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IGNOU BMTC 102 SOLVED ASSIGNMENT HINDI

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IGNOU BMTC 102 SOLVED ASSIGNMENT HINDI

~~Rs. 90~~
Rs. 15

Last Date of Submission of IGNOU BMTC-102 (BSCFMT) 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 BMTC 102 SOLVED ASSIGNMENT HINDI
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/Assignmentt-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 BMTC-102 (BSCFMT) 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. 90~~
Rs. 15

Questions Included in this Help Book

Ques 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)
$\quad\quad$ 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।

Ques 2.

निम्नलिखित सीमा ज्ञात कीजिए :
(i) $\quad \lim_{x \to +\infty} \left( \frac{x^2}{8x^2 - 3} \right)^{1/3}$
(ii) $\quad \lim_{x \to 0^+} (\sin x)^{\sin x}$

Ques 3.

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

के लिए इनका अस्तित्व होता हो।

Ques 4.

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।

Ques 5.

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।

Ques 6.

ग्रीन प्रमेय का कथन दीजिए और इसकी सहायता से

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।

Ques 7.

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

(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)

Ques 8.

$\int\limits_{C} (2x^2 + 3y^2) dx$ 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

Ques 9.

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।

Ques 10.

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

Ques 11.

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।

Ques 12.

) 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।

Ques 13.

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).

Ques 14.

Find the following limits:
$\quad$ (i) $\displaystyle \lim_{x \to +\infty} \left( \frac{x^2}{8x^2 - 3} \right)^{1/3}$
$\quad$ (ii) $\displaystyle \lim_{x \to 0^+} (\sin x)^{\sin x}$

Ques 15.

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}$

Ques 16.

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).

Ques 17.

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)$

Ques 18.

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

Find |2b - a + 3c|.

Ques 19.

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

Ques 20.

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

Ques 21.

tate Green’s theorem, and apply it to evaluate

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

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

Ques 22.

Find the extreme values of the function

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

Ques 23.

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}$

Ques 24.

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.

Ques 25.

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.$

Ques 26.

Use double integration of find the volume of the ellipsoid

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

Ques 27.

Find the values of a and b, if

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

Ques 28.

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

Ques 29.

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.

Ques 30.

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.

Ques 31.

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$

Ques 32.

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.

Ques 33.

Ques 34.

Ques 35.

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

के लिए इनका अस्तित्व होता हो।

Ques 36.

Ques 37.

Ques 38.

Ques 39.

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

Ques 40.

Ques 41.

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।

Ques 42.

Ques 43.

Ques 44.

Ques 45.

Ques 46.

Find the following limits:
$\quad$ (i) $\displaystyle \lim_{x \to +\infty} \left( \frac{x^2}{8x^2 - 3} \right)^{1/3}$
$\quad$ (ii) $\displaystyle \lim_{x \to 0^+} (\sin x)^{\sin x}$

Ques 47.

Ques 48.

Ques 49.

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)$

Ques 50.

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

Find |2b - a + 3c|.

Ques 51.

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

Ques 52.

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

Ques 53.

tate Green’s theorem, and apply it to evaluate

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

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

Ques 54.

Find the extreme values of the function

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

Ques 55.

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}$

Ques 56.

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.

Ques 57.

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.$

Ques 58.

Use double integration of find the volume of the ellipsoid

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

Ques 59.

Find the values of a and b, if

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

Ques 60.

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

Ques 61.

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.

Ques 62.

Ques 63.

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$

Ques 64.

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.

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IGNOU BMTC 102 SOLVED ASSIGNMENT HINDI

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