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 $\_\_\_\_$ .

148- In the equation $ax^2 + bx = 5$, coefficient $a$ is chosen randomly from the interval $[1,3]$ and coefficient $b$ is chosen randomly from the interval $[-3, 0]$. With which probability is the set of solutions of this equation more than $\dfrac{2}{3}$?
$$\frac{4}{9} \ (1) \hspace{2cm} \frac{5}{9} \ (2) \hspace{2cm} \frac{7}{12} \ (3) \hspace{2cm} \frac{5}{6} \ (4)$$

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) $\frac { 19 } { 36 }$

Q80. The coefficients $a , b , c$ in the quadratic equation $a x ^ { 2 } + b x + c = 0$ are chosen from the set $\{ 1,2,3,4,5,6,7,8 \}$ . The probability of this equation having repeated roots is :
(1) $\frac { 1 } { 128 }$
(2) $\frac { 1 } { 64 }$
(3) $\frac { 3 } { 256 }$
(4) $\frac { 3 } { 128 }$

Q90. Let $\mathrm { a } , \mathrm { b }$ and c denote the outcome of three independent rolls of a fair tetrahedral die, whose four faces are marked $1,2,3,4$. If the probability that $a x ^ { 2 } + b x + c = 0$ has all real roots is $\frac { m } { n } , \operatorname { gcd } ( \mathrm {~m} , \mathrm { n } ) = 1$, then $\mathrm { m } + \mathrm { n }$ is equal to $\_\_\_\_$

ANSWER KEYS

1. (2)	2. (2)
9. (3)	10. (1)
17. (1)	18. (2)
25. (36)	26. (25)
33. (2)	34. (2)
41. (1)	42. (2)
49. (3)	50. (1)
$57 . ( 8 )$	58. (2)
65. (1)	66. (4)
73. (2)	74. (3)
81. (9)	82. (1)
89. (39)	90. (19)

1. (4)
2. (2)
3. (3)
4. (160)
5. (1)
6. (1)
7. (164)
8. (3)
9. (1)
10. (1)
11. (9)
12. (4)
13. (2)
14. (3)
15. (10)
16. (4)
17. (2)
18. (4)
19. (4)
20. (3)
21. (3)
22. (32)
23. (2)
24. (2)
25. (3)
26. (200)
27. (1)
28. (4)
29. (2)
30. (3)
31. (2)
32. (2)
33. (25)
34. (4)
35. (2)
36. (100)
37. (15)
38. (1)
39. (2)
40. (82)
41. (2)
42. (1)
43. (4)
44. (14)
45. (2)
46. (3)
47. (1)
48. (3)
49. (20)
50. (17)
51. (4)
52. (3)
53. (4)
54. (3)
55. (4)
56. (4)
57. (4)
58. (3)
59. (2)
60. (3)
61. (3)
62. (1)
63. (1)
64. (2)
65. (1010)
66. (81)