MTE-07 Solved Assignment 2023

IGNOU MTE-07 Solved Assignment 2023 | MTE

Solved By – Narendra Kr. Sharma – M.Sc (Mathematics Honors) – Delhi University


Please read the following points before ordering this IGNOU Assignment Solution.

Share with your Friends

Details For MTE-07 Solved Assignment

IGNOU MTE-07 Assignment Question Paper 2023


Course Code: MTE-07

Assignment Code: MTE-07/TMA/2023

Maximum Marks: 100

1. State whether the following statements are true or false. Justify your answers.

(a) The function \(f(x, y)=\frac{(x+2)(y-2)}{x+y}\) is homogeneous on its domain.

(b) \(\left\{3 x+\frac{1}{2 x} \mid 0<x<1\right\}\) is bounded above.

(c) \(\mathrm{f}: \mathbf{R}^{2} \rightarrow \mathbf{R}, \mathrm{f}(\mathrm{x}, \mathrm{y})=\frac{1+\mathrm{x}}{\mathrm{x}}, \mathrm{x} \neq 0\) is continuous on \([5,10] \times \mathbf{R}\).

(d) The function \(\mathrm{f}: \mathbf{R}^{3} \rightarrow \mathbf{R}, \mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z})=\mathrm{e}^{2 \cos \pi}\) is integrable on the region bounded by the sphere, \(x^{2}+y^{2}+z^{2}=1\)

(e) If \(\mathrm{f}: \mathbf{R}^{2} \rightarrow \mathbf{R}, \mathrm{f}(\mathrm{x}, \mathrm{y})=10\) and \(D=[3,10] \times[-1,5]\), then \(\iint_{\mathrm{D}} \mathrm{f} \mathrm{dx} \mathrm{dy}=420\).

2) (a) Show that the following limits do not exist:
i) \(\lim _{(x, y) \rightarrow(0,0)} \frac{(x-y)^{2}}{x^{2}+y^{2}}\)
ii) \(\lim _{(x, y) \rightarrow(0,0)} \frac{x y}{|y|}\)

(b) Show that \(\mathrm{f}(\mathrm{x}, \mathrm{y})=4 \mathrm{xy}-\mathrm{x}^{2}-\mathrm{y}^{4}\) has a saddle point at \((0,0)\).

3) (a) Find the directional derivation of

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

at \((0,0)\) in the direction \(\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\).

(b) Verify the chain rule for the Jacobians for the functions,

& x=e^{3 u}, y=2 u+5 v-w, z=u+v \\
& u=p+6, v=q^{3}, w=3 r

4. (a) Identify the intermediate forma and evaluate:
i) \(\quad \lim _{x \rightarrow \pi / 2} \frac{\ln \sin x}{1-\sin x}\)
ii) \(\quad \lim _{x \rightarrow \infty} 2 x(\ln (x+1)-\ln x)\)

(b) Show that

f(x, y)= \begin{cases}x \sin \frac{1}{x}, & x \neq 0 \\ y \sin \frac{1}{y}, & y \neq 0 \\ 0, & x=0, y=0\end{cases}

is continuous at \((0,0)\), but \(\mathrm{f}_{\mathrm{x}}(0,0)\) does not exist.

5. (a) Show that the equation \(\mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{y}^{2}-\mathrm{yx} \mathrm{x}^{2}-2 \mathrm{x}^{5}=0\) defines a continuously differentiable function \(y=g(x)\), in the neighbourhood of \((1,-1)\). Also find the derivative of \(g\).

(b) Show that the following functions are functionally dependent:

& u=3 x+2 y-z \\
& v=x-2 y+z \\
& w=x(x+2 y+z)

(c) Find the values of \(a\) and \(b\), if

\lim _{x \rightarrow 0} \frac{x(1+a \cos x)-b \sin x}{x^{3}}=1

6. (a) Evaluate the integral by converting to polar coordinates:

& \iint_{D}\left(4-x^{2}-y^{2}\right) d x d y \text {, where } D \text { is the region in the } 1^{\text {st }} \text { quadrant and bounded by } \\
& x^{2}+y^{2}=2 x .

(b) Find the work done by a force \(\overline{\mathrm{F}}=\left(x^{2} y, x y^{2}\right)\) in moving a particle from \((0,0)\) to \((1,1)\) along \(\mathrm{y}=\mathrm{x}^{2}\), and then from \((1,1)\) to \((2,1)\) along \(\mathrm{y}=1\).

7. (a) Find the surface area of the portion of the paraboloid \(z=4-x^{2}-y^{2}\), that lies above the xy-plane.

