Please help me solve the math problems in junior high school!

Junior high school mathematics basic knowledge examination questions

School name score

1. Fill in the blanks (30 small questions in this question, 2 points for each small question, out of 60 points)

1, collectively called real number.

2. The solution of the equation -= 1 is.

3. The solution set of inequality group is.

4. There are *** 100 50-cent and 50-cent coins with a value of 2 3 yuan cents. If there are 50 cents X coins and 50 cents Y coins, you can get the equation.

5. Calculation: 28X6Y2 ÷ 7x3Y2 =.

6. Factorization: x3+x2-y3-y2 =.

7. When x, the score is meaningful; And when x, its value is zero.

8. Calculation:+=; (x2-y2)÷ =。

9, expressed by scientific notation:-0.00002008 =; 12 1900000= .

The square root of 10 is; The cube root of is.

1 1, calculation:-=; (3+2 )2= .

12, denominator is physicochemical: =; = .

13 A rectangular iron sheet with a length of 8cm and a width of 6cm, cut a small square with equal sides at each of the four corners to make a rectangular box without a cover, so that its bottom area is 24 cm2. If the side length of a small square is x cm, the following equation can be obtained.

14. If there are two unequal real roots about the x equation 2x2-4x+k = 0, then the value range of k is.

15. If x 1 and x2 are two roots of the equation 2x2+6x-1= 0, then+=.

16, a quadratic equation with roots of+1 and-1 is.

17, factorization in real number range: 3x2-4x- 1 =.

18, the solution of equation x+= 5 is.

19, the proportional function y = kx is known. When x = 5 and y = 7, then when x = 10, y =.

20. When k, if the inverse proportional function y = is in the quadrant where its image is located, the function value increases with the decrease of x 。

2 1. In the rectangular coordinate system, the analytical formula of straight lines passing through points (-2, 1) and (1, -5) is.

22. if k < 0, b > 0, then the image of linear function y = kx+b passes through the fourth quadrant.

23. If the circumference of an isosceles triangle is 24cm, the functional relationship between the waist length y(cm) and the base length x(cm) is.

24. The opening direction of the image with quadratic function y =-2x2+4x-3; The vertex is.

25. The analytical formulas of parabola passing point (1 3), (-1 7) and (-2,6) are as follows.

26. Translate the parabola y =-3 (x- 1) 2+7 by 3 units to the right and 4 units to the down, and the analytical formula of the parabola is.

27. One class in Liuying Middle School 18 students, age 14, 16, 15, 6 students, age 16. The average age of students in this class is 15 years old.

28. When a set of data has eight numbers arranged from small to large, the median of this set of data is.

29. A set of data * * * has 80 numbers, of which the largest number is 168 and the smallest number is 122. If the group distance in the histogram of frequency distribution is 5, this group of data can be grouped.

30. The standard deviation of samples 29, 23, 30, 27 and 3 1 is.

2. Fill in the blanks (30 small questions in this question, 2 points for each small question, out of 60 points)

3 1. If two parallel lines are cut by a third line, they are equal and complementary.

32. The topic of the proposition "Complementary angles of two parallel lines with internal angles" is,

The conclusion is.

33. If the lengths of the three sides of a triangle are 6, 1 1 and m respectively, the range of m is.

34. If the sum of the internal angles of a polygon is 2520, then the polygon is a polygon.

35, isosceles triangle, and overlap each other.

36. In △ABC, if ∠ A = 80 and ∠ B = 50, then △ABC is a triangle.

37. In Rt△ABC, ∠ C = 90, ∠ A = 60. If AC = 5 cm and AB= cm. ..

38. In RT △ ABC, ∠ c = 90, if AC = 3cm and BC = 4cm, then the height of AB side CD = cm.

39. If the difference between two adjacent angles of a parallelogram is 30, then the larger internal angle of the parallelogram is (degrees).

40. Two sets of quadrangles with opposite sides are parallelograms.

4 1. In a rhombic ABCD, if an inner angle is 120 and the shorter diagonal is 12cm, the circumference of the rhombic is cm.

