Mathematics (liberal arts)
This paper is divided into two parts: the first volume (multiple choice questions) and the second volume (non-multiple choice questions). The perfect score is 150. Examination time 120 minutes.
The first volume (multiple choice questions, ***60 points)
Matters needing attention
1. Before answering the first volume, candidates must scribble their names, test numbers and test subjects on the answer sheet with pencils.
2. After choosing the answer for each question, black the answer label of the corresponding question on the answer sheet with a pencil. If you need to change it, clean it with an eraser, and then choose to apply other answer labels. I can't answer on the test paper.
I. Multiple-choice questions: There are *** 12 small questions in this big question, with 5 points for each small question and 60 points for * * *. Of the four options given in each question, only one meets the requirements of the topic.
1. Let the corresponding point of a complex number on the complex plane be opposite to ().
A. first quadrant B. second quadrant C. third quadrant D. fourth quadrant
2. Description of function properties: ① The function image is symmetrical about the origin; ② The function image is symmetrical about Y axis; ③ This function has a maximum value and a minimum value. The correct number is ()
A.3 B.2 C. 1 D.0
3. A school has X students in Grade One, 900 students in Grade Two and Y students in Grade Three. If a sample with a capacity of 370 students is sampled by stratified sampling, then this middle school will sample 120 students from Grade One and 100 students from Grade Three.
1900
4. In positive geometric series, Sn is the sum of the first n terms. If S 10= 10 and S30= 130, the value of S20 is ().
A.50 B.40 C.30 D
5. ""means "straight lines are perpendicular to each other"
()
A. Sufficient and unnecessary conditions B. Necessary and insufficient conditions
C. Necessary and sufficient conditions D. It is neither a sufficient condition nor a necessary condition
6. Given the solution set of the function and inequality, the image of the function can be ().
7. In △ABC, the area of △ ABC is equal to ().
A.B. C. D。
8. If point A is a fixed point on circle O and point B is a moving point on circle O, and the included angle is, the probability is ().
A.B. C. D。
9. Let p and q be two non-empty sets, and define an operation "⊙" between the sets: p ⊙ q =
If so, then P⊙Q= ()
A.B.
C.[ 1，4] D.(4，+)
10. Let F 1 and F2 be the two focal points of hyperbola, and the straight line passing through F 1 intersects hyperbola at points A and B. If |AB|=m, the maximum circumference of △AF2B is ().
A.4-m B.4 C.4+m D.4+2m
1 1. Let three different planes, and m and n are two different straight lines. Fill in one of the following three conditions at the horizontal line of the proposition "and, then m//n" to make the proposition hold.
① ; ② ; ③ 。
The conditions that can be filled in are ()
A.① or ② B.② or ③ C.① or ③ D.① or ② or ③
12. If a function is set, the number of zeros of the function is ().
A.3 B.2 C. 1 D.0。
Volume 2 (multiple choice questions, ***90 points)
Precautions:
1. Volume II * * * Page 7, answer directly on the test paper with a pen or ballpoint pen.
2. Fill in the items in the sealing line clearly before answering the questions.
2. Fill in the blanks: There are 4 small questions in this big question, with 4 points for each small question, and *** 16 points. Fill in the answers on the lines of the questions.
13. It is known that the minimum values of real numbers x and y are.
14. The tangent at a given curve is L, and the equation of a straight line passing through point P (- 1 2) and perpendicular to L is.
15. A senior middle school in Liaocheng * * has m students, numbered 1, 2,3, …, m (m ∈ n *); The school * * * offers n elective courses numbered 1, 2, 3, …, n(n∈N*). Define tags; If not. My students chose the first place. J is of course =1; Otherwise = 0; If so, the practical significance of this equation is.
16. Give the following proposition:
① Sample variance reflects the deviation degree of all sample data from the sample average.
② If the random variable x ~ n (0.43,0.182), the normal curve reaches the peak at X~N(0.43.
③ In the regression analysis model, the smaller the sum of squares of residuals, the worse the fitting effect of the model.
④ When the municipal government investigated the relationship between citizens' income and citizens' willingness to travel in Jiangbei Shuicheng, 3000 people were randomly selected. After calculation, it is found that K2=6.023. According to this data, referring to the following table, the municipal government has 97.5% confidence that citizens' income is related to their desire to travel.
p(K2≥k)…0.25 0. 15 0. 10 0.025 0.0 10 0.005 0.00 1
k… 1.323 2.072 2.706 5.024 6.635 7.879 10.888
The serial number of the correct proposition is (note: fill in the serial number of all the propositions you think are correct. )
Third, answer: This big question is ***6 small questions, ***74 points. The solution should be written in words, proof process or calculation steps.
