Given the system of equations $\left\{ \begin{array} { l } k x + ( k + 1 ) y + z = 0 \\ - x + k y - z = 0 \\ ( k - 1 ) x - y = - ( k + 1 ) \end{array} \right.$ it is requested: a) (2 points) Discuss the system according to the values of the real parameter k. b) ( 0.5 points) Solve the system for $\mathrm { k } = - 1$.
a) (1.25 points) Let $f$ and $g$ be two differentiable functions for which the following data are known: $$f ( 1 ) = 1 ; f ^ { \prime } ( 1 ) = 2 ; g ( 1 ) = 3 ; g ^ { \prime } ( 1 ) = 4 :$$ Given $h ( x ) = f \left( ( x + 1 ) ^ { 2 } \right)$, use the chain rule to calculate $h ^ { \prime } ( 0 )$. Given $k ( x ) = \frac { f ( x ) } { g ( x ) }$, calculate $k ^ { \prime } ( 1 )$. b) (1.25 points) Calculate the integral $\int ( \operatorname { sen } x ) ^ { 4 } ( \cos x ) ^ { 3 } d x$. (You can use the change of variables $t = \operatorname { sen } x$.)
Given the points $\mathrm { A } ( 1,1,1 ) , \mathrm { B } ( 1,3 , - 3 )$ and $\mathrm { C } ( - 3 , - 1,1 )$, it is requested: a) (1 point) Determine the equation of the plane containing the three points. b) ( 0.5 points) Obtain a point D (different from $\mathrm { A } , \mathrm { B }$ and C ) such that the vectors $\overrightarrow { A B } , \overrightarrow { A C } , \overrightarrow { A D }$ are linearly dependent. c) (1 point) Find a point P on the OX axis, such that the volume of the tetrahedron with vertices A, B, C and P equals 1.
A company has carried out a personnel selection process. a) ( 1.25 points) It is known that $40 \%$ of the total number of applicants have been selected in the process. If among the applicants there was a group of 8 friends, calculate the probability that at least 2 of them have been selected. b) (1.25 points) The scores obtained by the applicants in the selection process follow a normal distribution, X, with mean 5.6 and standard deviation $\sigma$. Knowing that the probability of obtaining a score $\mathrm { X } \leq 8.2$ is 0.67, calculate $\sigma$.