IGNOU MTE 7 SOLVED ASSIGNMENT

MTE 7: Advanced Calculus

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Title Name	IGNOU MTE 7 SOLVED ASSIGNMENT
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	BACHELOR DEGREE PROGRAMMES
Course Code	BSC
Course Name	Bachelor in Science
Subject Code	MTE 7
Subject Name	Advanced Calculus
Year	2026
Session	-
Language	English Medium
Assignment Code	MTE 7/Assignment-1/2026
Product Description	Assignment of BSC (Bachelor in Science) 2026. Latest MTE 07 2026 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU BEGC-131 (BAG) 2025-26 Assignment is for January 2026 Session: 30th September, 2026 (for December 2025 Term End Exam). Semester Wise January 2025 Session: 30th March, 2026 (for June 2026 Term End Exam). July 2025 Session: 30th September, 2025 (for December 2025 Term End Exam).
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MTE 7 2025 - English

Course Code: MTE-07

Assignment Code: MTE-07/TMA/2025

Maximum Marks: 100

1. State whether the following statements are true or false. Give reasons for your answers.

(i) $lim_{x ightarrow 0}frac{x^2sinfrac{1}{x}}{sin x}=1$

(ii) A real-valued function of three variables which is continuous everywhere is differentiable.

(iii) The function $F:mathbf{R}^2 ightarrowmathbf{R}^2$ , defined by F(x, y) = ( y + ,2 x + y) at any (x, y) ∈ $mathbb{R}^{2}$

(iv) $f:[-1,1] imes[-2,2] ightarrowmathbf{R}$ ,defined by

$f(x,y)=egin{cases}x,& ext{if,,,,}y,,,, ext{is rational}\0,& ext{if,,,,}y,,,, ext{is not rational}end{cases}$

is integrable.

(v) The function $f:mathbf{R}^2 ightarrowmathbf{R}$ defined by $f(x,y)=1-y^2+x^2$ has an extremum at (0,0).

2) (a) Find the following limits:

(i) $lim_{x ightarrow+infty}left(frac{x^2}{8x^2-3} ight)^{1/3}$

(ii) $lim_{x ightarrow0^+}(sin x)^{sin x}quad ext{}$

(b) Find the third Taylor polynomial of the function f (x,y) = 1 + 5xy + 3²y at (1,2).

$f(x,y)=egin{cases}frac{x^2 y}{sqrt{x^2+y^2}},&(x,y) e(0,0)\0,& ext{}end{cases}$

3) (a) Let the function f be defined by

$f(x,y)=egin{cases}frac{3x^{2}y^{4}}{x^{4}+y^{8}},&(x,y) e(0,0)\0,&(x,y)=(0,0)end{cases}$

Show that f has directional derivatives in all directions at (0,0).

(b) Let x e^r = cosθ, y = e^rsin θ and f be a continuously differentiable function of x and y, whose partial derivatives are also continuously differentiable. Show that

$frac{partial^2 f}{partial r^2}+frac{partial^2 f}{partial heta^2}=(x^2+y^2)left(frac{partial^2 f}{partial x^2}+frac{partial^2 f}{partial y^2} ight)$

4. (a) Find the centre of gravity of a thin sheet with density δ(x, y) = y, bounded by the curves y = 4x² and x = 4.

(b) Find the mass of the solid bounded by z =1 and , z = x² + y² the density function being δ (z,y,x) = | x | .

5. (a) State Green’s theorem, and apply it to evaluate

$int_C(3x^2-4y)dx-(2x+y^3)dy$

Where C is the ellipse $4x^2+9y^2=36$

(b) Find the extreme values of the function

$f(x,y)=x^2+y$ on the surface $x^2+2y^2=1$

6. (a) State a necessary condition for the functional dependence of two differentiable functions f and g on an open subset D of . $mathbb{R}^{2}$ Verify this theorem for the functions f and g, defined by

$f(x,y)=frac{y-x}{y+x},quad g(x,y)=frac{x}{y}$

(b) Using the Implicit Function Theorem, show that there exists a unique differentiable function g in a neighbourhood of 1 such that g (1) = 2 and F(g( y), y) = 0 in a neighbourhood of (1,2), where

$F(x,y)=x^5+y^5-16xy^3-1=0$

defines the function F. Also find g′( y).

(c) Check the local inevitability of the function f defined by f(x,y) =(x²-y²,2xy) at (1,1) Find a domain for the function f in which f is invertible.

