A question is given, followed by three statements labelled I, II and III. Identify which of the statements is/are sufficient to answer the question.

Question:

What is the value of A when D = 7?

Statements:

I. A varies directly with the sum of B and C, B varies directly with D and C varies directly with $$\text D^2$$​.

II. When D = 3, A = 33.

III. When D = 4, A = 52.

Correct option is A

Given:

Question:

What is the value of A when D = 7?

Statements:

I. A varies directly with the sum of Band C,B varies directly with D and C varies directly with $$\text D^2$$​.

II. WhenD= 3, A = 33.

III. WhenD= 4, A = 52.

Solution:

$$\text{If a variable } A \text{ varies directly with } x, \text{ then: } \\A = k \cdot x \\\text{If } A \text{ varies with } B + C, \text{ and }\\ B \propto D,\\ \ C \propto D^2,\\ \text{ then:} \\B = mD,\quad C = nD^2 \\A = k(B + C) = k(mD + nD^2) = aD + bD^2 \\\textbf{Step 1: From Statement I:} \\A = aD + bD^2 \quad \text{(general form)} \\\\\ \\\textbf{Step 2: From Statement II:} \\\text{When } D = 3, \ A = 33 \Rightarrow 33 = a \cdot 3 + b \cdot 9 \\\Rightarrow 3a + 9b = 33 \Rightarrow a + 3b = 11 \quad \text{(i)} \\\ \\\textbf{Step 3: From Statement III:} \\\text{When } D = 4, \ A = 52 \Rightarrow 52 = a \cdot 4 + b \cdot 16 \\\Rightarrow 4a + 16b = 52 \Rightarrow a + 4b = 13 \quad \text{(ii)} \\\ \\\textbf{Step 4: Solve Equations (i) and (ii)} \\\text{Subtract (i) from (ii):} \\(a + 4b) - (a + 3b) = 13 - 11 \Rightarrow b = 2 \\\text{Put into (i):} \quad a + 3(2) = 11 \Rightarrow a = 5 \\\ \\\textbf{Step 5: Find A when D = 7} \\A = aD + bD^2 = 5 \cdot 7 + 2 \cdot 49 \\ = 35 + 98 = 133 \\$$

​Conclusion:
Statement I is required to form the algebraic relation.
Statements II and III are required to calculate the constants.
So, all three statements are jointly sufficient to answer the question.

Final Answer: (A) I, II and III