IGNOU MTE 7 SOLVED ASSIGNMENT HINDI

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IGNOU MTE 7 SOLVED ASSIGNMENT HINDI

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IGNOU MTE 7 SOLVED ASSIGNMENT HINDI

~~Rs. 80~~
Rs. 41

Last Date of Submission of IGNOU MTE-07 (BSC) 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 MTE 7 SOLVED ASSIGNMENT HINDI
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-07/Assignmentt-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 MTE-07 (BSC) 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. 80~~
Rs. 41

Questions Included in this Help Book

Ques 1.

swers.
(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).

Ques 2.

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}$

Ques 3.

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|.

Ques 4.

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 $x^2 + 2y^2 = 1$ .

Ques 5.

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.

Ques 6.

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.

Ques 7.

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

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

Ques 8.

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

Ques 9.

) Does the function

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

Ques 10.

Ques 11.

$$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 12.

(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

Find |2b - a + 3c|.

Ques 13.

$$\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 $x^2 + 2y^2 = 1$ .

Ques 14.

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

Ques 15.

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.

Ques 16.

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

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

Ques 17.

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

Ques 18.

) Does the function

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Course Name	Bachelor in Science
Course Code	BSC
Programm	BACHELOR DEGREE PROGRAMMES Courses
Language	English

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IGNOU MTE 7 SOLVED ASSIGNMENT HINDI

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