IGNOU MMT 4 SOLVED ASSIGNMENT

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IGNOU MMT 4 SOLVED ASSIGNMENT

~~Rs. 200~~
Rs. 123

IGNOU MMT 4 SOLVED ASSIGNMENT

~~Rs. 200~~
Rs. 123

Last Date of Submission of IGNOU MMT-04 (MSCMACS) 2026 Assignment is for January 2026 Session: 30th September, 2026 (for December 2026 Term End Exam).
Semester Wise
January 2026 Session: 30th March, 2026 (for June 2026 Term End Exam).
July 2026 Session: 30th September, 2026 (for December 2026 Term End Exam).

Title Name	IGNOU MMT 4 SOLVED ASSIGNMENT
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	MASTER DEGREE PROGRAMMES
Course Code	MSCMACS
Course Name	M.Sc. Mathematics with Applications in Computer Science
Subject Code	MMT 4
Subject Name	Real Analysis
Year	2026
Session
Language	English Medium
Assignment Code	MMT-04/Assignmentt-1//2026
Product Description	Assignment of MSCMACS (M.Sc. Mathematics with Applications in Computer Science) 2026. Latest MMT 04 2026 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU MMT-04 (MSCMACS) 2026 Assignment is for January 2026 Session: 30th September, 2026 (for December 2026 Term End Exam). Semester Wise January 2026 Session: 30th March, 2026 (for June 2026 Term End Exam). July 2026 Session: 30th September, 2026 (for December 2026 Term End Exam).

~~Rs. 200~~
Rs. 123

Questions Included in this Help Book

Ques 1.

1. State whether the following statements are true or false. Justify your answers. $5 \times 2 = 10$
a) The outer measure m^ of the set $A = \{x \in \mathbb{R} : x^2 = 1\} \cup [-3, 2]$ is 0.
b) A finite subset of a metric space is totally bounded.
c) A connected subspace in a metric space which in not properly contained in any other connected subspace is always open.
d) The surface given by the equation $x + y + z - \sin(xyz) = 0$ can also be described by an equation of the form $z = f(x, y)$ in a neighbourhood of the point (0, 0).
e) A real valued function f on [a, b] is continuous if it is integrable on [a, b].

Ques 2.

Find the interior, closure, the set of limit points and the boundary of the set

$$A = \{(x, y) \in \mathbb{R}^2 : y = 1\}$

in $\mathbb{R}^2$ with the standard metric.

Ques 3.

Find the interior, closure, the set of limit points and the boundary of the set

$$A = \{(x, y) \in \mathbb{R}^2 : y = 1\}$

in $\mathbb{R}^2$ with the standard metric.

Ques 4.

Consider $f : \mathbb{R}^3 \to \mathbb{R}^3$ given by

$$f(x, y, z) = (2x + 3y + z, xy, yz, xz)$

Find f(2, 0, -1).

Ques 5.

Does Cantor’s intersection theorem hold for the metric space $X = (0, 1]$ with the standard metric? Justify your answer.

Ques 6.

Obtain the second Taylor’s series expansion for the function given by

$$f(x, y) = xy^2 + 5xe^y \text{ at } \left(1, \frac{\pi}{2}\right).$

Ques 7.

Find the Lebesgue integral of the function f given by

$$f = \chi_{[0, 1]} + 2\chi_{(3, 5]} + \chi_{[6, 9]}.$

Ques 8.

Find and classify the extreme values of $(x, y) = xy$

Subject to the constraint

$$\frac{x^2}{8} + \frac{y^2}{2} - 1 = 0.$

Ques 9.

Let $f : \mathbb{R}^3 \setminus \{(0,0,0)\} \to \mathbb{R}^3 \setminus \{(0,0,0)\}$ be given by

$$f(x, y, z) = \left( \frac{x}{x^2 + y^2 + z^2}, \frac{y}{x^2 + y^2 + z^2}, \frac{z}{x^2 + y^2 + z^2} \right)$

Show that f is locally invertible at all points in $\mathbb{R}^3 \setminus \{(0,0,0)\}$ .

Ques 10.

For the equation $x^2 + y^2 + z^3 = 0$ , at which points on its solution set, can we assured that there is a neighbourhood of the point in which the surface given by the equation can be described by an equation of the form $z = f(x, y)$ .

Ques 11.

Find the Fourier series of $f(t) = t^2$ on $[-\pi, \pi]$ .

Ques 12.

