IGNOU BMTC 102 SOLVED ASSIGNMENT

BMTC 102: Multivariable Calculus

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Title Name	IGNOU BMTC 102 SOLVED ASSIGNMENT
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	BACHELOR DEGREE PROGRAMMES
Course Code	BSCFMT
Course Name	Bachelor of Science (Mathematics)
Subject Code	BMTC 102
Subject Name	Multivariable Calculus
Year	2026
Session	-
Language	English Medium
Assignment Code	BMTC 102/Assignment-1/2026
Product Description	Assignment of BSCFMT (Bachelor of Science (Mathematics)) 2026. Latest BMTC 102 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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BMTC 102 2026 - English

ASSIGNMENT

Course Code: BMTC-102

Assignment Code: BMTC-102/TMA/2026

Maximum Marks: 100

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

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

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

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

(b) 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) 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}$

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

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

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

$\quad$ (b) Use double integration of find the volume of the ellipsoid

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

8. (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$

$\quad$ (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:
(i) $\sqrt{x^2 + y^2}$
(ii) $\sqrt{4 - x^2 - y^2}$
9. a) Using polar coordinates, show that $\displaystyle \lim_{\substack{x \to 0 \\ y \to 0}} \frac{x^3 - y^3}{x^2 + y^2} = 0$ . Also, find the two repeated limits.

$\quad$ b) Write $\displaystyle \int_0^1 \int_0^{\sqrt{1 - x^2}} (\sqrt{1 - y^2}) \, dy \, dx$ as an integral over a region D. Sketch the region D and show that it is of both types 1 and 2. Reverse the order of integration and evaluate it.

10. (a) Check if the following integrals are independent of path and evaluate those which are independent.

$$\text{i) } \int_{(0,0)}^{(3,4)} (6xy - y^3) \, dx + (3x^2 - 3xy^2) \, dy$

$$\text{ii) } \int_{(-1,4)}^{(3,8)} (3x^2 - 2y^2) \, dx - 4xy \, dy$

$\quad$ (b) Evaluate $\displaystyle \iiint_S z^2 \, dx \, dy \, dz$ , where S is the solid region between the spheres $\rho = 1$ and
$\quad$ $\rho = 2$ , by using spherical coordinates.

BMTC 102 2026 - Hindi

सत्रीय कार्य

पाठ्यक्रम कोड: BMTC-102
सत्रीय कार्य कोड: BMTC-102/TMA/2026
अधिकतम अंक: 100

1. बताइए निम्नलिखित कथन सत्य हैं या असत्य। अपने उत्तरों के कारण बताइए। (10)

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

(ii) $\quad$ तीन चरों वाला एक वास्तविक-मान फलन, जो सर्वत्र सतत है, अवकलनीय होता है।

(iii) $\quad F(x, y) = (y + 2, x + y)$ से परिभाषित फलन $F: \mathbf{R}^2 \to \mathbf{R}^2$ किसी भी बिन्दु $(x, y) \in \mathbf{R}^2$ पर स्थानिकतः व्युत्क्रमणीय होता है।

(iv) Image ignouassignments-ignouacademy-com-ignou-bmtc-102-solved-assignment-html-p-assignment-78732
$\quad\quad$ से परिभाषित फलन $f: [-1,1] \times [-2,2] \to \mathbf{R}$ समाकलनीय होता है।

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

2) (क) निम्नलिखित सीमा ज्ञात कीजिए :

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

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

(ख) केवल परिभाषाओं को लागू करके f_xy(0, 0) और f_yx(0, 0) ज्ञात कीजिए, जबकि फलन

Image ignouassignments-ignouacademy-com-ignou-bmtc-102-solved-assignment-html-p-assignment-52029

के लिए इनका अस्तित्व होता हो।

3) (क) मान लीजिए
$$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}$ $
दिखाइए कि (0,0) पर सभी दिशाओं में f दिक् अवकलज होते हैं।

(ख) मान लीजिए $x = e^r \cos \theta, y = e^r \sin \theta$ और f, x और y का 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 \theta^2} = (x^2 + y^2) \left( \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} \right)$ $

