The second math survey and examination of senior three in Yancheng in 2008-2009 school year.

Yancheng City, Jiangsu Province, 2008-2009 school year, the second survey exam of senior three.

Math test questions

(Total score 160, examination time 120 minutes)

Reference formula:

The volume formula of the ball (for the radius of the ball).

Volume formula of cylinder (where the bottom area is the height).

The coefficient formula of linear regression equation is.

1. Fill in the blanks: this big question *** 14 is a small question, with 5 points for each small question and 70 points. There is no need to write the answer process. Please write the answer in the specified position on the answer sheet.

1. If a complex number is set, then = ▲.

2. The domain of a given function is a set and a set of natural numbers, then = ▲.

3. The necessary and sufficient condition for a line to be parallel is ▲.

4. Execute the pseudo code as shown in the figure, and the output result is ▲.

5. The three views of the geometry are as shown in the figure. The length and width of the two rectangles in the front view and the left view are 4 and 2 respectively, and the diameters of the two concentric circles in the top view are 4 and 2 respectively, so the geometric volume is equal to ▲.

6. The distance from the vertex of hyperbola to its asymptote is ▲.

7. Known, then = ▲.

8. A set of data between known data is as follows:

x 2 3 4 5 6

y 3 4 6 8 9

For the data in the table, the following fitting lines are given: ①, ②, ③, ④, and the line with the best fitting degree according to the least square idea is ▲ (fill in the serial number).

9. If the sequence satisfies the sum of the first n terms, then = ▲.

10. The international unit of measurement for diamonds is carats. As we all know, someone

The value of a diamond v (dollars) and its weight (carats)

In direct proportion to the square, if you cut the diamond to a certain weight.

Two diamonds, respectively, have no value.

Percentage = (cut

Except for weight loss), the maximum value loss percentage.

For ▲.

1 1. In the triangle number array shown in the figure, the number satisfying: (1) 1 row is1; (2) The first two numbers in the n(n≥2) row are n, and the rest are equal to the sum of the two numbers on its shoulder, so the second number in this row is ▲ (represented by n).

12. Given the function (the base of natural logarithm), if the real number is the solution of the equation, then ▲ (fill in ">", "≥", "

13. It is known that there are three points on the plane that are not * * * lines. Let it be any point on the middle perpendicular of the line segment. If, the value is ▲.

14. It is known that there are three different real number solutions to the equation about x, so the range of the real number k is ▲.

Second, the solution: this big question is ***6 small questions with a score of 90 points. The answer should be written in the necessary words to prove the process or calculation steps. Please write your answers in the designated area on the answer sheet.

15. (The full score of this small question is 14)

Take as many as possible, among them.

(i) When, the probability of finding a point to satisfy;

(ii) When, find the probability of point satisfaction.

16. (The full score of this small question is 14)

As shown in the figure, in a straight triangular prism, and are the midpoint of, and respectively.

(i) Verification:

(ii) Verification: aircraft.

17. (The full score of this small question is 14)

The edges of three known internal angles are,, and.

(i) Find out the size of the angle;

(2) Give three conditions: ①; ② ; ③ .

Try to choose the area of two conditions (note: you only need to choose one scheme to answer questions, and if you answer questions with multiple schemes, you will be given points according to the first scheme).

18. (Full score for this small question 16)

It is known that the right focus of the ellipse is f, the right directrix is, and the straight line intersects at point A.

(i) If ⊙C passes through O, F and A, find the equation of ⊙C;

(ii) When it changes, verify that ⊙C passes through another fixed point b except the origin o;

(iii) If yes, find out the range of ellipse eccentricity.

19. (Full score for this small question 16)

Let the sum of the first few terms of the positive series and the first term as be a non-zero constant, which is known to always hold true for any positive integer.

(1) Verification: the sequence is a geometric series;

(2) If unequal positive integers become arithmetic progression, try to compare the sum;

(3) If unequal positive integers become geometric series, try to compare the sum.

20. (Full score for this small question 16)

Known,

And ...

(i) When, find the tangent equation at;

(ii) When, let the length of the corresponding independent variable value interval be (closed interval

The length of is defined as), try to find the maximum value of;

(3) Is there such a thing that is timely? If it exists, the range of values obtained; If it does not exist, please explain why.

The Second Survey of Grade Three in Yancheng City in 2008/2009 School Year

Reference answers to math test questions

Fill-in-the-blank question: This big question is a *** 14 small question, with 5 points and 70 points for each small question.

1.2.3.4.25 5.6.

7.8.③ 9.6 10.50% (0.5 is correct)

11.12. <13.1214. Or

Second, the answer: this big question ***6 small questions, 90 points.

15. Solution: (i) If there are 28 points P * * and 19 satisfying points p,

So as to obtain an approximate

............................................. (7 points)

(ii) When the area of the rectangle formed by is and satisfies.

The area is, so the probability is ......................... (14).

16. certificate: (i) connection, connection.

∵ is the midpoint of∴‖ and ∴ quadrilateral is a rectangle.

∴ is the midpoint of the world population in .................................................... (3 points).

∵ is the midpoint of ∴‖ ................................................... (5 points).

