Surface Areas and Volumes Assertion–Reason PYQs Class 10 CBSE (With Answers)

Official CBSE answer · Explanation by Team TutorVertex

Assertion: True.
The cuboid has length l = 8 cm, breadth b = 4 cm and height h = 4 cm.
Total surface area = 2(lb + bh + hl).
= 2(8 × 4 + 4 × 4 + 4 × 8)
= 2(32 + 16 + 32)
= 160 cm².

Reason: False.
The correct total surface area is 2(lb + bh + hl).
The printed formula omits the factor 2.

Link: A gives the correct area, but R gives an incorrect formula. A is true and R is false. Answer: C.

Official CBSE answer · Explanation by Team TutorVertex

Assertion: False.
The new cuboid has dimensions 20 cm × 10 cm × 10 cm.
Total surface area = 2(lb + bh + hl).
= 2(20 × 10 + 10 × 10 + 10 × 20)
= 1000 cm².
The two touching faces are hidden, so 1200 cm² is not the exposed area.

Reason: True.
Each face is a square of side 10 cm.
Area = 10² = 100 cm².

Link: A counts the two hidden faces. R correctly gives the area of one square face. A is false and R is true. Answer: D.

Official CBSE answer · Explanation by Team TutorVertex

Assertion: False.
The resulting cuboid measures 24 cm × 12 cm × 12 cm.
Its surface area = 2(24 × 12 + 12 × 12 + 12 × 24).
= 1440 cm².
Twice the surface area of one cube = 2 × 6 × 12² = 1728 cm².
These are unequal because the two joining faces are hidden.

Reason: True.
A cuboid has two faces of each area lb, bh and hl.
Adding them gives 2(lb + bh + hl).

Link: R is the correct cuboid formula; applying it disproves A. A is false and R is true. Answer: D.

Official CBSE answer · Explanation by Team TutorVertex

Assertion: False.
The original flat circular face has area πr².
The new hemispherical cavity has curved area 2πr².
Increase in surface area = new curved area − removed flat area.
= 2πr² − πr²
= πr², not 2πr².

Reason: True.
The curved surface area of a hemisphere of radius r is 2πr².

Link: R gives the new cavity’s area. To find the increase, we must also subtract the circular face that it replaces. A is false and R is true. Answer: D.

Official CBSE answer · Explanation by Team TutorVertex

Assertion: True.
The two circular bases coincide at the join and are hidden.
Only the curved hemisphere and the curved cone are exposed.
Total surface area = curved area of hemisphere + curved area of cone.
= 2πr² + πrl, where r is their common radius and l is the cone’s slant height.

Reason: True.
The figure shows the hemisphere and cone joined along their equal circular plane faces.

Link: R explains why the bases are excluded and only the two curved surfaces remain. R explains A. Answer: A.

Official CBSE answer · Explanation by Team TutorVertex

Assertion: True.
Here r = 3 cm and the cylinder’s height h = 7 cm.
Volume of cylinder = πr²h = π × 3² × 7 = 63π cm³.
Volume of hemisphere = $\frac{\text{2}}{\text{3}}$πr³ = $\frac{\text{2}}{\text{3}}$ × π × 3³ = 18π cm³.
Total volume = 63π + 18π = 81π cm³.

Reason: True.
The cylinder and hemisphere meet only along a common circular face.
Their volumes therefore add.

Link: R gives the addition of component volumes used to obtain A. R explains A. Answer: A.

Official CBSE answer · Explanation by Team TutorVertex

Assertion: True.
Two matching hemispheres joined along their circular bases form one complete sphere.

Reason: False.
The total surface area of a sphere is 4πr².
The expression 3πr² is the total surface area of one solid hemisphere.

Link: A correctly describes the joined solid, while R uses the hemisphere’s total-area formula for a sphere. A is true and R is false. Answer: C.

Official CBSE answer · Explanation by Team TutorVertex

Assertion: True.
Let the radius be r cm.
Total surface area = 3πr².
462 = 3 × $\frac{\text{22}}{\text{7}}$ × r².
r² = $\frac{\text{462 × 7}}{\text{66}}$ = 49.
Since a radius is positive, r = 7 cm.

Reason: True.
Total surface area = curved area + area of circular base.
= 2πr² + πr² = 3πr².

Link: R gives the total-area formula used to calculate the radius in A. R explains A. Answer: A.

