Three brothers want to divide equally a total of 540 shares valued at 1560 euros, which correspond to three companies A, B and C. Knowing that the current stock market value of share A is three times that of B and half that of C, that the number of shares of C is half that of B, and that the current stock market value of share B is 1 euro, find the number of each type of share that corresponds to each brother.
Let the line $r \equiv \left\{ \begin{array} { l } - x - y + z = 0 \\ 2 x + 3 y - z + 1 = 0 \end{array} \right.$ and the plane $\pi \equiv 2 x + y - z + 3 = 0$. It is requested:\ a) ( 0.75 points) Calculate the angle formed by $r$ and $\pi$.\ b) (1 point) Find the symmetric point of the intersection of line $r$ and plane $\pi$ with respect to the plane $z - y = 0$.\ c) ( 0.75 points) Determine the orthogonal projection of line $r$ onto plane $\pi$.
The lifespan of individuals of a certain animal species has a normal distribution with a mean of 8.8 months and a standard deviation of 3 months.\ a) (1 point) What percentage of individuals of this species exceed 10 months? What percentage of individuals have lived between 7 and 10 months?\ b) (1 point) If 4 specimens are randomly selected, what is the probability that at least one does not exceed 10 months of life?\ c) ( 0.5 points) What value of $c$ is such that the interval ( $8.8 - c , 8.8 + c$ ) includes the lifespan (measured in months) of $98 \%$ of the individuals of this species?
Consider the following system of equations depending on the real parameter a: $$\left. \begin{array} { l }
a x - 2 y + ( a - 1 ) z = 4 \\
- 2 x + 3 y - 6 z = 2 \\
- a x + y - 6 z = 6
\end{array} \right\}$$ a) (2 points) Discuss the system according to the different values of $a$.\ b) ( 0.5 points) Solve the system for $a = 1$.
Consider the function $$f ( x ) = \left\{ \begin{array} { l l l }
\operatorname { sen } x & \text { if } & x < 0 \\
x e ^ { x } & \text { if } & x \geq 0
\end{array} \right.$$ a) ( 0.75 points) Study the continuity and differentiability of $f$ at $x = 0$.\ b) (1 point) Study the intervals of increase and decrease of $f$ restricted to ( $- \pi , 2$ ). Prove that there exists a point $x _ { 0 } \in [ 0,1 ]$ such that $f \left( x _ { 0 } \right) = 2$.\ c) (0.75 points) Calculate $\int _ { - \frac { \pi } { 2 } } ^ { 1 } f ( x ) d x$.
Let the planes $\pi _ { 1 } \equiv x + y = 1$ and $\pi _ { 2 } \equiv x + z = 1$.\ a) ( 1.5 points) Find the planes parallel to plane $\pi _ { 1 }$ such that their distance to the origin of coordinates is 2.\ b) ( 0.5 points) Find the line that passes through the point ( $0,2,0$ ) and is perpendicular to plane $\pi _ { 2 }$.\ c) ( 0.5 points) Find the distance between the points of intersection of plane $\pi _ { 1 }$ with the $x$ and $y$ axes.
An air quality measurement station measures levels of $\mathrm { NO } _ { 2 }$ and suspended particles. The probability that on a given day a level of $\mathrm { NO } _ { 2 }$ exceeding the permitted level is measured is 0.16. On days when the permitted level of $\mathrm { NO } _ { 2 }$ is exceeded, the probability that the permitted level of particles is exceeded is 0.33. On days when the level of $\mathrm { NO } _ { 2 }$ is not exceeded, the probability that the level of particles is exceeded is 0.08.\ a) ( 0.5 points) What is the probability that on a given day both permitted levels are exceeded?\ b) ( 0.75 points) What is the probability that at least one of the two is exceeded?\ c) ( 0.5 points) Are the events ``on a given day the permitted level of $\mathrm { NO } _ { 2 }$ is exceeded'' and ``on a given day the permitted level of particles is exceeded'' independent?\ d) ( 0.75 points) What is the probability that on a given day the permitted level of $\mathrm { NO } _ { 2 }$ is exceeded, given that the permitted level of particles has not been exceeded?