Liberal arts mathematics (compulsory+elective me)
Book one (multiple choice questions) and book two (non-multiple choice questions) are two parts of the test paper. Volume 1 volume 1-2 volume 3-4. Documents and answer sheets submitted after the exam.
A quantity
note:
? 1。 Before answering questions, candidates must use a black ink pen with a diameter of 0.5 mm, clearly fill in their name and admission ticket number, and have a good barcode and paste it on the answer sheet. Please carefully approve the admission ticket number, name and subject on the bar code.
2。 After choosing the answer for each question, use 2B pencil and eraser to clean the label on the answer sheet, and then choose to blacken the changes of other answer labels and the answers corresponding to the questions. The number of questions answered is invalid.
? 3。 This volume contains 12 small questions, with 5 points for each small question and 60 points for each small question. Of the four options given in each small question, only one is a subject that meets the requirements.
Reference formula:
If the events are mutually exclusive, the surface area formula of the ball
?
If these events are independent of each other, it indicates the radius of the ball.
What is the volume formula of the ball?
If the event is in the experiment, the probability of occurrence is
? Time independently repeats the probability, and the test time represents the radius of the ball or something.
?
multiple-choice question
(1) value
(A)(B)(C)(D)
[A] Inductive formula for small-scale investigation, trigonometric function value from special angle, and basic problems.
Solution: so choose a.
(2) Set A = {4 4,5,7,9}, B = {3 3,4,7,8,9}, complete set, and total elements in the set.
(A)(B)4 (C)5(D)6
[A] The aggregation of small-scale operations is a basic problem. (Also 1)
Solution: choose a. it can also be used for Morgan's law:
(3) Solution set
(A)(B)
(C)(D)
[A] Study and solve the inequality of the absolute value of small problems and basic problems.
Solution:
So choose D.
(4) given that tan = 4, then tan, stack =(A +)=
(A)(B)(C)(D)
[a] A small question examines the relationship between trigonometric function, formula of identical angle and tangent angle, and basic problems.
Solution: topic, choose B.
(5) Set the tangent of parabola as the asymptote of hyperbola, and the eccentricity of hyperbola is equal to
(A)(B)(C)(D)
[a] asymptote equation of hyperbola, eccentricity of straight line and conic hyperbola, positional relationship between basic problems, and minor investigation.
Solution: Substitute a hyperbolic asymptote equation into the parabolic equation, and sort out the problem that the parabola of the asymptote is tangent, so choose C.
The known function of the inverse function of (6),
(A)0(B) 1(C)2(D)4
[a] a small question examines the inverse function, the basic problem.
Solution: such a topic, so, so choose C.
4 (7) There are 5 boys and 3 girls in Group A, 6 boys and 2 girls in Group B, and both Groups A and B choose one for every two students. The newly elected girls choose the total number of students in different ways.
(a)150 (b)180 (c) 300 (d) 345 species.
[Answer] The principles of small classification, step-by-step counting and combination are fundamental problems.
Solution selected d: total number of problems.
(8) is a nonzero vector to satisfy.
150(B) 120(C)60(D)30
[a] A small question examines the vector operation of geometry and studies the basic problems of the combination of numbers and shapes.
Solution: the parallelogram rule of vector addition, knowledge can form two adjacent sides of a rhombus, and the diagonal length at the beginning is equal to the rhombus with side length, so choose B.
(9) It is known that the length of each side of the bottom surface of a triangular prism is equal, and the cosine of the angle value formed by straight lines of different planes at the midpoint of the projected bottom surface.
(A)(B)(C)(D)
[a] The nature of a small question is the basic problem of investigating prisms, angles and straight lines in different planes. (Same as 7)
Solution: Let our midpoint D, connecting line D, AD be easy to know, it is a straight line, to the angle known by both sides, trigonometric cosine law. I chose e.
(10) If the image of this function is symmetrical about the center of this point, the minimum value of.
