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UPSC NDA I 2025 Mathematics Paper

Question Numbers: 95-96

Consider the following for the two (02) items that follow:

The function f(x) satisfies

f(x),f(\frac{x}{y})=\frac{f(x)}{f(y)}.

for all positive real values of x and y, and f(2) = 3

Question : 96 of 120

Marks: +1, -0

What is f(1)f(4) equal to?

Solution:

Concept:

Use the given functional equation

f\left(\frac{x}{y}\right) = \frac{f(x)}{f(y)}

to find

f(1)

and

f(4)

from

f(2) = 3

Explanation:

To find

f(4)

, set

x = 4

and

y = 2

f\left(\frac{4}{2}\right) = f(2) = \frac{f(4)}{f(2)}

Thus

\frac{f(4)}{3} = 3

, so

f(4) = 9

To find

f(1)

, set

x = 2

and

y = 2

f\left(\frac{2}{2}\right) = f(1) = \frac{f(2)}{f(2)} = \frac{3}{3} = 1

Now multiply:

f(1) \cdot f(4) = 1 \times 9 = 9

Answer:

9 (Option C).

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