11 to 12 - GS
Mathematics Entrance Exam Time: 150 min
I. (5 pts.)
m
f is the function defined over 1 by f x ax b .
x 1
(C) is the graph of f in an orthonormal system of axes.
1) Find a, b and m so that (C) is tangent to the line y 2 at the point A with
abscissa x = 3 and the tangent to (C) at the point B 2, f 2 is parallel to the
line of equation y 3x .
4
In what follows, suppose that f x x 3 .
x 1
2) a) Find the limits of f(x) at the open boundaries of its domain. Deduce the
equation of the vertical asymptote to (C).
b) Prove that the line (d) with equation y x 3 is an oblique asymptote to
(C).
c) Discuss according to the values of x the relative positions of (C) and (d).
3) a) Show that f ' x
x 1 3 x .
x 1
2
b) Set up the table of variations of f.
c) Calculate f 2 , f 0 , f 2 and f 4 , then draw (C).
4) a) Find a second point E on (C) where the tangent is also parallel to y 3x .
b) The tangent at E to (C) intersects its asymptotes at L and F.
Prove that E is the midpoint of [LF].
II. (3 pts.)
The space is referred to a direct orthonormal system O; i , j , k .
Consider the points A(0, 1, 1), B(1, 2, 2), C(2, 1, 0), D(1, 6, 8) and E(1, , ).
1) a) Prove that A, B and C determine a triangle and that BAC ˆ is obtuse.
b) Calculate the area of triangle ABC.
c) Deduce the distance from A to (BC).
2) Determine the equation of the plane (ABC).
3) a) Prove that E is the orthogonal projection of D on (ABC).
b) Deduce the distance from D to (ABC).
1
III. (4 pts.)
Consider the two sequences (U n ) and (Vn ) such that :
2 3U n 2 Un
U0 1 ; U n 1 and Vn .
2 Un 1Un
1) Calculate the terms U1 and U 2 .
1
2) a) Prove that (Vn ) is a geometric sequence whose common ratio r and
4
whose first term is to be calculated.
b) Calculate Vn in terms of n.
c) Calculate the sum S V0 V1 V2 V3 Vn in terms of n.
d) Study the sense of variation of the sequence ( Vn )
3) a) Calculate U n in terms of n.
b) Prove that, for all n in IN , U n 2 .
IV. (2.5 pts.)
dx x dx
Let I and J ; x , 1 .
3 2x 3 2x
1) Determine I.
2) Find 3I 2 J . Deduce J.
6 x 3 2 x dx
3) Determine K .
3 2x
V. (3 pts.)
Consider a rectangle ABCD of center O. S
(SB) is perpendicular to plane (ABCD).
AB = SB = 4 and AD = 3.
Let M be the midpoint of [AS]. M
1) a) Prove that the triangle SAD is right.
b) Show that the planes (SAD) and (SAB) are perpendicular. A
B
2) Prove that (AB) and (SC) are orthogonal.
3) Calculate the angle of the two planes (ABC) and (SAD). O
4) Prove that (SA) is perpendicular to plane (BMC). D C
5) a) Justify that (OM) is parallel to (CS).
b) Calculate the angle that (OM) makes with the plane (BMC).
2
V. (2.5 pts.)
A bag contains 9 cards numbered 1 through 9.
Two cards are selected from this bag one after the other and without
replacement.
Calculate the probability of each of the following events:
A: "a number less than 4 and a number greater than 3 are obtained".
B: "the two numbers are even"
C: "only one of the numbers is even"
D: "at least one of the two numbers is odd"
E: "one of the numbers is less than 4 and the other is a multiple of 4".
