$1$ . Consider ise following $5$ statements: $\newline$ (i) Statement (ii) is true. $\newline$ (ii) At most one of these $5$ statements is true. $\newline$ (iii) All $5$ of these statements are true. $\newline$ (iv) $\newline$ (v) $\newline$ The last two statements are invisible. Which of the following is correct? $\newline$ (A) Only statement (i) is true. $\newline$ (B) Only statement (ii) is true. $\newline$ (C) Only one of statements (iv) or (v) is true. $\newline$ (D) Both statements (iv) and (v) are true. $\newline$ (E) Neither of statements (iv) and (v) are true.

Full solution

1

. Consider ise following

5

statements:

\newline

(i) Statement (ii) is true.

\newline

(ii) At most one of these

5

statements is true.

\newline

(iii) All

5

of these statements are true.

\newline

(iv)

\newline

(v)

\newline

The last two statements are invisible. Which of the following is correct?

\newline

(A) Only statement (i) is true.

\newline

(B) Only statement (ii) is true.

\newline

\newline

(D) Both statements (iv) and (v) are true.

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(E) Neither of statements (iv) and (v) are true.

Analyze statement (ii): Analyze statement (ii): ＂At most one of these $5$ statements is true.＂ If statement (ii) is true, then all other statements must be false to satisfy the condition of at most one true statement.
Check statement (i): Check statement (i) under the assumption that statement (ii) is true: Statement (i) says, ＂Statement (ii) is true.＂ This aligns with our assumption and does not contradict statement (ii) being the only true statement.
Check statement (iii): Check statement (iii) under the assumption that statement (ii) is true: Statement (iii) says, ＂All $5$ of these statements are true.＂ This contradicts statement (ii) because it would mean more than one statement is true. Therefore, statement (iii) must be false.
Check statements (iv) and (v): Since statements (iv) and (v) are not visible, we cannot directly analyze them. However, if statement (ii) is true, then both statements (iv) and (v) must be false to maintain that only one statement is true.
Conclusion: Conclusion: Given that statement (ii) being true makes all other statements false or aligns with them being false, and no contradictions arise from this assumption, statement $(ii)$ is the only true statement.