SECTION F — Assertion and Reasoning Type Questions [1 Marks Each]
Directions: Each of the following questions consists of two statements, namely, Assertion (A) and Reason (R). Select the correct option from the choices given below:
- (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.
-
Q1. Assertion (A): Total surface area of a solid hemisphere of radius $r$ is $3\pi r^2$.
Reason (R): Total surface area of a hemisphere includes the curved surface area ($\frac{1}{2}\text{CSA}$ of sphere) and the area of the circular base ($\pi r^2$). -
Q2. Assertion (A): If the radius of a sphere is doubled, its volume becomes $8$ times the original volume.
Reason (R): The volume of a sphere is directly proportional to the cube of its radius ($V \propto r^3$). -
Q3. Assertion (A): The curved surface area of a right circular cone of radius $r$ and slant height $l$ is $\pi r l$.
Reason (R): The total surface area of a right circular cone is $\pi r l + \pi r^2$. *(Note: Both statements are true formulas, but the total surface area formula does not explain why the curved surface area is $\pi rl$).* -
Q4. Assertion (A): When a solid cylinder is melted and recast into a solid cone, the volume of the cylinder is equal to the volume of the cone.
Reason (R): When a solid shape is converted into another shape, the total surface area remains unchanged. *(Note: Surface area changes during recasting, making Reason false).* -
Q5. Assertion (A): The height of a frustum of a cone whose radii of ends are $r_1$ and $r_2$ and slant height is $l$ is given by $h = \sqrt{l^2 - (r_1 - r_2)^2}$.
Reason (R): In a frustum, the slant height, vertical height, and the difference of radii form a right-angled triangle. -
Q6. Assertion (A): If the total surface area of a solid hemisphere is $462\text{ cm}^2$, then its radius is $7\text{ cm}$ (using $\pi = \frac{22}{7}$).
Reason (R): Total surface area of a hemisphere is given by the formula $2\pi r^2$. *(Note: Total surface area of a solid hemisphere is $3\pi r^2$, making Reason false while Assertion is true).* -
Q7. Assertion (A): A plumbline (sahul) is a combination of a cylinder and a cone.
Reason (R): A plumbline is shaped like a solid cone pointing downwards, attached to the bottom of a cylinder. -
Q8. Assertion (A): The volume of a right circular cylinder of radius $r$ and height $h$ is $\pi r^2 h$.
Reason (R): Volume of a cylinder is calculated as the area of the base multiplied by its height. -
Q9. Assertion (A): If the radii of the ends of a frustum of a cone are $3\text{ cm}$ and $4\text{ cm}$, and its height is $10\text{ cm}$, its volume can be found using the formula $\frac{1}{3}\pi h (r_1^2 + r_2^2 + r_1 r_2)$.
Reason (R): A frustum of a cone can be viewed as the difference of two similar cones. -
Q10. Assertion (A): The curved surface area of a cylinder of radius $r$ and height $h$ is $2\pi rh$.
Reason (R): The perimeter of the circular base of a cylinder is $2\pi r$. -
Q11. Assertion (A): The volume of a sphere with radius $3\text{ cm}$ is $36\pi\text{ cm}^3$.
Reason (R): The volume of a sphere is given by $\frac{4}{3}\pi r^3$. -
Q12. Assertion (A): If two identical solid cubes of side $a$ are joined end to end, the surface area of the resulting cuboid is $10a^2$.
Reason (R): Joining two cubes of side $a$ results in a cuboid of dimensions $2a \times a \times a$, whose total surface area is $2(l b + b h + l h)$. -
Q13. Assertion (A): The ratio of the volumes of two spheres whose radii are in the ratio $2:3$ is $8:27$.
Reason (R): The volume of a sphere is proportional to the cube of its radius. -
Q14. Assertion (A): A solid toy is in the form of a hemisphere surmounted by a right circular cone. The total surface area of the toy is the sum of the curved surface area of the cone and the curved surface area of the hemisphere.
Reason (R): The touching circular bases of the cone and the hemisphere are internal to the combined solid and are not exposed to the surface. -
Q15. Assertion (A): The total surface area of a solid cylinder of radius $r$ and height $h$ is $2\pi r(h + r)$.
Reason (R): Total surface area is the sum of curved surface area and the areas of the two circular flat ends. -
Q16. Assertion (A): If a solid metal sphere of radius $6\text{ cm}$ is melted to form small spherical balls of radius $2\text{ cm}$, then $27$ such balls can be made.
Reason (R): The number of small balls is equal to the volume of the big sphere divided by the volume of one small ball. -
Q17. Assertion (A): The curved surface area of a frustum of a cone is $\pi(r_1 + r_2)l$, where $l = \sqrt{h^2 + (r_1 - r_2)^2}$.
Reason (R): A frustum is formed by cutting a cone with a plane parallel to its base. *(Note: Both statements are true, but the definition of a frustum does not directly explain why its curved surface area equals the given formula).* -
Q18. Assertion (A): The diagonal of a cuboid of dimensions $l, b,$ and $h$ is $\sqrt{l^2 + b^2 + h^2}$.
Reason (R): The total surface area of a cuboid is $2(lb + bh + lh)$. *(Note: Reason talks about total surface area and does not explain the diagonal formula).* -
Q19. Assertion (A): The volume of a cone with base radius $r$ and height $h$ is one-third the volume of a cylinder with the same radius and height.
Reason (R): The formula for the volume of a cone is $V = \frac{1}{3}\pi r^2 h$. -
Q20. Assertion (A): Water flowing through a cylindrical pipe of internal radius $r$ at a speed $v$ covers a distance of $v$ units in one hour, delivering a volume of $\pi r^2 v$ cubic units per hour.
Reason (R): The volume of water flowing out per unit time depends only on the cross-sectional area of the pipe and the flow speed.
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English: These model solutions are structured to global academic standards for self-assessment and practice purposes. Students are advised to verify with official textbooks for board examinations.
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