BMTC-133 Solved Assignment 2023

IGNOU BMTC-133 Solved Assignment 2023 | B.Sc (G) CBCS

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

101.00

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

Share with your Friends

Details For BMTC-133 Solved Assignment

IGNOU BMTC-133 Assignment Question Paper 2023

 

Course Code: BMTC-133

Assignment Code: BMTC-133/TMA/2023

Maximum Marks: 100

1. Which of the following statements are true or false? Give reasons for your answers.

a) The singleton set \(\{x\}\) for any \(x \in \boldsymbol{R}\) is an open set.

(b) The series is \(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots\) is a convergent series.

c) The function \(f(x)=\left\{\begin{array}{cc}e^{-x}+e^{x}, & \text { when } x \neq 0 \\ 1, & \text { when } x=0\end{array}\right.\) is continuous on \([0,1]\).

d) The function \(f\) defined by \(f(x)=|x-\sqrt{2}| \forall x \in \boldsymbol{R}\) has a critical point at \(x=\sqrt{2}\).

e) If a function has finitely many points of discontinuities, then the function is not integrable.

2. a) Prove that the sequence \(\left\{a_{n}\right\}\) where \(a_{n}=\frac{2^{2}}{n^{2}+3^{2}}\), converges to 0 .

b) Find the following limit, if it exists:

\[
\lim _{x \rightarrow 0} \frac{x^{3} \sin x^{3}}{1-\cos x^{3}}
\]

c) Test the convergence of the following series.

i) \(\frac{1.2}{3^{2} \cdot 4^{2}}+\frac{3.4}{5^{2} \cdot 6^{2}}+\frac{5 \cdot 6}{7^{2} \cdot 8^{2}}+\cdots\)

ii) \(\sum \frac{\sqrt{n^{4+1}}-\sqrt{n^{4-1}}}{n}\)

3. a) Explain the order completeness property of \(\boldsymbol{R}\), and use it to show that the set \(S=\left\{\frac{n}{n+1} \mid n \in \boldsymbol{N}\right\}\) has a supremum as well as infimum in \(\boldsymbol{R}\).

b) Let \(f\) be the function defined by

\[
f(x)= \begin{cases}2 x-1, & \text { if } x \in] \infty, 1[ \\ \frac{3 x^{2}-2}{x}, & \text { if } x \in[1,2[ \\ (1+2 x)^{2}, & \text { if } x \in[2, \infty[\end{cases}
\]

Discuss the continuity of \(f\) on \(] \infty, \infty[\).

c) Check whether the following sets are open, closed or neither:

i) \(] 1,5[\cup[3,6]\)

ii) \([0,1] \cup\left\{\frac{5}{9}, \frac{3}{4}, \frac{10}{7}\right\}\)

iii) \(\{5 n: n \in N\}\) 4. a) Using the principle of mathematical induction, prove that 7 is a factor of \(3^{2 n-1}+2^{n+1}, \forall n \in \boldsymbol{N}\)

b) Show that the equation \(x^{3}-2 x^{2}+5 x-12=0\) has a root which is a positive real number.

c) Prove that the set \(\left\{\frac{3}{6}, \frac{3}{7}, \frac{3}{8}, \ldots\right\}\) is a countable set.

5. a) Show that the local maximum value of \(\left(\frac{1}{x}\right)^{x}\) is \(e^{1 / e}\).

b) Verify Cauchy Mean Value Theorem for the functions

\[
f(x)=x, g(x)=\frac{1}{x}, x \in[1,4]
\]

c) Show that \(1+x \leq e^{x}, \forall x \in[0, \infty[\). Does the inequality hold for \(x<0\) ? Justify your answer.

6. a) By showing that the remainder after \(n\)-terms tends to zero, find Maclaurin’s series expansion of \(\sin 2 x\).

b) Find the greatest value of the function \(f(x)=x^{4}-2 x^{3}-3 x^{2}+4 x+7\) over the interval \([0,1]\).

7. a) Consider the function \(f(x)=2 \cos x\) in the interval \(\left[0, \frac{\pi}{2}\right]\). Show that \(L\left(P_{1}, f\right) \leq L\left(P_{2}, f\right)\) and \(U\left(P_{2}, f\right) \leq U\left(P_{1}, f\right)\) where \(P_{1}=\left\{0, \frac{\pi}{3}, \frac{\pi}{2}\right\}\) and \(P_{2}=\left\{0, \frac{\pi}{6}, \frac{\pi}{3}, \frac{\pi}{2}\right\}\)

b) Show that the derivative \(f^{\prime}\) of the following function \(f\) given by

\[
f(x)=\left\{\begin{array}{cl}
x^{2} \sin \frac{1}{x} & \text { if } x \neq 0 \\
0 & \text { if } x=0
\end{array}\right.
\]

exists at \(x=0\) but \(f^{\prime}\) is not continuous at 0 .

8. a) Check whether the following function has a mean value in the interval \([2,5]\)

\[
f(x)=\left\{\begin{array}{lll}
1 & \text { if } & 2 \leq x<3 \\
3 & \text { if } & 3 \leq x \leq 5
\end{array}\right.
\]

Does this contradict the mean value theorem? Justify.

b) Find the limit as \(n \rightarrow \infty\), of the sum

\[
\frac{n}{3 n^{2}+1^{2}}+\frac{n}{3 n^{2}+2^{2}}+\frac{n}{3 n^{2}+3^{2}}+\cdots+\frac{1}{4 n} \text {. }
\]

c) Apply Weierstrass \(M\)-test to show that the series \(\sum \frac{10}{n^{4}+x^{4}}\) converges uniformly for all \(x \in \boldsymbol{R}\).

9. a) Using Riemann integration show that \(\int_{1}^{2}(3 x+1) d x=\frac{11}{2}\).

b) Show that the function \(f(x)=\frac{1}{x}\) is continuous on \(\left.] 0,1\right]\) but not uniformly continuous.

10. a) Give one example for the following. Justify your choice of examples.

i) A bounded set having no limit point.

ii) A bounded set having infinite number of limit points.

iii) A infinite compact set which is not an interval.

b) Prove that the function \(f\) defined by

\[
f(x)=\left\{\begin{array}{cc}
4, & \text { if } x \text { is rational } \\
-4, & \text { if } x \text { is irrational }
\end{array}\right.
\]

is discontinuous at each real number, using the sequential definition of continuity. (4)

\(sin\left(\theta -\phi \right)=sin\:\theta \:cos\:\phi -cos\:\theta \:sin\:\phi \)

BMTC-133 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.

\(sin\left(2\theta \right)=2\:sin\:\theta \:cos\:\theta \)

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