17. (The full score of this small question is 12)
The known vector m=.
(i) Find the monotone interval of the function;
(ii) If the image is first translated to the left by one unit, and then the ordinate remains unchanged, the abscissa becomes
Original multiple, get the image of the function, if it is an even function, find the minimum value.
18. (The full score of this small question is 12)
Three views of a geometric figure are shown in the figure, where p is the intersection of diagonal lines of square ABCD and g is the midpoint of PB.
(i) Drawing a direct view of the geometry according to the three views;
(2) In the orthographic drawing, ① it is proved that: PD// surface AGC；;
② Proof: surface PBD⊥AGC.
19. (The full score of this small question is 12)
Due to the limitation of production capacity and technical level, a factory will produce some defective products. The relationship between the defective rate of this product produced by this factory and the daily output satisfaction X (unit: pieces)
.
As we all know, every qualified product can make a profit of M yuan, but every defective product will lose RMB yuan.
(1) When the daily output X exceeds 94, can the production of this product be profitable? And explain the reasons;
(ii) When the daily output x does not exceed 94, express the daily profit y (yuan) of the product produced by the factory as a function of the daily output x; In order to get the highest daily output, how many pieces should the daily output be set?
20. (The full score of this short question is 12)
known function
(i) When k is what value, the function has no value;
(2) when k > 4, determine the value of k, so that the minimum value is 0.
2 1. (The full score of this small question is 12)
According to the program block diagram shown in the figure, the output X value and Y value are recorded as y 1, y2, …, yn, …, y2007 respectively.
(i) Find the general term formula of the sequence;
(2) Write y 1, y2, y3, y4, and guess the order {yn}
A general formula yn, and prove your conclusion.
22. (The full score of this short question is 14)
As shown in the figure, it is known that the positive semi-axis of circle O and Y axis intersect at points P, A (- 1 0) and B (1 0), the straight line L and circle O are tangent to point S (L is not perpendicular to X axis), and the parabola passes through points A and B with L as the alignment.
(1) When point S moves on the circumference, prove that the focus Q of parabola is always on an ellipse C, and find this.
Equation of ellipse c;
(ii) Let m and n be two different points on the ellipse C in (i) except the short axis end point,
Q: Is there a maximum area △MON? If it exists, find the maximum value; If it does not exist, please explain why.
Liaocheng City, Shandong Province in 2007 simulated college entrance examination questions.
Mathematics (liberal arts) reference answer
I. Multiple-choice questions: This big question is a * *12 small question, with 5 points for each small question and 60 points for * * *.
1.B 2。 B 3。 D 4。 B 5。 A six. B 7。 D 8。 C 9。 A 10。 D 1 1。 C 12。 A
2. Fill in the blanks: There are 4 small questions in this big question, with 4 points for each small question, and *** 16 points.
13. 14. (Li); (Text) No.65438 +05.3 Students take 5 courses;
16.①②④
Third, answer: This big question is ***6 small questions, ***74 points.
17. Solution: (1)
........................, two points.
pass by
The monotone increasing interval of ∴ is
pass by
The monotone decreasing interval of ∴ is 6 points.
(ii) Move the image to the left by one unit to obtain the function.
The image of ................. 7 points.
Then double the abscissa to get the function.
The image of .................... 9 points.
∵ is an even function,
∴ ∵ ,
When k=0, there is a minimum value of ......................... 12.
18. Solution: (i) The direct view of this geometry is shown in the figure. ........................, 3 points.
(2)① It is proved that connecting AC, BD intersects with point O, and connecting OG, because G is the midpoint of PB,
O is the midpoint of BD, so OG//PD. There are OG plane AGC, PD plane AGC,
So PD// AGC. …………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………
② Connect PO. From three views, PO⊥ faces ABCD, so AO⊥PO. And AO⊥BO,
So AO⊥ surface PBD. Because of AO surface AGC, surface AGC … 12, 9 points.
(Li) ③ Establish the coordinate system as shown in the figure. From the three views, we can see that PO=, AB=2, AC=2, AO=,
∴P(0,0，)B(0，，0)，a(，0，0)，
C(-，0，0)，
Let the normal vector of PBA be n=(x, y, z).
