Let $$x_0$$ be the point of local minima of $$\mathrm{f}(x)=\overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})$$ where $$\overline{\mathrm{a}}=x \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}, \overline{\mathrm{b}}=-2 \hat{\mathrm{i}}+x \hat{\mathrm{j}}-\hat{\mathrm{k}}, \overline{\mathrm{c}}=7 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+x \hat{\mathrm{k}}$$, then value of $$\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}$$ at $$x=x_0$$ is

Let the curve be represented by $$x=2(\cos t+t \sin t), y=2(\sin t-t \cos t)$$. Then normal at any point '$$t$$' of the curve is at a distance of ______ units from the origin.

Let $$\mathrm{B} \equiv(0,3)$$ and $$\mathrm{C} \equiv(4,0)$$. The point $$\mathrm{A}$$ is moving on the line $$y=2 x$$ at the rate of 2 units/second. The area of $$\triangle \mathrm{ABC}$$ is increasing at the rate of

The maximum value of the function $$f(x)=3 x^3-18 x^2+27 x-40$$ on the set $$\mathrm{S}=\left\{x \in \mathrm{R} / x^2+30 \leq 11 x\right\}$$ is