Types of geometric probability in the mathematical space of college entrance examination

(18) (the full score of this small question is 12)

A wholesale market makes statistics on the weekly sales volume (unit: tons) of a commodity, and the statistical results of the latest 100 week are shown in the following table:

(1) According to the above statistical results, find out the frequency of selling 2 tons, 3 tons and 4 tons per week respectively;

(2) It is known that the sales profit of the commodity per ton is 2000 yuan. Represents the sum of the two-week sales profits of this commodity (unit: thousand yuan). If the above frequency is taken as the probability, the weekly sales volume is independent of each other, how is it calculated? Distribution list and mathematical expectation.

Answer: (18) This small question mainly examines the basic knowledge such as frequency, probability and mathematical expectation, and examines the ability to solve practical problems by using probability knowledge. The perfect score is 12.

Solution: (1) The weekly sales volume is 2 tons, and the frequency of 3 tons and 4 tons is 0.2, 0.5 and 0.3 respectively. ..... 3 points

(Ⅱ)? The possible values of are 8, 10, 12, 14, 16 and.

p(？ =8)=0.22=0.04,

p(？ = 10)=2×0.2×0.5=0.2,

p(？ = 12)=0.52+2×0.2×0.3=0.37,

p(？ = 14)=2×0.5×0.3=0.3,

p(？ = 16)=0.32=0.09.

The distribution list is

8? 10? 12? 14? 16

p？ 0.04? 0.2? 0.37? 0.3? 0.09

..... 9 points

f？ = 8× 0.04+10× 0.2+12× 0.37+14× 0.3+16× 0.09 =12.4 thousand yuan)? ..... 12 point

(19) This small question mainly examines the basic knowledge of line-plane relationship, plane-plane relationship and triangle solution in space, and examines the spatial imagination and logical ability. The full score is 12.

Solution 1:

(i) Prove that in a cube, AD'? A'D,AD'⊥AB, which is also known to be available.

PF‖A′D，PH‖AD′，PQ‖AB，

So what? PH⊥PF,PH⊥PQ，

So what? PH⊥ aircraft PQEF.

So plane PQEF and plane PQGH are perpendicular to each other, ... 4 points.

(2) Evidence: According to (1)

The PQEF section and PQCH section are both rectangular, and PQ= 1, so the sum of the areas of PQEF section and PQCH section is

, is a fixed value.

Answer: (19) This small question mainly examines the basic knowledge of line-plane relationship, plane relationship and triangle solution in space, and examines the spatial imagination and logical ability, with a full score of 12.

Solution 1:

(i) Prove that in a cube, AD'? A'D,AD'⊥AB, which is also known to be available.

PF‖A′D，PH‖AD′，PQ‖AB，

So what? PH⊥PF,PH⊥PQ，

So what? PH⊥ aircraft PQEF.

So plane PQEF and plane PQGH are perpendicular to each other, ... 4 points.

(2) Evidence: According to (1)

The PQEF section and PQCH section are both rectangular, and PQ= 1, so the sum of the areas of PQEF section and PQCH section is

, is a fixed value. Eight points.

(III) Solution: Connect BC' intersection EQ at point M. 。

Because PH‖AD', PQ‖AB,

So plane ABC'D' and plane PQGH are parallel to each other, so the angle formed by D'E and plane PQGH is equal to

The angle formed by d ′ e and plane ABC ′ d ′ is equal.

In the same way as (i), we can prove the EQ⊥ plane PQGH and know the EM⊥ plane ABC'D', then the ratio of EM to D'E is a sine value.

Let AD' and PF intersect at n point and connect EN, which is known from FD = l-B.

Because AD'⊥ plane PQEF, we also know that D'E is equal to plane PQEF. Horn,

So what? d ' E =？ That is to say,

The solution shows that e is the midpoint of BC.

So EM= and D'E=? ,

Therefore, the sine value of the angle between D'E and plane PQCH is? .

Solution 2:

Taking D as the origin, rays DA, DC and DD' are the positive semi-axes of X, Y and Z axes respectively, and the spatial rectangular coordinate system D-x, Y and Z as shown in the figure is established from known DF-l-b, so

A( 1，0，0)，A′( 1，0， 1)，D(0，0，0)，D′(0，0， 1)，

P( 1，0，b)，Q( 1， 1，b)，E( 1，-b， 1，0)，？

F( 1-b，0，0)，G(b， 1， 1)，H(b，0， 1)。

(i) Prove that in the established coordinate system, we can get

Because? Is the normal vector of the plane PQEF.

Because? Is the normal vector of the plane PQGH.

Because? ,

So plane PQEF and plane PQGH are perpendicular to each other ... 4 points.

(2) Proof: Because? So, so PQEF is a rectangle, and PQGH is also a rectangle.

Can be obtained in the established coordinate system?

So what? ,

So what is the sum of the areas of section PQEF and section PQCH? , is a fixed value. Eight points.

(3) Solution: From the known? Horn again? free

that is

So the sine value of the angle between D'E and plane PQGH is

..... 12 point