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A sheet of paper has statements numbered from 1 to 70. For all values of n from 1 to 70. Statement n says At least n of the statements on this sheet are false. Which statements are true and which are false?
- a)The even numbered statements are true and the odd numbered are false.
- b)The odd numbered statements are true and the even numbered are false.
- c)The first 35 statements are true and the last 35 are false.
- d)The first 35 statements are false and the last 35 are false.
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
A sheet of paper has statements numbered from 1 to 70. For all values ...
1st Statement says: Atleast 1 statement is false.
2nd Statement says: Atleast 2 statement is false. (If atleast 2 statements are false, then atleast 1 statement has to be false, this makes statement 1 true.)
35th Statement says: Atleast 35 statement is false. (If atleast 35 statements are false, then atleast 34 statement has to be false, this makes statement 34 true.)
Now,
36th Statement says: Atleast 36 statement is false. (Now we already have that 35 statements are false according to statement 35 which is true and all statements above 35 are also true.)
Thus Statement 36 is false, hence every statement after 35 is also false.
Hence first 35 statements are true and the last 35 are false.
Most Upvoted Answer
A sheet of paper has statements numbered from 1 to 70. For all values ...
Solution:
Given, there are statements numbered from 1 to 70.
For all values of n from 1 to 70. Statement n says "At least n of the statements on this sheet are false."
We need to determine which statements are true and which are false.
Let's analyze the given statements:
Statement 1 says "At least 1 of the statements on this sheet is false." This statement is true because it is possible that only statement 1 is false and all other statements are true.
Statement 2 says "At least 2 of the statements on this sheet are false." This statement is false because if statement 2 is true, then statement 1 must also be true, which contradicts statement 2.
Similarly, we can analyze all the statements and conclude that:
Statements 1 to 35 are true.
Statements 36 to 70 are false.
Therefore, option C is the correct answer.
Explanation:
We can prove that statements 1 to 35 are true and statements 36 to 70 are false using mathematical induction.
Base case: Statement 1 is true.
Assumption: Statements 1 to k are true.
Induction step: Consider statement k+1. It says "At least (k+1) of the statements on this sheet are false." If statement k+1 is true, then there are at least (k+1) false statements, which means that statements 1 to k are false. But this contradicts our assumption that statements 1 to k are true. Therefore, statement k+1 must be false.
Using this proof, we can conclude that statements 1 to 35 are true and statements 36 to 70 are false.
Given, there are statements numbered from 1 to 70.
For all values of n from 1 to 70. Statement n says "At least n of the statements on this sheet are false."
We need to determine which statements are true and which are false.
Let's analyze the given statements:
Statement 1 says "At least 1 of the statements on this sheet is false." This statement is true because it is possible that only statement 1 is false and all other statements are true.
Statement 2 says "At least 2 of the statements on this sheet are false." This statement is false because if statement 2 is true, then statement 1 must also be true, which contradicts statement 2.
Similarly, we can analyze all the statements and conclude that:
Statements 1 to 35 are true.
Statements 36 to 70 are false.
Therefore, option C is the correct answer.
Explanation:
We can prove that statements 1 to 35 are true and statements 36 to 70 are false using mathematical induction.
Base case: Statement 1 is true.
Assumption: Statements 1 to k are true.
Induction step: Consider statement k+1. It says "At least (k+1) of the statements on this sheet are false." If statement k+1 is true, then there are at least (k+1) false statements, which means that statements 1 to k are false. But this contradicts our assumption that statements 1 to k are true. Therefore, statement k+1 must be false.
Using this proof, we can conclude that statements 1 to 35 are true and statements 36 to 70 are false.
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A sheet of paper has statements numbered from 1 to 70. For all values of n from 1 to 70. Statement n says At least n of the statements on this sheet are false. Which statements are true and which are false?a)The even numbered statements are true and the odd numbered are false.b)The odd numbered statements are true and the even numbered are false.c)The first 35 statements are true and the last 35 are false.d)The first 35 statements are false and the last 35 are false.Correct answer is option 'C'. Can you explain this answer?
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A sheet of paper has statements numbered from 1 to 70. For all values of n from 1 to 70. Statement n says At least n of the statements on this sheet are false. Which statements are true and which are false?a)The even numbered statements are true and the odd numbered are false.b)The odd numbered statements are true and the even numbered are false.c)The first 35 statements are true and the last 35 are false.d)The first 35 statements are false and the last 35 are false.Correct answer is option 'C'. Can you explain this answer? for Quant 2025 is part of Quant preparation. The Question and answers have been prepared according to the Quant exam syllabus. Information about A sheet of paper has statements numbered from 1 to 70. For all values of n from 1 to 70. Statement n says At least n of the statements on this sheet are false. Which statements are true and which are false?a)The even numbered statements are true and the odd numbered are false.b)The odd numbered statements are true and the even numbered are false.c)The first 35 statements are true and the last 35 are false.d)The first 35 statements are false and the last 35 are false.Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Quant 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A sheet of paper has statements numbered from 1 to 70. For all values of n from 1 to 70. Statement n says At least n of the statements on this sheet are false. Which statements are true and which are false?a)The even numbered statements are true and the odd numbered are false.b)The odd numbered statements are true and the even numbered are false.c)The first 35 statements are true and the last 35 are false.d)The first 35 statements are false and the last 35 are false.Correct answer is option 'C'. Can you explain this answer?.
A sheet of paper has statements numbered from 1 to 70. For all values of n from 1 to 70. Statement n says At least n of the statements on this sheet are false. Which statements are true and which are false?a)The even numbered statements are true and the odd numbered are false.b)The odd numbered statements are true and the even numbered are false.c)The first 35 statements are true and the last 35 are false.d)The first 35 statements are false and the last 35 are false.Correct answer is option 'C'. Can you explain this answer? for Quant 2025 is part of Quant preparation. The Question and answers have been prepared according to the Quant exam syllabus. Information about A sheet of paper has statements numbered from 1 to 70. For all values of n from 1 to 70. Statement n says At least n of the statements on this sheet are false. Which statements are true and which are false?a)The even numbered statements are true and the odd numbered are false.b)The odd numbered statements are true and the even numbered are false.c)The first 35 statements are true and the last 35 are false.d)The first 35 statements are false and the last 35 are false.Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Quant 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A sheet of paper has statements numbered from 1 to 70. For all values of n from 1 to 70. Statement n says At least n of the statements on this sheet are false. Which statements are true and which are false?a)The even numbered statements are true and the odd numbered are false.b)The odd numbered statements are true and the even numbered are false.c)The first 35 statements are true and the last 35 are false.d)The first 35 statements are false and the last 35 are false.Correct answer is option 'C'. Can you explain this answer?.
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