Discuss the static stability and dynamic stability for the following demand and supply functions

A patient is given a dose Q mg/ml of a drug at regular interval of time t. The concentration C, of the drug in the blood has been shown experimentally to obey the law 

i) if the first dose is administered at t = 0 hr. then find the concentration after hr. have elapsed.

ii) assuming an instantaneous rise in concentration whenever the drug is administered, find the concentration after the second dose and T hr. have elapsed again.

iii) show that the limiting value R of the concentration for doses of Qmg/ml repeated at interval T hr. is given by the formula

A cup of coffee at 95°C is kept in a room at 25°C . Two minutes later, the temperature of the coffee is 70°C . When will the temperature of the coffee reach 50°C ?

Consider the problem of heat conduction in a one dimensional slab (0

where u(x, 1) is the temperature.

Using the separation of variable method seek the solution ) u(x,t) = T t) X (x) and find

i) equation for T (t), X (x) and corresponding conditions on X (x) to solve these equations

ii) the eigenvalues and the corresponding eigenfunctions of the bvp obtained in i) above for ) X (x)

iii) the solution u(x,t) .

A stone is dropped vertically from a tower of height h. At the same time, another stone is thrown vertically upwards from the base of the tower with a velocity u. What is the minimum value of u so that the two stones will meet each other mid- air?

A raindrop starts falling from the clouds at a considerable height above the surface of the earth. During the fall, the raindrop experiences retardation due to air resistance, which is directly proportional to the instantaneous speed v(t) of the drop.

i) Write the model equations

ii) Is this system static or dynamic? Why?

iii) Obtain an expression for the speed v(t).

iv) Discuss the behaviour of v(t) as the time t changes.

Using dimensional analysis, show that the planets obey Kepler's third law.

Find the Boolean expression in the DNF form for the function defined in tabular form below:

Is there a 4-regular graph on 7 vertices? Justify your answer.

Solve the recurrence relation:

using generating function technique. Also find a₅ using your answer.

How many numbers from 0 to 759 are not divisible by either 3 or 7?

Define vertex connectivity and cut vertex set of any graph G. Find the vertex connectivity and cut vertex set for the following graph:

A box contains 6 red and 4 green balls. Four balls are selected from the box at random. What is the probability that two of the selected balls will be red and two will be green?

Show that if 7 colours are used to paint 50 bicycles and each bicycle is coloured with a single colour, at least 8 bicycles will have the same colour.

If K_m,n for m, n ≥ 2 is Hamiltonian, how are m and n related? Justify your answer.

Find a recurrence relation for a_n, the number of ways to arrange cars in a row with n spaces if we can use Maruti 800, Tata Safari or Scorpio. A Tata Safari or Scorpio requires two spaces, whereas a Maruti 800 requires just one space. Assume that you have unlimited number of each type of car and we do not distinguish between 2 cars of the same type.

Express x⁵ in terms of falling factorials and hence evaluate for m = 0,1,2,3,4,5.

write down and count all the partitions of the number 7. To verify your answer use the generating function for P, taking n =7 in Theorem 5 (of Unit 5, Block2).

In the June, 2021 Term-End Examination of MTE-13, it was asked to give a direct and an indirect proof of the following statement.

"If a,b ∈ Z such that a is even and a+b is even, then b is even."

One student gave an indirect proof as follows:

"Let b be m+1, which is an odd number. We already know a and a+b are even. If we substitute b=m+1 in a + b, it becomes a+m+1, which is an odd number. This contradicts the given statement. Hence b is an even number."

What is wrong with the above proof? Also give a correct direct and an indirect proof.

What do you understand by a subdivision of a graph? Is every subdivision of a Hamiltonian graph Hamiltonian? Justify.