7. (a) Check the continuity and differentiability of the function at (0,0) where

$f(x,y)=egin{cases}frac{2x^3y}{x^2+y^2},&(x,y) eq(0,0)\0,& ext{otherwise}end{cases}$

(b) Find the domain and range of the function f , defined by $f(x,y)=frac{2xy}{x^2+y^2}.$ Also find two level curves of this function. Give a rough sketch of them.

8. (a) Evaluate $int_C(2x^2+3y^2)dx$ where C is the curve given by

$x(t)=at^2,y(t)=2at,0le tle 1.$

(b) Use double integration of find the volume of the ellipsoid

$frac{x^2}{4}+frac{y^2}{9}+frac{z^2}{16}=1.$

9. (a) Find the values of a and b, if

$lim_{x ightarrowinfty}frac{x(1+acos x)-bsin x}{x^3}=1$

(b) Suppose S and C are subsets of $mathbb{R}^{3}$ . S is the unit open sphere with centre at the origin and C is the open cube = ${P(x,y,z)mid-1<x<1,-1<y<1,-1<z<1}.$

Which of the following is true. Justify your answer.

(i) S ⊂ C

(ii) C ⊂ S

(i) $sqrt{x^2+y^2}$

(ii) $sqrt{4-x^2-y^2}$

(iii) x − y

(iv) y / x

10. (a) Does the function

$f(x,y)=frac{x^2-y^2}{x^2+y^2},quad x e 0,y e 0$ satisfy the requirement of Schwarz’s theorem at (1,1) ? Justify your answer.

(b) Locate and classify the stationary points of the following:

$ext{(i)}quad f(x,y)=4xy+x^4-y^4$

$ext{(ii)}quad f(x,y)=xy+frac{2}{x}+frac{4}{y},quad x>0,y>0$

MTE 7 2026 - English

ASSIGNMENT

Course Code: MTE-07

Assignment Code: MTE-07/TMA/2026

Maximum Marks: 100

1. State whether the following statements are true or false. Give reasons for your answers.

(i) $\lim_{x \to 0} \frac{x^2 \sin \frac{1}{x}}{\sin x} = 1$

(ii) A real-valued function of three variables which is continuous everywhere is differentiable.

(iii) The function $F : \mathbb{R}^2 \to \mathbb{R}^2$ , defined by $F(x, y) = (y + 2, x + y)$ , is locally invertible at any $(x, y) \in \mathbb{R}^2$ .

(iv) $f : [-1, 1] \times [-2, 2] \to \mathbb{R}$ , defined by
$$f(x, y) = \begin{cases} x, & \text{if } y \text{ is rational} \\ 0, & \text{if } y \text{ is not rational} \end{cases}$
is integrable.

(v) The function $f : \mathbb{R}^2 \to \mathbb{R}$ , defined by $f(x, y) = 1 - y^2 + x^2$ , has an extremum at (0, 0).

2) (a) Find the following limits:

$\quad$ (i) $\lim_{x \to +\infty} \left( \frac{x^2}{8x^2 - 3} \right)^{1/3}$

$\quad$ (ii) $\lim_{x \to 0^+} (\sin x)^{\sin x}$

$\quad$ (b) Find the third Taylor polynomial of the function $f(x, y) = 1 + 5xy + 3^2y$ at (1, 2).

$\quad$ (c) Using only the definitions, find f_xy(0, 0) and f_yx(0, 0), if they exists, for the function

$$f(x, y) = \begin{cases} \frac{x^2 y}{\sqrt{x^2 + y^2}}, & (x, y) \neq (0, 0) \\ 0, & \text{otherwise} \end{cases}$

3) (a) Let the function f be defined by
$$f(x, y) = \begin{cases} \frac{3x^2 y^4}{x^4 + y^8}, & (x, y) \neq (0, 0) \\ 0, & (x, y) = (0, 0) \end{cases}$

$\quad$ Show that f has directional derivatives in all directions at (0, 0).

(b) Let $x = e^r \cos \theta$ , $y = e^r \sin \theta$ and f be a continuously differentiable function of x and y, whose partial derivatives are also continuously differentiable. Show that

$$\frac{\partial^2 f}{\partial r^2} + \frac{\partial^2 f}{\partial \theta^2} = (x^2 + y^2) \left( \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} \right)$

(c) Let $a = (1, 2, 3)$ , $b = (-5, 3, -2)$ , $c = (2, -4, 1)$ be three points in $\mathbb{R}^3$ .

