How to assign image on specified category and not show the image on other categories

It seems that you need to do this inside a post. For this, you can either use has_category() or in_category to test if a post belongs to the given category, and if the condition is true, apply your changes

Example:

if( has_category('name or I'd or slug of category to test) ) {
     echo '<img src="http://organeek.net/wp-content/uploads/2014/10/85.jpg' . $category->cat_ID . '.jpg" alt="' . $category->cat_name . '" />';
}else{
    //displays nothing if the post does not belongs to this category
}