Mathematics (science)
First, the topic (5 points for each small question, * * * 40 points)
1. Given the directional quantity (1), (,1), if the included angle with is, the value of the real number is.
A.B. C. D。
2. The positional relationship between the straight line x-y+ 1=0 and the circle (x+ 1)2+y2= 1 is ().
A. Tangency B. A straight line passes through the center of the circle C. A straight line passes through the center of the circle but intersects the circle D. It is separated.
3. In the plane rectangular coordinate system xOy, the coordinate of point P is (-1, 1). If the polar coordinate system is established with the origin o as the pole and the positive semi-axis of the X axis as the polar axis, then the polar coordinate of point P is not () in the following options.
A.()B.( ) C.( ) D .()
4. If p and q are simple propositions, then it is false ().
A. Sufficient and unnecessary conditions B. Necessary and insufficient conditions
C. Necessary and sufficient conditions D. It is neither a sufficient condition nor a necessary condition
5. The five test scores of two athletes A and B are shown in the figure below.
Nail shank b
7 7 8 6 8
8 6 2 9 3 6 7
Suppose that the standard deviation of the test scores of two athletes A and B and the average of the test scores of two athletes are respectively
A.，b，
C.，d，
6. If the function is known, the range of the real number x is ().
A.B. C. D。
7. Let f(x) and g(x) be differentiable functions on R, which are the derivative functions of f(x) and g(x) respectively, then, if there is ().
A.f(x)g(x)>f(b)g(b) B. f(x)g(a) > English (Arabic) case (10)
C.f(x)g(b)>f(b)g(x) D. f(x)g(x) > female (a) male (a)
8. As shown in the figure, in a triangular prism, points G and E are the midpoint of the sum of line segments, and points D and F are the moving points on line segments AC and AB, respectively. If so, the minimum length of the line segment DF is ()
A. 1。
Fill in the blanks (5 points for each small question, 30 points for * * *)
9. Execute the program block diagram as shown on the right, and the value of output result Y is _ _ _ _ _ _ _ _.
10. As shown in the figure below, AB is the diameter of the semicircle O, C is a point on the extension line of AB, CD cuts the semicircle to D, CD=4, AB=3BC, then the length of AC is.
1 1. The focus of the ellipse is that a straight line passing through F2 and perpendicular to the X axis intersects the ellipse at point P, then the value of |PF 1| is.
12. Known. If a point p is randomly projected on a region, then the probability that this point p falls into region A is.
13. As shown on the right, there is a 30m-high China mobile signal tower (BC) on the hillside with an inclination angle of 150(≈CAD = 150), and the elevation angle of Tower B measured at a is 450(≈BAD = 450), which is the distance from the top of the tower to the horizontal plane.
14. For an array of positive numbers with unequal numbers (positive integers not less than), if it exists, it is said that sum is an order of the array, and the number of all orders in an array is called the order of the array. For example, there are sequences "2,4" and "2,3" in the array.
Third, answer the question (this big question is ***6 small questions, ***80 points)
15.( 12 points) The known function f(x)= (where a >；; 0,) as shown.
(1) find one? And then what? The value of;
(2) If Tan? The value of =2.
16.( 14 point) In a regular quadrangle, e and f are the midpoint, g is any point, and the tangent of the angle between EC and the bottom ABCD is 4.
Verify agef
(2) Determine the position of G point to make the surface of AG CEF, and explain the reasons;
(iii) Find the cosine of the dihedral angle.
17.( 13 points) In a lucky draw, there are five white balls and five black balls in a pocket, and all the balls are the same except the colors. Take out two balls at a time, observe the color and put them back. If they are the same color, you will win the prize.
(1) Touch the ball only once to find the winning probability;
(2) Find the probability of not winning the prize just once after touching the ball for two consecutive times;
(3) Remember that the winning number of three consecutive touches is 0, and get the distribution list.
18.( 14 points) Known function.
