The third grade mathematics volume and the answer.

Selected Questions and Answers of the Second Round Review of Mathematics in Grade Three (I)

1. Make a ring on the equator of the globe with iron wire. Now the hoop radius is increased by 1 m, and it is necessary to add m meters of iron wire. Suppose there is an iron hoop on the equator of the earth, and the same radius increases by 1 m, so it is necessary to increase the iron wire with the length of n, and the relationship between m and n is ().

A, m > n b, m < n c, m = n d, uncertain.

2. The picture shows a simple movable dining table. Now measured OA=OB=30cm, OC=OD=50cm. Now the height of the table top from the ground is required to be 40cm, so the opening angle ∠COD of the two table legs should be ().

a . 100； b . 120； c . 135； D. 150。

3. A pair of triangular plates are stacked together as shown in the figure, so the degree of ∠ α in the figure is ().

75 (B)60 (C) 65 (D)55

4. There are four factories A, B, C and D evenly distributed on the circular road, and each factory has enough warehouses to store products. Now all products should be stored in a factory warehouse. It is known that the output ratio of A, B, C and D is 1: 2: 3: 5. If the freight is directly proportional to the distance and quantity of transportation, in order to make the selected.

A，A B，B C，C D，D

A decoration company should install a flashlight every 20cm along the five-pointed star as shown in the figure. If BC=-1m, you need to install a flashlight () a.100b.10/c.102d.65438+.

6. As shown in the figure, the fan-shaped OAB is the lateral expansion of the cone. If the sides of small squares are all 1cm, then the radius of the bottom of the cone is () cm.

A.B. C. D。

7. Place five squares with a side length of 2cm as shown in the figure. Points A, B, C and D are the center of a square, so the sum of the areas of the four shaded parts on the way _ _ _ _ _ _ _ _ cm2.

8. As shown in the figure, in the rectangular coordinate system, the OABC is folded in half along the OB, so that the point falls on the point A 1. Given OA=, AB= 1, the coordinate of point A 1 is _ _ _ _ _ _ _.

9. The picture shows the calendar of 2006 1 month. Li Gang will take part in 1 football match every week this month, and * * * will take part in it five times. According to the original arrangement, 1 time is Sunday, Monday and Saturday, and the second time is Wednesday. Then the total number of days for Li Gang to participate in the competition is _ _ _ _ _ _.

10. It is known that the coordinates of points A, B, C and D are the intersection of two dotted lines in the figure. If △ABC and △ADE are similar, the coordinates of the point are _ _ _ _ _ _ _ _ _ _ _ _.

1 1. Fold the rectangular piece of paper ABCD in Figure 1 in half, and points B and C just coincide and fall on point P on the side of AD (Figure 2). It is known that ∠MPN=, PM=3, PN=4, so the area ABCD of rectangular paper is.

12, the base BC of isosceles △ABC is 8 cm, the waist length AB is 5 cm, and the moving point P moves from point B to point C at a speed of 0.25cm/ s at the bottom edge. When the point P moves to the position where the PA is perpendicular to the waist, the time for the point P to move should be

_ _ _ _ _ seconds.

13. Suppose a hotel has 30 rooms and 30 numbers, numbered 1 ~ 30 respectively. Now it is necessary to engrave a number on the key of each room. The engraved number must make it easy for the waiter to identify which room key it is, and it is difficult for outsiders to guess. Now there is a coding method: engrave two numbers on each key. The number on the left is the remainder obtained by dividing the original room number of this key by 5, and the number on the right is the remainder obtained by dividing the original room number of this key by 7. Then the key engraved with the number 36 should correspond to this number in the original room.

14, (1) A pair of triangular plates are stacked as shown in the figure, and the ratio of the areas of the left and right shadow parts is equal to _ _ _ _ _ _ _.

(2) Place a pair of triangular plates as shown in the figure, and the ratio of the areas of the upper and lower triangular plates is equal to _ _ _ _ _ _.

