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BMTC 131: Calculus

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Title Name	IGNOU BAM BMTC 131 Calculus Solved Assignment English
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	BACHELOR DEGREE PROGRAMMES
Course Code	BAM
Course Name	Four Year Under Graduate Programmes/Bachelor of Arts
Subject Code	BMTC 131
Subject Name	Calculus
Year	2026
Session	-
Language	English
Assignment Code	BMTC 131/Assignment-1/2026
Product Description	Assignment of BAM (Four Year Under Graduate Programmes/Bachelor of Arts) 2026. Latest BMTC 131 2026 Solved Assignment Solutions
Format	Ready-to-Print PDF (.soft copy)

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BMTC 131 2026 - English

Assignment
(To be done after studying all the blocks)

Course Code: BMTC-131

Assignment Code: BMTC-131/TMA/2026

Maximum Marks: 100

1. Which of the following statements are true or false? Give reasons for your answer in the form of a short proof or a counter-example, whichever is appropriate. $\quad (10)$

a) The set $\{x \in \mathbf{R} : x^2 - 3x + 2 = 0\}$ is an infinite set.

b) The greatest interger function is continuous on $\mathbf{R}$ .

c) $\frac{d}{dx} \left[ \int_{3}^{e^x} \ln t \, dt \right] = x e^x - \ln 3$ .

d) Every integrable function is monotonic.

e) $a \oplus b = \sqrt{a + b}$ defines a binary operation on $\mathbf{Q}$ , the set of rational numbers.

2. a) Find the domain of the function f given by $f(x) = \sqrt{\frac{2 - x}{x^2 + 1}}$ .

b) The set $\mathbf{R}$ of real numbers with the usual addition (+) and usual multiplication $(\cdot)$ is given. Define (*) on $\mathbf{R}$ as:

$$a * b = \frac{a + b}{2}, \forall a, b \in \mathbf{R}.$
Is (*) associative in $\mathbf{R}$ ? Is $(\cdot)$ distributive over (*) in $\mathbf{R}$ ? Check.

3. a) If $|z - 1 + 2i| = 4$ , show that the point z + i describes a circle. Also draw this circle.

b) Express $\frac{x - 1}{x^3 - x^2 - 2x}$ as a sum of partial fractions.

4. a) Find the least value of $a^2 \sec^2 x + b^2 \csc^2 x$ , where a > 0, b > 0.

b) Evaluate:

$$\int \frac{x^2 \cot^{-1}(x^3)}{1 + x^6} dx$

c) For any two sets S and T, show that:

$$S \cup T = (S - T) \cup (S \cap T) \cup (T - S).$

Depict this situation in the Venn diagram.

5. a) Let f and g be two functions defined on $\mathbf{R}$ by:

$$f(x) = x^3 - x^2 - 8x + 12$

and $g(x) = \begin{cases} \frac{f(x)}{x + 3}, & \text{when } x \neq -3 \\ \alpha, & \text{when } x = -3 \end{cases}$

i) Find the value of $\alpha$ for which g is continuous at $x = -3$ .

ii) Find all the roots of $f(x) = 0$ .

b) Find the area between the curve $y^2(4 - x) = x^3$ and its asymptote parallel to y-axis.

6. a) If the revenue function is given by $\frac{dR}{dx} = 15 + 2x - x^2$ , x being the input, find the maximum revenue. Also find the revenue function R, if the initial revenue is 0.

b) Trace the curve $y^2(x + 1) = x^2(3 - x)$ , clearly stating all the properties used for tracing it.

7. a) Find the length of the cycloid $x = \alpha(\theta - \sin \theta), y = \alpha(1 - \cos \theta)$ and show that the line $\theta = \frac{2\pi}{3}$ divides it in the ratio 1 : 3.

b) Find the condition for the curves, $ax^2 + by^2 = 1$ and $a'x^2 + b'y^2 = 1$ intersecting orthogonally.

8. a) If $y = e^{m \sin^{-1} x}$ , then show that $(1 - x^2)y_2 - xy_1 - m^2y = 0$ . Hence using Leibniz's formula, find the value of (1 - x²)y_n+2 - (2n + 1)xy_n+1.

b) Find the largest subset of $\mathbf{R}$ on which the function $f : \mathbf{R} \to \mathbf{R}$ defined as:
$$f(x) = \begin{cases} 2x &, x > 5 \\ x + 5 &, 1 \leq x \leq 5 \\ |x| &, x < 1 \end{cases}$
is continuous.

9. a) Solve the equation:

$$x^4 + 15x^3 + 70x^2 + 120x + 64 = 0$

given that its roots are in G.P.

b) Evaluate:

$$\lim_{x \to 0} \frac{\tan x - \sin x}{x^3}$

10. a) If $I_{m,n} = \int x^m (\log x)^n dx$ , show that:

$$(m+1)I_{m,n} = x^{m+1}(\log x)^n - n I_{m,n-1}.$
Hence find the value of $\int x^4 (\log x)^3 dx$ .

b) Verigy Lagrange's mean value theorem for the function f defined by

$f(x) = 2x^2 - 7x - 10$ over [2, 5].

