The Islamic Institution For In his name
Education & Teaching Mid-year term Mathematics Department
February 2013
Subject: Mathematics Mark : ------ /
Grade 11(S) Duration: 150 minutes
I. ( points)
One, among the following answers, is true. Choose it after you justify your answer.
Question A B C
1 a & b are two real numbers such that :
3a 3b 8 8
8 x2 x 1 0
ab 1 x2 x 1 0 3x 8 x 3 0
2
3
3
then a & b are the roots of the quadratic
equation:
2 3 3 3x 6x
If h( x) then h '(x)=
x2 1 ( x 2 1) x 2 1 ( x 2 1) x 2 1 ( x 2 1) x 2 1
3 The value of m for which
f ( x) mx 3 (m 6) x 7 0 admits an extremum 7 1 9
1 6 2 2
at x0= is:
3
II. ( points)
Consider the second degree equation ( E ) : x 2 2mx m 2 m 5 0 with m is a real parameter.
1. Calculate or ' and discuss according to the values of m the existence of the roots of
Equation (E).
2. Without calculating the roots x1 & x2 ,determine m so that the roots of the equation (E)
Verify the relation ( x1 1)( x2 1) 12
III.
1. Calculate the following limits:
x 2 2x 3
a. lim 2
x 1 3 x 2 x 1
1 sin 1
b. lim
0
(cos2 x cos x)
c. lim
x 0 cos 2 x 3 cos x 2
x2 a2
if x 0
x2
2. Given f ( x)
x a 1 if x 0
2
Calculate a so that f is continuous at x0 = 0.
1
IV. ( points)
Un 3
Consider the sequences (Un) and (Vn) defined by U0= 1, Un+1= and Vn= Un + 3.
2
1. Calculate V0 , V1 and V2.
2. Prove that (Vn) is a geometric sequence whose common ratio is to be determined.
3. Express Vn and Un in terms of n.
n
4. Calculate S n Vi then deduce S ' n U 1 U 2 ........... U n .
i 0
V. ( points)
Given the circle (C) of equation: x 2 2 x y 2 4 y 4 0
1. Determine the center I and the radius R of circle (C).
2. Does line of equation ( y=-x) cuts circle ( C) ?If yes , determine the coordinates of the points of
intersection .
VI. ( points)
3
1. Let x be a real number such that x ]; [ with sinx = -0.8.
2
a. Calculate cosx and sin2x.
b. Find tan2x.
5
2. Given A ] , [ and (tanA. tanB) = -1 and (A + B) = .
2 6
3
a. Show that tan(A+B)=
3
3
b. Show that tan A + tan B= 2
3
3. Given a direct triangle ABC.
a. Show that sin(2A)+sin(2B)+sin(2C)= 4 sin A .sin B . sin C
cos A cos B cos C
b. Deduce that : 2
sin B. sin C sin A. sin C sin A. sin B
VII. ( points) y
4
Consider the curve (C ) of a function f
3
(C)
2 2
-2 -1 0 1 2 3 4 x
-1
Note: (C) cuts x'Ox at 0 and at 3
(C) admits a horizontal tangent at x = 0 and at x = 2 .
1. Set up the table of variations of f .
2. Solve graphically: f(x) <0; f(x)≥ 0 ; f ' (x) =0 .
3. Discuss graphically according to the values of
m the number of roots of the equation f(x)= m .
4. One of the graphs (H) or (G) below represents the
Derivative function f ' of f .
a. Determine the graph of f '. Justify your answer.
b. Write the equation of the tangent to (C ) at x= 1 .
y y
6
(G)
4
5
4 3
3 2
2 1
1
(H)
-2 -1 0 1 2 3 4 x
-1 0 1 2 3 x -1
-1
-2
-2
-3
-3
5. Assume that f is defined by: f(x) = ax3+bx2+cx .
Use (C ) to calculate a, b and c .
VIII. ( points)
1 4 3 2
Consider the function f defined over IR by f ( x) x x 2x 2
4 2
and let (C) be its representative curve.
1. Determine the limits of f at and .
2. Verify that f ( x) x 3 3 x 2 ( x 2)( x 1) 2
3. Set up the table of variation of f.
4. Deduce that f (x) =0 admits two distinct roots & .
5. Show that f admits two inflection points to be determined.
6. Given that ] - 3;-2.5[ & ] 1;0.5[ ;
Draw (C) in an orthonormal system.
7. Deduce (C') the graph of g where g(x) = f (x) .
GOOD WORK!
