IGNOU MMT 4 SOLVED ASSIGNMENT 2025

Ques 1.

State whether the following statements are true or false. Justify your answers.

a) The outer measure m* of the set

b) A finite subset of a metric space is totally bounded.

c) A connected subspace in a metric space which in not properly contained in any other connected subspace is always open.

d) The surface given by the equation x+y+z-sin(xyz) = 0 can also be described by an equation of the form z = f(x, y) in a neighbourhood of the point (0,0).

e) A real valued function f on [a,b] is continuous if it is integrable on [a,b].

Ques 2.

Find the interior, closure, the set of limit points and the boundary of the set

in R² with the standard metric.

Ques 3.

Consider 

f(x,y,z) = (2x+3y+z,xy,yz,xz)

Find ƒ (2,0,-1).

Ques 4.

Does Cantor's intersection theorem hold for the metric space X = (0,1] with the standard metric? Justify your answer.

Ques 5.

Obtain the second Taylor's series expansion for the function given by

Ques 6.

Find the Lebesgue integral of the function f given by

Ques 7.

Find and classify the extreme values of (x, y) = xy Subject to the constraint

Ques 8.

be given by

Show that f is locally invertible at all points in  \ {(0,0,0)}.

Ques 9.

For the equation , x² + y³ + z³ = at which points on its solution set, can we assured that there is a neighbourhood of the point in which the surface given by the equation can be described by an equation of the form z = f (x, y) .

Ques 10.

Find the Fourier series of  f (t) = t² on [−π,π].

Ques 11.

Prove that if an open set U can be written as the union of pariwise disjoint family V of open connected subsets, then these subsets must be the components of U. Use this theorem to find the components of the set D U E where

Ques 12.

Which of the following subsets of R are compact w.r.t. the metric given against them. Justify your answer.

i) A = (1,0) in of − with standard metric.

ii) A = [4,3] − with discrete metric.

iii) {(x, y) ∈ y > 0} − with standard metric

Ques 13.

If E is a subset of with standard metric, then show that 

Ques 14.

Show that a set A in a metric space is closed if and only if every convergent sequence in A converges to a point of A.

Ques 15.

Find the interior and closure of the set of rationals in with standard metric.

Ques 16.

Let F be the function from to defined by

F(x, y) = (x² + y² , xy)

Show that F is differentiable at (2,1) . Find the differential matrix of F.

Ques 17.

Show that the function f defined by

is not differentiable at (0,0). Does the partial derivatives of f exists at (0,0)? or any at any other point in R²? Justify your answer.

Ques 18.

Is the continuous image of a Cauchy sequence a Cauchy sequence? Justify.

Ques 19.

Find the directional derivative of the function    defined by

at the point (1,2,-1,-2) in the direction v = (1,0,-2,2).

Ques 20.

Suppose that is given by f(t) = (t,t²) and is given by g(x, y) = (x², xy, y²-x²). Compute the derivative of gof.

Ques 21.

 Find  in  where d is the metric given by

Ques 22.

Give an example of a family f₁ of subsets of a set X which has finite intersection property. Justify your choice of example.

Ques 23.

Verify the hypothesis and conclusions of the Fatou's lemma for the sequence {f_n} given by

Ques 24.

Let (X,d) be a metric space and A be a subset of X. Show that bdy(A) = Qif and only if A is both open and closed.

Ques 25.

Give an example of an algebra which is not a σ − algebra. Justify your choice of examples.

Ques 26.

If E is a measurable set and f is a simple function such that a , show that

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