1. The problem asks to determine whether each given statement is true or false. 2. Let's analyze each statement one by one: 3. (i) $xy = 0 \Rightarrow x = 0$ or $y = 0$. - This is a property of real numbers: if the product of two numbers is zero, then at least one of the numbers must be zero. - Therefore, this statement is **true**. 4. (ii) $(x - 1)(x + 2)^2 = 0 \Rightarrow x = 1$. - For a product to be zero, at least one factor must be zero. - Here, $(x - 1) = 0$ or $(x + 2)^2 = 0$. - $(x + 2)^2 = 0$ implies $x = -2$. - So, $x$ can be $1$ or $-2$. - The statement claims $x = 1$ only, so it is **false**. 5. (iii) All relations are functions. - A function is a special type of relation where each input has exactly one output. - Not all relations satisfy this. - Therefore, this statement is **false**. 6. (iv) $|x - a| \leq 1 \Rightarrow a - 1 \leq x \leq a + 1$. - By definition of absolute value inequality, this is true. - So, this statement is **true**. 7. (v) A sequence is convergent if its $n$th term tends to 0 in the limit. - A sequence converges if it approaches some finite limit. - If the limit is 0, the sequence converges to 0. - But a sequence can converge to other values as well. - The statement implies convergence only if limit is 0, which is not true. - So, the statement is **false**. **Final answers:** (i) True (ii) False (iii) False (iv) True (v) False