Let $\lambda$ be a real number and consider the matrices $A = \left( \begin{array} { c c c } \lambda & 1 & \lambda \\ 0 & \lambda & - 1 \end{array} \right)$ and $B = \left( \begin{array} { c c } 1 & \lambda \\ 0 & - 1 \\ 1 & - \lambda \end{array} \right)$. It is requested: a) ( 0.5 points) Determine whether there exists some value of $\lambda$ for which the matrix $AB$ does not have an inverse. b) (1 point) Study the rank of the matrix $BA$ as a function of the parameter $\lambda$. c) (1 point) For $\lambda = 1$, discuss the system $\left( A ^ { t } A \right) \left( \begin{array} { c } x \\ y \\ z \end{array} \right) = \left( \begin{array} { c } a ^ { 2 } \\ a ^ { 2 } \\ 2 a \end{array} \right)$ according to the values of $a$.
There are containers of three different sizes to fill a cistern. With six small containers and 2 L we exactly fill one medium container and one large one. With two large containers we fill two medium ones, one small one, and 1 L remains. The cistern is completely filled either with fourteen small containers plus six medium ones, or with five medium ones together with five large ones. It is requested to calculate the capacity of each type of container and, once known, that of the cistern.
Let the function $f ( x ) = \left\{ \begin{array} { l l l } x ^ { 2 } - 6 x + 11 & \text { if } & x < 2 \\ \sqrt { 5 x - 1 } & \text { if } & x \geq 2 \end{array} \right.$. a) ( 0.5 points) Study the continuity of the function in $\mathbb { R }$. b) (1 point) Study the relative extrema of the function in the interval ( 1,3 ). c) (1 point) Calculate the area enclosed by the function and the $x$-axis between $x = 1$ and $x = 3$.
Given the function $f ( x ) = \sin \left( \frac { \pi } { 2 } x \right)$, it is requested: a) ( 0.5 points) Study the parity of the function $g ( x ) = f ( x f ( x ) )$. b) (1 point) Calculate $\lim _ { x \rightarrow 0 } \frac { \sqrt { 4 + 3 f ( x ) } - 2 } { x }$. c) (1 point) Calculate $\int _ { 0 } ^ { 1 } x f ( x ) d x$.
Let the points $A ( 0,0,0 )$ and $B ( 1,1,1 )$, and the line $r \equiv ( x , y , z ) = ( \lambda , \lambda , \lambda + 1 ) , \lambda \in \mathbb { R }$. a) (1 point) Find an equation of the plane with respect to which the points $A$ and $B$ are symmetric. b) (1 point) Find an equation of the plane that contains the line $r$ and passes through the point $B$. c) (0.5 points) Find an equation of a line that is parallel to $r$ and passes through $A$.
Given the three planes $\pi _ { 1 } : - 2 x - 2 y + z = 0$; $\pi _ { 2 } : - 2 x + y - 2 z = 0$ and $\pi _ { 3 } : x - 2 y - 2 z = 0$, it is requested: a) (1 point) Determine the angle formed by the planes pairwise. Determine the intersection of the three planes. b) (1.5 points) Determine the point $P$ in space such that its orthogonal projection onto $\pi _ { 1 }$ is the point $Q _ { 1 } ( 1 / 3,4 / 3,10 / 3 )$ and its orthogonal projection onto $\pi _ { 2 }$ is the point $Q _ { 2 } ( - 1 / 3,8 / 3,5 / 3 )$. Determine the orthogonal projection $Q _ { 3 }$ of the point $P$ onto the plane $\pi _ { 3 }$.
According to data from the Community of Madrid, in the 2021-2022 season the coverage of the flu vaccine among people over 65 years old was 73.2%. a) ( 1.5 points) In the face of an epidemic outbreak situation, the authorities decide to restrict those gatherings in which the probability that there is more than one unvaccinated person is greater than 0.5. Assuming that attendees at a gathering constitute a random sample, should gatherings of 5 people over 65 years old be restricted? And gatherings of 7 people over 65 years old? b) ( 1 point) A random sample of 500 people over 65 years old is taken. Calculate, approximating by the appropriate normal distribution, the probability that at least 350 of them are vaccinated against the flu.