MTE-05 Solved Assignment 2023

IGNOU MTE-05 Solved Assignment 2023 | MTE

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

151.00

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

Share with your Friends

Details For MTE-05 Solved Assignment

IGNOU MTE-05 Assignment Question Paper 2023

 

Course Code: MTE-05

Assignment Code: MTE-05/TMA/2023

Maximum Marks: 100

1. Check whether the following statements are true or false. Justify your answer with a short explanation or a counter example.

\((2 \times 10=20)\)

(i) The numbers \(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{3}}, \frac{3}{4}\) are the direction cosines of a line.

(ii) The points \((1,2),(7,6)\) and \((4,4)\) are collinear.

(iii) The conic \(12 x^{2}+12 x y+3 y^{2}+2 x+y=0\) is degenerate.

(iv) Intersection of the ellipsoid \(\frac{x^{2}}{4}+\frac{y^{2}}{25}+\frac{z^{2}}{4}=1\) and the plane \(y=5\) is a circle.

(v) The conicoid \(3 x^{2}+y^{2}+2 x y+x-y-z+1=0\) is non-central.

(vi) The line \(y=x\) is a tangent to the parabola \(y^{2}=c x, c>0\).

(vii) The equation \(2 x^{2}+y+z+1=0\) represents a paraboloid.

(viii) Projection of a line segment on a line perpendicular to it is the length of the line segment.

(ix) The lines \(x=-y, z=2\) and \(x=y, z=0\) intersect each other.

(x) Every planar section of a cylinder is a circle.

2. (a) Trace the conic \(x^{2}-2 x y+y^{2}-3 x+2 y+3=0\).

(b) Prove that the conic passing through the points of intersection of two rectangular hyperbolas is also a rectangular hyperbola.

(c) Show that the line \(x=y\) touches the conic \(a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+\) \(c=0\), if \(f+g=0\).

3. (a) Let \(P\) be the midpoint of the line segment joining the points \(A(a+b, b)\) and \(B(a-b, a+b)\). Find the slope of the line passing through \(P\) and \(Q\left(b,-\frac{a}{2}\right)\). Under what conditions on \(a\) and \(b\), this line is parallel to the \(y\)-axis?

(b) (i) Show that \(\left|\begin{array}{ccc}x & y & 1 \\ 2 & 3 & 1 \\ -4 & 7 & 1\end{array}\right|=0\) represents the equation of a line passing through \((2,3)\) and \((-4,7)\)

(ii) Prove that the equation of a line through \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) can be expressed in the form \(\left|\begin{array}{lll}x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1\end{array}\right|=0\)

(c) Find the eccentricity, foci, centre and directrices of the ellipse \(\frac{x^{2}}{16}+\frac{y^{2}}{4}=1\). Also give a rough sketch of it.

(d) Prove that the length of the chord of a parabola which passes through the focus and which is inclined at \(30^{\circ}\) to the axis of the parabola is four times the length of the latus rectum.

4. (a) Find the equations of the line through \((1,3,4)\) and parallel to the line joining the points \((-4,5,3)\) and \((8,9,7)\).

(b) Find the equation of the plane which passes through the line of intersection of the planes \(3 x+4 y-5 z=9\) and \(2 x+6 y+6 z=7\) and which is perpendicular to the plane \(3 x+2 y-5 z+6=0\). (c) Find the distance of the origin from the plane which passes through \((2,1,8),(1,0,2)\) and \((-3,4,6)\).

5. (a) Show that the plane \(2 x+y+2 z=0\) is a tangent plane to the sphere \(x^{2}+y^{2}+\) \(z^{2}-2 x+2 y-2 z+2=0\).

(b) Find the equation of the sphere touching the plane \(8 x+5 y+3 z+1=0\) at \((3,-1,-1)\) and cutting the sphere \(x^{2}+y^{2}+z^{2}-2 x+y-z-6=0\) orthogonally.

(c) Find the angle between the lines of intersection of the cone \(4 x^{2}+y^{2}+4 z^{2}+\) \(4 y z+2 z x=0\) and the plane \(x+2 y+3 z=0\).

(d) Find the equation of the cylinder with base

\[
x^{2}+y^{2}+z^{2}-3 x-6 z+9=0, x-2 y+2 z-6=0
\]

6. (a) Show that the perpendiculars drawn from the origin to tangent planes to the cone \(x^{2}-y^{2}+5 z^{2}+4 x y=0\) lie on the cone \(x^{2}-y^{2}+z^{2}+4 x y=0\).

(b) Transform the equation \(x^{2}+2 y^{2}-6 z^{2}-2 x-8 y+3=0\) by shifting the origin to \((1,2,0)\) without changing the directions of the coordinate axes. What object does this new equation represent? Give a rough sketch of it.

(c) Show that the conicoid \(2 x^{2}+2 y^{2}+x y-y z+z x+2 x-y+5 z+1=0\) is central. Hence find its centre.

7. (a) Examine which of the following conicoids are central and which are non-central. Also determine which of the central conicoids have centre at the origin.

(i) \(x^{2}+y^{2}+z^{2}+4 x+3 y-z=0\)

(ii) \(2 x^{2}-y^{2}-z^{2}+x y+y z-z x=1\)

(iii) \(x^{2}+y^{2}-z^{2}-2 x y-3 y z-6 z x+x-2 y+5 z+4=0\)

(b) Find the transformation of the equation \(12 x^{2}-2 y^{2}+z^{2}=2 x y\) if the origin is kept fixed and the axes are rotated in such a way that the direction ratios of the new axes are \(1,-3,0 ; 3,1,0 ; 0,0,1\).

(c) Find the projection of the line segment joining the points \((1,-1,6)\) and \((4,3,2)\) on the line \(\frac{x-4}{3}=-y=\frac{z}{5}\).

8. (a) Identify and trace the conicoid \(y^{2}+3 z^{2}=x\). Describe its sections by the planes \(y=0\) and \(z=0\).

(b) Find the equation of tangent plane to the conicoid \(x^{2}+3 y^{2}=4 z\) at \((2,-4,13)\). Represent the tangent plane geometrically.

\(sin^2\left(\frac{\theta }{2}\right)=\frac{1-cos\:\theta }{2}\)

MTE-05 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.

\(2\:cos\:\theta \:cos\:\phi =cos\:\left(\theta +\phi \right)+cos\:\left(\theta -\phi \right)\)

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 (abstract4math@gmail.com) 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