42. A parallelogram with two diagonal lines is a square.

43. In the trapezoid, AB=DC, AD‖BC. If AB = DC, the equilateral angle is.

44. The figure obtained by connecting the midpoints of four sides of a diamond in turn is a shape.

45. In △ ABC, point D and point E are on the sides of AB and AC respectively. If DE‖BC, AD = 5, AB = 9, EC = 3, then AC =.

46. In △ ABC, point D and point E are on the sides of AB and AC respectively, AD = 2 cm, DB = 4 cm, AE = 3 cm, EC = 1 cm, so △ ABC ∽△ ade.

47. The three median lines AD, BE and CF of △ABC intersect at point G. If the area of △AEG is 12 square centimeter, then the area of △ ABC is square centimeter.

48. Replace a triangle with a similar triangle. If the side length is expanded to 10 times, the area will be doubled.

49. If ∠A is an acute angle and tgA= =, then CTGA =.

50. Calculation: SIN 30 =；; tg60 =。

5 1, in Rt△ABC, ∠ c = 90. If sinA= =, then ∠ b = (degrees).

52. If the plane overlooks a target on the ground at an altitude of 5000 meters and the depression angle is 30, then the distance between the plane and the target is meters.

53. If the slope is 1∶4 and the horizontal width is 20m, the vertical height is m. 。

54. In a circle with a radius of 10cm, the arc length subtended by a central angle of 20 is cm.

55. If the radii of two circles are 9cm and 4cm respectively, and the center distance is 5cm, the positional relationship between the two circles is.

56. If straight line AB passes through points C and OC⊥AB on ⊙O, then straight line AB is ⊙ O. 。

57. In △ABC, if AB = 9 cm, BC = 4 cm, CA = 7 cm, and its inscribed circle intersects AB at point D, then AD= cm. ..

58. In Rt△ABC, ∠ c = 90. If AC = 5cm and BC = 12cm, the radius of inscribed circle of △ABC is cm.

59. Two circles with a circumscribed radius of 5cm and 15cm have a circumscribed length of cm, and the acute angle between the connecting line and the circumscribed circle is (degrees).

Any regular polygon is a symmetrical figure, and so is a regular polygon with an even number of sides.

answer

First, 1, rational number; Irrational number .2, y = 3.3, x ≤-.4, .5, 4x3.6, (x-y) (x2+xy+y2+x+y) .7, ≦-; = 1 .8、 ; (x+y)2 .9 、- 2.008× 10-5； 1.2 19× 108 . 10、 3; - . 1 1、 ; 29+ 12 . 12、 ; ... > 0.2 13, (8-2x) (6-2x) = 24 (or x2-7x+6 = 0), y =-2x-3.22, one, two, four ... (1,-1) .25, y = 2x2+5x-4.26, y =-3 (x-4) 2+3.27, 14.7.28, the average of the fourth and fifth numbers .29.

2.3 1, congruent angle or internal angle; Internal angle of the same side. 32. Two straight lines are parallel; The internal angles are 0.33, 5 < m < 17.34, 16.35, and the vertex angles are bisectors of complementary angles; The center line on the bottom edge; Height .36, isosceles .37, 10.38, 2.4.39, 105.40, parallel (or equal) .41,48.42, vertical ∠A=∠D equals. .51,30.52, 10000.53, 5.54, π.55, tangent.56, tangent.57, 6.58, 2.59,10; 30.60, axis; Center.

Basic knowledge test of algebra

A fill-in-the-blank question (20 points for this question, 4 points for each question):

1. The side length of a square is 1 cm. If each side of the square is reduced by 1 cm, the area of the reduced square is

cm2

2.A, B, and C represent three rational numbers, and A, B, and C represent the additive associative law as:

3. The difference of 7 times between x and y is expressed as:

4. When, the value of algebraic expression is;

5. The solution of equation X-3 = 7 is.

Answer:

1.(a- 1)2；

2 . a+(b+ c)=(a+b)+c；

3.x-7y；

4. 1;

5. 10.

