Differential Equations Class 12 | Language Based Questions | Mathematics | NEB | New Course

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Questions

1. A water tank is filled in a such a way that the rate at which the depth of water increases is proportional to the square roots of the depth. Initially the depth is 'n' meters.

a) Write down a differential equation for h.

b) show that √h = kt / 2+ √n where k is a constant.

c) When h = 16 and t = 6 hours, prove that n = (4 - 3k)². [NEB 2079]

2. A college hostel accommodating 1000 students; one of them came from abroad with infection of corona virus, then the hostel was isolated. If the rate at which the virus spreads is assumed to be proportional to the product of the number 'N' of infected students and number of non-infected students and the number of infected students is 50 after 4 days.

a) Express the above information in the form of differential equation.

b) Solve the differential equation.

c) Show that more than 95% students will be infected after 10 days.

3. Water is drained from a vertical cylindrical tank by opening a valve at the base of the tank. It is known that the rate at which the water level drops is proportional to the square root of water depth y, where the constant of proportionality k > 0 depends on the acceleration due to gravity and the geometry of the hole. If t is measure in minutes and k=1/51 then what is the time taken to make the tank empty if water is 4 m deep at the beginning? [Ans: 60 min]

4. Water flows out of a tank through a hole in the bottom and, at time t minutes, the depth of water in the tank is x metres. At any instant, the rate at which the depth of water in the tank is decreasing is proportional to the square root of the depth of water in the tank

a) Write down a differential equation which models this situation.

b) When t = 0, x = 2; when t = 5, x = 1. Find t when x = 0.5, giving your answer correct to 1 decimal place. [Ans: 8.5 min]

5. The rate at which body temperature T falls is proportional to the difference between the body temperature T and the temperature T, of the surroundings. Find a differential equation relating body temperature, T, and time t.

6. In studying the spread of a disease, a scientist thinks that the rate of infection is proportional to the product of the number of people infected and the number of people uninfected. If N is the number infected at time t and P is the total number of people in the population, form a differential equation to summarize the scientist's theory.

7. The rate of increase of a population is proportional to its size at the time. Write down a differential equation to describe this situation. It is also known that when the population was 2 million, the rate of increase was 1,40,000 per day. Find the constant of proportionality in your differential equation. [Ans: 0.07]

8. A tank is draining in such a way that when the height of water in the tank is h cm, it is decreasing at the rate of 0.5√h cm/s. Initially the water in the tank is at a height of 25 cm.

a) Write down a differential equation which describes this situation.

b) Solve the differential equation to find h as a function of time.

c) What is the height of the water after 10 seconds?

d) How long does it take for the water to reach a height of 5 cm? [Ans: c) 6.25 cm d) 11.1 sec]

9. A stone falls through the air from rest and, t seconds after it was dropped, its speed v satisfies the equation dv/dt=10-0.2v

a) Show that v=50( 1-e^-0.2t ).

b) Calculate the time at which the stone reaches a speed of 20 m/s. [Ans: b) 2.55 sec]

10. The spread of a disease occurs at a rate proportional to the product of the number of people infected and the number not infected. Initially 50 out of a population of 1050 are infected and the disease is spreading at a rate of 10 new cases per day.

a) If n is the number infected after t days, show that : dn/dt=n(1050-n)/5000

b) Solve this differential equation to find the number of people infected after t days.

c) How long will it take for 250 people to be infected?

d) Explain why everyone in the population will eventually be infected. [Ans: c) 8.73 days]