Given the matrices $A = \left( \begin{array} { c c c } 2 & 1 & 0 \\ - 1 & 0 & 2 \end{array} \right)$ and $B = \left( \begin{array} { l l } b & 0 \\ 1 & b \end{array} \right)$, find:\ a) ( 0.5 points) Calculate the determinant of $A ^ { t } A$.\ b) ( 0.5 points) Calculate the rank of $B A$ as a function of $b$.\ c) (0.75 points) Calculate $B ^ { - 1 }$ for $b = 2$.\ d) ( 0.75 points) For $b = 1$, calculate $B ^ { 5 }$.
A team of engineers conducts fuel consumption tests for a new hybrid vehicle. The fuel consumption in liters per 100 kilometers as a function of speed, measured in tens of kilometers per hour, is $$c ( v ) = \left\{ \begin{array} { l l l }
\frac { 5 v } { 3 } & \text { if } & 0 \leq v < 3 \\
14 - 4 v + \frac { v ^ { 2 } } { 3 } & \text { if } & v \geq 3
\end{array} \right.$$ a) (1 point) If in a first test the vehicle must travel at more than 3 tens of kilometers per hour, at what speed should the vehicle travel to obtain minimum fuel consumption?\ b) (1.5 points) If in another test the vehicle must travel at a speed $v$ such that $1 \leq v \leq 8$, what will be the maximum and minimum possible fuel consumption of the vehicle?
Let the plane $\pi : z = 1$, the points $\mathrm { P } ( 1,1,1 )$ and $\mathrm { Q } ( 0,0,1 )$ and the line $r$ passing through points P and Q.\ a) ( 0.25 points) Verify that points P and Q belong to the plane $\pi$.\ b) (1 point) Find a line parallel to $r$ contained in the plane $z = 0$.\ c) (1.25 points) Find a line passing through P such that its orthogonal projection onto the plane $\pi$ is the line $r$, and it forms an angle of $\frac { \pi } { 4 }$ radians with it.
Knowing that $P ( A ) = 0.5 , P ( A / B ) = 0.625$ and $P ( A \cup B ) = 0.65$, find:\ a) ( 1.5 points) $P ( B )$ and $P ( A \cap B )$.\ b) (1 point) $P ( A / A \cup B )$ and $P ( A \cap B / A \cup B )$
Given the system $\left\{ \begin{array} { l } - 2 x + y + k z = 1 \\ k x - y - z = 0 \\ - y + ( k - 1 ) z = 3 \end{array} \right.$, find:\ a) ( 1.25 points) Discuss it as a function of the parameter $k$.\ b) ( 0.5 points) Solve it for $k = 3$.\ c) ( 0.75 points) Solve it for $k = 3 / 2$ and specify, if possible, a particular solution with $x = 2$.
Given the functions $$f ( x ) = 2 + 2 x - 2 x ^ { 2 } \text { and } g ( x ) = 2 - 6 x + 4 x ^ { 2 } + 2 x ^ { 3 } ,$$ find:\ a) (1 point) Study the differentiability of $h ( x ) = | f ( x ) |$.\ b) (1.5 points) Find the area of the region bounded by the curves\ $y = f ( x ) , y = g ( x ) , x = 0$ and $x = 2$.
Given the plane $\pi : x + 3 y + 2 z + 14 = 0$ and the line $r \equiv \left\{ \begin{array} { l } x = 2 \\ z = 5 \end{array} \right.$, find:\ a) ( 0.5 points) Find the point on the plane $\pi$ closest to the origin of coordinates.\ b) (1 point) Calculate the orthogonal projection of the OZ axis onto the plane $\pi$.\ c) (1 point) Find the line with direction perpendicular to $r$, contained in $\pi$, and intersecting the OZ axis.
65\% of 18-year-old university students who attempt the practical driving exam pass it on the first try. 10 randomly selected 18-year-old university students who have already passed the practical driving exam are chosen.\ Find:\ a) (0.75 points) Calculate the probability that exactly 3 of them needed more than one attempt to pass the practical driving exam.\ b) (0.75 points) Calculate the probability that at least one of them needed more than one attempt to pass the practical driving exam.\ c) (1 point) Using a normal distribution approximation, determine the probability that, given 60 of these university students, at least half passed the practical driving exam on the first try.