Mathematics (science)
This test paper is ***4 pages, 2 1 question, full mark 150. Examination time 120 minutes.
Reference formula: If the events are mutually exclusive, then.
Which is a positive integer.
First, multiple-choice questions: this big question is ***8 small questions, with 5 points for each small question, out of 40 points. Of the four options given in each question, only one meets the requirements of the topic.
1. It is known that the real part of a complex number is 1, so the value range of is ().
A.B. C. D。
2. Remember that the sum of the previous arithmetic series is, if, then ().
A. 16
Grade one, grade two, grade three
Girls 373
Boys 377
3. There are * * * 2,000 students in a school. See table 1 for the number of boys and girls in each grade. It is known that 1 student is randomly selected from the whole school, and the probability of drawing girls in the second grade is 0. 19. At present, 64 students are selected by stratified sampling in the whole school, so the number to be selected in the third grade is (C).
Table 1
4. If the variable is satisfied, the maximum value is ()
90 ad to 80 ad
5. Cut off three corners of a regular triangular prism (as shown in figure 1, which are the midpoints of three sides respectively) to obtain the geometric figure shown in figure 2, then the side view (or left view) of the geometric figure in the direction shown in figure 2 is ().
6. It is known that all rational numbers of the proposition are real numbers, and the logarithm of the positive number of the proposition is negative, then the true proposition in the following proposition is ().
A.B. C. D。
7. Let, if the function has extreme points greater than zero, then ()
A.B. C. D。
8. In a parallelogram, the intersection of and is the midpoint of the line segment, and the extension line of intersects with this point. If,, then ()
A.B. C. D。
Fill-in-the-blank question: There are 7 small questions in this big question, and the examinee answers 6 small questions, with 5 points for each small question, out of 30 points.
(1) Required questions (9~ 12)
9. Read the program block diagram in Figure 3. If is, output.
, .
(Note: The assignment symbol in the block diagram can also be written as ""or "").
10. In the known (positive integer) expansion, the coefficient of is less than.
Then 120.
1 1. The equation of a straight line passing through the center of the circle and perpendicular to the straight line is.
12. Given the function,, the minimum positive period is.
Second, choose to do the questions (13- 15 questions, candidates can only choose to do two questions)
13. (coordinate system and parameter equation are selected as questions) If the polar coordinate equation of the curve is known to be 0, the polar coordinate of the curve and the intersection point is 0.
14. It is known that if the equation has real roots, the value range of is.
15. (geometry proof lecture and multiple choice questions) The tangent of a known circle is,. Is the diameter of the circle, and when it intersects the circle, the radius of the circle is.
Third, the solution: this big question ***6 small questions, out of 80 points. You must write a solution to prove the process or calculus steps.
16. (The full score of this small question is 13)
It is known that the maximum value of the function is 1, and its image passes through this point.
Analytical formula of (1);
(2) The value of,, is known.
17. (The full score of this small question is 13)
200 pieces of a product from a factory were randomly selected. After quality inspection, there are 26 first-class products/kloc-0, 50 second-class products, 20 third-class products and 4 defective products. It is known that the profits of producing 1 piece of first-,second-and third-class products are 60,000 yuan, 20,000 yuan and 1 10,000 yuan respectively, while 65,438+.
(1);
(2) Find the average profit of 1 product (i.e. mathematical expectation);
(3) After technological innovation, there are still four grades of products, but the rate of defective products has decreased to and the rate of first-class products has increased to. If the average profit of 1 product is not less than 47300 yuan, what is the highest rate for third-class products?
18. (The full score of this small question is 14)
Let the elliptic equation be and the parabolic equation be. As shown in Figure 4, the intersection of the parallel line with this point as the axis and the parabola in the first quadrant is, and it is known that the tangent of the parabola at this point passes through the right focus of the ellipse.
(1) Find the elliptic equation and parabolic equation that satisfy the conditions;
(2) Let the left and right points of the long axis of the ellipse be respectively, and try to explore whether there is a point on the parabola to make it a right triangle. If yes, please indicate how many such points are there in * * *? And explain why (you don't need to find the coordinates of these points in detail).
19. (The full score of this small question is 14)
Suppose, function, and try to discuss the monotonicity of function.
20. (The full score of this short question is 14)
As shown in Figure 5, the bottom surface of a quadrangular pyramid is an inscribed quadrilateral of a circle with radius, where the diameter of the circle and the vertical bottom surface are points at the top respectively, and parallel lines passing through these points intersect.
