Verified Solution • 2025–26 Session moderate

QJEEM21Ma3728: JEE Main 2021 Mathematics - Class 12 JEEM21Ma3728 Solution: The probability of selecting integers a [ 5, 30] such that x 2…

The probability of selecting integers a ∈\in [ −- 5, 30] such that x2 + 2(a + 4)x −- 5a + 64 > 0, for all x ∈\in R, is :

Multiple Choice Options

  • A 16{1 \over {6}}
  • B 14{1 \over {4}}
  • C 736{7 \over {36}}
  • D 29{2 \over {9}}

Step-by-Step Verified Solution

D < 0

⇒\Rightarrow 4(a + 4)2 −- 4( −- 5a + 64) < 0

⇒\Rightarrow a2 + 16 + 8a + 5a −- 64 < 0

⇒\Rightarrow a2 + 13a −- 48 < 0

⇒\Rightarrow (a + 16) (a −- 3) < 0

⇒\Rightarrow a ∈\in ( −- 16, 3)

∴\therefore Possible a : { −- 5, −- 4, ............., 3}

∴\therefore Required probability = 836{8 \over {36}} = 29{2 \over {9}}

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