Eighth grade mathematics volume I final exam questions 1. Multiple choice question: This big question * *12 is a small one. Only one of the four options given in each question is correct. Please select the correct option. 1-8 small questions have 3 points for each small question, 9- 12 small questions have 4 points for each small question, and the wrong questions are wrong.
1. The following four numbers are energy-saving, water-saving, low-carbon and green food signs respectively. Of these four signs, the one that is symmetrical is ().
A.B. C. D。
2. The following operation is correct ()
A.a+a=a2 B.a3？ a2=a5 C.2 =2 D.a6？ a3=a2
3. The square root of is ()
A.2 B？ Two C. D.
4. Use scientific notation to express -0.00059 as ().
A.﹣59? 10﹣5 B.﹣0.59？ 10﹣4 C.﹣5.9？ 10﹣4 D.﹣590？ 10﹣7
5. The range of X that makes the score meaningful is ()
A.x？ 3 B.x？ 3 C.x？ Three-dimensional x=3
6. In quadrilateral ABCD, diagonal AC and BD intersect at point O, and the following conditions cannot determine that this quadrilateral is a parallelogram ().
A.AB∨DC, AD∨ BC B.AB=DC, AD = BC C.AO=CO, BO = DO D. AB∨DC, AD=BC.
7. If it makes sense, the value of is ()
A.B.2 C. D.7
8. Given that A-B = 1 and ab=2, the value of formula a+b is ().
A.3 B？ c？ 3 D？ four
9. As shown in the figure, the perimeter of parallelogram ABCD is 4a, and AC and BD intersect at point O, OE? AC passes through AD to e, then the circumference of △DCE is ()
a . a . b . 2a c . 3a d . 4a
10. Known xy
A.B. C. D。
1 1. As shown in the picture, young students fold a piece of paper with a right triangle. A and b overlap, and the crease is DE. If AC=4 and BC=3 are known,? C=90？ The length of EC is ()
A. 2 D BC.
12. If the fractional equation about x has no solution, the value of the constant m is ().
C.﹣ 1 D.﹣2
Fill-in-the-blank question: This big topic is ***4 small questions, and the score is *** 16. Only the final result is needed, and every small question is answered correctly.
13. Factorize xy-x+y- 1, and the result is.
14. The base length of an isosceles triangle with a waist length of 5 and a height of 3 is.
15. If x2 ~ 4x+4+= 0, the value of xy is equal to.
16. As shown in the figure, in quadrilateral ABCD, AB=20, BC= 15, CD=7, AD=24,? B=90？ And then what? A+？ C= degrees.
Third, answer: This big question is ***6 small questions, ***64 points. Write the necessary text description, proof process or calculation steps when solving problems.
17. As shown in the figure, write down the coordinates of each vertex of △ABC and the coordinates of △ A1b1each vertex of △ABC symmetric about X, and draw △A2B2C2 of △ABC symmetric about Y..
18. Simplify before evaluating:
(1) 5x2-(y+x)-(2x-y) 2, where x= 1 and y=2.
(2)( )? , where a=.
19. List equations and solve application problems.
A middle school customized a batch of cotton student clothes in Juxian garment factory, and Workshop A produced it alone for 3 days to complete the total amount. At this time, the weather forecast is coming soon, so we should speed up production. At this time, workshop B was added, and the two workshops produced together for 2 days, and the orders were all completed. How many days will it take if workshop B produces this batch of cotton student clothes alone?
20. The lengths of the three sides of △ ABC are A, B and C respectively, which satisfy A2-4A+B2-4C = 4B- 16-C2. Try to determine the shape of △ ABC and prove your conclusion.
2 1. As shown in the figure, the quadrilateral ABCD is a parallelogram, and? BCD= 120？ ，CB=CE，CD=CF。
(1) Verification: AE = AF
(2) Q? The degree of EAF.
22. Reading materials:
After learning the quadratic root, Xiao Ming found that some formulas with roots can be written as the square of another formula, such as 3+2 =( 1+ )2. Xiao Ming, who is good at thinking, made the following explorations:
Let a+b =(m+n )2 (where a, b, m and n are integers), then a+b = m.
