IGNOU BMTC131 Solved Assignment 2023  B.Sc (G) CBCS
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IGNOU BMTC131 Assignment Question Paper 2023
Course Code: BMTC131
Assignment Code: BMTC131/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 counterexample, 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:
\[
f(x)=\sqrt{\frac{1x}{x}}
\]
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:
\[
f(x)=\frac{x^{2}+x6}{x2}
\]
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 \(x^{3}7 x6=0\). Also, find the equation whose roots are \(\alpha+\beta \cdot \beta+\gamma\) and \(\alpha+\gamma\).
b) Evaluate: \(\lim _{x \rightarrow 0} \frac{e^{4 x}14 x}{x^{2}}\).
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 counterexample, 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 \((x1)\) upto four terms.
b) Verify Rolle’s theorem for the function \(f\), defined by \(f(x)=x(x2) e^{x}\) 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:
\[
y=\frac{x}{x1}
\]
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:
\[
\int \frac{(x2)}{x^{2}6 x+10} d x
\]
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) \(\int_{0}^{3 \pi / 2}\sin x d x\)
BMTC131 Sample Solution 2023
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