We have, maximise $$Z=11 x+7 y\quad\text{..... (i)}$$

Subject to the constraints

$$\begin{aligned} 2 x+y & \leq 6 \quad\text{.... (ii)}\\ x & \leq 2 \quad\text{.... (iii)}\\ x \geq 0, y & \geq 0 \quad\text{.... (iv)} \end{aligned}$$

We see that, the feasible region as shaded determined by the system of constraints (ii) to (iv) is $O A B C$ and is bounded. So, now we shall use corner point method to determine the maximum value of $Z$.

Hence, the maximum value of Z is 42 at (0, 6).

Maximise $Z=3 x+4 y$, Subject to the constraints

$$x+y \leq 1, x \geq 0, y \geq 0$$

The shaded region shown in the figure as $O A B$ is bounded and the coordinates of corner points $O, A$ and $B$ are $(0,0),(1,0)$ and $(0,1)$, respectively.

Hence, the maximum value of Z is 4 at (0, 1).

Maximise $Z=11 x+7 y$, subject to the constraints $x \leq 3, y \leq 2, x \geq 0, y \geq 0$.

The shaded region as shown in the figure as $O A B C$ is bounded and the coordinates of corner points are $(0,0),(3,0),(3,2)$ and $(0,2)$, respectively.

Hence, Z is maximum at (3, 2) and its maximum value is 47.

Minimise $Z=13 x-15 y$ subject to the constraints $x+y \leq 7,2 x-3 y+6 \geq 0, x \geq 0, y \geq 0$.

Shaded region shown as $O A B C$ is bounded and coordinates of its corner points are $(0,0)$, $(7,0),(3,4)$ and $(0,2)$, respectively.

Hence, the minimum value of Z is $(-30)$ at (0, 2).

As clear from the graph, corner points are $O, A, E$ and $D$ with coordinates $(0,0),(52,0)$, $(144,16)$ and $(0,38)$, respectively. Also, given region is bounded.

$$\begin{array}{lrl} \text { Here, } & Z & =3 x+4 y \\ \because & 2 x+y & =104 \text { and } 2 x+4 y=152 \\ \Rightarrow & -3 y & =-48 \\ \Rightarrow & y & =16 \text { and } x=44 \end{array}$$

Hence, $Z$ is at $(44,16)$ is maximum and its maximum value is 196 .