2. It is known at △, which is an acute angle. From the vertex to the edge or its extension line is vertical, and the vertical foot is; When the vertex is perpendicular to the edge or its extension line, the vertical foot is. When sum is a positive integer, what triangle is delta? And prove your conclusion.
3.1/a+1/b =1/a+b, and find the value of b/a+a/b.
2008 national junior high school mathematics competition Zhejiang division preliminary contest simulation test questions
(The full mark of this volume is 120, and the examination time is 120 minutes. Allow the use of scientific calculators. )
A, multiple-choice questions (***8 small questions, 5 points for each small question, 40 points. There are four answers to each question, code-named A, B, C and D, and only one of them is correct. Please write the code number in front of the question in brackets after it. No score is given for more, less or nothing. )
1. The roots of the equation ax2+bx+c=0 about X are 2 and 3, so the root of the equation AX2-BX-C = 0 is ().
Answer. -2，-3 B. -6， 1 C.2，-3 D. - 1，6
2. It is known that the moving point P moves uniformly along A-B-C-D on the side of a square with a side length of 2. X represents the distance traveled by point P from point A, and y represents the area of △APD, so the image of the functional relationship between y and x is roughly ().
A B C D
3. Reverse the order of a three-digit number and add the obtained number with the original number. If neither number in the obtained sum is even, it is called "odd sum number". So, how many of all three digits are "odd sums"? ( )
a . 200 b . 120 c . 160d . 100
Let a, b and c be positive numbers. If so, the relationship between numbers A, B and C is () A.C.
5. The opposite lengths of the three internal angles A, B and C of a triangle are A, B and C respectively (A, B and C are prime numbers), and a+B+C = 16 is satisfied. Let ∠A be the minimum internal angle. The value of cosA is ()
The situation of a.b.c.d is not enough to calculate.
6. Beauty is a feeling, and there should be no objective standard. However, in nature, the proportion of object shapes provides a symmetrical and harmonious aesthetic reference. In mathematics, this ratio is called the golden section. In the ratio of human torso (from the sole of the foot to the navel) to height, the navel is the most ideal golden section, that is to say, the closer this ratio is, the more aesthetic it will give others. If a woman is tall, the ratio of trunk to height, and she wants to use high heels to achieve this effect in pursuit of beauty, then the height of high heels she chooses is probably ()
A.B. C. D。
7. As shown in Figure 2, if one vertex of a right triangle with side lengths of 3cm, 4cm and 5cm coincides with the vertex B of the square, and the other two vertices are on the two sides of the square, the area of the square is ().
A.cm2.b.cm2.
C. square centimeter
8. Rotate the parabola y=2x2- 12x+22 around the point (5,2) 1800.
The intersection point of the new parabola and the coordinate axis is
( )
1 D.0
Fill in the blanks (***6 small questions, each with 5 points and 30 points. )
9. It is known that the centers of two circles are (2,0) and (0,2) respectively, and the radii are both 2. Then the area of the common part of these two circles is.
10. If three digits satisfy a < b < c or a > b > c, it is called "strictly ordered three digits". Then, the probability that one of all three digits happens to be a "strictly ordered three digit" is.
1 1. If it is the sum of all numbers (n is an arbitrary positive integer), for example, then; Remember, k is a positive integer, then.
12. It is known that the least common multiple of three positive integers x, y and z is 300, and it is satisfied, then the solution of this system of equations (x, y, z)=.
13. If the diameter of a circle and the length of a chord are A and B, respectively, and the chord center distance of this chord is a positive rational number, and it is known that A and B are both positive integers with two digits, then their ten digits and single digits are just exchanged. Then the value of a2+b2 is.
14. The abscissas of two intersections of parabola y = n (n+1) x2-(3n+1) x+3 and straight line Y =-NX+2 are x 1 and x2, respectively, and dn = ∣ x/kloc.
Three. Answer (***4 small questions, 15, 16, 17, each 12, 18 14, a total of 50 points. )
15. It is known that seven real numbers a 1, a2, a3, a4, a5, a6 and a7 satisfy the following three equations:
⑴a 1+4a 2+9 a3+ 16 a4+25 a5+36 a6+49a 7 = 2008，
⑵4a 1+9 a2+ 16 a3+25 a4+36 a5+49 a6+64 a7 = 208，
(3) 9a1+16a2+25a3+36a4+49a5+64a6+81a7 = 28. Try to find the values of the following algebraic expressions:
16a 1+25 a2+36 a3+49 a4+64 a5+8 1 a6+ 100 a7 .
