Rose was wild with joy_______the result of the examination.

不选as 是因为后面是一个名词词组,不符合搭配,而选 at 是因为这个结构：to be wild with joy at...为...而狂喜

on hearing=when he heard,表示在听到这个消息的时候、
with hearing 表示伴随状态,但是用法错了.应该是with he hearing the news

He cried with joy when got the news that he gained admission to Fudan University.

wept for joy 和 wept with joy 一个意思,wept for joy是因为快乐而哭了,wept with joy 如果字面上说不一定是因为快乐而哭.但这么说就太死板了,口语的话,没有区别.

at
at可表示“听到”或“看到”.
句意：听到考试的结果,罗斯欣喜若狂.

at the result of
罗斯对于考试的结果欣喜若狂
at对于

for joy
高兴得 ...
例如：
She swooned into his arms for joy.
她喜极而昏倒在他的怀里.
一般与jump连用,jump for joy“欢欣鼓舞”的意思.
例如：
I fairly jumped for joy.
我简直高兴得跳了起来.
They jumped for joy.
他们高兴地跳了起来.
with joy
adj. 高高兴兴地(高兴得 ... )
例如：
Her eyes sparkled with joy.
她高兴得两眼闪光.
His heart bounded with joy.
他高兴得心头砰砰直跳.
I felt as if my heart would burst with joy.
我觉得自己高兴得心花怒放.
希望帮到你,祝学习愉快!

B.at
罗斯对于考试的结果欣喜若狂
at对于

热情的打招呼
[兔葛锐特味子jiu椅]

On hearing
before

Happiness is a tree,（幸福是颗树）
With joy,sweetness and excitement as its fruits.（拥有欢快,甜蜜,激情作为它的果实）
Happiness is a house,（幸福是一座房）
With gentleness,comfort and love as its furniture.（拥有温柔,舒适,和爱作为它的家具）
Happiness is a teacher,（幸福是名教师）
Who teaches us to love,to sacrifire and own.（它教导我们去爱,去牺牲,去拥有）
Happiness is a hope plus the faith of you and me.（幸福是希望加上你我的信念）
Happiness is Spring's wind,（幸福是春天的风）
Summer's flowers,（夏天的花朵）
Autumn's fruits （秋天的水果）
and Winter's snow.（通天的雪）
Happiness is shapeless and invisible at all.（幸福是没有形状并看不到的）
We need to touch it with full hearts.（我们需要用心去碰触它）
Happiness is a perfume you cannot pour on others without getting a few drops on yourself（幸福是香水,一种你必须先在自己身上点几滴才能泼向别人的香水）