A=m2+2n2，b=2mn。 In this way, Xiao Ming found a way to turn a formula similar to a+b into a plane.
Please imitate Xiao Ming's method and try to solve the following problems:
(1) When a, b, m and n are all positive integers, if a+b =(m+n )2, a and b are represented by formulas containing m and n, so that a= and b =.
(2) Using the conclusion of inquiry, express it in a completely flat way: =.
Please simplify:
First, multiple-choice questions: this big question *** 12 is a small one. Only one of the four options given in each question is correct. Please select the correct option. 1-8 is 3 points for each small question, and 9- 12 is 4 points for each small question. Wrong choice.
1. The following four numbers are energy-saving, water-saving, low-carbon and green food signs respectively. Of these four signs, the one that is symmetrical is ().
A.B. C. D。
Axisymmetric figure of test point.
The analysis is based on the concept of axisymmetric graphics.
Solution: A, it is not an axisymmetric figure, so this option is wrong;
B, not axisymmetric graphics, so this option is wrong;
C, not axisymmetric graphics, so this option is wrong;
D is an axisymmetric figure, so this option is correct.
So choose D.
This topic examines the knowledge of axisymmetric graphics. The key to an axisymmetric figure is to find the axis of symmetry, and the two parts of the figure can overlap after being folded along the axis of symmetry.
2. The following operation is correct ()
A.a+a=a2 B.a3？ a2=a5 C.2 =2 D.a6？ a3=a2
The division of the power of the same base in the examination center; Merge similar projects; Multiplication with the same base; Addition and subtraction of quadratic root.
This analysis can be solved by combining similar terms, multiplication and division of the same base power.
Solution: A, a+a=2a, so it is wrong;
b、a3？ A2=a5, correct;
Therefore, C is wrong;
d、a6？ A3=a3, so it is wrong;
Therefore, choose: B.
On this topic, we study the multiplication and division of merging similar terms and the same base power. The key to solve this problem is to combine similar items in memory and same base powers's multiplication and division.
3. The square root of is ()
A.2 B？ Two C. D.
Square root of arithmetic in test center; square root
Thematic standard questions.
Simplify the analysis first and then solve it according to the definition of square root.
Solution: ∫= 2,
? What is the square root of? .
So choose D.
This topic examines the definitions of square root and arithmetic square root. Correct simplification is the key to solving problems, and this topic is easy to make mistakes.
4. Use scientific notation to express -0.00059 as ().
A.﹣59? 10﹣5 B.﹣0.59？ 10﹣4 C.﹣5.9？ 10﹣4 D.﹣590？ 10﹣7
Scientific counting of test sites? Represents a smaller number.
Positive numbers with absolute values less than 1 can also be expressed by scientific notation, and the general form is a? Different from the scientific notation of large numbers,10-n uses a negative exponential power, and the exponent is determined by the number of zeros before the first non-zero number on the left of the original number.
Solution: -0.00059 =-5.9? 10﹣4,
So choose: C.
Comment on this topic. Use scientific notation to represent smaller numbers. The general form is a? 1 0-n, where1? | a |< 10, n is determined by the number of zeros before the first non-zero number on the left of the original number.
5. The range of X that makes the score meaningful is ()
A.x？ 3 B.x？ 3 C.x？ Three-dimensional x=3
Conditions for the meaningful part of the test point.
The condition that the analysis score is meaningful is that the denominator is not equal to zero, so x-3? 0.
Answer: the score is meaningful,
? x﹣3？ 0.
Solution: x? 3.
So choose: C.
Comments on this issue mainly examine the conditions under which scores are meaningful. When the score is meaningful, the denominator of the score is not zero, which is the key to solving the problem.
6. In quadrilateral ABCD, diagonal AC and BD intersect at point O, and the following conditions cannot determine that this quadrilateral is a parallelogram ().
