Question

Evaluate the following expression: \[ \left( (|1| \ 2) - \left( 10^{1.3} \vee \log_{10} 0.1 \right) \right) \div (1 \vee 2) \]

• A. \( \frac{3}{8} \)
• B. \( \frac{4 \cdot \log_{10} 0.1}{8} \)
• C. \( \frac{4 + 10^{13}}{8} \)
• D. None of these

Hint:

When calculating with special operations, always follow the rules for the operations carefully: - Use the specific formulas for \( \ \) and \( \vee \) as defined in the question.

Answer 1

The Correct Option is A

We will evaluate each part of the expression step by step. Step 1: Evaluate \( |1| \ 2 \) Since \( |1| = 1 \) and both 1 and 2 are positive, we use the definition of the \( \ \) operation: \[ |1| \ 2 = 1 + 2 = 3. \] Step 2: Evaluate \( 10^{1.3} \vee \log_{10} 0.1 \) First, calculate \( 10^{1.3} \): \[ 10^{1.3} \approx 19.9526. \] Next, calculate \( \log_{10} 0.1 \): \[ \log_{10} 0.1 = -1. \] Now, calculate \( 10^{1.3} \vee \log_{10} 0.1 \). Since \( 10^{1.3} \times \log_{10} 0.1 = 19.9526 \times (-1) = -19.9526 \), which is negative, we use the alternative definition of \( \vee \), so: \[ 10^{1.3} \vee \log_{10} 0.1 = 1. \] Step 3: Evaluate \( 1 \vee 2 \) 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 4: Substitute into the expression Now, substitute the values into the original expression: \[ \left( 3 - 1 \right) \div 5 = \frac{2}{5}. \]