The Islamic Institution for
Math Department
Education & Teaching Mid-Year Exam
Al-Mahdi Schools
February 2010
Subject: Mathematics
Grade: 11th Scientific Duration: 150 minutes
I. ( 2 points)
In the table below, one of the answers is correct. Write the letter corresponding to the correct
Answer and justify.
Answers
Questions
A B C D
2 2 2 2 2
1) Knowing that: cos , then cos =
2
2
4 2 8 2 2 4 4
In an orthonormal system consider the
equation of circle (C): (1; 2) (1; 2) (1; 2) (1; 2)
2)
2x y 4 x 8 y 4 0. r 3 r 3
2 2
r= 3 r= 3
The center and the radius r of (C) are:
2x 3 2 2
3) lim 0 -
x
9x 1
2
3 3
II. (5.5 Points)
1. Calculate the derivative of the following functions:
3
a) f ( x ) x ² 1 defined on ]– ; – 1[ ]– 1 ; +[
x² 1
b) f (x) = sin² (x²) defined on IR.
2 x 2 3x 5
2. Determine the domain of the following function f ( x ) .
x2 x 2
3. Given the function f defined on IR by :
x2 4 x 3
x 2 3x 2 si x 1
f ( x ) k 2 k si x 1 ; k is a real constant
1 x
si x 1
1 x
a-Calculate lim f ( x ) and lim f ( x ) .
x 1 x 1
x 1 x 1
b- Find k so that f is continuous at x = 1
1
III. (3 points)
Consider the second degree equation in x (E): (m 1)x2 2(m 2)x + m 7 = 0 ; m IR .
1. Calculate m knowing that one of the roots is 4, and deduce the other root.
2. Calculate m so that (E) admits two distinct roots.
( x1 1)( x2 1)
3. Let F ,with x1 and x2 are the roots of (E) when they exist, without
x12 x2 2
2(1 m)
calculate x1 and x2 verify that : F .
m2 1
4. On an axis x’ox place the points M 1 ( x1 ) and M 2 ( x2 ) . Calculate m so that M 1 and M 2 are
3
symmetric with respect to the point I of abscissa .
2
IV. (3 points)
1. Prove that : sin 2 3a.cos 2 a cos2 3a.sin 2 a 2 cos 2a.sin 2 2a .
sin 2 3a.cos 2 a cos 2 3a.sin 2 a
2. Simplify: E .
cos3 2a cos 2 2a
E 2 tan a
3. Verify that: .
2 tan a 1 tan 2 a
V. (6 points)
u0 = 1
Define the sequence un by: 1
un+1 = 2 un + 2n - 1
1
1. Prove that u1 = − , then calculate u2 and u3. Is the sequence (un) arithmetic or geometric?
2
2. Let vn = un − 4n + 10. Calculate v0, v1, v2, v3.
3. Prove that the sequence (vn) is geometric; calculate its common ratio r and first term v0.
4. Deduce vn as a function of n.
n
1
5. Prove that an expression of un as a function of n is given by: un = 11 × + 4n − 10.
2
6. Tn = v0 + v1 + v2 + · · · + vn. Give an expression of Tn as a function of n.
2
VI. (2.75 points)
Let f be a function defined on IR. Let (G) be
the representative curve of f in an
orthonormal system O; i, j .
f(0) = 2, f(1) = 0, f(1) = 4,
lim f ( x ) , et lim f ( x ) . The
x x
figure at the right (C) is the curve of the
function f’, the derivative of f . using (C) :
f ( x ) f (0)
1. Determine: lim .
x 0 x
2. Write the equation of tangent (T) to
(G) at the point of abscissa 0.
3. Draw the table of variation of f.
4. Deduce the sign of f on IR.
VII. (7.75 points)
Let g be a function defined on IR by g ( x) x 4 6 x 2 8 x 20 , designate by (C) its representative
curve in an orthonormal system O; i; j .
1. Determine the limits of g at the open boundaries of its domain.
2. Show that x 3 3x 2 ( x 1) 2 ( x 2)
3. Calculate g '( x) , then deduce it's sign.
4. Draw the table of variations of g.
5. Prove that the equation g ( x ) 0 , admits on IR two roots and , and verify that
3 2 and that 2 1 .
6. Prove that g admits two points of inflection, to be determined.
7. Trace (C).
8. Let h( x) g x .
Explain how to deduce the graphical representation (C ') of h from (C) and draw (C ') in
the system O; i; j .
GOOD WORK
