Sketch the region bounded by the lines $$2 x+y=8, y=2, y=4$$ and the $$y$$-axis. Hence, obtain its area using integration.

(a) If the vectors $$\vec{a}$$ and $$\vec{b}$$ are such that $$|\vec{a}|=3,|\vec{b}|=\frac{2}{3}$$ and $$\vec{a} \times \vec{b}$$ is a unit vector, then find the angle between $$\vec{a}$$ and $$\vec{b}$$.

OR

(b) Find the area of a parallelogram whose adjacent side are determined by the vectors $$\vec{a}=\hat{i}-\hat{j}+3 \hat{k}$$ and $$\vec{b}=2 \hat{i}-7 \hat{j}+\hat{k}$$

Find the vector and the cartesian equations of a line that passes through the point $$A(1,2,-1)$$ and parallel to the line $$5 x-25=14-7 y=35 z$$.

$$\text { If } A=\left[\begin{array}{ccc} 1 & 2 & 3 \\ 3 & -2 & 1 \\ 4 & 2 & 1 \end{array}\right] \text {, then show that } A^3-23 A-40 I=O \text {. }$$