IGNOU MTM 7 SOLVED ASSIGNMENT

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IGNOU MTM 7 SOLVED ASSIGNMENT

~~Rs. 90~~
Rs. 15

IGNOU MTM 7 SOLVED ASSIGNMENT

~~Rs. 90~~
Rs. 15

Last Date of Submission of IGNOU MTM-07 (MTM) 2025 Assignment is for January 2025 Session: 30th September, 2025 (for December 2025 Term End Exam).
Semester Wise
January 2025 Session: 30th March, 2025 (for June 2025 Term End Exam).
July 2025 Session: 30th September, 2025 (for December 2025 Term End Exam).

Title Name	IGNOU MTM 7 SOLVED ASSIGNMENT
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	MASTER DEGREE PROGRAMMES
Course Code	MTM
Course Name	Master of Arts in Tourism Management
Subject Code	MTM 7
Subject Name	Managing Sales and Promotion in Tourism
Year	2025
Session
Language	English Medium
Assignment Code	MTM-07/Assignmentt-1//2025
Product Description	Assignment of MTM (Master of Arts in Tourism Management) 2025. Latest MTM 07 2025 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU MTM-07 (MTM) 2025 Assignment is for January 2025 Session: 30th September, 2025 (for December 2025 Term End Exam). Semester Wise January 2025 Session: 30th March, 2025 (for June 2025 Term End Exam). July 2025 Session: 30th September, 2025 (for December 2025 Term End Exam).

~~Rs. 90~~
Rs. 15

Questions Included in this Help Book

Ques 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 $F:\mathbb{R}^2\to\mathbb{R}^2,\text{defined by}F(x,y)=(y+2,x+y),$ is locally invertible at any $\left(x,y\right)\epsilon R^{2}$

(iv) $f:[-1,1]\times[-2,2]\to\mathbb{R},\text{defined by}$

$f(x,y)=\begin{cases}x,&\text{if}y\text{is rational}\\0,&\text{if}y\text{is not rational}\end{cases}$ is integrable.

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

Ques 2.

Find the following limits:

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

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

Ques 3.

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

Ques 4.

Using only the definitions, find $f_{xy}(0,0)\text{and}f_{yx}(0,0),$ 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 5.

Let the function f be defined by

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

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

Ques 6.

$\text{Let}x=e^r\cos\theta,y=e^r\sin\theta\text{and}$ 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 7.

$\text{Let}a=(1,2,3),b=(-5,3,-2),c=(2,-4,1)$ )1,4 be three points in . R³

Find |2 b − a + 3c l.

Ques 8.

Find the centre of gravity of a thin sheet with density δ(x, y) = y, bounded by the

curves $y=4x^2\text{and}x=4.$

Ques 9.

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

Ques 10.

State Green’s theorem, and apply it to evaluate

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

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

Ques 11.

Find the extreme values of the function

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

Ques 12.

State a necessary condition for the functional dependence of two differentiable functions f and g on an open subset D of . R²Verify this theorem for the functions f and g, defined by

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

Ques 13.

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

Ques 14.

Check the local inevitability 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 15.

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

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

Ques 16.

Find the domain and range of the function f , defined by $f(x,y)=\frac{2xy}{x^2+y^2}.\text{Also}$

find two level curves of this function. Give a rough sketch of them

Ques 17.

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

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

Ques 18.

Use double integration of find the volume of the ellipsoid

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

Ques 19.

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 20.

Suppose S and C are subsets of R³. 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

Ques 21.

Identify the level curves of the following functions:

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

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

(iii) x-y

(iv) y/x

Ques 22.

Does the function

$f(x,y)=\frac{x^2-y^2}{x^2+y^2},\quad x\neq 0,y\neq 0$ satisfy the requirement of Schwarz's theorem at

(1,1)? Justify your answer.

Ques 23.

Locate and classify the stationary points of the following:

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

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

Ques 24.

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 $F:\mathbb{R}^2\to\mathbb{R}^2,\text{defined by}F(x,y)=(y+2,x+y),$ is locally invertible at any $\left(x,y\right)\epsilon R^{2}$

(iv) $f:[-1,1]\times[-2,2]\to\mathbb{R},\text{defined by}$

$f(x,y)=\begin{cases}x,&\text{if}y\text{is rational}\\0,&\text{if}y\text{is not rational}\end{cases}$ is integrable.

