Mathematics Test
(Examination time 120, full mark 120)
Reference formula: ① Sector area formula, where is the radius, the degree of the central angle, and L is the arc length.
(2) The vertex coordinates of the quadratic function image are
First, multiple-choice questions (3 points for each question, out of 27 points. Of the four options given in each question, only one is correct;
After choosing the answer for each question, black the box corresponding to the answer label on the answer sheet with 2B pencil)
The reciprocal of 1.3 is ()
A.B. C. D。
2. If the picture on the right is three views of a geometric figure, then the geometric figure is ().
A. cubic
B. triangular column
C. cylinder
D. truncated cone
3. In a knowledge contest, six students in a class answered the questions correctly, and the number of correctly answered questions was 7, 5, 6, 8, 7 and 9. The average and mode of this set of data are () respectively.
A.7，7b . 6，8c . 6，7d . 7，2
4. According to the briefing held by the Yunnan Provincial Party Committee and the provincial government in May 1 1, 2065438, all levels and departments in the province have raised 3.2 billion yuan for drought relief, which is expressed as () by scientific notation.
Yuan RMB
5. The product of two roots of a quadratic equation is ()
A.- 1B。 -2c . 1d . 2
6. As shown in the figure, in △ABC, CD is the bisector of ∠ACB, ∠ A = 80, ∠ ACB = 60, then ∠BDC= ().
a . 80 b . 90 c . 100d . 1 10
7. In the following operations, the correct one is ().
A.B.
C.D.
8. As shown in the figure, it is known that the sector area of the cone-side development diagram is 65cm, and the arc length of the sector is 10 cm, so the length of the cone bus is ().
a . 5 CMB . 10cm
C. 12cm
9. As shown in the figure, at △ABC, AB = AC, AB = 8, BC = 12 respectively.
The diameters of AB and AC are semicircles, so the area of the shaded part in the figure is ().
A.B.
C.D.
Fill in the blanks (3 points for each small question, full score 18 points. Please write the answer in black carbon pen on the horizontal line after the corresponding question number on the answer sheet)
/kloc-the reciprocal of 0/0. -Six. Yes.
1 1. As shown in the figure, in △ABC, points D, E and F are the midpoint of AB, BC and CA respectively.
If the perimeter of △ABC is 10 cm, the perimeter of △DEF is cm.
12. Simplify:
13. Calculation: =.
14. The side length of a circle with radius r inscribed with a regular triangle is. (The result can keep the root symbol)
15. As shown in the figure, point A (x 1, y 1) and point B (x2, y2) are hyperbolas.
In, and,; Pass through point a and point b to point x respectively.
Axis and Y axis are vertical line segments, the vertical feet are C, D, E and F respectively, AC and BF intersect at G point, the area of quadrilateral FOCG is 2, and the area of pentagonal AEODB is 14, then the analytic formula of hyperbola is.
Three. Answer the question (*** 10, out of 75 points. Candidates should answer with a black charcoal pen in the answer area after the corresponding question number on the answer sheet, and must write down the operation steps, reasoning process or text description. The answer beyond the answer area is invalid. Special attention: when you draw, you must use a black charcoal pen to draw on the answer sheet)
16.(5 points) Calculation:
17.(6 points) As shown in the figure, points B, D, C and F are on a straight line, BC = FD, AB = EF.
(1) Please add only one condition (no auxiliary line) to make △ ABC △ EFD. The conditions you added are:
(2) After adding conditions, it is proved that △ ABC △ EFD.
18.(5 points) Solve the inequality group:
19.(7 marks) A school conducted a math ability test for ninth-grade students, and the results were divided into four grades: A, B, C and D (note: A, B, C and D respectively represent excellent, good, qualified and unqualified). The school randomly selected 50 students from the ninth grade for statistical analysis and drew a fan-shaped statistical chart (figure)
Answer the following questions according to the information given in the picture:
(1) What is the percentage of D-level students and the number of D-level students in the randomly selected math proficiency test?
(2) In this random sampling, where does the median of students' mathematical ability test fall?
(3) If there are 800 ninth-grade students in this school, please estimate the number of students who have obtained the above (inclusive) qualifications in this math proficiency test.
20.(8 points) In the rectangular coordinate system as shown, answer the following questions:
(1) Write the coordinates of point A and point B respectively;
(2) Rotate △ABC 90 degrees clockwise around point A to draw the rotated △ AB1c1;
(3) Find the resolution function of the straight line L where the line segment B 1A is located, and write the range of the independent variable x from B 1 to a on the straight line L. 。
2 1.(8 points) The detector of the hot air balloon shows that the elevation angle of the roof of a tall building is 45 with the hot air balloon A, the depression angle of the bottom of the tall building is 60, and the horizontal distance between A and the tall building is 60m. How high is this tall building? (The result is accurate to 0. 1m, reference data:)
22.(8 points) As shown in the figure, a circular turntable divided into three identical sectors is marked with the numbers 1, 3 and 6 respectively, and the position of the pointer is fixed. After rotating the turntable, it is allowed to stop freely, and one sector will just stop at the position pointed by the pointer (when the pointer points to the intersection of two sectors, rotate the turntable again).
