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kb3628com:金版新学案(人教版)高中数学选修1-1练习:1.4.1、2、3存在量词(含答案)

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第一章 1. 4.1 1. 4.2 1. 4.3 一、选择题(每小题 5 分,共 20 分) 1.下列语句不是特称命题的是( A.有的无理数的平方是有理数 B.有的无理数的平方不是有理数 C.对于任意 x∈Z,2x 是偶数 D.存在 x∈R,2x+1 是奇数 解析: A、B、D 中含有存在量词是特称命题,C 中含有全称量词是全称命题. 答案: C 2.将“x2+y2≥2xy”改写成全称命题,下列说法正确的是( A.? x,y∈R,都有 x2+y2≥2xy 2 B.? x0,y0∈R,使 x0 +y2 0≥2x0y0 ) ) C.? x>0,y>0,都有 x2+y2≥2xy 2 D.? x0<0,y0<0,使 x2 0+y0≤2x0y0 解析: 这是一个全称命题,且 x,y∈R,故选 A. 答案: A 3.命题“所有能被 2 整除的整数都是偶数”的否定是( A.所有不能被 2 整除的整数都是偶数 B.所有能被 2 整除的整数都不是偶数 C.存在一个不能被 2 整除的整数是偶数 D.存在一个能被 2 整除的整数不是偶数 解析: 原命题是全称命题,其否定是:存在一个能被 2 整除的数不是偶数. 答案: D 4.已知命题 p:? x∈R,ax2+2x+3>0.如果命题? p 是真命题,那么 a 的范围是( 1 A.a< 3 1 C.a≤ 3 1 D.a≥ 3 1 B.0<a≤ 3 ) ) 解析: ? p 为:? x∈R,使 ax2+2x+3≤0. ①当 a≤0 时,存在 x∈R,使 ax2+2x+3≤0 成立; ②当 a>0 时,Δ=4-12a≥0 时? p 成立; 1 ∴0<a≤ . 3 1 综上 a≤ . 3 答案: C 二、填空题(每小题 5 分,共 10 分) 5.下列命题中的全称命题是________;特称命题的是______. ①正方形的四条边相等; ②有些等腰三角形是正三角形; ③正数的平方根不等于 0; ④至少有一个正整数是偶数. 解析: ①③是全称命题,②④是特称命题. 答案: ①③ ②④ 6.给出下列四个命题: ①有理数是实数;②有些平行四边形不是菱形;③对任意 x∈R,x2-2x>0;④有一个 素数含有三个正因数. 以上命题的否定为真命题的序号是____________. 解析: 写出命题的否定,易知③④的否定为真命题,或者根据命题①、②是真命题, ③、④为假命题,再根据命题与它的否定一真一假,可得③④的否定为真命题. 答案: ③④ 三、解答题(每小题 10 分,共 20 分) 7.判断下列语句是全称命题,还是特称命题: (1)凸多边形的外角和等于 360° ; (2)有的向量方向不定; (3)对任意角 α,都有 sin2α+cos2α=1; (4)有一个函数,既是奇函数又是偶函数; (5)若一个四边形是菱形,则这个四边形的对角线互相垂直. 解析: (1)可以改写为“所有的凸多边形的外角和等于 360°”,故为全称命题. (2)含有存在量词“有的”,故是特称命题. (3)含有全称量词“任意”,故是全称命题. (4)含有存在量词“有一个”,故为特称命题. (5)若一个四边形是菱形,也就是所有的菱形,故为全称命题. 8.判断下列命题的真假,并写出它们的否定: (1)? α,β∈R,sin(α+β)≠sin α+sin β; (2)? x0,y0∈Z,3x0-4y0=20; (3)在实数范围内,有些一元二次方程无解; (4)正数的绝对值是它本身. 解析: (1)由于 α=β=0 时,sin(α+β)=sin α+sin β,所以命题为假命题, 否定为:? α0,β0∈R,sin(α0+β0)=sin α0+sin β0; (2)真命题,否定为:? x,y∈Z,3x-4y≠20; (3)真命题,否定为:在实数范围内,所有的一元二次方程都有解; (4)是全称命题,省略了量词“所有”,命题为真命题.否定为:有的正数的绝对值不是它 本身. 9.(10 分)已知命题“存在 x0∈R,ax2 0-2ax0-3>0”是假命题,求实数 a 的取值范围. 2 解析: 因为命题“存在 x0∈R,ax2 0-2ax0-3>0”的否定为“对于任意 x∈R,ax -2ax- 3≤0 恒成立”,由“命题真,其否定假;命题假,其否定真”可知该命题的否定是真命题. 事实上,当 a=0 时,对任意的 x∈R,不等式-3≤0 恒成立; 当 a≠0 时,借助二次函数的图象,数形结合,很容易知道不等式 ax2-2ax-3≤0 恒成立 的等价条件是 a<0 且其判别式 Δ=4a2+12a≤0,即-3≤a≤0; 综上知,实数 a 的取值范围是[-3,0].

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