15, some people in life like to fold a note sent by others into a digital shape. The folding process is as follows (the shaded part indicates the reverse side of the paper money):

(l) If the width of a rectangular paper strip folded by stationery is 2cm, how many centimeters should the paper strip have in order to ensure that it can be folded into a digital shape (that is, the two ends of the paper strip exceed the point P)? When a rectangular banknote is the smallest, what is its area?

(2) Assuming that the width of the paper strip folded into a T-shape is xcm, one end of the paper strip exceeds the point P by 2cm, and the other end exceeds the point P by 3cm, if the length of the rectangular paper strip folded into stationery is ycm, find the functional relationship of Y with respect to X, and express the area S of the paper strip folded into a plane figure as shown in the T-shape with an algebraic expression containing X;

(3) If it is desired that the lengths beyond point P at both ends of the paper strip in (2) are equal, that is, the final figure D is an axisymmetric figure, if

Y = 15cm, where should M be placed at the beginning of folding?

16, as shown in figure 1, one side of rectangle ODEF falls on one side of rectangle ABCO, the similarity ratio of rectangle ODEF∽ rectangle ABCO is 1: 4, the side of rectangle ABCO is AB = 4, and BC = 4.

(1) Find the area of the rectangular ODEF;

(2) Rotate the rectangle odef in Figure L counterclockwise by 90 degrees around the O point. If the tangent OF the included angle between of and OA (FOA in Figure 2) is X and the area of the overlapping part of two rectangles is Y, find the functional relationship between Y and X;

(3) Rotate the right-angle ODEF in figure 1 counterclockwise around point O. Is there a maximum or minimum value in the area connecting EC, e a and △ACE? If it exists, find the maximum or minimum value; If it does not exist, please explain why.

17. in the isosceles trapezoid ABCD, AB=DC=5, AD=4, BC= 10. Point E is on the bottom BC, and point F is on the waist AB.

(1) If EF bisects the circumference of the isosceles trapezoid ABCD, let the length of BE be x, and try to express the area of △BEF with an algebraic expression containing X;

(2) Is there a line segment EF that bisects the circumference and area of the isosceles trapezoid ABCD? If yes, find out the length of BE at this time; If it does not exist, please explain the reason;

(3) Is there a line segment EF that divides the circumference and area of the isosceles trapezoid ABCD into 1∶2 at the same time? If yes, find out the length of BE at this time; If it does not exist, please explain why.

18. A foreign language school needs to prepare some Christmas hats for a demonstration at Christmas. In order to cultivate students' practical ability, the school decided to make these Christmas hats by itself. If the specification of the Christmas hat (conical) is that the bus length is 42 cm and the bottom diameter is 16 cm.

(1) Find the degree of the central angle (accurate to the degree) of the side development diagram (sector) of the Christmas hat;

⑵ It is known that type A paper can make three Christmas hats, type B paper can make four Christmas hats, and 26 Christmas hats are needed for reporting performances. Write the functional relationship between the maximum and minimum values of X and Y sheets of A-paper and X sheets of B-paper; If you make your own paper, how many sheets of paper do you buy for each of the two specifications, A and B, so as not to waste paper?

⑶ There is a square piece of paper with a side length of 79 cm, and at most several Christmas hats of this specification can be made (the bonding place of Christmas hats is ignored). Please use the scale of 1: 15 to draw the cutting sketch of the Christmas hat on the square paper, and use your mathematical knowledge to illustrate its feasibility.

19, as shown in the figure, the side length AB of the regular triangle ABC is known to be 480mm. A particle D starts from point B and moves to point A in the direction of BA at the speed of 10mm per second.

(1) Establish an appropriate rectangular coordinate system, and express the coordinates of point D with the movement time t (seconds);

⑵ Make a rectangle DEFG in triangle ABC through point D, where EF is on BC side and g is on AC side. Finding the point D in the diagram makes the rectangle DEFG square (the expression method is required to reflect the process of finding the point D);

(3) Make points D, B and C into parallelograms. When t is what value, the area of the parallelogram composed of points C, B, D and F is equal to the area of the triangular ADC, and the coordinates of point F at this time are obtained.

