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UPSC NDA I 2016 Mathematics Solved Paper

For th next two (2) items that follow :

Consider the function

f(x)=x g(x)

and

g(x)=\frac{1}{[x]}

Where

[\cdot]

is the greatest integer function

[NDA I 2016]

Question : 16 of 120

Marks: +1, -0

\text{ What is } \int\limits_{\frac{1}{3}}^{\frac{1}{2}} g(x) \, dx \text{ equal to? }

Solution:

\int\limits_{\frac{1}{3}}^{\frac{1}{2}} g(x) dx

g(x)=\begin{cases} 2, & \text{if } \frac{1}{3} < x \leq \frac{1}{2} \\ 3, & \text{if } x=\frac{1}{3} \end{cases}

g(x)

is a greatest integer function so value of

g(x)

in integral limit will be

\int\limits_{\frac{1}{3}}^{\frac{1}{2}} g(x) dx = \int\limits_{\frac{1}{3}}^{\frac{1}{2}} 2 dx

=2[x]_{\frac{1}{3}}^{\frac{1}{2}}

=2\left[\frac{1}{2}-\frac{1}{3}\right]

=\frac{1}{3}

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