Which of the following is a linear equation in \( x, y \) and \( x^{-2}+y+8 z=5 \) \( \cos x-y+z=0 \) None of the given option \( \pi \sqrt{x^{4}}-z=1+\sqrt{2} y \) \( x^{-1}-5 y-10 z=11 \) .

To determine which of the given equations is a linear equation in \( x \), \( y \), and \( z \), we need to check if each equation can be expressed in the form \( ax + by + cz = d \), where \( a \), \( b \), and \( c \) are constants, and \( d \) is a constant as well. A linear equation must not contain any variables raised to a power other than 1, nor can it contain products of variables or functions of variables (like trigonometric functions). Let's analyze each equation: 1. **\( x^{-2} + y + 8z = 5 \)**: - The term \( x^{-2} \) is not linear because it involves \( x \) raised to the power of -2. Therefore, this is not a linear equation. 2. **\( \cos x - y + z = 0 \)**: - The term \( \cos x \) is a trigonometric function of \( x \), which makes this equation non-linear. Therefore, this is not a linear equation. 3. **\( \pi \sqrt{x^{4}} - z = 1 + \sqrt{2} y \)**: - The term \( \sqrt{x^{4}} \) simplifies to \( x^2 \), which is not linear because it involves \( x \) raised to the power of 2. Therefore, this is not a linear equation. 4. **\( x^{-1} - 5y - 10z = 11 \)**: - The term \( x^{-1} \) is not linear because it involves \( x \) raised to the power of -1. Therefore, this is not a linear equation. Since none of the equations provided are linear in \( x \), \( y \), and \( z \), the correct answer is: **None of the given options.**