Consider the following system of equations depending on the real parameter a: $$\left. \begin{array} { l }
x + a y + z = a + 1 \\
- a x + y - z = 2 a \\
- y + z = a
\end{array} \right\}$$ It is requested:\ a) (2 points) Discuss the system according to the different values of a.\ b) (0.5 points) Solve the system for $\mathrm { a } = 0$.
Given the functions $f ( x ) = x ^ { 3 } + 3 x ^ { 2 } - 1$ and $g ( x ) = 6 x$, it is requested:\ a) (0.5 points) Justify, using the appropriate theorem, that there exists some point in the interval [1,10] where both functions take the same value.\ b) (1 point) Calculate the equation of the tangent line to the curve $y = f ( x )$ with minimum slope.\ c) (1 point) Calculate $\int _ { 1 } ^ { 2 } \frac { f ( x ) } { g ( x ) } d x$
Given the lines $r \equiv \left\{ \begin{array} { l } x - y = 2 \\ 3 x - z = - 1 \end{array} \right. , s \equiv \left\{ \begin{array} { l } x = - 1 + 2 \lambda \\ y = - 4 - \lambda \\ z = \lambda \end{array} \right.$ it is requested:\ a) (1 point) Calculate the relative position of lines $r$ and $s$.\ b) (0.5 points) Find the equation of the plane perpendicular to line $r$ and passing through point $P ( 2 , - 1,5 )$.\ c) (1 point) Find the equation of the plane parallel to line $r$ that contains line s.
An amateur archer has 4 arrows and shoots at a balloon placed in the center of a target. The probability of hitting the target on the first shot is 30\%. In successive launches the aim improves, so on the second it is 40\%, on the third 50\% and on the fourth 60\%. It is requested:\ a) (1 point) Calculate the probability that the balloon has burst without needing to make the fourth shot.\ b) (0.5 points) Calculate the probability that the balloon remains intact after the fourth shot.\ c) (1 point) In an exhibition ten professional archers participate, who hit 85\% of their shots. Calculate the probability that among the 10 exactly 6 balloons have burst on the first shot.
Let the function $$f ( x ) = \left\{ \begin{array} { l l l }
( x - 1 ) ^ { 2 } & \text { if } & x \leq 1 \\
( x - 1 ) ^ { 3 } & \text { if } & x > 1
\end{array} \right.$$ a) (0.5 points) Study its continuity on $[ - 4 ; 4 ]$.\ b) (1 point) Analyze its differentiability and growth on [-4;4].\ c) (1 point) Determine whether the function $g ( x ) = f ^ { \prime } ( x )$ is defined, continuous and differentiable at $x = 1$.
Given the points $\mathrm { P } ( - 3,1,2 )$ and $\mathrm { Q } ( - 1,0,1 )$ and the plane $\pi$ with equation $x + 2 y - 3 z = 4$, it is requested:\ a) (1 point) Find the projection of $Q$ onto $\pi$.\ b) (0.5 points) Write the equation of the plane parallel to $\pi$ that passes through point $P$.\ c) (1 point) Write the equation of the plane perpendicular to $\pi$ that contains points $P$ and $Q$.
Consider two events $A$ and $B$ such that $P ( A ) = 0.5 , P ( B ) = 0.25$ and $P ( A \cap B ) = 0.125$. Answer in a reasoned manner or calculate what is requested in the following cases:\ a) (0.5 points) Let $C$ be another event, incompatible with $A$ and with $B$. Are events $C$ and $A \cup B$ compatible?\ b) (0.5 points) Are $A$ and $B$ independent?\ c) (0.75 points) Calculate the probability $P ( \bar { A } \cap \bar { B } )$ (where $\bar { A }$ denotes the event complementary to event A).\ d) (0.75 points) Calculate $P ( \bar { B } / A )$.