^2x- ^2x ^2x ^2x dx is equal to:

Statement $-I$ : Let $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}$ and $\vec{b}=2 \hat{i}+\hat{j}-\hat{k}$. Then the vector $\vec{r}$ satisfying $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{r}}=0$ is of magnitude $\sqrt{10}$.

Statement $-II$ : In a triangle $A B C, \cos 2 A+\cos 2 B$ $+\cos 2 \mathrm{C} \geq-\frac{3}{2}$

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The foci of the curve which satisfies the differential equation $(1 + y^2) dx - xy\, dy = 0$ and passes through the point $(1 , 0)$ are :

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$\int_{}^{} {\frac{{dx}}{{{e^x} + {e^{ - x}}}} = } $

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If $X$ is brother of the son of $Y\ 's$ son. How is $X$ related to $Y?$

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If x < 0, y < 0 such that xy = 1, then $\tan^{-1}\text{x}+\tan^{-1}\text{y}$ equals:

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If $f(x) = \left\{ {\begin{array}{*{20}{c}}
{x\left[ x \right],\,\,\,\,\,\,\,\,\,\,\,\,\,}&{0 \le x < 2}\\
{\left( {x - 1} \right)\left[ x \right]\,,\,\,\,}&{2 \le x \le 4}
\end{array}} \right.,$ where $[.]$ denotes greatest integer function, then

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