Find |2b - a + 3c|.

4. (a) Find the centre of gravity of a thin sheet with density $\delta(x, y) = y$ , bounded by the curves $y = 4x^2$ and $x = 4$ .

(b) Find the mass of the solid bounded by $z = 1$ and $z = x^2 + y^2$ , the density function being $\delta(x, y, z) = |x|$ .

5. (a) State Green's theorem, and apply it to evaluate

$$\int_C (3x^2 - 4y) dx - (2x + y^3) dy,$

Where C is the ellipse $4x^2 + 9y^2 = 36$ .

(b) Find the extreme values of the function

$f(x, y) = x^2 + y$ on the surface $x^2 + 2y^2 = 1$ .

6. (a) State a necessary condition for the functional dependence of two differentiable functions f and g on an open subset D of $\mathbb{R}^2$ . Verify this theorem for the functions f and g, defined by

$$f(x, y) = \frac{y - x}{y + x}, \quad g(x, y) = \frac{x}{y}.$

(b) Using the Implicit Function Theorem, show that there exists a unique differentiable function g in a neighbourhood of 1 such that $g(1) = 2$ and $F(g(y), y) = 0$ in a neighbourhood of (2, 1), where

$$F(x, y) = x^5 + y^5 - 16xy^3 - 1 = 0$

defines the function F. Also find g'(y).

(c) Check the local invertibility of the function f defined by $f(x, y) = (x^2 - y^2, 2xy)$ at (1, -1). Find a domain for the function f in which f is invertible.

7. (a) Check the continuity and differentiability of the function at (0, 0) where

$$f(x, y) = \begin{cases} \frac{2x^3 y}{x^2 + y^2}, & (x, y) \neq (0, 0) \\ 0, & \text{otherwise} \end{cases}$

(b) Find the domain and range of the function f, defined by $f(x, y) = \frac{2xy}{x^2 + y^2}$ . Also find two level curves of this function. Give a rough sketch of them.

8. (a) Evaluate $\int_C (2x^2 + 3y^2) dx$ , where C is the curve given by

$x(t) = at^2, y(t) = 2at, 0 \le t \le 1$ .

(b) Use double integration of find the volume of the ellipsoid

$\frac{x^2}{4} + \frac{y^2}{9} + \frac{z^2}{16} = 1$ .

9. (a) Find the values of a and b, if

$$\lim_{x \to 0} \frac{x(1 + a \cos x) - b \sin x}{x^3} = 1$

(b) Suppose S and C are subsets of $\mathbb{R}^3$ . S is the unit open sphere with centre at the origin and C is the open cube

$= \{P(x, y, z) \mid -1 < x < 1, -1 < y < 1, -1 < z < 1\}$ .

Which of the following is true. Justify your answer.

(i) $S \subset C$

(ii) $C \subset S$

(c) Identify the level curves of the following functions:

$\quad$ (i) $\sqrt{x^2 + y^2}$

$\quad$ (ii) $\sqrt{4 - x^2 - y^2}$

$\quad$ (iii) x - y

$\quad$ (iv) y / x

10. (a) Does the function

$$f(x, y) = \frac{x^2 - y^2}{x^2 + y^2}, x \neq 0, y \neq 0$

satisfy the requirement of Schwarz's theorem at (1, 1)? Justify your answer.

$\quad$ (b) Locate and classify the stationary points of the following:

$\quad$ (i) $f(x, y) = 4xy + x^4 - y^4$

$\quad$ (ii) $f(x, y) = xy + \frac{2}{x} + \frac{4}{y}, x > 0, y > 0$

MTE 7 2025 - Hindi

सत्रीय कार्य

पाठ्यक्रम कोड: एम टी इ-07

सत्रीय कार्य कोड: एम टी इ-07/ टी एम ए/2025

अधिकतम अंकः 100

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

(i) $lim_{x ightarrow0}frac{x^{2}sinfrac{1}{x}}{sin x}=1quad ext{(i)}$

(ii) तीन घरों वाला एक वास्तविक मान फलन, जो सर्वत्र संतत है, अवकलनीय होता है।

(iii)F(x, y) = ( y + ,2 x + y) से परिभाषित फलन $F:mathbf{R}^2 ightarrowmathbf{R}^2$ किसी भी बिन्दु (x, y) ∈R²पर स्थानिकतः व्युत्क्रमणीय होता है।

