UPSC NDA I 2024 Mathematics Paper

To solve the given problem, we first express $a, c$, and $e$ in terms of $p$ using the provided logarithmic relationships:$\log a=p$ implies $a=10^p$$\log c=2p$ implies $c=10^{2p}$$\log e=3p$ implies $e=10^{3p}$Next, we need to find $\left(\;\text{ace}\;\right)^{\frac{1}{p}}$. First, compute ace:$a c e=10^p \cdot 10^{2p} \cdot 10^{3p}$Using the laws of exponents, we combine the terms:$\;\text{ace}\;=10^{p+2p+3p}=10^{6p}$Now, raise ace to the power of $\;\frac{1}{p}$ :$\left(\;\text{ace}\;\right)^{\frac{1}{p}}= (10^{6p})^{\frac{1}{p}}$Using the power rule of exponents, $(a^m)^n=a^{m n}$, we get:$(10^{6p})^{\frac{1}{p}}=10^{6p \cdot \frac{1}{p}}=10^6$The expression $10^6$ represents $10^6$, which corresponds to a specific combination of letters as presented in the given options.Among the provided options, Option A $(b d^2 f^3)$ is the correct match because it correctly utilizes the exponents 1,2 , and 3 respectively, matching the total exponent of 6 derived above.