BMTC 131 2025 - English

Assignment

(To be done after studying all the blocks)

Course Code: BMTC-131

Assignment Code: BMTC-131/TMA/2025

Maximum Marks: 100

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

a) The set {S∈ R: x²-3x+2=0} is an infinite set.

b) The greatest interger function is continuous on R.

c) $frac{d}{dx}left[int_{3}^{e^{x}}ln t,dt ight]=xe^{x}-ln 3.$

d) Every integrable function is monotonic.

e) $aoplus b=sqrt{a+b}$ defines a binary operation on Q, the set of rational numbers.

a) Find the domain of the function f given by $f(x)=sqrt{frac{2-x}{x^2+1}}$

b) The set R of real numbers with the usual addition (+) and usual multiplication (+) is given. Define (*) on R as:

$a*b=frac{a+b}{2},forall a,binmathbb{R}.$

Is (*) associative in R? Is (.) distributive (*) in R? Check.

a) If z-1+2i|= 4, show that the point z+ i describes a circle. Also draw this circle.

b) Express $frac{x-1}{x^3-x^2-2x}$ as a sum of partial fractions.

a) Find the least value of a² sec² x + b² cosec²x, where a>0,b>0.

b) Evaluate:

$intfrac{x^2cot^{-1}(x^3)}{1+x^6}dx$

c) For any two sets S and T, show that:

Image ignouacademy-com-ignou-bam-bmtc-131-solved-assignment-html-p-bmtc-52886

Depict this situation in the Venn diagram.

a) Let f and g be two functions defined on R by:

f(x) = x³-x²-8x+12

and

$g(x)=egin{cases}frac{f(x)}{x+3},& ext{when}x eq-3\alpha,& ext{when}x=-3end{cases}$

i) Find the value of a for which f is continuous at x = -3.

ii) Find all the roots of f(x) = 0.

b) Find the area between the curve y²(4-x) = x³ and its asymptote parallel to y-axis.

a) If the revenue function is given by dR /dx 15+2x-x², x being the input, find the maximum revenue. Also find the revenue function R, if the initial revenue is 0.

b) Trace the curve y²(x+1)=x²(3-x), clearly stating all the properties used for tracing it.

a) Find the length of the cycloid $x=a( heta-sin heta),quad y=a(1-cos heta)$ and show that the line $heta=frac{2pi}{3}$ divides it in the ratio 1: 3.

b) Find the condition for the curves, ax² + by² =1 and a'x² + b'y² = 1 intersecting orthogonally.

a) If y = e™ sin-¹ x, then show that (1-x²)y₂-xy, -m²y = 0. Hence using Leibnitz's formula, find the value of (1-x²)y_n+2-(2n+1)xy _n+1

b) Find the largest subset of R on which the function f: R→R defined as:

$f(x)=egin{cases}2x,&x>5\x+5,&1leq xleq 5\|x|,&x<1end{cases}$

is continuous.

a) Solve the equation:

x² +15x³ +70x²+120x+64=0

given that its roots are in G.P.

b) Evaluate:

$lim_{x o 0}frac{ an x-sin x}{x^3}$

10.

a) If $I_{m,n}=int x^3(log x)^n dx$ , show that:

(m+1)I_^m,n = x^m+l(log x)" -nI_m, n-1.

Hence find the value of fx²(logx)³dx.

Verigy Lagrange's mean value theorem for the function f defined by

f(x) = 2x²-7x-10over [2, 5].

BMTC 131 2025 - Hindi

सत्रीय कार्य

(सभी ब्लॉकों का अध्ययन करने के बाद किया जाना है)

पाठ्यक्रम कोड: BTMC-131

सत्रीय कार्य कोड: BTMC-131/TMA/2025

अधिकतम अंक: 100

1. निम्नलिखित कथनों में से कौन-से कथन सत्य और कौन-से असत्य हैं? अपने उत्तर के पक्ष में एक संक्षिप्त उपपत्ति या प्रति-उदाहरण दीजिए।

a) समुच्चय S ∈ R : x² - 3x + 2 = 0} एक अपरिमित समुच्चय है।

b) अधिकतम पूर्णांक फलन, R पर सतत् होता है।

c) $frac{d}{dx}left[int_3^{e^x}ln t,dt ight]=xe^x-ln 3.$

d) प्रत्येक समाकलनीय फलन एकदिष्ट होता है।

e) $aoplus b=sqrt{a+b}$ परिमेय संख्याओं के समुच्चय Q, पर एक द्विआधारी संक्रिया है।

