MMT-004 Solved Assignment 2023

IGNOU MMT-004 Solved Assignment 2023 | M.Sc. MACS

Solved By – Narendra Kr. Sharma – M.Sc (Mathematics Honors) – Delhi University


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IGNOU MMT-004 Assignment Question Paper 2023

Course Code: MMT-004
Assignment Code: MMT-004/TMA/2023

1. State whether the following statements are True or False. Give reasons for your answers.

a) The function \(\varphi(x)=\frac{1}{\mathrm{x}}, 3 \leq \mathrm{x} \leq 4\) is not uniformly continuous.

b) A complete metric space is a countable collection of nowhere dense sets.

c) The function \(\mathrm{f}: \mathbf{R} \rightarrow \mathbf{R}^{2}\) given by \(\mathrm{f}(\mathrm{x}, \mathrm{y})=(\mathrm{x}, \mathrm{x}|\mathrm{x}|)\) is differentiable at 0 .

d) Any Lebesgue intergrable function is always Riemann integrable.

e) The image of any connected set in \(\mathbf{R}^{2}\) under the function \(\mathrm{f}: \mathbf{R}^{2} \rightarrow \mathbf{R}\) given by \(f(x, y)=x^{2}+y^{2}\) is connected.

2. a) Let \(\mathrm{A}\) and \(B\) be non-empty disjoint closed subsets of a metric space \((\mathrm{X}, \mathrm{d})\). Show that there exist open sets \(\mathrm{U} \supset \mathrm{A}\) and \(\mathrm{V} \supset \mathrm{B}\) such that \(\mathrm{U} \cap \mathrm{V}=\phi\)..

b) Define saddle points. Compute the saddle points of the function \(\mathrm{f}: \mathbf{R}^{2} \rightarrow \mathbf{R}\) given by \(f(x, y)=\left(y-x^{2}\right)\left(y-2 x^{2}\right)\).

c) State the Lebesgue dominated convergence theorem. Find \(\lim _{n \rightarrow \infty}^{\infty} \int_{0}^{\infty} \frac{\sin x}{1+n x^{2}} d x\).

3. a) Define components in a metric space. What are all the components of the set of all nonzero real numbers under the

i) usual metric on \(\mathbf{R}\), and

ii) the discrete metric on \(\mathbf{R}\) ?

b) Find the directional derivation of the function \(\mathrm{f}: \mathbf{R}^{4} \rightarrow \mathbf{R}^{4}\) defined by

f(x, y, z, w)=\left(x^{2} y, x y z, x^{2}+y^{2}, z w\right)

at \((1,2,-1,-2)\) in the direction \(~=(1,0,-2,2)\).

c) Define measurable sets in \(\mathbf{R}\). Prove that intervals are measurable.

4. a) If \(\mathrm{f}: \mathrm{X} \rightarrow \mathrm{Y}\) is a continuous map between metric spaces \(\mathrm{X}\) and \(\mathrm{Y}\) and \(\mathrm{K}\) is a compact subset of \(X\), then show that \(f(K)\) is compact.

b) Find the Taylor series expansion of the function \(\mathrm{f}\) given by

f(x, y)=x+2 y+x y-x^{2}-y^{2}

about the point \((1,1)\)

c) Let \(f, g \in L^{\prime}(\mathbf{R})\), define convolution \(f * g\) of \(f\) and \(g\). Show that if either \(f\) or \(g\) is bounded, then the convolution \(\mathrm{f} * \mathrm{~g}\) exists for all \(\mathrm{x}\) in \(\mathbf{R}\) and is bounded in \(\mathbf{R}\).

5. a) Let \(\left\{x_{n}\right\}\) and \(\left\{y_{n}\right\}\) be Cauchy sequences in a metric space \((X, d)\). Show that the sequence \(\left\{d\left(x_{n}, y_{n}\right)\right\}\) converges in \(\mathbf{R}\).

b) Consider the function \(\mathrm{f}: \mathbf{R}^{3} \rightarrow \mathbf{R}\) given by

f(x, y, z)=x^{2}+y^{3}-x y \sin z

Prove that the equation \(\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z})=0\) defines a unique continuously differentiable function of near \((1,-1)\) such that \(\mathrm{g}(1,-1)=0\).

c) Define and give an example for each of the following concepts in the context of signals and systems:

i) A stable system

ii) A time-varying system

\(cos^2\left(\frac{\theta }{2}\right)=\frac{1+cos\:\theta }{2}\)

MMT-004 Sample Solution 2023


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\(cos\left(\theta +\phi \right)=cos\:\theta \:cos\:\phi -sin\:\theta \:sin\:\phi \)

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