Long Answer TypeShow that the differential equation 
is homogeneous. Find the particular solution of this differential equation, given that x = 0 when y = 1.
Find the vector equation of the plane passing through three points with position vectors 
Also, find the coordinates of the point of intersection of this plane and the line 
A cooperative society of farmers has 50 hectares of land to grow two crops A and B. The profits from crops A and B per hectare are estimated as Rs 10,500 and Rs 9,000 respectively. To control weeds, a liquid herbicide has to be used for crops A and B at the rate of 20 litres and 10 litres per hectare, respectively. Further not more than 800 litres of herbicide should be used in order to protect fish and wildlife using a pond which collects drainage from this land. Keeping in mind that the protection of fish and other wildlife is more important than earning profit, how much land should be allocated to each crop so as to maximize the total profit? Form an LPP from the above and solve it graphically. Do you agree with the message that the protection of wildlife is utmost necessary to preserve the balance in environment?
Let the land allocated for crop A be x hectares and crop B be y hectares.
Maximum area of the land available for two crops is 50 hectares. 
Liquid herbicide to be used for crops A and B are at the rate of 20 litres and 10 litres per hectare respectively. Maximum amount of herbicide to be used is 800 litres. 
The profits from crops A and B per hectare are Rs 10,500 and Rs 9,000 respectively.
Thus, total profit = Rs (10,500x + 9,000y) = Rs 1500 (7x + 6y)
Thus, the linear programming problem is:
Maximize Z = 1500 (7x + 6y) subject to the constraints
The feasible region determined by constraints is represented by the shaded region in the following graph:
The corner points of the feasible region are O (0, 0), A (40, 0), B (30, 20) and C (0, 50). The value of Z at these corner points are
| Corner point | Z = 1500 (7x+6y) | |
| O(0, 0) | 0 | |
| A (40, 0) | 420000 | |
| B (30, 20) | 495000 | Maximum |
| C (0, 50) | 450000 |
Assume that the chances of a patient having a heart attack is 40%. Assuming that a meditation and yoga course reduces the risk of heart attack by 30% and prescription of certain drug reduces its chance by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options, the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation
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