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UPSC NDA I 2023 Mathematics Solved Paper

Question Numbers: 86-88

Consider the following for the next three (03) items that follow :

Let f(x) = | ln x | , x ≠ 1

Question : 86 of 120

Marks: +1, -0

x = 0

.5 ?

Solution:

Concept:

The derivative of

| \ln x |

depends on the sign of

\ln x

, which changes at

x=1

Explanation:

f(x) = | \ln x |

is defined piecewise:

f(x) = \ln x

for

x \ge 1

, and

f(x) = -\ln x

for

0 < x < 1

Since

x = 0.5

is less than

1

, we use

f(x) = -\ln x

Differentiate:

f'(x) = \frac{d}{dx}(-\ln x) = -\frac{1}{x}

Substitute

x = 0.5

f'(0.5) = -\frac{1}{0.5} = -2

Answer:

-2

(option A).

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