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

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

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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)

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

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

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