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:
log‌a=p implies a=10p
log‌c=2p implies c=102p
log‌e=3p implies e=103p
Next, we need to find (‌ ace ‌)‌

1

p

. First, compute ace:
ace=10p⋅102p⋅103p
Using the laws of exponents, we combine the terms:
‌ ace ‌=10p+2p+3p=106p
Now, raise ace to the power of ‌

1

p

:
(‌ ace ‌)‌

1

p

=(106p)‌

1

p

Using the power rule of exponents, (am)n=amn, we get:
(106p)‌

1

p

=106p⋅‌

1

p

=106
The expression 106 represents 106, which corresponds to a specific combination of letters as presented in the given options.
Among the provided options, Option A (bd2f3) is the correct match because it correctly utilizes the exponents 1,2 , and 3 respectively, matching the total exponent of 6 derived above.