Mathematics Test

This paper is divided into two parts: the first volume and the second volume. Volume one is a multiple-choice question, and volume two is a non-multiple choice question.

The full mark of this volume is 120, and the examination time is 120 minutes.

Volume 1 (multiple choice questions, ***20 points)

Note: 1. Before answering answer sheet 1, candidates must fill in their names, admission ticket numbers and subjects on the answer sheet. After the exam, the invigilator will take back the test paper and the answer sheet together.

2. After choosing the answer for each question, black the answer label of the corresponding question on the answer sheet with 2B pencil. The answer on the test paper is invalid.

First, multiple-choice questions (this big topic * *10 small topic; 2 points for each question, ***20 points. Only one of the four options given in each question meets the requirements of the topic)

The reciprocal of 1 Yes ()

Seventh century BC

2. As shown in figure 1, line A and line B intersect at point O. If ∠ 1 equals 40, ∠2 equals ().

A.50 B.60 C. 140 D. 160

3. According to CCTV's "Wen Chao Tian Xia" reported on May 27, 2007, the current cars in Beijing

The ownership is about 3 100 000. Then 3 100 000 is expressed as () by scientific notation.

a . 0.3 1× 107 b . 3 1× 105

c . 3. 1× 105d . 3. 1× 106

4. As shown in Figure 2, if the image of the inverse proportional function passes through point m (1), then the inverse proportional function

The expression is ()

A.B.

C.D.

In a black box, there is a ball of the same color, among which there are only three red balls. After mixing the balls evenly every time, you can touch a ball at random, write down the color and put it back in the black box. After a lot of repeated touching experiments, it is found that the frequency of touching the red ball is stable at 25%, so it can be inferred that A is about ().

A. 12

6. In Figure 3, EB is the diameter of semicircle O, and point A is on the extension line of EB.

AD tangent semicircle O is at point D, BC⊥AD is at point C, ab = 2, semicircle O.

If the radius of is 2, the length of BC is ()

A.2 B. 1

C. 1.5

7. In the hot summer, Team A installed 66 air conditioners for Community A and Team B installed 60 air conditioners for Community B. Both teams started construction and completed at the same time. Team A installs 2 more air conditioners every day than Team B, and Team B installs X air conditioners every day. According to the meaning of the question, the correct one in the following equation is ().

A.B.

C.D.

8. The ancient "river map" in China consists of 3×3 squares, each of which

There are different numbers of bitmaps, three in each row, column and diagonal.

The sum of the points of a vertex graph is equal.

Fig. 4 shows a partial dot diagram of the "River Map". Please calculate the corresponding point of point p.

The picture shows ()

9. Party A and Party B drive at a constant speed from A to B along the same route, and the distance between A and B ..

It's 20 kilometers ... the distance they travel is between s(km) and t(h) after A leaves.

The functional image of is shown in Figure 5. According to the image information, the following statement is true ()

The speed of A.A is 4 km/h and the speed of b is10 km/h.

C.B leaves A later than A 1 h and arrives at B 3 h later than B.

10.m, n, p and q respectively represent one of four simple geometric figures (line segment, regular triangle, square and circle).

Figure 6- 1- Figure 6-4 is the combination of two graphs in M, N, P and Q (the combination is indicated by "&").

Then, in the combination diagram below, P & amp； Q's is ()

General nuclear division

Entrance Examination for Junior High School Graduates in Hebei Province in 2007

Mathematics Test

Volume 2 (multiple choice questions, *** 100)

Note: 1. Fill in the items on the left side of the sealing line clearly before answering the second volume.

When answering the second question, write the answer directly on the test paper with a blue or black pen or ballpoint pen.

Title 23

19 20 2 1 22 23 24 25 26

score

Scoring reviewer

Second, fill in the blanks (this big question ***8 small questions; 3 points for each small question, 24 points for * * *. Put the answer

Write on the horizontal line of the question)

1 1. Calculation: =.

