IGNOU MMT004 Solved Assignment 2023  M.Sc. MACS
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IGNOU MMT004 Assignment Question Paper 2023
Course Code: MMT004
Assignment Code: MMT004/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 nonempty 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 \(f(x, y)=\left(yx^{2}\right)\left(y2 x^{2}\right)\).
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
\[
f(x, y)=x+2 y+x yx^{2}y^{2}
\]
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 timevarying system
MMT004 Sample Solution 2023
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