Let S$\left(\mathrm{\alpha }\right) = \left\{\left(\mathrm{x}, \mathrm{y}\right) : {\mathrm{y}}^2 \le \mathrm{x}, 0 \le \mathrm{x} \le \mathrm{\alpha }\right\}$ and $\mathrm{A}\left(\mathrm{\alpha }\right)$is area of the region $\mathrm{S}\left(\mathrm{\alpha }\right)$. If for a $\mathrm{\lambda }, 0 \mathrm{n}< \mathrm{\lambda } < 4, \mathrm{A}\left(\mathrm{\lambda }\right) : \mathrm{A}\left(4\right) = 2 : 5$, then $\mathrm{\lambda }$ equals :

• $2{\left(\frac{4}{25}\right)}^{\frac{1}{3}}$

• $4{\left(\frac{4}{25}\right)}^{\frac{1}{3}}$

• $2{\left(\frac{2}{5}\right)}^{\frac{1}{3}}$

• $4{\left(\frac{2}{5}\right)}^{\frac{1}{3}}$

12.

The vector equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x – y + z = 0 is :

• $\overrightarrow{\mathrm{r} }. \left(\hat{\mathrm{i}} - \hat{\mathrm{k}}\right) + 2 = 0$

• $\overrightarrow{\mathrm{r}} . \left(\hat{\mathrm{i}} - \hat{\mathrm{k}}\right) - 2 = 0$

• $\overrightarrow{\mathrm{r}} \times \left(\hat{\mathrm{i}} - \hat{\mathrm{k}}\right) + 2 = 0$