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National unified entrance examination for ordinary primary and secondary schools in 2000
Shanghai Mathematics Examination Paper (Science, Engineering, Agriculture and Medicine)
Note to candidates: this question is * * *, with a total of 22 questions, with a full score of 150.
1. Fill-in-the-blank question (the full score of this big question is 48 points) This big question * * has 12 questions, and only the scores are required to be filled directly. If you fill in every blank correctly, you will get 4 points, otherwise you will get 0 points.
1. The known vector (-1, 2), =(3, m), if ┴, then m=.
2. The domain of the function is.
3. The focal coordinates of the conical section are.
4. Calculation: =.
5. If the known image of the inverse function passes through a point, then =.
6. According to the work report of the municipal government at the Third Session of the 11th Shanghai Municipal People's Congress, 1999, the GDP in Shanghai reached 403.5 billion yuan, and it is estimated that the GDP in Shanghai will increase by 9% in 2000. The municipal party committee and municipal government proposed that the annual natural growth rate of the permanent population in this city should be controlled at 0.08%. If GDP and population increase at this rate, the annual per capita GDP of this city will reach or exceed 65433.
(According to: 1999, the total resident population in this city is about 1300).
7. Proposition A: The base is a regular triangle, and the triangular pyramid with the projection of the vertex on the base as the center is a regular triangular pyramid. The equivalent problem B of proposition A can be: the bottom is a regular triangular pyramid, and the triangular pyramid is a regular triangular pyramid.
8. Let the function be an even function with a minimum positive period of 2, and its image on the interval [0, 1] is a line segment as shown in the figure, then on the interval [1, 2] =.
9. In binomial expansion, the coefficient of the term with the smallest coefficient is, (the result is expressed by numerical value).
10. There are three flags of red, yellow and blue, and the three flags of each color are marked with the numbers 1, 2 and 3 respectively. Now the probability of taking out the three flags is that the colors and numbers are different.
1 1. In the polar coordinate system, if a straight line passing through point (3,0) and perpendicular to the polar axis intersects with two points of the curve, then.
12. In arithmetic progression, if, then there is an equation. By analogy with the above properties, win-win: in this series, if, then there is an equation.
Second, multiple-choice questions (the full score of this big question is 16) This big question has four questions, and each question gives four conclusions, code-named A, B, C and D, of which one and only one conclusion is correct. The code name of the correct conclusion must be written in parentheses after the question, and 4 points will be scored if it is right. There is more than one code name for not choosing, choosing wrong or choosing (whether or not).
13. Complex number
[Answer] ()
14. Using different straight lines and different planes,,, gives the following three propositions:
(1) If, then. (2) If,, then.
(3) If,, then.
Where is the correct number?
(1) 0. 1。 (C)2。 (D)3。
[Answer] ()
15. If the set is:
.
[Answer] ()
16. The correct proposition among the following propositions is
(a) If the point is a point on the edge of the corner terminal, then.
(b) Only one angle can be satisfied simultaneously.
(c) At that time, the value of was constant.
(d) The solution set of trigonometric equation is.
[Answer] ()
Third, answer the question (the full score of this big question is 86 points). There are 6 questions in this big question. You must write the necessary steps to answer the following questions.
17. (The full mark of this question is 12)
It is known that the focus of an ellipse is 6 and the length of its major axis is 6. Let the orthogonal ellipse be at two points and find the midpoint coordinates of the line segment.
[Solution]
18. (The full mark of this question is 12)
As shown in the figure, in tetrahedral ABCD, AB, BC and BD are perpendicular to each other, AB=BC=2, E is the midpoint of AC, and the angle formed by non-planar straight line AD and BE is, so find the volume of tetrahedral ABCD.
[Solution]
19. (The full score of this question is 14) There are two small questions in this question. The full score of the first 1 question is 6, and the full score of the second one is 8. Known function.
(1) When, find the minimum value of the function:
(2) If any constant is true, try to find the range of real numbers.
[Solution ]( 1)
[Solution] (2)
20. (The full score of this question is 14) There are two small questions in this question, and the full score of 1 small question is 4, and the full score of the second small question is 10.
According to the instructions, the robot can complete the following actions on the plane: first, rotate the angle in place (counterclockwise is positive, clockwise is negative-), and then walk a distance in a straight line in the direction it faces.
(1) Now the robot is at the coordinate origin of the rectangular coordinate system, facing the positive axis direction. Try to give the robot an instruction to move to point (4,4).
(2) After completing this instruction, the robot found a ball rolling in a straight line at (17,0). It is known that the rolling speed of the ball is twice that of the robot walking in a straight line. If you ignore the time required for the robot to rotate in situ, where can the robot intercept the ball at the earliest? And give the instruction that the robot needs to intercept the ball (the result is accurate to two decimal places).
[Solution ]( 1)
[Solution] (2)
2 1. (The full score of this question is 16) There are three small questions in this question, and the full score of 1 small question is 4, the full score of the second small question is 6, and the full score of the third small question is 6.
On the XOY plane, for each natural number and point, there are a series of points on the image of the function, which form an isosceles triangle with its vertices.
(1) An expression for finding the ordinate of a point.
(2) If each natural number and can form a triangle, as the side length, find the value range.
(3) Suppose that if the smallest integer in the range determined in (2) is taken, the number of terms of the largest term of the series is found.
[Solution ]( 1)
[Solution] (2)
[Solution] (3)
22. (The full score of this small question is 18) There are three small questions in this question. The full score of 1 small question is 5, the full score of the second small question is 5 and the full score of the third small question is 8.
