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2016_2017学年高中数学第1章计数原理5二项式定理第1课时二项式定理课后演练提升

2016-2017 学年高中数学 第 1 章 计数原理 5 二项式定理 第 1 课时 二项式定理课后演练提升 北师大版选修 2-3
一、选择题

? 2 1?5 4 1.在二项式?x - ? 的展开式中,含 x 的项的系数是( ?
x?
A.-10 C.-5 B.10 D. 5

)

? 2 1?5 r 2(5-r) r -r 解析: ∵?x - ? 的展开式的通项为 C5x (-1) x ?
x?
=(-1) C5x
r r 10-3r

.

令 10-3r=4,得 r=2, ∴x 项的系数为 C5=10. 答案: B 2.设 n 为自然数,则 Cn2 -Cn2 A.2
n
0 n 1 n-1 4 2

+…+(-1) Cn2 B. 0 D. 1

k k n-k

+…+(-1) Cn等于(

n n

)

C.-1 解析: 原式=(2-1) =1. 答案: D
n

3.在(1-x) -(1-x) 的展开式中,含 x 的项的系数是( A.-5 C.-10
5 3 3 5

5

6

3

)

B. 5 D.10
6 3 3 3

解析: (1-x) 中 x 的系数-C =-10,-(1-x) 中 x 的系数为-C6·(-1) =20, 故(1-x) -(1-x) 的展开式中 x 的系数为 10. 答案: D 2 n 2 4.如果(3x - 3) 的展开式中含有非零常数项,则正整数 n 的最小值为(
5 6 3

x

)

A.3 C.6 解析: 由展开式的通项公式得:
2 n-r r r n-r r 2n-5r Tr+1=Cr (- 3) =Cn3 (-2) x . n(3x ) x

B. 5 D.10

2

5 ∵展开式中含有非零常数项,∴2n-5r=0,即 n= r, 2 ∵n 为正整数,∴当 r=2 时,n 取最小值 5,选 B.

1

答案: B 二、填空题 5.x (1-x) 展开式中含 x 的系数为____________. 解析: 由题意知即求(1-x) 展开式中 x 的系数, 即 C6(-1) =15. 答案: 15 6.在(x-a) 的展开式中,x 的系数是 15,则实数 a=____________. 解析: (x-a) 中 x 的系数为 C10(-a) =15. 1 3 即-120a =15,∴a=- . 2 1 答案: - 2 三、解答题
10 7 3 3 10 7 2 2 6 2 2 6 4

? 7.求?2 x+ ? ?

1 ?4

x?

? 的展开式.
1 ?4

? 解析: ?2 x+
= C 4 (2 x ) ?
0 4

x?

?

? 1 ?0 1 ? 1? 2 ? 1 ?2 3 ? 1 ?3 3 2 1 ? + C 4 (2 x ) · ? ? + C 4 ·(2 x ) · ? ? + C 4 (2 x ) · ? ?+ ? x? ? x? ? x? ? x?

C4·(2 x) ·?
4 0

? 1 ?4 ? ? x?
x x

8 1 2 =16x +32x+24+ + 2. 8.(1)求(1+2x) 的展开式的第 4 项的系数;
7

? 1?9 3 (2)求?x- ? 的展开式中 x 的系数及二项式系数. ?
x?
解析: (1)(1+2x) 的展开式的第四项是 T3+1=C7(2x) =280x , ∴(1+2x) 的展开式的第四项的系数是 280.
7 3 7 3 3

? 1?9 ? 1?r r 9-r (2)∵?x- ? 的展开式的通项是 Tr+1=C9x ·?- ? ?
x?

? x?

=(-1) C9x

r r 9-2r



∴9-2r=3,r=3, ∴x 的系数(-1) C9=-84,x 的二项式系数 C9=84 ? 尖子生题库 ?☆☆☆ 9.求 2 -3 除以 7 的余数.
2
30 3 3 3 3 3

解析: 2 -3=(2 ) -3=(8) -3=(7+1) -3 =C107 +C107 +…+C107+C10-3 =7×(C107 +C107 +…+C10)-2. 又∵余数不能为负数(需转化为正数), ∴2 -3 除以 7 的余数为 5.
30 0 9 1 8 9 0 10 1 9 9 10

30

3 10

10

10

3


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