IGNOU MTE 9 SOLVED ASSIGNMENT 2025

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IGNOU MTE 9 SOLVED ASSIGNMENT 2025

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IGNOU MTE 9 Solved Assignment 2025

This is latest Solved Assignment of MTE 9 of BSC .

Latest 2025 Solved Assignment
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MTE 9 ( Real Analysis )
Real Analysis 2025 Solved Assignment
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Last Date of Submission of IGNOU MTE-09 (BSC) 2025 Assignment is for January 2025 Session: 30th September, 2025 (for December 2025 Term End Exam).
Semester Wise
January 2025 Session: 30th March, 2025 (for June 2025 Term End Exam).
July 2025 Session: 30th September, 2025 (for December 2025 Term End Exam).

Title Name	IGNOU MTE 9 Solved Assignment 2025
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	BACHELOR DEGREE PROGRAMMES
Course Code	BSC
Course Name	Bachelor in Science
Subject Code	MTE 9
Subject Name	Real Analysis
Year	2025
Session
Language	English Medium
Assignment Code	MTE-09/Assignmentt-1//2025
Product Description	Assignment of BSC (Bachelor in Science) 2025. Latest MTE 09 2025 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU MTE-09 (BSC) 2025 Assignment is for January 2025 Session: 30th September, 2025 (for December 2025 Term End Exam). Semester Wise January 2025 Session: 30th March, 2025 (for June 2025 Term End Exam). July 2025 Session: 30th September, 2025 (for December 2025 Term End Exam).

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Questions Included in this Help Book

Ques 1.

Are the following statements true or false? Give reasons for tour answers.

a) -2 is a limit point of the interval 1-3,2].

b) The series $\frac{1}{2}-\frac{1}{6}+\frac{1}{10}-\frac{1}{14}+\cdots$ is divergent.

The function, f(x)=sin² x is uniformly continuous in the interval [0, π).

d) Every continuous function is differentiable.

e) The function f defined on R by $f(x)=\begin{cases}0,&\text{x is rational}\\2,&\text{x is irrational}\end{cases}$

is irrational Is integrable in the interval [2,3].

Ques 2.

Prove that the union of two closed sets is a closed set. Give an example to show that union of an infinite number of closed sets need not be a closed set.

Ques 3.

Examine the function f : R→R defined by

$f(x)=\begin{cases}\frac{1}{6}(x+1)^3&x\ne 0\\\frac{5}{6}&x=0\end{cases}$

for continuity on R. If it is not continuous at any point of R, find the nature of discontinuity there. (x+1)x+0 9-59-1

Find lim $\lim_{x\to 0}\frac{1-\cos^2 x}{x^2\sin^2 x}$

Ques 4.

Using the principle of mathematical induction, prove that $\frac{n^5}{5}+\frac{n^3}{3}+\frac{7n}{15}$ is natural numbuer. $\forall n\in\mathbb{N}.$

Ques 5.

Show that there is no real number, & for which the equation, $x^4-3x^2+k=0$ has two distinct roots in the interval [2,3].

Ques 6.

Let :[-3,3]→ R be defined by $f(x)=5(x)+x^3,$ where [x] denotes the greatest integer ≤x. Show that this function is integrable.

Ques 7.

Prove that the function $f$ defined by

$f(x)=\begin{cases}2,&\text{if}x\text{is irrational}\\-2,&\text{if}x\text{is rational}\end{cases}$ l is discontinuous, $\forall x\in\mathbb{R},$ , using the sequential definition of continuity.

Ques 8.

Examine the convergence of the following series

(i) $\frac{3\times4}{5^2}+\frac{5\times6}{7^2}+\frac{7\times8}{9^2}+\cdots$

(ii) $1+4x+4^2 x^2+4^3 x^3+\cdots(x>0)$

Ques 9.

Prove that the set of integers is countable.

Ques 10.

Prove that

$\lim_{n\rightarrow\infty}\left[\frac{1}{\sqrt{2n-1^2}}+\frac{1}{\sqrt{4n-2^2}}+\frac{1}{\sqrt{6n-3^2}}+\cdots+\frac{1}{n}\right]=\frac{\pi}{2}$

Ques 11.

Prove that the sequence $\left\{\frac{a_n}{n}\right\}$ is convergent where ( $\alpha$ _n)is a bounded sequence.

Ques 12.

Prove that continuous function of a continuous function is continuous.

Ques 13.

Examine the function, $f(x)=(x+1)^3(x-3)^2$ for extreme values

Ques 14.

Show that the series $\sum_{n=1}^{\infty}\frac{x}{1+n^2x^2}$ is uniformly convergent in $[\alpha,1]$ for any a>0.

Ques 15.

Give an example of an infinite set with finite number of limit points, with proper justification.

Ques 16.

Show that

(i) $\lim_{x\rightarrow\infty}\left(\frac{x-3}{x-1}\right)^x=\frac{1}{e^2}$

(ii) $\lim_{x\rightarrow\frac{5}{3}}\frac{1}{(3x+5)^2}=\infty$

Ques 17.

