Given the system of equations $$\left\{ \begin{array} { l }
x + m y = 1 \\
- 2 x - ( m + 1 ) y + z = - 1 \\
x + ( 2 m - 1 ) y + ( m + 2 ) z = 2 + 2 m
\end{array} \right.$$ it is requested: a) (2 points) Discuss the system as a function of the parameter $m$. b) ( 0.5 points) Solve the system in the case $m = 0$.
a) (1.5 points) In a laboratory experiment, 5 measurements of the same object have been made, which gave the following results: $\mathrm { m } _ { 1 } = 0.92 ; \mathrm { m } _ { 2 } = 0.94 ; \mathrm { m } _ { 3 } = 0.89 ; \mathrm { m } _ { 4 } = 0.90$; $\mathrm { m } _ { 5 } = 0.91$. The result will be taken as the value of x such that the sum of the squares of the errors is minimized. That is, the value for which the function $E ( x ) = \left( x - m _ { 1 } \right) ^ { 2 } + \left( x - m _ { 2 } \right) ^ { 2 } + \left( x - m _ { 3 } \right) ^ { 2 } + \left( x - m _ { 4 } \right) ^ { 2 } + \left( x - m _ { 5 } \right) ^ { 2 }$ reaches its minimum. Calculate this value x . b) (1 point) Apply the integration by parts method to calculate the integral $\int _ { 1 } ^ { 2 } x ^ { 2 } \ln ( x ) d x$, where ln denotes the natural logarithm.
Given the planes $\pi _ { 1 } \equiv 4 x + 6 y - 12 z + 1 = 0 , \pi _ { 2 } \equiv - 2 x - 3 y + 6 z - 5 = 0$, it is requested: a) (1 point) Calculate the volume of a cube that has two of its faces in these planes. b) (1.5 points) For the square with consecutive vertices ABCD , with $\mathrm { A } ( 2,1,3 )$ and $\mathrm { B } ( 1,2,3 )$, calculate the vertices C and D , knowing that C belongs to the planes $\pi _ { 2 } \mathrm { and } \pi _ { 3 } \equiv x - y + z = 2$.
60\% of sales in a large department store correspond to items with reduced prices. Customers return 15\% of the discounted items they purchase, a percentage that decreases to 8\% if the items were purchased at full price. a) (1.25 points) Determine the overall percentage of returned items. b) ( 1.25 points) What percentage of returned items were purchased at reduced prices?