2019南方科技大学机试样题练习

2019机试样题练习第⼀单元 数学

⼀、单项选择题（每题所设选项中只有⼀个正确答案，多选、错选或不选均不得分；每题1.5分）1．观察下列操作： 第1次操作：1335233

=+ 第2次操作：55331333

=++ 第3次操作：2505533=+Λ

如此反复操作，则第2017次操作后得到数是（ ） A ．55 B ．133 C ．250 D ．2602．已知数列}{n a 满⾜11=a ，n a a n n 21-=++，则201S =（ ） A ．20100-B ．20200-C ．20199-D ．20099-3．在ABC ?中，,,a b c 分别是⾓,,A B C 对边，若cos 3,cos1a B b A ==，则c=

4．在平⾯直⾓坐标系中，点)0,5(),0,0(),6,3(C B A ，则=∠BAC （ ）

5．在△ABC 中，若42cos 52cos 322=+-CB A ，则

C tan 的最⼤值为( ) A ．－34 B ．－43 C ．－24 D ．－22

6．ABC ?的外接圆的圆⼼为O ，半径为1，2AO AB AC =+u u u r u u u r u u u r，且||

||OA AB =u u u r u u u r ，则向量AC在向量CB u u u r⽅向上的投影为（ ）A ．

12 B ．23 C ． 12- D ．23-7．已知a 、b 为⾮零向量，则下列命题中真命题个数为（ ） ①若b a b a +=+，则a 与b ⽅向相同； ②若b a b a -=+，则a 与b⽅向相反； ③若b a b a -=+，则a 与b 有相等模；④若b a b a -=-，则a 与b ⽅向相同．A. 0B. 1C. 2D. 3

8．对任意向量a 、b ，下列关系式中不恒成⽴的是( ) A ．b a b a ||≤?B ．b a b a ||-≤-C ．22

||b a b a -≤-）（ D ．22

))((b a b a b a -=-+

9．如图，三棱锥A -BCD 的顶点A 、B 、D 为正⽅体的顶点，

C 为正⽅体棱的中点，现在正⽅体的⽹格纸上⽤粗线 画出的它的三视图，正确的是A B C D10．已知圆222

:()()2(0)C x a y a a a -+-=>及其外⼀点)2,0(A .若圆C 上存在点T 满⾜4π=

∠CAT ，则实数a 的取值范围是（ ）A. ()1,∞-B. [31,1)-C. [31,1]-D. [31,)-+∞

11．双曲线142222=-b

y a x 的右焦点F 与抛物线)0(42

>=p px y 的焦点重合，且在第⼀象限的交点为M ，MF 垂直于x 轴，则双曲线的离⼼率是 A ．222+ B ．22C ．12+D ．22+12.抛物线C ：2

8y x =的焦点为F ，准线为l ，P 是l 上⼀点，连接PF 并延长交抛物线C正前⽅ABCDAO C于点Q ，若4||||5

PF PQ =，则||QF =（ ）A ．3B ．4C ．5D ．613．椭圆22221(0)x y a b a b

+=>>的左、右焦点分别是12,F F ，点P 在椭圆上，21F PF ?的⾓平分线PM 交轴于点M ，若2

122||||||PM PF PF =，则该椭圆的离⼼率为( )．提⽰：内⾓平分线定理为|||

|||||2121M F M F PF PF =．A ．22 B ．2

C ．4D ．4

14.有6名同学参加校园歌⼿总决赛，观众中有四名同学猜测结果，其中甲同学猜测：4号

或5号选⼿得第⼀名；⼄同学猜测：3号选⼿不可能得第⼀名；丙同学猜测：1,2,6号选⼿中的⼀位获得第⼀名；丁同学猜测：4,5,6号选⼿都不可能获得第⼀名．⽐赛后发现没有并列名次，且甲、⼄、丙、丁中只有1⼈猜对⽐赛结果，此⼈是( ) A.甲、 B.⼄、 C.丙、 D.丁

15．在极坐标系中，曲线,sin 2:1θρ=C θρcos 32:2=C ，则1C 与2C 交点的坐标为（ ）A ．），（3π3 B ．），（6π3 C ．），（00D ．），（3π

3或），（00

16. 若复数z 满⾜|34|)i 43(i z +=-，则z 的虚部为( ).（A ）4-（B ）54-（C ）4 （D ）5

4 17．将离⼼率为1e 的双曲线1C 的实半轴长a 和虚半轴长()b a b ≠同时增加(0)m m >个单位长度，得到离⼼率为2e 的双曲线2C ，则（ ） A ．对任意的,a b ，12e e >B ．当a b >时，12e e >；当a b <时，12e e D ．当a b >时，12e e <；当a b <时，12e e >

⼆、不定项选择题（每题所设选项中⾄少有⼀个正确答案，多选、少选、错选或不选均不得分；每题2分）1．如图1，⼀个正四棱柱形的密闭容器底部镶嵌了同底的正四棱锥形实⼼装饰块，容器内

盛有a 升⽔时，⽔⾯恰好经过正四棱锥的顶点P ．如果将容器倒置，⽔⾯也恰好过点P （图2）．则下列四个命题正确的是

A ．正四棱锥的⾼等于正四棱柱⾼的53

B ．将容器侧⾯⽔平放置时，⽔⾯也恰好过点P

C ．任意摆放该容器，当⽔⾯静⽌时，⽔⾯都恰好经过点PD ．若往容器内再注⼊a 升⽔，则容器恰好能装满2. 已知函数xx f 2)(=，ax x x g +=2

)(（其中R a ∈）.对于不相等的实数21,x x ，设2121)()(x x x f x f m --=，2121)

()(x x x g x g n --=.则下列命题是真命题是（ ）.（A ）对于任意不相等的实数21,x x ，都有0>m ；

（B ）对于任意的a 及任意不相等的实数21,x x ，都有0>n ； （C ）对于任意的a ，存在不相等的实数21,x x ，使得n m =；（D ）对于任意的a ，存在不相等的实数21,x x ，使得n m -=.

