09 Haidian high school final liberal arts mathematics examination questions

The final exercise of the second semester of grade three in Haidian District, Beijing

Mathematics (liberal arts)

Precautions:

1. Fill in the school, class and name clearly before answering the question.

2. After choosing the answer for each question in Book 1, black the answer label of the corresponding question on the answer sheet with a pencil. Write the answers to the questions in Volume 2 directly on the test paper with a pen or ballpoint pen.

1. Multiple-choice question: This big question has ***8 small questions, each with 5 points and ***40 points. Choose one of the four options listed in each small question that meets the requirements of the topic.

1. If set, b∨ () equals ().

a.{5} b.{ 1，2，5} |？ ~Y0`QHa')2d~V: [This material comes from the treasure house of the college entrance examination channel of your state learning network] |? ~Y0`QHa')2d~V:

c.{ 1，2，3，4，5} d。

2. arithmetic progression tolerance d

a.b.

c.d.

3. If the inverse function of the function+1 is, the image of the function is roughly ().

a.b. c. d。

4. The focal length of hyperbola is 10, so the value of real number m is ().

a.- 16 b . 4 c . 16d . 8 1

5. If α and β are two different planes, and M and N are two different straight lines, the following proposition is incorrect ().

A. Then

M⊥α, then n⊥α.

C.n ‖ α, n⊥β, and then α ⊥β.

D.α ∩ β = m, and n is equal to the angle formed by α and β, then m ⊥ n.

6. If, the following inequality must be true ().

a.b.

c.d.

7. There are four boys and two girls in a science and technology group. Now choose three students to take part in the competition, at least one of whom is a girl.

The number of species selected by different selection methods is ()

a.b. c. d。

8. If is, then ""is ""

( )

A. Necessary and sufficient conditions B. Sufficient and unnecessary conditions

C. Necessary and insufficient conditions D. Neither sufficient nor necessary conditions

Fill-in-the-blank question: This big question is ***6 small questions, each with 5 points and ***30 points. Fill in the answers on the lines in the questions.

9. The solution set of inequality is.

10. After the circle is translated according to the vector =( 1, -2), a circle c' is obtained, the radius of the circle c' is, and the center coordinate is.

1 1. At the same time, for the same area, the probability of accurate weather forecast by two meteorological stations in the city and district is,

The probability of accurate forecast by two meteorological stations does not affect each other, so the probability of accurate forecast by at least one meteorological station at the same time is.

12. As shown in the figure, a dihedral angle D-AB-F consists of a square abcd with two sides, so the distance from point D to point F is, and the distance from point D to plane abef is.

13. If the domain of the function is r,

The value of is.

14. Use the following method to "split" the n power of a natural number m greater than or equal to 2.

Similarly, the largest number in the "split" of 52 is that if the smallest number in the "split" of 52 is 2 1, then the value of m is.

Third, the solution: this big question is ***6 small questions, and ***80 points. The solution should be written in proof process or calculus steps.

15. (This small question *** 13 points)

known function

(1) Find the minimum positive period and maximum value of the function;

(2) How can the image of a function be translated and expanded through the image of a new function?

Arrive?

16. (This small question *** 13 points)

Given a function, the tangent of the image of this function at point (2,) is parallel to the X axis.

(1) n is expressed by an algebraic expression about m;

(2) When m= 1, find the monotone interval of the function.

17. (This small question *** 14 points)

As shown in the figure: in the triangular pyramid P-abc, pb⊥ bottom abc, ∠ BAC = 90, Pb = ab = AC = 4°, and point E is the midpoint of pa.

(1) verification: ac⊥ plane pab；;

(2) Find the distance between the straight line be and ac;

(3) Find the included angle between the straight line pa and the plane pbc.

18. (This small question *** 13 points)

In the plane rectangular coordinate system, O is the coordinate origin, two fixed points A (1, 0) and b(0,-1) are known, and the moving point P () satisfies:

(1) Find the trajectory equation of point P;

(2) Let the locus of point P intersect the hyperbola at two different points m and n.. if

A circle with a diameter of mn passes through the origin, and the eccentricity of hyperbola C is equal to the equation of hyperbola C.

19. (This small question *** 13 points)

The sum of the first n terms of a sequence is true for anyone, where m is a constant and m.

(1) Verification: Series is geometric series;

(2) Remember that the common ratio of the sequence is q, and let the sequence satisfy;

). Verification: The series is arithmetic progression;

(3) Under the condition of (2), let the sum of the first n terms of the series be. Verification:

20. (This small question * *14 points)

The domain of the function is r, which satisfies the following conditions:

(1) For anyone, there is;

(2) For any, any;

③

( 1);

(2) Verification: It is a monotone increasing function on R;

(3) If yes, verify that:

The final exercise of the second semester of grade three in Haidian District, Beijing

Mathematics (liberal arts) answer

First, multiple-choice questions (this big topic ***8 small questions, 5 points for each question, ***40 points)

1.b 2.d 3.a 4.c 5.d 6.a 7.c 8.b

2. Fill in the blanks (6 small questions in this big question, 5 points for each small question, 30 points for * * *)

9. 10.(2 points) (0,0) (3 points) 1 1.0.98

12.2 (2 points) (3 points) 13. -6 14.9 (2 points) 5(3 points)

Third, answer the question (this big question is ***6 small questions, ***80 points)

1 5. (* *13 points) Solution: (1) .............................................................................................................................