(i) When a=0, find the tangent equation of the image of function f(x) at point A (1, f (1));
(ii) If f(x) is monotonic on r, find the range of a;
(iii) If, find the minimum value of the function f(x).
19.( 13 points) It is known that the sum of the first n items in the series is,,, arithmetic progression,,, and becomes a geometric series.
(a) to find the general formula of series;
(Ⅱ) Find the sum of the first n items in the series.
20.( 13) It is known that the focus of parabola is that the moving straight line passing through the focus and not parallel to the X axis intersects with parabola at two points, and the tangent of parabola at two points intersects with this point.
(1) Verification: the abscissa of three points is arithmetic progression;
(Ⅱ) Let a straight line intersect a parabola at two points to find the minimum area of a quadrilateral.
Fengtai District 20 10 Senior Three Unified Exercise (2)
Mathematics (science)
First, multiple-choice questions (5 points for each small question, ***40 points)
Title 1 2 3 4 5 6 7 8
Answer c b d b b c c c
Fill in the blanks (5 points for each small question, 30 points for * * *)
9. 1 ; 10.8 ; 1 1.； 12.； 13.40.5 ; 14.6.
Third, answer the question (this big question is ***6 small questions, ***80 points)
15.( 12 points) The known function f(x)= (where a >；; 0,) as shown.
(1) find one? And then what? The value of;
(2) If Tan? The value of =2.
Solution: (1) From the diagram, we know that a = 2.
T=2( )=？ ,
∴? = 2, ... 3 points.
∴f( x)=2sin(2x+？ )
∫= 2 sin(+？ )=2,
∴sin( +？ )= 1,
∴ +? = ,? = +，(k？ z)
∵ ,∴? =............................6 points.
According to (I), f(x)=2sin(2x+),
= 2 symplectic (2? + )=2cos2？ =4cos2？ -2 ...9 points.
Tan? =2，∴sin？ =2cos？ ,
∫ sin2 again? +cos2？ = 1，∴cos2？ = ,
∴ = ................... 12 points.
16.( 14 point) In a regular quadrangle, e and f are the midpoint, g is any point, and the tangent of the angle between EC and the bottom ABCD is 4.
Verification: argef
(2) Determine the position of G point to make the surface of AG CEF, and explain the reasons;
(iii) Find the cosine of the dihedral angle.
Solution: ∵ is a regular quadrangular prism.
∴ABCD is a square, and its side length is 2a. ECD is the angle between EC and the seabed surface. And then what? ECD=？ CEC 1, ∴ CC1= 4EC1= 4A ............................................1min.
With A as the origin, straight lines of AB, AD and AA 1 are respectively the X-axis, Y-axis and Z-axis, and a rectangular coordinate system as shown in the figure is established.
Then a (0 0,0,0), b (2a 2a,0,0), c (2a 2a,2a,0), d (0 0,2a,0),
A 1(0，0，4a)，B 1(2a，0，4a)，C 1(2a，2a，4a)，D 1(0，2a，4a)，
E(a, 2a, 4a), F(2a, a, 4a), let g (2a, 2a, b) (0
(ⅰ)=(2a，2a，b)，=(a，-a，0)，=2a2-2a2+0=0，
∴ Age 6 points.
(ii) According to (i), only AG Ce is needed to make the AG surface CEF,
Just =(2a, 2a, b)? (-a，0，4a)=-2a2+4ab=0，
∴b= a, that is, CG= CC 1, CEF on the surface of AG. ................. 10 point
(iii) As can be seen from (ii), when G(2a, 2a, a) is the normal vector of the plane CEF,
From the meaning of the question, it is the normal vector of the plane CEC 1
Let the size of dihedral angle be? ,
Then cos? = = = ,
The cosine of the dihedral angle is 14 minute.
(Give points accordingly by comprehensive method)
17.( 13 points) In a lucky draw, there are five white balls and five black balls in a pocket, and all the balls are the same except the colors. Take out two balls at a time, observe the color and put them back. If they are the same color, you will win the prize.