(A)(B)(C)(D)
[A] A small question examines the images of trigonometric functions and the nature of basic problems.
Solution: the role of central symmetry of image points
? It is selected, so it is easy to get.
It is known that the dihedral angle is 600( 1 1), and the distances between the moving points p and q on the plane p are q, p and q respectively, and the minimum distance between the two points is taken as
[Answer] The small problem of distance, dihedral angle and space and the comprehensive problem of maximum research. (Similarly, 10)
?
Solution: respectively
? , even if
Well,
and
If and only if, that is, coincidence takes the minimum value. So, the answer is C.
(12) The ellipse of the known focus is F, aligned to the right, and the point and line segment C are at point B. If AF crosses, then =
(A)(B)(C)(D)3
In [A], the carrier, the definition of ellipse and the basic problems used to study the alignment problem of small ellipse are studied.
Solution: through point b, at the intersection with x axis? Easy to recognize FN = 1 M and right pair sets. The meaning of the problem, so. From the second definition of ellipse, we get such a chosen one.
Unified national college entrance examination in 2009
Liberal arts mathematics (compulsory elective course 1)
Volume II
note:
1。 Before answering the question, the examinee fills in his name and admission ticket number on the answer sheet with a black ink pen with a diameter of 0.5 mm, and then sticks a bar code. Please carefully approve the admission ticket number, name and subject on the bar code.
2。 Volumes 2 and 7, please answer each question on the answer sheet with a black ink pen with a diameter of 0.5 mm. The number of questions answered is invalid.
3。 This article * * * is divided into 10, with a score of ***90.
Fill-in-the-blank question: This big question has four small questions, each with 5 points and ***20 points. Just fill in the answer to this question.
(Note: the number of questions answered is invalid)
Extension (13), and the sum of the coefficients is equal to the coefficient in the sum of _ _ _ _.
[a] a small question examines two extended general terms and basic problems. (Similarly, 13)
Solution: So
(14) Set the sum of arithmetic series in the previous paragraph. If, that year _ _ _ _ _ _ _.
[a] The nature of arithmetic progression The first project, the basic problem, is investigated in a small problem. (Similarly, 14)
Solution: arithmetic progression,
?
(15) In the cross section of a sphere, the area of the sphere circle at the midpoint and the vertical plane is called the radius. If? The surface area of a circle? The ball is equal to _ _ _ _ _ _ _ _ _ _ _.
[A] What are the properties of spherical cross section and surface area? Ball, basic problem.
Solution: let the radius of the ball be the radius m of the circle, that is, the problem obtained in the following way, so.
(16), if it is the length of the line segment between two parallel lines of a straight line, then the inclination angle can be
①②③④⑤
The correct answer to the serial number is (write the serial number of all the correct answers)
[a] A small question examines the slope of a straight line, the inclination of a straight line and the distance between two parallel lines, and examines the idea of combining numbers with shapes.
Solution: Given straightness and angle, inclination and the distance between two parallel lines of a graph, the inclination of a row is equal to or. Therefore, fill in ① ⑤.
III. Answer to the question: This big question is ***6 small questions, with a score of ***70. The answer should be written in pictures, proof process or calculus steps.
(17) (full mark 10) Small question (note: the answer sheet is invalid)
In the arithmetic progression specified in the preceding paragraph, the common ratio is the general formula of positive geometric progression.
[A] Make a small study on the basic problems in the general formula of arithmetic progression and geometric progression specified in the preceding paragraph.
Solution: Set the common ratio of the tolerance zone series.
From ①
? ②
① ② Solution
Therefore, the required general formula.
(18) (The full mark of this small question is 12) (Note: the answer on paper is invalid)
It is known that the length of the side of the inner corner is calculated.
[A] Sine theorem and cosine theorem are examined in small questions.
Solution: through cosine theorem
In addition,
Well,
This 1
According to sine theorem
Through well-known and obtained
Well,
So, ②
So ① ② Solution
?
