1. How do you explain her eccentric behaviors? (x)
How do you explain her eccentric behavior't.(√)
behavior表“行为”，是不可数名词。
2. Your behavior is like an idiot. (x)
Your behavior is like an idiot's. (√)
You behave like an idiot. (√)
英文的语言逻辑:某人可以像白痴(因白痴也是人).但行为是“物”，不能像白痴，故无法对应。上述两个正确的例句中，your behavior 可以和 an idiot's(省略behavior)对应; you可以和 an idiot对应。
3. He srore to find out the truth on the back of the rumor.(x)
He swore to find out the truth behind the rumor. (√)
on the back of sth是指“具体物体的背面”。the truth / motive behind sth是指“某事/物背后的真相/动机”，为固定用法。
4. I believe in George this time because he seems to be telling the truth.(x)
I believe George this time because he seems to be telling the truth.(√)
believe in sb / sth是“相信某人/物存在”，如:I believe in God / ghosts.(我相信上帝/鬼魂存在着。) believe sb是“相信某人所说的话”。另外believe in sb也有“信赖某人(的能力/人格)”的意思，如:We want a President that we can believe in.(我们要的是一个可以信赖的总统。)
5. Quite a few people still believe capital punishment.(x)
Quite a few people still believe in capital punishment. (√)
believe sth是“相信某事/物是真的”，如:I believe nothing she says.(我不相信她说的任何一句话。) believe in sth是“相信某事/物有益或有效”，所以有“主张”的意思，如:As far as losing weight is concerned, I believe in taking exercise. (就减肥而言，我主张运动。)
6. Can you believe in your eyes?(x)
Can you believe your eyes?(√)
can / could (not / hardly) believe your eyes / ears，即“能不能相信自己的亲眼所见或亲且所闻”，常用于否定句或疑问句中，为固定用法。
7. I deeply believe that nature will bite back.(x)
I firmly believe that nature will bite back.(√)
deeply believe是中文式的说法.英文无此说法。要说“对某事/物深信不疑”，正确的说法是 firmly / sincerely / strongly / truly believe (that).....
8. The dictionary belongs to that shelf.(x)
The dictionary belongs on that shelf. (√)
belong to sb是“属于某人”的意思，如:The encyclopedia belongs to me.(这套百科全书属于我。) belong on / in是“(该)归位在某个地方”的意思。
9. Please go back where you belong to. (x)
Please go back where you belong. (√)
belong here / there / where (不加to)，是固定用法。
10. When you get off, don't forget your personal belonging. (x)
When you get off, don't forget your personal belongings. (√)
belongings指“某人的财物或随身带的东西”，用复数。belonging(不可数名词)则见于sense of belonging这个词组，是“归属感”的意思，如:I have no sense of belonging working there.(在那里工作我没有归属感。)