Question:

Step 1: Evaluation of Statement-1. Statement-1 asserts that for all natural numbers $n$, $2 \cdot 7^n - 3 \cdot 5^n + 5$ is divisible by 24. Let $P(n) = 2 \cdot 7^n - 3 \cdot 5^n + 5$. To test this statement, we evaluate $P(n)$ for $n=1$: $$P(1) = 2 \cdot 7^1 - 3 \cdot 5^1 + 5$$ $$P(1) = 14 - 15 + 5$$ $$P(1) = 4$$ Since 4 is not divisible by 24, Statement-1 is false. Alternatively, consider $P(n)$ modulo 3: $$P(n) = 2 \cdot 7^n - 3 \cdot 5^n + 5$$ Since $7 \equiv 1 \pmod 3$ and $5 \equiv 2 \pmod 3$, and $3 \equiv 0 \pmod 3$: $$P(n) \equiv 2 \cdot (1)^n - 0 \cdot (2)^n + 2 \pmod 3$$ $$P(n) \equiv 2 + 2 \pmod 3$$ $$P(n) \equiv 4 \pmod 3$$ $$P(n) \equiv 1 \pmod 3$$ Since $P(n)$ is always congruent to 1 modulo 3, it is never divisible by 3. Consequently, $P(n)$ cannot be divisible by 24. Thus, Statement-1 is false. Step 2: Evaluation of Statement-2. Statement-2 asserts: "If $f(x)$ is divisible by $x$, then $f(x+1) - f(x)$ is divisible by $x+1$, for all $x \in \mathbb{N}$." To determine the truth value of this statement, we can attempt to find a counterexample. Let $f(x) = x^2$. For any natural number $x$, $f(x) = x^2$ is divisible by $x$. Now, we examine the expression $f(x+1) - f(x)$: $$f(x+1) - f(x) = (x+1)^2 - x^2$$ $$f(x+1) - f(x) = (x^2 + 2x + 1) - x^2$$ $$f(x+1) - f(x) = 2x + 1$$ According to Statement-2, $2x+1$ must be divisible by $x+1$ for all $x \in \mathbb{N}$. Let's test this for $x=1$: For $x=1$, $f(1) = 1^2 = 1$, which is divisible by 1. Then, $f(1+1) - f(1) = 2(1) + 1 = 3$. For Statement-2 to be true, 3 must be divisible by $1+1=2$. However, 3 is not divisible by 2. Therefore, Statement-2 is false.