Question

Find \( 2 \circledast 3 \), where \( 2 \circledast 3 \) need not be equal to \( 3 \circledast 2 \).

I. \( 1 \circledast 2 = 3 \)
II. \( a \circledast b = \dfrac{a+b}{a} \), where a and b are positive.

• A. The question can be answered with the help of any one statement alone but not by the other statement.
• B. The question can be answered with the help of either of the statements taken individually.
• C. The question can be answered with the help of both statements together.
• D. The question cannot be answered even with the help of both statements together.

Hint:

Check whether each statement defines the operation completely. A single value like 1 \(\circledast\) 2 = 3 cannot fix the rule, but a formula can.

Answer 1

The Correct Option is A

Step 1: Check statement II:
Statement II defines the operation for all positive a and b: \( a \circledast b = \dfrac{a+b}{a} \). So \( 2 \circledast 3 = \dfrac{2+3}{2} = 2.5 \). Statement II alone is sufficient.

Step 2: Check statement I:
Statement I only says \( 1 \circledast 2 = 3 \). Many rules fit this: \( a \circledast b = a + b \) gives \( 2 \circledast 3 = 5 \), \( a \circledast b = \dfrac{a+b}{a} \) gives 2.5, and \( a \circledast b = 3a \) gives 6. So \( 2 \circledast 3 \) is not fixed. Statement I alone is not sufficient.

Step 3: Combine:
Statement II is enough on its own, and statement I is not. (Statement I is also consistent with II, since \( 1 \circledast 2 = \dfrac{3}{1} = 3 \).)

Final Answer:
The correct option is (1): The question can be answered with the help of any one statement alone but not by the other statement.