20 10 Guangzhou first liberal arts mathematics answer

Reference answers and grading standards of mathematics (liberal arts) test questions

Note: 1. Reference answers and grading standards point out the main knowledge and ability to be tested in each question, and give one or several solutions for reference. If the candidate's answer is different from the reference answer, the corresponding score can be given according to the knowledge points and abilities mainly examined in the question type.

2. For the calculation questions in the analytical questions, if the examinee answers incorrectly in a certain step, if the content and difficulty of the subsequent part of the question are not changed, the score of the subsequent part can be determined according to the degree of influence, but the score given shall not exceed half of the score because of the correct answer of this part; If there are serious mistakes in the subsequent answers, no more points will be given.

3. Answer the score on the right hand side, which means that the candidate should get the accumulated score if he does this step correctly.

4. Only integer scores are given, and multiple-choice questions and fill-in-the-blank questions are not given intermediate scores.

First, multiple-choice questions: this big question mainly examines basic knowledge and basic operations. * * * 10 small questions, 5 points for each small question, out of 50 points.

The title is 1 23455 6789 10.

Answer a a a b c c b d c

Fill-in-the-blank question: This big question mainly examines basic knowledge and basic operation. There are 5 small questions in this big question, and the candidates answer 4 small questions, with 5 points for each small question, out of 20 points. Among them, 14 ~ 15 is optional, and candidates can only choose one question.

1 1.300 12. 13.32 14. 15.

Third, the solution: this big question ***6 small questions, out of 80 points. The solution must be written in words, proof process and calculation steps.

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

(This small topic mainly examines the properties of trigonometric functions, the basic relationship of trigonometric functions with the same angle, the double angle formula and other knowledge, and examines the mathematical thinking method and operational solving ability of reduction transformation. )

(1) solution:

..... 2 points

..... 3 points

..... 4 points

The minimum positive period of ∴ is, and the maximum value is ... 6 points.

(2) Solution: ∫∴...7 points.

... 8 points

∵ is an acute angle, that is ∴.

∴ ... 10 point

∴ ... 12 point

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

(This small topic mainly examines the knowledge of stem leaf diagram, sample mean, sample variance, probability, etc. The possibility and necessity of mathematical thinking method, as well as data processing ability, operational solving ability and application consciousness are investigated. )

(1) solution:, ... 1 fraction

, ... 2 points

, ... 3 points

, ... 4 points

Five points for,, ...

∴ The weight of products in Workshop A is relatively stable. ...............................................................................................................................................................

(2) Solution: 2 samples are randomly selected from 6 samples in Workshop B, * * * There are 15 different methods:

, ... 8 points

Suppose the random event "the difference between the weights of two samples taken is not more than 2g", then there are four basic events:

, ... 10 point

So the probability is ... 12 points.

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

(This small topic mainly examines the relationship between a straight line and a plane in space, the volume of a cone and other knowledge, as well as the mathematical thinking method of combining numbers with shapes and transformation, as well as the ability of spatial imagination, reasoning and argumentation, and operational solution. )

(1) proof: connected, set, intersecting point, connected,

∵ quadrilateral is a parallelogram,

The point is.

Yes, the midpoint,

∴ is the center line of the delta,

∴.. 3 points

Aircraft, aircraft,

∴ Plane ... 6 points

(2) Solution 1:∫ plane, plane,

Aircraft, aircraft, and aircraft, aircraft.

Because, the vertical foot is, then the plane, ... 8 points.

∵ , ,

In rt delta,,,

..... 10 point

∴ The volume of four pyramids ... 12 points

The volume of the four pyramids is ... 14.

Solution 2:∫ plane, plane,

∴ .

∵ ,

∴ .

∵ ,

∴ Plane ... 8 points

Take the midpoint of, connect, and then,

Airplane.

The volume of triangular prism is, ... 10 minute.

Then,.

..... 12 point

Besides,

∴ .∴ .

The volume of the four pyramids is ... 14.

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

(This small question mainly examines the knowledge of finding curve trajectory equations, straight lines, circles and parabolas, and examines the mathematical thinking methods of number-shape combination, conversion transformation, functions and equations, as well as reasoning and argumentation ability, operational solving ability and innovative consciousness. )

(1) solution 1: let the coordinates of the moving point be, according to the meaning of the question,

That is to say, ... two points

Simplify it,

The equation of the curve is ... 4 points.

Solution 2: Since the distance between moving points is equal to its distance to a straight line,

According to the definition of parabola, the trajectory of a moving point is a parabola with a point as the focus and a straight line as the alignment.

..... 2 points

The equation of the curve is ... 4 points.

(2) Solution: the coordinate of the point is and the radius of the circle is.

A point is a moving point on a parabola,

∴ ( ).

∴.. 6 points

∫∴, then when, the minimum value, ... 8 points.

According to the meaning of the question,

Both sides are squares,

Solution or (irrelevant, give up) ... 10.

namely

The coordinates of the center of the circle are.

The circle intersects the axis at two points,

∴ .

∴ ... 12 point

Distance from point to straight line,

∴: This line is separated from the circle ... 14 point

20. (The full score of this short question is 14)

(This small topic mainly examines the mathematical thinking and method knowledge of sequence, inequality, transformation, classification and integration, as well as abstract generalization ability, reasoning and argumentation ability, operation and solution ability and innovation consciousness. )

(1) Solution: ∵ Series is a arithmetic progression with the first term and tolerance.

∴ .

∴.. 2 points

When,;

When.

Also suitable for the above formula.

∴.. 4 points

(2) Solution:

..... 6 points

∴

..... 8 points

Therefore, to make the inequality hold for any n,

That is, it applies to any n,

It holds for any n ... 10 points.

Let's order then.

∴ ...12 point

∴ .

The range of real numbers is ... 14 points.

[Alternative]:

∴ ...12 point

∴ .

The range of real numbers is ... 14 points.

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

(This small topic mainly examines the quadratic function, the nature of the function, the zero point of the function, the piecewise function and other knowledge. , and examined the mathematical thinking method of function and equation, classification and synthesis, as well as the ability and application consciousness of abstract generalization, reasoning, operation and solution. )

(1) solution: ∞, ∴... 1.

∫ For any r,

The symmetry axis of the function is, that is, you get ... 2 points.

That is, it applies to any r,

and .. / with ..

∵ , ∴ .

∴.. 4 points

(2) Solution: ... 5 points.

① When the symmetry axis of the function is,

If, that is, the function is monotonically increasing; ..... 6 points

In other words, the value of the function monotonically increases and the value monotonically decreases.

..... 7 points