(19) (the full mark of this small question is 12) (Note abstract: the number of questions with invalid answers)
As shown in the figure, a quadrangular pyramid, points on the side of a bottom surface with a rectangular bottom surface,
(i) Proof: the midpoint of the side;
(2) Find the size of dihedral angle. (Similarly, 18)
Solution:
㈠
Intersect at point e, plane SAD
Connecting AE, quadrilateral ABME trapezoid
For pedal F, then AFME rectangle.
Settings ",and then,
?
get through
The solution is
That is to say, this
Therefore, for the side of the midpoint
(ii) Similarly, it is an equilateral triangle,
The midpoint of (I) m sc is known.
? , so
Take the midpoint g and connect the midpoint of BG and SA? The connecting rod GH is therefore called the plane angle of dihedral angle.
At the time of ligation, in the culture medium,
?
therefore
The size of the dihedral angle is
Solution:
D is the coordinate origin and the axis of ray DA, and the Cartesian coordinate system of D-xyz represented by the positive X axis is established.
Set ",and then
(1) Settings,
?
and
therefore
should
Solution,
M is the side SC of the midpoint.
㈡
AM of possible midpoint
and
?
therefore
So it is equal to the plane angle of dihedral angle.
?
So the size of the dihedral angle.
(20) (The full mark of this small question is 12) (Note: the answer to the question is invalid)
A and B won the first three games in one breath, and the game was over. Suppose the winning probability of the game is 0.4, and the results of the game are independent of each other. Before the second game, A and B won 1 game.
(1) Find the probability of the end of two games;
(2) Find the probability of winning the game.
[A] The topic of this small problem is simultaneous, independent, mutual and comprehensive probability detection of probability events in mutually exclusive events.
Solution: In the case of "victory" in the first game, the first game is a win-win situation.
(a) Set up "Game 2 Game over" activities.
Because the results of each game are independent of each other,
?
?
?
(2) Remember, "win a victory in this game", the victory of this game of A and B in event B is only if and only if the first victory, the second victory and the first two victories in the game behind A,
Because the results of each game are independent of each other,
?
?
(2 1) (The full mark of this small question is 12) (Note: the answer to the test paper is invalid)
? Known function.
(i) Monotonicity of discussion;
(ii) When set at the point p on the curve, the curve is tangent to the point p of the demand equation through the coordinate origin.
[A] The application of derivative in the synthesis of small questions and monotonous questions.
Solution: (1)
Order;
Order or
Therefore, as an increasing function in the time interval; Time intervals and fewer functions.
(ii) Set a point through the origin, and the equation
Therefore,
That is to say,
arrange
Solution or
Therefore, the equation tangent or
(22) (The full mark of this small question is 12) (Note: the answer to the question is invalid)
As we all know, parabola and circle intersect at four points: a, b, c and d.
(A) the scope of demand
When the area of the intersection of coordinates P(II) is? Quadrilateral ABCD, required diagonal AC, BD.
Solution: Substitute the parabola of (Ⅰ) into the circular equation and delete it.
Finishing ①
Necessary and Sufficient Conditions of Four Intersections of "AND": Equation ① Two Same Roots
Therefore,
The solution is
and
So the range
The coordinates of the four intersections of (II) are,, respectively.
By (i) according to Vieta's theorem
then
?
Make ",the following maximum demand.
Methods 1: 3 average:
?
? If and only if, that is, when the maximum. At this time, when you check, you should conform to the meaning of the question.
Method 2: Let the coordinates of the four intersections be
The linear AC and BD equations are
?
Coordinates of point p
Settings, and (i) yes.
Because the quadrilateral ABCD is an isosceles trapezoid, its area
?
then
Generation, and so on, also.
Well,
∴,
Have it, or (truncate)
When, when, when.
Therefore, if and only if the coordinates of the point P with the maximum value are needed, that is, the maximum ABCD of the quadrilateral.