The college entrance examination in China

The college entrance examination in China (2014) Mathematics(Science) Full Points: 150. Time: 2 hours. Mustn’t use calculator
1.Choice (50 Points)
(1)Provide 2 sets M={0，1，2}，N={x|x2-3x+2 0} M∩N= A．{1} B．{2} C．{0，1} D．{1，2}

(2) Complex number Z1,Z2, In complex plane, the two points are symmetry of imaginary axis. Z1=2+i Z1Z2= A．-5 B．5 C．-4+i D．-4-i

(3)There are Euclidean vector a,b. And |a+b|= 10 ,|a-b|= 6 a·b= A．1 B．2 C．3 D．5
1 2

(4)A acute triangle ABC, the area is , AB=1,BC= 2 AC= A．5 B． 5 C．2 D．1

(5)In a city, The probability of rain for one day is 0.75, and for continuously two days is 0.6. Now we know some day the city is rain, what is the probability for next day rain. A．0.8 B．0.75 C．0.6 D．0.45

(6)Curve y=ax-ln(x+1), The tangent equationis y=2x at (0,0) a= A．0 B．1 C．2 D．3

?x ? y ? 7 ? 0 ? (7) x,y have ? x ? 3 y ? 1 ? 0 , so what is the maximumvaluefor z=2x-y ?3 x ? y ? 5 ? 0 ?

A．10

B ．8

C ．3

D．2

(8)Parabola C：y2=3x, It has focus F. A line cross F and the tilt angle is 30° . Parabola C and The line, those crossover point are A andB. O is a point located at the origin of the coordinate plane. So the area of △OAB is A．
3 3 4

B．

9 3 8

C．

63 32

D．

9 4

(9)Inright angletriangular prismABC-A1B1C1, BAC=90°, Mis midpoint in A1B1,Nis midpoint in A1C1,BC=CA=CC1,What is cosine value in angle of BM and AN. A．
1 10

B．

4 5

C．

3 10 10

D．

2 2

(10)f(x)= 3 sin

?x 2 ,If f(x)extrema point x0has x0 ? [ f ( x)]2 ? m2 m

What is m’s range A． （-∞,-6）∪ (6,+ ∞) B． （-∞,-4）∪ (4,+ ∞) C． （-∞,-2）∪ (2,+ ∞) D． （-∞,-2）∪ (2,+ ∞)

2.Fill in the blank.
(11)In the expansion of (x+a)10,the multinomial coefficient of x7 is 15, so a=_______(numbers).(5 Points) (12)The function, f(x)=sin(x+2φ )-2sinφ cos(x+φ ), which the maximum value is_________.(5 Points) (13)Let an even function f(x) is monotone decreasing in [0, ??) ,and f(2)=0, if f(x-1)>0, so the interval of x is_________.(5 Points) (14)Let a point M(x0,1), if there is a point N on the circle O: x2+y2=1, and ∠ OMN=45°,so the interval of x0 is__________.(5 Points)

3.Solve questions.
(15)Let a sequence {an} , and a1=1, an+1=3an+1. (12 Points) ①Please prove the sequence {an+ } is Geometric sequence, and find the formula of {an}; ②Please prove
1 1 1 1 3 ? ? ??? ? . a1 a2 a3 an 2
1 2

(16)Look at the picture, In the rectangular pyramid P-ABCD,

the

underside plane ABCD is a rectangle, and PA⊥plane ABCD , and E is the midpoint of the line segment PD. (12 Points) ①Please prove PB//plane AEC ②Let the dihedral angle D-AE-C is equal to 600.AP=1 and AD= 3 , Calculate the volume of triangular pyramid E-ACD.

(17)Here is a data sheet about a family’s income(y) from 2007 to 2013 (12Points)
Year Year of code: t Income :y 2007 1 2.9 2008 2 3.3 2009 3 3.6 2010 4 4.4 2011 5 4.8 2012 6 5.2 2013 7 5.9

①Find the equation of linear regression about Y and T. ②Use the equation of linear regression from ①, please analyze the situation about the change of this family’s income. And predict the income in 2015. (Add: The way to calculate the equation of linear regression’s slope and intercept

?? is b

? (t
i ?1

n

i

? t )( yi ? y )
2 i

? (t ? t )
i ?1

n

? ? y ? bt ? ) ,a

(18)Let

F1 , F2 are the left focal point and right focal point of a ellipse C:

x2 y 2 ? ? 1(a ? b ? 0) , M is a point in the ellipse and MF2⊥x axis, the line MF1 a 2 b2

has another intersection on C called N.(12 Points) ①If the slope of line MN is equal to
3 ,Calculate the eccentricity of ellipse C; 4

②If the intercept in y axis of line MN is equal to 2, and MN=5F1N. Calculate a and b.

(19)Let

f(x)=ex-e-x-2x (12 Points)

①Discuss the monotony of this function;
②Let g(x)=f(2x)-4bf(x), when x>0, g(x)>0, find the maximum of b;

③We know 1.4142< 2 <1.4143, Please estimate the approximate value of ln2 (accurate to 0.001)

Please choose a question to solve from the three questions
(20)Look at this picture, P is a point on ⊙O, PA is a tangent line and A is point

of tangency. The secant line PBC have two intersections with ⊙O, PC=2PA, D is midpoint of PC. The line AD has another intersection with ⊙O called E. Please prove(10 Point) ①BE=EC ②AD?DE=2PB2

(21)In the rectangular coordinate systemxOy, The origin is polar and x axis is positive axis to making a polar coordinate system. The polar equation of a semicircle named C is ? ? 2 cos ? ,? ? [0, ] .(10 Point)
2

?

①Find the parameter equation of C ②If D is on C, the tangtant line of C at D is vertical to the line: y ? 3x ? 2 .Find the coordinate of D.

(22)Letf(x)=|x+ |+|x-a| (a>0) ①Prove: f(x)≥2 ②If f(3)<5, find the value range of a.

1 a

(10 Points)

Should the college entrance examination be changed ...

The college entrance examination should be changed in China Nearly millions of Chinese students undergo a fierce competition of the entrance examination to ...

the college entrance examination

I think the college entrance examination should be abolished by Chinese Education Department, because it gives students too much pressure, influences students...

...of entrance exams to university in China and US

In addition, The College Entrance Examination has been around in China for nearly 20 years. It helps many people change their life, especially poor ...

The advantage of the college entrance examination

The advantage of the college entrance examination_英语学习_外语学习_教育专区...In addition, The College Entrance Examination has been around in China for ...

My view on the College Entrance Examination

But it is the most efficient way in China now. China has a large number of populations. So the College Entrance Examination appears. It makes us to ...

Abolishment of college entrance examination

Abolishment of college entrance examination From Steven0717 It’s probably not arbitrary to assert that college entrance examination in China is a nightmare ...

Chinese College Entrance Exam System

Chinese College Entrance Exam System CHUNXI TAN Class 3 11210120 The college entrance examination system with Chinese characteristics is only based on the ...

...in the National College Entrance Examination

Cancelling English subject in the National College Entrance Examination is not acceptable to everyone in China. As we all know, in the Chinese contemporary...

My View on Entrance Examination to College

My View on Entrance Examination to College_英语学习_外语学习_教育专区。My...in China care nothing but the enrollment rate to Universities and dedicate ...

My View On the College Entrance Examination

Under such a system, we can only be “the students made in China” but...To make a conclusion, the College Entrance Examination is just the tool of...