A.AB∨DC, AD∨ BC B.AB=DC, AD = BC C.AO=CO, BO = DO D. AB∨DC, AD=BC.
Determination of parallelogram of test center.
The analysis is judged according to the parallelogram judgment theorem.
Solution: a, by? AB∨DC, AD∨ BC? It can be seen that the two opposite sides of the quadrilateral ABCD are parallel, so the quadrilateral is a parallelogram.
B. By whom? AB=DC, AD = BC? It can be seen that the two opposite sides of the quadrilateral ABCD are equal, so the quadrilateral is a parallelogram.
C. By whom? AO=CO，BO=DO？ It can be seen that the two diagonals of the quadrilateral ABCD are bisected, so the quadrilateral is a parallelogram.
D. By who? AB∨DC，AD=BC？ It can be seen that one set of opposite sides of the quadrilateral ABCD is parallel and the other set of opposite sides is equal, so it cannot be judged that the quadrilateral is a parallelogram.
So choose D.
This topic reviews the judgment of examining parallelogram.
(1) Two groups of parallelograms with parallel opposite sides are parallelograms.
(2) Two groups of quadrangles with equal opposite sides are parallelograms.
(3) A set of quadrilaterals with parallel and equal opposite sides is a parallelogram.
(4) Two groups of quadrangles with equal diagonals are parallelograms.
(5) The quadrilateral whose diagonal lines bisect each other is a parallelogram.
7. If it makes sense, the value of is ()
A.B.2 C. D.7
A meaningful condition for testing the quadratic root of the center.
The value of x can be calculated according to the concept of arithmetic square root.
Solution: from the meaning of the question, x? 0,﹣x？ 0,
? x=0，
Then =2,
Therefore, choose: B.
This topic examines the meaningful conditions of quadratic square root and the concept of arithmetic square root. Mastering the number of square roots of quadratic roots must be nonnegative, which is the key to solve the problem.
8. Given that A-B = 1 and ab=2, the value of formula a+b is ().
A.3 B？ c？ 3 D？ four
Complete square formula of test center.
Thematic calculation problems; Algebraic expression.
Analyze the square of both sides of A-B = 1, simplify it with the complete square formula, substitute ab=2 to find the value of a2+b2, and then use the complete square formula to find the value of the formula.
Solution: both sides of the square A-B = 1, and we get: (A-B) 2 = A2+B2-2AB = 1
Substituting ab=2 to get a2+b2=5,
? (a+b)2=a2+b2+2ab=5+4=9，
So a+b=? 3,
So choose C.
This topic reviews the complete square formula, and mastering the complete square formula is the key to solve this problem.
9. As shown in the figure, the perimeter of parallelogram ABCD is 4a, and AC and BD intersect at point O, OE? AC passes through AD to e, then the circumference of △DCE is ()
a . a . b . 2a c . 3a d . 4a
The nature of parallelogram in examination center.
Analysis basis? The circumference of ABCD is 4a, and AD+CD=2a and OA=OC can be obtained. AC, according to the nature of the perpendicular line, it can be proved that AE=CE, and then the circumference of △DCE =AD+CD can be obtained.
Solution: ∵? The circumference of ABCD is 4a,
? AD+CD=2a，OA=OC，
∵OE？ Communication,
? AE=CE，
? The circumference of △DCE is: CD+DE+CE=CD+DE+AE=CD+AD=2a.
Therefore, choose: B.
This topic reviews the properties of the vertical line in parallelogram and line segment. Note that the perimeter of △DCE =AD+CD is the key.
10. Known xy
A.B. C. D。
Properties and simplification of quadratic roots in inspection center.
First, we should find the range of x and y, and then simplify it according to the nature of square root.
Solution: To be meaningful, you must? 0,
Solution: x? 0,
∵xy & lt； 0,
? y & lt0,
? y =y？ =﹣ ,
So choose a.
This paper investigates the application of the properties of quadratic roots, and the key to solve this problem is to simplify correctly according to the properties of quadratic roots.