12. Comparison size: 7. (Fill in ">", "=" or "

13. As shown in Figure 7, if □ABCD and □EBCF are symmetrical about the line where BC is located, ∠ Abe = 90,

Then ∠ f = 0.

14. If is, the value of is.

15. In Figure 8, every square marked with numbers is a reversible wooden sign, and only two wooden signs are marked with winning marks on the back, so the probability of randomly flipping a wooden sign to win the prize is _ _ _ _ _ _ _ _.

16. As shown in Figure 9, in the grid diagram of 10×6 (the side length of each small square is 1 unit), the radius ⊙A is 1 and the radius ⊙ B is 2, so it is necessary to make ⊙A and.

Yao ⊙A needs to be translated to the right by one unit length from the position shown in the figure.

17. It is known that when n= 1, a1= 0; = 0; When n=2, A2 = 2；; When n=3,

a3 = 0； ... then the value of a 1+a2+a3+a4+a5+a6 is.

18. Figure 10- 1 shows three geometric bodies with the same shape standing on a horizontal plane (the bottom surface is a circular surface, unit: cm). When they are combined into a new geometry as shown in figure 10-2, the volume of the new geometry is cm3 ... (the calculation results are retained).

Third, answer questions (this big question ***8 small questions; ***76 points. The solution should be written in words, proving the process or calculation steps)

Scoring reviewer

19. (Full score for this small question)

Given the value of,,.

Scoring reviewer

20. (The full score for this short question is 7)

On the straight-line speed-limited expressway, the maximum speed of cars is not allowed to exceed 60 km/h (that is, m/s). The traffic management department has set up a speed monitoring point A at a distance of 100 m from the expressway. In the coordinate system shown in Figure 1 1, point A is located on the Y axis, velocity measuring section BC is located on the X axis, and point B is located 60 northwest of point A..

(1) Please draw the ray AC representing the direction of 45 northeast in figure 1 1 and mark the position of point C;

(2) The coordinate of point B is, and the coordinate of point C is;

(3) It takes 15 s for the car to travel from point B to point C. Please judge whether the car is speeding on the speed-limited highway through calculation. (In this short question)

Scoring reviewer

2 1. (Full score for this small question 10)

Two basketball teams, Team A and Team B, played five games during the training. After counting the game results, draw a statistical chart, as shown in figure 12- 1 and figure 12-2.

(1) Draw a dotted line in figure 12-2, indicating the changes of team B's performance in five games during training;

(2) It is known that Team A scored an average of 90 points in five games. Please calculate the average score of team B in five games.

(3) For these five games, calculate the difference between the two teams respectively;

(4) If you choose a team from Team A and Team B to participate in the basketball championship, on the basis of the above statistics, try to make a brief analysis from four aspects: average score, disconnection trend, winning times and range. Which team do you think can get better results?

Scoring reviewer

22. (The full score for this short question is 8)

As shown in figure 13, the image of known quadratic function passes through point A and point B. 。

(1) Find the expression of quadratic function;

(2) Write the symmetry axis and vertex coordinates of parabola;

(3) both point P(m, m) and point q are on the function image (where m > 0), and these two points are symmetrical about parabola. Find the value of m and the distance from Q point to X axis.

Scoring reviewer

23. (Full score for this small question 10)

In figure 14- 1- 14-5, the side length of square ABCD is a, the hypotenuse AE of isosceles right triangle FAE is 2b, and the sides AD and AE are on the same straight line.

Operation example

When 2b < a, as shown in figure 14- 1, select point g on BA, make BG = b, connect FG and CG, cut off △FAG and △CGB, and splice them to the positions of △FEH and △CHD respectively to form quadrilateral FGCH. ..

Thinking and discovering

After Xiao Ming's operation, he found that the shear splicing method is to rotate △FAG counterclockwise around point F by 90 to the position of △FEH, so it is easy to know that EH and AD are on the same line. The connection CH, DH=BG can be obtained by shear splicing method, so △CHD △△ CGB can rotate 90 clockwise around point C to the position of △ CHD. For the quadrilateral FGCH obtained by cutting and splicing (as shown in figure 14- 1), the FM⊥AE (abbreviated) whose point F is point M can be judged by SAS axiom, and FH=HC=GC=FG and ∠ FHC = 90 can be easily obtained.

