The following is the question and answer of science mathematics in Grade 09.
1. Multiple-choice question: (There are 8 small questions in this big question, with 5 points for each small question and 40 points for each small question. Only one of the four options given in each small question meets the requirements of the topic. )
1. Let the complete set U = R and A =, then UA = ().
A.{ x | x & gt0} C.{x | x≥0} D. ≥0
2. It is "the minimum positive period of the function is" ().
A. sufficient and unnecessary conditions B. necessary and insufficient conditions C. necessary and sufficient conditions D. neither sufficient nor necessary conditions
3 1, 2, 2, 3, 3, 4, 4, ... in the sequence, the 25th item is ().
A.25 B.6 C.7 D.8
4. Let two non-zero vectors not be * * * lines. If they are not * * * lines, then the value range of real number k is
( ).
A.B.
C.D.
5. If the intersection of the curve on the right side of the Y axis and the straight line is marked as P 1, P2, P3, ..., then |P2P4| is equal to ().
A.B.2 C.3 D.4
6. The image on the right is a function image, where m and n are constants.
Then the following conclusion is correct ().
A.& lt0，n & gt 1 B . >0，n & gt 1
C.& gt0，0 & ltn & lt 1d . & lt； 0，0 & ltn & lt 1
7. A pool has two water inlets and 1 water outlets. The water inlet and outlet speeds are shown in Figures A and B. On a certain day, from 0: 00 to 6: 00, the pool should
The water storage capacity of the pool is shown in Figure C (at least one water inlet is open).
Give the following three conclusions:
(1) From 0 o'clock to 3 o'clock, only water enters and no water exits; (2) From 3 o'clock to 4 o'clock, there is no water, only water; (3) From 4 o'clock to 6 o'clock, there is no water. Then you can be sure that the correct conclusion is
A.① B.①② C.①③ D.①②③
8. The output result after executing the following program is (C)
a 、- 1 B、0 C、 1 D、2
Fill-in-the-blank question: (This big question consists of ***6 small questions, each with 5 points and ***30 points. Write the answer on the line).
9, a city high school mathematics sampling examination, score above 90 points (including 90 points) statistics, its frequency distribution.
As shown in the figure, if the number of people in 130- 140 is 90, then the number of people in 90- 100 is
10.。
1 1. It is known that I and J are mutually perpendicular unit vectors, a = I–2j, b = i+λj, and the included angle between A and B is acute, so the range of real numbers is.
12 known function, satisfying the sum of arbitrary real numbers.
Then.
The symbol 13 represents the largest integer, for example, no more than defining a function,
Then the correct serial number in the following proposition is.
The domain of the (1) function is r, and its range is; (2) The equation has many solutions;
(3) The function is a periodic function; (4) The function is incremental.
14. In the plane rectangular coordinate system, the curve c:, () is known.
Then the curve equation that curve C is symmetrical about y=x is
Third, the solution: this big problem ***6 small questions, out of 74 points, the solution should be written in the proof process or calculus steps.
15. (Full score of this question) Known,
(i) the value of; (ii) the value of.
16. In a box, there are three cards with labels. Now, take out two cards from this box, labeled, and respectively.
(1) Find the maximum value of random variables and the probability of "getting the maximum value" of events;
(Ⅱ) Find the distribution table and mathematical expectation of random variables.
17. (Full score of this question) As shown in the figure, it is known that the side length of the bottom of a regular triangular prism is the midpoint of the side, and the angle formed by a straight line and the side is.
(i) finding out the side length of the regular triangular prism;
(ii) Find out the size of dihedral angle;
Find the distance from a point to a plane.
18. (The full mark of this small question is 14) A beam of light starts from a point, reflects from a point on a straight line, and just passes through that point.
(i) finding the coordinates of a symmetrical point about a point of a straight line;
(2) Find the elliptic equation with, as the focus and passing through this point;
(3) Let the two directrix of a straight line and an ellipse intersect at two points, which are moving points on the line segment, and find the minimum value of the ratio of the distance from this point to the right directrix of the ellipse, and the coordinates of the point where the minimum value is located.
19. (Full score for this question) The known series satisfies: and
.
(i) Find the value of the general term formula of,, and series;
(ii) Set and find the sum of the previous items in the sequence;
20. (Full score for this question) If the function and point are known, the two tangents of the curve are,, and respectively.
(i) Assume and try to find the expression of the function;
(2) Whether it exists or not, make a three-point * * * line. If it exists, it is the calculated value; If it does not exist, please explain why.
(iii) Under the condition of (i), if there are always real numbers in the interval of any positive integer, the inequality holds and the maximum value of is obtained.
Comprehensive examination paper (1) science answers
First, multiple-choice questions:
1. Answer: C. {x | x ≥ 0}, so C.
2.C
3. (Rational) For the middle, when n = 6, there is no 25th item, so it is 7. Choose C.
4.D
5 . a . *
= ,
∴ Make a function diagram according to the meaning of the question.
6. answer: D. when x= 1, y = m, and m
7.A
8.C
Second, fill in the blanks:
9.8 10
10. Answer:.
1 1. Answer:.
12.
13.(2)、(3)
14.
15. (Full score for this question)
Known,
(i) the value of;
(ii) the value of.
Solution: (1) 2 points for ................
..........................., five points.
(2) Original formula =
................... 10 point
................... 12.
16. (Full score for this question)
In a box, there are three cards with labels. Now, take out two cards with labels from this box.
(1) Find the maximum value of random variables and the probability of "getting the maximum value" of events;
(Ⅱ) Find the distribution table and mathematical expectation of random variables.
