🎬 CLIPURI
\[
\textbf{1.}
\text{Dacă } x_1,x_2 \text{ sunt soluțiile reale ale ecuației }
x^2+3x+1=0,
\text{atunci valoarea expresiei }
\left(\frac{x_1}{x_2+1}\right)^2+
\left(\frac{x_2}{x_1+1}\right)^2
\text{ este ?}
\]
Rezolvare
\[
\textbf{2. }\;
\text{Fie } a,b,c,d\in\mathbb{N}^* \text{ astfel încât }
\log_a b=\frac{4}{3} \text{ și } \log_c d=\frac{5}{6}.
\text{Dacă } c-a=37,\text{ atunci determinați } b-d.
\]
\[
\begin{aligned}
\textbf{Rezolvare.}\\
\log_a b=\frac{4}{3} &\Rightarrow a^{\frac{4}{3}}=b \Rightarrow a^4=b^3 \Rightarrow a \text{ cub perfect}.\\
\text{Fie } a=x^3 &\Rightarrow (x^3)^4=b^3 \Rightarrow x^{12}=b^3 \Rightarrow b=x^4.\\
\log_c d=\frac{5}{6} &\Rightarrow c^{\frac{5}{6}}=d \Rightarrow c^5=d^6.\\
\text{Fie } c=y^6 &\Rightarrow (y^6)^5=d^6 \Rightarrow y^{30}=d^6 \Rightarrow d=y^5.\\
c-a=37 &\Rightarrow y^6-x^3=37,\quad x,y\in\mathbb{N}^*.\\
&2^6-3^3=64-27=37 \Rightarrow y=2,\; x=3.\\
&b=3^4=81,\qquad d=2^5=32.\\
&b-d=81-32=49.
\end{aligned}
\]
3. Fie sistemul de ecuații
\[
\left\{
\begin{aligned}
2mx+y+(m+1)z&=2m+1\\
(m+2)x+(m+1)y+(m+2)z&=2\\
3mx+y+(2m+1)z&=1
\end{aligned}
\right.
\]
\(m\in\mathbb{R}\). Determinați \(m\in\mathbb{R}\) pentru care sistemul este incompatibil.
\[
\begin{gathered}
\textbf{Rezolvare}
\\[8pt]
A=
\begin{pmatrix}
2m&1&m+1\\
m+2&m+1&m+2\\
3m&1&2m+1
\end{pmatrix}
\\[10pt]
\text{sistem incompatibil }\Leftrightarrow
\begin{cases}
\det A=0\\
\Delta_c\neq0
\end{cases}
\\[12pt]
\det A=
\begin{vmatrix}
2m&1&m+1\\
m+2&m+1&m+2\\
3m&1&2m+1
\end{vmatrix}
=
\begin{vmatrix}
m-1&1&m+1\\
0&m+1&m+2\\
m-1&1&2m
\end{vmatrix}
\\[12pt]
=(m-1)
\begin{vmatrix}
0&m+1&m\\
1&m+1&1\\
1&1&2m
\end{vmatrix}
=
(m-1)
\begin{vmatrix}
0&0&-m\\
1&m+1&1\\
1&1&2m
\end{vmatrix}
\\[12pt]
=(m-1)m
\begin{vmatrix}
1&m+1\\
1&1
\end{vmatrix}
=(m-1)m(m+1)
\\[12pt]
\det A=0
\Rightarrow m\in\{-1,0,1\}
\\[14pt]
d_p=
\begin{vmatrix}
2m&1\\
m+2&m+1
\end{vmatrix}
=2m(m+1)-(m+2)
=2m^2+2m-m-2
=2m^2+m-2
\\[10pt]
d_p\neq0 \text{ pentru } m\in\{-1,0,1\}
\\[12pt]
r=2,\qquad s=3 \Rightarrow s-r=1
\\[12pt]
\Delta_c=
\begin{vmatrix}
2m&1&2m+1\\
m+2&m+1&2\\
3m&1&1
\end{vmatrix}
=
\begin{vmatrix}
2m&1&2m\\
m+2&m+1&1-m\\
3m&1&0
\end{vmatrix}
\\[12pt]
=
\begin{vmatrix}
0&1&2m\\
2m+1&m+1&1-m\\
3m&1&0
\end{vmatrix}
\\[12pt]
=
-\begin{vmatrix}
2m+1&1-m\\
3m&0
\end{vmatrix}
+2m
\begin{vmatrix}
2m+1&m+1\\
3m&1
\end{vmatrix}
\\[12pt]
=3m(1-m)+2m(2m+1-3m^2)
\\[8pt]
=3m-3m^2+2m(-3m^2-m+1)
\\[8pt]
=-6m^3-3m^2+m
\\[8pt]
=m(-6m^2-3m+1)
\\[12pt]
\Delta_c\neq0
\Rightarrow m\neq0
\\[10pt]
\boxed{m\in\{-1,1\}}
\end{gathered}
\]