) ........................................ 4 points.

∴ t = ... 6 points.

(2) First, move (the image) to the left by one unit to obtain an image; Then the abscissa of the image is changed to half of the original, and the ordinate is unchanged, and the obtained image is obtained. ...................................................................................................................................................

Or first, change the abscissa of the image to half of the original, and the ordinate remains unchanged to get the function.

Image of; Move the image to the left by one unit, and the obtained image ............................................................. 13 points.

16. (* *13 points) Solution: (1) ...................................................................................................................., 2 points.

According to the known conditions: ∴ 3m+n = 0.....................4 points ∴ n =-3m...6 points.

(2) If m= 1, then n =-3 ................................ scored 7 points.

, making ...................... 8 points.

Or 12 points.

The monotonic increasing interval of ∴ is (-∞, 0), (2,+∞).

The monotone decreasing interval of ∴ is (0,2) ....................................................13.

17. (* *14 points)

Solution 1: (1)∵ triangular pyramid P-abc, pb⊥ bottom abc, ∠ BAC = 90.

Ba ⊥∴ Pb ⊥ AC ................................ 4 points.

∫Pb∩ba = b∴ac⊥ Plane PAB ... 4 o'clock.

(2)∵pb=ba=4, and point E is the midpoint of pa.

∴ Yes ⊥ ................................................................................................................................................................

According to (1), AC ⊥ ea .......................... scored 6 points.

∴ea is the common vertical segment of straight lines be and ac on different planes. ..........................................................................................................................................................

∵pb⊥ab ∴△pba is a right triangle with .......................... 8 points.

∴ EA = PA =× 4 = 2 ∴ The distance between BE and ac is 2.......................9 minutes.

(3) Take the midpoint D of bc and connect ad, with PD∶ab = AC = 4 and ∠ BAC = 90.

∴bc⊥ad Advertisement =2 ∵pb⊥ abc at the bottom, abc at the bottom of the advertisement.

∴ Pb ⊥ AD: Pb ∩ BC = B ∴ AD ⊥ Plane PBC ................11min.

∴pd is the projection of pa on pbc plane, ∴∠∠ APD is the angle .............................................. 12 minute formed by pa and pbc plane.

In rt△adp.

∴∠ APD = 30 ....................14 minutes ∴∴ The angle between PA and pbc plane is 30.

Solution 2: (1) is the same as solution 1. .............................. scores 4 points.

(2) 9 points for the same solution.

(3) If the intersection point A is ad//pb, then the ad⊥ plane abc

As shown in the figure, the coordinate origin is used to establish a spatial rectangular coordinate system.

Then a (0 0,0,0), b (-4,0,0), c (0 0,4,0),

P (-4,0,4) 4)................. 10/0.

................ 1 1 min

Let the normal vector of the plane pbc.

.................... 12 point

=( 1,-1, 0) = (4, 0, -4), let the angle formed by the straight line pa and the plane pbc be

Sin = cos<,> ........................13.

The angle formed by the straight line pa and the plane pbc is 30 ...........................14 minutes.

1 8. (* *13 points) Solution: (1) .............................................................................................................................

That is, the trajectory equation of point P is

(2) From: =0

The locus of point p intersects with hyperbola c at two different points m, n,

and

Okay, ..............., six.

∫ A circle with a diameter of mn passes through the origin, that is:

that is

That is, ① 8 points.

② 10.

∴ The solutions of ① and ② conform to the formula (*).

The equation of hyperbola c is ........................ 13.

19.(* * * 13) Proof: (1) When n= 1

(1) (2) 2 points for .....................................

①-② Score: ... 3 points.

..............................., 4 points.

∴ Series is a geometric series whose first term is 1, and the common ratio is 4 points.

27 points.

9 points ... 9 points.

∴ Series {} is a arithmetic progression, the first term is 1, and the tolerance is 1.

(3) If you get n from (2), then ... 10 point ... 1 1 minute.

.................. 12 point

........................ 13.

20. (* * 1 4) Option 1: (1) Place an order, and score: ...................................................................................................

..............................., 3 points.

(2) Take,, and. Set rules.

............................., 4 points.

On r, it is a monotonically increasing function ... 10 score.

(3) From (1)(2)

..... 1 1 min

And ... 14.

Solution 2: (1)∵ For any x, y∈r, there is

..... 1 point ∴ add 2 points when appropriate.

∵ Any x∈r, x ……………………………………………………………………………….

(2) 6 points.

Is a monotone increasing function on R, that is, a monotone increasing function on R; ...... 10 point