IN HIS NAME
The Islamic Institution For
Math Department
Education & Teaching
February 2015
Al-Mahdi Schools
Subject: Mathematics Mark: 30 points
Class: Grade 11 - Scientific Mid-Year Exam Duration: 150 minutes
I- (6 points)
Remark: The three parts of this question are independent.
1) Determine the derivative function of each of the following functions.
a) g(x) = ( ) .
b) h(x) = .
√
3 cos x 1
2) Consider the function f defined, on , by: f ( x ) .
x2 2
4 2
a) Show that: f (x) 2 .
x 2
2
x 2
b) Deduce lim f ( x ) .
x
3) Given the function f defined, on , by: f(x) = , where b is a real number.
√
{
1
a) Prove that lim f ( x ) lim f ( x ) .
x 4 x 4 6
b) Find the value of b so that f is continuous at x = 4.
II- (2.5 points)
Consider the second degree equation (E): x 2 2(m 1) x m 2 2 0 , where m is a real parameter.
1) Determine m so that "–3" is a root of the equation (E).
2) Determine the set of values of m so that the equation (E) admits two distinct real roots x and x.
m m
3) Determine the set of values of m if 0.
x ' x"
III- (4 points)
U 0 1
Let (Un) be a sequence defined, for every natural number n, by: 1 .
U n 1 U n n
2
1) Calculate U1 and U2.
2) Show that the sequence (Un) is neither arithmetic nor geometric.
3) Calculate Un+1 – Un. Deduce that the sequence (Un) is strictly increasing.
4) Consider the sequence (Vn ) defined by Vn = Un+1 – Un.
a) Verify that (Vn) is a geometric sequence whose common ratio r and first term V0 are to be
determined.
b) Express Vn in terms of n, then deduce the value of V10.
c) Calculate, in terms of n, the sum S V0 V1 V2 ...... Vn .
1
IV- (3 points)
Consider, in an orthonormal system of axes ( ⃗ ⃗), the circle (C) of equation: x 2 y 2 4 x 6 y 0 .
1) Determine the center I and the radius R of the circle (C).
2) Let B(1, –2). Determine the distance BI, then deduce the position of point B with respect to (C).
3) Given the line (d m ) : mx y m 2 0 , where m is a real parameter.
a) Verify that (d m ) passes through B.
b) For which values of m is (d m ) tangent to (C)?
c) Deduce the equations of the tangent lines to (C) that pass through B.
V- (5 points)
Remark: The four parts of this question are independent.
1) Calculate, without using the calculator, ( ) ( ) ( ) ( ).
( )
2) Show that tan(2x) – tan(x) = .
( )
√
3) Given: sin(x) = , where . Calculate, without using the calculator, cos(2x), then verify
that cos(4x) = sin(x)
4) Let ABCD be a direct square of center O. Give a measure of each of the following oriented angles:
(⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗), (⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗), and (⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗).
VI- (2 points)
3x 2 ax b
Let f be a function defined by: f ( x ) , where a and b are two real numbers. Let (C) be the
x2 1
representative curve of f in an orthonormal system ( ⃗ ⃗).
A and B are two points such that A(0, 3) and B(–1, 1).
1) Write f (x) in terms of a and b.
2) Find the values of a and b, knowing that the straight-line (AB) is tangent to (C) at A such that A (C).
VII- (7.5 points)
Let f be a function defined, on , by: f(x) = x3 – 6x2 + 9x + 1. Let (C) be the representative curve of f in an
orthonormal system ( ⃗ ⃗).
1) Determine the limits of f(x) at the boundaries of its domain of definition.
2) Set up the table of variations of f.
3) Show that the equation f(x) = 0 has a unique root . Verify that –0.2 < < –0.1 .
4) Write an equation of (T), the tangent to (C) at the point of abscissa 0.
5) Prove that (C) has an inflection point I of abscissa 2. Find the coordinates of I.
6) Draw (T) and (C).
7) Solve, graphically, the inequality f(x) > 0.
8) Let g(x) = f( | x | ).
a) Verify that g is an even function.
b) Deduce the construction of (G), the representative curve of g, in the same previous system ( ⃗ ⃗).
GOOD WORK