Practical inquiry

The area of (1) square FGCH is; (expressed by a formula containing a and b)

(2) Compared with the cutting and splicing method in figure 14- 1, please draw a schematic diagram of cutting and splicing a new square in three cases: figure 14-2- figure 14-4.

Lenovo development

Xiao Ming found that when b≤a, all these figures can be cut into squares, and the position of the selected point G moves up in the direction of BA with the increase of B.

When b > a, can the figure shown in figure 14-5 be cut into squares? If you can, please draw a schematic diagram of cutting and spelling in the picture; If not, briefly explain the reasons.

Scoring reviewer

24. (Full score for this small question 10)

In △ABC, AB=AC, the extension line of BA intersects at point G. Place an isosceles right-angled triangular ruler as shown in figure 15- 1, with the right-angled vertex of the triangular ruler being f, one right-angled side being in a straight line with the AC side and the other right-angled side passing through point B. 。

(1) In the figure 15- 1, please observe and measure BF and CG.

Length, guess and write the quantitative relationship between BF and CG,

Then prove your guess;

(2) When the triangular ruler translates to the position shown in Figure 15-2 along the AC direction,

One right-angle side is still on the same line as the AC side, and the other

The right-angled edge intersects the BC edge at point D, and the intersection point D is DE⊥BA.

At point e, please observe and measure DE, DF and CG at this time.

The length of, guess and write the satisfaction between DE+DF and CG.

Quantitative relationship, and then prove your guess;

(3) When the triangular ruler continues to be straight along the AC direction on the basis of (2)

Move to the position shown in figure 15-3 (point f is on the AC line,

And point f does not coincide with point c), does the conjecture in (2) hold?

Is it still valid? (No need to explain why)

Scoring reviewer

25. (The full score of this short question is 12)

A mobile phone dealer plans to buy three mobile phones of a certain brand ***60, each with at least 8 sets, and just used up the payment of 6 1 1,000 yuan. Suppose you buy X mobile phones and Y mobile phones. The purchase price and pre-sale price of the three mobile phones are as follows:

Mobile phone models a, b and c

Purchase price (unit: yuan/department) 90012001100.

Pre-sale price (unit: yuan/department)120016001300

(1) Use the formula containing x and y to express the purchase quantity of Class C mobile phones;

(2) Find the functional relationship between y and x;

(3) Assuming that all the purchased mobile phones have been sold, mobile phone dealers need to pay various extra fees in the process of buying and selling these mobile phones *** 1500 yuan.

(1) Find the functional relationship between the expected profit p (yuan) and x (part);

(Note: Estimated profit P = total pre-sale-purchase price-various expenses)

Find out the maximum estimated profit and write down how many mobile phones to buy at this time.

Scoring reviewer

26. (The full score of this short question is 12)

As shown in figure 16, in the isosceles trapezoid ABCD, AD‖BC, AB=DC=50, AD=75, BC = 135. Point P starts from point B and moves at a uniform speed of 5 units per second along the dotted line BA-AD-DC to point C; Point q starts from point c and moves in the direction of CB at a constant speed of 3 units per second. After passing through point Q, the light QK⊥BC goes up, and the intersecting line segment CD-DA-AB starts to move at point E, and points P and Q start to move at the same time. When point P coincides with point C, point Q also stops. Set points p and q move for t seconds (t > 0).

(1) When point P reaches the end point C, find the value of t and point out the length of BQ at this time;

(2) When point P moves to AD, why does the value of t make PQ‖DC?

(3) Let the area of light QK sweeping the trapezoidal ABCD be S, and find out the functional relationship between S and T when point E moves to CD and DA respectively; (Don't write the range of T)

(4) Can △ PQE become a right triangle? If yes, write the range of t; If not, please explain why.

