1. Multiple-choice question: This big question is a small question of *** 10, with 5 points for each small question and 50 points for * * *. Only one of the four options given in each small question meets the requirements of the topic.
(1) If A= and B=, then =
(A)(- 1，+∞) (B)(-∞，3) (C)(- 1，3) (D)( 1，3)
Answer: C analysis: It is easy to know by drawing a few axes.
(2) If known, then i( )=
(A) (B) (C) (D)
Answer: b analysis: direct calculation.
(3) If the vector is 0, the following conclusion is correct.
(A) (B)
(C) (D) and vertical
Answer: D analysis: using formula calculation and using exclusion method.
(4) The equation of the straight line passing through point (1, 0) and parallel to the straight line x-2y-2=0 is
x-2y- 1 = 0(B)x-2y+ 1 = 0(C)2x+y-2 = 0(D)x+2y- 1 = 0
Answer: a analysis: using the point oblique equation.
(5) Let the sum of the first n items in the series {} =, and the value is
(A) 15 (B)
Answer: A analysis: Use =S8-S7, that is, the sum of the first 8 items MINUS the sum of the first 7 items.
(6) let ABC > 0, and the image of quadratic function f(x)=ax2+bx+c may be
Answer: D analysis: Using the relationship between opening direction A, symmetry axis position and intercept point C on Y axis, combined with ABC > 0, it is easy to know the contradiction by exclusion method.
(7) Let a=, B = and C =, then the relationship between A, B and C is
(A)A > C > B(B)A > B > C(C)C > A > B(D)B > C > A
Answer: a analysis: compare a and c by constructing a power function, and then compare b and c by constructing an exponential function.
(8) Assuming that X and Y satisfy the constraint conditions, the maximum value of the objective function z=x+y is
(A)3 (B) 4 (C) 6 (D)8
Answer: C analysis: It is easy to draw the feasible region.
(9) The figure shows three views of a geometry with a surface area of
372 (C)292
360 (D)280
Answer: B analysis: It can be understood as the combination of a cuboid with a length of 8, a width of 10 and a height of 2 and a cuboid with a length of 6, a width of 2 and a height of 8. Note that 2×6 overlaps twice, so you should subtract it.
(10) A randomly selects two vertices from four vertices of a square to form a straight line, and B randomly selects two vertices from four vertices of a square to form a straight line, so the probability that two straight lines are perpendicular to each other is
(A) (B) (C) (D)
Answer: C analysis: All possibilities are 6×6, and the two straight lines obtained are 5×2.
Mathematics (Liberal Arts) (Anhui Volume)
Volume 2 (multiple choice questions *** 100)
Fill-in-the-blank question: This big question contains ***5 small questions, each with 5 points and ***25 points. Fill in the answers in the corresponding places on the answer sheet?
(1 1) The negation of the proposition "x∈R exists, so x2+2x+5=0" is
Answer: For any X∈R, there is X2+2X+5≠0.
Analysis: the negation of "existence" is "any, arbitrary", which is easy to know.
(12) The focal coordinate of parabola y2=8x is
Answer: (2,0) Analysis: It is easy to know by definition.
(13) As shown in the figure, the output value of the program block diagram (algorithm flow chart) is x=
Answer: 12 analysis: the values of x sequence during operation are: 1, 2, 4, 5, 6, 8, 9, 10, 12.
(14) There are 100000 households in a certain place, including 99,000 ordinary households and 0/00 high-income households. 990 households were randomly selected from ordinary families, and 100 households were randomly selected from high-income families for investigation, and it was found that * * had 120 households. 70 high-income families. According to these data and statistical knowledge, do you think the reasonable estimate of the proportion of families with three or more houses in this area is.
Answer: 5.7% Analysis: Easy to know.
(15) if A >；; 0, b>0, a+b=2, then the following inequality holds for all A and B that meet the conditions. Write the numbers of all the correct propositions.
①ab≤ 1； ② + ≤ ; ③a2+B2≥2； ④a3+B3≥3；
Answer: ①, ③, ⑤ Analysis: ① and ⑤ are the same after simplification, so a=b= 1 excludes ② and ④, and it is correct to reuse ③.
