Page 58 - RMGO 2

58 Florea BADEA

Clasa a X-a

∗
1. Fie funct , iile f, g : R → R, f(x) = ax + b s , i g(x) = bx + a, a, b ∈ R . S˘a
se arate c˘a reprezent˘arile grafice ale funct , iilor fog −1 s , i gof −1 nu pot fi
perpendiculare.
Mihai Florea Dumitrescu, Potcoava

2. S˘a se determine suma numerelor z ∈ C, |z| = 1 care satisfac condit , ia
2
2
|z + z · z + z | = 1.
* * *
x
x
3. (a) S˘a se determine num˘arul real m, astfelˆıncˆat ecuat , ia 9 −m·3 −m+8 = 0
s˘a aib˘a o singur˘a solut , ie real˘a.
x
x
(b) S˘a se rezolve ecuat , ia 9(27 + 27 −x ) − 73(3 + 3 −x ) = 0.
Iuliana Tras , c˘a, Margineni

4. S˘a se demonstreze identitatea
2
k k+1 C k C k ·C k+1
1 + (k + 1) · C n +C n − n−k+1 · n+1 = (n − k + 1) · (k + 1) · n+1 n+1 .
k 2
C n k k+1 C k+1 (C n )
n+1
Octavian Marinescu-Ghemeci

Clasa a XI-a

1 1 1 1
ˆ
1. In mult , imea M 2 (R) se consider˘a matricele A = s , i B = .
0 1 0 3
∗
n
n
Ar˘atat , i c˘a matricea A + B este inversabil˘a pentru orice n ∈ N .
Mihai Florea Dumitrescu, Potcoava

471 600
3
2. Determinat , i matricea A ∈ M 2 (R), s , tiind c˘a A = , tr (A) = 9
75 96
2
s , i tr (A ) = 69.
Daniela Haret, Br˘aila, Supliment G.M.-B nr. 12/2013

x ln(x + e), x ∈ (0, ∞)
∗
3. Consider˘am funct , iile f, g, h : R → R, f(x) = ,
[2x] x ∈ (−∞, 0)

x ln x, x ∈ (0, ∞) 2x sin x
g(x) = , h(x) = , unde [x] reprezint˘a partea
[x] + 1 x ∈ (−∞, 0) 1 + x 2
ˆıntreag˘a a num˘arului x.
(a) Studiat , i existent , a limitelor funct , iilor f, g, h ˆın x 0 = 0.
(b) Calculat , i lim (f(x) − g(x)) s , i lim (f(x) − g(x)).
x→∞ x→−∞