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2017-2018 高三上学期期中考试数学(理科)试卷 考试时间 120 分钟 分值 150 分 第Ⅰ卷 一.选择题:本大题共 12 小题,每小题 5 分,在每小题给出的四个选项中,只有一项是符合题 目要求的. 1 设集合 A={y|y=2x,x∈R},B={x|x2-1<0},则 A∪B=( ) A.(-1,1) C.(-1,+∞) B.(0,1) D.(0,+∞) ) 2 在等差数列{an}中,a1=0,公差 d≠0,若 am=a1+a2+…+a9,则 m 的值为( A.37 C.20 B.36 D.19 π 3 在△ABC 中,内角 A,B,C 所对的边分别是 a,b,c.若 c2=(a-b)2+6,C= ,则△ABC 3 的面积是( ) 9 3 3 3 A.3 B. C. D.3 3 2 2 4 命题 p:任意 x∈R,ax2+ax+1≥0,若非 p 是真命题,则实数 a 的取值范围是( A.(0,4] C.(-∞,0]∪[4,+∞) B.[0,4] D.(-∞,0)∪(4,+∞) ) 5.已知向量 a , b 满足 a ? 1 , a ? b ,则向量 a ? 2b 在向量 ? a 方向上的投影为( A.0 B .1 C. 2 D. ?1 ) 6.中国古代数学著作《算法统宗》中有这样一个问题:“三百七十八里关,初步健步不为难, 次日脚痛减一半,六朝才得到其关,要见次日行里数,请公仔细算相还.”其大意为:“有 一个人走 378 里路,第一天健步行走,从第二天起脚痛每天走的路程为前一天的一半,走了 6 天后到达目的地.”则该人最后一天走的路程为( ) A.24 里 B.12 里 C.6 里 D.3 里 7 向量 a ? ? cos25?,sin 25?? , b ? ?sin 20?,cos20?? ,若 t 是实数,且 u ? a ? tb ,则 u 的最小值 为( ) C. 2 1 D. 2 2 A . 2 B. 1 π? π 8 若函数 f(x)=sin? ?ωx+6?(ω>0)的图象的相邻两条对称轴之间的距离为2,且该函数图象关于 π? 点(x0,0)成中心对称,x0∈? ) ?0,2?,则 x0=( 5π π π π A. B. C. D. 12 4 3 6 9 已知函数 y=f(x) (x∈R)且在[0,+∞)上是增函数,g(x)=f(|x|) ,若 g(2x﹣1)<g (2) ,则 x 的取值范围是( A. (﹣ , ) C. ( ,+∞) ) B. (﹣∞, ) D. (﹣∞, )∪( ,+∞) 10 如图所示,在边长为 1 的正方形 OABC 中任取一点 P,则点 P 恰好 取自阴影部分的概率为( 1 A. 4 1 1 B. C. 5 6 ) 1 D. 7 <0 恒成立,则不等式 11.设 f(x)是定义在 R 上的奇函数,且 f(2)=0,当 x>0 时,有 x f(x)>0 的解集是 ( A.(-2,0)∪(2,+∞) C.(-∞,-2)∪(2,+∞) 2 ) B.(-2,0)∪(0,2) D.(-∞,-2)∪(0,2) 12 若 a,b 是函数 f(x)=x2-px+q(p>0,q>0)的两个不同的零点,且 a,b,-2 这三个数可 适当排序后成等差数列,也可适当排序后成等比数列,则 p+q 的值等于( A.6 B.7C.8 D.9 ) 第Ⅱ卷 二:填空题 每题 5 分,共计 20 分 13 已知 tan ? ? 3, 则 sin 2? ? 2 cos 2 ? ? _____________ 14.已知函数 f(x)=ln(1+x2),则满足不等式 f(2x-1)<f(3)的 x 的取值范围是________. π ?x+π?的图象分别交于 M, x+ ?, 15 若动直线 x=a(a∈R)与函数 f(x)= 3sin? g ( x ) = cos N 两点, ? 6? ? 6? 则|MN|的最大值为________. 16.对于任意实数 a,b 定义运算 a*b=(a+1)(b+1)-1,给出以下结论: ①对于任意实数 a,b,c,有 a*(b+c)=(a*b)+(a*c); ②对于任意实数 a,b,c,有 a*(b*c)=(a*b)*c; ③对于任意实数 a,有 a*0=a,则以上结论正确的是.(写出你认为正确的结论的所有序号) 三:解答题 共计 70 分 1 17(本题 10 分)已知数列{an}是公差为 3 的等差数列,数列{bn}满足 b1=1,b2= , 3 anbn+1+bn+1=nbn. (1)求{an}的通项公式; (2)求{bn}的前 n 项和. cos A cos B sin C 18 (本题 12 分)在△ABC 中,角 A,B,C 所对的边分别是 a,b,c,且 + = . a b c (1)证明:sin Asin B=sin C; 6 (2)若 b2+c2-a2= bc,求 tan B. 5 19(本题 12 分)已知二次函数 f(x)的最小值为-4,且关于 x 的不等式 f(x)≤0 的解集为 {x|-1≤x≤3,x∈R}. (1)求函数 f(x)的解析式; f?x? (2)求函数 g(x)= -4ln x 的零点个数. x 20. (本题 12 分) 已知函数 f(x)=2 sin ω xcos ω x+2cos ω x(ω >0),且 f(x)的最小正周期 为π . (1)求ω 的值及 f(x)的单调递减区间; (2)将函数 f(x)的图象向右平移 个单位长度后得到函数 g(x)的图象,求当 x∈[0, ]时 g(x)的 最大值. 2 → → → 21. (本题 12 分)已知△ABC 的周长为 6,|BC|,|CA|,|AB|成等比数列,求: (1)△ABC 面积 S 的最大值; → → (2)BA· BC的取值范围. 22. (本题 12 分)已知函数 f(x)=aln x(a>0)

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