CBSE Guess > Papers > Important Questions > Class XII > 2010 > Maths > Mathematics By Mr. M.P.Keshari

Linear Programming

Q.1. David wants to invest at most Rs12,000 in Bonds A and B. According to the rule, he has to invest at least Rs2,000 in Bond A and at least Rs4,000 in Bond B. If the rate of interest in bonds A and B respectively are 8% and 10% per annum, formulate the problem as L.P.P. and solve it graphically for maximum interest. Also determine the maximum interest received in a year.

Solution :

Fig.

Let Rs x be invested in bond A and Rs y in bond B.
The L.P.P. is
Maximize : I = Interest = 0.08 x + 0.10 y
Subject to the constraints
x ≥ 2000
y ≥ 4000
x + y ≤ 12,000
x ≥ 0, y ≥ 0.
The lines x = 2000 ------------------------ (1)
y = 4000 ------------------------- (2)
and x + y = 12,000 ----------------------- (3)
are drawn on the same graph paper.
The shaded region ABCA represents the feasible region.
Now,
(I)A = 0.08 × 2000 + 0.10 × 4000 = Rs560.
(I)B = 0.08 × 8000 + 0.10 × 4000 = Rs1040.
(I)C = 0.08 × 2000 + 0.10 × 10,000 = Rs1160.
Therefore, the interest I is maximum at C(2000, 10000) and the maximum interest is Rs1160. [Ans.]

Q.2. A dealer wishes to purchase a number of fans and sewing machines. He has only Rs5,760 to invest and has space for at most 20 items. A fan and sewing machine cost Rs360 and Rs240 respectively. He can sell a fan at a profit of Rs22 and sewing machine at a profit of Rs18. Assuming that he can sell whatever he buys, how should he invest his money in order to maximize his profit? Translate the problem into LPP and solve it graphically.

Solution:

Let x fans and y machines are purchased by the dealer. The profit function P is defined as P = 22 x + 18 y.
The two variables x and y satisfy the constraints :
360 x + 240 y ≤ 5760
ó 3x + 2y ≤ 48.
x + y ≤ 20
x ≥ 0, y ≥ 0.
The lines 3x + 2y = 48 -------------------- (1)
And x + y = 20 -------------------- (2) are drawn.

Fig.

The above lines intersect at R(8, 12).
The shaded region OARD represents the feasible region.
Now (P)O = 0,
(P)A = 22×16 +18×0 = 352,
(P)R = 22×8 + 18×12 = 392,
(P)D = 22×0 + 18×20 = 360.
Thus profit is maximum at R(8, 12).
When 8 fans and 12 sewing machines are purchased and sold, the profit is maximum and is Rs392. [Ans.]

Paper By Mr. M.P.Keshari
Email Id : [email protected]
Ph No. : 09434150289