(1) Touch the ball only once to find the winning probability;
(2) Find the probability of not winning the prize just once after touching the ball for two consecutive times;
(3) Remember that the winning number of three consecutive touches is 0, and get the distribution list.
Solution: (1) Let the probability of winning the prize by touching the ball only once be P 1, then P 1 = = ......
(ii) If you touch the ball twice in a row (put it back after each touch), and the probability of not winning the prize just once is P2, then
P2 =……7 ... 7 points.
The value of (III) can be 0, 1, 2, 3.
=( 1- )3= ,
= = ,
= = = ,
= =
So the distribution list is as follows.
0 1 2 3
P
.................................. 13.
18.( 14 points) Known function.
(i) When a=0, find the tangent equation of the image of function f(x) at point A (1, f (1));
(ii) If f(x) is monotonic on r, find the range of a;
(iii) If, find the minimum value of the function f(x).
Solution:
(I) when a=0
, ,
The tangent equation of the image of function f(x) at point A (1, f( 1)) is y-3e=5e(x- 1).
That is 5ex-y-2e = 0: 04.
(Ⅱ) ,
Considering that the constant holds and the coefficient is positive,
∴f(x) is monotone and is equivalent to a constant on R. 。
∴(a+2)2-4(a+2)？ 0,
∴-2? Answer? 2, that is, the value range of A is [-2,2], and the value range of ... is
(If the value range of A is (-2,2), 1 min can be deducted)
(iii) When,
..................................... 10.
Make, get or x,
Make, get or x,
Order, get.
The changes of x and f (x) are shown in the following table.
X
1 )
+ 0 - 0 +
f(x)
maximum
minimum value
Therefore, the minimum value of the function f(x) is f (1) = ...............................14 minutes.
19.( 14 minutes) It is known that the sum of the first n terms of the series is,,, arithmetic progression,, and again, it becomes a geometric series.
(a) to find the general formula of series;
(Ⅱ) Find the sum of the first n items in the series.
Solution: (I)∫,
∴ ,
∴ ,
∴ ,
∴ ........................................... 2 points.
And, ⅷ
∴ series is a geometric series with 1 as the first term and 3 as the common ratio.
.........................................., 4 points.
∴ ,
In arithmetic progression, ∫.
Because of the geometric series, let the tolerance of arithmetic progression be d,
∴ () ........................................... 6 points.
The solution is d=- 10, or d=2, ∫∴ Cut off d=- 10, and take d=2, ∴b 1=3.
∴bn=2n+ 1, 8 points.
(ii) Starting from (i)
①
② 10.
①-② Acquisition
................. 12 point
,
∴ ........................................14.
20.( 13) It is known that the focus of parabola is that the moving straight line passing through the focus and not parallel to the X axis intersects with parabola at two points, and the tangent of parabola at two points intersects with this point.
(1) Verification: the abscissa of the three points is marked as arithmetic progression;
(Ⅱ) Let a straight line intersect a parabola at two points to find the minimum area of a quadrilateral.
Solution: (1) From the known situation, it is obvious that the slope of a straight line exists and must not be 0.
Then the equation of the straight line can be set to (),,,
Through elimination, it is obvious.
So, two points.
By, by, so,
So, the slope of the straight line is,
So, the equation of this straight line is,
So the equation of the straight line is ①. .................................., 4 points.
Similarly, the equation of a straight line is ②. Five points.
②-① According to the abscissa of point M,
That is, the abscissa of the three points is arithmetic progression. Seven points
(Ⅱ) Y =-1 is easily obtained from ① and ②, so the coordinate of point M is (2k,-1) ().
So,
The equation of the straight line MF is
Let c (x3, y3) and d (x4, y4)
By excluding, obviously,
So ... nine points.
and
. .............. 10 point
. ................... 1 1 min.
Because, therefore,
So,,,
If and only if the quadrilateral area is the minimum. .................... 13 o'clock