For the function, $f(x)=x^2-2$ defined over [1,5], verify: $L(P,f)\le U(-P,f)$ where P is the partition which divides [1.5] into four equal intervals.

Ques 18.

Let (a) be a sequence defined as $a_1=3,a_{n+1}=\frac{1}{5}a_n$ show that (a_n) converges

Ques 19.

Using the sequential definition of the continuity, prove that the function $f$ defined

by: $f(x)=\begin{cases}3,&\text{if}x\text{is irrational}\\-3,&\text{if}x\text{is rational}\end{cases}$ is discontinuous at each real number.

Ques 20.

Show that on the curve, $y=3x^2-7x+6,$ the chord joining the points whose abscissa are x=1 and x=2. is parallel to the tangent at the whose abscissa is $a x\frac{3}{2}.$

Ques 21.

Give an example of a series ∑ $a_{n}$ such that ∑ $a_{n}$ is not convergent but the $(a_{n)$ sequence converges to 0.

Ques 22.

Test the series:

$\sum_{n=1}^{\infty}(-1)^{n-1}\frac{\sin nx}{n\sqrt{n}}$ for absolute and conditional convergenc

Ques 23.

Check whether the function $f$ given by:

$f(x)=(x-4)^3(x+1)^2$ has local maxima and local minima.

Ques 24.

Check, whether the collection $G'=\left\{\left]\frac{1}{n+2},\frac{1}{n}\right[:n\in\mathbb{N}\right\}$ given by: open cover of $]0,1[$

Ques 25.

State Bonnet's mean value theorem for integrals. Apply it to show that:

$\left|\int_{3}^{5}\frac{\cos x}{x}dx\right|\le\frac{2}{3}$

Ques 26.

Show that the sequence (a), where $a_n\frac{n}{n^2+4}$ is monotonic. Is $(a_n)a$ Cauchy sequence Justify your answer.

Ques 27.

Check whether the intervals [2.5] and [7,10] are equivalent or not.

Ques 28.

बताइए निम्नलिखित कथन सत्य हैं या असत्य। अपने उत्तरों के कारण बताइए।

क) - 2 अंतराल ]-3,2] का सीमा बिन्दु है।

ख) श्रेणी $\frac{1}{2}-\frac{1}{6}+\frac{1}{10}-\frac{1}{14}+\cdots$ अपसारी है।

ग) फलन f (x) = sin²x अंतराल [0, $\pi$ ] पर एकसमानतः सतत है।

घ) प्रत्येक सतत फलन अवकलनीय है।

ड.) $f(x)=\begin{cases}0,&\text{x is rational}\\2,&\text{x is irrational}\end{cases}$ द्वारा $\mathbb{R}$ पर परिभाषित फलन ƒ अंतराल [2, 3] में समाकलनीय है।

Ques 29.

सिद्ध कीजिए कि दो संवृत समुच्चयों का सम्मिलन संवृत समुच्चय है। यह दिखाने के लिए एक उदाहरण दीजिए कि संवृत समुच्चयों की परिमित संख्या का सम्मिलन संवृत समुच्चय नहीं भी हो सकता है।

Ques 30.

R पर सातत्य के लिए

$f(x)=\begin{cases}\frac{1}{6}(x+1)^3&x\ne 0\\\frac{5}{6}&x=0\end{cases}$

द्वारा परिभाषित फलन f: IR IR की जाँच कीजिए। यदि R के किसी बिन्दु पर सतत नहीं है तब असातत्य का प्रकार ज्ञात कीजिए

Ques 31.

$\lim_{x\to 0}\frac{1-\cos^2 x}{x^2\sin x^2}$ ज्ञात कीजिए।

Ques 32.

गणितीय आगम नियम द्वारा सिद्ध कीजिए कि $\frac{n^5}{5}+\frac{n^3}{3}+\frac{7n}{15}$ एक प्राकृत संख्या है, $\forall n\in\mathbb{N}.$

Ques 33.

दिखाइए कि ऐसी कोई वास्तविक संख्या नहीं है जिसके लिए समीकरण x⁴-3x²+k=0 के अंतराल [2,3] में दो अलग-अलग मूल होते हैं।

Ques 34.

मान लीजिए $f:[-3,3]\to\mathbb{R},\quad f(x)=5(x)+x^3$ द्वारा परिभाषित है, जहाँ [x] महत्तम पूर्णांक $\le x$ को निरूपित करता है। दिखाइए कि यह फलन समाकलनीय है।

Ques 35.

सातत्य की अनुक्रमिक परिभाषा द्वारा सिद्ध कीजिए कि

$f(x)=\begin{cases}2,&\text{if x is irratioanl}\\-2,&\text{if x is rational}\end{cases}$

द्वारा परिभाषित फलन f असतत है, $\forall x\in\mathbb{R}$ I

Ques 36.