(1) Please draw a tree diagram or list (only choose one) to show all the results of the number of sectors pointed by the pointer after the turntable rotates for two free stops;
(2) Find the probability of the arithmetic square root irrational number of the sum of numbers in the sector pointed by the pointer after the turntable rotates twice and stops freely.
23.(7 points) Since the autumn of last year, there has been a once-in-a-century drought in Yunnan Province, and there has been no effective precipitation for more than eight months. For drought relief, an army plans to build a new canal of 3600 meters for resident villagers. In order to put the canal into use as soon as possible, the actual work efficiency is 1.8 times of the original plan. As a result, the task of canal repair was completed 20 days ahead of schedule. How many meters of canals were planned to be built every day?
24.(9 points) As shown in the figure, in trapezoidal ABCD, AD∨BC, ∠ DCB = 90, E is the midpoint of AD, P is the moving point on the side of BC (not coincident with B), and EP and BD intersect at O 。
(1) When point P moves on the edge of BC, verify: △ bop ∽△ doe;
(2) Let the similarity ratio in (1) be AD︰BC = 2︰3. Please explore: k What is quadrilateral ABPE in the following three situations? (1) When = 1, yes; ② When = 2, yes; ③ When = 3, yes. And prove the conclusion when = 2.
25.( 12 points) In the plane rectangular coordinate system, the parabola passes through O (0 0,0), A (4 4,0) and B (3 3,0).
(1) Find the analytical expression of this parabola;
(2) With the center point M of OA as the center and the length of OM as the radius ⊙M, whether there is such a point P on the parabola in (1), and the tangent L of ⊙M passes through the point P, and the angle between L and X axis is 30, if there is, then the coordinates of point P at this time are found; If it does not exist, please explain why. (Note: the result of this question can retain the root number)
Kunming 20 10 Ordinary High School (Technical Secondary School) Unified Entrance Examination
Reference answers and grading standards of mathematics test papers
1. Multiple choice questions (3 points for each small question, out of 27 points. There is only one correct answer to each small question, wrong choice, no choice, zero score for multiple choice questions).
Title 1 234 56789
Answer c a a c b b d d d
Fill in the blanks (3 points for each small question, full mark 18)
Title:10112131415.
Answers 6 and 5
r
Iii. Answering questions (out of 75)
16.(5 points) solution: original formula = ......
= ……………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………
(Note: In the first step, each pair of items gets 1 point. )
17.(6 points) (1)∠B = ∠F or AB∨EF or AC = ed ................................................................. 2 points.
(2) Proof: When ∠B = ∠F
At △ABC and △EFD
…………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………
∴△ ABC△ EFD (SAS) .............................. scored 6 points.
(Other proofs of this question are given with reference to this standard)
18.(5 points) Solution: Solve inequality ① and get: x ≤ 3 .....................1minute.
The score comes from ②: .......................... scored 2 points.
Simplified: 3 points.
Solution: 4 points.
∴ The solution set of the original inequality group is
19.(7 points) Solution: (1) ∫1-30%-48%-18% = 4%, and the percentage of people with grade ∴D is 4% .......
∫4%×50 = 2, and the number of students with grade ∴D is 2 ........................... and 2 points.
(2) ∵ The number of A-level students is 30%×50 = 15, and the number of B-level students is 48%×50 = 24.
The number of students in grade C is 18%×50 = 9, and the number of students in grade D is 4%×50 = 2. .............................................................................................................................
The median of ∴ falls in Grade B. ...................................................................................................................................................................
(3) The number of qualified persons above = 800× (30%+48%+ 18%) = 768. .........................................................................................................................
About 768 people scored 7 points above the standard ..................................................................................
20.(8 points) Solution: (1) A (2,0), B (- 1,-4) ........................................... 2 points.
(2) Whether the drawings are correct ........................................... 4 points.
(3) Let the analytical formula of the straight line L where the line segment B 1A is located be:
∫b 1(-2，3)，A(2，0)
∴ ............................... 5 points.
.......................... scored six points.
∴ The analytical formula of the straight line L where the line segment B 1A is located is: …………………………………………………………………………………………………………………………………………………………………………………………………………………………………………….
