a) Find the feasible region determined by the following inequalities x + 2y 7, x 0, y 0 . [2] b) Convert the decimal numeral 20345 into hexadecimal number. [2] c) Write condition for the system of…

a) Feasible region for

$$\{\begin{array}{l}x+2y\le 7\\ x\ge 0\\ y\ge 0\end{array}$$

1. Plot the boundary lines

• $x+2y=7$ → $y=\frac{7-x}{2}$ (a straight line with intercepts $(0,3.5)$ and $(7,0)$).
• $x=0$   → the $y$-axis.
• $y=0$   → the $x$-axis.

2. Determine the side of each line that satisfies the inequality

• For $x+2y\le 7$: test the origin $(0,0)$.
  $0+2\cdot 0=0\le 7$ → the region below the line $x+2y=7$ is admissible.
• For $x\ge 0$: the region to the right of the $y$-axis.
• For $y\ge 0$: the region above the $x$-axis.

3. Intersection of the three half‑planes

The common part is the triangle bounded by the three lines. Its vertices are obtained by pairwise intersections:

Hence the feasible region is the closed triangle with vertices $(0,0),(0,3.5),(7,0)$.

Result: the set $\{(x,y)\mid x\ge 0,y\ge 0,x+2y\le 7\}$ is the triangular region shown above.

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b) Decimal $20345_{10}$ → hexadecimal

The conversion proceeds by repeated division by 16, recording remainders (0–15) and then reading the remainders in reverse order.

Reading the remainders from the last division to the first gives 4 F 7 9.

$$\boxed{20345_{10}=4F79_{16}}$$

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c) Condition for the linear system

$$\{\begin{array}{l}ax+by=p\\ cx+dy=q\end{array}$$

1. Determinant of the coefficient matrix

$$\Delta =ad-bc$$

If $\Delta \ne 0$ the system has a unique solution (well‑defined).

2. Ill‑defined cases ($\Delta =0$)

When $\Delta =0$ the two equations are linearly dependent. Two possibilities arise:

Thus the necessary and sufficient condition for the system to be ill‑defined is

$$\boxed{ad-bc=0}$$

together with the proportionality test on the constant terms to decide whether the inconsistency or infinite‑solution case occurs.

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Complete code (Python) for part b – decimal to hexadecimal conversion

def dec_to_hex(n: int) -> str:
    """Return hexadecimal representation of a non‑negative integer."""
    if n == 0:
        return "0"
    hex_digits = "0123456789ABCDEF"
    res = []
    while n > 0:
        n, r = divmod(n, 16)      # divide by 16, get remainder
        res.append(hex_digits[r]) # map remainder to hex digit
    return ''.join(reversed(res))

# Example
print(dec_to_hex(20345))   # prints 4F79

The program repeatedly divides the decimal number by 16, stores each remainder, maps it to the corresponding hexadecimal character, and finally reverses the list to obtain the correct order. The printed output for the given input is 4F79.