a) In graph shade the feasible region under the constraints. [2] 2x + y 40 , x + 2y 50 , x 0 , y 0 . b) Convert the decimal number 31923 into hexadecimal number. [2] c) Examine whether the system of…

a) Graphical Representation of the Feasible Region

Step 1: Rewrite the inequalities in equality form to find boundary lines The given constraints are:

1. $2x+y\le 40$
2. $x+2y\le 50$
3. $x\ge 0$
4. $y\ge 0$

We convert these inequalities into equations to plot the boundary lines:

1. $2x+y=40$
2. $x+2y=50$

Step 2: Find intercepts for each line For $2x+y=40$:

• When $x=0$, $y=40$
• When $y=0$, $x=20$

For $x+2y=50$:

• When $x=0$, $y=25$
• When $y=0$, $x=50$

Step 3: Plot the boundary lines

• Draw the line joining $(0,40)$ and $(20,0)$ for $2x+y=40$.
• Draw the line joining $(0,25)$ and $(50,0)$ for $x+2y=50$.

Step 4: Determine the feasible region The feasible region is the area that satisfies all constraints simultaneously. Since $x\ge 0$ and $y\ge 0$, we only consider the first quadrant. The feasible region is bounded by the intersection of the lines and the axes.

Step 5: Find the intersection point of the two lines Solve the system: $$2x+y=40\text{(1)}$$ $$x+2y=50\text{(2)}$$

Multiply equation (1) by 2: $$4x+2y=80\text{(3)}$$

Subtract equation (2) from equation (3): $$(4x+2y)-(x+2y)=80-50$$ $$3x=30⟹x=10$$

Substitute $x=10$ into equation (1): $$2(10)+y=40⟹y=20$$

So, the intersection point is $(10,20)$.

Step 6: Shade the feasible region The feasible region is the area bounded by:

• The x-axis ($y=0$),
• The y-axis ($x=0$),
• The line $2x+y=40$,
• The line $x+2y=50$.

The feasible region is the triangular area with vertices at $(0,0)$, $(20,0)$, and $(10,20)$.

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b) Conversion of Decimal Number 31923 to Hexadecimal

Step 1: Divide the number by 16 and record the remainders We repeatedly divide the number by 16 and keep track of the remainders.

$$\begin{array}{rl}31923÷16& =1995\text{remainder }3\\ 1995÷16& =124\text{remainder }11(\text{which is }B\text{ in hexadecimal})\\ 124÷16& =7\text{remainder }12(\text{which is }C\text{ in hexadecimal})\\ 7÷16& =0\text{remainder }7\end{array}$$

Step 2: Write the remainders in reverse order The remainders, read from bottom to top, give the hexadecimal number: $7CB3$.

Final Answer: The hexadecimal equivalent of $31923$ is 7CB3.

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c) Examination of Diagonal Dominance

Definition: A system of linear equations is diagonally dominant if for each row, the absolute value of the diagonal element is greater than the sum of the absolute values of the other elements in that row.

Given system: $$\{\begin{array}{l}3x+12y-z=28\\ x+4y+7z=2\\ 10x+4y-2z=20\end{array}$$

Step 1: Check diagonal dominance for each row

Row 1: $3x+12y-z=28$

• Diagonal element: $|3|$
• Sum of other elements: $|12|+|-1|=12+1=13$
• Check: $|3|>13$? No, because $3\ngtr 13$.

Since the first row is not diagonally dominant, the system is not diagonally dominant.

Verification for completeness:

Row 2: $x+4y+7z=2$

• Diagonal element: $|4|$
• Sum of other elements: $|1|+|7|=1+7=8$
• Check: $|4|>8$? No, because $4\ngtr 8$.

Row 3: $10x+4y-2z=20$

• Diagonal element: $|4|$
• Sum of other elements: $|10|+|-2|=10+2=12$
• Check: $|4|>12$? No, because $4\ngtr 12$.

Conclusion: Since none of the rows satisfy the diagonal dominance condition, the system is not diagonally dominant.