Multiple choice questions (30 points for this question, 6 points for each small question):

1. The following types are algebraic types: ............................................. ().

S =πr (B)5>3 (C)3x-2 (D)a 0。 3a-4b Positive or Negative?

3a-4b < 0。 Answers 6a-4b.

12. If += 0, then X = _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.

What do tips and each mean? [The arithmetic square root of X-8 and Y-2 must be non-negative.] What conclusion can you draw? 【X-8 = 0，Y-2 = 0。 ] Answer 8, 2.

The physical and chemical factors of 13.3-2 are _ _ _ _ _ _.

Prompt (3-2) (3+2) =- 1 1. Answer 3+2.

14. When < x < 1,-= _ _ _ _ _ _.

Prompt x2-2x+1= () 2; -x+x2=( )2。【x- 1； -X.] when < x < 1, are x- 1 and -X positive or negative respectively? [X- 1 is negative, and X is negative] Answer-2x ..

15. If the simplest secondary root is the same secondary root, then a = _ _ _ _ _ _ _ _ _ _ _ _

b=____________。

What is the root index of the quadratic root? [3b- 1 = 2。 What is the relationship between A+2 and 4b-a? Are the two expressions the same quadratic root? [a+2 = 4 B- a]

Answer 1, 1.

(3) Multiple choice questions: (3 points for each small question, *** 15 points)

16. In the following variants, the correct one is ... () (a) (2) 2 = 2× 3 = 6 (b) =-

Comment on this topic to examine the properties of quadratic roots. Note (b) is incorrect because = |-| =; (c) Incorrect because there is no formula =.

17. In the following categories, () (a) = A+B (b) = A2+ 1 must be true.

Comment on this topic to investigate the conditions for the establishment of quadratic radical properties. (a) is incorrect, because a+b is not necessarily negative, (c) must be a≥ 1, and (d) must be a≥0 and b > 0.

18. If the formula -+ 1 is meaningful, the value range of x is ........................... ().

(a) x ≥ (b) x ≤ (c) x = (d) or more is incorrect.

Tip To make a formula meaningful, you must

Answer C.

19. When a < 0, b < 0, it is reduced to the simplest quadratic root, and …………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………….

(A) (B)- (C)- (D)

Tip = =. Answer B.

Comment on this question. Nature = | a | and denominator are rational numbers. Note (a) The reason for the error is that the number is not considered when using nature.

20. when a < 0, the result of simplifying | 2a-| is ... () (a) a (b)-a (c) 3a (d)-3a.

Prompt to simplify first, ∫a < 0, ∴ =-a. Then simplify | 2a-| = | 3a |. Answer D.

(d) Factorization in the real number range: (4 points for each small question, ***8 points)

2 1.2x 2-4； It is suggested to extract 2 first, and then use the square difference formula. Answer 2 (x+) (x-).

22.x4-2x2-3。 It is suggested that x2 should be regarded as a whole and decomposed by x2+px+q = (x+a) (x+b), where a+b = p and ab = q, and then decomposed by the square difference formula. The answer is (x2+ 1) (x+) (.

(5) Calculation: (5 points for each small question, ***20 points)

23.( - )-( - );

It is suggested that each quadratic radical should be transformed into the simplest quadratic radical first, and then similar quadratic radicals should be merged.

24.(5 + - )÷ ;

Solution formula = (20+2-) × = 20×+2×-

=20+2- × =22-2 .

25.+ -4 +2( - 1)0; The solution formula is = 5+2 (- 1)-4×+2× 1.

=5 +2 -2-2 +2=5 .

26.( - +2 + )÷ .

It is suggested that division be converted into multiplication, multiplied by the distribution law and then simplified.

Solution formula = (-+2+)?

= ? - ? +2 ? + ? = - +2+ =a2+a- +2。

It is complicated to simplify the items in brackets and then multiply them by the distribution law.

