Linear Programming: Class 12 Important Questions with Answers

Practice set: test yourself

Try every question before looking at the answers below. 64 or more out of 80 means you are ready for the board.

C. Very short answer (1 mark each)

1. Max $Z=2x+3y$ at corners (0, 0), (4, 0), (0, 3).
2. Min $Z=x+y$ at corners (2, 3), (1, 5), (4, 1) of a bounded region.
3. Does (2, 2) satisfy $x+2y\le 6$?
4. Corners of $x\le 3$, $y\le 2$, $x,y\ge 0$.
5. $Z=4x-y$ at (3, 2).
6. Non-negativity constraints for $x$ chairs, $y$ tables.
7. Is $x+y\ge 4$, $x,y\ge 0$ bounded?
8. Max $Z=5x+2y$ on $x+y\le 3$, $x,y\ge 0$.
9. Intersection of $x+y=6$ and $x-y=2$.
10. Min $Z=3x+4y$ on $x+y\ge 1$, $x,y\ge 0$.

D. Short answer I (2 marks each)

1. Maximise $Z=4x+y$ subject to $x+y\le 50$, $3x+y\le 90$, $x,y\ge 0$.
2. Minimise $Z=2x+3y$ subject to $x+y\ge 4$, $x,y\ge 0$.
3. Find the corner points of $2x+y\le 8$, $x+2y\le 10$, $x,y\ge 0$.
4. Maximise $Z=3x+2y$ on the region of Q3.
5. Corners (0, 2), (3, 0), (6, 0), (6, 8), (0, 5) of a bounded region: max and min of $Z=4x+6y$.
6. Corners (0, 0), (5, 0), (3, 4), (0, 5): find $p$ if $Z=px+y$ is maximum at both (3, 4) and (5, 0).
7. Why has $Z=x+y$ no maximum on $x+y\ge 2$, $x,y\ge 0$?
8. Corners (0, 0), (4, 0), (2, 3), (0, 4) of a bounded region: max of $Z=3x+4y$.

E. Short answer II (3 marks each)

1. Maximise $Z=10x+6y$ subject to $3x+y\le 12$, $2x+5y\le 34$, $x,y\ge 0$.
2. Show that $Z=3x+2y$ subject to $x+y\ge 8$, $3x+5y\le 15$, $x,y\ge 0$ has no feasible solution.
3. Minimise $Z=20x+10y$ subject to $x+2y\le 40$, $3x+y\ge 30$, $4x+3y\ge 60$, $x,y\ge 0$.
4. Maximise $Z=2x+5y$ subject to $x+4y\le 24$, $3x+y\le 21$, $x+y\le 9$, $x,y\ge 0$.
5. Minimise and maximise $Z=3x+9y$ subject to $x+3y\le 60$, $x+y\ge 10$, $x\le y$, $x,y\ge 0$.
6. Minimise $Z=5x+7y$ subject to $2x+y\ge 8$, $x+2y\ge 10$, $x,y\ge 0$.

F. Long answer (5 marks each)

1. Product A needs 1 hour on machine I and 2 hours on machine II; product B needs 3 hours on machine I and 1 hour on machine II. Machine I is free for 15 hours and machine II for 10 hours a day. Profit is ₹4 on A and ₹5 on B. Formulate and solve graphically for the maximum profit.
2. Food X has 2 units of vitamin A and 1 unit of vitamin B per kg; food Y has 1 unit of A and 2 units of B per kg. A diet needs at least 8 units of A and 10 units of B. X costs ₹4 and Y ₹3 per kg. Find the cheapest diet.

G. HOTS (higher-order thinking)

1. Why does the optimum of a linear $Z$ occur at a corner?
2. Why is the half-plane test needed only for unbounded regions?
3. When can an LPP have infinitely many optimal solutions?
4. What does an empty feasible region mean in a real problem?

