Question

The following operations are defined for real numbers. \[ a \ b = a + b, \, \text{if} \, a \, \text{and} \, b \, \text{both are positive, else} \, a \ b = 1 \] \[ a \vee b = (a \times b) + a + b, \, \text{if} \, a \times b \, \text{is positive, else} \, a \vee b = 1 \] What is \[ \frac{(2 \ 1)}{(1 \vee 2)}? \]

• A. \( \frac{1}{8} \)
• B. 1
• C. \( \frac{3}{8} \)
• D. 3

Hint:

Always check if the numbers in the operation are positive before applying the defined operations. For operations with multiplication, ensure the product is positive to proceed as defined.

Answer 1

The Correct Option is C

First, let's solve the individual operations \( 2 \ 1 \) and \( 1 \vee 2 \). Step 1: Solve \( 2 \ 1 \) Since both 2 and 1 are positive, we use the definition of the \( \ \) operation: \[ 2 \ 1 = 2 + 1 = 3. \] Step 2: Solve \( 1 \vee 2 \) Next, for the \( \vee \) operation, we check if the product \( 1 \times 2 \) is positive. Since \( 1 \times 2 = 2 \) (which is positive), we use the definition of the \( \vee \) operation: \[ 1 \vee 2 = (1 \times 2) + 1 + 2 = 2 + 1 + 2 = 5. \] Step 3: Calculate the expression Now, substitute the results into the expression \( \frac{(2 \ 1)}{(1 \vee 2)} \): \[ \frac{(2 \ 1)}{(1 \vee 2)} = \frac{3}{5}. \] Thus, the value of the expression is \( \frac{3}{5} \).