1) A straight line crosses the focus of the parabola y^2 = 4x, and intersects it at two points A and B. A and B's x-coordinates add to 5. How many such straight lines are there:
A) 1 and only 1.
B) 2 and only 2.
C) Infinitely many.
D) 0.
2) A "level" cross-section is made on a sphere, the cross section (a circle) has a perpendicular distance of 1 unit from the center of the sphere. And the cross section has an area of Pi. What is the surface area of the sphere:
3) Line L crosses the point (-2, 0), and intersects the circle (x^2 + y^2 = 2x) at 2 points. What is the range of possible values for the slope of L:
4) In the picture below, ABCD is a square with side length 1. Triangles ADE, BCF are both equilateral triangles. EF || AB, and EF=2.
What is the volume of the ABCDEF:
5) Given hyperbola x^2/a^2 - y^2 = 1, a>0. One of its directrix coincides with the directrix of y^2 = -6x. What is the eccentricity of that hyperbola:
6) When 0 < x < Pi/2, what is the minimum value of [1 + cos(2x) + 8sin^2(x)] / sin(2x):
7) Let b > 0. The graph of the equation y = ax^2 + bx + a^2 - 1 matches one of the following:
What is the value of a:
8) Let 0 < a < 1, f(x) = log[base a](a^(2x) - 2(a^x) - 2). What is the range of possibe values for x, such that f(x) < 0:
9) The set of inequalities { y >= x-1, y <= -3*abs(x) + 1} represents a region on the xy plane. What is the area of that region?
( abs(x) means the absolute value of x):
10) In triangle ABC, given that tan((A+B)/2)) = sin(C), there are 4 propositions:
Which are the correct propositions (note: cot = co-tangent)?
II) Filling the blanks (20 points): 4 questions, 5 points each.
1) m is a positive integer, and:
10^(m-1) < 2^512 < 10^m
m=________ (log[base 10](2) = 0.3010)
2) When [ 2x - 1/(sqrt(x)) ]^9 is expanded, the constant term is________
3) Given a function f(x) = sin(x) + 2*abs[sin(x)], 0 <= x <= 2*Pi. ( abs[sin(x)] means absolute value of sin(x). )
The graph of f(x) intersects the line y = k at two and only two different points. What is the possible range of values for k: __________
4) Which of the following propositions about tetrahedrons are correct:
I) If there's a tetrahedron whose base is an equilateral triangle, and its three side-surfaces form three equal dihedral angles with the triangular base, then that is a regular tetrahedron. (dihedral angle = an angle formed by two planes)
II) If there's a tetrahedron whose base is an equilateral triangle, and its three side-surfaces are all isosceles triangles, then that is a regular tetrahedron.
III) If there's a tetrahedron whose base is an equilateral triangle, and its three side-surfaces have equal areas, then that is a regular tetrahedron.
IV) If there's a tetrahedron whose 3 side-edges form three equal angles with the triangular base, and its 3 side-surfaces also form three equal dihedral angles with its triangular base, then that is a regular tetrahedron.
Write the correct proposition number(s) : ___________.
III) Free Response (60 points) : 4 questions, 15 points each. You must show steps to get full credits. Partial credits are given.
Question 1:
Let f(x) = sin(2x + k), -Pi < k < 0.
The graph of y=f(x) is symmetrical about the line x = Pi/8. (Note: Do not assume this is the only line of symmetry for f(x). Of course it's possible that it's the only line).
I) Find k.
II) Find the region(s) of y = f(x) in which f(x) is monotonously increasing. (A function is monotonically increasing if m <= n implies f(m) <= f(n).)
III) Prove that the line 5x - 2y + c = 0 is not tangent to the graph of y=f(x).
Question 2:
In the following picture: ABCD is a right-angle trapezoid. AB || DC. Angle DAB = 90 degrees. PA is perpendicular to ABCD. PA=AD=DC=(1/2)*AB=1. M is the middle point of PB.
I) Prove: plane PAD is perpendicular to plane PCD, along the line PD.
II) Find the angle formed by AC and PB, due to their different directions.
III) Find the angle formed by the planes AMC and BMC, along the line MC.
Question 3:
There's a geometric sequence {A(n)}, n=1,2,3,... Its common ratio is q. Let the sum of the first n terms of A(n) be denoted as Sn. Sn > 0.
I) Find the range(s) of possible values for q.
II) Let B(n) = A(n+2) - (3/2)*A(n+1). Let the sum of the first n terms of the series {B(n)} be denoted as Tn. Determine whether Tn > Sn, or Tn < Sn, or Tn=Sn. If there's more than one possibility, describe the conditions for each possibility.
Question 4:
There are 9 seeds being planted in 3 pots, each pots receiving 3 seeds. The probability of each seed blossoming into a flower is 0.5. If a pot has no seed becoming flowers, then that pot needs re-planting (putting in 3 new seeds again). If a pot has at least one seed becoming a flower, then that pot does not need re-planting.
There can be maximum of 1 re-planting for each pot, and each re-planting for each pot costs 10 Yuan.
Let C represent the total cost in Yuan of re-planting for all three pots.
I) List the probability distribution for C (list all possible values of C and their respective probability), and find its expected value. Be accurate to 0.01.