# IGNOU BMTC-131 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-131 Solved Assignment

## IGNOU BMTC-131 Assignment Question Paper 2023

Course Code: BMTC-131

Assignment Code: BMTC-131/TMA/2023

Maximum Marks: 100

\section{PART – A (40 marks)}

1. Which of the following statements are true, and which are false? Give a short proof or a counter-example, whichever is appropriate in support of your answer.

i) A cubic equation with real coefficients has at least one real root.

ii) If $$\mathrm{A}$$ and $$\mathrm{B}$$ are two sets, then:

$\mathrm{A} \cup \mathrm{B}=\mathrm{B} \cap(\mathrm{A} / \mathrm{B})$

iii) The greatest integer function is continuous on $$\mathbf{R}$$.

iv) The maximum possible domain of a function $$\mathrm{f}$$, given by:

$f(x)=\sqrt{\frac{1-x}{x}}$

is $$] 0,1[$$.

v) $$\quad \lim _{x \rightarrow \infty}\left(\frac{1}{2^{x}}-1\right)=-1$$

2. a) Find $$\frac{d y}{d x}$$ for the following cases:

i) $$\quad y=\left[x+\left(x+\sin ^{2} x\right)^{3}\right]^{4}$$

ii) $$x^{4}+y^{4}=16$$

b) Find $$\frac{d y}{d x}$$, when $$y=x^{x}+x^{x}$$.

3. a) Let:

$f(x)=\frac{x^{2}+x-6}{|x-2|}$

Find:

i) $$\quad \lim _{x \rightarrow 2^{+}} f(x)$$

ii) $$\lim _{x \rightarrow 2^{-}} f(x)$$

iii) Does $$\lim _{x \rightarrow 2} f(x)$$ exist? Why, or why not?

iv) Sketch the rough graph of $$h$$.

b) Is:

$\left[\left(\frac{2-\mathrm{i}}{1+\mathrm{i}}-\frac{\mathrm{i}}{2+\mathrm{i}}\right) 3 \mathrm{i}\right]$

a purely imaginary number? Give reasons for your answer. Also, represent this number in an Argand plane.

4. a) Find all the roots $$\alpha, \beta, \gamma$$ of the cubic equation $$x^{3}-7 x-6=0$$. Also, find the equation whose roots are $$\alpha+\beta \cdot \beta+\gamma$$ and $$\alpha+\gamma$$.

b) Evaluate: $$\lim _{x \rightarrow 0} \frac{e^{4 x}-1-4 x}{x^{2}}$$.

c) For which values of the constant $$\mathrm{C}$$ is the function $$\mathrm{f}$$ continuous on $$\mathbf{R}$$, where $$\mathrm{f}$$ is defined by:

$f(x)= \begin{cases}C x^{2}+2 x, & \text { if } x<2 \\ x^{3}-C x, & \text { if } x \geq 2\end{cases}$

\section{PART – B (40 marks)}

5. Which of the following statements are true, and which are false? Give a short proof or a counter-example, whichever is appropriate in support of your answer.

i) A critical point of a function is its extremum point.

ii) Curve $$\mathrm{y}\left(\mathrm{x}^{2}+1\right)=3$$ has an oblique asymptote.

iii) $$\frac{d}{d x}\left(\sin \left(x^{2}\right)\right)=\frac{d}{d x}\left(\sin ^{2} x\right)$$

iv) $$\frac{d}{d x}\left(\int_{1}^{x^{4}} \sec t d t\right)=4 x^{2} \sec \left(x^{4}\right)$$.

v) The function $$f$$, defined by $$f(x)=\frac{1}{1+x^{2}}$$, is integrable on every finite subinterval in $$\mathbf{R}$$.

6. a) Expand $$\mathrm{e}^{2 x}$$ in powers of $$(x-1)$$ upto four terms.

b) Verify Rolle’s theorem for the function $$f$$, defined by $$f(x)=x(x-2) e^{-x}$$ on the interval $$[0,2]$$.

7. a) If:

$y=x^{3} \cos x$

then find the $$n$$th derivative of $$y$$.

b) Check whether the relation:

$\mathrm{R}=\{(\mathrm{x}, \mathrm{y}) \mid \mathrm{xy} \text { is the square of an integer, } \mathrm{x}, \mathrm{y} \in \mathbf{N}\}$

is and equivalence relation or not. 8. Trace the curve:

$y=\frac{x}{x-1}$

stating all the properties you use to trace if.

\section{PART – C (20 marks)}

9. a) Find the perimeter of the cardioids $$r=1+\sin \theta$$.

b) Using the $$\in-\delta$$ definition of limit, prove that:

$\lim _{x \rightarrow 1} x^{3}-2 x=-1$

10. a) Evaluate the following integral:

$\int \frac{(x-2)}{x^{2}-6 x+10} d x$

b) Evaluate the following integrals:

i) $$\int_{1}^{9} \frac{\left(2 \mathrm{t}^{2}+\mathrm{t}^{2} \sqrt{\mathrm{t}}-1\right)}{\mathrm{t}^{2}} d t$$.

ii) $$\int_{0}^{3 \pi / 2}|\sin x| d x$$

$$cos\:2\theta =1-2\:sin^2\theta$$

## BMTC-131 Sample Solution 2023

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$$cos\:2\theta =cos^2\theta -sin^2\theta$$

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