The range of the independent variable x of line segment B 1A is: -2 ≤ x ≤ 2...8 points.
2 1.(8 points) Solution: the intersection point A is the vertical line of BC, and the vertical foot is the ..................................................................................................... 1 minute of point D.
Judging from the meaning of the question: ∠ CAD = 45, ∠ Bad = 60, AD = 60m.
In Rt△ACD, ∞∠CAD = 45, AD⊥BC.
∴ CD = AD = 60 ...................................... 3 points.
In Rt△ABD,
....................................., 4 points.
∴ BD = AD？ Tan ∠ bad = 60 ............................. 5 points.
∴BC = CD+BD
= 60+60 ........................................................................................................................................................................
A: This tall building is about 163.9 meters.
22.(8 points) Solution: (1)
The list is as follows: The tree diagram is as follows:
1 3 6
1 ( 1 , 1) ( 1 ,3) ( 1 ,6)
3 (3 , 1) (3 ,3) (3 ,6)
6 (6 , 1) (6 ,3) (6 ,6)
Remarks: 4 points for this small question, 2 points for drawing a table 1 (or drawing1), and 2 points for writing correctly.
Table 1: Figure 1:
1 3 6
1
three
six
(2) The sum of numbers is: 2, 4, 7, 4, 6, 9, 7, 9, 12.
The arithmetic square root is 0, 2, 0, 2, 0, 3, 0, 3, 3.
Let the arithmetic square root of the sum of two numbers be an irrational number, which is the event A ∴ ................................. 8 points.
23.(7 points) Solution: Assume that the original plan was to build the canal x meters every day. ...................................................................................................................................................
According to the meaning of the question: ... What I want to ask is
Solution: x = 80 ... 80 ... 80 ... 80 ... 80 ... 80 ... 80.
It is verified that x = 80 is the solution of the original fractional equation.
A: It was originally planned to build an 80-meter canal every day, and ..................................... scored 7 points.
24.(9 points) (1) Proof: ∫AD∨BC
∴∠ OBP = ∠ ODE .........................1min.
In △△BOP and △△DOE.
∠OBP =∠ ode
∠ BOP = ∠ DOE ................................................ 2 points.
∴△BOP∽△DOE (there are two angles corresponding to two.
Triangle similar) .......................... 3 points.
(2) (1) parallelogram .................................... 4 points.
(2) right-angled trapezoid ........................................ 5 points.
③ ........................... of isosceles trapezoid scored 6 points.
It is proved that when ∵k = 2,
∴ BP = 2DE = AD
AD present BC = 2 present BC = AD 3 years.
PC = BC - BP = AD - AD = AD = ED
Ed∑PC, ∴ Quadrilateral PCDE is a parallelogram.
∫∠DCB = 90
∴ Quadrilateral PCDE is a rectangle with ............................. 7 points.
∴∠ EPB = 90 ..................................... 8 points.
Also in right-angled trapezoidal ABCD
A.D. ∨ B.C., AB and DC were not parallel.
∴AE∨BP, AB and EP are not parallel.
The quadrilateral ABPE is a 9-point right-angled trapezoid.
(Other proofs of this question are given with reference to this standard)
25.( 12 minutes) Solution: (1) Let the analytical formula of parabola be:
Judging from the meaning of the problem
Solution: 2 points.
The analytical formula of parabola is
(2) Existence ... 4 points
The vertex coordinates of the parabola are,
Make a parabola and ⊙M (as shown in the figure),
Let the tangent L that meets the conditions intersect the X axis at point B and tangent to ⊙M at point C.
Connect MC and pass C as the CD⊥ x axis on D.
mc = om = 2，∠CBM = 30，CM⊥BC
∴∠BCM = 90，∠BMC = 60，BM = 2CM = 4，∴B (-2，0)
Rt△CDM ∠ DCM = ∠ CDM-∠ cmd = 30.
∴DM = 1，CD = = ∴ C ( 1，)
Let the analytical formula of tangent L be:, and with points B and C on L, we can get:
Solution:
∴ The analytical formula of tangent BC is:
Point p is the intersection of parabola and tangent.
From the solution:
∴: The coordinates of point P are: ... The coordinates of point P are: ... The coordinates of point P are:
The symmetry axis of parabola is a straight line.
This parabola and ⊙M are both axisymmetric figures with straight lines.
Then make a symmetrical straight line L' of the tangent L about the straight line (as shown in the figure).
The symmetry points of B and C about straight lines B 1 and C 1 are obtained.
L' Meet the requirements in the question, and get the symmetry point of P 1 and P2 about the straight line from the symmetry:
, that is, the point of seeking.
There are four such points P***:, ... 12 points.
(For other solutions to this question, please refer to this standard score. )