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

Ques 25.

Find the following limits:

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

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

Ques 26.

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

Ques 27.

Using only the definitions, find $f_{xy}(0,0)\text{and}f_{yx}(0,0),$ 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 28.

Let the function f be defined by

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

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

Ques 29.

$\text{Let}x=e^r\cos\theta,y=e^r\sin\theta\text{and}$ 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 30.

$\text{Let}a=(1,2,3),b=(-5,3,-2),c=(2,-4,1)$ )1,4 be three points in . R³

Find |2 b − a + 3c l.

Ques 31.

Find the centre of gravity of a thin sheet with density δ(x, y) = y, bounded by the

curves $y=4x^2\text{and}x=4.$

Ques 32.

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

Ques 33.

State Green’s theorem, and apply it to evaluate

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

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

Ques 34.

Find the extreme values of the function

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

Ques 35.

State a necessary condition for the functional dependence of two differentiable functions f and g on an open subset D of . R²Verify this theorem for the functions f and g, defined by

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

Ques 36.

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

Ques 37.

Check the local inevitability 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 38.

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 39.

Find the domain and range of the function f , defined by $f(x,y)=\frac{2xy}{x^2+y^2}.\text{Also}$

find two level curves of this function. Give a rough sketch of them

Ques 40.

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

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

Ques 41.

Use double integration of find the volume of the ellipsoid

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

Ques 42.

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 43.

Suppose S and C are subsets of R³. 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

Ques 44.

Identify the level curves of the following functions:

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

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

(iii) x-y

(iv) y/x

Ques 45.

Does the function

$f(x,y)=\frac{x^2-y^2}{x^2+y^2},\quad x\neq 0,y\neq 0$ satisfy the requirement of Schwarz's theorem at

(1,1)? Justify your answer.

Ques 46.

Locate and classify the stationary points of the following:

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

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

Ques 47.

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 $F:\mathbb{R}^2\to\mathbb{R}^2,\text{defined by}F(x,y)=(y+2,x+y),$ is locally invertible at any $\left(x,y\right)\epsilon R^{2}$

(iv) $f:[-1,1]\times[-2,2]\to\mathbb{R},\text{defined by}$

$f(x,y)=\begin{cases}x,&\text{if}y\text{is rational}\\0,&\text{if}y\text{is not rational}\end{cases}$ is integrable.

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

Ques 48.

Find the following limits:

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

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

Ques 49.

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

Ques 50.

Using only the definitions, find $f_{xy}(0,0)\text{and}f_{yx}(0,0),$ 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 51.

Let the function f be defined by

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

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

Ques 52.

$\text{Let}x=e^r\cos\theta,y=e^r\sin\theta\text{and}$ 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 53.

$\text{Let}a=(1,2,3),b=(-5,3,-2),c=(2,-4,1)$ )1,4 be three points in . R³

Find |2 b − a + 3c l.

Ques 54.

Find the centre of gravity of a thin sheet with density δ(x, y) = y, bounded by the

curves $y=4x^2\text{and}x=4.$

Ques 55.

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

Ques 56.

State Green’s theorem, and apply it to evaluate

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

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

Ques 57.

Find the extreme values of the function

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

Ques 58.

State a necessary condition for the functional dependence of two differentiable functions f and g on an open subset D of . R²Verify this theorem for the functions f and g, defined by

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

Ques 59.

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

Ques 60.

Check the local inevitability 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 61.

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 62.

Find the domain and range of the function f , defined by $f(x,y)=\frac{2xy}{x^2+y^2}.\text{Also}$

find two level curves of this function. Give a rough sketch of them

Ques 63.

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

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

Ques 64.

Use double integration of find the volume of the ellipsoid

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

Ques 65.

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 66.

Suppose S and C are subsets of R³. 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

Ques 67.

Identify the level curves of the following functions:

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

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

(iii) x-y

(iv) y/x

Ques 68.

Does the function

$f(x,y)=\frac{x^2-y^2}{x^2+y^2},\quad x\neq 0,y\neq 0$ satisfy the requirement of Schwarz's theorem at

(1,1)? Justify your answer.

Ques 69.

Locate and classify the stationary points of the following:

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

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

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Course Name	Master of Arts in Tourism Management
Course Code	MTM
Programm	MASTER DEGREE PROGRAMMES Courses
Language	English

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