Directions: Each of the questions given below consists of a question followed by three statements. You have to study the question and the statements and decide which of the statement(s) is/are necessary to answer the question. A solid metallic cone is melted and recast into a sphere. What is the radius of the sphere? I. The radius of the base of the cone is 2.1 cm. II. The height of the cone is four times the radius of its base. III. The height of the cone is 8.4 cm.

### Concept & Logic When a solid is melted and recast into another shape, their volumes remain equal. $$ \text{Volume of Cone} = \text{Volume of Sphere} $$ $$ \frac{1}{3} \pi r_c^2 h_c = \frac{4}{3} \pi r_s^3 $$ To find the radius of the sphere ($r_s$), we fundamentally need the radius of the cone ($r_c$) and the height of the cone ($h_c$). ### Step-by-Step Solution - **Statement I:** Gives $r_c = 2.1$. (Insufficient alone) - **Statement II:** Gives the relationship $h_c = 4r_c$. (Insufficient alone) - **Statement III:** Gives $h_c = 8.4$. (Insufficient alone) - **Combining Statements:** - **I and II:** I gives $r_c$. II allows finding $h_c$. Both variables are known. (Sufficient) - **I and III:** I gives $r_c$. III gives $h_c$. Both variables are known. (Sufficient) - **II and III:** III gives $h_c$. II allows finding $r_c$. Both variables are known. (Sufficient) - Any two of the three statements provide the necessary values to calculate $r_c$ and $h_c$, which subsequently yields the radius of the sphere. ### Exam Strategy & Shortcut We are given three parameters regarding the cone: $r_c$, $h_c$, and a direct proportional relationship between them. Knowing any two of these will define all necessary dimensions to find the volume of the cone. ### Common Pitfall A common mistake is choosing "All I, II and III" by assuming more information is better. Data Sufficiency strictly asks for the minimum required information to answer the question. ### Final Answer Therefore, the correct answer is **Any two of the three**.