Given the matrices $A = \left( \begin{array} { c c c c } 1 & 3 & 4 & 1 \\ 1 & a & 2 & 2 - a \\ - 1 & 2 & a & a - 2 \end{array} \right)$ and $M = \left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array} \right)$; find: a) ( 1.5 points) Study the rank of A as a function of the real parameter a. b) ( 1 point) Calculate, if possible, the inverse of the matrix AM for the case a $= 0$.
Given $f ( x ) = \frac { \ln ( x ) } { x }$, where ln denotes the natural logarithm, defined for $\mathrm { x } > 0$, find: a) ( 0.5 points) Calculate, if it exists, a horizontal asymptote of the curve $\boldsymbol { y } = \boldsymbol { f } ( \boldsymbol { x } )$. b) (1 point) Find a point on the curve $\boldsymbol { y } = \boldsymbol { f } ( \boldsymbol { x } )$ where the tangent line to the curve is horizontal and analyze whether this point is a relative extremum. c) (1 point) Calculate the area of the bounded region limited by the curve $\boldsymbol { y } = \boldsymbol { f } ( \boldsymbol { x } )$ and the lines $\boldsymbol { y } = \mathbf { 0 }$ and $\boldsymbol { x } = \boldsymbol { e }$.
Given the line $r \equiv \frac { x - 1 } { 2 } = \frac { y - 3 } { - 2 } = z$ and the line s that passes through the point $( 2 ; - 5 ; 1 )$ and has direction $( - 1 ; 0 ; - 1 )$, find: a) (1 point) Study the relative position of the two lines. b) (1 point) Calculate a plane that is parallel to r and contains s. c) ( 0.5 points) Calculate a plane perpendicular to the line r and that passes through the origin of coordinates.
The probability that a fish of a certain species survives more than 5 years is 10\%. Find: a) (1 point) If in an aquarium we have 10 fish of this species born this year, find the probability that at least two of them are still alive in 5 years. b) ( 1.5 points) If in a tank of a fish farm there are 200 fish of this species born this same year, using an approximation by the corresponding normal distribution, find the probability that after 5 years at least 10 of them have survived.