Third, the solution: this big question is ***6 small questions. * * * 75 points. The solution should be written in words, proof process or calculation steps. The answer should be written in the designated area on the answer sheet.
(16)△ABC has an area of 30, with internal angles A, B and C, opposite sides A, B and C, and cosA=.
(1) is looking for
(2) If c-b= 1, find the value of a. 。
(The full mark of this small question is 12) This question examines the basic relationship of the triangle homoangle function, the triangle area formula, the product of vectors, and the ability to solve triangles by using cosine theorem and operation.
Solution: From cosA= 12 13, sinA= =5 13.
12 Sina =30, ∴bc= 156.
( 1) =bc cosA= 156？ 12 13 = 144.
(2)a2 = B2+C2-2bc cosA =(c-b)2+2bc( 1-cosA)= 1+2？ 156? ( 1- 12 13 )=25,
∴a=5
(17) ellipse e passes through point a (2,3), the axis of symmetry is the coordinate axis, the focus F 1, F2 is on the x axis, and the eccentricity.
(1) find the equation of ellipse e;
(2) Find the equation of the straight line where the bisector of f1a2 lies.
(The full mark of this small question is 12) This topic examines the basic knowledge such as the definition of ellipse, the standard equation and simple geometric properties of ellipse, the point-oblique equation and general equation of straight line, and the distance formula from point to straight line, and examines the basic idea and comprehensive operation ability of analytic geometry.
Solution: (1) Let the equation of ellipse e be e= 12, and we can get ca = 12, b2=a2-c2 =3c2. If a (2,3) is substituted, the equation of ellipse E is c=2.
(ii) F1(-2,0) and F2 (2,0) are known from (i), so the equation of straight line AF 1 is y=34 (X+2).
That is 3x-4y+6=0. The equation of the straight line AF2 is x=2. From the diagram of ellipse e,
∠ F 1A2 angle bisector has a positive slope.
Let P(x, y) be any point on the straight line where the bisector of F1A2 is located.
Then there is
If 3x-4y+6=5x- 10 and x+2y-8=0, the slope is negative, so it doesn't matter.
So 3x-4y+6=-5x+ 10, which means 2x-y- 1=0.
Therefore, the equation of the straight line where the bisector of f1a2 is located is 2x-y- 1=0.
18, (the full score of this small question is 13)
The test data of air pollution index in a city from April/KLOC-0 to April 30, 2065 are as follows (the main pollutants are inhalable particles):
6 1,76,70,56,8 1,9 1,92,9 1,75 ,8 1,88,67, 10 1, 103,95,9 1,
77,86,8 1,83,82,82,64,79,86,85,75,7 1,49,45,
(i) Completion of the frequency distribution table;
(ii) making a frequency distribution histogram;
(3) According to national standards, when the pollution index is between 0 and 50, the air quality is excellent; When it is between 5 1 and 100, it is better; When it is between 10 1~ 150, it is light pollution; When it is between 15 1~200, it is lightly polluted.
Please make a brief evaluation of the air quality in this city according to the given data and the above standards.
(The full score of this small question is 13) This question examines frequency, frequency and frequency distribution histogram, and examines the ability to solve simple practical problems, data processing ability and application consciousness by using statistical knowledge.
Solution: (1) Frequency distribution table:
Packet frequency number frequency rate
[4 1,5 1) 2 230
[5 1,6 1) 1 130
[6 1,7 1) 4 430
[7 1,8 1) 6 630
[8 1,9 1) 10 1030
[9 1, 10 1) 5 530
[ 10 1, 1 1 1) 2 230
(ii) Frequency distribution histogram:
(iii) Answer one of the following two questions correctly:
(1) Within one month, the number of days when the air pollution index in this city is at an excellent level is 2 days, accounting for 1 15 of the number of days in that month. There are 26 days in a good level, accounting for 13 15 of the days of the month. There are 28 days in excellent or good level, accounting for 146544 of the days of the month.
(2) There are two days of light pollution, accounting for 1 15 of the days of the month. The near light pollution with pollution index above 80 is 15 days and * * * has 17 days, accounting for 1730 days of the month, exceeding 50%.
