 # 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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