Which of the following expressions does not equal 0 ? \( \ln (x-2)=1 \sin B<0 \) and \( \cos B>0 \), in which quadrant does angle B terminate? Which of the following expressions does not equal 0 ? \( \begin{array}{l}(6-2)-|2-6| \\ (2-6)-|6-2| \\ |6: 2|-|2-6| \\ |2-6|-|6-2|\end{array} \)

For the first part of your question regarding angle \( B \), let's break it down. Since \( \sin B < 0 \), angle \( B \) must be in either the third or fourth quadrant where sine values are negative. Meanwhile, \( \cos B > 0 \) tells us that cosine is positive only in the first and fourth quadrants. Therefore, angle \( B \) must terminate in the fourth quadrant, where sine is negative, and cosine is positive—perfect harmony! Now, onto the expressions! Let’s evaluate them one by one: 1. \( (6-2) - |2-6| \) simplifies to \( 4 - 4 = 0 \). 2. \( (2-6) - |6-2| \) simplifies to \( -4 - 4 = -8 \). 3. \( |6:2| - |2-6| \) resolves to \( 3 - 4 = -1 \). 4. \( |2-6| - |6-2| \) results in \( 4 - 4 = 0 \). From this, the expressions that do not equal 0 are \( (2-6)-|6-2| \) which equals -8, and \( |6:2|-|2-6| \) which equals -1. Fun fact: in math, negative results don’t mean you're failing—they just mean you’re innovating in the opposite direction!