20 MCQs

1. The objective function of an LPP is
   (a) quadratic (b) cubic (c) linear (d) constant
2. The feasible region is the
   (a) common region of the constraints (b) union of the constraints (c) graph of $Z$ (d) complement
3. On a bounded region the optimum occurs at
   (a) an interior point (b) no point (c) the origin only (d) a corner point
4. Corners (0, 0), (2, 0), (0, 3), (1, 2): max $Z=3x+2y$
   (a) 6 (b) 7 (c) 3 (d) 9
5. Corners of $x+y\le 3$, $x,y\ge 0$
   (a) (0, 0), (3, 3) (b) (0, 0), (3, 0), (0, 3) (c) (3, 0), (0, 3) (d) (1, 1)
6. (2, 1) satisfies
   (a) $x+y>3$ (b) $2x+y<4$ (c) $x-y>2$ (d) $x+y\le 3$
7. Non-negativity constraints
   (a) $x,y\ge 0$ (b) $x,y\le 0$ (c) $x+y=0$ (d) $x=y$
8. Corners (0, 0), (30, 0), (20, 30), (0, 50): max $Z=4x+y$
   (a) 110 (b) 50 (c) 120 (d) 0
9. $Z$ is maximum at two corners A and B; then it is also maximum at
   (a) A only (b) no other point (c) the origin (d) every point of AB
10. The region $x+y\ge 4$, $x,y\ge 0$ is
    (a) bounded (b) unbounded (c) empty (d) a point
11. $x\ge 3$ and $x\le 2$ give
    (a) a line (b) an unbounded region (c) an empty region (d) a bounded region
12. Bounded region with corners (0, 4), (2, 1), (5, 0), (5, 4): min $Z=x+y$
    (a) 3 (b) 4 (c) 5 (d) 0
13. Bounded region with corners (0, 0), (4, 0), (0, 4), (2, 3): max $Z=x+2y$
    (a) 4 (b) 8 (c) 6 (d) 10
14. $x+2y=8$ and $3x+2y=12$ meet at
    (a) (3, 2) (b) (4, 2) (c) (2, 3) (d) (0, 4)
15. $Z=px+qy$ is maximum at both (3, 0) and (1, 1) if
    (a) $p=2q$ (b) $p=3q$ (c) $p=q$ (d) $q=2p$
16. Max of $Z=-x+2y$ on $x\ge 3$, $x+y\ge 5$, $x+2y\ge 6$, $y\ge 0$
    (a) does not exist (b) 1 (c) 0 (d) 6
17. The region $x-y\le -1$, $-x+y\le 0$, $x,y\ge 0$ is
    (a) empty (b) bounded (c) unbounded (d) the first quadrant
18. $Z=5x+3y$ at $\left(\frac{20}{19},\frac{45}{19}\right)$
    (a) 12 (b) 9 (c) 10 (d) $\frac{235}{19}$
19. The half-plane containing the origin
    (a) $x+y\ge 2$ (b) $x+y\le 2$ (c) $x\ge 1$ (d) $y\ge 3$
20. Min of $Z=3x+5y$ on $x+3y\ge 3$, $x+y\ge 2$, $x,y\ge 0$
    (a) 9 (b) 10 (c) 7 (d) 6

1 (c) · 2 (a) · 3 (d) · 4 (b) · 5 (b) · 6 (d) · 7 (a) · 8 (c) · 9 (d) · 10 (b) · 11 (c) · 12 (a) · 13 (b) · 14 (c) · 15 (d) · 16 (a) · 17 (a) · 18 (d) · 19 (b) · 20 (c)

Assertion–Reason (6)

(a) Both true, R explains A. (b) Both true, R does not explain A. (c) A true, R false. (d) A false, R true.

1. A: On a bounded feasible region, $Z=ax+by$ has a maximum. R: A linear function on a bounded region attains its maximum at a corner point.
2. A: $Z=x+y$ has no maximum on $x+y\ge 2$, $x,y\ge 0$. R: $Z$ takes the value $2n$ at the feasible point $(n,n)$ for every $n$.
3. A: The origin is not in the region $x+y\ge 1$. R: $0+0<1$.
4. A: In NCERT 12.1 Q7 the maximum 600 occurs at infinitely many points. R: The line $5x+10y=600$ contains an edge of the region.
5. A: An LPP always has a unique optimal solution. R: An optimal value, if it exists, occurs at a corner point.
6. A: The region $x+y\le 4$, $x,y\ge 0$ is bounded. R: $Z=3x+4y$ is linear.

Key: 1 (a) · 2 (a) · 3 (a) · 4 (a) · 5 (d) · 6 (b)

Answers to the practice set