CBSE-Class-10-Maths-Assertion-and-Reason-Chapter-13-Surface-Areas-and-Volumes

Class 10^th Maths Chapter Assertion and Reason Questions

Question 1

Assertion (A): The surface area of a cube is calculated using the formula $6a^2$ .
Reason (R): A cube has six faces, each of which is a square with a side length of $a$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The surface area of a cube is derived from the total area of its six square faces.

Question 2

Assertion (A): The volume of a cylinder is given by the formula $V = \pi r^2 h$ .
Reason (R): The volume of a cylinder is the area of the base multiplied by its height.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The volume formula correctly represents the multiplication of the base area by the height.

Question 3

Assertion (A): The surface area of a sphere is given by the formula $4\pi r^2$ .
Reason (R): A sphere is a three-dimensional object with no edges or vertices.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The formula for the surface area of a sphere arises from its geometric properties, which include its smooth surface.

Question 4

Assertion (A): The volume of a cone can be calculated using the formula $V = \frac{1}{3}\pi r^2 h$ .
Reason (R): The volume of a cone is one-third the volume of a cylinder with the same base and height.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The formula for the volume of a cone accurately reflects its relationship with a cylinder of identical dimensions.

Question 5

Assertion (A): The lateral surface area of a cylinder is given by $2\pi rh$ .
Reason (R): The lateral surface area is the area around the sides of the cylinder without the top and bottom bases.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The formula for lateral surface area effectively describes the area of the cylinder's curved surface.

Question 6

Assertion (A): The volume of a sphere is expressed as $V = \frac{4}{3}\pi r^3$ .
Reason (R): The volume of a sphere is proportional to the cube of its radius.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The formula demonstrates how the volume scales with changes in radius.

Question 7

Assertion (A): The total surface area of a hemisphere is given by $3\pi r^2$ .
Reason (R): This formula accounts for both the curved surface area and the base of the hemisphere.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The total surface area of a hemisphere combines its curved area with the area of the circular base.

Question 8

Assertion (A): The height of a cylinder can be calculated if the volume and radius are known.
Reason (R): The formula $h = \frac{V}{\pi r^2}$ can be used to determine height.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The formula accurately expresses how to isolate height from the volume equation.

Question 9

Assertion (A): The surface area of a rectangular prism can be calculated using the formula $2(lw + lh + wh)$ .
Reason (R): This formula sums the areas of all six rectangular faces of the prism.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The formula comprehensively accounts for the area of each face of the rectangular prism.

Question 10

Assertion (A): The volume of a cuboid can be expressed as $V = l \times w \times h$ .
Reason (R): This formula measures the amount of space inside the cuboid.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The volume formula accurately represents the three-dimensional space occupied by the cuboid.

Question 11

Assertion (A): The slant height of a cone can be found using the Pythagorean theorem.
Reason (R): The slant height forms the hypotenuse of a right triangle formed by the height and radius of the cone.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The relationship between the slant height, radius, and height forms a right triangle, enabling the application of the Pythagorean theorem.

Question 12

Assertion (A): The curved surface area of a cone is given by the formula $\pi r l$ , where $l$ is the slant height.
Reason (R): The curved surface area represents the area of the cone excluding its base.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The formula for the curved surface area correctly describes the surface excluding the base.

Question 13

Assertion (A): The height of a prism can be calculated using the formula $h = \frac{V}{A}$ , where $A$ is the area of the base.
Reason (R): This formula relates volume, base area, and height, allowing for height calculation.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The formula accurately shows the relationship among volume, base area, and height.

Question 14

Assertion (A): A hemisphere has a total surface area of $3\pi r^2$ .
Reason (R): This includes both the curved surface area and the base area of the hemisphere.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The total surface area formula includes the area of the circular base along with the curved area.

Question 15

Assertion (A): The total surface area of a cube can be found using the formula $6a^2$ .
Reason (R): A cube has six identical square faces.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The surface area formula is derived by calculating the area of each of the six square faces.

Question 16

Assertion (A): The volume of a cylinder increases if either the radius or the height is increased.
Reason (R): Volume is a function of both the base area and height.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: An increase in radius or height will result in a larger volume due to the direct relationship described by the formula.

Question 17

Assertion (A): The total surface area of a cone is given by $\pi r (r + l)$ .
Reason (R): This formula accounts for the area of the base and the curved surface.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The total surface area includes both the base area and the curved area of the cone.

Question 18

Assertion (A): The lateral surface area of a cone can be found using the formula $\pi r l$ .
Reason (R): The lateral surface area represents the area of the cone excluding the base.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The formula is designed to calculate only the curved surface area without including the base.

Question 19

Assertion (A): The surface area of a cylinder can be expressed as $2\pi rh + 2\pi r^2$ .
Reason (R): This formula includes both the lateral surface area and the areas of the two circular bases.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The formula for surface area accurately accounts for all contributing surfaces of the cylinder.

Question 20

Assertion (A): The volume of a cuboid is dependent on the lengths of its sides.
Reason (R): The volume is calculated as $l \times w \times h$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The volume directly corresponds to the product of the lengths of the three dimensions.

Question 21

Assertion (A): The surface area of a right prism can be calculated as $2B + P \cdot h$ , where $B$ is the area of the base and $P$ is the perimeter of the base.
Reason (R): This formula includes the areas of the bases and the lateral area around the prism.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The formula comprehensively covers all surface areas of the prism.

Question 22

Assertion (A): The radius of a sphere affects its surface area and volume significantly.
Reason (R): Both surface area and volume are functions of the radius raised to a power.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: Changes in radius have a squared effect on surface area and a cubed effect on volume.

Question 23

Assertion (A): The curved surface area of a hemisphere is given by $2\pi r^2$ .
Reason (R): This formula calculates only the curved part, excluding the base.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The formula specifically addresses the curved surface area while excluding the base of the hemisphere.

Question 24

Assertion (A): To find the height of a cone when the volume and radius are given, one can rearrange the volume formula $V = \frac{1}{3}\pi r^2 h$ .
Reason (R): Rearranging allows solving for height using known values.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The formula can be manipulated to isolate height, demonstrating the relationship between volume and dimensions.

Question 25

Assertion (A): The surface area of a cylinder can be found using the formula $2\pi r(h + r)$ .
Reason (R): This formula combines the areas of the circular bases and the lateral surface.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The surface area formula indeed covers both the lateral and base areas, confirming the relationship.

Chapter 13 Surface Areas and Volumes

Class 10th Maths Chapter Assertion and Reason Questions