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MTE-09 Solved Assignment 2023
Solved By – Narendra Kr. Sharma – M.Sc (Mathematics Honors) – Delhi University

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IGNOU MTE-09 Assignment Question Paper 2023

 

Course Code: MTE-09

Assignment Code: MTE-09/TMA/2023

Maximum Marks: 100

1. Are the following statements true or false? Give reasons for your answer.

a) \(\frac{1}{2}\) is a limit of the interval ] \(-2.5,1.5[\)

b) Every function differentiable on \([a, b]\) is bounded on \([a, b]\)

c) The function \(f\) defined by \(f(x)=\left|x-\frac{5}{2}\right|, x \in \mathbf{R}\) has a local maxima of \(x=\frac{5}{2}\).

d) If \(\lim _{n \rightarrow \infty} u_{n}=0\), then the series \(\sum_{n=1}^{\infty} u_{n}\) is convergent.

e) A Riemann integrable function is not necessarily differentiable.

2. a) If \(\mathrm{a} \in \mathbf{R}\) is such that \(\mathrm{o} \leq \mathrm{a} \leq \varepsilon \forall \varepsilon>0\), then show that \(\mathrm{a}=0\).

b) Using the principle of mathematical induction, show that

\[
\frac{1}{1 \times 2}+\frac{1}{2 \times 3}+\ldots .+\frac{1}{n(n+1)}=\frac{n}{n+1}
\]

c) Show that the union of a countable collection of open sets is open.

d) Check whether the set of integers is compact or not.

3. a) Check whether the following functions are continuous or not at \(x=0\). Also, find the nature of discontinuity at that point, if it exists.

(i) \(f(x)=\left\{\begin{array}{cl}\frac{\sqrt{2-x}-\sqrt{2+x}}{x} & , x \neq 0 \\ \frac{1}{\sqrt{2}} & , x=0\end{array}\right.\)

(ii) \(f(x)=\left\{\begin{array}{cl}x^{2}+\frac{1}{3} & , x \leq 0 \\ -\left(x^{3}+\frac{1}{3}\right) & , x>0\end{array}\right.\)

b) Evaluate the following limit if it exists:

\[
\lim _{x \rightarrow 0} \frac{4 x^{3}}{\tan ^{3} x+\tan x-x}
\]

c) Show that the function \(f\) given by

\[
\left.\mathrm{f}(\mathrm{x})=\frac{1}{(\mathrm{x}+2)^{3}}, \forall \mathrm{x} \in\right]-2,2[
\]

is continuous but not bounded in \(]-2,2[\).

4. a) For the following sequences, find two subsequences which are convergent:

(i) \(\mathrm{a}_{\mathrm{n}}=\mathrm{n}\left[1+(-1)^{\mathrm{n}}\right]\).

(ii) \(\mathrm{a}_{\mathrm{n}}=\sin \left(\frac{\mathrm{n} \pi}{3}\right)\).

b) Check whether the following sequences \(\left\{s_{n}\right\}\) are Cauchy, where

(i) \(\mathrm{s}_{\mathrm{n}}=1+2+3+\ldots+\mathrm{n}\)

(ii) \(s_{n}=\frac{4 n^{3}+3 n}{3 n^{3}+n^{2}}\)

c) Give an example of an infinite set with finite number of limit points, giving justification.

d) Evaluate: \(\lim _{x \rightarrow \infty}\left(\sqrt{2 x^{2}+3 x-2}-\sqrt{2 x^{2}-3 x+2}\right)\)

5. a) Test for convergence the following series:

(i) \(\sum_{n=1}^{\infty} \frac{7^{n}}{(3 n+1) !}\)

(ii) \(\sum_{n=1}^{\infty} \frac{1}{n \sqrt{\log n}}\)

b) Test the following series for absolute and conditional convergence:

(i) \(\sum_{n=1}^{\infty}(-1)^{\mathrm{n}} \frac{5}{3 n+1}\)

(ii) \(\sum_{\mathrm{n}=1}^{\infty} \frac{\sin \mathrm{nx}}{\mathrm{n}^{3}}\)

c) Show that the set \(B=\left\{x \mid x^{2}>2\right\}\) is non-empty and bounded below. Is it bounded above? Justify.

