问题 解答题
在△ABC中,角A,B,C所对边分别为a,b,c,且向量
m
=(sinA,sinB),
n
=(cosB,cosA),满足
m
n
=sin2C.
(1)求角C的大小;
(2)若sinA,sinC,sinB成等差数列,且
AC
•(
AC
-
AB
)=18
,求边c的长.
答案

(1)

m
n
=sinAcosB+sinBcosA=sin(A+B)

对于△ABC,A+B=π-C,0<C<π,∴sin(A+B)=sinC

m
n
=sinC

又∵

m
n
=sin2C,

∴sin2C=2sinCcosC=sinC,即cosC=

1
2
,又C∈(0,π)

C=

π
3

(2)由sinA,sinC,sinB成等差数列,得2sinC=sinA+sinB

由正弦定理得2c=a+b,

AC
•(
AC
-
AB
)=18,

AC
BC
=18,

得abcosC=18,即ab=36,

由余弦定理c2=a2+b2-2abcosC=(a+b)2-3ab,

∴c2=4c2-3×36,即c2=36,

∴c=6.

阅读理解

Reading comprehension.

     I've loved my mother's desk since I was just tall enough to see above the top of it as mother sat doing

letters. Standing by her chair, looking at the ink bottle, pens and white paper, I decided that the act of writing

must be the most wonderful thing in the world.

     Years later, during her final illness, mother kept different things for my sister and brother. "But the desk,"

she said again, "is for Elizabeth."

     I never saw her angry, never saw her cry. I knew she loved me; she showed it in action. But as a young

girl, I wanted heart-to-heart talks between mother and daughter.

     They never happened. And a gulf opened between us. I was "too emotional (易动感情的)". But she lived

"on the surface".

     As years passed and I had my own family. I loved my mother and thanked her for our happy family. I

wrote to her in careful words and asked her to let me know in any way she chose that she did forgive me.

     I posted the letter and waited for her answer, none came.

     My hope turned to disappointment, then little interest and, finally, peace. It seemed that nothing happened.

I couldn't be sure that the letter had even got to Mother. I only knew that I had written it, and I could stop

trying to make her into someone she was not.

     Now the present of her desk told me that she was pleased that writing was my chosen work though she'd

never been able to. I cleaned the desk carefully and found some papers inside-a photo of my father and a one-

page letter, folded (折叠) and refolded many times.

     Give me an answer, my letter asks, in any way you choose, Mother, you always chose the act that speaks

louder than words.

1. The writer began to love her mother's desk _____.

A. after Mother died

B. before she became a writer

C. when she was a child

D. when mother gave it to her

2. The passage shows that _____.

A. Mother was cold on the surface but kind in her heart to her daughter

B. Mother was too serious about everything her daughter had done

C. Mother cared much about her daughter in words

D. Mother wrote to her daughter in careful words.

3. The world "gulf" in the passage means _____.

A. deep understanding between the old and the young

B. different ideas between the mother and the daughter

C. free talks between mother and daughter

D. part of the sea going far in land

4. What did Mother do with her daughter's letter asking for forgiveness?

A. She had never received the letter.

B. For years, she often talked about the letter.

C. She didn't forgive her daughter at all in all her life.

D. She read the letter again and again till she died.

5. What's the best title of the passage?

A. My letter to Mother

B. Mother and Children

C. My Mother's Desk

D. Talks between Mother and me

单项选择题