Q243 / 270 · NDA 2024 (Phase 1) · Mathematics
Let $f(x)=|x-1|, g(x)=[x]$ and $h(x)=f(x) g(x)$ where [.] is greatest integer function.
What is $\int_{-1}^0 h(x) d x$ equal to?
- Correct answerA$-\frac{3}{2}$
- B-1
- C0
- D$\frac{1}{2}$
Solution
$\begin{array}{ll}
\text { Given, } & f(x)=|x-1| \qquad ...(i) \\
& g(x)=[x] \qquad ...(ii) \\
and & h(x)=f(x) \cdot g(x) \\
\Rightarrow & h(x)=h(x)-1 \mid \cdot[x] \qquad ...(iii) \\
\end{array}$
$\begin{aligned} \int_{-1}^0 h(x) d x & =\int_{-1}^0 f(x) \cdot g(x) d x \\ & =\int_{-1}^0|x-1| \cdot[x] d x \\ & =\int_{-1}^0(1-x) \cdot(-1) d x \\ & =\int_1^2(x-1) d x=\left[\frac{x^2}{2}-x\right]_1^2 \\ & =\left[0-\left(\frac{1}{2}+1\right)\right]=-\frac{3}{2}\end{aligned}$
Hence, the correct answer is option: 'A'.