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2017 届高二第一学期周考数学理科试题(三)
一、选择题
1.在△ABC 中,若 a = 2 , b ? 2 3 , A ? 30 , 则 B 等于(
?

16. 0 ? x ?

?
2

, sin x ?

8 的最小值为 sin x

三、解答题
) D. 30 或 150
? ?

A. 60

?

B. 60 或 120

?

?

C. 30

?

17. (12 分)已知数列 {an } 的前 n 项和 Sn ? n2 ? 48n 。 (1)求数列的通项公式; (2)求 Sn 的最大或最小值。

1 2.在等比数列{ an }中,已知 a1 ? , a5 ? 9 ,则 a3 ? ( ) 9 A.1 B.3 C. ?1 D.±3 3.等比数列 ?an ? 中, a2 ? 9, a5 ? 243 , 则 ?an ? 的前 4 项和为(

18. (14 分)设数列 ?an ? 的前项 n 和为 S n ,若对于任意的正整数 n 都有 S n ? 2an ? 3n . ) (1)设 bn ? an ? 3 ,求证:数列 ?bn ? 是等比数列,并求出 ?an ? 的通项公式。

A. 81 B.120 C.168 D.192 4.已知{an}是等差数列,且 a2+ a3+ a8+ a11=48,则 a6+ a7= ( ) A.12 B.16 C.20 D.24 5.等差数列 {an } 的前 m 项和为 30,前 2 m 项和为 100,则它的前 3m 项和是( A.130 B.170 C.210 D.260 6.2.在△ABC 中,sin A∶sin B∶sin C=4∶3∶2,则 cos A 的值是 ( 1 A.- 4 1 B. 4 2 C.- 3 ) 。 ). 2 D. 3

)

19. 已知数列{an}的前 n 项和为 Sn, 且 an 是 Sn 与 2 的等差中项, 数列{bn}中, b1=1, 点P (bn, bn+1) 在直线 x-y+2=0 上. (1)求 a1 和 a2 的值; (2)求数列{an},{bn}的通项 an 和 bn; (3)设 cn=an·bn,求数列{cn}的前 n 项和 Tn. π 4 20.在△ ABC 中,角 A,B,C 的对边分别为 a,b,c,B= ,cosA= ,b= 3. 3 5 (1)求 sinC 的值; (2)求△ ABC 的面积. 21.(12 分)已知不等式 ax2-3x+6>4 的解集为{x|x<1 或 x>b}, (1)求 a,b; (2)解不等式 ax2-(ac+b)x+bc<0.

7.设 a ? b , c ? d ,则下列不等式成立的是(

a d A. a ? c ? b ? d B. ac ? bd C. ? D. b ? d ? a ? c c b 8.如果方程 x 2 ? (m ? 1) x ? m 2 ? 2 ? 0 的两个实根一个小于?1,另一个大于 1,那么实数
m 的取值范围是( A. (? 2,2 ) ) B. (-2,0) C. (-2,1) D. (0,1)

9.若 ?3 ? x ? 0 ,则 y ? x 9 ? x 2 的最小值为( )

9 3 1 C. ? D. 2 2 2 2 2 2 10.已知集合 A={x| x ? a ? 0 ,其中 a ? 0 },B={x| x ? 3x ? 4 ? 0 },且 A ? B = R,则实数 a 的
A. ? B. ) B. a ? ?4 C. a ? 4 D. 1 ? a ? 4 A c-b 11.在△ ABC 中,sin2 = (a、b、c 分别为角 A、B、C 的对应边),则△ ABC 的形状为( 2 2c A.正三角形 B.直角三角 C.等腰直角三角形 D.等腰三角形 12. x 、 y ? 0 ,且 A.最大值 16 取值范围( a?4 A.

9 2

22.某港口 O 要将一件重要物品用小艇送到一艘正在航行的轮船上,在小艇出发时,轮船位于港口 O 北偏西 30° 且与该港口相距 20 海里的 A 处, 并正以 30 海里/时的航行速度沿正东方向匀速行 驶.假设该小艇沿直线方向以 v 海里/时的航行速度匀速行驶,经过 t 小时与轮船相遇. (1)若希望相遇时小艇的航行距离最小,则小艇航行速度的大小应为多少?

)

(2)为保证小艇在 30 分钟内(含 30 分钟)能与轮船相遇,试确定小艇航行速度的最小值.

1 4 ? ? 1 ,则 xy 有( ) x y 1 1 B.最小值 16 C. 最大值 D. 最小值 16 16

二、填空题
1 ,则 ?ABC 的外接圆的半径为 _____. 2 ? 1 1? 2 14.若不等式 ax ? bx ? 2 ? 0 的解集是 ? ? , ? ,则 a ? b 的值为________。 ? 2 3?
13.在 ?ABC 中, 若 a ? 3, cos A ? ? 15.如图所示,为测一树的高度,在地面上选取 A、B 两点,从 A、B 两点分别测得树尖的仰角为 30° ,45° ,且 A、B 两点之间的距离为 60 m,则树高的高度为________.


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