The following observation is made about the scores obtained by 100 students in…
2026
The following observation is made about the scores obtained by 100 students in an exam: ‘For each student, there exists another student in the class such that their scores are at most ten marks away.’ If the above statement is false, which one of the following statements is necessarily true?
Answer: B. There exists at least one student in the class for whom the scores of all the other students are more than 10 marks away. — The given observation has the form \(\forall x\,\exists y\neq x\,(|s_x-s_y|\le 10)\). Its negation is \(\exists x\,\forall y\neq x\,(|s_x-s_y|>10)\), meaning…
- A.
For each student, the scores of all the other students are more than 10 marks away.
- B.
There exists at least one student in the class for whom the scores of all the other students are more than 10 marks away.
- C.
There is exactly one student in the class for whom the scores of some students are more than 10 marks away.
- D.
For each student, the score of exactly one other student is more than 10 marks away.
Attempted by 16 students.
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Correct answer: B
The given observation has the form \(\forall x\,\exists y\neq x\,(|s_x-s_y|\le 10)\). Its negation is \(\exists x\,\forall y\neq x\,(|s_x-s_y|>10)\), meaning at least one student is more than 10 marks away from every other student.