fskilnik wrote:
GMATH practice exercise (Quant Class 3)
If \({\left( {2x + y - z} \right)^2} + {\left( {x - y} \right)^2} + {\left( {z - 3} \right)^2} = - {\left( {3y - z} \right)^2}\) , what is the value of \(3x + 2y + z\) ?
(A) 8
(B) 9
(C) 10
(D) 11
(E) 12
\(? = 3x + 2y + z\)
\({\left( {2x + y - z} \right)^2} + {\left( {x - y} \right)^2} + {\left( {z - 3} \right)^2} = - {\left( {3y - z} \right)^2}\,\,\,\,\,\left( * \right)\)
\(\left. \matrix{\\
{\left( {2x + y - z} \right)^2} + {\left( {x - y} \right)^2} + {\left( {z - 3} \right)^2}\,\,\, \ge 0\,\, \hfill \cr \\
- {\left( {3y - z} \right)^2}\,\, \le 0 \hfill \cr} \right\}\,\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\,{\left( {2x + y - z} \right)^2} + {\left( {x - y} \right)^2} + {\left( {z - 3} \right)^2} = 0\,\,\,\,\,\,\left[ { = - {{\left( {3y - z} \right)}^2}} \right]\)
\({\left( {2x + y - z} \right)^2} + {\left( {x - y} \right)^2} + {\left( {z - 3} \right)^2} = 0\,\,\,\,\,\,\, \Rightarrow \,\left\{ \matrix{\\
\,x - y = 0\,\,\,\,\, \Rightarrow \,\,\,\,\,x = y\,\,\, \hfill \cr \\
\,z - 3 = 0\,\,\,\,\, \Rightarrow \,\,\,\,\,z = 3\,\,\,\, \hfill \cr \\
\,2x + y - z = 0\,\,\,\,\mathop \Rightarrow \limits^{{\rm{both}}\,{\rm{above}}} \,\,\,\,\,3x - 3 = 0\, \hfill \cr} \right.\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\,\left( {x,y,z} \right) = \left( {1,1,3} \right)\)
\(? = 3x + 2y + z = 3\left( 1 \right) + 2\left( 1 \right) + 3 = 8\)
The correct answer is (A).
We follow the notations and rationale taught in the
GMATH method.
Regards,
Fabio.