(ग) मान लीजिए $a = (1, 2, 3), b = (-5, 3, -2), c = (2, -4, 1), \mathbf{R}^3$ के तीन बिन्दु हैं।
|2b - a + 3c| ज्ञात कीजिए।

4. (क) वक्रों $y = 4x^{2}$ और $x = 4$ से परिबद्ध और $\delta(x,y) = y$ के घनत्व वाले एक पतली शीट का गुरुत्व केन्द्र ज्ञात कीजिए।

(ख) $z = 1$ और $z = x^{2} + y^{2}$ से परिबद्ध ठोस घनाकृति का द्रव्यमान ज्ञात कीजिए, जबकि घनत्व फलन, $\delta(x,y,z) = |x|$ हो।

5. (क) ग्रीन प्रमेय का कथन दीजिए और इसकी सहायता से
Image ignouassignments-ignouacademy-com-ignou-bmtc-102-solved-assignment-html-p-ignou-82982
जहाँ C, दीर्घवृत्त $4x^{2} + 9y^{2} = 36$ है।

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

6. (क) (0,0) पर निम्नलिखित फलन f के सांतत्य और अवकलनीयता की जाँच कीजिए, जहाँ

Image ignouassignments-ignouacademy-com-ignou-bmtc-102-solved-assignment-html-p-solved-40237

(ख) फलन $f(x, y) = \frac{2xy}{x^2 + y^2}$ से परिभाषित फलन f का प्रांत और परिसर ज्ञात कीजिए। इस फलन के दो स्तर वक्र भी ज्ञात कीजिए। (4)

7. (क) $\int\limits_{C} (2x^2 + 3y^2) dx$ के मान निकालिए, जहाँ C $x(t) = at^2, y(t) = 2at, 0 \le t \le 1$ से प्राप्त वक्र है।

(ख) द्विशः समाकलन का प्रयोग करके दीर्घवृत्तज
$$\frac{x^2}{4} + \frac{y^2}{9} + \frac{z^2}{16} = 1$ $
का आयतन ज्ञात कीजिए।

8. (क) यदि $\lim_{x \to 0} \frac{x(1 + a \cos x) - b \sin x}{x^3} = 1$ तो a और b के मान ज्ञात कीजिए।

(ख) मान लीजिए कि S और C $\mathbf{R}^3$ के उपसमुच्चय हैं। S मूल–बिन्दु पर केन्द्र वाला एकक विवृत गोलक है तथा C 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) $S \subset C$
(ii) $C \subset S$

(ग) निम्नलिखित फलनों के स्तर वक्र ज्ञात कीजिए :
(i) $\sqrt{x^2 + y^2}$
(ii) $\sqrt{4 - x^2 - y^2}$

9. क) ध्रुवीय निर्देशांकों का प्रयोग करते हुए दिखाइए कि $\lim_{\substack{x \to 0 \\ y \to 0}} \frac{x^3 - y^3}{x^2 + y^2} = 0$ है। दो पुनरावृत्ति सीमाएँ ज्ञात कीजिए।
$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$

ख) प्रदेश D पर एक समाकल के रूप में $\int\limits_{0}^{1} \int\limits_{0}^{\sqrt{1 - x^2}} \left( \sqrt{1 - y^2} \right) dy \, dx$ लिखिए। प्रदेश D का चित्र बनाइए और दिखाएं कि यह टाईप 1 और टाईप 2 दोनों है। समाकल के क्रम को उलटिए और इसका मूल्यांकन कीजिए।
$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$

10. क) जाँच कीजिए कि निम्नलिखित समाकलन स्वतंत्र पथ है और जो स्वतंत्र है उनका मूल्यांकन कीजिए
i) $\int\limits_{(0,0)}^{(3,4)} (6xy - y^3) \, dx + (3x^2 - 3xy^2) \, dy$ .
ii) $\int\limits_{(-1,4)}^{(3,8)} (3x^2 - 2y^2) \, dx - 4xy \, dy$ .
$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$

ख) निर्देशांकों का प्रयोग करते हुए $\iiint\limits_{S} z^2 dx \, dy \, dz$ का मूल्यांकन कीजिए जहाँ S गोले $\rho = 1$ और $\rho = 2$ के बीच ठोस प्रदेश है।
$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$

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