Who has the 2003 national college entrance examination mathematics (science) volume?

In 2003, the national unified examination for enrollment of ordinary colleges and universities (national volume)

Mathematics (science, engineering, agriculture and medicine)

Precautions:

1. Before answering the first volume, candidates must scribble their names, admission ticket numbers and exam subjects on the answer sheet with pencils.

2. After choosing the answer for each question, black the corresponding answer label on the answer sheet with a pencil. If you need to change it, clean it with an eraser and choose another answer. You can't answer them on the test paper.

At the end of the exam, the invigilator will take this paper back with the answer sheet.

Reference formula:

Sum and difference formula of trigonometric function: lateral area formula of regular prism and frustum.

Among them, respectively

The circumferences of the upper and lower bottom surfaces indicate the inclined height or bus length.

Volume formula of sphere:, where r

Represents the radius of the ball.

This paper is divided into two parts: the first volume (multiple choice questions) and the second volume (non-multiple choice questions).

The first volume (multiple choice questions ***60 points)

1. Multiple-choice question: This topic is entitled *** 12, with 5 points for each question and 60 points for each question. Only one of the four options given in each small question meets the requirements.

1. Known, 0), then ()

(A) (B) (C) (D)

2. The directrix equation of conic curve is ()

(A) (B) (C) (D)

3. Set a function. If, the value range of is ().

(A)(， 1) (B)(，)

(C)(，)(0，)(D)(，)( 1，)

4. The maximum value of this function is ()

(A) (B) (C) (D)2

5. Given a circle c: () and a straight line:, when the chord length of the straight line cut by c is, then ().

(A) (B) (C) (D)

6. It is known that the radius of the cone bottom is r and the height is 3R. In all its inscribed cylinders, the largest area is ().

(A) (B) (C) (D)

7. It is known that the four roots of the equation form a arithmetic progression, and the first term is, then ().

(A) 1 (B) (C) (D)

8. It is known that the center of the hyperbola is at the origin and a focus is f (,0), a straight line intersects it at two points, m and n, and the abscissa of the midpoint of MN is, then the equation of hyperbola is ().

(A) (B) (C) (D)

9. Inverse function of function ()

(A)， 1] (B)， 1]

(C)， 1] (D)， 1]

10. It is known that four vertices of a rectangle, A (0 0,0), B (2 2,0), C (2, 1), D (0, 1), a particle starting from the midpoint of AB, hits a point on BC in the direction of the included angle with AB.

(A)(， 1) (B)(，)(C)(，)(D)(，)

1 1.( )

Article 3 (2) (3) (iv) 6

12. All sides of a tetrahedron are equal, and all four vertices are on the same sphere, so the surface area of some spheres is ().

(A) (B) (C) (D)

In 2003, the national unified examination for enrollment of ordinary colleges and universities (national volume)

Mathematics (science, engineering, agriculture and medicine)

Volume 2 (non-multiple choice questions ***90 points)

Fill-in-the-blank question: This big question is ***4 small questions, with 4 points for each small question, and *** 16 points to fill in the answers on the horizontal lines in the questions.

Coefficients in 13 extension. be

14. The established value range is

15. As shown in the figure, an area is divided into five administrative regions, and now the map is in color, which requires adjacent areas not to use the same color. There are four colors to choose from, so there are different coloring methods (answer with numbers).

16. In the following five cubic graphs, it is a diagonal of the cube, and points M, N and P are the midpoints of their sides respectively. It can be concluded that the graphic serial number of MNP is (write all the graphic serial numbers that meet the requirements).

① ② ③ ④ ⑤

Third, the solution: this big question is ***6 small questions, and the score is ***74. The answer should be written in proof process or calculus steps.

17. (The full score of this small question is 12)

It is known that the radiation angle of a complex number is 0, which is the mean term of the equal ratio of sum.

18. (The full score of this small question is 12)

As shown in the figure, in a straight triangular prism, the base is an isosceles right triangle, the sides D and E are the midpoint of sum respectively, and the projection of point E on the plane ABD is the center of gravity G of △ABD.

(i) Find the included angle with plane ABD (the result is expressed by the inverse trigonometric function value).

(II) Find the distance from this point to the planar AED.

19. (Full score for this small question 12) is known and set.

P: the function decreases monotonically on r q: the solution set of inequality is R.

If p and q have one and only one correct value, the value range of is found.