6. a) Show that the function \(\mathrm{f}\) defined on \(\mathbf{R}\) by

\[
f(x)=\left\{\begin{array}{cc}
3 x^{2} \cos \frac{1}{2 x}, & \text { when } x \neq 0 \\
0, & \text { when } x=0
\end{array}\right.
\]

is derivable on \(\mathbf{R}\) but \(\mathrm{f}^{\prime}\) is not continuous at \(\mathrm{x}=0\). b) Find the least and the greatest values of the function \(f\) defined by

\[
f(x)=3 x^{4}-4 x^{3}+6 x^{2}+36 x-5
\]

on the interval \([0,2]\).

c) Using Taylor’s Theorem, prove that

\[
\cos x \leq 1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !} \forall \mathrm{x} \in \mathbf{R}
\]

7. a) Test the following series for convergence:

\[
\frac{2.4}{3.5}+\frac{2.4 .6}{3.5 .7} x+\frac{2.4 .6 .8}{3.5 .7 .9} x^{2}+\ldots(x>0)
\]

b) Show that the function \(\mathrm{f}:[0,1] \rightarrow \mathbf{R}\) defined by

\[
f(x)= \begin{cases}1 & \text { when } x \text { is rational } \\ 2 & \text { when } x \text { is irrational }\end{cases}
\]

is not Riemann integrable.

c) Evaluate the limit as \(\mathrm{n} \rightarrow \infty\) of the sum

\[
\frac{1}{n}\left[\sin \frac{\pi}{n}+\sin \frac{2 \pi}{n}+\ldots+\sin \frac{2 n \pi}{n}\right]
\]

8. a) Compute the Riemann integral of the function \(f(x)=|x|\) on the interval \([-1,1]\).

b) Suppose that \(\mathrm{f}\) is a non-negative continuous function on \([\mathrm{a}, \mathrm{b}]\) and

\[
\int_{a}^{b} f(x) d x=0 \text { Prove that } f(x)=0 \forall x \in[a, b] \text {. }
\]

c) Check whether the function \(\mathrm{f}(\mathrm{x})=[\mathrm{x}]+\mathrm{e}^{\mathrm{x}}\) is integrable in \([0,3]\).

9. a) Verify the second mean value theorem of integrability for the functions \(f\) and \(g\) defined on \([1,2]\) by \(f(x)=3 x\) and \(g(x)=5 x\).

b) Show that the sequence \(\left(f_{n}\right)\) where \(f_{n}(x)=\frac{x}{1+2 n x^{2}}, x \in[1, \infty[\) is uniformly convergent in \([1, \infty[\)

c) Verify Inverse function theorem for finding the derivative at a point \(y_{0}\) of the domain of the inverse function of the function \(f(x)=\cos x, x \in[0, \pi]\). Hence, find the derivative of the inverse function at \(\mathrm{y}_{0}\). 10. a) Find the upper and lower integrals of the function \(\mathrm{f}\) defined by

\[
\mathrm{f}(\mathrm{x})=\frac{7}{2}-2 \mathrm{x}, \forall \mathrm{x} \in[1,3]
\]

Is f integrable over the interval \([1,3]\) ? Justify.

b) Check whether the function \(f(x)=\sin \frac{1}{x}(x \neq 0)\) is uniformly continuous in the interval ]0,2[.Is it continuous? Justify.

c) Find the value of \(a \in \mathbf{R}\) for which

\[
\lim _{x \rightarrow \infty} \frac{(3 x+4)(x-1)(2 x+1)}{a x^{3}+x-4} \text { exists. }
\]

[stray-random categories=trigonometry]

MTE-09 Sample Solution 2023

 

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[stray-random categories=trigonometry]

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