(6) Evaluation: (6 points for each small question, *** 18 points)

27. Given a =, b =, find the value of-.

It is suggested to simplify the quadratic root first and then substitute it for evaluation.

Solve the original formula = = =.

When a = and b =, the original formula = = 2.

Comments If the values of A and B are directly substituted into the calculation, the calculation process is complicated and calculation errors are easy to occur.

28. Given x =, find the value of x2-x+.

It is suggested that this question be simplified after X, and then substituted for evaluation.

Solution ∵ x = =.

∴x2-x+=(+2)2-(+2)+= 5+4+4--2+= 7+4。

If we can notice that X-2 = thus (X-2) 2 = 5, we can also change X-2 =-X+ into about.

The quadratic trinomial of x-2 can be solved as follows:

∫x2-x+=(x-2)2+3(x-2)+2+=()2+3+2+= 7+4。

Obviously, the operation is convenient, but the constant deformation of the formula is very demanding.

29. Given += 0, find the value of (x+y) x 。

Hints are all arithmetic square roots, so they are all non-negative. What is the conclusion that the sum of two non-negative numbers is equal to 0?

Solution: ≥0, ≥0,

And += 0,

∴ (x+y) x = (2+ 1) 2 = 9。

(7) solving problems:

30.(7 points) It is known that the hypotenuse length of a right triangle is (2+) cm and the side length of a right triangle is (+2 )cm. Find the area of this right triangle.

What do you need to find the area of a right triangle in this question? [Another right-angled edge. ] How did you find it? 【 Using Pythagorean Theorem. ]

The solution is in a right triangle. According to Pythagorean theorem:

The length of the other right angle = 3 (cm).

The area of a right triangle is:

S= ×3×( )= (cm2)

Answer: The area of this right triangle is () square centimeters. ..

3 1.(7 points) Given | 1-x |-= 2x-5, find the value range of x. 。

The hint is given by the known | 1-x |-x-4 | = 2x-5. When was this formula established? [1-x ≤ 0 and x-4 ≤ 0. ]

Since the solution is known, the left side of the equation = |1-x |-=|1-x |-x-4, and the right side = 2x-5.

Only when | 1-x | = x- 1, | x-4 | = 4-x, left side = right side. At this point, the solution is 1 ≤ x ≤ 4. The value range of ∴ x is 1 ≤ x ≤ 4.

Basic test of binary linear equation

(a) fill in the blanks (2 points for each question, ***26 points):

1. It is known that the binary linear equation = 0, and X is expressed by an algebraic expression containing Y, then X = _ _ _ _ _ _ _ _ _ _ _ _；;

When y =-2 and X = _ _ _ _ _ It is suggested to take y as a known number to solve x, and the answer is X =；; x=。

2.( 1), (2) and (3), _ _ _ is the solution of equation x-3y = 9, _ _ _ is the solution of equation 2x+y = 4, and _ _ is the solution of equations. It is suggested to substitute three groups of values respectively. ( 1),(3); (1). Notes about the solutions of equations must be the same solution of each equation in the equation.

3. It is known that it is the solution of the equation x+2my+7 = 0, then m = _ _ _ _ _ It is suggested to substitute into the equation to find m, and the answer is-.

4. If the solution of the equations is 0, then A = _ _, and B = _. It is suggested that the original equations be transformed into binary linear equations about A and B, and then solved. The answer is A =-5 and B = 3.

5. The equation Y = KX+B is known. When X = 2, Y =-2; When x =- and y = 3, then k = _ _ _ _, and b = _ _ _.

It is suggested that the corresponding values of x and y be substituted to obtain the binary linear equations about k and b.

The answer is k =-2 and b = 2. It is a common method to solve undetermined coefficients by establishing equations.

6. If | 3a+4b-c |+(c-2b) 2 = 0, then a: b: c = _ _ _ _ _.

It is suggested that 3a+4b-c = 0 and c-2b = 0 from the non-negative nature. Then a and c are represented by an algebraic expression containing b, and the values of a=- b and c can be obtained. The answers are a =-b and c = 2b. A: B: C =-2: 3: 6.