Answer: (I) Possible values are,,,
, ,
And when or, ................. 3 points.
Therefore, the maximum value of a random variable is.
There are various situations in which two cards are drawn back.
.
Answer: The maximum value of the random variable is 0, and the probability of the event "getting the maximum value" is 0.
All values of (ii) are.
Only in this case,
There are four situations, or or or or,
Sometimes, there are one or two situations.
1 1.
The distribution list of random variables is:
Therefore, the mathematical expectation is ......................................................................... 13.
17. (Full score for this question)
As shown in the figure, it is known that the side length of the bottom surface of the regular triangular prism is, which is the midpoint of the side, and the angle formed by the straight line and the side is.
(i) finding out the side length of the regular triangular prism; (ii) Find out the size of dihedral angle;
Find the distance from a point to a plane.
Solution: (i) Let the side length of a regular triangular prism be. Take the midpoint and connect.
This is a regular triangle.
And the intersection line is.
Side.
Even number, then the angle formed by the line and the edge is .................................................................................................................................................................
In the middle, the solution is ........................................................................................................................................................................
The side length of the regular prism is
Note: The side length can also be calculated by vector method.
(2) solution 1: over-acting, even,
Side.
It is the plane angle of dihedral angle, ........................................................................................................................................, 6 points.
In the middle again.
, .
and
In ..............................., 8 points.
So the size of the dihedral angle is 9 points.
Solution 2: (Vector method, see below)
(3) Solution 1: According to (2), the plane, the plane and the intersection line are; if there are too many, the plane is. ........................................................................................................................
Yes, 12 points.
Is the midpoint, and the distance from the point to the plane is 13 minutes.
Solution 2: (Thinking) Take the midpoint, connect, and get the plane easily, and the intersection line is. If the intersection is at, the length of is the distance from the point to the plane.
Solution 3: (thinking) Equal volume transformation: from available.
Solution 4: (Vector method, see below)
Vector solutions of problems (Ⅱ) and (Ⅲ);
(2) Scheme 2: Establish a spatial rectangular coordinate system as shown in the figure.
Then.
Suppose it is the normal vector of the plane.
I see.
Six points for ...................
The normal vector of the plane ................... 7 points.
........................., eight.
As shown in the figure, dihedral angle is 9 points.
(III) Solution 4: Starting from Solution 2 (II)
Distance from point to plane =. 13 minutes.
18. (The full score of this small question is 14)
A beam of light starts from a point, reflects from a point on a straight line, and just passes through this point.
(i) finding the coordinates of a symmetrical point about a point of a straight line;
(2) Find the elliptic equation with, as the focus and passing through this point;
(3) Let the two directrix of a straight line and an ellipse intersect at two points, which are moving points on the line segment, and find the minimum value of the ratio of the distance from this point to the right directrix of the ellipse, and the coordinates of the point where the minimum value is located.
Solution: (i) If the coordinate is, then
Therefore, the coordinate of solving this point is ......................................................... 4 points.
(ii) According to the definition of ellipse,
Get, get, get, get.
, .
The required elliptic equation is 7 points.
(iii) The straightness equation of the ellipse is 8 points.
The coordinate of this point is, that is, the distance from this point to the right directrix of the ellipse.
Then,.
, ....................... 10.
So, order,
When,,,.
Get the minimum value in .....................................................................................................................................................................
Therefore, the minimum value is =, and the coordinates of the point at this time are ..........................................................................................................................................................
Note: The minimum value of can also be obtained by discriminant method and substitution method.
Note: The obtained point is the tangent point, and the minimum value is the eccentricity of the ellipse.
19. (Full score for this question)
Known sequences satisfy: and,.
(i) Find the value of the general term formula of,, and series;
(ii) Set and find the sum of the previous items in the sequence;
Solution: (1) After calculation, …
When it is odd, that is, the odd term of the sequence becomes arithmetic progression,
When it is an even number, that is, the even term of the sequence becomes a geometric series,
.
So the general formula of the series is.
(Ⅱ) ,
……( 1)
…(2)
(1) and (2) are subtracted,
get
.
.
20. (Full score for this question)
Given a function and a point, the intersection point is two tangents of the curve, and the tangents are,, and respectively.
(i) Assume and try to find the expression of the function;
(2) Whether it exists or not, make a three-point * * * line. If it exists, it is the calculated value; If it does not exist, please explain why.
(iii) Under the condition of (i), if there are positive integers, there are always real numbers in the interval.
Make the inequality hold, and find the maximum value of.
Answer: (i) Suppose the abscissa of two points is,
The equation for the tangent is,
A little off topic again, yes,
That is to say, ..................................... (1) ...2 points.
Similarly, from the tangent to the point, you get ............ (2)
Two equations can be obtained from (1) and (2).
............................................................................, 4 points.
Substituting into the formula (*),
So the expression of the function is 5 points.
(ii) When the dot and the * * * line,, =,
That is =, simplify, get,
.............. (3) ............................... 7 points.
Substitute the formula (*) into the formula (3) to obtain.
There is a * * * line between the make point and the three point, and ...................... scores 9 points.
(iii) Solution: It is easy to know that it is increasing function in the interval,
Then.
According to the meaning of the question, the inequality applies to all positive integers.
In other words, positive integers apply to everything.
, ,
.
Because it is a positive integer, 13 points.
When, exist, and meet all conditions.
Therefore, the maximum value of is ....................... 14.
Solution: According to the meaning of the question, when the interval length is the smallest, the maximum value obtained is the value obtained.
The interval with the smallest length is, 1 1.
When it is the same as the solution, it is concluded that,
solve .........................