2004 Chinese National Undergraduate Entrance Exam - Math Subject
Total Time: 2 Hours. No Calculators allowed. Total 120 points

Part I – Multiple Choice. 10 questions, 5 points each. Total 50 points.

1) If the sets M = {x | x^2<4 }, N= {x | x^2-2x-3<0 }, then the set M intersect N =

A) {x | x < -2}
B) {x | x > 3}
C) {x | -1 < x < 2 }
D) {x | 2 < x < 3 }

2) lim (x->1) [ (x^2+x-2)/(x^2+4x-5) ] =

A) 1/2
B) 1
C) 2/5
D) 1/4

3) The complex number w = -1/2 + [sqrt(3)*i]/2. 1 + w =

A) -w
B) w^2
C) -1/w
D) 1/(w^2)

4) Given circle C and Circle D, where D is (x-1)^2 + y^2 = 1. And C is symmetrical with D about y = -x. The equation of C is:

A) (x+1)^2 + y^2 = 1
B) x^2 + y^2 = 1
C) x^2 + (y+1)^2 = 1
D) x^2 + (y-1)^2 = 1

5) If the graph of y = tan(2x+b) passes through the point (Pi/12,0), then a possible value of b is:

A) -Pi/6
B) Pi/6
C) -Pi/12
D) Pi/12

6) The graph of y = -e^x is:

A) symmetrical to y=e^x about the y axis.
B) symmetrical to y=e^x about the origin.
C) symmetrical to y=e^(-x) about the y axis.
D) symmetrical to y=e^(-x) about the origin.

7) Given a sphere whose radius is 1. A,B,C are points on the surface of the sphere, and the surface distance between any 2 of those points is Pi/2. What is the distance from the center of the sphere to the plane ABC?

A) 1/3
B) sqrt(3)/3
C) 2/3
D) sqrt(6)/3

8) In an xy cartesian coordinate system, an A-line is defined as being 1 unit away from the point (1,2), and at the same time 2 units away from the point (3,1). How many A-lines are there?

A) 1
B) 2
C) 3
D) 4

9) What is the smallest positive period of y = sin^4(x) + cos^2(x)?

A) Pi/4
B) Pi/2
C) Pi
D) 2*Pi

10) Of all the distinct 5-digit numbers that are made from the combinations of 1,2,3,4,5, how many of them are greater than 23145 and less than 43521 ?

A) 56
B) 57
C) 58
D) 60

Part II - Filling the blanks. 5 points each. 4 questions. Total 20 points.

1) Given a bag with 3 red balls and 2 white balls inside. If 2 balls are randomly taken from the bag (without replacing), and if there are x red balls from those 2 balls. Then what is the probability distribution of the random variable x.

x: 0
P: ________
x: 1
P: ________
x: 2
P: ________

2) If x,y satisfy the following conditions:

x >= 0, x >= y, 2x-y <= 1,
What is the maximum value of z=3x+2y: __________.

3) Given an oval centered on the origin and a hyperbola 2x^2-2y^2=1. They have common foci. Also, their eccentricities are mutually inverse numbers. (like x and 1/x). The equation of the oval is: ___________________.

4) Let P be a moving point on the curve y^2=4(x-1). And let x be the sum of the distances from P to (0,1) and from P to the y axis. The smallest value of x is _______________.


Part III) - Free Response. 4 questions, 12 points for the first 3 and 14 points for the last one. Total 50 points.
Show your steps/explanations. Partial credits given..

. 1) 12 points.
In an acute angle triangle ABC, sin(A+B)=3/5, sin(A-B)=1/5.

I) Prove tan (A) = 2*tan(B).
II) Let AB=3, find the height on AB, with AB considered the base.

2) 12 points.
Given 8 teams, and 3 of them are weak teams. The 8 teams are randomly divided into 2 groups, A and B, with 4 teams each. Find:

I) Of groups A and B, the probability that one group has exactly two weak teams.
II) The probability that group A has at least two weak teams.

3) 12 points.
Given a parabola C: y^2=4x, F is the focus of C. Line L passes through F and intersects C at points A,B.

I) Let the slope of L be 1. Find the angle between the vectors OA and OB (O is the origin).
II) Let vector FB = m*(vector AF). If m belongs to [4,9], find the range of values of the intercept of L on the y axis.

4) 14 points.
Given f(x) = ln(1+x)-x and g(x) = x*ln(x)

I) Find the maximum value of f(x).
II) Let 0 < a < b. Prove: 0 < g(a)+g(b)-2*g((a+b)/2) < (b-a)*ln(2).