The child ______ a week ago hasn't yet been found.

主句是 the child ... hasn't yet been found
lost a week ago 是分词做 child 的后置定语,可以认为是省略了 who was 的定语从句（the child who was lost a week ago）,因此要选C.
不选 D 的理由是在一个句子中,除非是并列关系,否则不能同时有两个谓语,选D的话就变成一个句子有 was lost 和 hasn't been found 两个谓语了.如果在 hasn't 前加一个 and 变成并列的关系,就应该选D,句子为：
The child was lost a week ago and hasn't yet been found.

LOST走失 不是被动语态，
被动语态应为get lost： The child got lost a week ago and hasn't yet been found.

后面有个ago,强调的是，过去发生的一个动作。所以要用lost, be lost 的话，代表的是一种状态，而且含有被动语态在里面。所以选C

因为有ago,这是过去式的标志，所以选C
你可以翻译一下D：一周前被丢的孩子还没有被找到。意思是孩子被自己丢了。是不是很别扭？不合语法也不合实际了。