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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 ?


B .8

C .3


(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


9 3 8


63 32


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


4 5


3 10 10


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,


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



? t )( yi ? y )
2 i

? (t ? t )
i ?1


? ? y ? bt ? ) ,a


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.


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)


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)


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