16. Please use the distance formula between any two points (x 1, y 1) and (x2, y2) on the rectangular coordinate plane to answer the following questions:
It is known that the images of inverse proportional function and positive proportional function intersect at points A and B (A is in the first quadrant), and points F 1(-2, -2) and F2(2, 2) are on a straight line. Let point P (x0, y0) be any point on the inverse proportional function image, and remember that the difference between the distance of point P and F 1 and F2 is d = | pF 1-pF2 |. Try to compare the length of line segment AB with the size of d, and then get an important definition of hyperbola (expressed in concise language).
17. The inscribed circle of △ ABC cuts BC, CA and AB on D, E and F respectively, and G is a point on EF, DG⊥EF, and verification: DG bisects ∠ BGC.
18. Observe the following figure:
① ② ③
If you follow this rule until the nth number, please explore the following questions:
⑴ Let the number of all triangles in the nth graph and the n- 1 graph be an and an- 1 respectively, and ask: What is the quantitative relationship between them? Please write this relationship.
Please use an algebraic expression containing n to represent an and prove your conclusion.
2008 national junior high school mathematics competition Zhejiang division preliminary contest simulation test questions
Reference answers and grading standards
A, multiple-choice questions (***8 small questions, 5 points for each small question, 40 points. Every small question has only one correct answer, and you can't score if you choose more, less or not. )
1. the roots of the equation ax2+bx+c=0 about x are 2 and 3, so the roots of the equation AX2-BX-C = 0 are
(2)
Answer. -2，-3 B. -6， 1 C.2，-3 D. - 1，6
2. It is known that the moving point P moves uniformly along A-B-C-D on the side of a square with a side length of 2. X represents the distance traveled by point P from point A, and Y represents the area of △APD, so the functional relationship image of Y and X is roughly (a).
A B C D
3. Reverse the order of a three-digit number and add the obtained number with the original number. If neither number in the obtained sum is even, it is called "odd sum number". So, how many of all three digits are "odd sums"? (4)
a . 200 b . 120 c . 160d . 100
Let a, b and c be positive numbers. If so, the relationship between numbers A, B and C is (a) A.C.
5. The opposite lengths of the three internal angles A, B and C of a triangle are A, B and C respectively (A, B and C are all prime numbers), and a+B+C = 16 is satisfied, and ∠A is the minimum internal angle. The value of cosA is (c)
The situation of a.b.c.d is not enough to calculate.
6. Beauty is a feeling, and there should be no objective standard. However, in nature, the proportion of object shapes provides a symmetrical and harmonious aesthetic reference. In mathematics, this ratio is called the golden section. In the ratio of human torso (from the sole of the foot to the navel) to height, the navel is the most ideal golden section, that is to say, the closer this ratio is, the more aesthetic it will give others. If a woman is tall, the ratio of trunk to height, and she wants to achieve this effect with high heels in pursuit of beauty, then the height of high heels she chooses is probably (C).
A.B. C. D。
7. As shown in Figure 2, if one vertex of a right triangle with side lengths of 3cm, 4cm and 5cm coincides with the vertex B of the square, and the other two vertices are on the two sides AD and DC of the square, the area of the square is (D).
A.cm2.b.cm2.
C. square centimeter
8. Rotate the parabola y=2x2- 12x+22 around the point (5,2) 1800.
The intersection point of the new parabola and the coordinate axis is
(2)
1 D.0
Fill in the blanks (***6 small questions, each with 5 points and 30 points. The correct answer to each small question should be unique, and if you fill in more questions, you will not be given points; If the score is not reduced or converted to decimal, give points. )
9. It is known that the centers of two circles are (2,0) and (0,2) respectively, and the radii are both 2. Then the area of the common part of these two circles is.
10. If three digits satisfy a < b < c or a > b > c, it is called "strictly ordered three digits". Then, the probability that one of all three digits happens to be a "strictly ordered three digit" is.
1 1. If it is the sum of all numbers (n is an arbitrary positive integer), for example, then; Remember that,,, and k are positive integers, then 1 1.
12. It is known that the least common multiple of three positive integers x, y and z is 300, and it is satisfied, then the solution of this system of equations (x, y, z) = (20,60, 100).
13. If the diameter of a circle and the length of a chord are A and B, respectively, and the chord center distance of this chord is a positive rational number, and it is known that A and B are both positive integers with two digits, then their ten digits and single digits are just exchanged. Then the value of a2+b2 is 736 1.
14. The abscissas of two intersections of parabola y = n (n+1) x2-(3n+1) x+3 and straight line Y =-NX+2 are x 1 and x2, respectively, and dn = ∣ x/kloc.