(iv) $f(x,y)=egin{cases}x,& ext{if,,,,}y,,,, ext{is rational}\0,& ext{if,,,,}y,,,, ext{is not rational}end{cases}$ है फ(एक्स,य)= 10, यदि वाइ परिमेय नहीं है. से परिभाषित फलन $f:[-1,1] imes[-2,2] ightarrowmathbf{R}$ समाकलनीय होता है।

(v) $f(x,y)=1-y^2+x^2$ से परिभाषित फलन $f:mathbf{R}^2 ightarrowmathbf{R}$ का (0,0)पर एक चरम मान होता है।

2. (क) निम्नलिखित सीमा ज्ञात कीजिए:

(i) $lim_{x ightarrow+infty}left(frac{x^2}{8x^2-3} ight)^{1/3}$

(ii) $lim_{x ightarrow0^+}(sin x)^{sin x}quad ext{}$

(ख) बिन्दु (1,2) पर फलन $f(x,y)=1+5xy+3^2 y$ का तृतीय टेलर बहुपद ज्ञात कीजिए।

(ग) केवल परिभाषाओं को लागू करके f_xy (0,0) और f_yx, (0,0) ज्ञात कीजिए, जबकि फलन $f(x,y)=egin{cases}frac{x^2 y}{sqrt{x^2+y^2}},&(x,y) e(0,0)\0,& ext{}end{cases}$ अन्यथा

के लिए इनका अस्तित्य होता हो।

3) (क) मान लीजिए

$f(x,y)=egin{cases}frac{3x^{2}y^{4}}{x^{4}+y^{8}},&(x,y) e(0,0)\0,&(x,y)=(0,0)end{cases}$ दिखाइए कि (0,0) पर सभी दिशाओं में ई दिक् अवकलज होते हैं।

(ख) मान लीजिए

$x=e^tcos heta,quad y=e^tsin heta$ और f xऔर yका एक संततः अवकलनीय फलन है जिसके आंशिक अवकलज भी संततः अवकलनीय हैं। दिखाइए कि $frac{partial^2 f}{partial r^2}+frac{partial^2 f}{partial heta^2}=(x^2+y^2)left(frac{partial^2 f}{partial x^2}+frac{partial^2 f}{partial y^2} ight)$

(ग) माग जीजिए $a=(1,2,3),quad b=(-5,3,-2),quad c=(2,-4,1)inmathbf{R}^3$ के तीन बिन्दु हैं। 126-a+31 ज्ञात कीजिए।

4 (क) वक्रो y = 4x² और 4 से पसिद्ध और $delta$ (x, y) के गात वाजे एक पतली सीट का गुरुत्व केन्द्र ज्ञात कीजिए। (5)

(ख) z=1 और z=x+y² से पषिद्ध ठोस घनाकृति का इमान ज्ञात कीजिए जबकि धनन्त्तव $delta(x,y,z)=|x|$ हौ

5. (क) ग्रीन प्रमेय का कथनदीजिए और इसकी सहायता से

$int_C(3x^2-4y)dx-(2x+y^3)dy$ मान निकालिए जहाँ सी. दीर्घवृत $4x^2+9y^2=36$ है।

(ख) पृष्ठ $x^2+2y^2=1$ पर फलन $f(x,y)=x^2+y$ के चरम माग ज्ञात कीजिए।

6 (क) R²के एक वितृत उपसमुचय D पर दो अवकलनीय फलनों F और g है की फजनक आश्रितता का आरक प्रतिबंध बताने वाले प्रमेय का कब दीजिए। निम्नलिखित फलनों f तथा g परिभाषित इस प्रमेय को सत्यापित कीजिए।

$f(x,y)=frac{y-x}{y+x},quad g(x,y)=frac{x}{y}$

(ख) अस्पष्ट फल प्रमेय की सहायता सेवा दिखाइए कि 1 के प्रति में एक ऐसा वीयफजन होता है, जिससे कि (2,1) के प्रतिवेश में g(1)= 2 और $F(g(y),y)=0$ जह $F(x,y)=x^5+y^5-16xy^3-1=0$ परिभाषित है।g' (y) भी ज्ञात कीजिए।

