348. Subiectele de la BAC 2012 /Mate-Info.

by Virgil Nicula, Jul 5, 2012, 9:25 PM

Bacalaureat 2012 - mate-info.

Subiect I.

1. Calculati modulul numarului complex $(1+i)^2$ . Proof

2. Determinati coordonatele punctelor de intersectie ale graficelor functiilor $\left\{\begin{array}{c} f(x)=x^2+2x\\\ g(x)=-x-2\end{array}\right\|$ , unde $x\in\mathbb R$ . Proof

3. Rezolvati inecuatia $2^{x+1}\le 4$ , unde $x\in\mathbb R$ . Proof

4. Calculati probabilitatea ca alegand la intamplare una dintre submultimile cu trei elemente ale multimii $A=\{1,2,3,4,5\}$ si care sa fie termenii consecutivi ale unei progresii aritmetice. Proof

5. Se considera vectorii $u=\overrightarrow i-2\overrightarrow j$ and $v=a\overrightarrow i-\overrightarrow j$ . Determinati $a\in\mathbb R$ pentru care $u\cdot v=3$ . Proof

6. Calculati cosinusul unghiului $A$ al triunghiului $ABC$ in care $AB=c=4\ ,\ AC=b=5\ ,\ BC=a=7$ . Proof


Subiect II.


1. Se considera sistemul omogen $\left\{\begin{array}{ccc} 2x+y+3z & = & 0\\\\ x+2y+3z & = & 0\\\\ x+y+mz & = & 0\end{array}\right\|$ , unde $m\in\mathbb R$ .

a) Sa se calculeze determinantul matricii sistemului. Proof

b) Determinati $m\in\mathbb R$ ca sistemul sa aiba solutie unica. Proof

c) In cazul $m=2$ determinati solutia $\left(x_0,y_0,z_0\right)$ a sistemului pentru care $x_0>0$ si $x_0^2+y_0^2+z_0^2=3$ . Proof

2. Se considera matricea $A=\left(\begin{array}{cc} 3 & -2\\\ 3 & -2\end{array}\right)\in \mathcal M_2(\mathbb R)$ si multimea $\mathcal G=\left\{\left| X(p)=I_2+pA\right|p\in\mathbb R\ ,\ p\ne -1\right\}$ .

a) Aratati ca $X(p)\cdot X(q)\in \mathcal G$ pentru orice $p\ne -1$ si $q\ne -1$ . Proof

b) Admitem ca $(\mathcal G,\cdot )$ este grup comutativ cu element neutru $X(0)$ . Gasiti inversul elementului $X(p)$ in acest grup. Proof

c) Rezolvati ecuatia $X^3(p)=I_2+7A$ , unde $p\ne -1$ . Proof


Subiect III.


1. . Se considera functia $f(x)=x^3-12x$ , unde $x\in\mathbb R$.

a) Aratati ca functia $f$ este crescatoare pe $[2,\infty )$ . Proof

b) Calculati $\lim_{x\to\infty}\frac {e^x}{f(x)}$ . Proof

c) Determinati $a\in\mathbb R$ pentru care ecuatia $f(x)=a$ are trei solutii reale distincte. Proof

2. Se considera functia $f(x)=\frac {2x+3}{x+2}$ , unde $x\in \mathbb R\ ,\ x>-1$ .

a) Sa se arate ca orice primitiva a functiei $f$ este strict crescatoare pe $(-1,\infty )$ . Proof

b) Calculati $\int_0^1\frac {f(x)}{x+1}\ \mathrm{dx}$ . Proof

c) Calculati $\lim_{x\to\infty}\frac 1x\cdot\int_x^{2x}f(t)\ \mathrm{dt}$ . Proof
This post has been edited 18 times. Last edited by Virgil Nicula, Nov 17, 2015, 1:35 PM

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