Have users answer a question before viewing content
I'm not aware of a module that does this out of the box. But depending on your details there are a few ways to try to solve this problem.

One solution would be to create your form using Webform or Entity form (assuming your on Drupal 7 a contact form could be used in Drupal 8), place that form in a hidden block on the pages in question. On page load use JavaScript to put the form into a lightbox and set some kind of CSS effect over the rest of the page. When the user completes the form set a cookie so that on the next page load they aren't bothered by the popup. This is a common solution used by News papers trying to enable paywalls but it often means users can bypass the effect by disabling JS.