 # IGNOU MTE-09 Solved Assignment 2023 | MTE

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

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Details For MTE-09 Solved Assignment

## 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

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

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

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$$b^2=c^2+a^2-2ac\:Cos\left(B\right)$$

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