And then what? ........................................................................... (7 points)

(Note: Proved by plane parallelism, similar to scribing)

(ii) In a straight triangular prism, ⊥ bottom surface, ∴.

∵∵, that is ∴ ∴ ⊥.....................(9 points).

And face, ∴⊥ ................................ (12 points)

Again, ∴ Aircraft ......................................... (14 points)

17. Solution: (1) From, from

So ................................. (4 points)

So, so ..................................... (7 points)

(2) Scheme I: Scheme ① ③.

∫a = 30, a = 1, 2c-(+ 1) b = 0, so according to the cosine theorem,

Yes, if the solution is b=, then c = ................ (11).

∴ ......................... (14 points)

Scheme 2: Scheme ② ③. Can be converted into option ③, similar to giving points.

(Note: Triangle cannot be determined by selecting ① ②)

18. solution: (I), that is,

, alignment, ................................... (2 points)

Let the equation of ⊙C be, and substitute the coordinates of o, f and a to get:

, the solution is ...................................... (4 points)

∴⊙C's equation is ........................................ (5 points).

(ii) If the coordinates of point B are:

This is true for any real number. .............................................. (7 points)

∴, solution or,

Therefore, when changing, ⊙C passes through another fixed point, ........................... (10 minute), except the origin O.

(3) from B,,,

∴, the solution is .............................. (12 points)

Once again, ∴ ........................................ (14 points)

Eccentricity of ellipse () ................... (15)

∴ The eccentricity range of ellipse is ................................ (16 points).

19.(I) Proof: Because it is always true for any positive integer,

Make, get, and then .............................. (1 min)

Make, get (1), thus (2),

(2)-(1), ..................................... (3 points)

To sum up, so the series is geometric series .................................... (4 points).

(ii) If a positive integer becomes arithmetic progression, then, therefore,

Then ...................................... (7 points)

(1) dang, .......................................... (8 points)

(2) When, ...................... (9 points)

(3) When, .................. (10)

(iii) If a positive integer becomes a geometric series, then,

So, ............ (13)

(1) When, that is, when, ............................. (14 points)

② Dang, that is, Dang, ...................... (15)

(3) when, that is, when, ...................... (16)

20. solution: (1) when,

Because when,,,

Besides,

So when, and ............................ (3 points)

Because, therefore, once again,

Therefore, the tangent equation is,

Namely .............................................. (5 points)

(2) Because, therefore, then

(1) when, because,

So from, solve,

So when, ................................ (6 points)

(2) When, because,

So from, solve,

So when, .................................. (7 points)

③ When, because,

Therefore, .......................................... can never be established (8 points).

All in all, if and only if,

So .................................. (9 points)

So when, the maximum value is ............................. (10).

(3) "Dang" is equivalent to "Truth is established",

That is, "(*) applies to the constant" ..................... (1 1).

(1) when, then when, then, then (*) can be changed into.

That is to say, when,

Therefore, it is suitable for the meaning of ............................................... (12 points).

(2) when,

(1) when, (*) can be changed to, that is, and,

Therefore, ................................................ (13) is needed at this time.

(2) When, (*) can be changed to,

Therefore, at this time, only .............................. (14) must take the exam.

(3) When, (*) can be changed to, that is, while,

Therefore, ............................................... (15) is needed at this time.

According to (1) (2) (3), it must meet the requirements of the problem.

Comprehensive ① ② Knowledge satisfies the existence of the meaning of the question, and its value range is .............................. (16 points).

Mathematics additional problem part

2 1. a solution: because PA is tangent to the circle at point a, m is the midpoint of PA.

So PM=MA, then.

Again, so, so ................... (5 points)

In, by,

So, that is to say,

Therefore, ............................................ (10)

B. Solution: So =.........................(5 points)

That is, under the transformation of matrix, there is the following process,

Then, the analytical formula of the curve under matrix transformation is ... (10).

C. solution: from the title, the center of the circle is known, so the rectangular coordinate equation of the tangent line is obtained.

For ................................................ (6 points)

Therefore, the polar coordinate equation of the tangent line is ……………………… (10/0 point).

D. Proof: Because, by using Cauchy inequality, you get .......................... (8 points).

That is, ........................................... (10)

22. Solution: (1) Establish a spatial rectangular coordinate system A-XYZ with A as the origin and AB, AC and AP as the X-axis, Y-axis and Z-axis respectively.

Then a (0 0,0,0), b (2 2,0,0), c (0 0,2,0), e (0, 1, 0), p (0 0,0, 1),

So, ......................... (4 points)

Therefore, the cosine of the angle formed by the non-planar straight line BE and PC is (5 points).

(ii) Make PM⊥BE BE (or extension line) in M and CN ⊥ be (or extension line) in N,

Then the real numbers m, n, so, that is

Because, therefore,

Solution, so ........................... (8 points)

Therefore, it is the cosine ...................................................................................................... of the dihedral plane angle (10 minute).

23. Solution: (i) If, so the coefficient is,

So, the solution is .............................................. (4 points).

(2) ① Derived from.

( ≥ ).

Order, obtain,

That is, in the same way,

.................................... (7 points)

③ Integrate both sides on [0,2],

OK,

According to the basic theorem of calculus,

That is, it can be obtained in the same way.

Therefore, .......................... (10)