(1) Find the sine value of the included angle with the plane;
(2) Prove that it is a right triangle;
(3) When, find the area.
2 1. (The full score of this small question is 12)
Let it be a real number and two real roots of the equation, and the sequence satisfies,, (…).
(1) Proof:
(2) Find the general term formula of the sequence;
(3) If,, find the sum of the preceding items.
In 2008, the national unified entrance examination for ordinary colleges and universities (Guangdong volume)
Reference answers to mathematics (science)
First, multiple-choice questions: C D C A D B B
1.c analysis, that is,
2. Therefore, D analysis,
According to the meaning of the question, we know that there are 380 girls in grade two, so the number of students in grade three should be, that is, the proportion of students in all grades in the whole is, so the number of students in grade three should be selected by stratified sampling.
4.C 5。 A
6.d analysis is not difficult to judge that the proposition is true and the proposition is false, so the above statement is only true.
7.b analysis, if the function has an extreme point greater than zero, it has a positive root. Sometimes, obviously. At this time, we can get the range of parameters immediately.
8.B
Second, fill in the blanks:
9. Analysis of the operation of ending the program requires the conditional operation of divisibility and divisibility at the same time, so the minimum value of should be the least common multiple of the sum of 12, that is, there is.
10. The general term of binomial theorem expansion is that the coefficient we know is, that is, it is a positive integer, so we can only take 1.
1 1. It is easy to know that point C is vertical. We assume that the equation of the straight line to be solved is, and the value of the parameter can be obtained immediately by substituting the coordinates of point C, then the equation of the straight line to be solved is.
12. Analysis, so the minimum positive period of the function.
Second, choose to do the questions (13- 15 questions, candidates can only choose to do two questions)
13. The analysis is obtained from the solution, that is, the intersection of two curves is.
14.
15. According to the meaning of the question, we know that we have the nature of similar triangles, namely.
Third, the solution: this big question ***6 small questions, out of 80 points. You must write a solution to prove the process or calculus steps.
16. solution: (1) according to the meaning of the question, then, replace it with points, and, therefore;
(2) According to the meaning of the question,
,
.
17. Solution: (1) All possible values are 6,2, 1,-2; ,
,
Therefore, the distribution list is:
6 2 1 -2
0.63 0.25 0. 1 0.02
(2)
(3) Let the rate of third-class products after technological innovation be, then the average profit of 1 product at this time is
According to the meaning of the problem, that is, the solution
So the rate of third-class products is the highest.
18. Solve: (1) Derive,
When the coordinates of the G point are,
, ,
The tangent equation through the G point is,
So, the coordinates of this point are,
The coordinates of the points obtained from the elliptic equation are, that is,
That is, the equations of ellipse and parabola are summed separately;
(2) There is only one intersection point between the vertical line of the overtravel axis and the parabola,
There is only one right angle, and there is only one right angle.
If we take a right angle, the coordinates of the set point are, and the coordinates of the two points are and respectively.
.
The quadratic equation about has one solution greater than zero and two solutions, that is, there are two right-angled solutions.
So there are four points on the parabola that make it a right triangle.
19. Solution:
Because,
When, the function is an increasing function;
When the function is a decreasing function in the world and a increasing function in the world;
Because,
When the function is a subtraction function;
When, the function is a decreasing function on the ground and a increasing function on the ground.
20. solution: (1) in the middle,
,
While that vertical bottom surface ABCD of PD,
,
In, it is a right triangle with right angles.
The distance from the set point to the surface is,
Youyou,
That is to say,
;
(2) and,
That is,,, is a right triangle;
(3),
That is to say,
Area of
2 1. Solution: (1) By finding the root formula, we might as well set and get it.
,
(2) Settings, and then, by
Get, eliminate, get, is the root of the equation,
According to the meaning of the question,
(1) when, when the solution of the equation is recorded as
That is, geometric series, whose common ratios are,
It can be obtained from the properties of geometric series,
Subtract two expressions to get.
, ,
,
, that is,
② When, that is, the equation has multiple roots,
That is, you might as well set it up and know it from ①.
, ,
In other words, if you separate the two sides of the equation at the same time, you will get, that is
This sequence is an arithmetic series with a tolerance of 1.
All in all,
(3) Substitution, acquisition and solution