2. a) $f(x)=sqrt{frac{2-x}{x^2+1}}$ द्वारा परिभाषित फलन एफ का प्रांत ज्ञात कीजिए।

b) वास्तविक संख्याओं का समुच्चय R और उस पर सामान्य जोड़ (+) तथा सामान्य गुणनफल

(.) दिए गये हैं। (*), R पर निम्नलिखित से परिभाषित है :

$a*b=frac{a+b}{2},forall a,bin R.$

क्या (*), R सहयोगी है? क्या (.), आर में (*) पर वितरित है? जाँच कीजिए।

3. a) यदि $|z-1+2i|=4$ है, तो दर्शाइए कि बिन्दु z + i एक वृत्त निरूपित करता है। इस वृत्त को खींचिए।

b) $frac{x-1}{x^3-x^2-2x}$ को आंशिक भिन्नों के योग में व्यक्त कीजिए।

4. a) $a^2sec^2 x+b^2operatorname{cosec}^2 x$ , जहाँ $a>0,b>0$ हैं, का न्यूनतम मान ज्ञात कीजिए।

b) $intfrac{x^2cot^{-1}(x^3)}{1+x^6}dx$ का मान ज्ञात कीजिए।

c) दो समुच्चयों S और T के लिए दर्शाइए किः

$Scup T=(S-T)cup(Scap T)cup(T-S)$

है। वेन आरेख में भी स्थिति दर्शाइए।

5. a) R पर f(x) = x³ -x² -8x +12 और

और $g(x)=egin{cases}frac{f(x)}{x+3},& ext{if}x e-3\alpha,& ext{if}x=-3end{cases}$

द्वारा परिभाषित दो फलन f और g लीजिए।

i) $alpha$ का वह मान ज्ञात कीजिए जिसके लिए f , x = -3 पर सतत् है।

ii) f(x) = 0 के सभी मूल ज्ञात कीजिए।

b) वक्र y² (4-x) = x³ और इसकी y-अक्ष के समांतर अनंतस्पर्शी के बीच का क्षेत्रफल ज्ञात कीजिए।

6. a) डॉ यदि एक आय फलन $frac{dR}{dx}=15+2x-x^2$ द्वारा दिया गया है, जहाँ x निवेश है, तो अधिकतम आय ज्ञात कीजिए। यदि प्रारम्भिक आय 0 है, तो आय फलन R भी ज्ञात कीजिए।

b) वक्र y²(x+1) = x²(3 - x) का आरेखण कीजिए और ऐसा करने के लिए प्रयोग किए गये गुणधर्म भी लिखिए।

7. a) चक्रज $x=alpha( heta-sin heta),quad y=alpha(1-cos heta)$ की लम्बाई ज्ञात कीजिए और दर्शाइए कि रेखा $heta=frac{2pi}{3}$ इसे 1:3 के अनुपात में विभक्त करती है।

b) वह प्रतिबंध ज्ञात कीजिए कि वक्र ax² + by² = 1 और a'x² + b'y² = 1 एक-दूसरे को लम्बवत् प्रतिच्छेद करते हैं।

8. a) यदि y = e^m sin^-1 x है, तो दर्शाइए कि $(1-x^2)y_2-xy_1-m^2y=0$ है। इस प्रकार लाइब्नित्ज के सूत्र का प्रयोग करके $(1-x^2)y_{n+2}-(2n+1)xy_{n+1}$ का मान निकालिए। न+ल

b) $f(x)=egin{cases}2x,&x>5\x+5,&1le xle 5\|x|,&x<1end{cases}$ द्वारा परिभाषित R फलन f: R → R के जिस भी सबसे बड़े समुच्चय पर सतत् है वह निकालिए।

9. a) समीकरण x⁴ + 15x³ +70x² + 120x + 64 = 0 हल कीजिए, जिसके सभी मूल G.P. में हैं

b) $lim_{x o 0}frac{ an x-sin x}{x^3}$ ज्ञात कीजिए।

10. a) यदि $I_{m,n}=int x^3(log x)^n,dx$ , है, तो दर्शाइए कि :

$(m+1)I_{m,n}=x^{m+1}(log x)^n-nI_{m,n-1}$

है। इस प्रकार $int x^4(log x)^3,dx$ ज्ञात कीजिए।

b) f(x) = 2x² - 7x -10 द्वारा परिभाषित फलन f के लिए अंतराल [2, 5] पर लैग्रांज माध्यमान प्रमेय सत्यापित कीजिए।

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Subject: BMTC 131 - Calculus
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Product Information

Course Name	Four Year Under Graduate Programmes/Bachelor of Arts
Course Code	BAM
Subject Code	BMTC 131
Subject	Calculus
Programme	BACHELOR DEGREE PROGRAMMES Courses
Language	English
Material Type	Solved Assignment – Soft Copy / PDF

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