Prove that if an open set U can be written as the union of pariwise disjoint family V of open connected subsets, then these subsets must be the components of U. Use this theorem to find the components of the set $D \cup E$ where

$$D = \{(x, y) \in \mathbb{R}^2 : \|(x+1, y)\| = 1\} \text{ and}$

$$E = \{(x, y) \in \mathbb{R}^3 : \|x-1, y\| = 1\}.$

Ques 13.

Which of the following subsets of $\mathbb{R}$ are compact w.r.t. the metric given against them. Justify your answer.
i) $A = (0, 1)$ in $\mathbb{R}$ of $-\mathbb{R}$ with standard metric.
ii) $A = [3, 4] - \mathbb{R}$ with discrete metric.
iii) $\{(x, y) \in \mathbb{R}^2 \quad y > 0\} - \mathbb{R}^2$ with standard metric.

Ques 14.

If E is a subset of $\mathbb{R}$ with standard metric, then show that $(\bar{E})^c = (E^c)^\circ$ .

Ques 15.

Show that a set A in a metric space is closed if and only if every convergent sequence in A converges to a point of A.

Ques 16.

Find the interior and closure of the set $\mathbb{Q}$ of rationals in $\mathbb{R}$ with standard metric.

Ques 17.

Let F be the function from $\mathbb{R}^2$ to $\mathbb{R}^2$ defined by

$$F(x, y) = (x^2 + y^2, xy).$

Show that F is differentiable at (1, 2). Find the differential matrix of F.

Ques 18.

) Show that the function f defined by
$$f(x, y) = \begin{cases} \frac{xy}{x^2 + y^2}, & \text{if } (x, y) \neq (0, 0) \\ 0, & \text{if } (x, y) = (0, 0) \end{cases}$
is not differentiable at (0,0). Do the partial derivatives of f exist at (0,0)? or at any other point in $\mathbb{R}^2$ ? Justify your answer.

Ques 19.

Is the continuous image of a Cauchy sequence a Cauchy sequence? Justify.

Ques 20.

Find the directional derivative of the function $f : \mathbb{R}^4 \to \mathbb{R}^4$ defined by

$$f(x, y, z, w) = (x^2y, xyz, x^2 + y^2, zw^2)$

at the point (1, 2, -1, -2) in the direction $v = (1, 0, -2, 2)$ .

Ques 21.

Suppose that $f : \mathbb{R} \to \mathbb{R}^2$ is given by $f(t) = (t, t^2)$ and $g : \mathbb{R}^2 \to \mathbb{R}^3$ is given by

$g(x, y) = (x^2, xy, y^2 - x^2)$ . Compute the derivative of $g \circ f$ .

Ques 22.

Find $B \left[ 2, \frac{1}{2} \right]$ in $\mathbb{R}$ where d is the metric given by $d(x, y) = \min\{1, |x - y|\}$ .

Based on the image provided, here is the transcription of the final questions:

Ques 23.

Give an example of a family f_i of subsets of a set X which has finite intersection property. Justify your choice of example.

Ques 24.

Verify the hypothesis and conclusions of the Fatou’s lemma for the sequence $\{f_n\}$ given by

$$f_n(x) = 2n \text{ for } x \in \left( \frac{1}{2n}, \frac{1}{n} \right)$

$$= 0 \text{ for } x \in \left( 0, \frac{1}{2n} \right] \cup \left[ \frac{1}{n}, 1 \right]$

Ques 25.

Let (X, d) be a metric space and A be a subset of X. Show that $bdy(A) = \emptyset$ if and only if A is both open and closed.

Ques 26.

Give an example of an algebra which is not a $\sigma$ -algebra. Justify your choice of examples

Ques 27.

If E is a measurable set and f is a simple function such that $a \leq f(x) \leq b \text{ } \forall x \in E$ , show that

$$am(E) \leq \int_{E} f \, dm \leq m(E)b.$

Ques 28.

Ques 29.

Find the interior, closure, the set of limit points and the boundary of the set

$$A = \{(x, y) \in \mathbb{R}^2 : y = 1\}$

in $\mathbb{R}^2$ with the standard metric.

Ques 30.

Find the interior, closure, the set of limit points and the boundary of the set

$$A = \{(x, y) \in \mathbb{R}^2 : y = 1\}$

in $\mathbb{R}^2$ with the standard metric.

Ques 31.

Consider $f : \mathbb{R}^3 \to \mathbb{R}^3$ given by

$$f(x, y, z) = (2x + 3y + z, xy, yz, xz)$

Find f(2, 0, -1).

