Linear Programming: Graphical Method, Feasible Regions and Optimal Solution

In a linear programming problem, the common region determined by all the constraints, including the non-negative constraints, is called the:

1. ainfeasible region
2. bfeasible region
3. cobjective function
4. doptimal solution

The corner points of the feasible region given by \(2x + 3y \le 12\), \(x \ge 0\), \(y \ge 0\) are:

1. a\((0, 0)\), \((6, 0)\) and \((0, 4)\)
2. b\((0, 0)\), \((4, 0)\) and \((0, 6)\)
3. c\((0, 0)\), \((12, 0)\) and \((0, 12)\)
4. d\((0, 0)\), \((6, 0)\), \((0, 4)\) and \((6, 4)\)

Which of the following points lies in the feasible region given by \(x + 2y \le 8\), \(3x + y \le 9\), \(x \ge 0\), \(y \ge 0\)?

1. a\((1, 4)\)
2. b\((3, 1)\)
3. c\((-1, 2)\)
4. d\((1, 2)\)

1. I. Any point outside the feasible region of an LPP is an infeasible solution.
2. II. An optimal feasible solution is a point of the feasible region that gives the optimal (maximum or minimum) value of the objective function.
3. III. A feasible region that cannot be enclosed within a circle is called an unbounded region.

Which of the statements above is/are correct?

1. aOnly I
2. bOnly I and II
3. cI, II and III
4. dOnly II and III

The corner points of the bounded feasible region of an LPP are \((0, 0)\), \((5, 0)\), \((3, 4)\) and \((0, 6)\). The maximum value of \(Z = 3x + 2y\) is:

1. a15
2. b17
3. c12
4. d18

Assertion (A): The point \((1, 1)\) is a feasible solution of the LPP whose constraints are \(x + 2y \le 6\), \(3x + y \le 9\), \(x \ge 0\), \(y \ge 0\).

Reason (R): Every feasible solution of an LPP is a corner point of its feasible region.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

Which of the following is NOT a corner point of the feasible region given by \(x + 2y \le 10\), \(x + y \le 6\), \(x \ge 0\), \(y \ge 0\)?

1. a\((6, 0)\)
2. b\((2, 4)\)
3. c\((0, 5)\)
4. d\((10, 0)\)

Match each system of constraints in List I with its feasible region in List II:

1. a(i)-R, (ii)-S, (iii)-P, (iv)-Q
2. b(i)-S, (ii)-R, (iii)-P, (iv)-Q
3. c(i)-S, (ii)-R, (iii)-Q, (iv)-P
4. d(i)-Q, (ii)-R, (iii)-P, (iv)-S

Maximise \(Z = 3x + 4y\) subject to \(2x + y \le 10\), \(x + 2y \le 8\), \(x \ge 0\), \(y \ge 0\). The maximum value of \(Z\) is:

1. a20
2. b16
3. c15
4. d24

Assertion (A): On the bounded feasible region with corner points \((0, 0)\), \((4, 0)\), \((3, 2)\) and \((0, 3)\), the maximum value of \(Z = 2x + 3y\) is 12.

Reason (R): If the feasible region of an LPP is bounded, the maximum value of the objective function occurs at a corner point of the region.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

Minimise \(Z = 2x + 3y\) subject to \(x + y \ge 2\), \(x \le 4\), \(y \le 3\), \(x \ge 0\), \(y \ge 0\). The minimum value of \(Z\) is:

1. a0
2. b4
3. c6
4. d17

1. I. No point satisfies all these constraints, so the LPP has no feasible solution.
2. II. The feasible region of these constraints is unbounded.

For the constraints \(2x + 3y \le 6\), \(x + y \ge 4\), \(x \ge 0\), \(y \ge 0\), which of the statements above is/are correct?

1. aOnly I
2. bOnly II
3. cBoth I and II
4. dNeither I nor II

Minimise \(Z = 3x + 2y\) subject to \(2x + y \ge 6\), \(x + y \ge 4\), \(x \ge 0\), \(y \ge 0\). The minimum value of \(Z\) is:

1. a8
2. b12
3. c10
4. d9

Assertion (A): If the feasible region of an LPP is unbounded, the objective function \(Z = ax + by\) always has a maximum value on it.

Reason (R): On an unbounded feasible region, the maximum or the minimum value of the objective function may not exist.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

1. I. The minimum value of \(Z = x + y\) on this region is 5.
2. II. The point \((2, 2)\) is a feasible solution.
3. III. The minimum value of \(Z = 2x + y\) on this region is 5.

The feasible region of an LPP is the unbounded region in the first quadrant on or above both the lines \(x + y = 5\) and \(x + 3y = 9\); its corner points are \((0, 5)\), \((3, 2)\) and \((9, 0)\). Which of the statements above is/are correct?

1. aOnly I
2. bOnly II and III
3. cI, II and III
4. dOnly I and III

The feasible region of an LPP is the quadrilateral with corner points \(O(0, 0)\), \(A(3, 0)\), \(B(3, 2)\) and \(C(0, 5)\). Which statement about the maximum of \(Z = x + y\) is correct?

1. aThe maximum is 5, attained only at \(C(0, 5)\).
2. bThe maximum is 5, attained only at \(B(3, 2)\).
3. cThe maximum is 5, attained at every point of the segment \(BC\).
4. dThere is no maximum, because two corner points give the same value.

The feasible region of an LPP is the pentagon with corner points \((0, 0)\), \((4, 0)\), \((4, 2)\), \((1, 5)\) and \((0, 5)\). Match each objective function in List I with its maximum value on this region in List II:

1. a(i)-Q, (ii)-P, (iii)-R, (iv)-S
2. b(i)-S, (ii)-P, (iii)-R, (iv)-Q
3. c(i)-S, (ii)-R, (iii)-P, (iv)-Q
4. d(i)-P, (ii)-S, (iii)-R, (iv)-Q

A workshop makes table lamps and ceiling fans. A lamp needs 1 hour of assembly and 3 hours of wiring; a fan needs 2 hours of assembly and 1 hour of wiring. At most 40 hours of assembly and 45 hours of wiring are available in a week. The profit is ₹ 50 on a lamp and ₹ 60 on a fan. By the corner point method, the greatest weekly profit is:

1. a₹ 1,400, by making 10 lamps and 15 fans
2. b₹ 1,200, by making 20 fans only
3. c₹ 2,000, by making 40 lamps only
4. d₹ 2,700, by making 45 fans only

The feasible region of an LPP is the quadrilateral with corner points \(O(0, 0)\), \(A(6, 0)\), \(B(4, 3)\) and \(C(0, 5)\). The maximum of \(Z = px + qy\), where \(p, q > 0\), occurs at both \(A\) and \(B\). Then:

1. a\(3p = 2q\)
2. b\(p = q\)
3. c\(6p = 5q\)
4. d\(2p = 3q\)

A hostel mixes two foods F and G. Each kg of F has 2 units of protein and 1 unit of iron; each kg of G has 1 unit of protein and 2 units of iron. The mix must have at least 8 units of protein and at least 7 units of iron. F costs ₹ 20 per kg and G costs ₹ 30 per kg. The least cost of such a mix is:

1. a₹ 80
2. b₹ 140
3. c₹ 120
4. d₹ 240