Let the function $f ( x ) = x \sqrt [ 3 ] { \left( x ^ { 2 } - 1 \right) ^ { 2 } }$. a) ( 0.75 points) Find $\lim _ { x \rightarrow 1 } \frac { f ( x ) } { ( x - 1 ) ^ { 2 / 3 } }$. b) ( 1.75 points) Find the area, in the first quadrant, between the line $y = x$ and the graph of the function $f ( x )$.
Let the line $r \equiv \left\{ \begin{array} { l } x = \lambda \\ y = 0 \\ z = 0 \end{array} \right.$ and the plane $\pi : z = 0$. a) (1 point) Find an equation of the line parallel to the plane $\pi$ whose direction is perpendicular to $r$ and passes through the point $( 1,1,1 )$. b) (1.5 points) Find an equation of a line that forms an angle of $\frac { \pi } { 4 }$ radians with the line $r$, is contained in the plane $\pi$ and passes through the point ( $0,0,0$ ).
The Spanish national team will compete in the 2023 FIFA Women's World Cup. In the first two matches of the group stage, which consists of three matches, the probability of winning each one is 80\%. However, due to increased morale among the players, if they win the first two matches, the probability of winning the third rises to 90\%. Otherwise, the probability of winning the third match remains at 80\%. It is requested: a) ( 0.5 points) Determine the probability that the Spanish national team does not win any match during the group stage. b) (1 point) Calculate the probability that the national team wins the third match of the group stage. c) (1 point) If we know that the national team has won the third match, determine the probability that they did not win one of the two previous matches.
Consider the real matrices $A = \left( \begin{array} { c c c } m & 1 & 1 \\ 0 & m & 3 \end{array} \right)$ and $B = \left( \begin{array} { c c } 1 & m \\ 0 & m \\ 0 & 1 \end{array} \right)$. It is requested: a) ( 0.75 points) Study whether there exists some value of $m$ for which the matrix $B A$ has an inverse. b) ( 0.75 points) Study the rank of the matrix $A B$ as a function of the parameter $m$. c) (1 point) For $m = 1$, discuss the system $\left( A ^ { t } A \right) \left( \begin{array} { l } x \\ y \\ z \end{array} \right) = \left( \begin{array} { c } a \\ a \\ a ^ { 2 } \end{array} \right)$ according to the values of $a$.
Given the real function of a real variable $f ( x ) = x - \frac { 4 } { ( x - 1 ) ^ { 2 } }$, it is requested: a) ( 0.75 points) Find the domain of definition of $f ( x )$ and determine, if they exist, the equations of the asymptotes of its graph. b) (1 point) Determine the relative extrema of the function, as well as its intervals of increase and decrease. c) ( 0.75 points) Calculate the equation of a tangent line to the graph of $f ( x )$ that is parallel to the line with equation $9 x - 8 y = 6$.
Given the points $A ( 0,0,1 ) , B ( 1,1,0 ) , C ( 1,0 , - 1 ) , D ( 1,1,2 )$, it is requested: a) ( 0.75 points) Verify that the points $A , B , C$ and $D$ are not coplanar and find the volume of the tetrahedron they form. b) ( 0.75 points) Find the area of the triangle formed by the points $B , C$ and $D$ and the angle $\hat { B }$ of the same. c) (1 point) Find one of the points $E$ in the plane determined by $A , B$ and $C$ such that the quadrilateral $A B C E$ is a parallelogram. Find the area of said parallelogram.
In a sample space there are two mutually exclusive events, $A _ { 1 }$ with probability 0.5 and $A _ { 2 }$ with probability 0.3, and $A _ { 3 } = \overline { A _ { 1 } \cup A _ { 2 } }$ is considered. Of a certain event $B$ with probability 0.4 it is known that it is independent of $A _ { 1 }$ and that the probability of the event $A _ { 3 } \cap B$ is 0.1 . With this data it is requested: a) ( 1 point) Calculate the probability of $A _ { 3 }$. b) ( 1.5 points) Decide whether $B$ and $A _ { 2 }$ are independent.