Consider the set of non-zero relative integers between $-30$ and $30$; this set can be written as follows: $\{-30; -29; -28; \ldots -1; 1; \ldots; 28; 29; 30\}$. It contains 60 elements. We choose from this set successively and without replacement a relative integer $a$ then a relative integer $c$.

1. How many different pairs $(a; c)$ can we obtain?

Consider the event $M$: ``the equation $ax^2 + 2x + c = 0$ has two distinct real solutions'', where $a$ and $c$ are the relative integers previously chosen.

1. Show that event $M$ occurs if and only if $ac < 1$.
2. Explain why the opposite event $\bar{M}$ contains 1740 outcomes.
3. What is the probability of event $M$? Round the result to $10^{-2}$.

A number $p$ is randomly chosen from the interval $[ 0,5 ]$. The probability that the equation $x ^ { 2 } + 2 p x + 3 p - 2 = 0$ has two negative roots is $\_\_\_\_$ .

The probability of selecting integers $a \in [ - 5,30 ]$ such that $x ^ { 2 } + 2 ( a + 4 ) x - 5 a + 64 > 0$, for all $x \in R$, is:
(1) $\frac { 7 } { 36 }$
(2) $\frac { 2 } { 9 }$
(3) $\frac { 1 } { 6 }$
(4) $\frac { 1 } { 4 }$

The coefficients $a , b$ and $c$ of the quadratic equation, $a x ^ { 2 } + b x + c = 0$ are obtained by throwing a dice three times. The probability that this equation has equal roots is:
(1) $\frac { 1 } { 72 }$
(2) $\frac { 1 } { 36 }$
(3) $\frac { 1 } { 54 }$
(4) $\frac { 5 } { 216 }$

If the numbers appeared on the two throws of a fair six faced die are $\alpha$ and $\beta$, then the probability that $x ^ { 2 } + \alpha x + \beta > 0$, for all $x \in R$, is
(1) $\frac { 17 } { 36 }$
(2) $\frac { 4 } { 9 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 19 } { 36 }$