20. (The full score of this short question is 12)

There are typhoons on the sea near coastal cities. According to the monitoring, at present, the typhoon center is located on the sea surface 300km east of O city (as shown in the figure), and it moves to the northwest at the speed of 20 km/h. The typhoon attack scope is the circular area with the current radius of 60km, and the speed is increasing at10 km/h. How many hours later did the city begin to be attacked by the typhoon?

2 1. (The full score of this small question is 14)

Given a constant, in a right-angle ABCD, 0, O is the midpoint OF AB, points E, F and G move on BC, CD and DA respectively, and P is the intersection of GE and of (as shown in the figure). Do you have two fixed points, so that the sum of the distances from P to these two points is a fixed value? If it exists, find out the coordinates of these two points and this fixed value; If it does not exist, please explain why.

22. (Full score for this small question 12, additional question 4)

(i) Let it be a set in which all the numbers are arranged in descending order, namely,,,, …

Write each item of the series in the triangle table below according to the principle of "up small and down big, left small and right big":

three

5 6

9 10 12

— — — —

…………

(1) Write the numbers in the fourth and fifth lines of this triangle table;

(2) looking for

(2) this small problem is an additional problem. If the answer is correct, add 4 points, but the total score of the whole paper does not exceed 150 points)

Let it be a set in which all the numbers are arranged in a sequence from small to large.

In 2003, the national unified examination for enrollment of ordinary colleges and universities (national volume)

Mathematics (science, engineering, agriculture and medicine) answers

First, multiple-choice questions: This question examines basic knowledge and basic operations. 5 points for each question, out of 60 points.

1.D 2。 C 3。 D 4。 A 5。 C 6。 B 7。 C 8。 D 9。 D 10。 C 1 1。 B 12。 A

Fill in the blanks: This question examines the basic knowledge and basic operation. 4 points for each small question, full score 16.

13. 14.(- 1,0) 15.72 16.①④⑤

Third, the solution: this big question is ***6 small questions, and the score is ***74. The solution should be written in proof process or calculus steps.

17. solution: set, then the complex number is set by the topic.

18. (1) solution: connect BG, then BG is the projection of BE in ABD, that is, ∠EBG is the angle formed by A 1B and plane ABD.

Let f be the midpoint of AB, connecting EF and FC.

(2) Solution:

19. solution: the function decreases monotonically on R.

inequality

(None of the above methods are used in the Xinjiang test area. Most of them use the method of solving inequalities, and some use the mirror image method. )

20. Solution: Establish a coordinate system as shown in the figure, with O as the origin and the east direction as the positive direction of the X axis.

When: (1), the coordinates of typhoon center p () are

At this time, the area hit by the typhoon is

Among them, if the city o is hit by a typhoon at time t, there are

that is

A: 12 hours later, the city began to be hit by a typhoon.

2 1. According to the conditions, the equation satisfied by the coordinates of point P is obtained first, and then it is judged whether there are two fixed points, so that the sum of the distances from point P to two points is a fixed value.

According to the meaning of the question, there are A (-2,0), b (2 2,0), c (2 2,4a) and D (-2,4a).

Examples are (-2,4ak), f G(-2-4k,4a) and g (-2,4a-4ak).

The linear equation of is: ①

The equation of the straight line GE is: ②

By eliminating the parameter k from ① and ②, it is found that the coordinates of point P(x, y) satisfy the equation.

If arranged properly, the trajectory of point P is an arc, so there are no two points that meet the meaning of the question.

At that time, the trajectory of point P was a part of an ellipse, and the sum of the distances from point P to the focus of the ellipse was a fixed length.

At that time, the sum of the distances from point P to the two focal points of the ellipse was a constant value.

At that time, the sum of the distances from the point P to the two focal points (0,0) of the ellipse is a fixed value of 2.

22. (Full score for this small question 12, additional question 4)

(a) solution: expressed by (t, s), the law in the table below is

3((0, 1)= )

5(0,2) 6( 1,2)

9(0,3) 10( 1,3) 12(2,3)

— — — —

…………

(i) Line 417 (0,4)18 (14) 20 (2,4) 24 (3,4)

The fifth line 33 (0,5) 34 (1,5) 36 (2,5) 40 (3,5) 48 (4,5)

(i i) Solution 1: Because100 = (1+2+3+4+...+13)+9, (8,14) =/kloc-.