1. The minimum value of this function is (c).
A.0 B. 1 C.2 D.3
[Solution] Therefore, when
If and only if the above equations are equal and this equation has a solution, the minimum value in is 2.
2. Let, if, then the range of real numbers is (d)
A.B. C. D。
[Solution] Because there are two real roots.
, ,
So it's equivalent to and, that is
Besides,
Get a solution.
3. When Party A and Party B play table tennis, it is agreed that the winner will get 1 point and the loser will get 0 point. When one of them has 2 points more than the other or has played 6 rounds, the game will stop. Assuming that the probability of Party A winning each game is 0, the probability of Party B winning each game is 0, and the outcome of each game is independent, then the expected number of games when the game stops is (b).
A.B. C. D。
[Answer 1] According to the meaning of the question, all possible values of are 2, 4 and 6.
Let every two games be a round, then the probability that the game will stop at the end of the round is
.
If the game will continue at the end of this round, both parties must score one point in this round. At this point, the result of this round has no effect on whether the next round of competition will stop.
,
,
,
Therefore.
[Solution 2] According to the meaning of the question, all possible values of are 2, 4 and 6.
The order is that A wins the first game and B wins the second game.
Through independence and mutual incompatibility
,
,
,
Therefore.
4. If the sum of the surface areas (unit: cm) of a cube whose three sides are integers is 564 cm2, the sum of the volumes of these three cubes is (a).
A.764 cubic centimeters or 586 cubic centimeters.
C.586 cubic centimeters or 564 cubic centimeters deep 586 cubic centimeters
[Solution] If the side lengths of these three cubes are respectively 0, then there is, and it may be set, so, therefore, we can only take 9, 8, 7 and 6.
If, then, it is easy to know and get a solution.
If, then, but, therefore, it is still 5. If, then no solution, if, then no solution. At this time, there is no solution.
If there is only one solution.
If, then, at this time, therefore, but, therefore, there is no solution at this time.
To sum up, * * * has two solutions or.
The volume is cm3 or cm3.
5. The number of rational number solutions of the equation is (B)
A. 1 B. 2 C. 3 D. 4
[Solution] If, then the solution is or.
If there is, you will get it.
Jed. ②
Replace ③ with ②.
From ① to ③, simplify.
It is easy to know that there is no rational number root, so it is contradictory to get from ① and ②, so the system of equations * * * has two rational number solutions or.
6. If the opposite side of the inner angle is a geometric series, the range of values is
(3)
A.B.
C.D.
[Solution] Set the common ratio, and then, and
.
Therefore, only a range of values is needed.
Because the largest side of a geometric series can only be or, one and only one inequality group is needed to form three sides of a triangle.
that is
solve
Therefore, the range of values sought is.
2. Fill in the blanks (the full score of this question is 54 points, and each small question is 9 points)
7. Let, where is a real number,,, If, then 5.
[Solution] Judging from the meaning of the question,
,
Therefore, from …
8. Then, set the minimum value to.
[Solution]
,
(1), the minimum value;
(2) When the minimum value is 1;
(3), the minimum value.
Or, the minimum value of cannot be,
So, solve (give up).
9. If 24 volunteer places are allocated to 3 schools, each school will have at least one place, and there are 222 different allocation methods.
[Solution 1] Use the gap between four sticks to represent three schools, and use the number of delegates. take for example
It means that the first, second and third schools have 4 18 and 2 places respectively.
If each ""and each ""are regarded as a position, because the left and right ends must be | ",different distribution methods are equivalent to an" occupation method "in which a position (excluding both ends) is occupied by two|".
The division of "at least one quota per school" is equivalent to selecting two gaps from 23 gaps among 24 ""and inserting "|", so there are two kinds.
In the method of "at least one quota per school", the allocation method of "at least two schools have the same quota" is 3 1.
To sum up, there are 253-3 1 = 222 eligible distribution modes.
[Solution 2] If the number of places allocated to three schools is 0, then the score of at least one place in each school is an indefinite equation.
.
The number of positive integer solutions of, that is, the number of non-negative integer solutions of the equation is equal to 2 1 element recombination from three different elements:
.
In the method of "at least one quota per school", the allocation method of "at least two schools have the same quota" is 3 1.
To sum up, there are 253-3 1 = 222 eligible distribution modes.
10. Let the sum of the first few terms of the series satisfy:, then the general term =.
[Solution],
Namely 2
= ,
This leads to 2.
Order, (),
Yes, so, so.
1 1. Let the function be defined in, if and for any, satisfy.
, then =
[Solution 1] We know from the conditions of the topic.
,
Therefore, there is, therefore.
.
[Solution 2] So, the order
,
,
That is to say,
Therefore,
It must be a periodic function with a period of 2,
So ...
12. A ball with a radius of 1 can move freely in all directions in a regular tetrahedron container with an inner wall length of, so the inner wall area of the container that the ball can never touch is.
[Solution] If the answer is 12, figure 1, consider the situation that the ball is squeezed into the corner. Let the radius of the ball be//plane and tangent to this point, then the center of the ball is the center of the regular tetrahedron and the vertical foot is the center.
because
,
Therefore, therefore.
Remember that the tangent point of the ball and the surface is connected at this time, then
.
Considering that the ball is tangent to a surface of a regular tetrahedron, it is easy to know that the trajectory of the tangent point of the ball closest to the edge on the surface is still a regular triangle, which is recorded as, for example, the answer 12 Figure 2. Record regular tetrahedron.
The side length of is too long.
Because, yes, the side length of a small triangle.
The area of the part of the ball that does not touch the surface is (for example, the shaded part in Figure 2 in the answer 12).
.
Again, so
.
By symmetry and regular tetrahedron * * *, the area of the inner wall of the container that the ball can't touch is * * *.
Third, solve the problem (the full score of this question is 60 points, and each small question is 20 points)
13. It is known that there are only three intersections between the image of the function and the straight line, and the maximum abscissa of the intersection is, so it is verified that:
.
The three intersections of the ID image and the straight line are shown in the answer 13, and they are tangent inside, and the tangent point is.
... five points
Because, so, that's ... 10.
therefore
... 15 point
... 20 points
14. Solving inequalities
.
[Solution 1] Because there is increasing function in the world, the original inequality is equivalent to
.
That is ... five points.
Grouping decomposition
, … 10 integral
So,
... 15 point
So, this is either.
Therefore, the solution set of the original inequality is ... 20 points.
[Solution 2] Because there is increasing function in the world, the original inequality is equivalent to
... five points
that is
,
, … 10 integral
Order, inequality is
,
Obviously, it is an increasing function in the world, so the above inequality is equivalent to
, … 15 point
The solution (discard),
Therefore, the solution set of the original inequality is ... 20 points.
15. As shown in figure 15, it is the moving point on a parabola, the point is on the axis, and the circle is inscribed, so as to find the minimum area.
[Solution] Set, you might as well set.
Linear equation:
Simplify it.
The distance from the center of the circle is 1,
, ... 5 points
Therefore,
It is easy to know that the above formula is simplified,
Similarly, there are ... 10 points.
So, then,
.
Because it is a point on a parabola, if there is, then
, ... 15 o'clock
therefore
.
When, the above formula takes the equal sign, at this time.
So the minimum value is 8.