Entrance Examination for Junior High School Graduates in Hebei Province in 2007

Reference answers and grading standards of mathematics test questions

Description:

1. In the marking process, if candidates have other correct answers, they can refer to the grading standard and give points step by step as appropriate.

2. Adhere to the principle of reviewing each question to the end. When the examinee gives an incorrect answer in a certain step, which affects the subsequent part, if the answer after this step does not change the content and difficulty of this question, the score of the latter part can be determined according to the degree of influence, but it shall not exceed half of the score of the subsequent part; If there is a serious mistake in the answer after this step, no extra points will be given.

3. Solve the score marked on the right, indicating the cumulative score of correctly completing this step. Only integer points are given.

First, multiple-choice questions (2 points for each small question, 20 points for * * *)

The title is 1 23455 6789 10.

Answer A C D B A B D C C B

Fill in the blanks (3 points for each small question, 24 points for * * *)

1 1 . a3； 12. ,

This car is speeding on the speed limit highway. ................................................................................................................................................................

2 1. solution: (1) as shown in figure 2; ........................ (2 points)

(2) =90 (points); .................... (3 points)

(3) The difference of Team A is 18.

Team b performed 30 points worse; .................... (5 points)

(4) From the point of average score, the average score of the two teams is the same, and their strength is roughly equal;

Judging from the trend of the broken line, the performance of team A is rising, while that of team B is rising.

Competition results showed a downward trend; Judging from the number of wins, Team A won three games.

Team b won two games, and team a scored better;

From a very poor point of view, the performance of team A is less volatile than that of team B, and the performance of team A is relatively stable ... (9 points)

To sum up, it is better to choose Team A for ............................................ (10).

22. Solution: (1) x=- 1, y =-1; Replace X=3 and y=-9 respectively.

Solution ......................... (3 points)

The expression of the quadratic function is ............................................ (4 points).

(2) The symmetry axis is; Vertex coordinates are (2,-10) ........................... (6 points).

(3) Substituting (m, m) to obtain,

The solution is. ∵ m > 0, ∴ does not meet the meaning of the question, so discard it.

∴ m = 6 ......................................... (7 points)

Point p and point q are symmetrical about the symmetry axis,

∴ The distance from Q to X axis is 6 ....................................................... (8 points).

23. Practical inquiry (1) A2+B2; .................................... (2 points)

(2) The splicing method is shown in Figure 3- Figure 5. (2 points for each picture) ................... (8 points)

Lenovo can expand; ........................................ (9 points)

The cutting and spelling methods are shown in Figure 6 (BG = DH = B in the figure) and ........................... (10).

(Note: Figure 6 can be spliced into a square with an area of a2+b2 by other splicing methods and divided equally. )

24.( 1)BF = CG； ............................................. (1 min)

It is proved that at △ABF and △ACG,

∠∠F =∠G = 90，∠FAB=∠GAC，AB=AC，

∴△ABF≌△ACG(AAS)，

∴ BF = CG .................................... (4 points)

(2)DE+DF = CG； ............................. (5 points)

It is proved that DH⊥CG intersects with D at point H (as shown in Figure 7) ... (6 points)

∵DE⊥BA at point E, ∠ G = 90, DH ⊥ CG,

∴ quadrilateral EDHG is a rectangle, ∴DE=HG, DH ∴ BG. ∴∠ GBC = ∠ HDC。

ab = ac，∴∠ FCD = ∠ GBC = ∠ HDC。 And ∵∠F =∠DHC = 90°, CD=DC,

∴△FDC≌△HCD(AAS),∴DF=CH.

