2018届高考数学二轮复习第一部分专题五立体几何1.5.3空间向量与立体几何限时规范训练理

Lisa's birthday is on October 18th. It's on Wednesday. She is going to have a birthday party. She wants to invite Peter, Gao Wei and Li Yan to her birthday party. They can celebrate after school. Today Lisa's parents go shopping with Lisa. They want to buy a My dream is to work in a hospital.I thin that a(n) __1__ is one of the most important jobs in the world.__2__ people are busy working every day 限时规范训练 空间向量与立体几何

限时45分钟，实际用时

分值81分，实际得分

一、选择题(本题共6小题，每小题5分，共30分)

1．(2017·山东青岛模拟)已知正三棱柱ABC-A1B1C1的侧棱长与底面边长相等，则AB1与侧面

ACC1A1所成角的正弦值等于( )

A.C.6 42 2

B.D.

10 43 2

解析：选A.如图所示建立空间直角坐标系，设正三棱柱的棱长为2，则O(0,0,0)，B(3，0,0)，→→

A(0，－1,0)，B1(3，0,2)，则AB1＝(3，1,2)，则BO＝(－3，0,0)为侧面ACC1A1的法向量，

→→|AB1·BO||－3|6

故sin θ＝＝＝.→→22×34|AB1||BO|

2．在直三棱柱ABC-A1B1C1中，AA1＝2，二面角B-AA1-C1的大小为60°，点B到平面ACC1A1的

距离为3，点C到平面ABB1A1的距离为23，则直线BC1与直线AB1所成角的正切值为()

B.6D．2

A.7 C.5

解析：选A.由题意可知，∠BAC＝60°，点B到平面ACC1A1的距离为3，点C到平面ABB1A1的→→→距离为23，所以在三角形ABC中，AB＝2，AC＝4，BC＝23，∠ABC＝90°，则AB1·BC1＝(BB1

→→→

－BA)·(BB1＋BC)＝4，

→→

|AB1|＝22，|BC1|＝4，

→→

→→AB1·BC12cos〈AB1，BC1〉＝＝，

→→4|AB1|·|BC1|

→→

故tan〈AB1，BC1〉＝7.

Lisa's birthday is on October 18th. It's on Wednesday. She is going to have a birthday party. She wants to invite Peter, Gao Wei and Li Yan to her birthday party. They can celebrate after school. Today Lisa's parents go shopping with Lisa. They want to buy a My dream is to work in a hospital.I thin that a(n) __1__ is one of the most important jobs in the world.__2__ people are busy working every day 3．如图所示，在三棱锥P-ABC中，PA⊥平面ABC，D是棱PB的中点，已知PA＝BC＝2，AB＝

4，CB⊥AB，则异面直线PC，AD所成角的余弦值为()

B．－

3053010

A．－C.

3010305

D.

解析：选D.因为PA⊥平面ABC，所以PA⊥AB，PA⊥BC.过点A作AE∥CB，又CB⊥AB，则AP，AB，AE两两垂直．

如图，以A为坐标原点，分别以AB，AE，AP所在直线为x轴，y轴，z轴建立空间直角坐标

系，

则A(0,0,0)，P(0,0,2)，B(4,0,0)，C(4，－2,0)．

因为D为PB的中点，所以D(2,0,1)．

→→

故CP＝(－4,2,2)，AD＝(2,0,1)．

→→

→→AD·CP－630

所以cos〈AD，CP〉＝＝＝－.

→→105×26|AD|×|CP|

设异面直线PC，AD所成的角为θ，→→30

则cos θ＝|cos〈AD，CP〉|＝.

10

4．(2017·山西四市联考)在空间直角坐标系O-xyz中，已知A(2,0,0)，B(2,2,0)，C(0,2,0)，

D(1,1，2)．若S1，S2，S3分别是三棱锥D-ABC在xOy，yOz，zOx坐标平面上的正投影图形的面

积，则( )

A．S1＝S2＝S3

B．S2＝S1且S2≠S3C．S3＝S1且S3≠S2D．S3＝S2且S3≠S1