∴
Let x= 1 get y= 1 and z= 1.
∴n=( 1, 1, 1)
Let the normal vector of the surface PBC be)
∴
manufacture
∴m=( 1,- 1,- 1)。
Let the angle between PAB and PBC be θ,
rule
Therefore, the angle between PAB and PBC is the cosine of 12.
19. solution: (I) when x >; 94 o'clock, p= ∴ Produce X pieces of qualified products and X pieces of defective products every day.
∴ Qualified products can gain RMB yuan, and defective products can lose RMB yuan.
∴, if the daily output exceeds 94 pieces, and the profit and loss are balanced, it will not be profitable. ........................................................................................................................... scored 4 points.
(ii) When the batch of products is produced on the same day,
Every day, the qualified product is one, and so is the defective product.
∴ ……7
... 9 points
Orders, available (shed). .............. 10 point
∵
When ∴x=84, y has a maximum value.
In order to get the highest daily profit, the daily output should be set at 84 pieces. .............. 12 point
20. Solution: (Ⅰ) √.
∴ ...............2 points (3 points)
∫ infinite value,
Heng was established.
* Same number.
The quadratic coefficient of ∵ is -2,
∴ ≤0 constant holds, so k=4.
K = 4, infinite value
(ii) When k≠4, make ....................................................................................................................................... (7 (7).
(1) When k
x()
( ,2)
2 (2,+∞)
- 0 + 0 -
↘ Minimum ↗ Maximum
Order, ∴ k = 0 ...................................... (Richard) 9 points.
2 when k >; 4 o'clock, that is, > 2 o'clock, yes.
x()
2 (2, )
( ,+∞)
- 0 + 0 -
↘ Minimum ↗ Maximum ↘
∴ order k = 8 ...................................................................................................................................................................
∴ When k=0 or k=8, there is a minimum value of 0. ....................................................................................................................................................
∴ When k=8, there is a minimum value of 0. ..........................................................................................................................................................
2 1. solution: (I) know the order from the block diagram.
∴ ...............3 points (4 points)
(ⅱ)y 1 = 2，y2=8，y3=26，y4=80 .
So, guess what.
Proof: We know from the block diagram that in the sequence {yn}, yn+ 1=3yn+2.
∴
........................... (8 points)
∴ Sequence {yn+ 1} is a geometric series with 3 as the first term and 3 as the common ratio.
∴ + 1=3? 3n- 1=3n
∴ = 3n-1() ........................... 8 points (12 points).
(ⅲ) (Li) zn=
= 1×(3- 1)+3×(32- 1)+…+(2n- 1)(3n- 1)
= 1×3+3×32+…+(2n- 1)？ 3n-[ 1+3+…+(2n- 1)]
Remember Sn =1× 3+3× 32+…+(2n-1)? 3n，①
Then 3sn =1× 32+3× 33+…+(2n-1) × 3n+1②.
①-②，-2sn = 3+2？ 32+2? 33+…+2? 3n-(2n- 1)？ 3n+ 1
=2(3+32+…+3n)-3-(2n- 1)？ 3n+ 1
=2×
=
∴
And 1+3+…+(2n- 1) = N2.
∴ ............ 12 o'clock
22.(I) prove that if Q(x, y) is aa ′, bb ′ is perpendicular to the straight line L, and a ′ and b ′ are vertical feet, connecting AQ, BQ and OS, then OS ⊥ L.
∫OS is the center line of right-angled trapezoid AA'b'b,
∴|aa′|+|bb′|=2|os|
From the definition of parabola, we know that |AA'|=|AQ|, |BB'|=|BQ|.
∴|qa|+|qb|=|aa′|+|bb′|=2|os|=4>； 2=|AB|, ... 3 points.
According to the definition of ellipse, it is concluded that focus Q is the ellipse of focus A and B.
, while 2a=4, 2c=2, ∴b2=3.
∴ The equation of ellipse C is
(Ⅱ)∵
∴P, m, n three-point * * * line ................................... 6 points.
Judging from the meaning of the question, the slope of the straight line PN exists. Let the equation of the straight line PN be y=kx+2.
Substituting into the elliptic equation, you get
Start with.
Set, by Vieta's theorem, we get,
∴
The distance from the origin o to the straight line PN is 10 minute.
∴
.................. 13 o'clock
Take the equal sign if and only if, that is, k =+
The area of ∴△MON has the maximum value of ................ 14.