3. 已知m ，n 是两条不同直线，α，β是两个不同平⾯，则下列命题正确是（ ）. （A ）若α，β垂直于同⼀平⾯，则α与β平⾏（B ）若m ，n 平⾏于同⼀平⾯，则m 与n 平⾏

（C ）若α，β不平⾏，则在α内不存在与β平⾏的直线 （D ）若m ，n 不平⾏，则m 与n 不可能垂直于同⼀平⾯第⼆单元 物理

⼀、单项选择题（每题所设选项中只有⼀个正确答案，多选、错选或不选均不得分；每题1.5分）

1. 质量和电荷量都相等的带电粒⼦和，以不同的速率经⼩孔垂直进⼊匀强磁场，运⾏的半圆轨迹如图附3-1-1中虚线所⽰，下列表述正确的是（ ） A.带负电，带正电 B.的速率⼩于的速率 C.洛伦兹⼒对、做正功 D.的运⾏时间⼤于的运⾏时间图12图

2. ⼀⽔平抛出的⼩球落到⼀倾⾓为的斜⾯上时，其速度⽅向与斜⾯垂直，运动轨迹如图2-3-8中虚线所⽰.⼩球在竖直⽅向下落的距离与在⽔平⽅向通过的距离之⽐为（）A． B. C.tanθ D.2tanθ

⼆、不定项选择题（每题所设选项中⾄少有⼀个正确答案，多选、少选、错选或不选均不得分；每题2分）1. 如图所⽰，物体P静⽌于固定的斜⾯上，P的上表⾯⽔平.现把物体Q轻轻地叠放在P上，则（）

A.P向下滑动B.P静⽌不动

C.P所受的合外⼒增⼤D.P与斜⾯间的静摩擦⼒增⼤

2. 如图，左侧调压装置可视为理想变压器，负载电路中55 Ω，A、V为理想电流表和电压表，若原线圈接⼊如图（b）所⽰的正弦交变电压，电压表⽰数为110V，下列表述正确是（）A.电流表的⽰数为2AB.原、副线圈匝数⽐为1∶2C.电压表的⽰数为电压的有效值D.原线圈中交变电压的频率为100Hz

第三单元英语

Reading Comprehension (4 passages,20 scores)

Directions: There are 4 passages in this part. Each passage is followed by 4

questions or unfinished statements. For each of them there are four choices marked A, B, C and D. You should decide on thebest choice and write the corresponding letter on the Answer Sheet.【Sample】

Daniel Anderson, a famous psychologist, believes it’s important to distinguish television’s influences on children from thoseof the family. We tend to blame TV, he says, for problems it doesn’t really cause, ignoring our own roles in shaping children’sminds.

One traditional belief about television is that it reduces a child’s ability to think and to understand the world. While watchingTV, children do not merely absorb words and images (影像). Instead, they learn both clear and hidden meanings from whatthey see. Actually, children learn early the psychology of characters in TV shows. Furthermore, as many teachers agree,children understand far more when parents watch TV with them, explaining new words and ideas. Yet, most parents use aneducational program as a chance to put their kids in front of the set and do something in another room.

Another argument against television is that it replaces reading as a form of entertainment. But according to Anderson, theamount of time spent watching television is not related to reading ability. TV doesn’t take the place of reading for most

children; it takes the place of similar sorts of recreation, such as listening to the radio and playing sports. Things like parents’educational background have a stronger influence on a child’s reading. “A child’s reading ability is best predicted by howmuch a parent reads.” Anderson says.

Traditional wisdom also has it that heavy television-watching lowers IQ (智商) scores and affects school performance. Buthere, too, Anderson notes that no studies have proved it. In fact, research suggests that it’s the other way around. “If you’resmart young, you’ll watch less TV when you’re older,” Anderson says. Yet, people of lower IQ tend to be lifelong televisionviewers.

For years researchers have attempted to show that television is dangerous to children. However, by showing that televisionincreases none of the dangerous effects as believed before, Anderson suggests that television cannot be criticized withoutconsidering other influences.

1. An educational program is best watched by a child _________.A. on his ownB. with other kidsC. with his parentsD. with his teachers

2. Which of the following is most related to children’s reading ability?A. Radio-listeningB. Television-watchingC. Parents’ reading list

D. Parents’ educational background3. Anderson believed that _________.

A. the more a child watches TV, the smarter he isB. the younger a child is, the more he watches TV

C. the smarter a child is, the less likely he gets addicted to TVD. the less a child watches TV, the better he performs at school4. What is the main purpose of the passage?A. To advise on the educational use of TV.B. To describe TV’s harmful effects on children.C. To explain traditional views on TV influences.D. To present Anderson’s un traditional ideas.