AP TET Paper 1 Model Test 2

Concept:When exponential expressions share the same base, we can equate their exponents. Also, rewrite $\frac{9}{25}$ as $(\frac{3}{5})^2$ to simplify the equation.Explanation:First, convert $(\frac{9}{25})^{8}$ into base $\frac{3}{5}$: $(\frac{9}{25})^{8} = ((\frac{3}{5})^{2})^{8} = (\frac{3}{5})^{16}$.The given equation becomes $(\frac{3}{5})^{3p-2} = (\frac{3}{5})^{16} \times (\frac{3}{5})^{-3}$.Combine the terms on the right using the product rule: $(\frac{3}{5})^{16} \times (\frac{3}{5})^{-3} = (\frac{3}{5})^{16-3} = (\frac{3}{5})^{13}$.Now we have $(\frac{3}{5})^{3p-2} = (\frac{3}{5})^{13}$. Since the bases are equal, equate the exponents: $3p - 2 = 13$.Solve for $p$: $3p = 15$, so $p = 5$.Finally, compute $2p + 1 = 2 \times 5 + 1 = 11$.Answer:11 (Option B)