BMTC-131 Solved Assignment 2023

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

\(a=b\:cos\:C+c\:cos\:B\)

BMTC-131 Sample Solution 2023

 

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