Iman High School – Saida Scholastic Year: 2023-2024
Department: Mathematics Date: 16/01/2024
Midterm Test Time: 120 min.
Name: _________________ Grade: [ 12 LS ]
Section/s: ☒A ☒B
Note: This test is composed of three questions on two pages.
A: ☒Write your name on all the papers D: ☒You must hand in all papers (questions, answers & draft)
B: ☒Don't use red ink E: ☒Using calculators is allowed
C: ☒Borrowing is prohibited as well as using the correction pen F: ☒Graph paper is needed
I. (4 points)
In the table below, only one among the proposed answers to each question is correct.
Choose, with justification, the correct answer.
Questions Answers
A B C
1 The function 𝑔 defined on ] − ∞, −1[∪]1, +∞[
𝑥−1 neither even nor
by 𝑔(𝑥) = 𝑙𝑛( 𝑥+1 ) is even odd
odd
2 The number of ways [to arrange 4 Math books, 2
Chemistry books & 5 Arabic books on a shelf, if
34560 5760 240
the books of the same type are grouped together]
is:
3
3 2
The equation: 𝐶𝑛−1 + 𝐶𝑛−1 = 𝑛(𝑛 − 1) has two solutions one solution three solutions
4 Consider the function 𝑓 defined on ]0, +∞[ by
𝑓(𝑥) + 𝑙𝑛2 𝑓(𝑥) + 𝑙𝑛16 2𝑓(𝑥)
𝑓(𝑥) = 4𝑙𝑛(3𝑥). Then 𝑓(2𝑥) =
II. (10 points)
Part A: Let 𝑔 be a function defined over ]0; +∞[ by 𝑔(𝑥) = 𝑥 3 + 2 ln 𝑥 − 3.
1) Calculate lim 𝑔(𝑥) and lim 𝑔(𝑥).
𝑥→0 𝑥→+∞
2) Determine 𝑔′(𝑥) and set the table of variations of 𝑔.
3) Prove that the equation 𝑔(𝑥) = 0 has a unique root 𝛼 .
4) Study, according to the values of x, the sign of the function 𝑔 .
ln 𝑥 − 1
Part B: Given a function 𝑓 defined over ]0; +∞[ by 𝑓(𝑥) = 𝑥 + 1 − .
𝑥2
Let (C) be its representative curve in an orthonormal system (O; 𝑖⃗, 𝑗⃗).
1) Calculate lim 𝑓(𝑥). Deduce an asymptote to (C).
𝑥→0
2) a. Calculate lim 𝑓(𝑥).
𝑥→+∞
b. Show that the line (D) of equation: 𝑦 = 𝑥 + 1 is an asymptote to (C).
c. Study the relative position of (C) and (D).
𝑔(𝑥)
3) Verify that 𝑓 ′ (𝑥) = then draw the table of variation of 𝑓.
𝑥3
4) Draw (D) and (C). (take 𝛼 = 1.35).
III. (11 points)
Consider the function 𝑓 defined over ℝ by 𝑓(x) = 2x + 1 – xex-1.
Designate by (C) its representative curve in an orthonormal system (𝑂, 𝑖⃗, 𝑗⃗) .
1) Calculate 𝑙𝑖𝑚 𝑓(𝑥) & 𝑙𝑖𝑚 𝑓(𝑥)
𝑥→−∞ 𝑥→+∞
2) a. Show that the straight line (d) of equation y = 2x + 1 is an asymptote to (C).
b. Study the relative position of (C) and (d).
3) The table below is the table of variations of the function 𝑓′ the derivative of 𝑓.
x - -2 1 +
𝑓 ‘‘(𝑥) + 0 – –
-3
2+e
𝑓 ’(𝑥) 2 0 -
a. Show that 𝑓 admits a point of inflection I whose coordinates are to be determined.
b. Study the sign of 𝑓′ and draw the table of variations of 𝑓.
−1
4) Show that the equation 𝑓(𝑥) = 0 admits two roots and and verify that–1 < < and 1 < < 2.
2
5) Draw (d) and (C).
6) Determine the image of the interval [– 1 , 2] by the function 𝑓.
x-1
7) F is the function defined on ℝ by F(x) = x2 + x + (ax + b) e
Calculate the real numbers a and b so that F is an anti-derivative of 𝑓.
GOOD WORK
