Given the following system of linear equations dependent on the real parameter $m$ : $$\left. \begin{array} { l }
x - 2 m y + z = 1 \\
m x + 2 y - z = - 1 \\
x - y + z = 1
\end{array} \right\}$$ a) (2 points) Discuss the system as a function of the values of $m$. b) ( 0.5 points) Solve the system for the value $m = \frac { 1 } { 2 }$
Let the function $f ( x ) = \left\{ \begin{array} { l l l } x ^ { 3 } e ^ { - 1 / x ^ { 2 } } & \text { if } & x \neq 0 \\ 0 & \text { if } & x = 0 \end{array} \right.$ a) (1 point) Study the continuity and differentiability of $f ( x )$ at $x = 0$. b) ( 0.5 points) Study whether $f ( x )$ presents any type of even or odd symmetry. c) (1 point) Calculate the following integral: $\int _ { 1 } ^ { 2 } \frac { f ( x ) } { x ^ { 6 } } d x$
With a laser device located at point $\mathrm { P } ( 1,1,1 )$ it has been possible to follow the trajectory of a particle that moves along the line with equations $r \equiv \left\{ \begin{array} { l } 2 x - y = 10 \\ x - z = - 90 \end{array} \right.$. a) ( 0.5 points) Calculate a direction vector of $r$ and the position of the particle when its trajectory intersects the plane $z = 0$. b) (1.25 points) Calculate the closest position of the particle to the laser device. c) ( 0.75 points) Determine the angle between the plane with equation $x + y = 2$ and the line $r$.
According to the National Institute of Statistics, during the last quarter of 2020, the percentage of women belonging to the set of Boards of Directors of companies that make up the Ibex-35 was 27.7 \%. Ten of these board members were gathered. a) ( 0.75 points) Find the probability that half were women. b) ( 0.75 points) Calculate the probability that there was at least one man. c) (1 point) Determine, approximating by a normal distribution, the probability that at a congress of two hundred board members of these companies there would be at least 35 \% female representation.
Three cousins, Pablo, Alejandro and Alicia, are going to share a prize of 9450 euros in direct proportion to their ages. The sum of the ages of Pablo and Alejandro exceeds by three years twice the age of Alicia. Furthermore, the age of the three cousins together is 45 years. Knowing that in the distribution of the prize Pablo receives 420 euros more than Alicia, calculate the ages of the three cousins and the money each one receives from the prize.
Let the function $f ( x ) = \frac { x } { x ^ { 2 } + 1 }$ a) ( 0.5 points) Check whether $f ( x )$ satisfies the hypotheses of Bolzano's Theorem on the interval $[ - 1,1 ]$ b) (1 point) Calculate and classify the relative extrema of $f ( x )$ in $\mathbb { R }$. c) (1 point) Determine the area between the graph of the function $f ( x )$ and the x-axis on the interval $[ - 1,1 ]$.
Let the plane $\pi \equiv x + y + z = 1$, the line $r _ { 1 } \equiv \left\{ \begin{array} { l } x = 1 + \lambda \\ y = 1 - \lambda \\ z = - 1 \end{array} , \lambda \in \mathbb { R } \right.$ and the point $P ( 0,1,0 )$. a) ( 0.5 points) Verify that the line $r _ { 1 }$ is contained in the plane $\pi$ and that the point P belongs to the same plane. b) ( 0.75 points) Find an equation of the line contained in the plane $\pi$ that passes through P and is perpendicular to $r _ { 1 }$. c) (1.25 points) Calculate an equation of the line, $r _ { 2 }$, that passes through P and is parallel to $r _ { 1 }$. Find the area of a square that has two of its sides on the lines $r _ { 1 }$ and $r _ { 2 }$.
From a basket with 6 white hats and 3 black hats, one is chosen at random. If the hat is white, a handkerchief is taken at random from a drawer that contains 2 white, 2 black and 5 with white and black checks. If the hat is black, a handkerchief is chosen at random from another drawer that contains 2 white handkerchiefs, 4 black and 4 with white and black checks. It is requested: a) (1 point) Calculate the probability that the handkerchief shows some color that is not the color of the hat. b) (0.5 points) Calculate the probability that in at least one of the accessories (hat or handkerchief) the color black appears. c) (1 point) Calculate the probability that the hat was black, knowing that the handkerchief was checked.