TS EAMCET 10 May 2024 Shift 2 Solved Paper

Given the expression $3^{2 n+2}-8 n-9$, we want to determine the maximum power of 2 that divides this expression for all $n \in \mathbb{N}$.First, we can rewrite $3^{2 n+2}$ as $9^{n+1}$ :$9^{n+1}-8 n-9$Considering $9=1+8$, we can apply the binomial expansion to $(1+8)^{n+1}$ :$(1+8)^{n+1}={}^{n+1}C_0+{}^{n+1}C_1 \cdot 8+{}^{n+1}C_2 \cdot 8^2+{}^{n+1}C_3 \cdot 8^3+\cdots$Simplifying, we get:$1+8(n+1)+8^{2 n+1} C_2+{}^{n+1}C_3 \cdot 8^3+\cdots$Subtracting $8 n+9$ from this sequence, we have:$1+8 n+8+8^{2 n+1} C_2+\cdots-8 n-9$Which simplifies to:$8^2 \left( {}^{n+1}C_2+8^{n+1}C_3+\cdots \right)$The expression $8^2$ can be further evaluated as $2^6$. Thus, the binomial expansion shows that after simplification, the given expression is a multiple of $2^6$.Therefore, the maximum value of $p$ such that the expression is divisible by $2^p$ is:6