Three. Answer (***4 small questions, 15, 16, 17, each 12, 18 14, a total of 50 points. The scores of each question are given in strict accordance with the grading standards, and there is no need to give intermediate scores step by step; Other solutions are also scored according to this standard. )
15. It is known that seven real numbers a 1, a2, a3, a4, a5, a6 and a7 satisfy the following three equations:
⑴a 1+4a 2+9 a3+ 16 a4+25 a5+36 a6+49a 7 = 2008，
⑵4a 1+9 a2+ 16 a3+25 a4+36 a5+49 a6+64 a7 = 208，
(3) 9a1+16a2+25a3+36a4+49a5+64a6+81a7 = 28. Try to find the values of the following algebraic expressions:
16a 1+25 a2+36 a3+49 a4+64 a5+8 1 a6+ 100 a7 .
Solution: By observing the coefficients in front of the seven unknowns a 1, a2, a3, a4, a5, a6 and a7 in the algebraic formula (1) (2) (n+ 1), (2+2) and (3+2), we can find out each one.
The value of the algebraic expression = (1)-3 (2)+3 (3) = 2008-3× 208+3× 28 =1468.
(6 points)
16. Please use the distance formula between any two points (x 1, y 1) and (x2, y2) on the rectangular coordinate plane to answer the following questions:
It is known that the images of inverse proportional function and positive proportional function intersect at points A and B (A is in the first quadrant), and points F 1(-2, -2) and F2(2, 2) are on a straight line. Let point P (x0, y0) be any point on the inverse proportional function image, and remember that the difference between the distance of point P and F 1 and F2 is d = | pF 1-pF2 |. Try to compare the length of line segment AB with the size of d, and then get an important definition of hyperbola (expressed in concise language).
Solution: By solving the equations formed by and, the coordinates of point A and point B can be obtained as follows.
(,), (,), line segment length AB =4 (2 points)
Point P(x0, y0) is the point on the inverse proportional function image, ∴ y0=
∴P F 1= = =∣ ∣，
Pf2 = = ∣∣, (3 points)
∴d=|P F 1- P F2|=∣∣ ∣-∣ ∣∣，
When x0 > 0, d = 4；; When x0 < 0, d=4. (3 points)
Therefore, regardless of the position of point P, the lengths of line segments AB and D must be equal. (2 points)
It can be seen that the point set (locus) whose distance difference (positive value) to two fixed points is constant is a hyperbola. The meaning is right, and the rigor of expression is not required. ) (2 points)
17. The inscribed circle of △ ABC cuts BC, CA and AB on D, E and F respectively, and G is a point on EF, DG⊥EF, and verification: DG bisects ∠ BGC.
It is proved that connecting DF and DE, let n and k be the midpoint of DF and DE respectively, and connecting BN and CK, then:
Rt△BFN∽Rt△DEG, (2 points)
Rt△CEK∽Rt△DFG, (2 points)
∴BF？ GE= DF？ DE=CE？ FG (4 points)
∴, and ∠BFG=∠CEG (2 points)
∴△BFG∽△CEG, so ∠ BGF = ∠ CGE. * DG⊥ef，∴∠ BGD = ∠ CGD。
In other words, DG shares ∠ BGC. (2 points)
18. Observe the following figure:
① ② ③
If you follow this rule until the nth number, please explore the following questions:
⑴ Let the number of all triangles in the nth graph and the n- 1 graph be an and an- 1 respectively, and ask: What is the quantitative relationship between them? Please write this relationship.
Please use an algebraic expression containing n to represent an and prove your conclusion.
Solution: (1) According to the arrangement of numbers in the question, an=3an- 1+2 (3 points).
(2) obtained from (1): an=3an- 1+2, an- 1=3an-2+2, and the two expressions are subtracted:
An-an-1= 3 (an-1-an-2) ① (3 points)
When n takes 3, 4, 5, …, n respectively, the following (n-2) equations can be obtained from the formula 1:
a3-a2=3(a2-a 1)，a4-a3=3(a3-a2)，a5-a4=3(a4-a3)，…，
an-an- 1 = 3(an- 1-an-2)。 (2 points)
Obviously, an-an- 1≠0, multiply the left and right sides of the above (n-2) equations respectively and remove the same term, then an-an-1= 3n-2 (A2-A1) ② (3 points).
∫A2-a 1 = 17-5 = 12。 We can know an-1 = (an-2) from (1). Substituting them into (2) gives an=2×3n- 1. (3 points)