It is a common and effective method to express the remaining unknowns by algebraic expressions of unknowns.

7. When m = _ _ _ _ _ _ _, the equation x+2y = 2, 2x+y = 7, and MX-y = 0 has a common * * * solution.

It is suggested to solve the equation first, substitute the values of x and y into the equation MX-y = 0, or solve the equation.

The answer, m =-. Comment on "common * * * solution" is the basis of establishing equation.

8. A three-digit number is _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

Prompt to multiply the number on each number by the corresponding number, and then sum.

The answer is 100x+ 10y+2 (x-y).

(2) Multiple choice questions (2 points for each small question, *** 16 points):

9. The following equations are known: (1), (2), (3), (4),

Among them, the number of binary linear equations is .............................................. ()

1 (B)2 (C)3 (D)4

It is suggested that equation group (2) contains three unknowns, and the degree of y in equation group (3) is not 1, so neither (2) nor (3) is a binary linear equation group.

10. Assuming that 2xB+5Y3A and -4x2AY2-4B are similar projects, the value of ba is ........................... ().

(A)2 (B)-2 (C) 1 (D)- 1

Prompt is defined by similar items, so ba = (- 1) 2 = 1. Answer C.

1 1. Suppose the solution of the equation is, then the values of m and n are ... ().

(A) (B) (C) (D)

It is suggested that the binary linear equations about m and n be substituted into the equation to solve. Answer D.

12. The solution of the ternary linear equations is .................................... ().

(A) (B) (C) (D)

It is suggested to add the two sides of the three equations separately to get x+y+z = 6 or substitute the options into the equations one by one for verification.

X+y = 1 (b) and (d) are all wrong; Then y+z = 5, which excludes (c), so (a) is correct, and the former solution is called direct method; The latter solution is called inverse verification method. Answer a.

Comments Because the multiple-choice questions in mathematics are mostly single-choice questions-there is only one correct answer, it has one more known condition than the general questions: one and only one of the multiple-choice questions is correct. Therefore, there are many special solutions to multiple-choice questions besides direct method. With the deepening of the study, we will introduce them to the students one by one.

13. If the values of the solutions x and y of the equation are equal, the value of a is .................... ().

(A)-4 (B)4 (C)2 (D) 1

It is suggested that X = Y be substituted into 4x+3Y = 14 to get X = Y = 2, and then substituted into the equation containing a. Answer C.

14. If the solution of the equation about x and y satisfies the equation 2x+3y = 6, then the value of k is ().

(A)- (B) (C)- (D)

It is suggested that k should be regarded as a known constant, and the values of x and y can be found, and then the values of x and y can be substituted into 2 x+3 y = 6 to get K. Answer B.

15. if the equation y = kx+b, when x and y are reciprocal, b is smaller than k 1, and x =, then the values of k and b are ..........................................................................................................

(a) 2, 1 (b), (c)-2, 1 (d),-the answer d can be given by the known x =, y =-.

16. Students in a class participate in activities in groups. If there are 7 students in each group, the remaining 4 students; If there are 8 people in each group, there are 3 people in each group. If there are X students in the class and they are divided into Y groups, we can get the equation ............................................................ ().

(A) (B) (C) (D)

It is suggested that the equivalence relation be drawn from the meaning of the question: (1)7 groups of people = total number-4; (2) Number of people in 8 groups = total number +3. Answer C.

(3) Solve the following equation (4 points for each small question, 20 points for * * *):

17. it is suggested to eliminate x by addition, subtraction and elimination. Answer.

18. It is suggested to sort out the equation first, turn it into an integral coefficient equation, and eliminate x by addition and subtraction.

19. It is suggested that x = y be obtained from the first equation, and the offspring should be sorted into the second equation; Or from the first equation, let x = 2 k, y = 5 k, and substitute it into another equation to find the value of k, the answer.