Official CBSE answer · Explanation by Team TutorVertex

Assertion: True.
Here r = 3.5 cm and slant height l = 4 cm.
Curved surface area = πrl.
= $\frac{\text{22}}{\text{7}}$ × 3.5 × 4
= 44 cm².

Reason: True.
The curved surface area of a cone is πrl, using its slant height l.

Link: R gives the formula used to obtain 44 cm². R explains A. Answer: A.

Official CBSE answer · Explanation by Team TutorVertex

Assertion: True.
Radius r = $\frac{\text{14}}{\text{2}}$ = 7 m and perpendicular height h = 24 m.
Slant height l = √(r² + h²).
= √(7² + 24²) = √625 = 25 m.
The canvas covers the curved surface; the base rests on the ground.
Required area = πrl = π × 7 × 25 = 175π m².

Reason: True.
The curved surface area of a cone is πrl.

Link: R gives the covering-area formula. With the slant height calculated from the given dimensions, it explains A. Answer: A.

Official CBSE answer · Explanation by Team TutorVertex

Assertion: True.
Here r = 7 cm and perpendicular height h = 12 cm.
Volume = $\frac{\text{1}}{\text{3}}$πr²h.
= $\frac{\text{1}}{\text{3}}$ × $\frac{\text{22}}{\text{7}}$ × 7² × 12
= 616 cm³.

Reason: True.
A cone’s volume is $\frac{\text{1}}{\text{3}}$πr²h, using its perpendicular height h.

Link: R gives the formula used to calculate 616 cm³. R explains A. Answer: A.

Official CBSE answer · Explanation by Team TutorVertex

Assertion: True.
The largest sphere has diameter equal to the cube’s side a.
Hence its radius r = $\frac{\text{a}}{\text{2}}$ cm.
Surface area = 4πr².
= 4π × ($\frac{\text{a}}{\text{2}}$)²
= πa² cm².

Reason: False.
The surface area of a sphere is 4πr².
The expression $\frac{\text{4}}{\text{3}}$πr³ is its volume.

Link: A uses the correct inscribed-sphere radius. R incorrectly calls the volume formula a surface-area formula. A is true and R is false. Answer: C.

Official CBSE answer · Explanation by Team TutorVertex

Assertion: False.
Let the radius be r cm.
4πr² = 616.
4 × $\frac{\text{22}}{\text{7}}$ × r² = 616.
r² = $\frac{\text{616 × 7}}{\text{88}}$ = 49.
r = 7 cm, so diameter = 2r = 14 cm.
The stated 7 cm is the radius, not the diameter.

Reason: True.
The surface area of a sphere is 4πr².

Link: R gives the correct surface-area formula, which gives diameter 14 cm. A is false and R is true. Answer: D.

Official CBSE answer · Explanation by Team TutorVertex

Assertion: False.
Let the radius be r cm.
4πr² = 616.
4 × $\frac{\text{22}}{\text{7}}$ × r² = 616.
r² = 49, so r = 7 cm.
Diameter = 2r = 14 cm, not 12 cm.

Reason: True.
The surface area of a sphere is 4πr².

Link: Applying R gives diameter 14 cm and disproves A. A is false and R is true. Answer: D.

Official CBSE answer · Explanation by Team TutorVertex

Assertion: True.
Let the radii be r₁ and r₂.
$\frac{\text{S₁}}{\text{S₂}}$ = $\frac{\text{4πr₁²}}{\text{4πr₂²}}$ = $\frac{\text{r₁²}}{\text{r₂²}}$ = $\frac{\text{16}}{\text{9}}$.
Since the radii are positive, $\frac{\text{r₁}}{\text{r₂}}$ = $\frac{\text{4}}{\text{3}}$.
$\frac{\text{V₁}}{\text{V₂}}$ = $\frac{\text{r₁³}}{\text{r₂³}}$ = ($\frac{\text{4}}{\text{3}}$)³ = $\frac{\text{64}}{\text{27}}$.
Therefore V₁ : V₂ = 64 : 27.

Reason: True.
Surface areas are proportional to the squares of the radii.
Volumes are proportional to the cubes of the radii.
Therefore the volume ratio is the surface-area ratio raised to the power $\frac{\text{3}}{\text{2}}$.

Link: R converts the given surface-area ratio into the volume ratio in A. R explains A. Answer: A.