As we all know, complex numbers are real numbers and imaginary units. For any complex number.
(1) Try to find the value, and write the relationship between and respectively;
(2) Let (,) be the coordinates of the point, and (,) be the coordinates of the point. The above relationship can be regarded as a transformation of points on the coordinate plane: it turns points on the plane into points on this plane.
When the point moves in a straight line, try to find the trajectory equation of the point obtained after transformation;
(3) Is there such a straight line: any point on it is still on the straight line after the above transformation? If so, try to find all these straight lines; If it does not exist, explain why.
[Solution ]( 1)
[Solution] (2)
[Solution] (3)
2000 National Unified Entrance Examination for Colleges and Universities
Key Points and Grading Criteria of Answers to Shanghai Mathematics Examination Paper (Science, Technology, Agriculture and Medicine)
explain
1. This solution lists one or more solutions to the problem. If the candidates' answers are different from the listed answers, they can be graded according to the spirit of the grading standard in the answers.
2. When correcting the test paper, you should insist that every question is corrected at the end, and don't interrupt the correction of the questions because of the mistakes in the candidates' explanations. When a candidate answers incorrectly in a certain step, which affects the subsequent part, but the answer after that step does not change the content and difficulty of the question, the score of the latter part will be determined according to the degree of influence. At this time, in principle, it should not exceed half of the score in the second half. If there are serious conceptual errors, do not score.
3. 17 to the score marked at the right end of question 22 indicates that candidates should correctly do this step of the cumulative score of this question, and the unit of giving points or deducting points is 1.
explain
1.( 1 to 12) Get 4 points for each question correctly, otherwise all will get zero points.
1.4.2.3.(-4,0),(6,0)。 4.。 5. 1.6.9.7. The sides are equal/the angle between the sides and the bottom is equal/…………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………… 10. 1 1. 12.
2. (Questions 13 to 16) If the first question is correct, give 4 points.
Title13141516
Code name C A A D
Three. (Question 17 to 22)
17. 【 Solution 】 Let the equation of ellipse C be
According to the meaning of the question
Because the discriminant of quadratic equation △ > 0, there are two different intersections between straight line and ellipse, …(8 points)
set up
18. [Solution 1] Establish the spatial rectangular coordinate system as shown in the figure ... (2 points)
According to the meaning of the question, there are A (0 0,2,0), C (2 2,0,0), E (1, 1, 0). Let the coordinates of point D be (0,0, z), then
and
[Solution 2] The parallel line that introduces BE through A, the extension line that intersects CB is at f, ∠DAF is the angle formed by non-planar straight line BE and AD,
∴∠ DAF =...(4 points)
E is the midpoint of AC, ∴B is the midpoint of CF,
Af = 2be = ... (6 points)
BF and BA are projections of DF and DA, respectively, and BF=BC=BA.
∴DF=DA。 ... (8 points)
Triangle ADF is an isosceles triangle,
Therefore, ... …( 10/0)
Say it again,
So the volume of tetrahedral ABCD is …( 12 minutes)
19. [Solution] (1) When,
Interval increasing function, ... (3 points)
The minimum value between regions is ... (6 points)
(2) [Scheme 1] Regarding the franchise,
Continuous establishment, …(8 points)
Settings,
Increment, ∴ When appropriate, …( 12 points)
So if and only if the function holds,
Therefore. ... (14 points)
(2) [Solution 2], when the function value is always positive, …(8 points)
When, the function increases, so when, …( 12 points)
So if and only if the function holds, so. ... (14 points)
20. [Solution] (1), the instruction is, …(4 points)
(2) Set the robot to intercept the ball at this point as soon as possible ... (6 points)
Then because the speed of the ball is twice that of the robot, there will be …(8 points).
That is to say, get or,
The robot is required to intercept the ball as soon as possible, that is, the ball has the shortest rolling distance.
So the robot can intercept the ball at this point at the earliest, (10)
The instruction given is, (14 points)
2 1. [Solution] (1) In terms of the meaning of the question,, …(4 points)
[Solution] (2)∵ Function decreasing,
∴ For each natural number n, if there is one, then the necessary and sufficient condition for forming a triangle with the side length is,
That is ... (7 points)
Solution or ∴, ... (10)
[Solution] (3) ∴∴...( 12 points)
The sequence is a decreasing positive sequence. For each natural number,
So when, when,,
Therefore, the number of terms in the largest term of a sequence satisfies the inequality sum.
22. [Solution] (1) According to the topic,
Therefore, ... (3 points)
Therefore,
Get the relationship ... (5 points)
[Solution] (2) If a point is set on a straight line, its transformation point satisfies.
, ... (7 points)
Eliminate, gain,
Therefore, the trajectory equation of this point is …( 10 point)
[Solution] (3) Assuming that there is such a straight line, the straight line parallel to the coordinate axis obviously does not meet the conditions.
∴ The required straight line can be set to …( 12 point)
[Solution1] ∫ Any point on a straight line, a point obtained by transformation.
Still on this straight line,
∴ ,
That is to say,
When the equation has no solution,
So such a straight line does not exist. ... (16 points)
When, by
OK,
Solve or,
Therefore, such a straight line exists, and its equation is or, …( 18 point).
[Solution 2] Take a point on a straight line, and the transformed point is still on the straight line.
∴ ,
Get, …( 14 points)
So a straight line is to take a point on a straight line, and the point obtained after transformation is still on a straight line.
∴, ...( 16 points)
That is to say, get or,
Therefore, such a straight line exists, and its equation is or, …( 18 point).