निम्नलिखित श्रेणी के अभिसरण की जाँच कीजिए:

i) $\frac{3\times 4}{5^2}+\frac{5\times 6}{7^2}+\frac{7\times 8}{9^2}+\cdots$

ii) $1+4x+4^2 x^2+4^3 x^3+\cdots\quad(x>0)$

Ques 37.

सिद्ध कीजिए कि पूर्णांकों का समुच्चय गणनीय है।

Ques 38.

सिद्ध कीजिए कि

$\lim_{n\rightarrow\infty}\left[\frac{1}{\sqrt{2n-1^2}}+\frac{1}{\sqrt{4n-2^2}}+\frac{1}{\sqrt{6n-3^2}}+\cdots+\frac{1}{\sqrt{2n(n)-n^2}}\right]=\frac{\pi}{2}$

Ques 39.

सिद्ध कीजिए कि अनुक्रम $\left\{\frac{a_n}{n}\right\}$ अभिसारी। है जहाँ {a_n} परिबद्ध अनुक्रम है।

Ques 40.

सिद्ध कीजिए कि संतत फलन का संतत फलन संतत होता है।

Ques 41.

चरम मानों के लिए फलन f(x) = (x+1)³ (x-3)² की जाँच कीजिए।

Ques 42.

दिखाइए कि किसी भी a > 0 श्रेणी $\sum_{n=1}^{\infty}\frac{x}{1+n^{2}x^{2}},\quad[\alpha,1]$ में एकसमानतः अभिसारी है।

Ques 43.

उचित पुष्टि के साथ सीमा बिन्दुओं की परिमित संख्या वाले एक अपरिमित समुच्चय का उदाहरण दीजिए।

Ques 44.

दिखाइए कि

i) $\lim_{x\rightarrow\infty}\left(\frac{x-3}{x-1}\right)^{x}=\frac{1}{e^2}$

ii) $\lim_{x\rightarrow\frac{5}{3}}\frac{1}{(3x+5)^{2}}=\infty$

Ques 45.

[1,5] पर परिभाषित फलन f(x) = x²-2 के लिए सत्यापित कीजिए L(P, f) ≤U(-P, f) जहाँ P एक विभाजन है जो [1,5] को चार बराबर अंतरालों में विभाजित करता है।

Ques 46.

मान लीजिए $a_1=3,\quad a_{n+1}=\frac{1}{5}a_n$ के रूप में परिभाषित अनुक्रम {a_n} है। दिखाइए कि {a_n} शून्य तक अभिसरण करता है।

Ques 47.

फलन के सांतत्य की अनुक्रमिक परिभाषा का प्रयोग करते हुए सिद्ध कीजिए कि

$f(x)=\begin{cases}3,&\text{if}\,\,x\text{\,\,\,is irrational}\\-3,&\text{if}\,\,x\text{\,\,is rational}\end{cases}$

द्वारा परिभाषित फलन f, प्रत्येक वास्तविक संख्या पर असंतत है।

Ques 48.

दिखाइए कि वक्र y = 3x²-7x+6, पर बिन्दुओं को, जिनकी भुज x = 1 और x = 2, हैं, मिलाने वाली जीवा उस बिन्दु पर खींची गई स्पर्श रेखा के समान्तर होती है जिसकी भुजा $x=\frac{3}{2}.$

Ques 49.

ऐसी श्रेणी $\sum a_n$ का उदाहरण दीजिए जिसके लिए $\sum a_n$ , अभिसारी नहीं है लेकिन अनुक्रम (a_n) 0 की ओर अभिसरण करता है।

Ques 50.

निरपेक्ष और सप्रतिबंध अभिसरण के लिए श्रेणी

$\sum_{n=1}^{\infty}(-1)^{n-1}\frac{\sin nx}{n\sqrt{n}}$

की जाँच कीजिए।

Ques 51.

जाँच कीजिए :

f (x) = (x-s)³ (x+1)²

द्वारा दिए गए फलन f का स्थानीय उच्चिष्ठ और स्थानीय निम्निष्ठ होता है या नहीं।

Ques 52.

जाँच कीजिए कि :

$G'=\left\{\left]\frac{1}{n+2},\frac{1}{n}\right[:n\in\mathbb{N}\right\}$

द्वारा दिया गया संग्रह G, ]0,1[. का विवृत आवरक है या नहीं।

Ques 53.

समाकलों के लिए बोनट के माध्य मान प्रमेय का कथन दीजिए

$\left|\int_{3}^{5}\frac{\cos x}{x}dx\right|\leq\frac{2}{3}$

दिखाने के लिए इसे लागू कीजिए।

Ques 54.

दिखाइए कि अनुक्रम (a_n), एकदिष्ट है, जहाँ $a_n\frac{n}{n^2+4}1$ उत्तर की पुष्टि कीजिए। । क्या (a_n) कॉशी अनुक्रम है? अपने

Ques 55.

जाँच कीजिए कि अंतराल [2,5] और [7,10] तुल्य हैं या नहीं।

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IGNOU MTE 9 Solved Assignment

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