jee-main 2025 Q90

jee-main · India · session2_08apr_shift1 Discriminant and conditions for roots Probability involving discriminant conditions
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)
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 $\_\_\_\_$

\section*{ANSWER KEYS}
\begin{center}
\begin{tabular}{|l|l|}
\hline
1. (2) & 2. (2) \\
\hline
9. (3) & 10. (1) \\
\hline
17. (1) & 18. (2) \\
\hline
25. (36) & 26. (25) \\
\hline
33. (2) & 34. (2) \\
\hline
41. (1) & 42. (2) \\
\hline
49. (3) & 50. (1) \\
\hline
$57 . ( 8 )$ & 58. (2) \\
\hline
65. (1) & 66. (4) \\
\hline
73. (2) & 74. (3) \\
\hline
81. (9) & 82. (1) \\
\hline
89. (39) & 90. (19) \\
\hline
\end{tabular}
\end{center}

\begin{enumerate}
  \setcounter{enumi}{2}
  \item (4)
  \item (2)
  \item (3)
  \item (160)
  \item (1)
  \item (1)
  \item (164)
  \item (3)
  \item (1)
  \item (1)
  \item (9)
  \item (4)
  \item (2)
  \item (3)
  \item (10)
  \item (4)
  \item (2)
  \item (4)
  \item (4)
  \item (3)
  \item (3)
  \item (32)
  \item (2)
  \item (2)
  \item (3)
  \item (200)
  \item (1)
  \item (4)
  \item (2)
  \item (3)
  \item (2)
  \item (2)
  \item (25)
  \item (4)
  \item (2)
  \item (100)
  \item (15)
  \item (1)
  \item (2)
  \item (82)
  \item (2)
  \item (1)
  \item (4)
  \item (14)
  \item (2)
  \item (3)
  \item (1)
  \item (3)
  \item (20)
  \item (17)
  \item (4)
  \item (3)
  \item (4)
  \item (3)
  \item (4)
  \item (4)
  \item (4)
  \item (3)
  \item (2)
  \item (3)
  \item (3)
  \item (1)
  \item (1)
  \item (2)
  \item (1010)
  \item (81)
\end{enumerate}