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Sets and Relations question

2021 · 26 Aug · Shift 1 · Q28
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  5. /2021 · 26 Aug · Shift 1 · Q28

Sets and Relations question

2021 · 26 Aug · Shift 1 · Q28

JEE MainMathematicsSets and RelationsMCQ+4 / −1
Out of all the patients in a hospital 89% are found to be suffering from heart ailment and 98% are suffering from lungs infection. If K% of them are suffering from both ailments, then K can not belong to the set :
  1. A
    {80, 83, 86, 89}
  2. B
    {84, 86, 88, 90}
  3. C
    {79, 81, 83, 85}
  4. D
    {84, 87, 90, 93}
View written solutionFree

Correct answer: C

  1. Let:

    • HHH = set of patients suffering from heart ailment
    • LLL = set of patients suffering from lungs infection

    Given: ∣H∣=89%,|H| = 89\%,∣H∣=89%, ∣L∣=98%.|L| = 98\%.∣L∣=98%.

    Let: ∣H∩L∣=K%.|H \cap L| = K\%.∣H∩L∣=K%.

  2. We use the inclusion-exclusion principle: ∣H∪L∣=∣H∣+∣L∣−∣H∩L∣.|H \cup L| = |H| + |L| - |H \cap L|.∣H∪L∣=∣H∣+∣L∣−∣H∩L∣.

    Since H∪LH \cup LH∪L is a subset of all patients, its percentage cannot exceed 100100100. Therefore, 89+98−K≤100.89 + 98 - K \le 100.89+98−K≤100.

  3. Solve for KKK: 187−K≤100187 - K \le 100187−K≤100 −K≤−87-K \le -87−K≤−87 K≥87.K \ge 87.K≥87.

  4. Also, intersection cannot exceed the smaller set: K≤min⁡(89,98)=89.K \le \min(89,98) = 89.K≤min(89,98)=89.

    Hence, 87≤K≤89.87 \le K \le 89.87≤K≤89.

  5. So the only possible values of KKK are in the interval: [87,89].[87,89].[87,89].

  6. Now check each option:

    • A: {80,83,86,89}\{80,83,86,89\}{80,83,86,89}

      • Contains 898989, which is possible.
      • So KKK can belong to this set.

    • B: {84,86,88,90}\{84,86,88,90\}{84,86,88,90}

      • Contains 888888, which is possible.
      • So KKK can belong to this set.

    • C: {79,81,83,85}\{79,81,83,85\}{79,81,83,85}

      • All values are <87<87<87, so none is possible.
      • Thus KKK cannot belong to this set.

    • D: {84,87,90,93}\{84,87,90,93\}{84,87,90,93}

      • Contains 878787, which is possible.
      • So KKK can belong to this set.

  7. Therefore, the set to which KKK cannot belong is: C.\boxed{\text{C}}.C​.

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