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Sets and Relations question

2020 · 5 Sep · Shift 1 · Q31
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  5. /2020 · 5 Sep · Shift 1 · Q31

Sets and Relations question

2020 · 5 Sep · Shift 1 · Q31

JEE MainMathematicsSets and RelationsMCQ+4 / −1
A survey shows that 73% of the persons working in an office like coffee, whereas 65% like tea. If x denotes the percentage of them, who like both coffee and tea, then x cannot be :
  1. A
    63
  2. B
    36
  3. C
    54
  4. D
    38
View written solutionFree

Correct answer: B

  1. Let:

    • CCC = set of people who like coffee
    • TTT = set of people who like tea

    Given: n(C)=73%,n(C)=73\%,n(C)=73%, n(T)=65%,n(T)=65\%,n(T)=65%, and n(C∩T)=x%.n(C\cap T)=x\%.n(C∩T)=x%.

  2. We use the basic set relation: n(C∪T)=n(C)+n(T)−n(C∩T).n(C\cup T)=n(C)+n(T)-n(C\cap T).n(C∪T)=n(C)+n(T)−n(C∩T). So, n(C∪T)=73+65−x=138−x.n(C\cup T)=73+65-x=138-x.n(C∪T)=73+65−x=138−x.

  3. Since C∪TC\cup TC∪T represents a percentage of people, it cannot exceed 100%100\%100%. Hence, 138−x≤100138-x\le 100138−x≤100 x≥38.x\ge 38.x≥38.

  4. Also, the number liking both cannot exceed either group: x≤73,x≤65.x\le 73, \quad x\le 65.x≤73,x≤65. Therefore, x≤65.x\le 65.x≤65.

  5. Thus the possible range of xxx is: 38≤x≤65.38\le x\le 65.38≤x≤65.

  6. Now check the options:

    • A: 636363 lies in [38,65][38,65][38,65] → possible
    • B: 363636 is less than 383838 → not possible
    • C: 545454 lies in [38,65][38,65][38,65] → possible
    • D: 383838 lies in [38,65][38,65][38,65] → possible

  7. Therefore, xxx cannot be: 36\boxed{36}36​ which is option B\boxed{B}B​.

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