(b) Show that the following integral is independent of path and evaluate it:

\int_{(0, \pi)}^{(1, \pi / 2)}\left(e^{x} \cos y d x-e^{x} \sin y d y\right)

8. (a) What are the domain and range of \(f: \mathbf{R}^{2} \rightarrow \mathbf{R}\) defined by \(f(x, y)=\ln x+\frac{1}{y}\). (b) Evaluate \(f_{x y}\) at a point \((x, y)\) for the function \(f\) defined by \(f(x, y)=x \tan ^{-1} y\). Using Schwarz’s Theorem, evaluate \(f_{y x}\) at the point \((x, y)\).

(c) Let \(\mathrm{f}\) be a function defined by

f(x, y)=\left(\frac{|x|}{1+|x|}, \frac{|y|}{1+|y|}\right)

Check whether the composition gof exists, where \(\mathrm{g}:\left\{(\mathrm{x}, \mathrm{y}): \mathrm{x}^{2}+\mathrm{y}^{2} \leq 4\right\} \rightarrow \mathbf{R}\) is defined by \(g(x, y)=x y\). Find gof.

9. (a) Find the volume below the plane \(\mathrm{z}=1-\mathrm{y}\) and inside the cylinder \(\mathrm{x}^{2}+\mathrm{y}^{2}=1\), \(0 \leq \mathrm{z} \leq 1\).

(b) Reverse the order of integration and integrate:

\int_{0}^{2} \int_{y / 2}(x+y)^{2} d x d y

10. (a) Find the second Taylor polynomial for \(f(x, y)=e^{x y} \cos x\) about \((0, \pi / 2)\).

(b) If \(f(x, y)=\left\{\begin{array}{l}x \sin \frac{1}{x}+y \sin \frac{1}{y}, x y \neq 0 \\ 0, x y=0\end{array}\right.\)

\(sin\:3\theta =3\:sin\:\theta -4\:sin^3\:\theta \)

MTE-07 Sample Solution 2023


Frequently Asked Questions (FAQs)

You can access the Complete Solution through our app, which can be downloaded using this link:

App Link 

Simply click “Install” to download and install the app, and then follow the instructions to purchase the required assignment solution. Currently, the app is only available for Android devices. We are working on making the app available for iOS in the future, but it is not currently available for iOS devices.

Yes, It is Complete Solution, a comprehensive solution to the assignments for IGNOU. Valid from January 1, 2023 to December 31, 2023.

Yes, the Complete Solution is aligned with the IGNOU requirements and has been solved accordingly.

Yes, the Complete Solution is guaranteed to be error-free.The solutions are thoroughly researched and verified by subject matter experts to ensure their accuracy.

As of now, you have access to the Complete Solution for a period of 6 months after the date of purchase, which is sufficient to complete the assignment. However, we can extend the access period upon request. You can access the solution anytime through our app.

The app provides complete solutions for all assignment questions. If you still need help, you can contact the support team for assistance at Whatsapp +91-9958288900

No, access to the educational materials is limited to one device only, where you have first logged in. Logging in on multiple devices is not allowed and may result in the revocation of access to the educational materials.

Payments can be made through various secure online payment methods available in the app.Your payment information is protected with industry-standard security measures to ensure its confidentiality and safety. You will receive a receipt for your payment through email or within the app, depending on your preference.

The instructions for formatting your assignments are detailed in the Assignment Booklet, which includes details on paper size, margins, precision, and submission requirements. It is important to strictly follow these instructions to facilitate evaluation and avoid delays.


Terms and Conditions

  • The educational materials provided in the app are the sole property of the app owner and are protected by copyright laws.
  • Reproduction, distribution, or sale of the educational materials without prior written consent from the app owner is strictly prohibited and may result in legal consequences.
  • Any attempt to modify, alter, or use the educational materials for commercial purposes is strictly prohibited.
  • The app owner reserves the right to revoke access to the educational materials at any time without notice for any violation of these terms and conditions.
  • The app owner is not responsible for any damages or losses resulting from the use of the educational materials.
  • The app owner reserves the right to modify these terms and conditions at any time without notice.
  • By accessing and using the app, you agree to abide by these terms and conditions.
  • Access to the educational materials is limited to one device only. Logging in to the app on multiple devices is not allowed and may result in the revocation of access to the educational materials.

Our educational materials are solely available on our website and application only. Users and students can report the dealing or selling of the copied version of our educational materials by any third party at our email address ( or mobile no. (+91-9958288900).

In return, such users/students can expect free our educational materials/assignments and other benefits as a bonafide gesture which will be completely dependent upon our discretion.

Scroll to Top
Scroll to Top