If $\bar{a}=2 \hat{i}+2 \hat{j}+3 \hat{k}, \quad \bar{b}=-\hat{i}+2 \hat{j}+\hat{k}$ and $\bar{c}=3 \hat{i}+\hat{j}$ are the vectors such that $\overline{\mathrm{a}}+\lambda \overline{\mathrm{b}}$ is perpendicular to $\bar{c}$, then value of $\lambda$ is

The area (in sq. units) bounded between the parabolas $x^2=\frac{y}{4}$ and $x^2=9 y$ and the line $y=2$ is

A plane makes positive intercepts of unit length on each of $X$ and $Y$ axis. If it passes through the point $(-1,1,2)$ and makes angle $\theta$ with the X -axis, then $\theta$ is

The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is 1000 at $\mathrm{t}=0$. The number of bacteria is increased by $20 \%$ in 2 hours. If the population of bacteria is 2000 after $\frac{\mathrm{k}}{\log \left(\frac{6}{5}\right)}$ hours, then $\left(\frac{\mathrm{k}}{\log 2}\right)^2$ is