20.(A and B are nonzero constants)

It is suggested that the left and right sides of the two equations should be added separately to get X+Y = 2A1, and both equations should be used to solve ① at the same time.

answer

Comment on superposition elimination method is a common method to solve the rotation equation of unknown system.

2 1.

It is suggested that the first equation should be merged with the other two equations, and y should be eliminated by addition.

answer

It is the key to choose an appropriate solution to the problem by commenting and analyzing the composition characteristics of the coefficients of the unknown items in the equations that make up the equations. Therefore, it is necessary to observe carefully before solving the problem, so as to find out the shortcut to solve the problem.

(4) Problem solving (6 points for each small question, *** 18 points):

22. Given that the sum of the solutions x and y of the equation is 12, find the value of n. 。

It is suggested to solve the known equation, express x and y with algebraic expression of n, and then substitute x+y = 12.

Answer n = 14.

23. Assuming that the equation and the solution are the same, find the value of A2+2AB+B2.

It is suggested to solve the equation first to get x and y, and then substitute it into the equation to get a and b.

The answer.

Note When the solutions of n equations are the same, any two equations can be combined into a new equation.

24. It is known that when x = 1 and x =-3, the algebraic expression x2+ax+b has values of 0 and 14, respectively. Find the algebraic value when x = 3.

It is suggested that the equations about a and b can be obtained from the meaning of the question. Find a and b, write this algebraic expression, and then find its value when x = 3.

Answer 5.

Comment on this example. After calculating the values of A and B by the undetermined coefficient method, you should write this algebraic expression, because this is the key step of evaluation.

(5) Solving problems in the application of equations (every 1 item 10, 20 points * * *);

Last year, there were 80 more boys than girls in the first grade of a school. This year, the number of girls increased by 20%, while the number of boys decreased by 25%. As a result, there are 30 more girls than boys. How many boys and girls were there in Grade One last year?

It is suggested that there were x boys and y girls in senior one last year, and the equation can be obtained.

The answer is x = 280 and y = 200.

26. The distance between A and B is 20 kilometers. A and B go in opposite directions from A and B at the same time and meet on the road two hours later. Then A returns to A and B moves on. When A returns to A, B is 2 kilometers away from A. Find the speed of A and B. 。

According to the meaning in the question, A walked for 2 hours before they met. "When A returned to A, B was 2 kilometers away from A." We can get another equality relationship of the equations: A and B walked in the same direction for 2 hours, with a difference of 2 kilometers. Let the velocities of A and B be X km/h and Y km/h respectively, then

The speed of answer A is 5.5 km/h, and the speed of answer B is 4.5 km/h. 。

Basic test of scores

Fill in the blanks (2 points for each small question, *** 10):

1. If v = v0+at (a is not zero), then t =；;

2. the equation about x MX = a (the solution of m is;

3. The root of the equation is:

4. If -3 is the root of the fractional equation, then a =；;

A car can walk x kilometers an hour, and at the same speed, it can walk several kilometers in b minutes.

Answer:

1.； 2.； 3.； 4.3; 5.。

Multiple choice questions (3 points for each small question, *** 12 points):

1. Known = 2. If y is represented by an algebraic expression containing X, it is ………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………

(A)y = 2x+8(B)y = 2x+ 10(C)y = 2x-8(D)y = 2x- 10

2. The following equation about X, which is not a fractional equation, is ....................................... ().

(A) (B)

3. A project is completed by Party A within one hour, and Party B within one hour. The number of hours required for both parties to complete the work together is ......................................................................... ().

(A)a+b (B) (C) (D)

4. The solution of the equation about x (m2-1) x = m2-m-2 (m2 ≠1) should be expressed as ............ ().

(A)x= (B)x=

Answer:

1.d； 2.c； 3.d； 4.B。

Third, solve the following equations (8 points for each small question, 32 points for * * *):

1.； 2.；

Solution:, solution:,

, ,

. .

After testing, = 1 is the root of the original equation, and = 2 is the root of the original equation.

3.；