of cuboid are l,b,h units.

$l b = x, b h = y, l h = z \Rightarrow l b \times b h \times l h = x y z \Rightarrow l^{2} b^{2} h^{2} = x y z \Rightarrow l b h = \sqrt{x y z} ∴ Volume of cuboid = \sqrt{x y z} ∴ Assertion (A) and Reason (R) both are true and Reason (R) is the correct explanation of Assertion (A) .$

Q 6 :

Assertion (A): If two metallic right circular cones of equal height and base radii 3 cm and 4 cm are melted and recast into a solid sphere of radius 5 cm, then the cones are of height 20 cm each.

Reason (R): If two right circular cones of same height h and base radii $r_{1}, r_{2}$ are melted and recast into a solid sphere of radius r, then $h = \frac{4 r^{2}}{r_{1} + r_{2}} .$

Both A and R are true, and R is the correct explanation of A.
Both A and R are true, but R is not the correct explanation of A.
A is true, but R is false.
A is false, but R is true

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(3)

When one solid shape is converted into another, the volume remains the same.

Sum of the volumes of two cones = volume of the sphere

$\Rightarrow \frac{1}{3} π r_{1}^{2} h + \frac{1}{3} π r_{2}^{2} h = \frac{4}{3} π r^{3} \Rightarrow h (r_{1}^{2} + r_{2}^{2}) = 4 r^{3} \Rightarrow h = \frac{4 r^{3}}{r_{1}^{2} + r_{2}^{2}} . . . (i) S o, R e a s o n (R) i s f a l s e . P u t t i n g r = 5, r_{1} = 3, r_{2} = 4 i n (i), w e o b t a i n h = \frac{4 \times 125}{25} = 20 cm . S o, A s s e r t i o n (A) i s t r u e .$

Q 7 :

Assertion (A): From a solid cylinder whose height is 12 cm and diameter is 10 cm, a conical cavity of same height and same diameter is hollowed out. Volume of the cone is $\frac{2200}{7} c m ³ .$

Reason (R): If a conical cavity of same height and same diameter is hollowed out from a cylinder of height h and base radius r, then volume of the cone is equal to 1/2 of the volume of the cylinder.

Both, Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A).
Both, Assertion (A) and Reason (R) are true but Reason (R) is not correct explanation for Assertion (A).
Assertion (A) is true but Reason (R) is false.
Assertion (A) is false but Reason (R) is true.

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(3)

$R a d i u s o f c o n e, r = \frac{10}{2} c m = 5 c m$

$H e i g h t o f c o n e, h = 12 c m$

$V o l u m e o f c o n e = \frac{π}{3} r^{2} h$

$= \frac{1}{3} \times \frac{22}{7} \times 5 \times 5 \times 12$

$= \frac{110 \times 60}{21} = \frac{6600}{21} = \frac{2200}{7} {cm}^{3} S o, A s s e r t i o n (A) i s t r u e .$

$A l s o v o l u m e o f c y l i n d e r = π r^{2} h a n d$

$c o n i c a l c a v i t y = \frac{1}{3} π r^{2} h$

$\Rightarrow \frac{1}{3} (volume of cylinder) = volume of cone .$

$H e n c e, A s s e r t i o n (A) i s t r u e b u t R e a s o n (R) i s f a l s e .$

Topic Question Set

Q 1 :

Assertion (A): Two cubes each of edge length 10 cm are joined together. The total surface area of newly formed cuboid is 1200 ${cm}^{2}$ .

Reason (R): Area of each surface of a cube of side 10 cm is 100 ${cm}^{2}$ .

Q 2 :

Assertion (A): Length of diagonal of a cube of side 7 cm is $7 \sqrt{3} c m .$

Reason (R): Length of diagonal of a cube of edge $x u n i t = \frac{x}{\sqrt{3}} u n i t s .$

Q 3 :

Assertion (A): Total surface area of a hemisphere of radius 2 cm is $4 π {cm}^{2}$

Reason (R): Total surface area of a hemisphere of radius $r = 3 π r^{2} .$

Q 4 :

Assertion (A): A solid hemisphere of radius 4 cm is to be painted. Then the total cost of painting at the rate of $R s 2 p e r c m ² i s R s 96 π .$

Reason (R): Total surface area of a hemisphere $= 3 π r^{2} .$

Q 5 :

Assertion (A): If the areas of three adjacent faces of a cuboid are x,y,z respectively, then the volume of the cuboid is $\sqrt{x y z} .$

Reason (R): Volume of a cuboid whose edges are $l, b, h u n i t s i s l b h$ cube units.

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