1 1. As shown in the picture, young students fold a piece of paper with a right triangle. A and b overlap, and the crease is DE. If AC=4 and BC=3 are known,? C=90？ The length of EC is ()
A. 2 D BC.
Test center folding transformation (folding problem).
If DE is the perpendicular line of AB side, AE=BE, AE=x, then the value of X can be obtained by using the right-angle pythagorean theorem △BCE, and then the length of EC can be obtained.
Solution: ∫DE vertically divides AB,
? AE=BE，
Let AE=x, then BE=x, EC = 4-X.
In the right angle △BCE, BE2=EC2+BC2, then x2 = (4-x) 2+9,
Solution: x=,
Then EC = AC-AE = 4-=.
So choose B.
This topic reviews the folding properties and Pythagorean theorem of graphs, and it is the key to correctly understand the middle vertical line that virtue is AB.
12. If the fractional equation about x has no solution, the value of the constant m is ().
C.﹣ 1 D.﹣2
Solving the score equation of test sites; Solve a linear equation with one variable.
Thematic calculation problems; Change ideas; Linear equation (group) and its application; Fractional equation and its application.
The fractional equation is named integral equation, and x=3 is obtained from the fractional equation without solution, and the value of m can be obtained by substituting it into the integral equation.
Solution: Multiply both sides of the equation by the simplest common denominator (x-3) to get:1= 2 (x-3)-m,
When x=3, the original fractional equation has no solution.
? 1 =-m, which means m =-1;
So choose C.
Comment on this topic mainly investigates the solution of fractional order equation, and the understanding of the concept of no solution of fractional order equation is the key to this topic.
Fill-in-the-blank question: This big topic is ***4 small questions, and the score is *** 16. Only the final result is needed, and every small question is answered correctly.
13. Factorizing xy﹣x+y﹣ 1, the result is ﹣ 1) (x+ 1).
Test site factor decomposition-grouping decomposition method.
The analysis first regroups, and then decomposes the factors by extracting common factors to get the answer.
Solution: xy-x+y- 1
=x(y﹣ 1)+y﹣ 1
=(y﹣ 1)(x+ 1).
So the answer is: (y- 1) (x+ 1).
This review mainly examines the factorization method of grouping decomposition, and correct grouping is the key to solving the problem.
14. The base length of an isosceles triangle with a waist length of 5 and a height of 3 is 8 or 3.
Test the nature of the central isosceles triangle; The trilateral relationship of a triangle.
The analysis is classified and discussed according to the height of 3 on different sides. The answer to this question can be obtained by Pythagorean theorem.
Solution: ① As shown in figure 1.
When AB=AC=5 and AD=3,
Then BD=CD=4,
So the base length is 8;
② As shown in Figure 2.
When AB=AC=5 and CD=3,
AD=4，
So BD= 1,
Then BC= =,
That is, the length of the bottom edge at this time is;
③ As shown in Figure 3.
When AB=AC=5 and CD=3,
AD=4，
So BD=9,
Then BC= =3,
That is, the base length is 3.
So the answer is: 8 or 3.
This topic examines the properties and pythagorean theorem of isosceles triangle. The key to solving the problem is to discuss it in three categories.
15. If x2 ~ 4x+4+= 0, the value of xy is equal to 6.
Test points for solving binary linear equations; The nature of non-negative number: even power; The nature of non-negative numbers: arithmetic square root; Application of matching method.
Thematic calculation problems; Linear equation (group) and its application.
After analyzing the deformation of the known equation, the equation is listed by using non-negative properties, and the values of X and Y are obtained by solving the equation, so as to determine the value of xy.
Solution: ∵x2﹣4x+4+ =(x﹣2)2+ =0,
? ,
Solution:
Then xy=6.
So the answer is: 6.
Comment on this question. It is the key to understand binary linear equations, the application and nonnegative properties of collocation method, and master the algorithm skillfully.
16. As shown in the figure, in quadrilateral ABCD, AB=20, BC= 15, CD=7, AD=24,? B=90？ And then what? A+？ C= 180 degrees.
Inverse theorem of Pythagorean theorem in test sites; Pythagorean theorem
Analyzing the inverse theorem of Pythagorean theorem is one of the methods to judge right triangle.