(ग) $f(x,y)=(x^2-y^2,2xy)$ द्वारा परिभाषित फलन f की। (1-1) के पर स्थानीय की लिए जाँच कीजिए फलन f के लिए एक प्रांत ज्ञात कीजिए जिसमें f व्युक्रमणीयता है।

7. (क) (0.0) पर निम्नलिखित फलन f के सांतत्य और अवकलनीयता की जाँच कीजिए, जहाँ

$f(x,y)=egin{cases}frac{2x^3 y}{x^2+y^2},&(x,y) e(0,0)\0,& ext{}end{cases}$

8.(क) $int_C(2x^2+3y^2)dx$ के मान निकालिए, जहाँ $C:quad x(t)=at^2,quad y(t)=2at,quad 0le tle 1$ से प्राप्त वक्र है।

(ख) दितः समाकलन का प्रयोग करके दीर्घवृज

$frac{x^2}{4}+frac{y^2}{9}+frac{z^2}{16}=1$ का आयतन ज्ञात कीजिए।

(क) यदि $lim_{x ightarrowinfty}frac{x(1+acos x)-bsin x}{x^3}=1$ तो a और b मान ज्ञात कीजिए। (5)

(ख) मान लीजिए कि S और C R³के उपसमुच्चय हैं। ऍस मूल-बिन्दु पर केन्द्र वाला एकक विवृत तथा क विवृत घन = ${P(x,y,z)mid-1<x<1,-1<y<1,-1<z<1}$

निम्नलिखित में से कौनसा कथन सत्य है? अपने उत्तर की पुष्टि कीजिए।

(i) S ⊂ C

(ii)C ⊂ S

(i) $sqrt{x^2+y^2}$

(ii) $sqrt{4-x^2-y^2}$

(iii) x − y

(iv) y / x

10. (क) क्या निम्नलिखित फलन $f(x,y)=frac{x^2-y^2}{x^2+y^2},quad x e 0,y e 0$ श्वार्ज-प्रमेय आवश्यकताओं को पर संतुष्ट करता है? अपने उत्तर की पुष्टि कीजिए।

((ख) निम्नलिखित के स्तब्ध बिन्दु निर्धारित करके उनका वर्गीकरण कीजिए :

$ext{(i)}quad f(x,y)=4xy+x^4-y^4$

$ext{(ii)}quad f(x,y)=xy+frac{2}{x}+frac{4}{y},quad x>0,y>0$

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MTE 7 Assignment 2026

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Download IGNOU MTE 7 Solved Assignment 2026

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Ignou Assignment Solution MTE 7

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Getting your assignments right is the first step toward a successful degree. At IGNOU Academy, we provide high-quality reference materials designed to simplify your academic journey. Here is why thousands of students trust us:

Latest Curriculum: All content is strictly based on the current IGNOU syllabus.
Perfect Formatting: Understand the ideal structure and layout to score better marks.
Concept Clarity: We break down complex topics into simple, easy-to-grasp explanations.
Exam-Ready: Our materials serve as excellent revision notes for your term-end exams.
Student-Centric Language: Written in clear, simple English/Hindi to ensure every learner understands.
Nationwide Trust: A preferred choice for IGNOU learners across India.

Disclaimer: These materials are intended as reference study guides to help you understand topics and formats. We encourage students to use these insights to prepare and write their own original assignments as per university guidelines.

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Click on the "IGNOU Assignment Question Papers" section.
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How to Submit Your IGNOU Assignments

Handwritten is Key: Use clean A4-size sheets and write neatly.
The Front Page: Ensure your first page clearly mentions your Name, Enrollment Number, Course Code, Subject, and Study Center Code.
Offline Submission: Visit your assigned Study Center, submit in person, and always collect your acknowledgment receipt.
Online Submission: If your center allows, scan each subject as a separate PDF. Submit via the official Google Form, Email, or Portal provided by your center. Keep a screenshot of the confirmation.

Tracking Your Submission Status

Want to know if your marks are updated?

Visit the Student Zone on the official IGNOU website.
Navigate to "Assignment Status."
Enter your Enrollment Number and Program Code.
View your submission dates, current status, and any remarks from the evaluator.

A Quick Tip for Success

Dear Students, remember that assignments carry 30% weightage in your final result. They aren't just a formality—they are a game-changer for your overall percentage. Regular study and timely submission are the keys to a high grade.

Success in IGNOU = Smart Study + Well-Prepared Assignments!

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