Ques 32.

Does Cantor’s intersection theorem hold for the metric space $X = (0, 1]$ with the standard metric? Justify your answer.

Ques 33.

Obtain the second Taylor’s series expansion for the function given by

$$f(x, y) = xy^2 + 5xe^y \text{ at } \left(1, \frac{\pi}{2}\right).$

Ques 34.

Find the Lebesgue integral of the function f given by

$$f = \chi_{[0, 1]} + 2\chi_{(3, 5]} + \chi_{[6, 9]}.$

Ques 35.

Find and classify the extreme values of $(x, y) = xy$

Subject to the constraint

$$\frac{x^2}{8} + \frac{y^2}{2} - 1 = 0.$

Ques 36.

Let $f : \mathbb{R}^3 \setminus \{(0,0,0)\} \to \mathbb{R}^3 \setminus \{(0,0,0)\}$ be given by

$$f(x, y, z) = \left( \frac{x}{x^2 + y^2 + z^2}, \frac{y}{x^2 + y^2 + z^2}, \frac{z}{x^2 + y^2 + z^2} \right)$

Show that f is locally invertible at all points in $\mathbb{R}^3 \setminus \{(0,0,0)\}$ .

Ques 37.

Ques 38.

Find the Fourier series of $f(t) = t^2$ on $[-\pi, \pi]$ .

Ques 39.

$$D = \{(x, y) \in \mathbb{R}^2 : \|(x+1, y)\| = 1\} \text{ and}$

$$E = \{(x, y) \in \mathbb{R}^3 : \|x-1, y\| = 1\}.$

Ques 40.

Ques 41.

If E is a subset of $\mathbb{R}$ with standard metric, then show that $(\bar{E})^c = (E^c)^\circ$ .

Ques 42.

Show that a set A in a metric space is closed if and only if every convergent sequence in A converges to a point of A.

Ques 43.

Find the interior and closure of the set $\mathbb{Q}$ of rationals in $\mathbb{R}$ with standard metric.

Ques 44.

Let F be the function from $\mathbb{R}^2$ to $\mathbb{R}^2$ defined by

$$F(x, y) = (x^2 + y^2, xy).$

Show that F is differentiable at (1, 2). Find the differential matrix of F.

Ques 45.

Ques 46.

Is the continuous image of a Cauchy sequence a Cauchy sequence? Justify.

Ques 47.

Find the directional derivative of the function $f : \mathbb{R}^4 \to \mathbb{R}^4$ defined by

$$f(x, y, z, w) = (x^2y, xyz, x^2 + y^2, zw^2)$

at the point (1, 2, -1, -2) in the direction $v = (1, 0, -2, 2)$ .

Ques 48.

Suppose that $f : \mathbb{R} \to \mathbb{R}^2$ is given by $f(t) = (t, t^2)$ and $g : \mathbb{R}^2 \to \mathbb{R}^3$ is given by

$g(x, y) = (x^2, xy, y^2 - x^2)$ . Compute the derivative of $g \circ f$ .

Ques 49.

Find $B \left[ 2, \frac{1}{2} \right]$ in $\mathbb{R}$ where d is the metric given by $d(x, y) = \min\{1, |x - y|\}$ .

Based on the image provided, here is the transcription of the final questions:

Ques 50.

Give an example of a family f_i of subsets of a set X which has finite intersection property. Justify your choice of example.

Ques 51.

Verify the hypothesis and conclusions of the Fatou’s lemma for the sequence $\{f_n\}$ given by

$$f_n(x) = 2n \text{ for } x \in \left( \frac{1}{2n}, \frac{1}{n} \right)$

$$= 0 \text{ for } x \in \left( 0, \frac{1}{2n} \right] \cup \left[ \frac{1}{n}, 1 \right]$

Ques 52.

Let (X, d) be a metric space and A be a subset of X. Show that $bdy(A) = \emptyset$ if and only if A is both open and closed.

Ques 53.

Give an example of an algebra which is not a $\sigma$ -algebra. Justify your choice of examples

Ques 54.

If E is a measurable set and f is a simple function such that $a \leq f(x) \leq b \text{ } \forall x \in E$ , show that

$$am(E) \leq \int_{E} f \, dm \leq m(E)b.$

~~Rs. 200~~
Rs. 123

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Course Name	M.Sc. Mathematics with Applications in Computer Science
Course Code	MSCMACS
Programm	MASTER DEGREE PROGRAMMES Courses
Language	English

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