20. as shown in figure 1, in △ABC, AB = AC =5, AD = 3 .. translate △ACD in the direction shown by the arrow to get △A'CD' (as shown in figure 2), where A'D' intersects with AB and A'C intersects with AB and AD respectively.

(1) Find the functional relationship between Y and X and the range of the independent variable X (regardless of the endpoint);

(2) When the length of BD' is what, is the area ⊙O equal to the area of △ABD? (π takes 3, and the result is accurate to 0. 1)

(3) Connect EF and find the value of X when EF is tangent to ⊙ O. 。

2 1, as shown in the figure, in the plane rectangular coordinate system, the hypotenuse AB of Rt△ABC is on the X axis, the vertex C is on the negative semi-axis of the Y axis, tan∠ABC=, the point P is on the line segment OC, and the lengths of PO and PC (PO < PC) are two of the equations x2- 12x+27=0.

(1) Find the coordinates of point P;

(2) Find the length of AP;

(3) Is there a point Q on the X-axis, so that a quadrilateral with points A, C, P and Q as its vertices is a trapezoid? If it exists, please write the analytical formula of straight line PQ directly; If it does not exist, please explain why.

As we all know, the side length of a square is L.

(1) As shown in Figure ①, you can calculate the diagonal length of a square, find the diagonal length of a rectangle composed of two squares side by side, and guess the diagonal of a rectangle composed of n squares side by side;

(2) According to Figure ②, verify that:

(3) From Figure ③, choose a correct conclusion from the following three conclusions to prove it: ①; ② ; ③ 。

23. As shown in the figure below, equilateral △ABC moves along the straight line L to the rhombus DCEF at a speed of 2m/s until AB and CD overlap, where ∠ DCF = 60, and when x s is set, the area of the overlapping part of triangle and rhombus is y m2.

(1) Write the expression of the relationship between y and x.

(2) When x = 0.5, 1, what are y respectively?

(3) When the area of the overlapping part is half of the diamond, how long does the triangle move?

24. It is known that in the plane rectangular coordinate system xOy, the image of a linear function intersects with the X axis at point A, and the parabola passes through two points, O and A. ..

(1) tries to represent b with an algebraic expression containing a;

(2) Let the vertex of the parabola be D, the circle with D as the center and DA as the radius be divided into two parts by the X axis: the lower arc and the upper arc. If the bad arc is folded along the X axis, the folded bad arc falls within ⊙D, and its circle is just tangent to OD, so as to find the length of ⊙D radius and the analytical formula of parabola;

(3) Set point B as the moving point on the optimal arc that satisfies the condition in (2). Is there such a point p in the part of the parabola above the x axis that? If it exists, find the coordinates of point P; If it does not exist, please explain why.

Selection of the second round of review questions in junior high school mathematics

(The first series of reference answers)

1、C 2、B 3、A 4、D 5、B 6、B 7

10, (4,3)11,144/512,7 or 25 13,13/kloc-0.

15、

16、

17 and (1) are obtained from known conditions: the circumference of the trapezoid is 12, the height is 4 and the area is 28.

If F is FG⊥BC in G and A is AK⊥BC in K, then FG = 12-X5× 4 can be obtained.

∴S△BEF= 12, right? FG =-25 x2+245 x(7≤x≤ 10)……3’

(2) Yes ... Yes ... Yes.

From (1):-25x2+245x =14, x 1=7 x2=5 (don't give up).

∴ There is a line segment EF that bisects the circumference and area of the isosceles trapezoid ABCD at the same time. At this time BE=7.

(3) There is no "1".

Assuming it exists, it is obvious that: S△BEF∶SAFECD= 1∶2, (be+BF) ∶ af+ad+DC) =1∶ 2 ...1'

Then it is -25x2+ 165x = 283, and the arrangement is: 3x2-24x+70 = 0, △ = 576-840.

There is no such real number X.

