EUROPEAN SCHOOL LUXEMBOURG PREBAC 2007 ...

EUROPEAN SCHOOL LUXEMBOURG PREBAC 2007
MATHEMATICS 5 PERIODS
This examination consists of 4 compulsory questions and 2 questions to be chosen
from 3 optional ones.
Marks
3 points
3 points
Compulsory question 1 : Analysis
2x
+ 2
f is the function defined on R by f ( x)
=
and F is its graph referred
x²
+ 2x
+ 2
to an orthonormal coordinate system
a) Find the domain of this function, points of intersection of F with the
coordinate axes, behaviour at the limits of the domain and the equation of
the asymptote.
b) Find f ′(x).
Deduce the coordinates of the extrema of f and the intervals
on which it is increasing and decreasing.
2 points
4 points
c) Sketch the graph of f.
d) Solve the equation ∫ f ( t ) dt = 0 for x.
x
−3
Marks
4 points
4 points
2 points
2 points
Compulsory question 2 : Analysis
The mass y kg of a radioactive substance is given at time t years by the
dy
equation y = f (t)
. f satisfies the differential equation . = −ky
, where k is a
dt
positive real constant.
a) Find the general solution of this differential equation.
b) The time T after which the mass of the substance has been reduced by
one half is called the half-life..
ln 2
i) Show that T = .
k
ii) The half-life of radium is 1600 years. Find k.
iii) The mass of radium is 5kg when t = 0 . Find the mass of radium
remaining after 100 years.

EUROPEAN SCHOOL LUXEMBOURG PREBAC 2007
Marks
3 points
3 points
3 points
4 points
MATHEMATICS 5 PERIODS
Compulsory question 3 : Geometry
In three dimensional space referred to an orthonormal coordinate
system, you are given the points
O(0, 0, 0) ; A ( 2, 0, 0)
; B ( 2, 2, 0)
; C ( 0, 2, 0)
and D ( 1, 1, 4)
a) Draw a sketch showing the coordinate axes and the pyramid
OABCD.
b) Find the area of the triangle ABD .
c) Find a cartesian equation of the plane ABD .
d) d is the line passing through B and D, and d’ the line passing
⎛ 3 ⎞
through the midpoint of CD with direction vector v =
⎜
−2
⎟
. Show that
⎜ 1 ⎟
⎝ ⎠
d and d’ are not coplanar.
Marks
Compulsory question 4 : Probability
Town council elections have been organised in a certain town. The results are
analysed according to age and to those electors who exercise their right to
vote (the voters). Participation rates are calculated and expressed in
number of voters
percentages:
100
number of electors ×
The electors are divided into three groups:
• The group A consisting of electors aged 35 years or less make up 38%
of the total number of electors.
• The group B consisting of those aged more than 35 and up to 60 years
make up 43% of the total number of electors.
• The group C consisting of those aged over 60 make up 19% of the total
number of electors.
The participation rate for each of these groups is as follows:
• Group A : 81%
• Group B : 84%
• Group C : 69%
4 points
3 points
3 points
3 points
a) Show that the probability that an elector chosen at random has actually
voted is 0,80.
b) A voting paper is chosen at random from those opened. Find the
probability that the vote was cast by an elector aged over 35 years.
c) Find the probability that, out of 10 electors aged over 60 years and
chosen at random, exactly 3 have not voted.
d) Consider n electors chosen at random from all electors in the town. Let
the probability that these n have all voted be p. Find the least value of n if
p is strictly less than 0.01.

EUROPEAN SCHOOL LUXEMBOURG PREBAC 2007
OPTIONAL QUESTIONS
MATHEMATICS 5 PERIODS
REMINDER : answer 2 of the three questions that follow
Marks
1 point
2 points
2 points
Optional question I: Analysis
x
⎪
⎧
2
Consider the function f given by f ( x)
= xe + a , x ≤ 0
⎨
⎪⎩ x(
x − 2) + 2 , x > 0
F is the graph of f in an orthonormal coordinate system.
a) Give the domain of definition of f.
b) Find the value of a for which f is continuous where x = 0.
c) Study the differentiability of f for x = 0.
Consider now a = 2
1 points
4 points
3 points
2 points
3 points
3 points
2 points
2 points
d) Find the point of intersection of F with the y axis. f has no
zeros.
e) Find the extrema of f and the intervals on which it is
increasing and decreasing.
f) Find out if there is a point of inflexion and, if so, find its
coordinates.
g) Examine the behaviour of f at the limits of its domain and
find any asymptotes
h) Sketch the graph of F.
x
x
2 2
i) Show that G( x)
= 2xe
− 4e
+ 2x
is a primitive of g ( x)
= xe 2
+ 2 .
j) For x ≤ 0 , calculate the area A (k)
defined by the graph F, the
y axis and the lines having equations y = 2 et x = k, k < 0 .
k) Calculate lim A(
k)
.
k→−∞
x

EUROPEAN SCHOOL LUXEMBOURG PREBAC 2007
MATHEMATICS 5 PERIODS
Marks
Optional question II : Probability
3 points
2 points
2 points
3 points
3 points
A
Two fair six-faced dice are thrown together.
i) Find the probability that the two scores add up to 8.
ii) Find the probability that one score divides exactly into the
other.
iii) Show that the probability of both i) and ii) holding is 1/12.
iv) If the dice are thrown 8 times, find the probability that both
i) and ii) hold on exactly 3 throws.
v) Find the most probable number of throws out of 8 on which
both i) and ii) hold. (i.e. the mode)
6 points
2 points
4 points
B A 6-faced die is weighted in such a way that the probability of
obtaining a score of n is given by P(n) = an + b (n = 1,2,3,4,5,6
and a and b are real numbers such that the expected mean score is 3).
i) Find a and b
ii) Given that a = -1/35 and b = 4/15 , show that the
probability of obtaining the expected score of 3 on this die
is 19/105.
iii) How many times must this die be thrown if the probability
of getting the expected score at least once is to be more
than 0.9?
Marks
3 points
4 points
3 points
3 points
4 points
4 points
4 points
Optional question III : Geometry
In three dimensional space referred to orthonormal axes, consider:
• points A ( 0; 2; 1)
; B ( −1; 0; 2)
and C ( 0; 1; 0)
• the line d defined by :
y
x − 2 = = 1−
z
2
a) Show that A,B and C are not collinear .
b) Find a cartesian equation of the plane π containing the points A, B
and C.
c) Show that d is parallel to π .
d) Find the distance from d to π .
e) Find a cartesian equation of the plane π ' containing d and B.
f) Find the parametric equations of the line of intersection of π and
π '.
g) Find to the nearest degree the acute angle between the planes π
and π '.