(19) (the full score of this small question is 13)
As shown in the figure, in the polyhedron ABCDEF, the quadrilateral ABCD is a square, AB=2EF=2, ef∨ab, EF ⊥ FB, ∠ BFC = 90, BF = FC, and H is the midpoint of BC.
(i) Verification: FH∑ plane EDB；;
(ii) Verification: AC⊥ aircraft edb
(3) finding the volume of tetrahedron b-def;
(Full score for this small question 13) This question examines basic knowledge such as parallelism between spatial lines and planes, perpendicularity between lines and planes, perpendicularity between planes, volume calculation, etc., and also examines spatial imagination and reasoning ability.
(1) Prove that if AC and BD intersect at point G, then G is the midpoint of AC. Even EG and GH, because H is the midpoint of BC, GH∨AB and GH= AB and EF∨AB and EF= AB.
∴EF∥GH. And EF=GH ∴ quadrilateral EFHG is a parallelogram.
∴EG∥FH and EG aircraft EDB, ∴FH∥ aircraft EDB.
(2) Prove that the quadrilateral ABCD is a square with AB⊥BC.
And ef∑ab, ∴ EF⊥BC. and EF⊥FB, ∴ EF⊥ airplane BFC, ∴ EF⊥FH.
∴ AB⊥FH.BF=FC H is the midpoint of fh⊥bc. BC province ∴ FH⊥ plane ABCD.
∴ FH⊥AC.FH∨eg，∴ AC ⊥ eg. AC⊥BD，EG∩BD=G，
∴ AC⊥ Aircraft Engineering Bureau.
(iii) Solution: EF⊥FB, BFC = 90, ∴ BF ⊥ plane CDEF.
∴ BF is the height of the tetrahedron B-DEF, BC=AB=2, ∴ BF=FC=
(20) (The full score of this small question is 12)
Let function f(x)= sinx-cosx+x+ 1, 0￠x￠2, and find the monotone interval and extreme value of function f(x).
(The full mark of this small question is 12) This question examines the operation of derivatives, studies the monotonicity and extreme value of functions by using derivatives, and examines the ability of comprehensively applying mathematical knowledge to solve problems.
Solution: f (x) = sinx-cosx+x+ 1, 0 ￠ x 0 ￠ x 2,
Know =cosx+sinx+ 1,
So = 1+ sin(x+)。
Let =0 so that sin(x+ )=-, x=, or x=32.
When x changes, f(x) changes as follows:
X (0，)
( ,32 )
32
(32 ,2 )
+ 0 - 0 +
F(x) monotonically increases by ↗ +2.
Monotone decreasing ↘ 32
Monotonic increasing↗
Therefore, as can be seen from the above table, the monotone increasing interval of f(x) is (0,) and (32, 2), the monotone decreasing interval is (0, 32), the minimum value is f(32 )=32, and the maximum value is f( )= +2.
(2 1) (the full score of this small question is 13)
Jean, ..., ... is a series of circles on the coordinate plane. Their centers are all on the positive semi-axis of the X axis, and they are all tangent to the straight line Y = X. For every positive integer n, the circles are tangent to each other, and the radii they represent are called increasing sequences.
(i) Certification: geometric progression;
(Ⅱ) Let = 1 and find the sum of the first n items in the series.
(The full mark of this small question is 13) This question examines the basic knowledge of geometric series, and uses basic methods such as dislocation subtraction and summation to examine his abstract ability and reasoning ability.
Solution: (1) If the inclination of the straight line y= x is written, then tan =, sin = 12.
Let the center of Cn be (0,0), then the meaning of the problem is = sin = 12, which is = 2; Similarly, if we substitute = 2, we get rn+ 1=3rn.
So {rn} is a geometric series with q=3.
(ii) because r 1= 1 and q=3, rn=3n- 1, so =n? ,
Remember Sn=, so Sn= 1+2? 3- 1+3? 3-2+…………+n？ . ①
= 1? 3- 1+2? 3-2+…………+(n- 1)？ +n？ ② ①-②, yes
= 1+3- 1+3-2+…………+-n？ = - n？ =–(n+)？
sn =–(n+)？ .