1.1 If \( p \in\{0 ; 1 ; 2 ; 3 ; 4\} \), determine the value of p such that \( \sqrt{\frac{16}{p-2}} \) 1.1 .1 is irrational 1.1 .2 is non real 1.1 .3 is undefined 0,

Let's analyze the expression \( \sqrt{\frac{16}{p-2}} \) for each value of \( p \) in the set \( \{0, 1, 2, 3, 4\} \). ### Step 1: Evaluate the Expression for Each \( p \) 1. **\( p = 0 \)** \[ \sqrt{\frac{16}{0-2}} = \sqrt{\frac{16}{-2}} = \sqrt{-8} \] - **Nature**: Non-real (since the square root of a negative number is not a real number). 2. **\( p = 1 \)** \[ \sqrt{\frac{16}{1-2}} = \sqrt{\frac{16}{-1}} = \sqrt{-16} \] - **Nature**: Non-real. 3. **\( p = 2 \)** \[ \sqrt{\frac{16}{2-2}} = \sqrt{\frac{16}{0}} \] - **Nature**: Undefined (division by zero is undefined). 4. **\( p = 3 \)** \[ \sqrt{\frac{16}{3-2}} = \sqrt{\frac{16}{1}} = \sqrt{16} = 4 \] - **Nature**: Rational number. 5. **\( p = 4 \)** \[ \sqrt{\frac{16}{4-2}} = \sqrt{\frac{16}{2}} = \sqrt{8} = 2\sqrt{2} \] - **Nature**: Irrational number. ### Step 2: Determine the Values of \( p \) for Each Case 1. **1.1.1 The expression is **Irrational**:** - **Solution**: \( p = 4 \) - **Reason**: \( \sqrt{8} = 2\sqrt{2} \) is irrational. 2. **1.1.2 The expression is **Non-real**:** - **Solutions**: \( p = 0 \) and \( p = 1 \) - **Reason**: Both \( \sqrt{-8} \) and \( \sqrt{-16} \) are non-real numbers. 3. **1.1.3 The expression is **Undefined**:** - **Solution**: \( p = 2 \) - **Reason**: Division by zero occurs, making the expression undefined. ### Summary of Solutions - **1.1.1 (Irrational):** \( p = 4 \) - **1.1.2 (Non-real):** \( p = 0 \) and \( p = 1 \) - **1.1.3 (Undefined):** \( p = 2 \)