Solution: Connect AC, according to Pythagorean theorem AC= =25.
∫AD2+DC2 = AC2 means 72+242=252,
? According to the inverse theorem of Pythagorean theorem, △ADC is also a right triangle. D=90？ ,
So what? A+？ C=？ D+？ B= 180？ , so fill in 180.
This topic examines Pythagorean theorem and its inverse theorem. It is a better topic to examine the two theorems in the same topic.
Third, answer: This big question is ***6 small questions, ***64 points. Write the necessary text description, proof process or calculation steps when solving problems.
17. As shown in the figure, write down the coordinates of each vertex of △ABC and the coordinates of △ A1b1each vertex of △ABC symmetric about X, and draw △A2B2C2 of △ABC symmetric about Y..
Test site map-axisymmetric transformation.
The position of each corresponding point is obtained by using the coordinate properties of axisymmetric points about X axis and Y axis respectively, and then the answer is obtained.
Solution: Axis symmetry of each vertex coordinate of △ABC and △A 1B 1C 1 each vertex coordinate of △ABC about x;
a 1(﹣3,﹣2),b 1(﹣4,3),c 1(﹣ 1, 1)，
As shown in the figure: △A2B2C2, which is what you want.
This paper mainly investigates the axisymmetric transformation, and concludes that the position of the corresponding point is the key to solving the problem.
18. Simplify before evaluating:
(1) 5x2-(y+x)-(2x-y) 2, where x= 1 and y=2.
(2)( )? , where a=.
Simplified evaluation of test center scores: mixed operation of algebraic expressions? Simplify the assessment.
Analysis (1) Firstly, the original formula is simplified according to the law of mixed operation of algebraic expressions, and then the values of x and y are substituted for calculation.
(2) Simplify the original formula according to the law of fractional mixed operation, and then substitute the value of a for calculation.
Solution: (1) Original formula = 5x2-x2+y2-4x2+4xy-y2.
=4xy，
When x= 1 and y=2, the original formula =4? 1? 2=8;
(2) The original formula =?
= ?
=a﹣ 1，
When a=, the original formula =- 1.
This question examines the simplified evaluation of fractions, and understanding the law of fractional mixed operation is the key to solve this question.
19. List equations and solve application problems.
A middle school customized a batch of cotton student clothes in Juxian garment factory, and Workshop A produced it alone for 3 days to complete the total amount. At this time, the weather forecast is coming soon, so we should speed up production. At this time, workshop B was added, and the two workshops produced together for 2 days, and the orders were all completed. How many days will it take if workshop B produces this batch of cotton student clothes alone?
Application of score equation of test center.
It takes x days for workshop B to produce this batch of cotton student clothes alone, so it can produce the total amount every day. If workshop a completes the total production for 3 days alone, it can make the total production for each day. According to the total workload, list the equations and solve 1.
Solution: it takes x days for workshop b to make this batch of cotton student clothes alone, so it can be done in total every day; If Workshop A completes the total output for three days alone, it can produce the total output for every day.
According to the meaning of the question, it is: +2? ( + )= 1,
The solution is x=4.5.
X=4.5 is the root of the original equation.
A: It will take 4.5 days for Workshop B to make this batch of cotton student clothes.
In this paper, the application of fractional equation is investigated. When solving practical problems with fractional equations, there will be two equal relationships in general problems. At this time, according to the problem to be solved in the problem, one of the equations should be selected as the basis of the equations, and the other is used to set the unknown.
20. The lengths of the three sides of △ ABC are A, B and C respectively, which satisfy A2-4A+B2-4C = 4B- 16-C2. Try to determine the shape of △ ABC and prove your conclusion.
Application of factorization of test sites.
Analysis According to the complete square formula, the sum of non-negative numbers can be zero, and each non-negative number can be zero, and the values of A, B and C can be obtained. According to the inverse theorem of Pythagorean theorem, we can get the answer.