∴GH+CH=DE+DF=CG, that is, DE+DF = CG ............................. (9 points)

(3) It is still valid ....................................................... (10 score)

(Note: this problem can also be proved by the area, such as the link advertisement in (2))

25. Solution: (1) 60-x-y; ......................................... (2 points)

(2) According to the meaning of the question, 900 x+1200 y+1100 (60-x-y) = 61000,

Y = 2x-50 ............................................... (5 points)

(3)① From the meaning of the question, p =1200x+1600y+1300 (60-x-y)-61000-1500,

P = 500 x+500 .................................................... (7 points)

② The number of Class C mobile phones purchased is 60-x-y = 1 10-3x. According to the set of inequalities, we get

The solution is 29 ≤ x ≤ 34.

The value range of ∴x is 29≤x≤34, and x is an integer. (Note: X is not an integer without deduction) ... (10)

∵P is a linear function of x, k = 500 > 0, and∴ p increases with the increase of x 。

When the maximum value of X is 34, P has a maximum value, which is 17500 yuan ..................... (1 1).

At this time, I bought 34 A-type mobile phones, 18 B-type mobile phones and 8 C-type mobile phones ............... (12 points).

26. solution: when (1) t = (50+75+50) ÷ 5 = 35 (seconds), point p reaches the end point C. ........................................................................( 1).

At this point, QC=35×3= 105, and the length of ∴BQ is135-105 = 30 ............................................. (2 points).

(2) as shown in fig. 8, if PQ‖DC, AD‖BC, it is a quadrilateral PQCD.

Is a parallelogram, so that PD=QC, from QC=3t, BA+AP=5t.

50+75-5t = 3t, and t = is the solution.

Upon examination, when t=, PQ ‖ DC ....................................... (4 points).

(3)① When point E moves on the CD, as shown in Figure 9, it passes through point A and point D respectively.

Let AF⊥BC be at point F, DH⊥BC be at point H, and then a quadrilateral.

ADHF is rectangular, while △ ABF △ DCH, therefore,

FH= AD=75, so BF = CH = 30. ∴ DH = AF = 40。

QC=3t, then QE=QC? tanC=3t？ =4t。

(Note: Similar triangles can also be used to solve)

∴S=S⊿QCE = quantitative easing? QC = 6t2 .................................... (6 points)

② When point E moves on DA, as shown in Figure 8. Let point D be DH⊥BC of point H, from ①, DH=40, CH=30, QC=3t, then ED = QH = QC-CH = 3t-30.

∴S= trapezoidal qcde = (ed+QC) DH =120t-600 ............................. (8 points).

(4)△PQE can be changed into a right triangle, and ........................................................................................................................................................ (9) scored 9 points.

When △PQE is a right triangle, the value range of t is 0 < t ≤ 25 and t≠ or t = 35...( 12 minute).

(Note: (4) If you don't answer t≠ or t=35 in the question, deduct 1 point, and the rest of the writing methods are given as appropriate. )

The following is the answer to question (4) for teachers' reference only:

(1) When point P is on BA (including point A), that is, 0 < t ≤ 10, as shown in Figure 9. If point p is PG⊥BC of point g, then PG=PB? SinB=4t, while QE=4t = PG, it is easy to get that the quadrilateral PGQE is a rectangle, and at this time △PQE can always become a right triangle.

② When both point P and point E are on AD (excluding point A but including point D), that is, 10 < t ≤ 25, as shown in Figure 8.

According to QK⊥BC and ad‖ BC, at this time △PQE is a right triangle, but points P and E cannot coincide, that is,

5t-50+3t-30 ≠ 75，t≦。

③ When point P is on DC (excluding point D but including point C),

That is, when 25 < t ≤ 35, as shown in figure 10, ed > 25× 3-30 = 45,

It can be seen that point P is outside the circle with QE = 40°, so

∠EPQ will not be a right angle.

From ∠PEQ < ∠ dirk, we can know that ∠PEQ must be an acute angle.

For ∠PQE, ∠PQE≤∠CQE, only when points p and c

Coincidence, that is, when t=35, as shown in figure 1 1, ∠ pqe = 90, △PQE.

This is a right triangle.

To sum up, when △PQE is a right triangle, the value range of t is 0 < t ≤ 25 and t≠ or t = 35.