"Answer this question."
Translation:इस सवाल का जवाब दो।
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No. That won't work because the compound verb जवाब देना ('answer') won't admit an object. In fact, you can consider 'इस सवाल का जवाब' (this question's answer) itself as the object of देना (give).
You can use को with a different verb though. For example, इस सवाल को सुलझाओ (Solve this question).
If you replace सवाल by प्रश्न or जवाब by उत्तर, the sentence would still be correct. However, as with many other 'तत्सम' words (words picked as is from Sanskrit into Hindi), प्रश्न and उत्तर are seen as more 'formal' in modern Hindi. So, you would typically use सवाल and जवाब in daily speech but प्रश्न and उत्तर in newspaper articles (of course, this is not a rule). There are cases you would always use one or the other though- for instance, a 'question paper' is always 'प्रश्न पत्र'.
इस is the oblique form of यह that is used when it is a part of a noun phrase (इस सवाल) and that noun phrase is the object of a postposition.
If the pronoun were alone, we'd use a different form of यह that indicates the case and forgoes the need for a postposition. Eg: इसका जवाब दो ('Answer this') where इसका is a genitive case form of यह.