Solution: △ABC is an isosceles right triangle.
Reason: ∵a2﹣4a+b2﹣4 c=4b﹣ 16﹣c2,
? (a2﹣4a+4)+(b2﹣4b+4)+(c2﹣4 c+8)=0，
Namely: (a﹣2)2+(b﹣2)2+(c﹣2 )2=0.
∵(a﹣2)2？ 0,(b﹣2)2？ 0,(c﹣2 )2？ 0,
? a﹣2=0,b﹣2=0,c﹣2 =0，
? a=b=2，c=2，
∵22+22=(2 )2,
? a2+b2=c2，
So △ABC is an isosceles right triangle with C as the hypotenuse.
This topic examines the application of factorization and the inverse theorem of Pythagorean theorem. Using the sum of non-negative numbers to get the values of a, b and c is the key to solve the problem.
2 1. As shown in the figure, the quadrilateral ABCD is a parallelogram, and? BCD= 120？ ，CB=CE，CD=CF。
(1) Verification: AE = AF
(2) Q? The degree of EAF.
Congruent triangles's judgment and nature; Properties of parallelogram.
Analyze (1) to find out the triangles containing AE and AF respectively, and get AE=AF by proving that the two triangles are the same.
2 in? Can you find the bad guys? EAF=？ BAD﹣(？ BAE+？ FAD), in (1), we prove that triangles are congruent, and will? Fashion to equal angle? AEB can solve this problem.
The solution (1) proves that ∵ quadrilateral ABCD is a parallelogram, but? BCD= 120？ ,
BCE=？ DCF=60？ ，CB=DA，CD=BA，？ ABC=？ ADC，
CB = CE，CD=CF，
? △BEC and △DCF are equilateral triangles,
? CB=CE=BE=DA，CD=CF=DF=BA，
ABC+？ CBE=？ ADC+？ CDF，
Namely:? ABE=？ United States Food and Drug Administration
In △ABE and △FDA, AB=DF,? ABE=？ FDA，BE=DA，
? △ABE?△FDA(SAS)，
? AE=AF。
(2) solution: ∫In△ABE,? ABE=？ ABC+？ CBE=60？ +60? = 120? ,
BAE+？ AEB=60？ ,
∵? AEB=？ Fashion,
BAE+？ FAD=60？ ,
∵? Bad =? BCD= 120？ ,
EAF=？ BAD﹣(？ BAE+？ FAD)= 120？ ﹣60? =60? .
A:? The degree of EAF is 60? .
This topic examines congruent triangles's judgment and nature. The key to solve the problem is to find a suitable congruent triangles, prove congruence by finding equivalence relation, and draw a conclusion.
22. Reading materials:
After learning the quadratic root, Xiao Ming found that some formulas with roots can be written as the square of another formula, such as 3+2 =( 1+ )2. Xiao Ming, who is good at thinking, made the following explorations:
Let a+b =(m+n )2 (where a, b, m and n are integers), then a+b = m.
A=m2+2n2，b=2mn。 In this way, Xiao Ming found a way to turn a formula similar to a+b into a plane.
Please imitate Xiao Ming's method and try to solve the following problems:
(1) When a, b, m and n are positive integers, if a+b =(m+n )2, a and b are expressed by formulas containing m and n respectively, then a= m2+3n2 and b= 2mn are obtained.
(2) Using the explored conclusions, express them in a completely flat way: = (2+ )2.
Please simplify:
Properties and simplification of quadratic roots in inspection center.
Theme reading type.
Analysis (1) Use the known direct brackets to get the values of a and b;
(2) Directly use the complete square formula and get the answer through deformation;
(3) Simply use the complete square formula to simplify the deformation.
Solution: (1)∵a+b =(m+n )2,
? a+b =(m+n )2=m2+3n2+2 mn，
? a=m2+3n2，b = 2mn
So the answer is: M2+3N2; 2mn
(2) =(2+ )2;
So the answer is: (2+) 2;
(3)∵ 12+6 =(3+ )2,
? = =3+ .