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2018年高三最新 北京市西城区2018届高三理科数学一模试题 精品


北京市西城区 2018 年抽样测试 高三数学试卷(理科) (2018.5) 参考公式: 三角函数的和差化积公式 s i an? s i ?n ? 2s i ? n ? ? c ? os ? ? 2 2 s i an? s i ?n ? 2c ? os ? ? s i ? n ? ? 2 2 c oas? c o?s ? 2c ??? os c ??? os 2 2 c oas? c o?s ? ?2s i ?n ? ? s i ?n ? ? 2 2 球体的体积公式 V球 ? 4 ?R3 3 其中 R 表示球的半径 一、选择题:本大题共 12 小题;每小题 5 分,共 60 分。在每小题给出的四个选项中, 只有一项是符合要求的。请选出正确答案。 1.已知集合 M ? ?x | x2 ? x ? 0}, N ? {x | x ? 1}, 则 M∩N= ( ) A. {x|x≥1} B.{x |x>1 } C. Φ D. {x|x<0 或 x>1} 2.已知 f (x) ? sin(x ? ? ) , g(x) ? cos(x ? ? ) ,则 f(x)的图象 ( ) 2 2 A.与 g(x)的图象相同 B.与 g(x)的图象关于 y 的轴对称 C.是由 g(x)的图象向左平移 ? 个单位得到的 2 D.是由 g(x)的图象向右平移 ? 个单位得到的 2 3.如果复数 3 ? i 的辐角主值为θ ,那么复数-2+i 的辐角主值是 ( ) 1?i A.-θ B. ? ? ? 2 C.? ?? D. 2π -θ 4.已知直线 l1 : y ? x ? sin a 和直线 l2 : y ? 2x ? c ,则直线 l1 与 l2 A.通过平移可以重合 B.不可能垂直 () C.可能与 x 轴围成等腰直角三角形 D.通过绕 l1 上某一点旋转可以重合 5.已知点 P 的极坐标为(1,π ),那么过点 P 且垂直于极轴的直线的极坐标方程为 ( ) A.ρ =1 B.ρ =cosθ C.ρ cosθ =-1 D. ρ cosθ =1 6.某乒乓球队共有男女队员 18 人,现从中选出男女队员各一人组成一对双打组合由于 在男队员中有两人主攻单打项目,不参与双打组合,这样一共有 64 种组合方式,则乒乓球 队中男队员的人数为 ( ) A. 10 人 B. 8 人 C. 6 人 D. 12 人 7.一个圆柱的轴截面是正方形,其体积与一个球的体积之比为 3:2。 则这个圆柱的侧 面积与这个球的表面积之比为 ( ) A. 1:1 B.1: 2 C. 2 : 3 D. 3:2 8 . 已 知 函 数 y=f(x) 的 图 象 如 图 甲 所 示 , y=g(x) 的 图 象 如 图 乙 所 示 。 则 函 数 y ? f (x) ? g(x) 的图象可能是 ( ) 9.把长为 12 厘米的细铁丝截成两段,各自围成一个正三角形,那么这两个正三角形 面积之和的最小值是 ( ) A. 3 2 3cm 2 B. 4cm2 C. 3 2cm2 D. 2 3cm2 10. (如图) 正方体 ABCD ? A1B1C1D1 中,点 P 在侧面 BCC1B1 及其边界上运动,并 且总是保持 AP?BD1 ,则动点 P 的轨迹是 ( ) A.线段 B1C B.线段 BC1 C. BB1 中点与 CC1中点连成的线段 D. BC 中点与 B1C1 中点连成的线段 11.对于抛物线 y 2 ? 2x 上任意一点 Q, 点 P(a,0)都满足|PQ|≥|a|,则 a 的取值范围是 () A. (-∞,0) B. (-∞,1] C. [0,1] D. (0,1) 12.已知 a>0 且 a ≠1, f (x) ? x 2 ? a x ,当 x ? (?1,1) 时,均有 f (x) ? 1 ,则实数 a 2 的取值范围是 ( ) A. (0, 1] ?[2,??] 2 B.[1 ,1) ? (1,4] 4 C.[1 ,1) ? (1,2] 2 D. (0, 1] ? [4,??) 4 二、填空题:本大题共 4 小题,每小题 4 分。共 16 分,把答案填在题中横线上。 13.在 (x ? 1 )6 的展开式中,常数项的值是__________________.(用数字作答) x 14.已知抛物线 y ? x 2 ? bx ? 1 ,当 b(b∈R)变化时,抛物线顶点轨迹的普通方程为 ______. 15.G 是正三角形 ABC 的中心,过 G 平行于 BC 的直线交 AB 于 E,交 AC 于 F,沿直 线 EF 将三角形折成直二面角 A-EF-C。则直线 AC 与平面 EFCB 所成的角的大小为 __________. 16.设函数 f (x) ? lg( x2 ? ax ? a ?1) ,给出下述命题: ①f(x)有最小值; ②当 a=0 时,f(x)的值域为 R; ③当 a>0 时,f(x)在区间[2,+∞)上有反函数; ④若 f(x)在区间[2,+∞)上单调递增,则实数 a 的取值范围是 a ≥-4. 则其中正确的命题是_______________.(要求:把正确命题的序号都填上) 三、解答题:本大题共 6 小题,共 74 分,解答应写出文字说明,证明过程或演算步骤。 17.(本小题满分 12 分) 已知函数 f (x) ? sin(x ? ? ) ? sin(x ? ? ) ? cosx ? a 6 6 (Ⅰ)求函数 f(x)的最小正周期; (Ⅱ)若 x ?[? ? , ? ] 时,f(x)的最大值为 1,求 a 的值。 22 18.(本小题满分 12 分) (a∈R, a 是常数), 数列{an }的前 n 项和 Sn ? a ? 2

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