That is, there is no perimeter and area of isosceles trapezoid ABCD with segment EF.

At the same time, it is divided into two parts: 1: 2 ...........................

18, (1) The side of the Christmas hat is fan-shaped, so the arc length of the fan-shaped is 16, and the central angle of the fan-shaped is.

(2) Starting from y≥0, the maximum value of x is 0 and the minimum value is 0.

Obviously, x and y must be integers to avoid wasting paper.

When x= 1,; When x=2, y = 6；; When x=3,;

When x=4 and x=5, y = 2；; When x=6,

Therefore, when buying 6, 2 or 2 or 5 sheets of paper of A and B specifications, the paper will not be wasted.

(3) Cut the sketch, as shown in the figure.

Let the arcs of two adjacent sectors intersect at point P, then PD = PC.

When DC and DC intersect at point m, the vertical line PM intersects with point p,

Then cm = DC = × 79 = 39.5, CP=42,

So,

So < (),

Another 42+42

19, (1) establish the rectangular coordinate system as shown in the figure, and then

(2) (1) Draw a square first, then use the potential diagram to find point D, and read the diagram in detail.

(2) Use regular triangles and rectangles as axisymmetric figures or use similar triangles.

The properties of DG = 480- 10t and DE =. Then from 480- 10t =

Find t = ≈ 25.7 (mm). So when the distance between point d and point b

When it is equal to 10t = ≈ 257mm, a rectangle is a square.

(3) When point F is in the first quadrant, this parallelogram is CBDF；; ;

When point F is in the second quadrant, this parallelogram is BCDF ";

When point F is in the third quadrant, this parallelogram is cdbf'.

But the area of the parallelogram' bcdf' and the area of the parallelogram' cdbf'.

Equal to the area of parallelogram CBDF (equal base, equal height).

The base BC of the parallelogram CBDF is 480, and the corresponding height is, so the area is.

; The bottom AD of triangular ADC is 480- 10t, and the corresponding height is 240.

The area is 120 (480- 10t).

T = 120 (480- 10t) is obtained.

So when t = 16 seconds, the plane composed of points c, b, d and f.

The area of the row quadrangle is equal to the area of the triangular ADC. At this time, the key is coming.

The coordinates of f are f (560,80), (400, -80).

f”(-400，80)

20, (omitted)

2 1, (1) Solve the equation x2- 12x+27=0, and get x 1=3, x2=9. (2 points) po < pc, ∴PO=3, ∴ p.

(2)∫po = 3, PC=9, ∴OC= 12.(4 points) ∴∠ABC=∠ACO.∴ .(5 points) ∴ OA = 9.

(3) Existence, the analytical formula of the straight line PQ is: or. (10)

22、

23、

(4)5S

24.( 1) The image of the solution 1: ∵ linear function intersects the X axis at point A.

∴ The coordinate of point A is (4,0) ∫ The parabola passes through two points, O and A.

.................. 1 point

Solution 2: ∵ The image of the linear function intersects the X axis at point A.

∴ The coordinate of point A is (4,0) ∫ The parabola passes through two points, O and A.

The symmetry axis of a parabola is a straight line ........................... 1 min.

(2) Solution: According to the symmetry of parabola, do = da ∴ point O is on ⊙D, and ∠ DOA = ∠ Dao.

From (1), we can also know that the analytical formula of parabola is ∴ the coordinate of point D is ().

(1) When,

As shown in figure 1, let ⊙D be divided by the x axis, and the bad arc obtained after folding along the x axis is, and the obvious circle is symmetrical with ⊙D about the x axis, and let its center be d'

Point d' and point d are also symmetrical about X.

∵ Point O is on ⊙D' and ⊙D is tangent to ⊙D'

∴ Point O is the tangent point. .......................... is 2 points.

∴D'O⊥OD

∴∠DOA=∠D'OA=45

∴△ADO is an isosceles right triangle.

..........................., 3 points.

The ordinate of point d is

∴ The analytical formula of parabola is

2 when,

Similarly:

The analytical formula of parabola is