On a library shelf there are essays, novels and biographies. Three out of every sixteen books on the shelf are essays. The biographies together with one third of the essays exceed the novels by two. If we removed half of the essays and one fifth of the novels, one hundred five books would remain. Calculate the number of books of each type on the shelf.
Let the function $$f ( x ) = \left\{ \begin{array} { l l l }
\frac { 2 x + 1 } { x } & \text { if } & x < 0 \\
x ^ { 2 } - 4 x + 3 & \text { if } & x \geq 0
\end{array} \right.$$ a) ( 0.75 points) Study the continuity of $f ( x )$ in $\mathbb { R }$. b) ( 0.25 points) Is $f ( x )$ differentiable at $x = 0$ ? Justify your answer. c) ( 0.75 points) Calculate, if they exist, the equations of its horizontal and vertical asymptotes. d) ( 0.75 points) Determine for $x \in ( 0 , \infty )$ the point on the graph of $f ( x )$ where the slope of the tangent line is zero and obtain the equation of the tangent line at that point. At the point obtained, does $f ( x )$ achieve any relative extremum? If so, classify it.
Let the plane $\pi \equiv z = x$ and the points $\mathrm { A } ( 0 , - 1,0 )$ and $\mathrm { B } ( 0,1,0 )$ belonging to the plane $\pi$. a) ( 1.25 points) If points A and B are adjacent vertices of a square with vertices $\{ \mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } \}$ that lies in the plane $\pi$, find the possible points C and D. b) ( 1.25 points) If points A and B are opposite vertices of a square that lies in the plane $\pi$, determine the other two vertices of it.
In an autonomous community, three out of every five second-year high school students are enrolled in Mathematics II. Six students are randomly selected from all second-year high school students. It is requested: a) ( 0.75 points) Calculate the probability that exactly four of them are enrolled in Mathematics II. b) (0.75 points) Calculate the probability that at least one of them is enrolled in Mathematics II. c) (1 point) If an institute has 120 students enrolled in second-year high school, calculate, approximating the binomial distribution by a normal distribution, the probability that more than 60 of these students are enrolled in Mathematics II.
Consider the real matrices $$A = \left( \begin{array} { c c c }
1 & - 1 & k \\
k & 1 & - 1
\end{array} \right) , \quad B = \left( \begin{array} { c c }
1 & 1 \\
1 & - 1 \\
1 & 0
\end{array} \right)$$ a) (1 point) Calculate for which values of the parameter $k$ the matrix AB has an inverse. Calculate the inverse matrix of AB for $k = 1$. b) (1 point) Calculate BA and discuss its rank as a function of the value of the real parameter $k$. c) ( 0.5 points) In the case $k = 1$, write an inconsistent system of three linear equations with three unknowns whose coefficient matrix is BA.
Let the function $$f ( x ) = \begin{cases} x & \text { if } x \leq 0 \\ x \ln ( x ) & \text { if } x > 0 \end{cases}$$ a) ( 0.5 points) Study the continuity and differentiability of $f ( x )$ at $x = 0$. b) (1 point) Study the intervals of increase and decrease of $\mathrm { f } ( \mathrm { x } )$, as well as the relative maxima and minima. c) (1 point) Calculate $\int _ { 1 } ^ { 2 } f ( x ) d x$.
Let the lines $r \equiv \left\{ \begin{array} { l } x + y + 2 = 0 \\ y - 2 z + 1 = 0 \end{array} \right.$ and $s \equiv \left\{ \begin{array} { l } x = 2 - 2 t \\ y = 5 + 2 t \\ z = t \end{array} , t \in \mathbb { R } \right.$. a) (1.5 points) Study the relative position of the given lines and calculate the distance between them. b) ( 0.5 points) Determine an equation of the plane $\pi$ that contains the lines $r$ and $s$. c) (0.5 points) Let P and Q be the points on the lines $r$ and $s$, respectively, that are contained in the plane with equation $z = 0$. Calculate an equation of the line passing through points $P$ and $Q$.
A company markets three types of products A, B and C. Four out of every seven products are of type A, two out of every seven products are of type B and the rest are of type C. For export, 40\% of type A products, 60\% of type B products and 20\% of type C products are destined. A product is chosen at random, it is requested: a) (1.25 points) Calculate the probability that the product is destined for export. b) (1.25 points) Calculate the probability that it is of type C given that the product is destined for export.