Similar to the usage of に決まっている、に違いない is simply translated most of the time as "definitely" or "absolutely". One problem with this is that the usage of both can be a little bit ambiguous and can leave a serious student up in the air of which to use.
A less spoken concept regarding any grammar involves the context and "feeling" that is generally understood by native Japanese, but may be difficult for foreign learners to grasp.
に決まっている has the connotation of narrowness of options such that whatever precedes this is an absolute.
例文） 地球は平らではないに決まっている。 The earth is *absolutely* not flat.
In contrast, usage of 違いない supposedly has a feeling of slight openness in terms of speculation or conditions.
例文） 死体に丸い傷があるので、拳銃でやられちゃったに違いない。 Since the corpse has circular wounds, he must have been done in by a handgun.
In ex #5796, I`ve changed 乱魔ﾈ to 乱暴. But I`m note sure if it is right...
i think in #5798 it should be yohodo instead of yohado: こんなことを信じるなんて彼はよほどの間抜けに違いない。
I intended to fix ex #5792 and #5794 following the above alerts. 乱魔ﾈ in ex #5796 seems to be 文字化け so I turned it suspicious. I don't understand 彼はよはどの間抜け in ex #5798.
In #5794, この作文の英語はうますぎる。彼女がそれを書いたに違いない has wrong English translation.It should be: The English of composition is too good. She must have written it herself. If the reverse was intended, then the Japanese text is wrong. It shold read: この作文の英語はうますぎる。それを彼女が書いていないのに違いない。
#5792 Looks typo. Japanese should be かれはここに来たに違いない。
The translations of the examples do not seem to match the meaning given above, it seems to me. The meaning of "no mistake" or "definitely" is much different from the meaning of "must have" or "must be". I'm wondering which meaning is correct.
I believe that example #5794 says 'She must have written it herself'...I would be interested to know if there is a way to say 'She must not have written it herself' using this grammar form...