MEC-203 English Medium Solved Assignments 2024-25

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MEC-203 English Medium Solved Assignments 2024-25 Available

PART I
Answer the following questions.
1. Consider an investor who at time t = 0 is endowed with initial capital of x(0)=x0 > 0. At any time t ∈ [0,T] where T > 0 is given he decides about his rate of consumption c(t) ∈ [0, c̄] wherec̄ >0 is a large maximum allowable rate of consumption thus his capital stock evolves according to ẋ=αx –c(t) where α>0 is a given rate of return the investor’s time-t utility for consuming at rate c(t) is u(c(t)) where u=P+ → P is his increasing strictly concave utility function. The investor’s problem is to find a consumption plan c(t), t ∈ [0,T]so as to maximize his discounted utility.
𝐽(𝑐) = ∫ 𝑒
−𝑟𝑡𝑢(𝑐(𝑡))𝑑𝑡
𝑇0
𝑢(𝑐(𝑡)) = ln(𝛽𝑡)

𝑥(0) = 𝑥0
𝑥𝑡 = 0
where r ≥ 0 is a given discount rate subject to the solvency constraint that the capital stock x(t)
must be positive for all t ∈ [0,T). [The value 𝛽 lies in [0, ∞).]
2. Consider a fishing optimal control problem which is defined by
Pt = a + bpt − xt
where Pt is fish population at time t. xt
is fishing intensity or catch (a & b are constants). If r is the discount rate and objective function
V(.) e u(x )dt,u(x ) t t
0rt − =is utility function of
consumption i.e.,
ct = xt
a) State the transversality condition

b) Find the optimal consumption,
xtift Lnxt u(x ) =
PART II
Answer the following questions.
3. a) Let J be the functional defined by
𝐽(𝑦) = ∫ (𝑦
′2 − 𝑦
2 + 2𝑡𝑦)𝑑𝑡=10
With boundary conditions y(0) = 0 and y (1) = 1 Find the extremal(s) in interval [0, 1] for
J(y).
b) Find extremals for
∫ {[𝑥
′(𝑡)]
2 + 10𝑥(𝑡).𝑡}
10 𝑑𝑡
subject to x(t0) = 2 x(t1) = 3
4. a) What is a standard error and why is it important?
b) In a random sample of 400 students of the university teaching departments, it was found that 300 students failed in the examination. In another random sample of 500 students of the affiliated colleges, the number of failures in the same examination was found to be
300. Find out the S.E of the difference between proportion of failures (i) in the university teaching departments and (ii) in the university teaching departments and affiliated colleges taken together.
5. a) Consider the following Lagrange problem:
Maximise
f (u,v) = u + 3v
subject to
( , ) 10.
2 2
g u v = u + av =
Use the envelope theorem to estimate the maximum value
(1.01)
*
f when
a =1.01,
andcheck this by computing the optimal value function
( )
*f a .
b) Maximise the function
2 2
f (u;a) = −u + 2au + 4a
with respect to
a  0 .
6. a) For the function f(x)= cos(x), find (i) linear and quadratic approximations, and (ii)
Maclaurin’s series expansion of the function.
b) Let f: R2→ R
2 be defined as f (x, y) = (e2xy, 2×2 +3 y2
), find the Jacobian Jf at the point
(2, 1).

7. Write short notes on following:
a) Homeogeneous and Homothetic functions
b) L’Hopital’s rule
c) Order of a difference equation
d) Cramer –Rao inequality