MCQ
The value of $\int_{\,0}^{\,\sqrt 2 } {[{x^2}]\,dx} ,$ where $[.]$ is the greatest integer function
- A$2 - \sqrt 2 $
- B$2 + \sqrt 2 $
- ✓$\sqrt 2 - 1$
- D$\sqrt 2 - 2$
$ = \int_{\,0}^{\,1} {[{x^2}]\,dx + } \int_{\,1}^{\,\sqrt 2 } {[{x^2}]\,\,dx} $
$ = \int_{\,0}^{\,1} {\,0\,dx + } \int_{\,1}^{\,\sqrt 2 } {\,dx} $
$ = [x]_1^{\sqrt 2 } = \sqrt 2 - 1$.
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Match List $I$ with List $II$ and select the correct answer using the code given below the lists :
| List $I$ | List $II$ |
| $P.$ $\quad$m= | $1.$ $\quad\frac{1}{2}$ |
| $Q.$ $\quad$Maximum area of $\triangle E F G$ is | $2.$ $\quad4$ |
| $R.$ $\quad y_0=$ | $3.$ $\quad2$ |
| $S.$ $\quad y_1=$ | $4.$ $\quad1$ |
Codes: $ \quad P \quad Q \quad R \quad S $
$f(x)\left\{ \begin{gathered} = 1\,,\,{\text{if}}\,\,\,x > 0 \hfill \\ = - 1\,,\,{\text{if}}\,\,\,x < 0 \hfill \\ = 0\,,\,{\text{if}}\,\,\,x = 0 \hfill \\ \end{gathered} \right.$ then ${\left. {\frac{{dy}}{{dx}}} \right|_{x = \frac{{5\pi }}{4}}}$ is